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Generalized Polynom ial Chaos: Applications to Random

Oscillators and Flow-Structure Interactions

by

Didier Lucor

Maitrise de Mecanique mention: Aeronautique, June 1995 DEA de Mecanique des Fluides et Transferts, September 1996 Sc.M. in Applied Mathematics, Brown University, May 2000

Thesis

Submitted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy

in the Division of Applied M athematics at Brown University

May 2004


UMI Number: 3134317

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by

Didier Lucor

2004


This dissertation by Didier Lucor is accepted in its present form by the Division of Applied M athematics as satisfying the

dissertation requirement for the degree of Doctor of Philosophy

Date / ^

Date / Q

Professor George Em Karniadakis, Director

\V ,

Professor C h a u -H ^ g Su, Co-Director

Recommended to the Graduate Council

Date

// //

Professor Michael St Triantafyllou, Dissertation Reader

Professor M artin Maxey, Dissertation Reader

Date'

Approved by the Graduate Council

Karen NewmanDean of the Graduate School


The Vita of Didier Lucor

Didier Lucor was born December 1st, 1971, in Marseiiie, France. After completing an undergrad

uate degree in Mechanical Engineering and Aeronautics, at the Institu t de Mecanique de Marseille

de rUniversite de la Mediterranee, Prance, in 1995, he pursued a graduate degree in Mechanical

Engineering at the Institu t National Polytechnique de Grenoble, France, in 1996. In August of

1998 he came to the Applied M athematics Division of Brown University for graduate study.

Education

B.A., Universite de la Mediterranee, France, 1995

Sc.M, Institut National Polytechnique de Grenoble, Prance, 1996

Sc.M, Brown University, USA, 2000

Ph.D, Brown University, USA, 2003

Publications and Presentations

P eer-R eview ed Journal Publications:

1. D. Lucor and G.E. Karniadakis, “Adaptive Generalized Polynomial Chaos for Nonlinear

Oscillators”, SIAM Journal on Scientific Computing (SISC), (submitted 2003).

2. D. Lucor and G.E. Karniadakis, “Predictability and Uncertainty in Flow-Structure Interac

tions”, European Journal of Mechanics B /Fluids (EJMB), (in press 2003).

3. D. Lucor, C.-H. Su and G.E. Karniadakis, “Generalized Polynomial Chaos and Random

Oscillators”, International Journal for Numerical Methods in Engineering, (in press 2003).

4. D. Lucor and G.E. Karniadakis, “Effects of Oblique Inflow in Vortex-Induced Vibrations”,

Flow, Turbulence and Combustion, (in press 2003).

5. D. Lucor, D. Xiu, C.-H. Su and G.E. Karniadakis, “Predictability and Uncertainty in CFD”,

International Journal for Numerical Methods in Fluids” , Vol. 43, 483-505, 2003.

iii


6 . D. Xiu, D. Lucor, C.-H. Su and G.E. Karniadakis, “Stochastic Modeling of Flow-Structure

Interactions using Generalized Polynomial Chaos”, Journal of Fluids Engineering, Vol. 124,

51-59, 2002.

7. C. Evangelinos, D. Lucor, C.-H. Su and G.E. Karniadakis, “Flow-Induced Vibrations of

Non-Linear Cables. Fart I: Models and Algorithms”, International Journal for Numerical

Methods in Engineering, Vol. 55, 535-556, 2002.

8 . D. Lucor, C. Evangelinos, L. Imas and G.E. Karniadakis, “Flow-Induced Vibrations o f Non-

Linear Cables. Part II: Simulations”, International Journal for Numerical Methods in En

gineering, Vol. 55, 557-571, 2002.

9. D. Lucor, L. Imas and G.E. Karniadakis, “Vortex Dislocations and Force Distribution of

Long Flexible Cylinders Subjected to Sheared Flows”, Journal of Fluids and Structures, Vol.

15, 641-650, 2001.

10. C. Evangelinos, D. Lucor and G.E. Karniadakis, “DNS-Derived Force Distribution on Flex

ible Cylinders Subject to V IV ”, Journal of Fluids and Structures, Vol. 14, 429-440, 2000.

P eer-R eview ed Conference P ublications:

1. P. Byrne, D. Simonsson, D. Lucor and E. Fontes, “A Model of the Anode from the Chlorate

Cell”, Fluid Mechanics and its Applications, Vol. 51: Transer Phenomena in Magneto-

hydrodynamic and Electroconducting Flows, Alemany, M arty and Thibault (eds), Kluwer,

Dordrecht, 1999.

2. D. Lucor, C. Evangelinos and G.E. Karniadakis, “DNS-Derived Force Distribution on Flex

ible Cylinders Subject to V IV with Shear Inflow”, Proceedings of Flow Induced Vibration,

Ziada & Staubli (eds), Balkema, Rotterdam , 281-287, 2000.

3. D. Xiu, D. Lucor and G.E. Karniadakis, “Modeling Uncertainty in Flow-Structure Interac

tions”, Computational Fluid and Solid Mechanics, Proceedings of the 1st MIT Conference,

Cambridge, Massachusetts, K.J. Bathe (ed.). Vol. 2, 1420-1423, 2001.

iv


4. D. Lucor, J. Foo and G.E. Karniadakis, “Correlation Length and Force Phasing of a Rigid

Cylinder Subject to V IV ”, Proceedings of the lUTAM Symposium on Fluid-Structure Inter

actions, June 2-6, 2003, New Brunswick, New Jersey, USA.

5. D. Lucor and G.E. Karniadakis, “Stochastic Flow-Structure Interactions”, Computational

Fluid and Solid Mechanics, Proceedings of the 2nd MIT Conference (June 17-20 2003),

Cambridge, Massachusetts, K .J. Bathe (ed.), 2003.

6 . D. Lucor, X. Ma, M.S. Triantafyllou and G.E. Karniadakis, “Vortex-Induced Vibrations of

Long Marine Risers in Sheared Flows”, Proceedings of the 4th ASM E/JSM E Joint Fluids

Engineering Conference, July 6-10, 2003.

7. S. Dong, D. Lucor, V. Symeonidis, J. Xu and G.E. Karniadakis, “Multilevel Parallelization

Models: Application to V IV ”, Proceedings of the 2003 Users Group Conference DoD/HPCM P,

June 9-13, 2003, Bellevue, Washington, USA.

8. D. Lucor and G.E. Karniadakis, “Direct Numerical Simulations of V IV ”, Proceedings of the

5th International Symposium on Cable Dynamics, September 15-18, 2003, Santa Margherita,

Italy.

O ther C onference P u blication s (E xtend ed A b stracts R efereed);

Presenter is listed first.

1. R.M. Kirby, Y. Du, D. Lucor, X. Ma, G-S Karamanos and G.E. Karniadakis, “Parallel DNS

and LES of Turbulence and Flow-Structure Interactions”, Proceedings of the DoD HPCM P

Users Group Conference, June 5-8, 2000, Albuquerque, New Mexico, USA.

2. D. Lucor and G.E. Karniadakis, “Effects of Oblique Inflow in Vortex Induced Vibrations”,

Proceedings of the lUTAM Symposium on Unsteady Separated Flows, April 8-12, 2002,

Toulouse, France.

Conference A bstracts:

Presenter is listed first.


1. D. Lucor, C. Evangelinos and G.E. Karniadakis, “Vortex Dislocations and Body Force in

Flow-Structure Interactions”, Presented at the 52nd Annual Meeting of the American Phys

ical Society’s (APS) Division of Fluid Dynamics, November 21-23, 1999, New Orleans,

Lousiana, USA.

2. D. Lucor and G. E. Karniadakis, “Vortex Dislocations, Vortex Splits and Forces istribution

in Flows past B luff Bodies”, Presented at the lUTAM Symposium on Bluff Body Wakes and

Vortex-Induced Vibrations, June 13-16, 2000, Marseille, France.

3. D. Lucor and G. E. Karniadakis, “DNS-Derived Force Distribution on Flexible Cylinders

Subject to V IV with Shear Inflow”, Presented at the 7th International Conference on Flow-

Induced Vibration, June 19-22, 2000, Lucerne, Switzerland.

4. D. Lucor, G.E. Karniadakis, A.H. Techet, F.S. Hover and M.S. Triantafyllou, “A Numerical

and Experimental Study of Vortex Splits in Flow-Structure Interactions”, Presented at the

53nd Annual Meeting of the American Physical Society’s (APS) Division of Fluid Dynamics,

November 19-21, 2000, Washington, D.C, USA.

5. D. Xiu, D. Lucor, M. Jardak, C.-H. Su and G.E. Karniadakis, “Polynomial Chaos Solutions

of Fluid Dynamics with Applications”, Presented at the Stochastic Numerics Conference,

ETH, February 19-21, 2001, Zurich, Switzerland.

6 . D. Lucor, D. Xiu and G.E. Karniadakis, “Spectral Representations o f Uncertainty in Simu

lations: Agorithms and Applications”, Presented at the ICOSAHOM-01, June 11-15, 2001,

Uppsala, Sweden.

7. D. Lucor and G.E. Karniadakis, “Effect of Oblique Inflow on V IV ”, Presented at the 54nd

Annual Meeting of the American Physical Society’s (APS) Division of Fluid Dynamics,

November 18-21, 2001, San Diego, California, USA.

8 . G.E. Karniadakis and D. Lucor, “Predictability and Uncertainty in Flow-Structure Interac

tions”, Presented at the Conference on Bluff Body Wakes and Vortex-Induced Vibrations

vi


(BBVIV3), December 17-20, 2002, Port Douglas, Queensland, Australia.

9. D. Lucor and G.E. Karniadakis, “Random Oscillators and Polynomial Chaos”, Presented

at SIAM Conference on Computational Science and Engineering (CSE03), February 10-13,

2003, San Diego, California, USA.

10. D. Lucor, J. Foo and G.E. Karniadakis, “Correlation lengths and structural damping re

sponse of rigid cylinders subject to V IV ”, Presented at the 56nd Annual Meeting of the

American Physical Society’s (APS) Division of Fluid Dynamics, November 23-25, 2003,

East Rutherford, New Jersey, USA.


A cknow ledgm ents

My thesis is dedicated to my parents Alain and Colette and to my brother Christophe for their

unconditional love and support. They never questioned my choice to work or study abroad but

they always encouraged me and tried to help me as much as they could. I know th a t all this time

when I have been away from home it has been difficult for them too. I also dedicate my thesis

to all the other members of my family, in particular to my grandparents; Marius and Henrietta,

Theo and Jacqueline and my great-cousins Maurice and Claudie.

I ’d like to give sincere thanks to my thesis advisor, Pr. George, Em Karniadakis. I will always

be grateful to him for giving me a chance and accepting me in the CRUNCH group. He was always

very accessible and attentive. His great physical intuition and his huge knowledge of numerical

methods were very helpful for my work. He consistently secured my funding over the years and

gave me access to the best computing facilities in the US. He has always given excellent exposure

to his students by sending them to prestigious conferences all around the world or by encouraging

them to submit papers in renowed international scientific journals. I hope tha t we will keep on

collaborating in the future.

I also would like to give sincere thanks to my thesis co-advisor, Pr. Chau-Hsing Su. He helped

me a lot by developing the theoretical aspect of the project and performed most of the complex

mathematical derivations and estimates. His im portant contributions to the project and more

specifically towards Part II of the thesis were very crucial. He has been a model of kindness,

patience and humbleness aU along and, in a certain way, inspired me to try to pursue a career in

academia.

I feel very lucky to have been directed by two complemental persons with a different approach

in their work, different background and personality but with the common goal of better under

standing of physical phenomena. Their interaction has shown me tha t there is not one way to do

research but instead, th a t each individual can bring his scientific contribution in its own way.

Fd like to thank Pr. Michael Triantafyllou for being very helpful and inspirational with his

viii


remarkable experimental work on the subject of flow-structure interactions.

Fd like to give big thanks to Dr. Costas Evangelinos who was the second generation of people

working on the subject of VIV in the group and was continuing from the pioneering work of Dr.

David Newman. He helped me to get started with the project and taught me a lot about the code

and all the subtle tricks th a t make a big difference and are not to be found in the books!!

Thanks and good lucks are due to all other past and present CRUNCH group members. In

particular, Fd like to thank the members with whom I ’ve strongly interacted for my research: Dr.

Xia Ma, Dr. Mike Kirby, Dr. Mohamed Jardak, Dr. Dongbin Xiu and Dr. Steve Dong. Fm

glad and thankful th a t they shared their knowledge of the code and the mathematical framework

behind it.

Fd like to acknowledge my readers, Pr. Michael Triantafyllou and Pr. M artin Maxey for their

help and useful suggestions about my lengthy thesis!

Fd like to acknowledge Dr. Leonard Imas with whom I collaborated for some parts of my work

and who suggested very interesting and challenging problems to me.

I would also like to thank Pr. Charles Williamson from Cornell University for being very helpful

and curious about my work with his vast expertise on the subject of flow-structure interactions.

Thanks are also due to Jasmine Foo who is the fourth generation of people working on this

project in the CRUNCH group. Jasmine helped me with some of the runs and analysis of Chapter

3 of P art I of my thesis. I wish her the best of luck and I hope th a t we will keep interacting and

collaborating in the future.

Being a graduate student at Brown was great because it gave me a chance to meet very different

people, with different backgrounds and cultures with whom I felt very close partly because it

seemed they were sharing the same experiences as me. I want to thank my friends who made my

life here more bearable and enjoyable: Sid, Caroline, the Tribe (Harsh, Mokshay, Ashwin, Ashok

and Dnyanesh), Candace, Vas, Don and my friends from volleyball and from the gym (Coree,

Charles, Jean, Chris, Jason, Carsten, Kamil and Amine). Other friends helped me in the past

to get where I am today or where I will be tomorrow, and without them my professional and


personal life would have probably taken another path: thanks to Dr. Sophie Nigen for “saving

me” at the end of my DEA in Grenoble, thanks to Dr. Eduardo Fontes, my boss when I was

working in Sweden, for giving me more confidence in myself and making me believe th a t I would

succeed in a PhD, thanks to Dr. Jennifer Godbee and thanks to Dr. Eric Climent for giving me

good advices and sharing his own professional experience.

Fd also like to thank the staff of the division and more particularly Madeline Brewster who

was so good with the editing of my papers and our kind system adm inistrator Jie Zhang.

Merci a Doi'na.

This project was supported by the Office of Naval Research, under the supervision of Dr.T.F.

Swean. The work was funded under the following grants:

• AFOSR: F49620-93-1-0090

# DOE: DE-FG02-95ER25239

* ONR: N00014-95-1-0256 and NUW C/ONR contract: N66604-98-C-1266

* NSF: CTS-9417520


Contents

1 In troduction 3

1 Predictability and Uncertainty in Flow-Structure In te ra c tio n s ...................................... 3

2 From Deterministic to Stochastic Direct Numerical Simulations of Flow-Structure

In te ra c tio n s ................................................................................................................................ 5

3 Toward Multi-Level Parallelization Models for Stochastic Flow-Structure Interactions 7

4 Thesis Outline ........................................................................................................................... 9

1 D eterm inistic Flow-Structure Interactions 12

2 T he C oupled F low -Structure Form ulation 13

1 Structural M o d e l ........................................................................................................................ 13

2 Formulation of the Mapping A p p ro ach ................................................................................ 17

3 Limitations of the Mapping A p p ro a c h ................................................................................ 21

3 U niform and O blique F low s past R igid C ylinders 22

1 Correlation Length and Force P h a s in g ................................................................................. 22

1.1 Introduction ................................................................................................................. 22

1.2 Simulation P aram e te rs ................................................................................................ 24

1.3 Complex Demodulation A n a ly s is ............................................................................. 28

1.4 Results and D iscussion................................................................................................ 28

XI


1.5 S u m m a ry ........................................................................................................................ 38

2 Vortex Synchronizations P a t te rn s .......................................................................................... 43

2.1 Flow Visualizations....................................................................................................... 43

2.2 Frequency R e s p o n s e .................................................................................................... 48

2.3 Vortex and Potential Force Decom position........................................................... 52

3 Validity of the Independence P r in c ip le ................................................................................ 54

3.1 Introduction ................................................................................................................. 54

3.2 Simulation Parameters and F o rm u la tio n ................................................................ 58

3.3 Force D istribu tions....................................................................................................... 59

3.4 Flow V isualizations........................................................................................................ 64

3.5 S u m m a ry ........................................................................................................................ 69

4 Sheared Flow s past R igid and F lex ib le C ylinders 72

1 In troduction ................................................................................................................................. 72

1.1 Flow Visualizations of Vortex Dislocations and Vortex S p l i t s ........................ 74

2 Vortex Split, Hybrid Mode and Vortex Shedding Mode Selection .............................. 78

2.1 Flow Visualizations........................................................................................................ 79

2.2 Forces Distribution and Frequency Response ...................................................... 81

3 Vortex-Induced Vibrations for Linear Sheared F lo w s ...................................................... 84

3.1 Displacement and Force D is tr ib u tio n s ................................................................... 87

3.2 Frequency R esponses.................................................................................................... 90

3.3 Effects of Averaged Reynolds N u m b e r ................................................................... 92

4 Vortex-Induced Vibrations for Exponential Sheared F lo w s ............................................ 94

4.1 Displacement and Force D is tr ib u tio n s ................................................................... 95

4.2 Frequency R esponses.................................................................................................... 97

4.3 Effects of Spanwise Grid Resolution and Initial C o n d itio n ............................... 98

5 S u m m a ry ........................................................................................................................................101

X ll


II Random Oscillators 103

5 R epresentation o f R a n d o m P rocesses 104

1 The Generalized Polynomial Chaos E xpansion ....................................................................... 104

2 The Karhunen-Loeve Expansion .............................................................................................107

3 Convergence Is s u e s ....................................................................................................................... I l l

4 Covariance K e rn e ls ........................................................................................................................118

4.1 One-Dimensional K e r n e ls ............................................................................................. 124

4.2 Two-Dimensional K e rn e ls ............................................................................................. 134

4.3 Three-Dimensional K e r n e l s .........................................................................................136

6 Linear O scillators 138

1 In troduction .................................................................................................................................... 139

2 Governing E q u a tio n s .................................................................................................................... 142

3 Random Forcing P rocesses.......................................................................................................... 144

4 Random Parametric and Forcing V ariables.............................................................................150

5 S u m m a ry ........................................................................................................................................153

7 N onlinear O scillators 157

1 In troduction .................................................................................................................................... 157

2 Duffing O scilla tor...........................................................................................................................159

2.1 Generalized Polynomial Chaos Representation .....................................................159

2.2 Periodic Excitation with Random A m p litu d e .........................................................161

2.3 Random Forcing Processes ......................................................................................... 167

3 S u m m a ry ........................................................................................................................................175

III Stochastic F lo w -S tru c tu re Interactions 176

8 G eneralized P olynom ial C haos Form ulation 177

XIU


1 Incompressible Navier-Stokes E q u a t io n s ................................................................................ 177

1.1 Governing E q u a tio n s .......................................................................................................177

1.2 Post-Processing.................................................................................................................180

2 The coupled Flow-Structure P ro b lem .......................................................................................181

2.1 Transformed Navier-Stokes E q u a t io n s ...................................................................... 181

9 Stationary C ylinders Sim ulations 186

1 Two-Dimensional Simulations with Random In f lo w .............................................................186

1.1 Random Force R esponse................................................................................................ 186

1.2 Random-Flow Visualizations .......................................................................................199

2 Three-Dimensional Simulations with Random In f lo w ......................................................... 205

2.1 Random-Force R esponse................................................................................................ 205

2.2 Random-Flow Visuahzations ...................................................................................... 214

10 M oving C ylinders Sim ulations 217

1 Two-Dimensional Simulations with Random Inflow and Forced Structural Motion . 217

1.1 Random-Flow Visualizations .......................................................................................218

1.2 Random Force R esponse................................................................................................ 218

2 Two-Dimensional Simulations with Deterministic Inflow and Random Structural

P a ra m e te r s ................................................................................................................................... 223

2.1 Random Structural Response and Forces D is tr ib u tio n ........................................ 225

A Parallel A lgorith m s a n d Im plem entation Issues 233

1 Spectral//ip Discretizations on Unstructured and Hybrid G r i d s ..................................... 233

1.1 Local coordinate S y s te m s .............................................................................................234

1.2 Spectral Hierarchical E x p an sio n s ................................................................................236

1.3 Time Integration A lg o rith m ..........................................................................................239

2 The M e k T oltT c o d e .................................................................................................................241

XIV


2.1 Fourier D ecom position.................................................................................................. 241

B B rief O verview o f A lgebra o f R andom V ariables 244

C S tochastic N ew m ark Schem e 252


List of Tables

3.1 Average one-tenth highest amplitude of response Am axlD versus nominal reduced

velocity Vm and Reynolds number Re. * indicates simulations performed with 64

Fourier modes along the spanwise direction. Other simulations were performed

using 32 Fourier modes.............................................................................................................. 29

4.1 Non-dimensional flow and structural parameters for linear sheared flow cases. . . . 86

4.2 Non-dimensional flow and structural parameters for exponential sheared flow cases. 95

5.1 Correspondence between the type of Wiener-Askey Polynomial Chaos and the type

of random inputs (iV > 0 is a finite integer)........................................................................... 107

6.1 Speed-up factors S based on relative mean error tmean of Generalized Polynomial

Chaos {(P + 1) terms) versus Monte-Carlo simulations {N events) for Gaussian and

Uniform distributions....................................................................................................................155

7.1 Number of unknown deterministic coefficients in the polynomial chaos representa

tion, as a function of the number of random dimensions n and the highest polyno

mial order p ..................................................................................................................................... 158

XVI


List of Figures

1 Cross-flow velocity spectrum at a centerline point seven diameters behind a cylinder. DNS

(thin line) versus experiment of Ong & Wallace (thick line); Re = 3,900, taken from [1]. . 4

1 The coordinate system is attached to the moving flexible cylinder, producing an

undeformed, stationary computational domain................................................................... 19

1 Meshl; 1018 elements hybrid grid in the x — y plane; [x x y] = [(—22P; 69H) x

(—22D; 22D)]. Fourier expansions are used in the periodic spanwise direction (per

pendicular to the X — y plane) 25

2 Mesh2: 2340 elements hybrid grid in the x — y plane; [a; x y] = [(—20H; SOD) x

(—SOD; 30D)]. Fourier expansions are used in the periodic spanwise direction (per

pendicular to the X — y plane) 25

3 Close up of the discretization of Mesh2 around the cylinder............................................ 26

4 Mesh2: Variable expansion order in the x — y plane for Re = 2000 and Re = 3000. 26

5 Streamwise, cross-flow and spanwise velocity autocorrelation functions in the near

wake at a centerline point {x/d = 3 ;y /d = 0) (left column) and an off-centerline

point {x/d = 3; y /d = 1) (right column), for six different reduced velocities; Re=1000. 32


wake at a centerline point (x /d = 3; y /d = 0) (left column) and an off-centerline

point (x /d = 3 ;y /d = 1) (right column), for four different reduced velocities;

Re=2000........................................................................................................................................ 33

xvii



wake a t a centerline point {xjd = 3 ;j//d = 0) (left column) and an off-centerline

point {x jd = 3 ;y /d = 1) (right column), for four different reduced velocities;

Re=3000........................................................................................................................................ 34

8 Force correlation coefficients for four different reduced velocities. Lift coefficient

autocorrelation function (left plot). Drag coefficient autocorrelation function (right

plot); J?e = 1000.......................................................................................................................... 35

9 Cylinder responses using different spanwise resolutions for Vm = 4.18; Re = 3000. 35

10 Amplitude response as a function of reduced velocity Vm for different Re .................. 36

11 Complex demodulation analysis of Case II, Re = 1000 and Vm = 4.18. Isocontours

of cross-flow displacement rj (a). Isocontours of amplitude of demodulated cross-

flow displacement (b). Isocontours of amphtude of demodulated Lift coefficient

i?Ci, (c)- Isocontours of Lift coefficient Cl (d). Isocontours of phase difference A $

between demodulated cross-flow displacement and demodulated lift coefficient (e). 39

12 Complex demodulation analysis of Case IV, Re = 1000 and Vm = 4.99. Isocontours

of cross-flow displacement r? (a). Isocontours of am phtude of demodulated cross-

flow displacement Rn (b). Isocontours of amplitude of demodulated Lift coefficient

R c^ (c). Isocontours of Lift coefficient C l (d). Isocontours of phase difference A $


13 Complex demodulation analysis of Case VI, Re = 1000 and Vm = 7.00. Isocontours

of cross-flow displacement f? (a). Isocontours of amplitude of demodulated cross-

flow displacement (b). Isocontours of amplitude of demodulated Lift coefficient

R c^ (c). Isocontours of Lift coefficient Cl (d). Isocontours of phase difference A #


X V lll


14 Pressure isocontour at value —0.15 in the wake of a free rigid cylinder at Re=1000

(Case IV; V^n = 4.99). View perpendicular to cylinder axis with inflow coming

from left to right (left picture). View perpendicular to cylinder axis with inflow

coming from right to left (right picture). The instantaneous cylinder position is

rj = -0 .3 2 ...................................................................................................................................... 42

15 Isocontours of the instantaneous pressure in the near wake of a moving rigid cylinder

subject to VIV at Re=3000 (Case IV: Vm = 4.99)............................................................ 44

16 Isocontours of span and phase-averaged pressure field in the near wake of a moving

rigid cylinder at Vm = 7.0, Re = 1000 and Am ax/D = 0.554. Upper-left: cylinder

bottom extreme position; Upper-right: cylinder median position, moving upward;

Lower-left: cylinder top extreme position; Lower-right: cylinder median position,

moving downward....................................................................................................................... 45

17 Isocontours of span- and phase-averaged spanwise vorticity field in the near wake

of a moving rigid cyhnder at Vm = 7.0, R e = 1000 and Am ax/D = 0.554. Blue

contours show clockwise vorticity, red anticlockwise vorticity. Upper-left: cylinder

bottom extreme position; Upper-right: cylinder median position, moving upward;

Lower-left: cylinder top extreme position; Lower-right: cylinder median position,

moving downward....................................................................................................................... 46

18 Isocontours of span- and phase-averaged pressure and spanwise vorticity fields in the

near wake of a moving rigid cylinder at Vm = 7.0, Re = 2000 and Amax/D = 0.54.

2P vortices have been circled in red. (a): cylinder bottom extreme position; (b):

cylinder median position, moving upward; (c): cylinder top extreme position; (d):

cylinder median position, moving downward....................................................................... 47

19 Isosurfaces of phase-averaged filtered spanwise vorticity fields in the near wake of

a moving rigid cylinder at Vm = 7.0, Re = 2000 and Am ax/D = 0.54. Vortex pairs

are easier to spot at some locations along the span than others.................................... 49


20 Structural frequency response f s / f n as a function of reduced velocity Vm for dif

ferent Rejoiolds numbers........................................................................................................... 49

21 Shedding frequency response f v / f n for Re = 1000 and Vm = 7.0 at 6 different

locations in the wake.................................................................................................................. 50

22 Shedding frequency response f v / f n for Re = 2000 and Vm = 7.0 at 6 different

locations in the wake.................................................................................................................. 51

23 Relationship between to tal lift force coefEcient {Ctotal), the potential added mass

force (Cpotentiai) and the vortex force (Cvortex) for three different reduced velocities;

Re=3000................................. 53

24 Cylinder response and span-averaged lift coefficients for reduced velocity in the

lower branch. Left: Vm = 7.00; R e = 1000. Right: Vm = 7.00; Re = 2000................ 54

25 Complex demodulation analysis for Re = 2000 and Vm = 7.00. Isocontours of

cross-fiow displacement 77 (a). Isocontours of amplitude of demodulated cross-flow

displacement Rp (b). Isocontours of amplitude of demodulated Lift coefficient R c^

(c). Isocontours of Lift coefficient Cl (d). Isocontours of phase difference A #

between demodulated cross-fiow displacement and demodulated lift coefficient (e).

The demodulation frequency is = 0.1561........................................................................ 55

26 Simulation setup. Left: view perpendicular to the cross-flow direction (17-direction).

Right: 3D view............................................................................................................................ 59

27 Mean (left plot) and rms (right plot) drag coefficients versus reduced velocity Vm

based on free-stream inflow v e lo c i ty . ................................................................................... 60

28 Rms lift coefficients (left plot) and rms and maximum cross-flow cylinder responses

(right plot) versus reduced velocity Vm based on free-stream inflow velocity. . . . . 62

29 Freely vibrating yawed cylinder {9 = —60°) at lock-in at Re = 1,000. Drag (top)

and Lift (bottom) coefficients along the span versus non-dimensional tim e................ 63

XX


30 Stationary yawed cylinder (6 — —70°) at i?e = 1,000. Pressure isocontour at value

-0.025. View almost perpendicular to the plane of the inflow. The arrows represent

the inflow coming from left to right....................................................................................... 65

31 Freely vibrating yawed cylinder (0 = —70°) at lock-in at Re = 1,000. Pressure

isocontour at value —0.025. View almost perpendicular to the plane of the inflow.

Inflow coming from left to right.............................................................................................. 66

32 Freely vibrating yawed cylinder (0 = —60°) at lock-in at Re — 1,000. Pressure

isocontours at value —0.05. Views almost perpendicular to the plane of the inflow.

Left: Front view with inflow coming from left to right. Right; Back view with

inflow coming from right to left............................................................................................... 67

33 Freely vibrating yawed cylinder (0 = -60°) at lock-in at Re — 1,000. Spanwise

vorticity isocontours at value ±0.25. View almost perpendicular to the plane of the

inflow. Inflow coming from left to right................................................................................ 68

34 Freely vibrating yawed cylinder (0 = —60°) at lock-in at Re = 1,000. Pressure

isocontours at value —0.05. View almost perpendicular to the plane of the inflow.

Inflow coming from left to right.............................................................................................. 70

1 Experimental (left) vs Numerical (right) results. The cylinder is a t the bottom of

the pictures, the views are perpendicular to the cylinder axis and the flow is upward.

Left: Photograph of Vortex Dislocation at Re = 100 for stationaiy cylinder using

dye flow visualization (courtesy of Williamson [2]). Right: Isocontours of spanwise

vorticity (w = ±0.18) and oblique fronts in the wake of a forced vibrating flexible

cylinder at Re — 100 with linear sheared inflow................................................................. 75

XXI


Linear shear: Iso-contours of crossflow velocity at Rem = 1000. The cylinder is at the

bottom of the pictures, the view is perpendicular to the cylinder axis and the flow is

upward. Only the part corresponding to the large inflow is shown. Dark color: v = —0.2,

Light color: v = 0.2. View normal to the (x, z) horizontal plane, where 80 < z / D < 400

and 0 < x / D < 12.5; flow is upward............................................................................................... 76

Exponential shear: Iso-contours of crossflow velocity at Rcm = 1000. The cylinder is at

the bottom of the pictures, the view is perpendicular to the cylinder axis and the flow is

upward. The large inflow is on the left side. Dark color v = —0.2. Light color v = 0.2.

View normal to the (x, z) horizontal plane, where 0 < z j D < 400 and 0 < x /D < 35; flow

is upward.............................................................................................................................................. 76

Experimental (left) vs Numerical (right) results. The cylinder is a t the bottom of

the pictures, the views are perpendicular to the cylinder axis and the flow is upward.

Vortex split is located in the red circles. Left: Photograph of vortex split in the

wake of a forced rigid tapered cylinder at Re^ = 400 with uniform inflow using lead

precipitation visualization (courtesy of Techet and al. [3]). Right: Isocontour of

pressure (p = —0.25) in the wake of a forced rigid straight cylinder at Refy — 400

with linear sheared inflow......................................................................................................... 77

DNS results; Instanteneous isocontour of pressure (p = —0.25) in the near wake.

The cylinder is vertical in the pictures. Views are almost perpendicular to the

plane of the inflow. Left: Front view with inflow coming from right to left. Right:

Back view with inflow coming from left to right. Black arrows point to the 2S-type

vortical structures and the red arrows point to the 2P-type vortical structures. . . 80

DNS results: Instanteneous isocontour of pressure (p = —0.25) in the near wake.

The cylinder is at the bottom of the picture, the view is almost perpendicular to the

cylinder axis and the flow is upward. Presence of a clear vortex split that connects

28 vortex shedding mode on the right side to a 2P vortex shedding mode on the

left side.......................................................................................................................................... 81

x x ii


7 Spatio-temporal isocontour distributions of Lift coefficient.............................................. 82

8 Time history and corresponding spectrum of Lift forces at different spanwise loca

tions. Left: 2 /d = 5.625. Left: 2/d = 21.09.......................................................................... 82

9 Correlation coefficients of Lift forces....................................................................................... 83

10 Time traces of spanwise correlation of Lift coefficient........................................................ 83

11 Normalized linear sheared inflow velocity profiles for Case-0 and Case-00.................... 86

12 Left: Crossflow-displacement {rms values - horizontal axis) of the beam along

the span normalized with the cylinder diameter. Right: Corresponding power

spectral density (x-axis: frequency non-dimensionalized with maximum velocity,

2/-axis: power spectral density and 2-axis: span of the cy linder.)................................. 88

13 Left: Distribution of (C'o)mean along the span. Right: Distribution of (Ci)rm«

along the span. The local inflow velocity is used in the normalization......................... 89

14 Comparison of time-histories of drag force at different locations along the span:

(top) z /D = 248.06; (middle) z /D = 283.5; and (bottom) z /D = 416.39.................... 90

15 Spanwise (z) power spectral density of the u-component of the velocity field, (x-

axis denotes frequency non-dimensionalized with maximum, velocity y-axis denotes

power spectral density and z-axis denotes the span of the cy linder.).......................... 91

16 Frequency distribution of wake and structure along the span for Case-0. Circles denote

dominant frequency of crossflow velocity and triangles dominant frequency of the structure

response. The maximum inflow velocity is used in the normalization of the frequency. . . 92

17 Isocontours of spatio-temporal cylinder crossflow displacements Y for Case 00-

a: Re-max = 1000 (top). Case 00-b: Remax = 1500 (middle) and for Case 00-c:

Remax = 2000 (bottom )............................................................................................................ 93

18 Time-averaged power spectral density of cylinder crossflow displacement versus

modes number n for Case 00-a, Case 00-b and Case 00-c................................................ 94

19 Normalized exponential sheared inflow velocity profiles for Case-I and Case-II. . . 96

xxm


20 Time-history of the distribution of crossflow displacement along the span. A mixed

standing-traveling wave pattern prevails unlike the linear shear case...................................... 96

21 Left: Crossflow displacement {rms values) along the span (normalized with the cylinder

diameter). Right: Corresponding spectrum showing the (non-dimensional) frequency re

sponse (range: 0 — 0.3) along the span. The frequency is normalized with the maximum

inflow velocity (x-axis: frequency non-dimensionalized with maximum velocity, y-axis:

power spectral density and z-axis: span of the cylinder.) ....................................................... 96

22 Frequency distribution of wake and structure along the span showing cellular shedding

and multi-mode response. Left: linear shear; Right: exponential shear. Circles denote

dominant frequency of crossflow velocity and triangles dominant frequency of the structure

response. The maximum inflow velocity is used in the normalization of the frequency. . . 98

23 Isocontours of spatio-temporal cylinder crossflow displacements Y for Casell-a (left)

and for Case 0-b (right)............................................................................................................. 99

24 Spanwise distribution of rm s values of crossflow displacement (left) and rm s values

of lift forces (right) for Casell-a and C a s e l l - b ..................................................................... 100

25 Span-averaged power spectral density of cylinder crossflow displacement versus

Strouhal frequency for Casell-a and Casell-b (left). Time-averaged power spectral

density of cylinder crossflow displacement versus modes number n (right). Casell-a:

64 Fourier modes; Casell-b; 32 Fourier modes.......................................................................100

1 (a): Diagonal terms of the exponential covariance kernel R h h ij) = for

<T = 1.0, A = 5 and T = 100 for different number of Karhunen-Loeve random

dimensions n. (b); Minimum number of Karhunen-Loeve random dimensions n

times the prescribed accuracy e versus T jA ............................................................................110

xxiv


2 Estimates of required number of functionals versus the product of standard devi

ation with time for different pd f’s, (a): Laguerre-Chaos/Exponential distribution.

(b): Hermite-Chaos/Gaussian distribution, (c): Legendre-Chaos/Uniform distri

bution................................................................................................................................................ 119

3 Covariance kernels c{x,y) with \x\ < L j2 and \y\ < I//2. (a); K e rn e l l with n =

1; fc = 27t; a = 1/fc; (b): K ern e ll with n = 2.5; k — Stt; A = 1/fe; (c): K ernel2

with n = 2.5; k = Stt; A = 1/A:; (d): K ernel2 with n = 3;k = Gtt; A = 1/fc. i = 1

for all cases.......................................................................................................................................130

4 Eigenvalues A* for various values of the correlation length, here given by n, with

n = 2 and L = 1. Left: K ern ell. Right: K ernel2 .........................................................131

5 First 4-pairs eigenfunctions fi{x) with |x| < L(2 and L — 1. Left: K e rn e l l with

n = 2.5. Right: K e rn e l2 with n = 2.5.....................................................................................131

6 Approximation of Covariance kernels of c{x,y), see Figure (3-(b)). (a): 5-term

approximation of K ern e ll with n = 2.5; k = 27t; A = 1/fc; (b): 5-term absolute

error of K ern e ll approximation with n = 2.5; fc — 27t;A = 1/fc; (c): 21-term

approximation of K ern ell with n = 2.5; k = Stt; A = 1/fc; (d): 21-term absolute

error of K ern ell approximation with n = 2.5; fc = 27t; A = 1/fc....................................... 132

7 Approximation of Covariance kernels of c(x,y), see Figure (3-(c)). (a): 5-term

approximation of K e rn e l2 with n = 2.5; k = 2tx;A = 1/fc; (b): 5-term absolute

error of K ernel2 approximation with n = 2.5; fc = 27t; A = 1/fc; (c): 21-term

approximation of K ernel2 with n = 2.5; fc = 5tt; A = 1/fc; (d): 21-term absolute

error of K ernel2 approximation with n == 2.5; fc = 27t; A = 1/fc....................................... 133

8 Eigenvalues of KL decomposition with Bessel correlation function (90), 6 = 20. . . 135

9 Eigenfunctions of the KL decomposition with the Bessel correlation function (90),

h = 20; Left: first eigenfunction, Right: second eigenfunction. (Dashed lines denote

negative values.) ..........................................................................................................................135

XXV


10 Eigenfunctions of the KL decomposition with the Bessel correlation function (90),

b = 20; Left: third eigenfunction, Right; fourth eigenfunction. (Dashed lines denote

negative values.) ..........................................................................................................................136

1 Time evolution of second-order moment (Case I) for different number of random

dimensions n (a). Convergence rate of second-order moment Fajj, of the output

(Case I and II) and second-order moment F / / of the input (Case I and II) versus

the number of random dimensions n at u>ot = 20 (b)........................................................... 149

2 Time evolution of the relative pointwise Too error of the second-order moment

response Txx for different number of random dimensions n {Uniform distribution). 150

3 (a): Fifth-order {p = 5) Wiener-Askey Polynomial Chaos (P -f 1 = 5 6 terms).

Solution of the dominant random modes versus time for Case I. (b): Time evolution

of the variance of the solution for Case I and II .................................................................... 152

4 Convergence rate Tco error of the mean and variance of the solution versus the

order of the Wiener-Askey Polynomial Chaos expansion p. (a): Case I: Gaussian

input, (b): Case II: Uniform input............................................................................................154

1 Time evolution of the random modes solution for Case I ( Gaussian) using a GPC

expansion of 6 term s (p = 5); e = 1.0........................................................................................163

2 Time evolution of the random modes solution (Case II). Case of a GPC expansion

of 4 terms (p = 3); e = 5...............................................................................................................164

3 Phase projections of deterministic solutions and stochastic (uniform distribution)

solutions............................................................................................................................................166

4 Case I {Gaussian): Comparison of second-order moment response obtained by

adaptive GPC (AGPC) and Monte-Carlo simulation (MC) (500,000 events). o;o =

1.0; C = 0.1; A = 1.0; e = 1.0........................................................................................................... 170

5 Case II: Comparison of second-order moment response obtained by adaptive GPC

and Monte-Carlo simulation (500,000 events), loq = 1-0; C = 0.1; A — 1.0 ; e = 1.0 . 171


6 Comparison of second-order moment response obtained by adaptive GPC and

Monte-Carlo simulation (1,000,000 events), loq = 1.0; C = 0.02; A = 1.0; e = 0.1,

(Case I; Gaussian)......................................................................................................................... 172


Monte-Carlo simulation (1,000,000 events). u>o = 1.0; C = 0.02; A = 1.0; e = 0.1,

(Case II; Uniform)......................................................................................................................... 172

8 L2-norm of the random adaptive GPC modes with no reordering and with reorder

ing. (Uo = 1.0; C = 0.02; A = 1.0; e = 0.1. Relates to V: adaptive GPC (K=10,

M=10, p = 3 ).................................................................................................................................... 173


Monte-Carlo simulation (1,000,000 events), loq = 1.0; ( = 0.02; A = 1.0; e = 0.1

(Case I: Gaussian)......................................................................................................................... 174

10 Absolute value of second-order moment pointwise error obtained by adaptive GPC

and Monte-Carlo simulation (1,000,000 events), ujq = 1.0; C = 0.02; A = 1.0;

e = 0.1 (Case II: Uniform) ......................................................................................................175

1 MeshS: 412 elements triangular grid in the x — y plane; [x x y] = [(—loH ; 25D) x

i -9 D ;9 D )] 188

2 Time history of deterministic and mean lift (left) and drag (right) coefficients for

different values of p. Inflow velocity: u = u + cr-a ; u = 0; tt = 1.0 and <t„ = 0.05;

Re = 50.............................................................................................................................................188

3 Time history of the variance of lift (left) and drag (right) coefficients for different

values of p. Inflow velocity: u = u + (Jui\ u = 0; w = 1.0 and = 0.05; Re = 50. . 189


different values of cr„ at p — 15. Inflow velocity: u = u + (Jui\ v = 0; u = 1.0;

Re = 50.............................................................................................................................................189

X X V ll



values of cr a t p = 15. Inflow velocity; u = u + u = 0; u = 1.0; i?e = 50. . . . 190

6 Convergence ra te of lift (square symbols) and drag (circle symbols) coefficients at

different times {ti = 1021; t 2 == 1046; = 1071). Inflow velocity: u = u + cr„^;

u = 0; 12 = 1.0 and cr„ = 0.05; i?e = 50.....................................................................................190


different values of p. Inflow velocity: u = u + au^’, v = 0; u = 1.0 and au = 0.05;

l e = 100...........................................................................................................................................191


values of p. Inflow velocity: u = u + u = 0; 12 — 1.0 and cr„ = 0.05; Re = 100. 191

9 Comparison of normalized mean (a) and variance (b) lift coefficients for different

values of Re with same (j« = 0.05 and p = 15........................................................................ 192

10 Left: Strouhal-Reynolds number dependence for Re G [47; ISO]. Right: models of

mean (a) and variance (b) of lift coefficients for (t„ = 0.035 for two different cases:

(Re = 5 0 ; ^ = 0.1217) and (Re = 100; Rt = 0.1647)........................................................... 197

11 Time evolution of the pdf of Cl given by Equation (4) for Ap^ = 0.06, cr/„ = 0.05

and /„ = 0.13. is the deterministic shedding period....................................................... 197

12 Time evolution of the pdf of Cp given by the full scale simulation. Inflow velocity:

u = u + (Jui', u = 0; 22 = 1.0 and <Tu = 0.05; Re = 50 and p = 7. is the

deterministic shedding period..................................................................................................... 198

13 Instantaneous spatial distribution of deterministic value of vorticity at tU /D = 1096.199

14 Instantaneous spatial distribution of mean value of vorticity &t tU /D = 1096; p = 2.199

15 Instantaneous spatial distribution of mean value of vorticity at tU /D = 1096; p = 8 .200

16 Instantaneous spatial distribution of mean value of vorticity at tU /D = 1096; p = 15.200

17 Instantaneous spatial distribution of rms value of vorticity at tU jD = 1096; p = 2. 201

18 Instantaneous spatial distribution of rms value of vorticity &t tU /D = 1096; p = 8 . 201

19 Instantaneous spatial distribution of rms value of vorticity at tU /D = 1096; p = 15. 202

xxviii


20 Instantaneous spatial distribution of rms value of vorticity at 4 different instants

within one period; p = 15............................................................................................................ 202

21 Instantaneous spatial distribution of deterministic value of streamwise velocity at

t v ID = 1096................................................................................................................................. 202

22 Instantaneous spatial distribution of mean value of streamwise velocity at tV jD =

1096; p = 15..................................................................................................................................... 203

23 Instantaneous spatial distribution of rms value of streamwise velocity at tUJD =

1096; p = 15.....................................................................................................................................203

24 Instantaneous spatial distribution of deterministic value of crossflow velocity at

t U j D = i m & ....................................................................................................................................203

25 Instantaneous spatial distribution of mean value of crossflow velocity at tU /D =

1096; p = 15..................................................................................................................................... 204

26 Instantaneous spatial distribution of mean value of crossflow velocity at tUJD =

1096; p = 15.....................................................................................................................................204

27 Mesh4: 708 elements triangular grid in the x ~ y plane; [x x y\ = [(—22D;55jD) x

(-22H ;22D )]...................................................................................................................................206

28 Time evolution of span-averaged deterministic and mean lift coefficient Cl solutions

for different p. Re = 300; Uu = 0.05....................................................................................... 206

29 Time evolution of span-averaged deterministic and mean drag coefficient Cd solu

tions for different p. Re = 300; ~ 0.05............................................................................. 207

30 Time evolution of span-averaged deterministic and mean spanwise force coefiicient

Cz solutions for different p. Re = 300; (t„ == 0.05............................................................... 207

31 Comparison between deterministic (a) and mean (b-c-d) isocontours of lift coeffi

cient Cl solutions for different p; (b): p = 2; (c): p = 4; (d); p = 6.................................208

32 Comparison between deterministic (a) and mean (b-c-d) isocontours of drag coef

ficient Cd solutions for different p; (b): p = 2; (c): p = 4; (d): p = 6 ..............................209

XXIX


33 Comparison between deterministic (a) and mean (b-c-d) isocontours of spanwise

force coefiicient solutions for different p; (b): p = 2; (c): p = 4; (d): p — 6. . . . 210

34 Time evolution of spanwise standard deviation deterministic and mean lift coeffi

cient Cl solutions for different p. Re = 300; i7„ = 0.05.......................................................211

35 Span-averaged variance of lift force coefficient Cl solutions for different p. Re =

300; = 0.05................................................................................................................................. 212

36 Span-averaged variance of drag force coefficient Cd solutions for different p. Re =

300; Gu 0.05................................................................................................................................. 212

37 Isocontours of variance of lift (a), drag (b) and spanwise (c) force coefficient solu

tions for p = 2 ................................................................................................................................. 213

38 Isocountours of variance of lift (a), drag (b) and spanwise (c) force coefficient

solutions for p = 4..........................................................................................................................213

39 Isocontours of variance of lift (a), drag (b) and spanwise (c) force coefficient solu

tions for p = 6 ...........................................................................................................................214

40 Isocontour of the deterministic pressure pdet field in the near wake pdet = —0.15. . 215

41 Isocontour of the mean pressure po field in the near wake po = —0.15. Re = 300;

= 0.05.......................................................................................................................................... 215

42 Isocontour of deterministic spanwise vorticity in the near wake ^ 1................. ^16

43 Isocontour of mean spanwise vorticity in the near wake ± 1 . Re = 300; (j„ = 0.05.216

44 Isocontour of rms of crossflow velocity in the near wake Vrma = 0 .5 . i?e = 300;

(Ju = 0.05.......................................................................................................................................... 216

1 Comparison between deterministic (Top-left) and mean instanteneous vorticity

fields for different levels of noise (t„ at identical time to; = 0.1 (Top-right);

= 0.2 (Bottom-left); cr„ = 0.25 (Bottom-right). Re — 140; p = 10..............................219

XXX



fields for different levels of noise cr„ at identical time to + ^ ; au = 0-1 (Top-right);

(T„ = 0.2 (Bottom-left); au = 0.25 (Bottom-right). Re = 140; p = 10..............................219


fields for different levels of noise cr„ at identical time to + au = 0.1 (Top-right);

au = 0.2 (Bottom-left); <j„ = 0.25 (Bottom-right). Re = 140; p = 10..............................220

4 Comparison between deterministic (bottom) and mean (top) instanteneous vortic

ity field at identical time; (t„ = 0.1; Re = 400; p — 10....................................................... 220

5 Time evolution of deterministic and mean lift coefficients for different noise level

o-„; = 140; p = 10.....................................................................................................................221

6 Time evolution of deterministic and mean drag coefficients for different noise level

au, R e = 140; p = 10. . .........................................................................................................221

7 Comparison between deterministic and mean spectrums of lift coefiicientfor au =

0.25; Re = 140; p = 10..................................................................................................................222

8 Comparison between deterministic and mean spectrums of base pressure coefficient

for (j„ = 0.25; Re = 140; p — lQ................................................................................................. 223

9 Time evolution of the leading random modes for the cylinder cross-flow response p

(top) and the lift coefficient Cl (bottom )............................................................................... 225

10 Time evolution of the variance of the structural solution. Variance of the cylinder

cross-flow response p (top) and variance of the lift coefficient C i (bottom )................... 226

11 Instantaneous spatial distribution of rm s (gray scale) and mean (white line) of

cross-flow v e lo c ity ....................................................................................................................... 227

12 Instantaneous spatial distribution of rm s (gray scale) and mean (white line) of

vo rtic ity ...........................................................................................................................................227


13 Polar plots of pressure distribution on the cylinder surface relative to the cylinder

mean cross-flow position at four different instants within one shedding cycle. De

terministic pressure solution (dashed line); Stochastic pressure solution (solid line

and shaded area)............................................................................................................................ 228

14 Probability distribution function for the pressure at the rear stagnation point at

different instants within a shedding cycle................................................................................230

15 Probability distribution function for the amplitude of the crossflow oscillation rj at

different instants within a shedding cycle................................................................................230

16 Upper: Time variation of the mean base pressure (with error bars) versus the

deterministic solution. Lower: Probability distribution function of base pressure at

five time instants marked in the top plot................................................................................ 231

17 Upper: Time variation of the mean amplitude of cylinder oscillation (with error

bars) versus the deterministic solution. Lower: Probability distribution function of

amplitude at five time instants marked in the top plot....................................................... 232

1 Triangle to rectangle tra n s fo rm a tio n ...................................................................................... 234

2 Hexahedron to tetrahedron transfo rm ation ............................................................................ 235

3 The local coordinate systems used in each of the hybrid elements and the planes

described by fixing each local coordinate.................................................................................236

4 In the spectral/hp method the solution domain is decomposed into elements of

characteristic size h and then a polynomial expansion of order N is used within

every element. On the left we see a cuboid decomposed into 3072 tetrahedral

elements within which we use a polynomial expansion of order 4 as indicated by

the mode shapes on the right. (Courtesy of S.J.S. S h e rw in ) ...........................................239

5 Solution process in AfsK.'TotrJ^ ............................................................................................. 243

xxxii


List o f Sym bols

A = y/D : amplitude ratio

Ca : potential added-mass parameter

C*L = 1 --h -K - lift force coefficientipfU^D-

Cd = drag force coefficient

Cz = Xp^ui'D' force coefficient

e = y C y p A P : cable phase velocity

D: cylinder diameter

Fl : transverse lift force per unit length

Fd - inline drag force per unit length

Fz- longitudinal spanwise force per unit length

fs'. structural oscillation frequency

f n ' . natural frequency in vacuum

fy-. vortex shedding frequency

f p - . base pressure frequency

/ = f s / f n - frequency ratio

/jT,; transverse lift force frequency

/ d '- inline drag force frequency

k = 2tt : wave number response

L^: cylinder length

L = L zID : aspect ratio


m = mass ratio

(m + Ca )C'- mass-damping parameter

n: mode number response

Re = Reynolds number

S t = fyD /U : Strouhal number

U: free stream velocity

Vrn = U/fnD: reduced velocity

V — ixjpf fluid kinematic viscosity

7 = y j E I / psU^D'^: beam phase velocity

LJn = \ J ^ - natural circular frequency in vacuum

C = P- : damping ratio2 y / k p x

Pf. fluid density

Ps'. structure’s linear density

4>totoi'- to tal phase

4>vortex- vortex phasc

^ = {XjV)' non-dimensional cylinder displacement


Chapter 1

Introduction

1 Predictability and Uncertainty in Flow-Structure Inter

actions

Computational Fluid Dynamics (CFD) is a m ature discipline today. After more than forty years

of intense research efforts, starting with the seminal work of Harlow & Fromm in simulating

unsteady flow past a cylinder (1965) [4], finite differences and finite element/volume methods

are employed routinely in three-dimensional unsteady flow simulations. High-order methods and

methods for complex-geometry domains have been advanced considerably, and numerical accuracy

is adequately quantified in many simulations. While fully-adaptive simulations are limited to some

demonstration examples at the moment, at least the algorithmic framework and mesh generation

technology exist for routine adaptive CFD in the near future.

In turbulence simulations, in particular, there has been some dramatic progress, and direct

numerical simulation (DNS) has been the prevailing tool for analyzing accurately the physics of

turbulence at all scales albeit at modest Reynolds numbers. This development, which coincides

with the birth of supercomputing in the mid 1970s and has greatly benefited from it, has proved

erroneous the pessimistic forecast that simulations of turbulence based on first principles could


never be achieved [5, 6]. Fast solvers and terafiop speeds of today have allowed accurate DNS not

only in canonical domains, such as turbulent channel flows [7], bu t also in spatially developing

flows involving more complex-geometry domains.

An example of such DNS is shown in Figure 1, where we plot the results of the energy spectrum

predicted by a spectral discretization involving about 100 million degrees of freedom [1]. In par

ticular, the flow past a circular cylinder is simulated at Re = 3,900 corresponding to a turbulent

wake. In the figure we plot the one-dimensional wave number spectrum of the cross-flow velocity

at a centerline location seven diameters behind the cylinder. Both axes are normalized with Kol

mogorov scaling at R e = 3,900, and a comparison of the DNS predictions with the experimental

results of Ong & Wallace is included [8]. Very good agreement is obtained in the inertial range

and in capturing the vortex shedding and shear layer frequencies, i.e. the two pronounced peaks

in the plot of figure 1. Surprisingly, the agreement is less satisfactory in the large scales (i.e., low

wave numbers) which are clearly fully resolved.

Figure 1: Cross-flow velocity spectrum at a centerline point seven diameters behind a cylinder. DNS (thin line) versus experiment of Ong & Wallace (thick line); Re = 3,900, taken from [1].

The large scales are influenced by the domain size and by the specific form of the boundary

conditions imposed on the truncated domain. More tests with larger domains are required, but

at such high resolutions systematic tests cannot be easily performed. Moreover, even with the

larger domain it is not clear th a t matching the outer flow with the experimental conditions can


be achieved. There are raany uncertainties associated with the experiment itself th a t have not

been quantified; in fact, some uncertainties are irreducible and cannot be precisely quantified.

This example raises the issue of uncertainty in the boundary conditions and the experimental

input but there are many more sources of uncertainty th a t can be present in simulating fluid flows,

laminar or turbulent, in a simple setting or in multi-disciplinary problems. In addition to boundary

and initial conditions, there is still an uncertainty component associated with the physical problem,

and specifically with such diverse factors as constitutive laws, transport coefficients, source and

interaction terms, geometric irregularities (e.g. roughness), etc. W ith the CFD field reaching now

some degree of m aturity and with petafiop resources within the horizon, it is timely to pose the

more general question of how to model uncertainty and stochastic input, and how to formulate

algorithms in order for the simulation output to reflect accurately the propagation of uncertainty.

Assuming we can quantify numerical accuracy, the new objective is to model uncertainty from

the beginning of the simulation and not simply as an afterthought!

2 Prom D eterm inistic to Stochastic D irect Numerical Sim

ulations of Flow-Structure Interactions

In the mid-1990s, our group initiated a research program in direct numerical simulation (DNS)

of three-dimensional general flow-structure interactions. The main focus has been on simulating

vortex-induced vibrations (VIV) for flows past cylindrical rigid or flexible structures. We started

at low Reynolds number (of the order of 100 to 200) and relatively small aspect ratio (of the order

of 10). However, today based on new algorithms and faster parallel processors we simulate VIV

at Reynolds number of the order of a few thousands and flexible cylinders of aspect ratio of the

order of 1000. Progress has also been made with respect to the type of structures and inflows we

can simulate. Our initial efforts involved linear structures but more recent work has focused on

nonlinear structures [9, 10] with sag and vibrations in all three direction, i.e. including the axial

direction. We are now able to simulate complex and severe inflows as well, such as linear and


6

exponential inflows with large shear param eter [11, 12, 13] or oblique inflows [14] with angle of

yaws as large as 70°.

Our new path of research attem pts to model the various uncertainties associated with flow-

structure interaction problems. There exists numerous methods to represent stochastic processes

and random fields and to treat the problems associated with computational stochastic mechanics

[15]. The Monte-Carlo approach, which states th a t the stochastic differential equations governing

the system can be interpreted as an infinite set of deterministic differential equations, can be

employed but it is computationally expensive and is only used as the last resort. The sensitivity

method is an alternative more economical approach, based on moments of samples from which

it is possible to derive estimates of the probability distribution function (pdf) of the solution.

However, it is less robust than the Monte-Carlo simulations and depends strongly on the modehng

assumptions or closure approximations (i.e., gaussianity or non-gaussianity of the solution) [16].

W ith this method, the complexity of the moment equations dramatically increases with the order

of the closure. Markov methods are based on the theory of continuous multi-dimensional Markov

processes. The state transition probability function for such a process is governed by a linear

partial differential equation: the Fokker-Planck-Kolmogorov (FPK) equation. The solution of

this equation gives an indication of the diffusion of probability “mass” in state space [17]. The

“drift” or “diffusion” coefficients from the FPK can be directly related to the parameters of the

system. However, the class of non-linear random problems for which the FPK equation can be

solved exactly is limited. There are other more suitable methods for physical applications, and

there has already been good progress in other fields, most notably in seismology and structural

mechanics. A number of papers and books have been devoted to this subject, e.g. [18, 19, 20, 21,

22, 23, 24, 25]. The most popular technique for modeling stochastic engineering systems is the

perturbation method where all stochastic quantities axe expanded around their mean via a Taylor

series. This approach, however, is limited to small perturbations and does not readily provide

information on high-order statistics of the response.

A more effective approach pioneered by Ghanem & Spanos in the context of finite elements for


solid mechanics is based on a spectral representation of the uncertainty [21]. This allows high-order

representation, not just first-order as in most perturbation-based methods, at high computational

efficiency. It is based on the original ideas of Wiener (1938) on Homogeneous or Polynomial Chaos

(PC) and employs Hermite polynomials basis [26] as a means of solving efficiently problems with

Gaussian inputs.

We adopt this approach for representing the uncertainty spectrally, and extend it to the

Generalized Polynomial Chaos (GPC) representation. This generalization of the original PC is

not restricted to Gaussian cases and includes a broad class of polynomial basis from the Askey

scheme family. This approach is more efficient to model problems with general non-Gaussian

random inputs. Moreover, the method is convenient as it leads to a deterministic coupled set of

partial differential equations, which can be solved using standard algorithms for temporal and

spatial discretization. This is one of the most attractive features of this method, namely th a t the

deterministic solvers tha t have already been developed can be used directly in the new context.

The generalization of this stochastic approach could lead to a new generation of non-sterilized

simulations of VIV, where uncertainties in flow conditions, structural properties and support are

modeled explicitly.

3 Toward M ulti-Level Parallelization M odels for Stochastic

Flow-Structure Interactions

Realistic simulations of deterministic or stochastic flow past a flexible cylinder subject to VIV

require high resolutions in all directions of the computational domain. If a Fourier decomposition

is used along the homogeneous spanwise direction, then a large number of Fourier modes along the

cylinder span is needed and high resolutions in the streamwise and the cross-flow directions are

needed as well. Parallel computations employing a single-level parallelism for this type of prob

lems have clear performance limitations th a t prevent effective scaling to the large processor count

on modern supercomputers. Stochastic flow-structure interaction problems demonstrate natural


hierarchical structures when discretized with polynomial chaos [27] and spectral/hp element [28]

algorithms. The stochasticity is represented as a truncated series of appropriately weighted ran

dom basis functions. As a result, the Navier-Stokes equations are reduced to a set of equations for

the deterministic expansion coefficients, called random modes. The random modes axe de-coupled

from one another except for the non-linear terms. Each random mode of the flow quantities is

a deterministic three-dimensional field, and is discretized with the spectral/hp element method.

The flow domain is partitioned into spectral elements of various types. Element-wise the random

mode is expanded in terms of Jacobi polynomials. Operations across the spectral elements include

the Poisson solve of pressmre and the Helmholtz solve of velocity. Sub-elemental operations are at

still another level of the hierarchy. These include transforms between modal and physical spaces

and the derivative computations. This inherent hierarchy suggests a multi-level parallelization

strategy. At the top-most level are groups of the MPI (Message Passing Interface) processes.

Each group computes one random mode. At the next or second level, the three-dimensional flow

domain of each random mode is decomposed, via a Fourier discretization along the spanwise direc

tion, into a series of two-dimensional sub-domains of space and time, each consisting of a number

of spectral elements. Each MPI group from the second level computes one of these sub-domains.

At the third level, multiple threads (OpenMP) or MPI processes can be deployed to share the

computations within the sub-domain. Compared with the flat message-passing paradigm, the

multi-level parallelization strategy reduces the network latency overhead because a greatly re

duced number of processes are involved in the communications at each level yet with increased

message sizes. This strategy enables the application to scale to a large number of processors more

easily.

For instance, for a stochastic turbulent flow at Re = 10,000 past a freely flexible cylinder of

aspect ratio An with 35 random modes, 256 Fourier modes, with each Fourier plane decomposed in

8 subdomains, the to tal number of processors would be 71,680. This M PI/M PI/M P I three-level

parallelism with such a number of processors will enable PetaFLOPS performance.

Multi-level parallelization models already exist for deterministic flow-structure applications. Two


paradigms for multi-level parallelization have been shown to be effective in ehminating the per

formance limitation of the single-level parallelism [29]. Indeed, there exists a limitation in the

efficiency of the decomposition of the three-dimensional problem to a series of two-dimensional

problems using multiple one-dimensional FFTs: the number of Fourier planes in the homoge

neous spanwise direction imposes an upper limit on the number of precessors th a t can be em

ployed. When this maximum number is used, the performance is solely determined by the speed

of computations of the two-dimensional non-homogeneous plane. In this case, the problem comes

from the fact th a t the cost increase of grid refinement cannot be further balanced by increasing

the number of processors accordingly. Dong et al. [29] compared two multi-level paralleliza

tion paradigms to overcome this limitation. They compared an M PI/M PI two-level parallelism

with an M PI/OpenM P hybrid parallelism. W ith this approach they were able to simulate three-

dimensional turbulent flow at Re = 10,000 past a stationary cylinder of aspect ratio tt [30]. They

used 64 Fourier modes (128 spectral element planes) along the homogeneous spanwise z-direction

and a triangular grid of 6272 elements for each x-y plane . Each plane was decomposed in 8 sub-

domains and the polynomial order for each spectral element within each subdomain was p = 8 .

This computation included a to tal of about 173.4 million DOFs and ran on 512 processors.

Multi-Level parallelization models are very promising. They seem to be the answer to the

always increasing computational cost due to larger Reynolds number as well as the computational

cost associated with the higher dimensionality of the stochastic problem.

4 Thesis Outline

The research presented in P art I of this thesis represents the continuation of the effort of Blackburn,

Newman, Evangelinos and Karniadakis ([31], [32], [33], [34], [35], [36], [37]). Then, a new path

of research is presented in Parts II and III in which we consider the effect of uncertainty in

flow-structure interactions.

It focuses on:


10

• The extension of the direct numerical simulation (DNS) calculations employing linear struc

tural models to turbulent regime up to Reynolds number Re = 3000 with uniform or complex

inflow conditions, presented in Part I.

• The development of direct solvers for stochastic linear and nonlinear structural models,

presented in P art II.

• The development and coupling of these structural solvers with two and three-dimensional

DNS flow solvers for full stochastic flow-structure interaction problems, presented in P art

III.

P art I contains:

® Chapter 3: A study of uniform and oblique flows past rigid cylinders in the turbulent regime

for difiierent reduced velocities and low mass-damping parameter.

• Chapter 4: A study of linear and exponential sheared flows past rigid and long flexible cables

and beams in the turbulent/transition regime.

Part II contains:

• Chapter 5: A description of the Generalized Polynomial Chaos and Karhunen-Loeve repre

sentations.

• Chapter 6: A study of stochastic linear oscillators subject to random forcing processes and

random param etric and forcing variables.

• Chapter 7: A description of the Adaptive Generalized Polynomial Chaos method and a

study of stochastic nonlinear oscillators such as Dufflng oscillator subject to random forcing

variables and processes.

P art III contains:

® Chapter 8 : A description of the Generahzed Polynomial Chaos formulation in the context

of three-dimensional stochastic flow-structure interactions.


11

• Chapter 9: Two and three-dimensional studies of random flows past stationary cylinders in

the laminar and post-transition regime.

• Chapter 10: Two-dimensional studies of random flows past forced or free moving cylinders,

with or without random parametric forcing, in the laminar and post-transition regime.


Part I

D eterm inistic Flow-Structure

Interactions

12


Chapter 2

The Coupled Flow-Structure

Formulation

We consider the problem of deterministic flow-structure interaction for incompressible flow past

a circular, rigid or flexible, stationary or free cylinder. The equations th a t describe these prob

lems are the coupled system of the Navier-Stokes equations and a set of equations modeling the

structure. Only linear equations for the structural part will be considered in this part.

1 Structural M odel

A major factor affecting the coupled flow-structure response is the structural characteristics, i.e.

damping and bending stiffness. W hen the frequency of the structural mode of vibration is close

to the Strouhal frequency S t = f D / U of the wake, the vortex shedding frequency shifts to the

structure’s natural frequency of vibration. This phenomeilon is called “lock-in” and generally

coincides with large hydrodynamic forces and vibration amplitudes. The case of zero structural

damping results in the maximum response at resonairt conditions. The elasticity and stiffness

of the structure, on the other hand, determines the excited modal shape. Therefore, a tension-

dominated cable with zero bending stiffness will respond differently than a beam with no tension

13


14

and finite bending stiffness even at lock-in conditions. For turbulent flows, even if the first mode

or low modes dominate the response, excitation of higher modes of the structure is possible, giving

rise to significant differences in the topology and dynamics of the near-wake.

Under a small angle approximation assumption (sin0 w tan 6 and cos 9 w 1), for a flexible

cylinder under tension T, possessing non-negligible bending stiffness E l , “elastically anchored”

[38] with a spring constant k and allowed to oscillate in a transverse and inline directions only,

the dimensional equation of motion is the following [38]:

d ^ X ( Z , t ) d f d X { Z , t ) \ f d ‘ X { Z , t ) \

+ k { Z ) X { Z , t ) + R [ Z ) ^ ^ ^ ^ = F {Z ,t ) = \p fD U ^ C F { Z ,t ) (1)

where X { Z , t ) = { X { Z , t ) ,Y { Z , t ) ) is the cylinder displacement where X { Z , t ) is the inline dis

placement (along the X-direction) and Y {Z , i) is the crossflow displacement (along the Y-direction).

The variables (Y, Y, Z) represent the frame of reference in the dimensional space, ps is the mass

per unit length of the cylinder. Prom Equation (1), we see th a t the dimensional forcing per unit

length of the cylinder F {Z , t) can also be written as a function of some non-dimensional coeffi

cients C p which represent the to tal non-dimensional force coefficients. We will be more specific

about those coefficients once we will have the non-dimensionalized form of Equation (1).

We non-dimensionalize the flow problem using the fluid density p, the cylinder diameter D

and the free stream velocity U. This gives us a Reynolds number Re = where v = p /p j is the

kinematic viscosity of the fluid and z = Z jD is the non-dimensional spanwise coordinate. We use

the same non-dimensionalization for the structure. Henceforth, all flow and structure variables

employed can be assumed to be non-dimensional.

Moreover, if we suppose tha t the structural parameters are homogeneous along the span of

the cylinder and if we have;

R = 2psUinC and ^ = ^nPs with = 27t/„, (2)


15

then, the Equation (1) becomes:

( A-nQ\ d^{z,t) f 2t t ^ I C p iz^ t)dt^ dz^ d z ‘ V Vrn ^ dt \ Vrn ’ 2 m ’

where ^{z , t) = {x{^jt),ri{z,t)) is the cylinder displacement with x (2 , t) being the inline displace

ment and r]{z,t) being the crossflow displacement, ( is the damping fraction, and = U/ fn D is

the reduced velocity based on the free-stream velocity U and the natural frequency /„ in vacuum

of the structure. The mass ratio (non-dimensional linear density) is m = p g /p fD ‘ where pu is

the structure’s linear density. The coefScients C p {z , t ) = (Cj:){z,t),CL{z,t)) are the to tal non-

dimensional forces, i.e., including both pressure and viscous contributions. They represent the

drag coefficient Co for the inline motion and the lift coefficient Cl for the crossflow motion of the

structure. They are obtained through the flow solver. Also, c = ^/T /psU ^ and 7 = ^jEI/psU'^D'^

are the non-dimensional cable and beam phase velocity, respectively. Practically, the structural

nature of the body is defined based on the simple criterion developed by Vandiver [39]). The

structure is said to be a cable if it is tension-dominated, i.e.

^ > 3 0 , (4)Elk"^

otherwise it is said to be a beam and its bending stiffness should be taken into account. Here k is

the wave number describing the excitation mode.

If the cylinder is rigid (or in the case of a two-dimensional simulation) and is only allowed

to move in the crossflow direction, its motion has no spanwise z-dependence. Its displacement is

the solution of a single degree of freedom viscously damped second-order oscillator subject to the

external hydrodynamic forcing, i.e.

where rj represents the crossflow cylinder response and Chit) is the spanwise-averaged local lift


16

coefficient.

The representation of Equation (3) in Fourier space becomes:

. ( 0 ’ « M ) = i S h p ) . ,e ,

where the wave number k = and L z is the length of the structure in the spanwise direction.

For the n-th Fourier mode of the structure, we have:

The nice feature of this representation is th a t the homogeneous parts of the equations for the

modes are decoupled (any coupling comes from the external force term ). In the absence of an

external forcing, the natural frequency of the oscillation for the n-th Fourier mode is:

If the structure is pinned at both ends (zero Dirichlet: ^ = 0), then a Fourier sine series can

be used instead and the equations become:

(i) i t ) ®for the n-th sine Fourier mode (sin{kz)) of the structure where the wave number k = This

is accomplished by projecting the forces into a Fourier sine-series th a t gives zero contributions

at the two ends. This time, in the absence of an external forcing, the natural frequency of the

oscillation for the n-th sine Fourier mode is:


17

In the case of flow past a flexible cylinder th a t we are considering, the lift forces display

oscillations of frequency ~ S t (Strouhal frequency) and so resonance (lock-in) requires th a t »

2-nSt.

For the initial boundary values problem of Equation (1) we need initial conditions for the

displacement and the velocity of our structure. For uniform inflows past cylinders with reduced

velocity close to the Strouhal frequency, the forcing term dominates the behavior of the system

and the initial conditions usually only affect an initial period of transient response. However, in

the case of shear inflows, or in the case of reduced velocities away from the classical lock-in region,

we will see (see Chapter 4) th a t initial conditions can be crucial in the response of the system.

We now examine the relevant boundary conditions for Equation (3) and its derived equations.

We have a choice of several boundary conditions for strings (wave equation - see [38]) for which

7 == 0.0. We can use pinned or fixed (zero Dirichlet: ^ = 0), free (zero Neumann: = 0),

springy ( ^ -|- = 0) or damped i T ^ + = 0) boundary conditions. If a fourth derivative

in space term exists (7 / 0 .0 ), we need additional boundary conditions on higher derivatives, for

a simply supported (pinned) beam (^ = = 0), a clamped beam (^ = lif = 0) and a free beam

( | ^ = = 0)-

In previous works ([32], [33], [35]) and [36] most of the simulations used (free) periodic bound

ary conditions. These seem to favor the traveling wave response for a flexible cylinder in VIV.

In this work, we wiU consider free rigid structm es (periodic boundary conditions) or free flexible

structures with fixed pinned ends (zero Dirichlet: ^ = 0 and zero curvature: ^ = 0).

2 Formulation of the Mapping Approach

We use a Newmark integration scheme [40] for solving numerically Equations (7). Since the

equations for the Fourier modes are decoupled this will be the same ODE solver as the one used

in the case of the 2D flow-structure interaction problem.

- I *To update the state of this system at timestep 1, (^„ ^$,n) using the forcing at time level


we use the following scheme:

18

1+1 - l+l sn — 1? — m

1 5t^n + 2

' X *■

r -• ■■ i+i-(11a)

snU

i+i-

+ Cn 2

(11b)

^ I“ $n F St €n A Ctj

2(11c)

or more compactly

A

- - + 1 - I -

L i n Fn

= B in1

H----m 0

i n i n 0

i+i

( 12)

The 3 x 3 m atrix A is easily invertible analytically. The Newmark scheme is unconditionally

stable, second order accurate. If instead of is used (because we are lagging the flow

solver), an order 0{5t^) error is introduced for - this is less than the order of the scheme,

incurs an extra 0{5t^) error and only the extra error in {0{5t)) is higher than second order.

In a stationary, Cartesian coordinate system {x ' ,y ',z ') the non-dimensionalized incompressible

Navier-Stokes equations (in convective form) are:

0 u 'dt'

+ (u' • V) u'

V - u ' = 0,

(13a)

(13b)

where Re = U d/v is the Reynolds number based on the free-stream velocity U and the cylinder

diameter d; v is the kinematic viscosity.

We now consider a coordinate system attached to the moving cylinder. This maps the time-

dependent and deforming problem domain to a stationary and non-deforming one as shown in


19

►

Figure 1; The coordinate system is attached to the moving flexible cylinder, producing an undeformed, stationary computational domain.

Figure 1. A convenient mapping is described by the following transformation;

X =

y'

t = t'.

(14a)

(14b)

(14c)

(14d)

which changes our partial derivative operators as following:

d ddx' dxd d

dy' dyd d d x d drj d

dz' dz dz dx dz dyd d drj d

dV ^ dt~~ dt dx dt dy

(15a)

(15b)

(15c)

(15d)

Accordingly, the velocity components and pressure are transformed as follows:

> d x

, drj

,d xdz

,drjd z '

(16a)

(16b)

w = w \ (16c)


20

p = p (16d)

The Navier-Stokes equation and continuity equation are transformed to:

du~E

-f (u ■ V )u = — Vp + Re V u + A{Re, u, p, ^ ) ,

V ■ u 0 ,

(17a)

(17b)

where the forcing term A(i?e, u ,p , ^) is the extra acceleration term introduced by the transfor

mation, consisting of both inviscid and viscous contributions. In 2D flow, A{Re, u ,p , £) has a

very simple form:

A = (18)

which is not x or y dependent. For 3D flow A{Re, u ,p , ^) has a more complicated form:

92A-x

A y

A .

dD Re

d S J _dfi ^ Re

d '^u 9 x „ 2

9z^ dz dtdz'^

d'^v dp 2 , ,3 1I -V w +

9^2 dtdz^

d x dp dp dp d^w d^w dz''^ dz'^dz dx dz dy

where for a more compact form of the equations we denote:

ddt

d d dt dx

'2 _ 92 92xy 9a;2 ^ 9p2

9 9

=

(19a)

(19b)

(19c)

(20a)

(20b)

Note that the incompressibility condition is unchanged in both 2D and 3D:

du' dv' dw' ^ du dv dwdx' ^ dy' dz' d x ^ dy~^ d z '

This is because the mapping produces no mesh divergence, f | + |^ = 0

(21)


21

3 Limitations of the M apping Approach

This particular mapping assumes th a t no deformation of the mesh occurs in the z— (spanwise)

direction which means th a t we disregard any longitudinal oscillations of the flexible cylinder.

Moreover no rotation can be handled and therefore it cannot be used when any torsional effects

for the structure need to be taken into account. Because this mapping is a translation in the x —

and 2/—directions the intersection of the cylinder with the x ~ y plane will always be circular. As

a cylinder flexes though, its circular cross-section is preserved only in a plane perpendicular to

the tangent to its deformed centerline. The corresponding intersection with the x — y plane is

actually an ellipse with axes of size:

(22a)

So long as and remain small the difference between an ellipse and a circle is small enough

to be neghgible for our purposes. This however is the same small angle/small amplitude approxi

mation that is required for Equations (1) to hold. Another assumption of the mapping approach

is tha t the computational domain is large enough so th a t the boundary conditions at the far field

do not enter anywhere in the formulation. As the un-mapped computational mesh moves in space,

it covers spatial regions th a t it did not contain at previous times and for which primitive variables

information is missing.

The transformed Navier-Stokes/structure dynamics equations are discretized in space using an

spectral/hp element based method [28], th a t employs a hybrid grid in the x — y plane. The method

uses Fourier complex exponentials along the spanwise z—direction and is covered in Appendix

A. The time integration algorithm uses the three-step time-splitting strategy for advancing the

Navier-Stokes equations in time using a stiffly stable time integration scheme.


Chapter 3

Uniform and Oblique Flows past

Rigid Cylinders

1 Correlation Length and Force Phasing

1.1 Introduction

The apparently simple case of VIV of an elastically mounted rigid cylinder constrained to move

tranversely to a uniform flow remains of practical and theoretical importance. The distinction

between the different types of response as a function of the nominal reduced velocity Vm depending

on whether the cylinder has a high or low mass-damping param eter has been described in the

literature [41, 42, 43] and shows good agreement. In the case of a low mass-damping, it seems th a t

there exists three different branches of response as a function of the reduced velocity Vm- For low

reduced velocities, there exists an initial hiaxich associated with a 2S vortex wake mode (two single

vortices shed per cycle). For intermediate and larger reduced velocities there exists an upper and a

lotcer branch associated with a 2P vortex wake mode [41, 42] (two pairs of vortices shed per cycle).

The complex mechanisms tha t induce the mode transitions between the different branches remain

to be explained. In particular, the mechanisms tha t induce the well-known ’phase-jump’ (jump in

22


23

the phase between the cylinder displacement and the lift force) th a t occurs during the transition

from the upper to the lower branch are not well understood. Other experiments indicate the

existence of a reduced velocity region containing the Strouhal frequency, for which a sharp drop

in the spanwise correlation of the flow quantities in the wake and the forces is observed [43]. This

region stands mainly on the right side end of the upper branch (large amplitude response) near the

transition between the upper and the lower branch (small amplitude response). Therefore, this

decrease in the spanwise correlation does not preclude a large response from the structure. The

study of the forces and phasing between cylinder displacement and forces in this region are key to

the understanding of the mode transition. We propose to investigate the existence of this region

of poor correlation using correlation length computations and phase analysis of DNS results.

The fluctuations of the lift forces acting on a free rigid cylinder subject to VIV depend on the

degree of three dimensionality present in the near wake of the body. The spanwise correlation

length of the flow gives an accurate measure of this three dimensionality and consequently gives

some indications of the magnitude of the cylinder response amplitude. However, this information

is incomplete as it does not supply a measure of the phase difference between structure motion

and lift forces for instance.

Experimental studies have been limited so far in terms of force measurements. They are limited

to measurements at both ends of the cylinder [42, 43] or on an elemental slice of the cylinder [44, 45]

(stationary cylinder). Direct numerical simulations provide us with an alternative tool capable of

accurate spatial and temporal representations of both the pressure and viscous force contributions.

It is therefore possible to compute the correlation length of the forces along the spanwise direction

and to relate it to the correlation length of the flow quantities in the near wake.

In studies of VIV the value of spanwise correlation length is very im portant as many empirical

models rely on it [46]. It is also im portant in numerical simulation studies as it provides guidance

for the choice of spanwise numerical resolution; mesh refinements along the spanwise direction

are often overlooked as the strongest flow gradients occur along the streamwise and crossflow

directions. However, related studies and experimental measurements of correlation length are


24

relatively few [47, 48, 49, 50]. They generally agree with the idea th a t increasing the amplitude

of motion, either through forced or free vibrations, increases spanwise coherence. For stationary

cylinders, experimental measurements of correlation length based on the autocorrelation function

were obtained only recently [51, 52]. For moving, rigid, free cylinders, detailed measurements of

cross-correlation between lift forces measured at the two ends of the cylinder can be found in [43].

The influence of nominal reduced velocity on the correlation length and phasing of the near wake

flow remains to be explored.

1.2 Simulation Param eters

Here, we present DNS results of vortex-induced vibrations of a smooth rigid cylinder with aspect

ratio L /d = 26 and mass ratio m = 2. The Reynolds number is taken to be Re = U d /v —

1000, 2000, 3000. It is in the subcritical range resulting in a turbulent near-wake.

This increase in Reynolds number was made possible by the advances in the computer hardware

(an increase of the Re by a factor 2 would approximately require one order of magnitude increase

in CPU ressources [6]) and by use of a larger and more refined grid (Mesh2, see Figures 2 and

3) compared to the grid (Meshl, see Figure 1) used in previous work ([36], [37]). Results for

Re = 1000 were obtained using M eshl with variable spectral order such as described in [36, 37].

Results for Re = 2000 and 3000 were obtained using Mesh2 and corresponding p-refinement in

space, see Figure 4. Computations for R e = 3000 were done with 32 or 64 Fourier modes along

the spanwise direction in order to access the sensitivity of the response to a change in the spanwise

resolution. Mesh2 is conceptually very similar to Meshl. However, it is worth mentioning th a t

Mesh2 contains more than twice the number of elements than M eshl. Moreover, in the near

wake of the cylinder for Mesh2, the transition in element size between the triangular unstructured

elements and the quadrilaterals structured elements is much smoother than the transition for

M eshl. Four layers for Mesh2, instead of three for Meshl, of thin quadrilaterals are used next

to the cylinder. The thickness of the first layer is d = 1.25%D for Mesh2 instead of d = 3.5%D

for Meshl. This refinement (combined with a p-refinemant) is sufficient to resolve the cylinder’s


25

boundary layer within the first layer.

y::: g \ / | V v M v i \

Figure 1: M eshl: 1018 elements hybrid grid in the x — y plane; [x x y] — [(—22D;69D) x (—22D; 22£>)]. Fourier expansions are used in the periodic spanwise direction (perpendicular to the x — y plane

Figure 2: Mesh2: 2340 elements hybrid grid in the x — y plane; [x x y] = [(—20D ;80il) x (-30H ; 30jD)]. Fourier expansions are used in the periodic spanwise direction (perpendicular to the x — y plane).

We set the structural damping ( to be zero for all numerical simulations of Sections 1 and

2 of Chapter 3 and 4 as we are interested in the maximal response of the system. We consider

that this choice of mass ratio and structural damping puts us in the low mass-damping parameter

range. We only consider the dominant motion in the cross-fiow direction (y-direction) and we

preclude any motion of the structure in the streamwise direction (x-direction). The cylinder is

rigid and thus its motion has no spanwise 2-dependence, see Equation (8), Chapter 2.

We test different reduced velocities for the natural frequency of the oscillator. We choose

our reference reduced velocity W = We/ = 4.18 to be based on the Strouhal frequency of the


26

Figure 3: Close up of the discretization of Mesh2 around the cylinder.

Expansion order

-20 -10 0 10 20 30 40 50 60 70 80x/D

Figure 4: Mesh2: Variable expansion order in the x — y plane for R e = 2000 and Re = 3000.


27

two-dimensional stationary cylinder wake [36]. We choose the other reduced velocities to be above

and below the reference value.

We define our autocorrelation function as follows:

= ( A S f . . - M ) | ^M ■

M T y f = i U ^ { x , y , z , t )

The autocorrelation function is therefore computed by shifting the signal to obtain u(x, y , z —

l ,t), multiplying it by the unshifted sequence u { x ,y , z , t ) then summing all the values of the

product and normalizing. The bar denotes the final averaging over time. The signal u{x, y, z, t)

is the fluctuation obtained after we subtract the mean quantity, i.e. averaged value of u{x, y, z, t)

in time for each spanwise (z) location a t the (x, y) point. We allow the shift I to be:

MI = [0 ,dz /d ,2dz /d , . . . , — dz/d] with dz — L /M . (2)

We have M = 64 points in the z-direction. Spatial spanwise periodicity of the quantity u(x, y, z, t)

is used to wrap up the signal in order to keep the same number of terms in the summation

for all shifts. W ith our definition, the autocorrelation is symmetric around I = 0. We use

this autocorrelation function to compute hydrodynamic force correlations on the structure as

well as velocity correlations at two locations in the near-wake. One point is on the centerline

(x /d ~ 3.0; y /d = 0) and the other one cylinder diameter above it {x/d = 3.0; y /d = 1.0).

Taking the time average of the correlation coefficients might mask some of the fluctuations

of the flow quantities in time. Similarly, computing the spectrum of the velocity components in

the wake will not provide a good description of the signals. In particular, in the case of multi

frequency response or beating phenomena, short time integration of the correlation coefficient or

phase drift angle analysis might be necessary [50].


28

1.3 Com plex D em odulation Analysis

Multi-frequency responses and beating phenomena demand a time-varying description of the phase

difference between the cylinder displacement and the lift force. To quantify the phase difference we

employ complex demodulation analysis, which is a more general approach than harmonic analysis

in dealing with non-exact periodic time series [53]. A complex demodulation of a time series

Ci^{tn,z) (lift coefficient time series a t location 2; in the spanwise direction) with a dominant

frequency component Xc^ (obtained by taking the F FT of the span-averaged signal) will give a

time varying amplitude (f, z) and phase (t, z) such that:

CL(tn,z) « (3)

Time-dependent amplitude and phase of the signal at time t are determined only by the signal

in the neighborhood of t. The procedure uses a linear filtering th a t can be tuned by choosing

some free parameters th a t control the width and the shape of the filter. This process is repeated

for the time-series at each z-location along the span. Similarly, we would have for the cylinder

displacement;

However, here we have no ^-dependence because the cylinder is rigid. We define the phase

difference A $ as being:

A ^ ( t n , z ) = - ^CL(^n,z)- (5)

1.4 Results and Discussion

We present numerical results for Be = 1000,2000,3000. We investigate a to tal of seven different

reduced velocities Vm = [3.76,4.18,4.62,4.99,6.0,7.0,8.0] for the oscillator, see Table 3.1. These

values should be immediately adjacent (from below) or included in the region of poor correlation

[43] or included in the lower branch response. The idea is to start from the region of high


29

correlation with Vm = 3.76 and increase the value of Vm toward Vm = 4.99, referred as CaselV,

and see if we experience a drop in the spanwise correlation of the flow. Then, continuing to

increase the reduced velocity toward the lower branch will show if we recover larger correlation

values. These nominal reduced velocities are defined based on the natural frequency / of the

structure in vacuum. The reference reduced velocity Vm = 14-e/ = 4.18, referred as Casell, was

already investigated [36] but for a somewhat shorter cylinder {L/d = Vk).

Re=1000 Re=2000 Re=3000Vrn Amax/D Amax/D Amax / D

Casel 3.76 0.599Casell 4.18 0.736 0.7458 « 0.7*; 0.481C asein 4.62 0.766CaselV 4.99 0.76 0.805 0.832*; 0.87CaseV 6.00 0.539 0.706 0.709CaseVI 7.00 0.554 0.54 0.546*Case VII 8.00 0.536

Table 3.1: Average one-tenth highest amplitude of response Amax/D versus nominal reduced velocity Vm and Reynolds number Re. * indicates simulations performed with 64 Fourier modes along the spanwise direction. Other simulations were performed using 32 Fourier modes.

V elocity C orrelation C oefficients

The autocorrelation function \Ruu\ (see Figure 5, first row) for the centerline point becomes

very small and close to zero around d z /d = 7.0 for all cases except for Vm = 6.00 (CaseV)

and Vrn = 7.00 (CaseVI). The function Run becomes negative for d z /d > 7.0 for the case of

Vrn = 3.76 (Casel). For the off-centerline point, only Casel-V-VI exhibit a larger correlation than

the other cases, with an almost constant value between 0.3 and 0.4. The other cases drop quickly

to small values, and Casell is the only one to present negative values of for shifts in the range

dz/d G [1.8; 3.5]. This is consistent with the results by Evangelinos [36].

The autocorrelation function (see Figure 5, second row) indicates a high degree of cor

relation which is expected for a rigid cylinder subject to VIV with reduced velocities close to

the Strouhal frequency. It is very clear for both the centerline and off-centerline points th a t an

increase of the reduced velocity, in this case from Vm = 3.76 to Vm = 4.99, drastically decreases

the spanwise correlation of the cross-flow velocity in the wake of the cylinder. For the centerline


30

point, Casel shows higher correlation than Casell across the entire domain but the difference is

more pronounced for d z jd < 6 . The autocorrelation function for Caselll (corresponding to

Vrn = 4.62) and CaselV show similar decay for shifts smaller than d z /d 4. Then they deviate

considerably, and R w for CaselV takes negative values for d z /d € [4.0; 9.0]. Eventually the two

functions reach a similar value of = 0.1 for d z /d 11. The other main finding is th a t we

recover large correlations when we increase the reduced velocity from 4.99 to 6.00 and then 7.00.

This is true for both the centerline and off-centerline points. Interestingly, it seems th a t correla

tions for cases from the lower branch are slightly larger than correlations for cases from the initial

branch or left side of the upper branch regions.

Finally, the values for \Ry]w\ (see Figure 5, last row) are comparatively much smaller for all

reduced velocity cases, especially in the case of the centerline point where they drop to very small

values after d z /d > 1.0. For the case of the off-centerline point, Rww becomes negative for all

tested reduced velocities for d z /d e [1; 5] before going back to very small values around zero. This

was not the case with a shorter cylinder for Casell, see [36].

Figure 6 shows the autocorrelation functions for five different reduced velocities for Re = 2000.

The results are very similar to the results for Re = 1000 at the same reduced velocities. The only

striking difference is for the case of Vm = 6.0 (CaseV). For this reduced velocity with Re = 2000,

there is a sharp drop (but somewhat milder) in the correlations as in CaselV. This could indicate

th a t the location of the region of poor correlation depends on the Reynolds number. In this

particular case, it seems th a t the region is shifted to the left for R e = 1000 compared to Re = 2000.

A reduced velocity of Vm = 6.0 would be part of the lower branch for Re = 1000, which would

explain the large correlations. However, it would still be part of the right-hand side of the upper

branch for Re = 2000, which would explain the low correlations. This discrepancy needs to be

investigated further as the difference could also be due to an insufficient spanwise resolution in the

case of Re = 2000. The sensitivity of the response to the spanwise resolution will be demonstrated

later in the case of Re = 3000.

Overall, these results are in good qualitative agreement with experimental results of wake cor


31

relation (F. S. Hover, private communication). Also, the results for Casell are in good agreement

with the experimental results for oscillating rigid cylinders [47].

Force C orrelation Coefficients

We compute the average one-tenth highest amplitude of response Amax I diox the different reduced

velocities and Reynolds numbers, see Table 3.1 and Figure 10.

For the case of Re = 1000, the response is noticeably larger for the values of reduced veloci

ties larger than Casel. Surprisingly, the response remains large for Caselll-IV even if the wake

correlation coefficients present a sharp drop as described in the previous section. Regarding the

magnitude of the lift forces, we notice a decrease of the fluctuations of the span-averaged lift

coefficient as we increase the reduced velocity. The time evolutions of the average lift force for

Casel and Casell are almost equivalent but the forces for Caselll and particularly CaselV present

a clear weakening and exhibit a beating phenomena. The maximum instantaneous lift amplitude

is obtained for Casell and the minimum for Casel. Generally, if the cylinder is long compared to

the typical length over which the correlation coefficients remain large, not all vortices cause forces

in phase with each other, and the net exciting force is smaller. Consequently, we also compute

the force correlation coefficients in a similar manner as the wake correlation coefficients. Figure

8 shows the force correlation coefficients. The left plot shows the lift correlation coefficients and

the right plot shows the drag correlation coefficients. Forces are integrated quantities of the flow;

therefore we expect to have larger and smoother correlation coefficients than in the case of ve

locity correlation computed at some pinpoint location in the wake. The results are very similar

to the ones for the correlation in the near wake, in the sense th a t there exists a clear drop in the

spanwise correlation of both drag and lift forces as the reduced velocity is increased from Casel to

CaselV. Then, the force correlations return to large values when the Vm is increased from CaselV

to CaseVI.

The amplitude responses for Re = 2000 are generally similar to the ones for Re = 1000; the

only noticeable difference being for Vm = 6.0 where the response is much lower for Re = 1000.


32

C enterline point; x/d=3.0, y/d=0

C enterline point; x/d=:3,0, y/d=0

Off centerline point: x/d=3.0. y/d=l.O

rfi ill ift A i. ■

Off centerline point; x /d=3,0, y /d=1 ,0

Cenlertine point: x/d=3,0, y/d=0Off centerline point; x /d=3.0, y/d=1.0

Figure 5: Streamwise, cross-flow and spanwise velocity autocorrelation functions in the near-wake at a centerline point {x/d = 3; y /d = 0) (left column) and an off-centerline point {x /d = 3; y /d = 1) (right column), for six different reduced velocities; Re=1000.


33

C e n te rlin e po in t: x /d = 3 .0 , y /d=0 O ff c e n te rl in e po in t: x /d = 3 .0 . y /d= 1 .0

Vn=4.18V ^ = 4 .9 9

V ^ = 6 .0

v|"=7.0v '"= 8.0

Vr^=4.18 V%,99 V ^ = 6 .0

v|| =:7.0 V = 8 .0

10 12

C e n te rlin e po in t; x /d = 3 .0 , y /d= 0 O ff c e n te rl in e po in t: x /d = 3 .0 , y fd= 1 .0

V ^ = 4 .1 8

V ,„=8.0

d z /d d z /d

C e n te rlin e point: x /d = 3 .0 , y /d= 0 O ff c e n te rl in e po in t: x /d = 3 .0 . y /d = 1 .0

V"=4 3,0

Vr =7-0 V =8,0

V. =6 .0

V =7 .0

V = 8 .0

Figure 6 ; Streamwise, cross-flow and spanwise velocity autocorrelation functions in the near-wake at a centerline point {x/d = 3 ;y /d = 0) (left column) and an off-centerline point { x /d = " i \ y /d = 1) (right column), for four different reduced velocities; Re=2000.

Reproduced witfi permission of tfie copyrigfit owner, Furtfier reproduction profiibited witfiout permission.

34

C e n te rlin e po in t: x /d= 3 .0 , y /d= 0 O ff c e n te r l in e po in t; x /d = 3 .0 , y/'d=1.0

IQ-’

3 10'

V =4,i

d z /d d z /d

C e n te rlin e po in t: x /d= 3 .0 , y /d=0 Off c e n te ilin e poin t: x /d= 3 .0 , y /d = 1 .0

10 '

d z /d

C e n te rlin e po in t: x/ds^3.0, y /d= 0 O ff c e n te r l in e poin t: x /d = 3 .0 , y /d= l.O

v| "=s.ov' =7.0

Vmv e.ov' =7.0

Figure 7; Streamwise, cross-flow and spanwise velocity autocorrelation functions in the near-wake at a centerline point (x /d = 3; y /d = 0) (left column) and an off-centerline point (x /d = 3;y /d = 1) (right column), for four different reduced velocities; Re=3000.


35

R e = l 0 0 0 ; Lift C oeffiderrt R e = 1 000; D rag C o e ffid sn t

0.3

Figure 8 : Force correlation coef&cients for four different reduced velocities. Lift coefficient autocorrelation function (left plot). Drag coefficient autocorrelation function (right plot); Re = 1000.

This is consistent with our hypothesis mentioned earlier about the Reynolds number dependence

of the region of poor correlation. We see th a t there is a small increase of the response with

Reynolds number for Casell and CaselV. Some of the simulations for Re = 3000 are carried out

with different spanwise resolutions (see Table 3.1). An increase in the spanwise resolution does

not seem to affect strongly the response except for Casell. In this case, the time integration of

our simulation w ith high spanwise resolution was not long enough to produce accurate statistics,

but the trend is clear as seen in Figure 9.

Figure 9: Cylinder responses using different spanwise resolutions for Vm = 4.18; Re = 3000.

Overall, these results are in good qualitative agreement with experimental results of cross-


36

S

A v e ra g e o n e - te n th h ig h e s t a m p iitu d e of r e s p o n s e

0.8

0 .7

0 .5

A v erag e + R e 2 0 0 0

+ RelOOO0 .4

e x tra p o la tio n0 .3

Figure 10; Amplitude response as a function of reduced velocity Vm for different Re.

correlation coefficients between forces measured at the two ends of a free rigid cylinder in the

same reduced velocity range [43].

P hasing A nalysis

We perform complex demodulation analysis of the cylinder displacement and lift force signals for

Casell and CaselV for Re = 1000 and CaseVI for Re = 1000,2000. Our goal is to establish a

relationship between force amplitude, cylinder displacement and phase difference between the two

signals. This method turns out to be very useful in particular for multi-frequency response system

presenting beating phenomena.

Figure 11 and 14 show the results for Casell. Figure ll-(a ) and ll-(b ) show the isocontours of

the cylinder cross-flow displacement r){t, z) and its corresponding demodulated amplitude, respec

tively. The demodulation frequency is A,, = 0.19672. We see th a t the method isolates the region

of larger response but the fluctuations are small in this case as the beating is not very pronounced.

Similarly, Figure ll- (d ) and ll-(c ) represent the isocontours of the lift coefficient CL{t,z) and its

corresponding demodulated amphtude signal respectively. The demodulation frequency remains

the same with Ac^ = 0.19672. The correspondence between the two plots is striking and we can

see regions of almost zero amplitude in Figure ll-(c ) tha t correspond to regions of very small


37

forcing on Figure ll-(d ). For instance, there are two spots with low (blue color) amplitude values,

around (t « 400, z /d w 9) and around {t » 435, z /d « 10) th a t m atch the same locations on

Figure 11-(d). Finally, Figure 11-(e) shows the phase difference (in radians mod 2tt) between

the two demodulated signals. For most of the domain, the cylinder displacement and lift forces

are in phase and A ^ { t , z ) « 0, except at three locations, and in particular at the location of

the two spots mentioned above. At these locations, the phase difference is smaller than vr and a

positive value means th a t the lift force signal is laging the cylinder displacement. The complex

demodulation proves th a t there is in general a good phasing between displacement and lift force

for Casell.

Figure 12 shows the results for CaselV. For this reduced velocity, we expect the three-

dimensionality of the flow to be more developed and consequently, the forces to be less organized

along the span. Figure 12-(a) and 12-(b) represent the isocontours of the cylinder cross-fiow

displacement r]{t,z) and its corresponding demodulated amplitude, respectively. The demodu

lation frequency is A,, = 0.1844. The method isolates the alternation between regions of larger

and smaller response, and fluctuations are larger in this case as the beating is more pronounced.

Similarly, Figure H -(d) and ll-(c ) represent the isocontours of the lift coefficient CL{t,z) and

its corresponding demodulated amplitude signal respectively. The demodulation frequency re

mains the same. This time, we see th a t the maximum lift coefficient is subject to very large

modulation. We see a very clear correspondence between alternated streaks of small and large

amplitude in Figure ll-(c ) corresponding to regions of small and large forcing on Figure H-(d).

Finally, Figure 11-(e) shows the phase difference (in radians mod 2tt) between the two demodu

lated signals. Again, regions of high lift amplitude forces correspond to regions where the cylinder

displacement and lift forces are in phase and A $ (t, z) « 0. More interestingly, regions of low lift

amplitude forces correspond to regions where the cylinder displacement and lift forces are out of

phase and A $ (t, z) tt, which explains the overall drop of spanwise correlation. Indeed, strong

three-dimensionality in the wake (see Figure 14) influences the topology of the vortices inducing

forces that are not in phase with each other. The net exciting force becomes smaller as well as


38

the cylinder motion.

The complex demodulation analysis shows th a t regions of low lift force coincide with regions of

poor phasing. Moreover, it establishes a close relationship between those forces and the cylinder

displacement. Regions of large forces occur at the time or slightly before regions of large cylinder

displacement. More generally, the method gives a striking visual demonstration of the simultaneity

of the loss of force correlation and the phase angle transition and relates it to the loss of wake

correlation.

1.5 Summ ary

We presented DNS of uniform flow at Reynolds number Re = 1000,2000,3000 past a flexibly-

mounted rigid cylinder subject to VIV. We investigated different nominal reduced velocities near

or in the region of maximum amplitude response for a low mass-damping with zero structural

damping. We focused in particular on the correlation length of the flow quantities in the near wake

and relate it to the force correlations along the cylinder. We also performed complex demodulation

analysis to quantify the phase difference between structural displacement and forces.

Those results are in good agreement with the experimental results by Hover [43]. We confirmed

tha t there exists a reduced velocity range, near the mode transition from the upper to the lower

hysteretic branch, and very close to the Strouhal frequency, for which a severe drop in the spanwise

correlation of the flow quantities in the near wake and the forces is observed. Surprisingly, however,

the cylinder response remains large in comparison to the response in the initial or lower branches.

Both forces and cyhnder response demonstrate beating. There exists a correspondence between

a strongly three-dimensional wake with low correlation and phase angle transition, and a loss of

force correlation. The topology of the wake vorticity in the wake is very complex and consists of

multiple irregular cells inducing forces tha t are not in phase with each other. Large forces exist

where the phase difference between cylinder displacement and lift force is close to zero and small

forces exist where the phase difference is close to tt. Otherwise, some noticeable differences exist

between the cases with Re = 1000 and Re = 2000 — 3000. In particular, the average one-tenth


20

10

20

10

20

10

39

(a)

360 380 400 420 440 460(b)

0.5

0

- 0.5

0.74

0.72

0.7

360 380 400 420 440 460

360

P i l llllaiiiiM li l i l l l360 380 400 420 440 460

(e)

360 380 400fU/d

420 440 460

Figure 11: Complex demodulation analysis of Case II, Re = 1000 and Vm = 4.18. Isocontours of cross-flow displacement rj (a). Isocontours of amplitude of demodulated cross-flow displacement Rrj (b). Isocontours of amplitude of demodulated Lift coefficient (c). Isocontours of Lift coefficient Cl (d). Isocontours of phase difference A $ between demodulated cross-fiow displacement and demodulated lift coefficient (e).


40

20'

10

620 630 640 650 660 670 680 690 700 710 720(b)

20

10

620 630 640 650 660 670 680 690 700 710 720

620 630 650 660 670 680 690 700 710 720(d)

20

10

20

10

620 630 640 650 660 670 680 690 700 710 720tU/d

0.5

- 0.5

0.75

0.7

0.65

0.6

620 630 640 650 660 670 680 690 700 710 720(e)

B

Figure 12; Complex demodulation analysis of Case IV, Re = 1000 and Vm = 4.99. Isocontours of cross-flow displacement i] (a). Isocontours of amplitude of demodulated cross-flow displacement

(b). Isocontours of amplitude of demodulated Lift coefficient R c^ (c). Isocontours of Lift coefficient C l (d). Isocontours of phase difference A $ between demodulated cross-flow displacement and demodulated lift coefficient (e).


41

770 780 790 800 810 820

(c)

........

: i t W I f c i iM n . “ ^ ^

„ ' J . _ . _ , 1 - l l ^ M ^ j : . K - ^ Z770 780 790 800 810 820

(d)

0.4

0.2

0

- 0.2

- 0.4

0.55

0.54

0.53

0.5

- 0.5

0.5

0.4

0.3

0.20.1

^ | c

Figure 13: Complex demodulation analysis of Case VI, Re = 1000 and Vm = 7.00. Isocontours of cross-flow displacement r] (a). Isocontours of amplitude of demodulated cross-flow displacement Rn (b). Isocontours of amplitude of demodulated Lift coefficient Hci, (c)- Isocontours of Lift coefficient Cl (d). Isocontours of phase difference A # between demodulated cross-flow displacement and demodulated lift coefficient (e).


42

Figure 14; Pressure isocontour at value —0.15 in the wake of a free rigid cylinder at Re=1000 (Case IV: Vm = 4.99). View perpendicular to cylinder axis with inflow coming from left to right (left picture). View perpendicular to cylinder axis with inflow coming from right to left (right picture). The instantaneous cylinder position is rj = —0.32.


43

highest amplitude of response increases with Reynolds number for cases from the upper branch

region. The average amplitude is about 10% higher for Re = 3000 than Re = 1000 (see Table

3.1) and becomes closer to experimental values [42, 43].

2 Vortex Synchronizations Patterns

As reported in [42], there has been some debate concerning the vortex formation modes th a t

might be associated with the different response branches for the case of free transverse vibrations.

If the 2S vortex formation mode associated with the initial branch is well documented for 2D and

3D computations ([54], [55], [36, 37]). However, to our knowledge, there does not exist a clear

numerical result th a t establishes the existence of the 2P vortex mode as a steady state pattern for

any response outside the initial branch region. We propose to look at flow visualizations of the

near wake and frequency response for cases with reduced velocity outside of the initial branch.

More specifically, we investigate the existence of a Reynolds number effect (as proposed by [42]),

th a t would explain the discrepancy between experimental and numerical results. We present flow

visualizations and frequency response results for R e = 1000, Re = 2000 and Re = 3000.

2.1 Flow Visualizations

Figures 16, 17 and 18 show phase- (over 3 cycles) and span-averaged isocontours of the pressure

and spanwise vorticity fields in the near wake of the cylinder. I t is worth mentioning tha t these

isocontours are not accurate and representative of the flow features in the imediate vicinity of the

cylinder for |r | = y^(x/D)^ + (yfD)'^ < 1. This is due to the post-processing isocontour routine.

Moreover, each three-dimensional spanwise vorticity field has been gently fUtered, using a three-

dimensional convolution filter, in order to get rid of the smaller weak structures resulting from

interm ittent small scale three-dimensionality of the flow. For Figures 16 and 17, each of the plots

is separated by a 1/4 period. Top and bottom left images show the cylinder wake when the body

is a t its extreme postions and hits the bottom or top of its stroke respectively. Top and bottom


44

right images show the wake when the cylinder is a t the center of its lateral oscillation and the

body is moving up or down respectively. Figure 18 captures two different locations of the body

motion that are separated by 1/2 a period. These two locations do not represent the extrema of

the cylinder motion.

Figure 15: Isocontours of the instantaneous pressure in the near wake of a moving rigid cylinder subject to VIV at Re=3000 (Case IV: V/n = 4.99).

Before describing and analyzing the images, it is crucial to point out the limitations of the

averaging process tha t we use in our simulations. In our computations, each flow field is saved at

regular time intervals. The sampling frequency in the case of a free motion is difficult to predict

a priori. Ideally, it should be equal to the frequency of oscillation fg of the body. In our case, we

use the natural frequency /„ of the body. As we will see later, those two frequencies are distinct

in most cases. Therefore, the difference prohibits long time averaging as the different frames over

which we phase-average shift in space. Obviously, averaging over 3 cycles is not enough to obtain

explicit visualizations (10 cycles would be more appropriate [42]). We estimate th a t there is a

10% error in the averaged location of the cylinder in Figures 16, 17 and 18. Another factor which


45

I

.I

Figure 16: Isocontours of span and phase-averaged pressure field in the near wake of a moving rigid cylinder at Vm — 7-0, Re = 1000 and Amax/D = 0.554. Upper-left: cylinder bottom extreme position; Upper-right: cyhnder median position, moving upward; Lower-left: cylinder top extreme position; Lower-right: cylinder median position, moving downward.


46

affects the clarity of the results is the three-dimensionality of the wake combined with the span-

averaging. Span-averaged values of strongly three-dimensional flow structures get smeared and

can mask the presence of the 2P vortex shedding. This is particularly true for the middle-range

wake (8 < x j D < 13 in our pictures) where inherent three-dimensionality takes place.

I■ I I I

ISil

T> i

I,a

10 12 IFigure 17: Isocontours of span- and phase-averaged spanwise vorticity field in the near wake of a moving rigid cylinder at l^-n = 7.0, Re = 1000 and AmaxjD = 0.554. Blue contours show clockwise vorticity, red anticlockwise vorticity. Upper-left: cylinder bottom extreme position; Upper-right: cylinder median position, moving upward; Lower-left: cylinder top extreme position; Lower-right: cylinder median position, moving downward.

Prom Figures 16, 17 and 18, it is clear th a t we are in presence of a 2P vortex shedding which

means th a t two pairs of vortices are formed per cycle. As the cylinder reachs its lowest position

(Figure 17, top-left image), the lower anticlockwise vorticity structure created by the lower shear

layer starts stretching and is finally pinched to split into two parts. As the cylinder moves up,

the upper part forms a weaker anticlockwise red vortex beside the stronger clockwise blue vortex


47

Figure IS: Isocontours of span- and phase-averaged pressure and spanwise vorticity fields in the near wake of a moving rigid cylinder at Vm = 7.0, Re = 2000 and AmaxlD — 0.54. 2P vortices have been circled in red. (a): cylinder bottom extreme position; (b): cylinder median position, moving upward; (c): cylinder top extreme position; (d): cylinder median position, moving downward.


48

(Figure 17, top-right image). These two vortices form a pair. The same mechanism takes place

on the other side as the cylinder reaches its highest position and then moves down (Figure 17,

bottom-left and right images). The deformation, stretching and splitting of the main vorticity as

described by Govardhan et al. [42] has been clearly observed by looking at movies of the wake at

Re = 1000 for instance. The strength of the second vortex of each pair is much weaker than the

first vortex and remains somewhat attached to the vortex of the opposite sign shed next. The

weaker circulation strength of the second vortex has also been documented by Govardhan et al.

[42]. As we look downstream, it is sometimes difficult to clearly identify the location of the second

vortex as it weakens along the streamwise direction, see Figure 19. We have circled in red the

locations of the vortices for the case of Re = 2000 on the two-dimensional Figure 18. Figure 19

shows isosurfaces of phase-averaged (over 3 cycles) filtered spanwise vorticity field corresponding

to the same situation as the top-right image of Figure 18.

Next, we pursue our quest of the existence and persistence of the 2P vortex shedding in the

lower branch region by looking at the frequency response of the structure and the wake. We also

investigate the relevance of the vortex and potential force decomposition.

2.2 Frequency R esponse

The classic lock-in phenomenon known as a synchronization of the vortex shedding frequency /„

and the cylinder oscillation frequency fs with the natural frequency fn of the body induces a

ratio f s l f n close to one over some range of reduced velocity Vm- This is particularly true for

large to moderate mass ratios but does not hold well for low mass ratios. Instead, Khalak &

Williamson [41] suggest another definition of the synchronization as a ratio f v / f s close to one,

which implies th a t the fluid force-frequency must match the shedding frequency. Moreover, the

range or synchronization regime over which lock-in happens widens in the case of low mass ratios.

Figure 20 shows the structural frequency response, which we have taken to be the ratio between the

frequency of oscillation fs and the natural frequency /„ of the body, for three different Reynolds

numbers. We see th a t we obtain, to some extent, the same response for all three Reynolds


49

i

Figure 19; Isosurfaces of phase-averaged filtered spanwise vorticity fields in the near wake of a naoving rigid cylinder at Vm = 7.0, R e = 2000 and Am ax/D = 0.54. Vortex pairs are easier to spot at some locations along the span than others.

R e=3000 ^ R e=2000; C=0.0Q333

Figure 20: Structural frequency response f s / f n as a function of reduced velocity V m for different Reynolds numbers.


50

numbers. The response for Vm = 4.18 and Re — 3000 stands out as it corresponds to the case

with insufficient spanwise resolution, see Table 3.1. The agreement between the different cases is

very good for reduced velocities Vm < 6 . In this case, the response grows almost linearly and the

ratio reaches a value close to unity for Vm ~ 6 for all Reynolds numbers. This corresponds to

the jum p in to tal phase as the frequency of oscillation /., passes through the natural frequency in

vaccum /„ [42]. For higher reduced velocities the response remains almost constant for all cases

(including the case for Re = 2000 with an additional small structural damping C) except for a

lower value at Re = 1000. These results are quantitatively in excellent agreement with the results

of Hover et al. [43].

x/D=1, y/D=0 x/D=1, y/D=1~ x/D=3; y/D=0 x/D=3; y/D=1— x/D=5; y/D=0— x/D=5; y/D=1

Q

82 3 4 5 6 7 90 1f

Figure 21: Shedding frequency response / „ / / „ for Re = 1000 and Vm = 7.0 a t 6 different locations in the wake.

Figures 21 and 22 present the shedding frequency response, which we have taken to be the

ratio between the frequency of vortex shedding /„ and the natural frequency /„ of the body, for

Re = 1000 and Re = 3000 respectively. The frequency fv is measured a t six different locations

in the near wake of the cylinder. The cross-flow velocity component is sampled at these points

for a long time integration and we compute its spectrum to obtain the shedding frequency. The

coordinates of the six points are: { x / D = 'i-,y/D = 0); {x / D = l , y / D = 1); {x / D = 3 ,y /D =


51

0); { x j D = ^ , y / D ~ 1); {x /D = 'o,y/D = 0); { x / D = ^ , y j D = 1). Unfortunately, no history

points were placed at locations corresponding to negative values of y / D. This is based on the

assumption of symmetry of the wake which might not be valid in the case of a P+S vortex shedding

for instance.

10 ' — ■ x/D=1; y/D=0 x/D=1;y/D=1 x/D=3; y/D=0 x/D=3; y/D=1~ x/D=5; y/D=0 — x/D=S; y/D 1

10“

10“

2 3 4 6 7 8 90 1 5f.A

Figure 22: Shedding frequency response f v / f n for Re = 2000 and Vm = 7-0 at 6 different locations in the wake.

Both cases illustrate the importance of the choice of those points. For both Reynolds numbers,

we see tha t the shedding frequency response for points placed along the y = 0 axis at a reasonable

distance away from the body, peak at a single frequency where the ratio is close to unity. These

results do not hint for a 2P shedding mode but for a 28 shedding mode where only one pair of

vortices is shed per cycle. However, points placed along the y = 1 axis, away from the axis of

symmetry of the body, also clearly exhibit a peak at twice the frequency and the ratio is in this

case close to two. This indicates two pairs of vortices shed per cycle which is the tradem ark of

a 2P shedding mode. It seems th a t the center locations of the two vortices shed per half-cycle

do not remain aligned with the y = 0 axis as they move downstream. This would explain why

the second vortex is not captured by the history points lying on this axis. The two points placed

very close to the cylinder give a peak close to one as well as other super harmonics. In particular.


52

the point located in the upper shear layer ( x / D = l , y j D = 1) provides multiple super harmonics

with decaying power. It holds as much power in the main frequency as in the first superharmonic.

Based on these studies it seems reasonable to say th a t we have obtained a steady-state peri

odic 2P vortex wake mode with formation of two same-sign vortices within each half-cycle. The

strength of the second vortex of each pair is weaker than the first vortex. Moreover, it is rather dif

ficult to clearly identify the 2P mode in the wake as three-dimensionality of the vortical structures

develops along the spanwise direction as they travel downstream.

2.3 Vortex and Potential Force D ecom position

We use the decomposition first introduced by Lighthill [56] and also used with success by Williamson

[42]. The to tal lift force can be decomposed into a “potential force” (due to the potential added-

mass force), and a “vortex force” (due to the vorticity dynamics). If we write the decomposition

in term s of the non-dimensional span-averaged lift forces, we would have:

C v o r t e x i i ) — C t o t a l ( t ) ^ p o t en t ia l ( t ) where C t o t a l i ^ ) ~ ( ^ L ( t ) (b)

The instanteneous potential added-mass coefficient is given by:

C , o t e n t i a l ( t ) =

where = n p f D ^ L z j i is the displaced fluid mass and Ca is the potential added-mass coefficient

(Ca ~ 1-0 for a circular cylinder). The time variations of the acceleration of the structure y(t)

and the total lift coefficient Ctotal (t) are both directly computed by the code so we do not need

to use any ad hoc models to evaluate the value of the vortex force coefficient Cvortex- The

instantaneous potential force is generally in phase with the cylinder motion. However, any jumps

in the vortex force, which is closely related to the vorticity dynamics, would indicate a sudden

change in the vortex formation process. We plot the span-averaged force decomposition as the well


53

as the reponse of the structure in Figure 23 for three different reduced velocities corresponding

to the three different branches of response. The Reynolds number is Re = 3000. The similarity

with the experimental results obtained by Williamson ([42]; Figure 6) is striking. It indicates

tha t there exists two distinct phases; a vortex phase (phase between the vortex force and the

cylinder motion) and a to tal phase (phase between the total lift force and the cylinder motion).

By comparing the amplitudes and the phases of each component of the force decomposition (see

Figure 23), one can conclude th a t there exists a large jump in vortex phase at the initial-upper

transition and a large jum p in to tal phase at the upper-lower transition. The jum p in vortex

phase is associated with a switch in the timing of vortex shedding. Similar results were obtained

for Re — 1000 and R e = 2000. In particular, the jum p of tt in the to tal phase at the upper-lower

transition was obtained as well for Re = 1000 (see Figure 24-left and Figure 13-e) and R e = 2000

(see Figure 24-right and Figure 25-e).

Re=3000:V =4.18 Re=3000;V =4.99rn

Re=3000; V „=7.0

1

t 0

1310 320 330 340 350

1

0

1380 400 420

1

0

1450 460 470 480 490

2

§ 0o

•2310 320 330 340 350

-2-2400 450 460 470 480 490380 420

S 0O m m m

A A A ^ A A /A // V ' J y V

310 320 330 340 350 380 400 420 450 460 470 480 490

2

0O'

-2310 320 330 340 350

4 h. A A. A r\ A a AI\

Time380 400

Time420

-2450 460 470 480 490

Time

Figure 23; Relationship between to tal lift force coefficient (Ctotai)^ the potential added mass force {^potential) and the vortex force (Cportex) ^or three different reduced velocities; Re=3000.


54

nWfimmmI I 1 I * 1 i ' , i i i t ' •

Figure 24: Cylinder response and span-averaged lift coefficients for reduced velocity in the lower branch. Left: Vm = 7.00; Re = 1000. Right: Vm = 7.00; Re = 2000.

3 Validity of the Independence Principle

3.1 Introduction

When a cylinder is placed at an angle with the respect to the main flow, the hydrodynamic forces

and correspondingly the cylinder response may change compared to the normal-incidence case.

Despite several theoretical and numerical studies of the stability of three dimensional boundary

layer on a yawed circular cylinder, there is very little published on the vortex shedding of yawed

cylinder placed in a steady current or the effect of the angle of yaw on the force distributions and

shedding frequency. Conflicting reported results are related to such basic characteristics as the

drag coefficient, the base pressure and the shedding frequency.

In particular, to our knowledge, there are no direct numerical simulations of free rigid yawed

cylinders subject to VIV with large angle of yaw. Both theoretical and most of experimental works

have shown, at least in the subcritical range, th a t the yawed cylinder is similar to the normal-

incidence case through the use of the component of the free-stream velocity normal to the cylinder

axis [57, 58]. This is known as the Cross f low or Independence Principle (IP) and is also referred

as the Cosine Law. Several investigators reported deviations from the predictions based on that

principle [59, 60, 61, 62] and in particular at large yaw angles [63, 64]. Surry & Surry [65] found

the Strouhal number based on the normal velocity component remains approximately constant in


55

450 460 470 480 490

450 460 470 480 490(c)

.............................................. ........20 .ic »

10# ♦ * - - i '

..... . m m # iiP !|i ............. ..........oil............................ ......................... ..

# % i

i i l l l .laiBlI

r t i i

450 460 470 480 490

450 460 470 480 490

0.4

0.2

0

- 0.2

-0.4

500

500

0.5

0.48

0.46

0.5

0

-0 .5

500

500

Figure 25: Complex demodulation analysis for Re = 2000 and Vm = 7.00. Isocontours of crossflow displacement ry (a). Isocontours of amplitude of demodulated cross-flow displacement (b). Isocontours of amplitude of demodulated Lift coefficient i?Cx, (c). Isocontours of Lift coefficient Cl (d). Isocontours of phase difference between demodulated cross-flow displacement and demodulated lift coefficient (e). The demodulation frequency is \ = 0.1561.


56

the wake of stationary yawed cylinders for Reynolds numbers in the range 4,000 < Re < 63,000

up to angles 9 of 60° to 70°. Here (and in the following), 0 = 0° corresponds to cross flow and

6 = 90° corresponds to axial flow. They found th a t the energy in the Strouhal peak disperses and

decreases significantly with increasing inclination; it is virtually submerged in the general wake

turbulence spectrum for 9 > 60°. Snarski & Jordan [64] measure the wall pressure spectra on a

stationary cylinder for various angles of incidence at sub-critical Reynolds number. They show

th a t the variation in narrowband (corresponding to periodic vortex shedding) and broadband

(corresponding to lam inar/transitional boundary layer) spectra levels with angle of incidence is

not monotonic. They also conclude th a t there is a fundamental shift in the separation mechanism

for 9 ss 55° between Re = 7250 and Re = 14500. Kozakiewicz et al. [66] show tha t the IP can be

applied to stationary cylinders in the vicinity of a plane wall in the subcritical range for an angle

of yaw 0° < 0 < 45°.

However, the validity of the IP remains questionable when the angle of yaw becomes very

large and the flow direction is almost parallel to the cylinder axis. Hanson [60] studied vortex

shedding from vibrating yawed hot wires in an air stream for low Reynolds number. He validated

the IP for 9 < 68° but he found discrepancies in shedding frequencies for an angle of yaw 9 = 72°.

Van A tta [61] investigated the region of an apparent discontinuity for angle of yaw in the range

50° < 9 < 75°. He concluded th a t the discontinuity observed by Hanson was not due to the large

angle of yaw but was due to the existence of locked-in modes depending on the value of the wire

tension. He also reported th a t for a given tension, the wire does not necessarily vibrate with the

frequency of the harmonic tha t is nearest to the natural shedding frequency, but always locks-in

to the frequency th a t is lower than the natural shedding frequency. In the case of non-vibrating

yawed cylinders Van A tta showed th a t for 9 < 35° the vortex shedding frequency decreases nearly

like the Cosine Law, whereas for larger angles the decrease with increasing angle of yaw 9 is

slower than the proposed Cosine Law.

Koopman [67] carried out tests specifically designed to study flow-induced oscillations of yawed

cylinders. He noticed a drastic decrease in the cyhnder periodic response as well as a drop in the


57

correlation of the lift forces along the span for angles of yaw larger than IS”. King [68] investigated

VIV of yawed circular cylinders for Reynolds number in the range 2,000 < Re < 20,000 and

for yaw angles —45° < 0 < 45°. He measured drag forces and fluctuating in-line and cross-

flow cylinder responses. He observed tha t the maximum cross-flow amplitude corresponds to

reduced velocity Vm (based on the normal component of velocity of the inflow) in the range 5.8 <

Vrn < 7.0. The fact tha t the profiles of instability regions collapse was presented as experimental

justification of the generalisation of the Cosine Law to the case of the oscillating yawed cylinders.

He attributed the increase in crossflow response with yaw angle to a corresponding decrease in

the reduced damping. He also recorded sustained oscillations at yaw angles 9 = ±65° and showed

th a t the cylinder response is virtually identical for positive and negative angles of yaw.

Finally, Ramberg [62] studied the effect of yaw angle (0 = 0° — 60°) and end-conditions for

stationary and forced vibrating circular cylinders (with aspect ratio 20 — 100) in the Reynolds-

number range 150 — 1,000. He determined that the results were very sensitive to end-conditions

especially at the lower Reynolds numbers. He showed tha t slantwise shedding at angles other than

the cylinder yaw angle is intrinsic to stationary inclined cylinders in the absence of end-effects.

He reported th a t the IP fails in the case of stationary yawed cylinders because the shedding

frequency is always greater than expected from the IP, while the shedding angle, the vortex-

formation length, the base pressure and the wake width are all less than expected. However, he

concluded tha t locked-in vortex wakes of vibrating yawed cylinders can be described successfully

by the IP. In this case, frequency lock-in between the vortex wake and the cylinder motion was

accompanied by vortex shedding parallel to the cylinder axis.

In the current chapter we investigate the validity of the IP for fixed yawed circular cylinders

and free yawed circular rigid cylinders subject to vortex-induced vibrations (VIV) at subcritical

Reynolds number. Our goal is to verify if large cylinder responses are possible for large angles of

yaw (i.e. > 60°). We also examine time- and span-averaged forces on the cylinder and compare

them with the values predicted by the IP.


58

3.2 Sim ulation P a ra m e te rs and Formulation

We report here simulation results at constant Reynolds number Re — Ud/ u = 1,000 based on the

free-stream velocity for stationary and vibrating rigid yawed cylinders. The angle of yaw 9 (see

Figure 26), defined by the direction normal to the free-stream velocity direction and the cylinder

axis, is such th a t a zero value corresponds to a cylinder normal to the free-stream velocity. We

consider mainly two angles of yaw 9 = —60° and 9 = —70°. The negative signs come from the

orientation of the frame of reference tha t is commonly used in the litterature (see Figure 26).

By changing 9 we modify the normal Reynolds number Re„ = {Ucos9)d/v based on the normal

component of the inflow.

In all vibrating cases we only allow vertical motions in the crossflow j/-direction, i.e. we do not

allow any motion in the streamwise ar-direction. The cylinder is rigid and thus its motion has no

spanwise z-dependence. The mass ratio of the system (cylinder mass over displaced fluid mass)

is taken to be m = 2 and the structural damping coefficient to be ^ = 0.003.

The governing equations are the incompressible Navier-Stokes equations coupled with the struc

tural dynamical equation. The cylinder is represented by a single degree of freedom viscously

damped second-order oscillator subject to the external hydrodynamic forcing, i.e.

f)(t)-f ^ » ) ( t ) + ^ r ? ( t ) = ry(0) = 770 and 7?(0) = ?)o (8)

where Vm — is the reduced velocity based on the free-stream velocity U. The reduced velocity

is based on the normal component of the inflow.

The computational domain is Mesh2 (see Figure 2) and the aspect ratio is L z / D = 22. We use

a fourth order p = 4 polynomial expansion per element. Also, 64 z-planes (32 Fourier modes)

are used along the spanwise direction. Periodic boundary conditions are imposed at the two ends

along the cylinder axis. This is equivalent to trea t the structure as infinitely long, and then

employ (free) periodic boundary conditions on a piece of finite length. By doing so, strong three-

dimensional end effects th a t are usually associated with oblique inflows past cylinders are avoided.


59

Figure 26: Simulation setup. Left: view perpendicular to the cross-flow direction (y-direction). Right: 3D view.

In the case of very large angles of incidence, or the extreme case of an axial flow, the flow can be

thought as a fully developed, three-dimensional boundary layer flow along the spanwise direction

of the cylinder surface.

3.3 Force D istributions

We consider stationary and rigidly moving yawed cylinders (SYC and RYC) and investigate the

distribution of hydrodynamic forces acting on the body both in time and along the span. We

normalize these forces based on the component of the inflow normal to the cylinder axis to obtain

drag and lift coefficients. In order to present more concise information, we will mainly present here

time- and span-averaged quantities for the different cases, instead of instantaneous quantities.

In Figures 27 and 28, we plot mean, maximum and rms (root mean square) values of drag

coefficient Cjo, lift coefficient Cl , and cross-flow cylinder response ry against the reduced velocity

Vrn used in our simulations. All these statistical quantities correspond to averaged values over

both the time and space domains. Here is the definition we use to compute our rms results (for


60

® RYCVIVO = 60°:R e = 1000) A RYC VIV (0 = 70°; Re = 1000)

• • • SYC (e = 70°; Re = 1000) RNC VIV (Re = 1000), V=4.202

- RNC VIV (Re = 300); V=4.87

RYC VIV (9 = 6 (f; Rs = 1000) A RYCVIV® = 70°;R a = 1000)- SYC (6 = 70“; Re = 1000)

RNC VIV (Re = 1000); V=<1.202 RNC VIV (Re = 300); V=4.87

Reduced Velocity V9 10 11 12 13 14 15

Reduced Velocity V

Figure 27: Mean (left plot) and rms (right plot) drag coefficients versus reduced velocity F)-, based on free-stream inflow velocity.

the drag coefficient for instance):

COrT L i { C o i z , k )

1/2

(9)

where the bar denotes averaging over space and n is the number of term s in our time series for

each spanwise location z. Another definition, where we first compute the rms values along the

space direction and then average in time, for the lift coefficient will be used as well and described

later on.

Due to the cost of DNS, we have chosen a few reduced velocities for each angle. The reduced

velocities for which we find maximum cylinder response and a “capture” of the natural vortex

shedding by the cylinder motion are in the lock-in range given by Ramberg, i.e. Vrncos ~ 4.5 to

6.0 [62] and King, i.e. Vrncos = 5 -- 6 [68]. We also include for reference some of the corresponding

values for rigid with normal (inflow: 6 = 0°) cylinders (RNC) subject to VIV at lock-in (numerical

results from [36]). It is worth mentioning th a t these reference results were obtained with a different

aspect ratio o f L / d = 4tt. Each one of these reference cases corresponds to a single reduced velocity

(mentioned in the legend). Nevertheless, for convenience reason, it has been represented by a line

in the plots. Otherwise, circle symbols refer to cases with an angle of yaw 9 = —60°, and triangle

symbols refer to 9 — —70°.


61

Figure 27 displays the drag coefBcient (Cd ) results. We first examine the case of the stationary

yawed cylinder with 0 = —70° and i?e„ = 342. By comparing with the case of a stationary

cylinder with normal inflow at the same Reynolds number, we can check the validity of the IF.

The numerical study of this case was not available to us for comparison. Instead, we considered

one of our studies where Re = 300 and we also compared with the base pressure coefficient

results for Rsn = 342 by Ramberg [62]. It appears that the ratio between the mean base pressure

coefficient of the yawed cylinder and the base pressure coefficient of the normal cylinder is greater

than unity (in fact equal to 1.2). This is consistent with Ramberg’s findings [62]; it confirms that

the base pressure on the yawed cylinder is lower than the value predicted by the IP. Therefore, it

produces a drag significantly greater (Cd ~ 1.34) than predicted by the IP (Cjj ss 1.2).

If we now look at the moving yawed cylinders with 6 = —70°, we see th a t the cases outside

the lock-in region give very similar results to the stationary case. In particular, mean and rms

values of the drag coefficient for reduced velocities Wn = 7.55 and 9.55 are almost identical to

the stationary case for the same angle of yaw. Reduced velocity Vm = 14.5 or Wrtcos 4.9593

(referred as Case I) leads to lock-in. We see th a t the drag quantities are larger than predicted

by the IP. Indeed, they are quite larger than the reference case of the free cylinder at normal

incidence at Re = 300 (again, an exact value of Re = 342 was not available for comparison).

They are closer to the results for the free cylinder at normal incidence at i?e = 1,000.

Results for free yawed cylinders with 6 = —60° exhibit a very clear trend. As the reduced

velocity increases toward the lock-in region, around Vm = 9.55 or Vm^^e ~ 4.775 (referred as Case

II) , mean and rms values of the drag force increase and approach the case of the free cylinder at

normal incidence at Re = 1,000.

Figure 28 displays the lift coefficient Cl and cross-how cylinder response ry results. Different

rms values of the lift coefficient are presented. Filled symbols and reference lines follow the

previous dehnition. Hollow symbols (with an added star sign in the legend) follow a different


62

® RYC VIV § = ■A RYC VIV ^ = 70°; Re = 1000) O RYC' VIV (6 = 60°; Re = 1000) A RYC‘ V lV ^ = 70°:R e = 1000)• •• SYC (6 = 70°; Re =1000)„ RNC VIV (Re = 1000): V.=4,202 p RNC VIV (Re = 300); V^=4.87

Reduced Velocity VReduced

® \^ fO f 'R '^ C V Iv e = k ) ‘' ; R e = 1000)A 7 ) ^ for RYC VIV p = 70°; Re =1000)O ’1 to rR '< 'C V lV p = 50°;Be=1DO0) A n f^ fo fR Y C V lV ^ = 70°,R e=1000)

- - ' rrra V=4.202~ T )^ for RNC VIV (Re = 300); V=4.87

for RNC VIV (Re = 1000); V=4.202 T|^“ for RNC VIV (Re = 300); V=4,B7

9 10 11 12 13 14 15Reduced Ve}ocity V

Figure 28; Rms lift coefScients (left plot) and rms and maximum cross-flow cylinder responses (right plot) versus reduced velocity Vm based on free-stream inflow velocity.

definition, i.e.:

Cl ^'E L i

1 /2

p(10)

where, this time, the bar denotes averaging over time and p is the number of ^;-planes along

the cylinder axis. W ith the latest definition, low values of rms clearly indicate a good spatial

correlation of the lift forces.

For both 9 = —60° and 9 ~ —70°, the maximum cylinder response increases as the cylinder gets

closer to lock-in. For the Case I, it reaches the value given by the IP [p = 0.52). For Case II, no

comparison was available for the IF, but the value {rj = 0.63) remains lower than the value for

the free cylinder at normal incidence at Re = 1,000 (r? = 0.75). We notice tha t the maximum

amplitude at lock-in decreases with a yaw angle increase. This is in disagreement with King’s

work [68]. Nevertheless, King attributed the increase in crossflow response with yaw angle to a

corresponding decrease in the reduced damping (due to a change in the immersion length). This

explanation is not obvious as pointed out by Ramberg in his review of the paper. Moreover, King

found differences (up to 50 %) between his main and subsidiary test responses that he attributed

to some three-dimensional effects in the main test cylinder configuration. This bottom end effect

was not quantified by the author. It would appear th a t there are im portant Reynolds number,

yaw, aspect ratio and end effects tha t cannot be accounted for through the reduced velocity.


63

For both angles, {r})rms follow the same trend and increase with the reduced velocity. Maximum

values are obtained at lock-in; value for Case I matches the predicted value given by the IP

(hrms = 0.36). For d = —60°, the rms values of the lift coefficient (see hollow symbols) peak at

the reduced velocity corresponding to the largest cylinder response (see Case II). Therefore, it

seems that there is a loss of spanwise coherence and a drop in the correlation of the hft forces

when the cylinder response amplitude is maximum. For 9 = —70°, the situation is different and

the maximum rms value of the lift coefficient is reached at Vm = 7.55. The correlation of the lift

forces increase as we approach the lock-in region. However, this is of less importance in th a t case

as the force amplitudes are small.

The study of spatio-temporal variation oi Cd and C i for Case II (see Figure 34) shows regions

corresponding to moderate forces alternating with regions corresponding to large forces. These

regions look like inclined braids th a t follow a travelling wave pattern (see Cl distribution). The

angle of inclination does not depend on time. At the present time, we can not explain the existence

of those braids but we will relate it to near-wake flow visualizations in the next section.

SO!

640 760 780660 700 720 740 800 820

640 660 700 720 740 760 780 800 820tU/d

Figure 29: Freely vibrating yawed cylinder {9 = —60°) at lock-in at Re = 1,000. Drag (top) and Lift (bottom) coefficients along the span versus non-dimensional time.


64

3.4 Flow V isualizations

In two-dimensional bluff-body flows, irrotational fluid from outside the wake region is swept

into the vortex street. However, the yawed cylinder flow in the base region is inherently three-

dimensional due to the additional spanwise vorticity and the changing direction of the free-stream

flow as it approaches the cylinder.

Stationary Yawed C ylinders

Flow visualizations for large values of yaw are useful in this case to prove the existence of a

vortex-shedding regime in the near-wake of the cylinder as opposed to a steady trailing vortex

shedding regime. Moreover, it might also illustrate the slantwise shedding at angles other than

the cylinder yaw angle. We present here results for the most extreme case of 6 = —70° from our

simulations. We plot in Figure 30 pressure isocontours at Re = 1,000. We do notice a vortex

shedding regime in the near-wake. This is consistent with the results from Ramberg [62] for this

range of Reynolds number and similar aspect ratio but a somewhat lower yaw angle (around 60°)

for flat end-shapes. Moreover, the shedding is parallel (25 pattern) and slantwise (with a shedding

angle smaller than the yaw angle) as found by Ramberg [62]. The shedding angle a (defined with

the same convention as the yaw angle) approaches a value of a = —58° for this case and it is

stable in space and time. Measured shedding frequencies in the wake have shown th a t the ratio

St / S t n is greater than cos9, which is consistent with a < 9, see Ramberg [62]. Final results for

9 — —60° exhibit a similar trend with a shedding angle value th a t becomes closer still smaller to

the yaw angle.

Freely V ibrating Yaw ed C ylinders at Lock-in

We present flow patterns for freely vibrating yawed cylinders {9 = —70° and 9 = —60°) at lock-in

(see cases I and II in section 3.3). In Figures 31 and 32 we plot instantaneous pressure isocontours

of the flow. In both cases, we notice a vortex-shedding regime in the near-wake. This time, the

vortex shedding is parallel to the cylinder axis. This extends the findings by Ramberg [62] for


65

VN 12

Figure 30: Stationary yawed cylinder {9 = -70°) at Re = 1,000. Pressure isocontour at value —0.025. View almost perpendicular to the plane of the inflow. The arrows represent the inflow coming from left to right.


66

1Figure 31: Freely vibrating yawed cylinder {6 = —70°) at lock-in at Re = 1,000. Pressure isocontour at value —0.025. View almost perpendicular to the plane of the inflow. Inflow coming from left to right.

forced vibrating yawed cylinders with similar Reynolds number, aspect ratio and somewhat lower

yaw angle {6 about 50°) to the case of freely vibrating yawed cylinders. Parallel shedding to the

cylinder axis was encountered in most of our cases as long as the cylinder crossflow amplitude was

not too small. In the case of reduced velocity value for which the cyhnder response was almost

zero (therefore mimicking a stationary cylinder), the vortex shedding had the tendency to slant.

Figure 31 shows the flow past a vibrating cylinder at lock-in for an angle of yaw 9 = —70°. It

corresponds to the instant of time when the cross-flow cylinder displacement reaches its minimum

value {rj = —0.506). In this plot, we see th a t the flow structures are much more contorted than

in the stationary case. The shedding pattern of the von Karman vortices is of type 2S bu t we

notice strong streamwise vortices winding up around the spanwise vortices. The latter seem to

be connected via “braids” consisting of streamwise-oriented counter-rotating vortex pairs. Down

stream in the wake, those streamwise vortices significantly distort and pinch the von Karman

vortices. Streamwise vortex tubes axe not perpendicular to the cyhnder axis but they seem to


67

X 15

1 5 -

1 0 -

Figure 32: Freely vibrating yawed cylinder (B — —60°) at lock-in at Re = 1,000. Pressure isocontours at value —0.05. Views almost perpendicular to the plane of the inflow. Left: Front view with inflow coming from left to right. Right: Back view with inflow coming from right to left.

follow helical paths around the spanwise vortical structures. Streamwise vortices appearing on the

front side of Figure 31 are slanted along the direction of the inflow. However, streamwise vortices

from the other side (not visible on Figure 31) are slanted along the direction perpendicular to

the inflow. This suggests tha t the free-stream flow direction may have a strong influence which

is in disagreement with the IP. The steady trailing regime, as mentioned in several publications,

would correspond to the extreme case where the spanwise vortices would bend and loop over to

the point where they would eventually detach from the shedding vortex filaments and form a wavy

line streaming out behind the cylinder.

Figure 32 presents two different views of the same instantaneous flow past a vibrating cylinder at

lock-in for an angle of yaw B = —60°. It shows the front side and the back side of the pressure

isocontours in the near-wake. Figure 33 shows smoothed spanwise vorticity isocontours. Specif

ically, the spanwise vorticity has been filtered several times using a three dimensional Gaussian

convolution kernel [ 3 x 3 x 3 ] with standard deviation a ~ 0.65 in order to isolate the large struc

tures from the noise. Both Figures 32 and 33 are taken at the same time {tU/d = 748.75). The

cross-flow cylinder displacement is 77 = 0.262 and the cylinder is travelling on its way down to its


y ^

Figure 33: Freely vibrating yawed cylinder {0 = —60”) at lock-in at Re = 1,000. Spanwise vorticity isocontours at value ±0.25. View almost perpendicular to the plane of the inflow. Inflow coming from left to right.

minimum position (t] —0.6). It shows the front side and the back side of the pressure isocon

tours in the near-wake. Figure 34 presents another view of the pressure isocontours in the wake at

a different time ( tU/d = 753.75). The corresponding cross-flow displacement is rj ~ —0.278 and

the structure is travelling on its way up to its maximum position {rj w 0.6). These instants are

chosen in order to relate to Case II (see section 3.3) and examine how the flow pattern correlates

with the force distribution. Spanwise vortices close (and away) from the cylinder are distorted

(see Figure 33). The last (see green arrows on Figure 33) and the next to last (see yellow arrows

on Figure 33) vortices shed from the cylinder could be described as making a X oi H shape where

the central part of the vortical structures in the spanwise direction remains closer to the cylinder

body while the two ends of the structure (toward the cylinder ends) stand further downstream

(see yellow arrows on Figure 33). The dynamics of this vortical pattern can be opposed to those

of the generic “chevron” pattern where vortex shedding near the cylinder ends is delayed. In our

case, the gap left between those two legs and the cylinder body is filled by the spanwise vortex of


69

opposite sign shed on the other side of the body (see green arrows on Figure 33). Therefore, close

to the cylinder and toward the ends of the body, a 2P vortex shedding mode takes place within

the 2S mode.

Further downstream, it seems tha t there is a strong interaction between spanwise and streamwise

vortical structures. The streamwise vorticity sheets, shed from the upper and lower part of the

body, wrap around the Von Karman vortices (see red arrows on Figure 34). They also induce a

local spanwise velocity component opposed to the spanwise component of the free-stream flow.

This could result in the instability of the Karman vortices th a t become constricted and distorted;

they loop and take an helical shape along the z-direction (see arrows on left side of Figure 32).

From this view, we see th a t the half bottom part of the last roll shed from the back of the cylinder

describes a left-handed helical form tha t coils counter-clockwise (see red arrows). The top bottom

part describes a right-handed form th a t coils clockwise (see yellow arrows). This mismatch in the

rotation orientation constrains the bottom structure curvature to change and this explains the

aforementioned vortex loops.

Other instants in time for this case were also investigated and flow visualizations provided

the same type of vortical topology. Moreover, we observed th a t the part of the “chevron” pat

tern attached to the cylinder was moving in time. For increasing time, it moves at a constant

speed in the positive z-direction along the cylinder in the main direction of the inflow. This

phenomenon seems to be strongly correlated to the travelling-like wave seen in the lift coefficient

spatio-temporal distribution of Case II described in section 3.3.

3.5 Summary

We have investigated the validity of the independence principle (IF) for fixed yawed circular

cylinders and free yawed circular rigid cylinders subject to VIV at subcritical Reynolds number

{Re = 1,000) using DNS. We considered here two angles of yaw 9 = —60° and 6 = —70°.

For stationary cylinders, we have shown tha t the IP must not be valid for large angle of attack.

We confirmed th a t slantwise shedding at angles other than the cylinder yaw angle is intrinsic to


70

$

Figure 34: Freely vibrating yawed cylinder {9 = —60°) at lock-in at Re = 1,000. Pressure isocontours at value —0.05. View almost perpendicular to the plane of the inflow. Inflow coining from left to right.

stationary inclined cylinders in the absence of end-effects. The shedding angle is smaller than the

yaw angle in the case of 0 = —70°. The base pressure is lower than the value predicted by the IP,

and hence, the drag coeflicient is higher than the value predicted by the IP.

For freely moving cylinders at lock-in, we showed th a t we get large cylinder cross-flow am

plitude even for large angle of yaw {9 = —60° and 9 = —70°). The mean and rms values of

the cylinder cross-flow amphtude decrease for increasing angles of yaw. The values of the corre

sponding reduced velocities are in the range given by King [68] and Ramberg [62]; the shedding

is parallel to the cylinder axis. The mean and rms values of the drag coefficient are larger than

the values predicted by the IP as they approach the VIV values for a flow at normal incidence

(0 = 0°) at R e = 1,000. For 9 = —70°, lift forces are more correlated for reduced velocities in the

lock-in range but their magnitude is small. For 9 = —60°, different rms values seem to indicate

a loss of spanwise coherence and a drop in the correlation of the lift forces along the cyhnder as

the reduced velocity gets closer to the lock-in region where the cylinder response is maximum.


71

Near-wake flow visualizations for 6 = —60° at lock-in provided a vortical topology tha t we related

to the spatio-temporal distribution of forces.


Chapter 4

Sheared Flows past Rigid and

Flexible Cylinders

1 Introduction

The offshore industry is moving to ever-increasing water depths, producing presently oil in depths

of up to 2,500m, requiring detailed fatigue calculations for the risers and tendons used in the

floating structures, see [69, 70]. Currents in the ocean are invariably highly sheared, hence the

modes that can potentially be excited are many, see [71]. The calculation of how many and which

modes are excited, can affect fatigue life very significantly, but there are no guidelines presently

available for conducting this calculation.

Prediction of vortex induced vibrations has been a semi-empirical discipline until recently.

Most of the models employed in industrial applications (e.g., SHEAR7, VIVA, etc.) involve

eigensolutions of the structure but the required flow input (e.g., lift coefficients, added mass,

correlation length, etc.) is obtained empirically. When semi-empirical models predict several

modes as possible to respond, it is not certain tha t the riser will respond in all of these modes;

or that it will respond simultaneously in aU modes - it may transition from one mode to another

72


73

for example, or from a set of modes to another set of modes. A major missing element, therefore,

in our present capability is a reliable procedure to determine how many and which of the modes

will be predicted in a given shear current. It is known tha t in some cases a few modes, or even

a single mode, will dominate the response, while in other cases several modes will be excited, see

[72]. Also, non-stationary response may be obtained.

There is great difficulty in conducting experiments to establish either multi- frequency re

sponse, or its absence: Multi-frequency response occurs in long risers within a highly sheared

current, hence the facility must be very wide to accomodate the riser length, while the incoming

velocity must have a controlled form of shear. By necessity, in any of the available facilities, the

Reynolds number will be much smaller than full scale in order to model correctly the diameter

scaling; while several screens must be designed to provide the desirable current velocity shape.

Full scale experiments could be a great source of information, but they are expensive, and by

necessity extremely limited in the scope of currents to study. In such a phenomenon, where fun

damental information is missing on the controlling mechanisms and parameters, we have to start

from a more global investigation and then fine-tune through full-scale experiments.

To this end, direct and large eddy simulations offer a very accurate approach to study funda

mentally VIV. Here, we present DNS results of vortex-induced vibrations of long flexible cylinders

with aspect ratio of the order of 1000, subjected to linear and exponential sheared flows. The max

imum Reynolds number Rem is of the order 1000, resulting in a turbulent near-wake. Although

this range is below the Reynolds numbers of reahstic industrial applications, the correspond

ing Strouhal number is representative of higher speeds and extrapolations can be made. Indeed,

most of the predictions agree quantitatively with experimental measurements (M.S. Triantafyllou,

private communication).

Previous experimental studies of vibrating cylinders subjected to sheared flows have shown the

existence of cellular shedding patterns [73],[74], [75]. The size and stability of such cells have some

subtle differences with similar structures encountered in stationary cylinders, as synchronization

(lock-in) and multi-mode response compete directly with frequency mismatching along the span.


74

The latter is the primary reason for the cell formation in either sheared flow or uniform flow

past tapered cylinders [76], [77], [78]. The results from the experimental work suggest th a t the

size of the cells are proportional to the amplitude of vibration and inversely proportional to the

shear parameter /3. This parameter is defined as p = [d/U)\du/dz\ , where z denotes the spanwise

(cylinder axis) direction, d is the cylinder diameter, and tJ the span-averaged freestream velocity.

If the shear is linear, then the slope d u j d z of the velocity profile is constant along the span.

However, if the shear is exponential, the span-averaged or the maximum value of the slope can

be used instead in the calculation of the shear parameter. It is possible, therefore, to find cells of

constant shedding frequency of forty diameters or more unlike the stationary cylinder where such

cells are not longer than approximately ten diameters [75].

The experimental work has primarily focused on frequency and point measurements. However,

measurements of forces on the cylinders are needed to establish the effect of vortex dislocations

(strong localized distortions of a spanwise vorticity and its connections with more than one vor

tex). Such data are currently missing with the exception of recent work by Triantafyllou and

collaborators who investigated the effect of vortex splits [43]. In numerical work, the first three-

dimensional simulations of VIV of flexible cylinders subjected to sheared flow has been reported

in [35].

1.1 Flow Visualizations of V ortex D islocations and Vortex Splits

The most prominent feature observed in shear flow over a bluff body is the shedding of vortices

in cells of constant frequency. Because of the mismatch in frequency, vortex dislocations are

generated between these cells. The presence of these vortex dislocations in wakes contributes to

the transition to turbulence. Vortex dislocations are encountered in transitional and turbulent

wakes as well as in laminar wakes, but in a more ordered fashion.

Visualization of vortex dislocations are more clear at low Reynolds number. To this end,

we first simulated shear flow past a flexible beam subject to forced and free vibrations for a

pinned cylinder of aspect ratio T /d = 567 with a mean Reynolds number Re = 80.35 and a shear


75

parameter /? « 8.8 x 10^^; these results and corresponding visualizations were first reported

in [11]. The frequency reduces as the cosine of the shedding angle (this angle gets steeper as

the inflow velocity decreases); similar results were reported for a stationary cylinder in [79]. A

localized lock-in of the left part of the beam, which corresponds to the side experiencing the large

inflow, is obtained. This is similar to the lock-in cell observed in the experiments of [75] which

extended over 44 cylinder diameters in a cylinder with aspect ratio 107. Here the size of the first

cell is larger than th a t as both the amplitude of the vibration is larger and the shear parameter

is smaller than the experiment (see also [75]). In addition, a significant increase of drag and lift

forces was observed in this region of the structure.

Figure 1: Experimental (left) vs Numerical (right) results. The cylinder is a t the bottom of the pictures, the views are perpendicular to the cylinder axis and the flow is upward. Left: Photograph of Vortex Dislocation at Re = 100 for stationary cylinder using dye flow visualization (courtesy of Williamson [2]). Right: Isocontours of spanwise vorticity (w = ±0.18) and oblique fronts in the wake of a forced vibrating flexible cylinder at Re = 100 with linear sheared inflow.

Figure 1 (right) shows instantenous isocontours of spanwise vorticity for this case. We observe

some vortex dislocations and vortex splits very similar to the ones obtained in [2] (see Figure 1:

left).

The structure, size and dynamics of the dislocations in the case of uniform inflow past a

stationary cylinder have also been described by Williamson in more recent experimental work

[78]. He obtained a relationship between the different vortex structures across the boundary

between two cells, and explained the interactions between these structures. He also established,

in the case of transitional wakes, th a t the beating frequency of the dislocation between a cell of

frequency f i and a cell of frequency /2 is f u = f i — f 2 - This has been successfully verified in


76

our simulation for a moving cylinder, suggesting a universality of vortex dislocations. As regards

the fluctuation of cell boundaries, our results confirm the experimental results of [73, 75] that

vibrations have a stabilizing effect.

Figure 2: Linear shear; Iso-contours of crossflow velocity at Rem = 1000. The cylinder is at the bottom of the pictures, the view is perpendicular to the cylinder axis and the flow is upward. Only the part corresponding to the large inflow is shown. Dark color: v = —0.2, Light color: v = 0.2. View normal to the {x, z) horizontal plane, where 80 < z f D < 400 and 0 < x / D < 12.5; flow is upward.

. - . . M - >

Figure 3: Exponential shear: Iso-contours of crossflow velocity at Rcm = 1000. The cylinder is at the bottom of the pictures, the view is perpendicular to the cylinder axis and the flow is upward. The large inflow is on the left side. Dark color v = —0.2. Light color v = 0.2. View normal to the (x, z) horizontal plane, where 0 < z / D < 400 and 0 < x j D < 35; flow is upward.

Similar visualizations were obtained for a freely moving cylinder. At higher Reynolds number,

however, it is more difficult to discern such a clear pattern although a similar picture emerges. For

example, for the linear shear flow at Rcm = 1000 we use the cross-flow velocity to qualitatively

capture instantaneous vortex dislocations in Figure 2. We can see a V-shaped vortex dislocations


77

close to the cylinder slightly to the right of the middle of the picture. In Figure 3 we plot the

crossflow velocity contours in the near-wake for the exponential shear case. We observe a structure

much more complex than the linear shear case but w ith distinct pockets of vortex dislocations,

qualitatively similar to the structures we observed in laminar wakes and in linear shear. It is also

clear from this plot tha t the flow does not correlate well along the span of the cylinder, and this

invalidates the assumption of full-span correlation employed in the semi-empirical models such as

SHEAR? [80].

In Figure 4-(right), we present another example of complex flow physics successfully captured

by DNS. In this case, we properly resolve a vortex split th a t connects two vortical patterns (28 and

2P) along a rigid cylinder forced to move in the cross-flow direction with a prescribed amplitude

and frequency, see also [3]. In contrast to vortex dislocations, this hybrid mode is periodic and

repeatable with the location of the vortex split remaining stable. We will explain in more details

this case in the next section.

Figure 4: Experimental (left) vs Numerical (right) results. The cylinder is at the bottom of the pictures, the views are perpendicular to the cylinder axis and the flow is upward. Vortex split is located in the red circles. Left: Photograph of vortex split in the wake of a forced rigid tapered cylinder at i?e j = 400 with uniform inflow using lead precipitation visualization (courtesy of Techet and al. [3]). Right: Isocontour of pressure {p = -0 .25) in the wake of a forced rigid straight cylinder at Rep = 400 with linear sheared inflow.


78

2 Vortex Split, Hybrid M ode and Vortex Shedding M ode

Selection

Rigid cylinders subject to VIV caused by uniform inflow at relatively small Reynolds number

exhibit different responses depending on whether the combined mass-damping parameter is high or

low [42]. Nevertheless, the amplitude responses present different branches, separated by hysteretic

or interm itting switching transition regimes, depending on the value of the reduced velocity of

the system. Maximum amplitude responses between the different branches can be very different

and consequently the vortical topology of the wake is strongly affected (2S-type vortex shedding

mode versus 2P-type).

In the case of flexible structures, the question of which shedding mode prevails in the wake arises.

The value of the reduced velocity can be changed by a change in the characteristic length (for

instance, radius) of the body, natural structural frequency or incoming inflow velocity of the

system. If the bluff body is not homogeneous along the spanwise direction or if the incoming

inflow is not uniform, it is necessary to predict which mode will dominate in the wake. In the

case of sheared flows past flexible structures, the inflow velocity varies along the cylinder. Even

in the event of uniform inflows past flexible structures with in-line flow-induced vibrations, one

can imagine th a t the angle of yaw of the structure relative to the incoming inflow velocity would

also vary along the structure. This will induce a spanwise variation of the normal component of

the inflow velocity. This variation might affect the local reduced velocity of the structure. It is

therefore of great theoretical and practical interest to address the question of the predominant

vortical pattern behind a free oscillating body with varying local reduced velocity.

As a first attem pt, Techet et al. [3], studied the flow behind a forced oscillating tapered cylinder

at Reynolds number from Re = 400 to Re = 1500. It was shown that within the lock-in region

and above a threshold amplitude, no cells formed bu t a single mode of response, either 2S or

2P dominated the entire span. More surprisingly, within specific parametric ranges for forcing

amplitudes and frequencies, a hybrid mode was observed, consisting of a 2S pattern along the side


79

with larger diameter and a “2P ” pattern along the side with smaller diameter. A distinct vortex

split connecting the two patterns was observed. This hybrid mode was periodic and repeatable

and the location of the vortex split was stable.

In this context, we use DNS to check if these findings can be extended to the case of a linear

shear flow past a forced oscillating straight cylinder. We also evaluate the effects of the vortex

split on the load distribution on the structure. We make sure to properly define our inflow sheared

velocity profile such th a t we conserve an equivalent local Reynolds number along the structure

compared to the experiment. In our case, the Reynolds number varies along the span of the

cylinder, not due to its radius variation as for the case of the tapered cylinder, but due to the

linear sheared inflow velocity distribution. We choose the case of an averaged Reynolds number

Rep = 400, based on the averaged inflow velocity. The aspect ratio of the cylinder is L = 26.

As in the experiment, we force the cylinder to oscillate only in the crossflow direction and we

take the forcing amplitude and forcing frequency of the harmonic motion to be A /D = 0.5 and

/* = f s D /U = 0.198 respectively, see [3].

2.1 Flow Visualizations

As in the experiment, we do obtain a periodic and persistent hybrid mode 2S-2P connected by a

vortex split, see Figures 5 and 6 . This time, the 2S mode is on the side of the low inflow velocity

while the 2P mode is on the side of the large inflow velocity. The location of the split somewhat

oscillates a b it in time around the midspan position. Figure 7 shows spatio-temporal distributions

of the lift coefficient using two different color thresholds and the presence and persistence of the

vortex split is very clear in the bottom picture with the 2S mode in the top half and the 2P

mode in the bottom half. This is because the forces distribution on the structure strongly reflects

the topology of the near wake. Figure 8 shows time history (top) and corresponding spectrum

(bottom) of lift force signals at two different spanwise locations. The left time series (Figure

8-left) is taken on the side of the low inflow velocity and exhibits a single frequency value around

0.2, characteristic of a 2S shedding mode at this Strouhal number. The right time series (Figure


80

Figure 5: DNS results; Instanteneous isocontour of pressure (p = --0.25) in the near wake. The cylinder is vertical in the pictures. Views are almost perpendicular to the plane of the inflow. Left: Front view with inflow coming from right to left. Right: Back view with inflow coming from left to right. Black arrows point to the 2S-type vortical structures and the red arrows point to the 2P-type vortical structures.


81

Figure 6: DNS results: Instanteneous isocontour of pressure (p = —0.25) in the near wake. The cylinder is at the bottom of the picture, the view is almost perpendicular to the cylinder axis and the flow is upward. Presence of a clear vortex split th a t connects 28 vortex shedding mode on the right side to a 2P vortex shedding mode on the left side.

8-right) is taken on the side of the large inflow velocity and exhibits a doubling of the frequency

with a value around 0.4, corresponding to the 2P shedding mode.

2.2 Forces D istribution and Frequency R esponse

Figure 9 shows the absolute value of the correlation coelEcients of the lift forces along the span.

We see th a t the correlation drops to zero and becomes negative for a length corresponding to

about half of the length of the cylinder. This is consistent with the observation th a t the average

location of the split lies at the midspan. Figure 10 shows time traces of the spanwise correlation of

the lift coefficient for different spanwise lengths. The horizontal time axes have been normalized

by the the shedding time period Tg of the flow. This result gives us a more detailed and local

information concerning the large modulation of the spanwise lift force correlations on the side

of the 2P mode. It also states th a t three-dimensional effects are stronger on the side of the 2P

mode which might indicate that the 2P mode is the most precarious, and the 2S mode is the more

robust.


■ 1"^m 125E-'

50oe-o'q.qoe+oo

Figure 7: Sp »tlo -«p ord isocontourdistributions of Lift coefficients

730

Figure 8 : Time history and corresponding spectrum Left: z l i ^ h M h . Left: z j d ^ 2 \ m .

+ different spanwise locations, of Lift forces at ditterei

Reproduced wittipetmissionOf the copyright owner. Further

seproduCiooprobWedvritbou,permission.

83

Figure 9: Correlation coefficients of Lift forces.

0.8

0 .4

0.2

- 0.2

0 105 15 20 25

0.4

0.2dz/d=<

- 0.2

0 5 10 15 20 25tn-,

Figure 10: Time traces of spanwise correlation of Lift coefficient.


84

3 Vortex-Induced Vibrations for Linear Sheared Flows

Numerical simulations treating the case of the response of free long slender cylindrical structures

subject to shear inflows in the turbulent regime without any ad hoc models for the flow are

very few. We are not aware, to the best of our knowledge, of accurate simulations representing

realistic situations corresponding to experimental and field conditions [81, 82]. Detailed DNS

results of linear shear flow past stationary cylinders with short aspect ratio L = 24; 48 in the

laminar regime {Re — 131.5) have been provided by Vanka and collaborators [79, 83]. Another

study is the work by Meneghini and collaborators [84]. They compute the response of different

risers subject to uniform or linear shear with moderate aspect ratio and they investigate one case

with high aspect ratio L = 4600 and piecewise linear shear at critical regime. The hydrodynamic

forces are evaluated in two-dimensional strips using a Discrete Vortex Method (DVM) and a

three-dimensional scheme is used for the structural response evaluation. However, it is very likely

tha t the DVM method under-resolves the wake at this range of Reynolds number th a t correspond

to highly three-dimensional flows.

The formulation and numerical methods used to solve this kind of problem are identical to

the ones described in Chapter 2 of P art I. Obviously, spatial resolution in the spanwise direction

becomes a critical issue because of the shear which injects vorticity in the far field, upstream of

the body. Enough Fourier modes have to be used in this direction in order to capture the scale of

the correlation length of the flow. Furthermore, the Fourier decomposition implies periodicity of

the solution and of the imposed sheared velocity profile at the inflow. Therefore, there must exist

a confined region or buffer region in the spanwise direction in which the velocity adjusts itself to

satisfy periodicity.

In table 4.1 we list the values of the parameters th a t we use in those simulations. We note

that unlike most of the previous studies where the aspect ratio of the cyhnder was of the order of

100 or less (typically 20 to 50), here we consider a very large value of aspect ratio (> 500). Our

focus is on long structures with pinned and hinged ends tha t are only allowed to oscillate in the


85

crossflow direction. We perfom two sets of experiments corresponding to two different situations.

In the second set of experiments, see Case-00 in table 4.1, our goal is to increase the complexity

of the simulation by getting closer to more realistic applications, see [81] and [82]. This is done by

increasing the aspect ratio and the shear parameter by about a factor two compared to the Case-0.

The number of Fourier modes along the span is doubled accordingly from 32 modes for Case-0

to 64 modes for Case-00, in order to keep the same spanwise resolution. The averaged Reynolds

number is also larger for each Case-00 simulations compared to the Case-0 simulation. However, it

is important to mention tha t the shear parameter is kept constant for all the Case-00 simulations.

The linear sheared inflows (including the buffer regions) for both cases are shown on Figure 11.

The buffer regions are represented by third-order polynomials with properly chosen coefficients

in order to insure periodicity of the profile, its slope and its curvature at the end points of the

domain. The structure for Case-00 is also more complex than the structure for Case-0, which is a

pure beam, as it is a mixed cable-beam structure (see table 4.1). We refer to the type of structure

as a beam if 7 0 and as a cable if c ^ 0 (see equation structure C hapterl). Moreover cable and

beam phase velocities for Case-00 are tuned to produce higher modal response. We note th a t the

mass ratio is quite different between the two experiments but this should not affect dramatically

the response as the mass-damping parameter is kept very low or zero in both cases. In our study

of Case-00, we are also more critical as we try to isolate and quantify the limitations due to the

method. The numerical method and mesh is similar to the one used in [37] with a spanwise

resolution sufficient to capture the larger structures only. The corresponding eigenspectrum of a

beam-cable structure pinned at both ends is determined by

where n is the mode number.

For the first simulation the cylinder to fluid mass ratio is m = 2, the structural damping is set to

( = 0.0 in order to obtain a maximum response, and the beam phase speed is set to 7 = 4487.92 for


86

FLOW Linear shear inflow Case-0

Linear shear inflow Case-00

Inflow profile Linear Linear

Reynolds number (max^min) Re = Ud/v:

Re — 1000 ~ 600 Turbulent Regime

Re = 2000 ~ 100 Turbulent/

Transition RegimeAveraged Reynolds number: Re Re w 800 Case-OO-a: Re « 550

Case-OO-b: Re « 825 Case-OO-c: Re w 1100

STRUCTURE Beam Beam /CableBoundary Conditions Pinned and hinged ends

Constraints No streamwise motion (x-direction)Free Transverse motion (y-direction)

Aspect ratio: L X = 567 L = 1000Mass Ratio: m m = 2 m = 8

Cable Phase Velocity: c c = 0.0 c = 5.27Beam Phase Velocity: 7 7 = 4487.92 7 = 373.0

Damping fraction: ( C = 0 .0% C = 0 .0%Shear Parameter: j3 ,5 » 8 . 8 X 10-4 /3rs 1.6 X lO-'^

Table 4.1; Non-dimensional flow and structural parameters for linear sheared flow cases.

0

-200

-400

0 .4 0.6/m e an ,

Figure 11: Normalized linear sheared inflow velocity profiles for Case-0 and Case-00.


87

an anticipated lock-in to the third mode; this corresponds to a natural non-dimensional frequency

for the beam of 0.1973. For this Imear shear inflow, the maximum Reynolds number (at z jD =- 0.0)

and minimum Reynolds number (at z j D = 567) are 1000 and 607, respectively, with a mean value

of 803.5. The shear param eter is /3 ~ 8.8 x 10“ ^, smaller than most of the values used in previous

experiments where /3 > 0.005 (e.g., see [75]). These values are close to the ones used in the

experiments reported in [81] with the exception of Reynolds number, which is lower in the current

simulations.

3.1 D isplacem ent an d Force D istributions

In [11] we first reported results for this case; a standing wave response was obtained with the third

mode excited. The location of the nodes, however, moves somewhat in time, which explains the

small but non-zero rm s values of the cross-flow displacement. Specifically, a slight shift of these

nodes towards the side of the low inflow in the shear inflow case compared to the uniform inflow

cases is observed [85]. The standing wave partitions the span of the cylinder in three different

cells. The maximum structural response of the beam is reached on the side of the high inflow.

Figure 12 displays the standard deviation or rm s values of the vertical displacement of the beam,

(normalized by the cylinder diameter), and the spectrum of (r?)^^^. The maximum rm s

value of the cross-flow displacement of the beam occurs within the first cell. The rm s structural

responses obtained in the second and the th ird cell are equal even though the beam experiences

different inflow velocity. These amplitudes are about 20% lower than the maximum amplitude of

the first cell.

The natural frequency of the beam was set to = 0.1973, which is the frequency response

of a rigid cylinder subject to VIV at a Reynolds number of Re = 1000 [37]. If we represent the

cross-flow motion in the spectral space, we see th a t only one mode is excited. The structure

frequency response is fs = 0.183 (see Figure 12, right). The spectral density is maximum in each

cell between the nodes. The wake frequency based on the Reynolds number associated with the

mean inflow velocity from the shear inflow for a stationary cylinder is around fy = 0.21 [86].


i

Figure 12: Left: Crossflow-displacement {rms values - horizontal axis) of the beam along the span normalized with the cylinder diameter. Right: Corresponding power spectral density (x- axis: frequency non-dimensionalized with maximum velocity, y-axis: power spectral density and ^;-axis: span of the cylinder.)

Therefore, the coupled flow-structure system has a frequency response th a t deviates towards a

smaller value from the imposed frequency, here /„ = 0.197, [see also [37]].

In Figure 13 we plot the mean values of the drag coefficient, {CD{z))mean, and rm s values of the

lift coefficient along the span of the cylinder. The lift and drag coefficients represented along the

span are normalized by the local inflow velocity. Due to the shear inflow the local Reynolds number

along the span varies from 1000 to 600. The mean value of C d { z ) is about 1.66. The maximum

value of C d {z ) takes place at the midspan and its value is 13% larger than the maximum C d {z )

in the case of the uniform inflow (2.1 versus 1.82, see [85]) and 29% larger than the Co{z) at the

same location in the case of the uniform inflow (2.1 versus 1.55, see [85]). The local minimum

values of the C d { z ) are located at the nodes, in agreement with the uniform inflow results [85].

The plot of {CL{z))rms exhibits a large value in the third cell and a small value in the first

cell. The response in the central cell is more intriguing and is split into two zones. The overall

{CL{z))rms along the span has a mean value of 1.12. A maximum value of 2.22 is reached at

z /D = 505 and a minimum value of 0.35 is achieved at the midspan.

The plot of the lift coefficient can be related to the cross-flow motion. The apparent inconsis

tency between the plots above could be explained by a phase analysis. The spectral density plot

(not shown here) of the Cl signal as a function of the frequency shows th a t only two frequencies are


89

------ - -

Figure 13: Left: Distribution of {Co)mean along the span. Right: Distribution of {CL)rms along the span. The local inflow velocity is used in the normalization.

primarily excited. The first one corresponds to the frequency of the beam oscillation (/„ = 0.183,

see also Figure 12), while the second one is an incomensurate higher frequency ( / = 0.296), which

we have not been able to relate to other frequencies. Clearly, the dominant frequency of Cl

along the cylinder is / = 0.183, especially within the third cell where large spectral densities are

obtained. We have observed th a t the Cl signal is in-phase with the beam motion in this cell.

This might also explain why the displacement in the third cell is comparable to the displacement

of the second cell even though they experience different inflows.

FinaUy, in Figure 14 we plot the time histories of the drag force at three different locations

along the beam. These positions have been identified after a spectrum analysis of the crossflow

u-velocity and streamwise n-velocity (not shown here) in order to locate the positions of the main

vortex dislocations. The first plot shows the drag force at z j D = 248.06, which is the location

of the main vortex dislocation of this flow. The second graph is used as a reference and does not

reside within a vortex dislocation. Finally, the third graph shows values of drag forces at another

vortex dislocation of smaller intensity. There exists a significant low frequency modulation at the


90

10.8

0.40.2

I O.S 0.6

f e . 0-4jSr 0.2

0.6

0.4

0.2

-A i\ j]

V v - .AiA

V v A *, j i i \

\ j v S III J ' \J ■ V V

560 570 580 590 600 610 620 630 640 650 660

mii\ !\ S

k f uliHii ii

4 uH IM

jiiiykilI k i ! ifWi A fi s fi 11P Pkimm

560 570 580 590 600 610 620 630 640 650 660

'

. f\ A^^JV 1 v \ / \ / P' j" V\j

fi A p . V \ AA f\ A A

560 570 580 590 600 610 620 630 640 650 660W l d

Figure 14: Comparison of time-histories of drag force at different locations along the span: (top) z / D = 248.06; (middle) z jD = 283.5; and (bottom) z j D — 416.39.

first and the third locations compared to the second one. These low frequencies are the same as

the leading frequencies of the streamwise u-velocity at these positions.

3.2 Frequency R esponses

In Figure 15 we plot frequency spectra of the crossflow velocity along the beam. These frequencies

can be interpreted as Strouhal numbers based on the maximum inflow velocity of the flow {Um =

1.0). The most distinguishing feature observed in Figure 15 is the shedding of vortices at constant

frequencies. It can be shown th a t all these peak frequencies are, in fact, linear combinations of

the harmonics of f i = 0.1996 with the harmonics of /2 == 0.2082, and they can be written in this

case as

(m — 3(n — 1))/1 + (4(n — 1) — m )/2 (2 )

where n = 2 and m = 1,2,...16. It is interesting to note th a t / i is very close to the natural

frequency for the beam (/„ = 0.1973) and /2 corresponds to the Strouhal number of the flow past

a stationary circular cylinder at Re = 620 [86]. We recall th a t 607 is the Reynolds number th a t

corresponds to the minimum inflow velocity of our shear inflow.


91

Figure 15: Spanwise (z) power spectral density of the u-component of the velocity field, (x-axis denotes frequency non-dimensionalized with maximum, velocity y-axis denotes power spectral density and z-axis denotes the span of the cylinder.)

Barring end-effects th a t may be induced by possible numerical artifacts, we can distinguish

several different frequency cells or frequency ranges. These cells do not match the cells defined

by the beam displacement (see Figure 12). The length of these cells is approximately as follows;

There is a large first cell from z j D = 50 to z j D ~ 250 with a frequency of / = 0.183, which is the

frequency of the crossflow motion of the beam. Then, the second cell lies between zJD = 250 to

z jD = 370 with two frequencies / = 0.183 and / = 0.165. A third large cell is between z j D = 370

to z / D = 425, and the last cell fills the gap between z / D = 430 to the end of the beam span.

In this case, the spectrum is not very sharp, and it is therefore more difficult to identify the

characteristic frequency. Between the cells we notice buffer zones with small spectral density, low

energy, e.g. at the midspan, or to tal energy distributed over a larger number of modes, e.g. a t the

second-node zone. Clearly, these cells are much larger than the cells encountered in stationary

cylinders but this is expected for vibrating cylinders, especially since the shear param eter is very

small in our case, and the aspect ratio is very large. The current results are certainly consistent

with the experimental results and conclusions reported in [75].

Figure 16 shows the frequency response of the wake and the structure as well as the natural

frequency of the structure. The crossflow velocity at a point {x/D = "i.y/D = 0) along the span


92

200 300z/d

Figure 16: Frequency distribution of wake and structure along the span for Case-0. Circles denote dominant frequency of crossflow velocity and triangles dominant frequency of the structure response. The maximum inflow velocity is used in the normalization of the frequency.

is analyzed to obtain the shedding frequency. The most distinguishing feature is the shedding of

vortices at constant frequencies, as it is evident by the values of the Strouhal number which are

on parallel lines. We note th a t the beam locks on to a wake frequency which dominates on the

side of the high inflow. This frequency is lower than the natural frequency of the beam for mode

n — 3.

3.3 Effects o f A veraged R eynolds Num ber

W ith Case-00 (Case 00-a, Case 00-b and Case 00-c) we pursue a param etric study of the effect

of the averaged Reynolds number on a linear shear inflow past a flexible structure with potential

high-modal response. Figure 17 shows isocontours of non-dimensional spatio-temporal cylinder

crossflow displacements for different Reynolds number. For all cases, we obtain a mixed response,

with coexistence of standing and traveling waves. It can be seen from those plots tha t we obtain

high-modal response for all cases. Low modes are also sustained for a long time integration, see

Figure 17. Moreover, it seems th a t the change in Reynolds number affects the low modes in the

structure response. This is confirmed on the next figure where we plot the time-averaged power

spectral density of the cylinder crossflow displacement versus modes number n for Case 00-a, Case


93

00-b and Case 00-c respectively. We see that the average energy in the low modes increase as we

increase the maximum Reynolds number. The variance of the energy, th a t gives an indication of

the fluctuations along the span, also increases as the Reynolds number grows. Here, we mention

th a t the value of the variance has been multiplied by ten on the graph for clarity. On the other

hand, the energy present in the high modes (n = 20) does not seem to depend strongly on the

Reynolds number. Those results can be surprising at first. However, all cases started from initial

conditions with energy already present in the low modes. Therefore, a possible explanation would

be th a t the hydrodynamic damping of a turbulent flow affects the high modes of the structural

response first before to affect the low modes. This effect seems to become more prononced as

the averaged Reynolds number is increased. Nevertheless, it is clear th a t an excitation of the low

modes can be sustained by the system for a long time is they are originally present in the initial

condition. From our experience, the presence of low modes in the response is also very sensitive

to the magnitude of the inflow velocity on the side of the low inflow.

-10008 50 89 0 SCO 91 0 9 2 0 930 940860 r .

HR0 60 4

..0

- 0 4

-100085 0 86 0 87 0 8 80 8 9 0 9 0 0 9 10 9 2 0 9 3 0 9 40

-5 0 0

9 00 9 20 9 40

.1 ^

0,5

0

-0 .5

0

i

9 8 0 1000 1 020 1040

Figure 17: Isocontours of spatio-temporal cylinder crossflow displacements Y for Case 00-a; R^max = 1000 (top), Case 00-b: Rcmax = 1500 (middle) and for Case 00-c: Remax = 2000 (bottom).


94

PSD of the cable response (vs) mode number

0.07

0.06

Time-averaged values over 100 time unite Soiid lines represent mean values and the variance is represented by error bars0.05

S 0.04

0.03

“2 0.02

0.01

Re=2000

modes

Figure 18: Time-averaged power spectral density of cylinder crossflow displacement versus modes number n for Case 00-a, Case 00-b and Case 00-c.

4 Vortex-Induced Vibrations for Exponential Sheared Flows

Currents in the ocean are invariably highly sheared and exponential shear profiles with long tails

axe the most often encountered. Here, we are earring out two different experiments where we

have similar shear parameters, but where the inflow velocity distribution is different. We list the

values of the parameters tha t we use in those simulations in table 4.1. The specific form of the

shear profiles th a t we impose at the inlet for Case-I and Case-II is a combination of exponential

functions. This will insure smoothness and compatibility with the Fourier decomposition employed

in the spanwise direction. Figure 19 shows the exponential sheared inflows (including the buffer

regions) for both cases. In the table, the shear parameters /3mean and f3max are based on the

span-averaged value or maximum value along the span of the slope of the profile respectively. We

notice th a t the shear parameters for both cases are very similar even if the shape of the profiles

are noticeably different. For instance, the specific shear profile tha t we impose at the inlet for

Case-I takes the form:

i7 (2;/I)) = f / / = 0.3; ^ = 5 .0 x lG -^ 5 = 7.5x10^

(3)


95

FLOW Exponential shear profile: Case-I

Exponential shear profile: Case-II

Reynolds number (max~irdn) Re = Ud/v.

Re = 1 0 0 0 ~ 3 0 0

Turbulent/ Transition Regime

Re = 1 0 0 0 ~ 2 0 0

Turbulent/ Transition Regime

STRUCTURE Beam /Cable Beam /CableBoundary conditions: Pinned and hinged ends

Constraints: No streamwise x-motionFree Transverse y-motion

Aspect ratio: L = 9 1 4 . 4 L = 1 8 4 3 . 3 6

Mass Ratio: m = 2 m = 2

Cable Phase Velocity: c = 2 5 . 8 c = 6 6 . 2 2

Beam Phase Velocity: 7 = 3 4 5 . 2 4 8 7 = 7 7 8 . 2 5

Damping fraction: C C = 0 . 0 % C = 0 . 0 %

Shear Parameter: j3 ^ m e a n « 2 . 0 X 1 0 “ ^

/? m a x « 1 . 0 X 1 0 - 2

P m ea n » 2 . 0 X 10-= ^

P m a x » 0 . 8 X 1 0 - 3

Table 4.2: Non-dimensional flow and structural parameters for exponential sheared flow cases.

with the high inflow velocity located at z / D = 0. But the slope of the inflow profile for Case-II

is more complex with a less fiat profile in the median region of the domain. The main difference

between the two profiles remain the value of the velocity on the side of the low inflow {U = 0.3

for Case-I versus U = 0.2 for Case-II).

In inflows with large shear the possibility exists for excitation of high modes and of a multi-

mode response [87, 39]. This is evident in the first case we consider here tha t corresponds to

y = 345.248 and c = 25.8 (see table 4.2). For instance, substituting the structural parameters of

Case-I in the Equation (1) of the eigenspectrum of the structure, we obtain a non-dimensional

frequency of 0.193 for mode n = 12. Given th a t the Strouhal number at Re ~ 0(1000) is about

S t 0.2, we verify th a t indeed the possibility exists for such high modes to be locked-in to the

wake.

4.1 Displacem ent and Force D istributions

In Figure 20 we plot the time-history of the crossflow displacement along the spanwise direction.

We see tha t a mixed response is established, which can be characterized as hybrid between a

standing wave and a traveling wave. The rms values of the crossflow displacement along the


96

1.20.6

Figure 19: Normalized exponential sheared inflow velocity profiles for Case-I and Case-II.

w .4 t f i

h Z . ..t „., I ..,...L..

y/d

2.67E-01 2.00E-01 1 .33E-01 S .67E-02 O.OOE+00

-G.Q7E-02 -I3 3 E -0 1 -2.00E-01 -2.67E-01 -3.33E-01 4 .0 0 E -0 1

Figure 20: Time-history of the distribution of crossflow displacement along the span. A mixed standing- traveling wave pattern prevails unlike the linear shear case.

3 IS 02 o.a

Figure 21: Left: Crossflow displacement (rm s values) along the span (normalized with the cylinder diameter). Right: Corresponding spectrum showing the (non-dimensional) frequency response (range: 0 — 0.3) along the span. The frequency is normalized with the maximum inflow velocity (i-axis: frequency non-dimensionalized with maximum velocity, y-axis: power spectral density and a-axis: span of the cylinder.)


97

span of the cylinder are plotted in Figure 21 (left) along with the corresponding spectrum (right)

showing a multi-mode frequency response. The amplitude response is quite small with a maximum

response between 0.2 and 0.25 on the side of the large inflow. Unlike the linear shear Case-0 , here

the structure oscillates a t low frequencies and only little energy is present in the high modes.

To investigate further this multi-mode response we obtained the excited modes by analyzing two

different instantaneous responses in wavenumber space in (plot not shown here). The highest

contributing modes are n = 1 4 — 16 which agree with the results of VIVA, a code based on

empirical modeling of the flow and eigenfrequency analysis for the structure [88]. The span-

averaged value of the crossflow displacement predicted by the current simulation is 0.22 compared

to 0.243 obtained by VIVA. Also, comparison of the instantaneous response profiles with field

measurements (performed on a drilling riser with similar geometric properties and responding to

a current with similar shear parameter) between the locations 27 < z /D < 55, is very good [82].

4.2 Frequency R esponses

We have already seen th a t the beam-cable structure oscillates mainly at a low frequency corre

sponding to a modal response between n = 3 and n = 5 (see Figure 22). This type of very severe

exponential shear where the large inflow velocity range is comparatively small compared to the

extent of the span, makes any a-priori response prediction very difficult. Indeed, the excited high

est modes are in good agreement with the theoretical estimates, but most of the energy resides

in lower modes. The occurence and the persistence of those highest modes relative to the lower

modes is even more difficult to predict. Interestingly, the low frequency of the structure oscillation

is the same as the low frequency in the wake (plot not shown here) only the low inflow side. So we

have a lock-in of the wake on the side of the low inflow velocity instead of the large inflow velocity

as in Case-0. On the large inflow side, the wake frequency is very close to the Strouhal frequency.

In contrast, the lift follows a response similar to the wake on the side of the high velocity inflow.

Therefore, unlike the linear shear Case-0 where the frequency response of lift, the wake, and the

structure are the same, in this case the hft and the beam-cable response are not the same.


98

0 < m 3 < 3 Q OCOOO<3<3<XS OOOC

O C <j OOOOOOO O O ©<J<3<i«30 C<SOO<5<’(<<J

100 200 300 400 500 600 700 800

Figure 22: Frequency distribution of wake and structure along the span showing cellular shedding and multi-mode response. Left: linear shear; Right: exponential shear. Circles denote dominant frequency of crossflow velocity and triangles dominant frequency of the structure response. The maximum inflow velocity is used in the normalization of the frequency.

4.3 Effects o f Spanwise Grid R esolution and Initial Condition

We investigate the effect of the spanwise resolution for both the fluid and the structure. To

this end, Case-II represents two different cases (Casell-a, Casell-b) where all parameters and

initial conditions are kept identical, except the number of Fourier modes used along the spanwise

direction. We test 64 complex Fourier modes (128 planes) for Casell-a and 32 Fourier modes

(64 planes) for Casell-b. The maximum Reynolds number is Rcm = 1000 and thus a turbulent

near-wake is developed on the side of the high inflow. The exponential sheared inflow velocity is

shown in Figure 19.

We obtain a mixed response for both cases, with coexistence of standing and traveling waves,

see Figure 23. Amplitude and velocity of the traveling waves for Casell-b seem to cover a wider

range and they fluctuate more than Casell-a.

Figure 24 represents the spanwise distribution of rms values of the non-dimensional crossflow

displacement (left) and rms values of the lift forces (right). The values of the crossflow displace

ment for both cases are slightly larger on the side of the low inflow than on the side of the high

inflow {z /D = 0 on the plot). This might be due to the reflection of the traveling waves once they

reach the anchor point and bounce back toward the ocean surface. The relative error between


99

-200

-400

- 1 '

-1200

1020 1040 1060 1080 1100 1120 1140 1160 1180

I

-1200

-14001080 1100 1120 1140 1160 1180 1200 1220

m

Figure 23: Isocontours of spatio-temporal cylinder crossflow displacements Y for Casell-a (left) and for Case 0-b (right).


100

Figure 24: Spanwise distribution of rm s values of crossflow displacenaent (left) and rm s values of lift forces (right) for Casell-a and Casell-b

Casell-a and Casell-b is within 10%, and the agreement is particularly good on the side of the

low inflow where fewer modes are needed to properly capture the flow scales. The values of the

lift forces for both cases are larger on the side of the high inflow (high inflow corresponds to

z /D == 0 on the plot). The agreement between the two cases is relatively good and the relative

error between Casell-a and Casell-b is within 15%. Lift forces for Casell-b look more correlated

along the spanwise direction on the side of the high inflow. Prom Figure 25-(top), where we show

8

frequency

Figure 25: Span-averaged power spectral density of cylinder crossflow displacement versusStrouhal frequency for Casell-a and Casell-b (left). Time-averaged power spectral density of cylinder crossflow displacement versus modes number n (right). Casell-a: 64 Fourier modes; Casell-b: 32 Fourier modes.

the span-averaged value of the power spectral density of the crossflow displacement against the

structural frequency, we see tha t both CasesII-a and Il-b capture almost indentical subharmonic


101

frequencies. However, there exists a difference in the higher frequencies close to the Strouhal

frequency of the flow. Casell-b predicts a lower frequency response for the structure. Concerning

the modal response of the structure, we see from Figure 25-(bottom), tha t in addition to the low

modes, it is mainly mode number n = 14 for Casell-a and n = 13 for Casell-b, which are excited

in terms of oscillations of the structure around its mean position. In conclusion, low modes and

low oscillation frequencies are not affected by the Fourier spanwise resolution. The rms values of

the crossflow displacement are overpredicted by the low resolution case (32 modes: Casell-b) ver

sus the high resolution case (64 modes; Casell-a). However, a decrease in the spanwise resolution

causes a slight shift of the high modal response as well as of the structural Strouhal frequency

to lower values. This could be explained by a change in the frequency and spanwise correlation

of the hydrodynamical forces acting on the body. A coarser resolution in the spanwise direction

could induce a stronger spanwise correlation of the lift forces. Those would make the structure

less wavy in space and subsequently it would induce a lower oscillation frequency in time.

5 Summary

The presence of strong low modes in the structural response of Case 0 seems to indicate that

this feature is very sensitive to the magnitude of the inflow velocity on the side of the low inflow.

However, other simulations (not presented here) with similar low inflow velocity but different

initial conditions for the structure and for the flow did not produce any structural low modes

response. In conclusion, it seems tha t the excitation of low modes can be sustained by the system

for a long time if they are present in the initial condition.

Numerical buffer regions in CFD are very common and they are in general located at the

ouflow. They are usually acceptable as long as they do not affect or change radically the flow

close to the body of interest. In our case, because of the periodicity of the solution in the spanwise

direction (due to the Fourier decomposition) a buffer region for the flow is needed to accommodate

the shear. This buffer region is located on the low side of the inflow. Another concern is the


102

indirect effect of the buffer region on the structure. A larger buffer region means th a t relatively

more energy is injected to the structure on the side where the fluid should be almost stagnant.

Therefore, one would think that a shorter buffer region would minimize the interference with the

dynamics of the structure more effectively. However, additional simulations with different buffer

region size have shown that a shorter buffer region enhances the growth of the low modes in time

and thus they appear sooner in the system.

We have found tha t the nodes of the structure are not always located exactly at a boundary

between two cells. We also obtained cells of constant shedding frequency, which are much longer

than the cells in stationary cylinders reported in experimental work. However, their size is consis

tent with the experimental results and corresponding conclusions reached in the works of [73, 75].

It was reported in these works th a t the size of the cells scales proportionally to the amplitude

of crossflow displacement and inversely proportional to the shear parameter. Moreover, we have

seen in our simulations tha t the larger aspect ratio of the flexible cylinder allows for larger cellular

patterns. This too is consistent with the experimental results of Peltzer [89] if we extrapolate from

his range of aspect ratio {L ~ 20 to 100) to our values {L ~ 567 to 914).

W ith regards to force distribution on the structure, we have found tha t vortex dislocations

have a significant effect on the instantaneous force distributions along the span of the cylinder.

The location of vortex dislocations in the wake can best be obtained by searching for energetic

low frequencies of the streamwise velocity component. We have observed th a t there exists a

significant modulation of the forces on the body by these low frequencies at the spanwise locations

corresponding to the vortex dislocations. This demonstrates th a t strong vortex dislocations can

have a substantial effect on the forces acting on the body, and such effects have to be taken into

account when constructing low-dimensional predictive models.


Part II

Random Oscillators

103


Chapter 5

R epresentation of Random

Processes

In this section we briefly review the Wiener-Askey Polynomial Chaos expansion along with the

Karhunen-Loeve expansion; the la tter is useful for representing the input random processes.

Throughout this section, we will use the symbol ^ to denote a random variable with zero mean

and unit variance.

1 The Generalized Polynom ial Chaos Expansion

The method we adopt in this work is an extension of the classical polynomial chaos approach [26].

This representation is an infinite sum of multi-dimensional orthogonal polynomials of standard

random variables with deterministic coefEcients. Practically, only a finite number of terms in the

expansion can be retained as the sum has to be truncated. Consequently, the multi-dimensional

random space has a finite number of dimensions n and the highest order of the orthogonal polyno

mial is finite, denoted here by p. The Hermite-chaos expansion, which is the basis of the classical

polynomial chaos, is effective in solving stochastic differential equations with Gaussian inputs as

well as certain types of non-Gaussian inputs [21, 90, 18, 19]; theoretical justification is based on

104


105

the Cameron-Martin theorem [91]. However, for general non-Gaussian random inputs, the opti

mal exponential convergence ra te is not achieved, and in some cases the convergence rate is in

fact severely deteriorated, see [92].

The Generahzed Polynomial Chaos expansion is a generalization of the original Polynomial

Chaos, first proposed by Wiener [26]. It is well suited to represent more general random inputs.

The expansion basis is the complete polynomial basis from the Askey scheme family [92, 93]. A

general second-order random process X{0) is represented by:

X{6) = ao/o

+ f ^ c M C ^ A O ) )i i = i

i l = l 22 = 1

OO X 2

+ E E Eil~l 42 = 1 3 = 1

-f ■ ■ •, (1)

where IniCii > • ■ ■ j G,,) denotes the Wiener-Askey Polynomial Chaos of order n in terms of the ran

dom vector C, = (C*i, • • • >Cin) and oq, are deterministic coefficient functions tha t uniquely

specify the process X{9).

For example, one possible choice for are the Hermite poljoiomials which correspond to the

original Wiener-Hermite Polynomial Chaos JT„. The expression of the Hermite polynomials is

given by:

7 „ ( C n , . . . , C i J = H n i C n . - . • , G J = e K " C ( _ i ) » ( 2 )

where ^ denotes the vector consisting of n Gaussian random variables (Gi > • • • i G„)-

In the Wiener-Askey chaos expansion, the polynomials J„ are not restricted to Hermite polyno

mials but rather can be all types of the orthogonal polynomials from the Askey scheme. For


106

example, the expression of the Jacobi polynomials is given by:

. . . , o j = ■ • , C J = d c J ^ r d i Z

(3)

where ^ denotes the vector consisting of n Beta random variables {Qi, ■ ■ ■ ,Q„)-

For notational convenience, we rewrite Equation (1) as:

X { 9 ) ^ ' £ c j ^ j { C ) , (4)1=0

where there is a one-to-one correspondence between the functions IniCii ,■■■, C»„) and $j(C)- Since

each type of polynomials from the Askey scheme forms a complete basis in the Hilbert space

determined by their corresponding support, we can expect each type of Wiener-Askey expansion

to converge to any L 2 functional in the L 2 sense in the corresponding Hilbert functional space as

a generalized result of the Cameron-Martin theorem; see [91, 94]. The orthogonality relation of

the Wiener-Askey Polynomial Chaos takes the form

< > dy, (5)

where Jy is the Kronecker delta and < ■,• > denotes the ensemble average which is the inner

product in the Hilbert space of the variables C- We also have:

< f i C ) 9 ( C ) > = J f i O g i O W i O d C , ( 6 )

or

< f { C ) 9 i c ) > = Y , m 9 { C ) w { c ) (7) c

in the discrete case. Here W(C) is the weighting function corresponding to the Wiener-Askey

polynomials chaos basis {# 1}.


107

Most of the orthogonal polynomials from the Askey scheme have weighting functions tha t

take the form of probability function of certain types of random distributions. We then choose

the type of independent variables ^ in the polynomials {$*(C)} according to the type of random

distributions as shown in Table 5.1. Legendre polynomials, which are a special case of the Jacobi

polynomials with parameters a = /3 = 0, correspond to an im portant distribution — the Uniform

distribution.

Random inputs Wiener-Askey chaos SupportContinuous Gaussian

GammaBeta

Uniform

Hermite-ChaosLaguerre-Chaos

Jacobi-ChaosLegendre-Chaos

( — 00, oo) [0, oc) [a,h [a,5j

Discrete Poisson Binomial

Negative Binomial Hypergeometric

Charlier-Chaos Krawtchouk-Chaos

Meixner-Chaos Hahn-Chaos

{ 0 ,1 ,2 ,...} { 0 ,1 , . . . ,A} { 0 ,1 ,2 ,...}

{ 0 ,1 , . . . ,A}

Table 5.1; Correspondence between the type of Wiener-Askey Polynomial Chaos and the type of random inputs (A' > 0 is a finite integer).

Another issue with the polynomial chaos decomposition is the fast growth of the dimensionality

of the problem with respect to the number of random dimensions and the highest order of the

retained polynomial, see Table 7.1. This issue becomes critical if one deals with a very noisy input

(white noise) or a strongly nonlinear problem or both. Indeed, an accurate representation of a

noisy input requires using a large number of random dimensions while strong nonlinear dynamics

can only be captured accurately with the use of a high polynomial order.

2 The Karhunen-Loeve Expansion

The Karhunen-Loeve expansion is another way of representing a random process [95]. It is based

on the spectral expansion of the covariance function of the process. Let us denote the process as

h{t, 9) and its covariance function as Rhh{t i ,h ) , where t\ and t 2 are the two temporal coordinates.

By definition, the covariance function is real, symmetric and positive-definite. It has an orthogonal

set of eigenfunctions which forms a complete basis. The Karhunen-Loeve expansion then takes


108

the following form:OO

h{t, 6) = h{t) +(ThY2 (S)i = l

where h{t) denotes the mean of the random process, ah denotes the standard deviation of the

process and is a set of independent random variables with a given random distribution; they

form an orthonormal random vector. Also, 6i{t) and A, are the eigenfunctions and eigenvalues of

the covariance function, respectively, i.e..

J Rhh{tut2)(j)i{t2)dt2 = \i(t>i{ti) . (9)

Among many possible decompositions of a random process, the Karhunen-Loeve expansion is

optimal in the sense th a t the mean-square error resulting from a finite-term representation of

the process is minimized [21]. Its use, however, is limited as the covariance function of the

solution process is not known a priori. Nevertheless, the Karhunen-Loeve expansion still provides

a powerful means for representing input random processes when the covariance structure is known.

Obviously, truncated expansions will be used. Therefore, an im portant issue is how many terms

are needed to represent accurately the random input. Some numerical experiments regarding the

accuracy of truncated Karhunen-Loeve representation for different covariance kernels have been

reported in [96]. Here, we derive an asymptotic expression of the relationship between the different

factors affecting the convergence of the representation in the particular case of a random first-

order Markov process. We also present numerical results to validate the theory.

Let us assume th a t the random process is applied over the time interval [0, T] and it is specified

by its correlation function:

Rhhitut2) = , A > 0 (10)

where A is the correlation length. Knowing the form of its covariance kernel, we decompose

the process in its truncated Karhunen-Loeve representation up to order n. It is then possible

to use this decomposition to reconstruct the covariance kernel and thus investigate how well it


109

is represented with n term s in the expansion. To this end, we observe th a t the diagonal terms

of the kernel should be equal to the variance a \ of the process. For a fixed number of random

dimensions, we estimated a bound on the error e of the variance of the process. This bound is

accurate for large values of i (see Equation (8)). In particular, in this case it becomes very sharp

for the diagonal end-point (see figure l-(b)).

The variance of the process is defined as:

< h^{t, 0) > = h‘ {t) + c r l Y ^ (t) and < hl{t , 0) >= ^ (t). (11)

We define the relative error in variance as:

The eigenvalues A* and eigenfunctions (pi, when i is large, take the form:

2 T2and pi{t) rjcos (13)

where r? is a normalization factor (see below, Equation (15)) and T is the length of the time

domain. For large i, we have i] w \/2 /T - Using the above expressions, we obtain the following

error bound

~ 4T cos^ ( f ^ ) ^ 4 1 T^ i2 - n A ' ^

i —n + l

We see th a t it depends on the ratio ^ and it is inversely proportional to the number of retained

terms. The dependence in (A) is related to the rate of convergence of the Fourier decomposition

for this particular kernel which has a discontinuity in its first derivative at the origin.

In figure 1 we show the diagonal terms of the kernel reconstructed from the Karhunen-Loeve

decomposition. These coefficients should be equal to the variance cr of the process. The plot (a)

^Thanks to Pr. C.-H. Su, Division of Applied Mathematics, Brown University


110

n=100

0.4n=5

0,2

1004 0 7 020O t

— M idd le p o in t ■— E n d p o in t

7 0 0

6 0 0 S lo p e = 0 .4 0 4 5

400

3 0 0

Slope=a.2015

2000200 1000 1200T/A

Figure 1: (a): Diagonal terms of the exponential covariance kernel RhhiT) = ~A = 5 and T = 100 for different number of Karhunen-Loeve random dimensions n. (b); Minimum number of Karhunen-Loeve random dimensions n times the prescribed accurac}^ e versus T /A .


I l l

confirms the convergence of the representation when we increase the number of random dimensions

n. We notice tha t the error at the boundaries is larger compared to the middle-point. The plot

(b) summarizes results from multiple cases where we estimate the minimum number of terms n

needed to reach a prescribed accuracy e as a function of T/A . We present results for the middle-

point of the interval [0, T] (figure 1 (a): at = 50) and the right end-point (figure 1 (a): at = 100).

D ata are represented by crosses and are obtained for different values of T, T / A and e. Solid lines

represent least-square approximations of the data and we indicate the corresponding slopes. The

linear relationship between T / A and n for both points is verified; it is consistent with the estimate

of inequality (14).

3 Convergence Issues

The considerable speed-up obtained by the Polynomial Chaos approach versus Monte-Carlo sim

ulation (see [92] and examples in section 1), applies to stochastic input which is a t least partially

correlated and with a relatively low dimensionality. However, for a random process describing an

input close to white noise (very short correlation length), a high dimensional chaos expansion is

required and this is difficult to handle. Even if the input is a fully correlated random variable,

there exist some open issues regarding the resolution properties of truncated Polynomial Chaos

representations.

In this section, we demonstrate the dependence of the number of terms in the Polynomial

Chaos expansion to the time domain given a prescribed level of solution accuracy. Using the par

ticular example of a first-order linear ODE with random input, we first show th a t the Polynomial

Chaos solution in this case can be found more efficiently by solving an eigenvalue problem. We

then derive a theoretical expression of the number of terms (P + 1) required in the expansion in

order to reach a given time and a prescribed accuracy for the variance of the solution. Finally,

we verify numerically the validity of the proposed analysis.


112

Let us consider the first-order linear ODE:

^ = —kx with x{ t = 0) = xq and t € [0, T] (15)

where the decay rate coefficient k is considered to be a random variable k{^) = <r (with zero mean

and variance a^) and a certain probability distribution f {k) . The deterministic and stochastic

analytic solutions are, respectively;

x{t) = xqb^^^ and x{t,^{9)) = xqb~^^^ . (16)

where k is the mean value of the decay rate coefficient k. The Polynomial Chaos stochastic

solution will be called S in the remaining of the section. The mean and variance of the stochastic

solution are, respectively:

x{t) = f xoe“ ^‘/(fc)dfc and < x ‘ { t )> = ( {xoe~^^ — x{t))^ f {k)dk. (17)JQ, Jn

The integrations are performed over the support Q defined by the corresponding distribution.

The initial condition is taken to be one to simplify the analysis, so we have: x (t = 0) = xq = 1-

Here we focus on the Gaussian, Gamma and Beta continuous distribution functions, but this is

still applicable to other distributions. We can express the solution x{t ,^) as:

x { t , 0 = Y ^ O i j { t ) ^ j i 0 with P = - D (18)j=0

(see Equation (4)) where the functions ‘i ’j(C) can be Hermite (Gaussian distribution), Laguerre

(Gamma-Exponential distribution) or Legendre (Beta-Uniform distribution) polynomial function

als. The variables n and p are the number of random dimensions and the highest polynomial


113

order of the expansion, respectively. Equation (15) can be written as:

~ -1 kx = 0 with x { t = 0) = xq and t G [0, T ] . (19)

We can then express the second term as:

p

kx{ i , t ) = CT^x{i,t) = (20)j=0

Using the following recurrent relation, valid for this family of orthogonal polynomials we have:

= cr^^jiO = + Cj^j+iiO) (21)

with ao = 0. Therefore, the right-hand-side takes the form:

p

kx{^, t ) = a ^ x { ^ , t ) == cr Y ^a j{ t ) (22)j=o

with ao = 0. The left-hand-side can be w ritten as:

p= ^ d j ( f ) $ j ( C ) , (23)

j=0

This leads to:

do{t) + boao{t) + aiai{t ) = 0; j = 0

dj{t) + C j - ia j ^ i { t ) + b ja j { t ) + aj+iaj+i{t) = 0; j = l ,2 ,3 , . . . P (24)

The recurrent relationship and corresponding set of equations for the coefEcients for H e rm ite

polynom ials take the form:

kH jiO - = cr{jHj^ iiO + J? i+ i(0 ) , aj = j, bj = 0, c, = 1 (25)


114

and

d o { t ) + a i { t ) = 0 ; j = 0

dj{t) + a j - i { t ) + {j + l )aj+i{t ) = 0; j = l ,2 ,3 , . . . P (26)

Similarly, for Laguerre polynom ials, we have;

k L j iO = = o -(-jL j_ i(C )+ (2j+ l )£ j ( C ) - ( j+ l) i j+ i ( C ) ) , aj = - j , bj = 2 j+ l , 9 = - ( j + 1)

(27)

and

do{t) + ao{t) - ai{t) = 0 ; j = 0

d j { t ) - j a j ^ i { t ) + i2j + l ) a j i t ) - ( j + l )aj+i(t ) = 0; j = 1 ,2 ,3 ,. . . P (28)

Finally, for Legendre polynom ials we have:

kP . iO = b, = 0, c, = ^

(29)

do{t) + ai{ t ) / 3 = 0; j = 0

dj{t) + 2 j L = 0; j = l , 2 , 3 , . . . P (30)

In all cases, we are left with a coupled system of equations to solve th a t can be written as:

d{t) + A a { t ) = 0, a ( t = 0) = ao (31)

where a{t) represents the vector of the unknown random modes at time t and A is the m atrix of

size [P + 1 X P + 1] built from Equations (26), (28) or (30). The solution of this system can be


115

expressed as:p

= (32)i=o

with A and © being the eigenvalues and corresponding eigenvectors of A , respectively. The scaling

of the random modes corresponding to the coefficients Cj is determined by the initial condition

ccq, using the relation:p

= (33)j=o

This method gives us a very fast and efficient way to solve the system. I t can be used for large

systems with a total number of terms in the Polynomial Chaos expansion of the order of hun

dreds. This is particularly true for the case of the Gamma and the Beta distributions. However,

in the case of the Gaussian distribution (Hermite polynomial), the m atrix A becomes poorly

conditioned for large values of P and the inversion of the m atrix becomes difficult and inaccurate

without further effort.

Because of the simple exponential form of the solution, a theoretical expression of the conver

gence rate of the moments of the expansion S for the different distributions can be derived. In

particular, we derive convergence rate estimates for the variance of the solution. We first tu rn

our attention to the case where the decay param eter k has an Exponential distribution (with zero

mean and variance cr^), which is a particular case of the general Gamma distribution (with a = 0).

The following theoretical convergence rate estimates for the variance were derived and kindly pro

vided by Pr. C.-H. Su [97]. The associated polynomial used with the Exponential distribution

is the Laguerre polynomial Lj{^) and its generating function g{z,^) is as follows:

g { z , 0 = ^ = |z| < 1. (34)i=o

If we make a change of variables with z j i l — z) = at, where a is the standard deviation of k and

^Division of Applied Mathematics, Brown University


116

t represents time, we obtain:

i= i

Here, we do not include the mean solution in the expansion. Similarly, the truncated expansion

Sp becomes:

= r i b s ( i t s ) ’j=i /

and thus,

j=p+i ^ ''

We compute the norm:

■2 ■< > =f O O 1 1

/ = — --------- (38)Jo 1 + 2(7t (1 + n't)

And thus, we can estim ate the relative error for the variance of the solution as:

< { S - Sp)^ > _ f at ^< 5'2 > V 1 -b nt

(39)

If we call this error e, we End a relation tha t links time t, e, and the total number of terms (P + 1)

in the expansion, i.e.,

p + i = i _____ + 1 (40)21og((l + a t ) / ( a t ) ) ^ ^

We now consider the case where the decay param eter k has a Gaussian distribution (with zero

mean and variance a^). The associated polynomial used with the Gaussian distribution is the

Hermite polynomial Hj{^) and its generating function g{z,^) is as follows:

2 O C j

(41)1=0

If we make a change of variables with 2 = —at, where a is the standard deviation of k and t


117

represents time, we obtain;

OO\2 yS = y - (42)n\

n = \

Here, we do not include the mean solution in the expansion. To study the rate of convergence

of this expansion we consider the norm:

< 52 > = ( v ^ ) - i • (43)

We also denote by Sp the truncation expansion in Equation (42). We note here th a t Sp is not

exactly the Polynomial Chaos expansion but we will use it to obtain an approximate estimate.

To this end, we compute:

< ( S -S p )2 > = e ( " * )^ V (44)nln—P+l

and thus we can estimate the relative error from:

< { s - s p f > 1 ^ ____________< 52 > nl - 1 (P +l)!(l - (crf)2/(J’ + 1)) ’

where we have assumed that {at )^/{P + 1) < 1. If we call this error e, we find an error bound

th a t links time t, e, and the number of terms ( F + 1), i.e.,

^ + 1)’(1 - + I ) ) ) ” '- (46)e(rt)^ - 1

Given a prescribed accuracy e, this non-linear equation is solved numerically for (P + 1) using an

iterative method, which is a combination of bisection, secant, and inverse quadratic interpolation

methods. The numerical solution is used as our reference estimate for the computation using the


118

Polynomial Chaos method described previously.

Numerical results for the case of a Uniform distribution (Wiener-Legendre Chaos) are also

presented here. The generating function of the Legendre polynomial does not lead readily to the

calculation of the coefficients of the expansion. However, a recurrent formula for the coefficients

can be derived (not presented here) based on the Legendre polynomial recurrence formula in order

to compute the variance of the solution.

Prescribing the error e to a finite value, an iterative search can be employed to estimate what

is the minimum number of terms (P + 1) required in the truncated Polynomial Chaos expansion

for any given time t in order to keep the variance of the solution within the prescribed tolerance.

Figure 2 summarizes all the results for the three different distributions with a = 1 and for an

error e = lO'"’ . It compares the estimates with the results obtained by using the Generalized

Polynomial Chaos method.

Depending on the distribution, we see th a t for a long time integration, a very large number of

modes is required to sustain the error to a prescribed level. Also, we notice tha t the theoretical

estimates for the Exponential (Figure 2 plot (a)) and Gaussian distribution (Figure 2 plot (b))

provide a lower bound, i.e. it gives the least number of required modes to achieve the error level e.

As mentioned previously, in the case of the Gaussian distribution, the m atrix A becomes poorly

conditioned for large values of P and the inversion of the m atrix for P > 30 becomes difficult and

inacurate without further effort. Overall, the Legendre-Chaos seems to be the most robust and

requires the least number of modes for the same integration time.

4 Covariance Kernels

A brief overview of some useful algebra for random variables in given in Appendix B. The

notations and derivations in the following are based on this overview and on the work by Su [97].


119

P o ly n o m ia l C h a o s e s tim a te

ot

6 = 1 . 0 X 1 0 " ’

4 .5

O Recurrence formula cstinmtc + Polynomial Chaos estiaiate

(C)

o

o © +

6=1.0x10"

Figure 2; Estimates of required number of functionals versus the product of standard deviation with time for different pdf’s, (a): Laguerre-Chaos/Exponential distribution, (b): Hermite- Chaos/Gaussian distribution, (c): Legendre-Chaos/Uniform distribution.


120

K arhunen-Loeve R epresen tation o f a R andom P rocess

We consider a random process of one independent variable as denoted by n(t; where t is a scalar

variable, taken to be time, and ^ denotes the random nature of the dependent variable v. We take

the range of t being from a to b. One way to have an intuitive feeling about is to discretize

the independent variable t between a and b into n equal intervals A t = (6 — a) /n with

to = a

ti = to + *At for i = 1 , . . . n — 1 (47)

tfi ~ h

At each of these nodes tj we assign a random number of zero mean and unit variance. We

assume these generators are mutually independent i.e., — Sij. For normal random number

generators, are normal random numbers, and the same is true for other kinds of generators.

We now denote the function v{t, at the i**' node by u,: and express it in terms of the as

n

Vi=='^ai jCj , (48)3=0

where is an (n + 1) x (n + 1) matrix. It is obvious from Equation (48) th a t each element of

aij represent the effect of the random number generator at site j on the random variable v at site

i. When a is diagonal, Vi are mutually independent.

The covariance of v is defined as:

c = {viVj) = Uikik ^ aikUjk = aoF- (49)fc=0 1=0 k= 0

Now for normal random variables a linear combination of them is itself a normal random variable.


121

We consider a linear transformation of ^ in Equation (48) to a new base r] by:

U = ^ a-ijVj or C = at], (50)i=o

Under this change of base v = a^ = aarj = f3rj with (3 = aa and

c = /3/3^ = (aa){aa)^ = a a a ^ a ^ = a a ^ (51)

provided we choose aa^ = I i.e., a is orthogonal transformation or orthogonal matrices.

Given a symmetric real matrix of (n + 1) x (n + 1), there are (n + 1) real eigenvalues and

there are also (n + 1) linear independent normalized eigenvectors It can

be shown th a t the m atrix c can be represented in term s of this eigenspectrum as follows

= (52)fe=0

where denotes the z*** component of the eigenvector. The expression in the right side of

Equation (52) can be written as a product of a m atrix (3 with its transpose as where

/3’s T*' column is eigenvector (as a column vector) multiplying by the square root of the

eigenvalue, i.e., y/Xi. The random variable u* in Equation (48) can then be represented in this

new base rji asn n

= P i j V j = Y (53)Vij=0 j=0

By letting A t 0, defined in Equation (48), one gets the Karhunen-Loeve representation

’(i) = (54)1 = 0

The covariant m atrix c becomes a kernel function c(t, t i ) and the eigenvalue problem of the matrix


122

c becomes an integral equation

and the covariance kernel of Equation (52) becomes:

O O

c{t, h ) = ^ Afc^fe(t)^fe(ti), (56)fc=0

where and ipk are the eigenvalues and eigenfunctions of the m atrix c (Equation (51)),

respectively.

Fourier Series R epresentation o f th e Integral E quation

If the random process is stationary, i.e., if it is translational invariant with respect to t, then the

covariance satisfies:

c{t, t i ) = {v{t)v{ti)) = c{t - ti). (57)

The corresponding integral Equation (55) for the eigenfunctions and eigenvalues used in the

Karhunen-Loeve representation can be cast into a spectral form by taking a Fourier series repre

sentation of the kernel function c{t — ti) and the eigenfunction il it).

The Fourier series representation provides a natural way of solving the problem if one is to

deal with a periodic problem, i.e. both the kernel function c and the eigenfunctions ip are periodic

functions with period T. In this case, the integral Equation (55) can be solved exactly as follows.

It is worthwhile to note th a t this approach can also be used as an approximation for non-periodic

cases if the domain of the problem, as denoted by T is much larger than the correlation length A

of the process.

The range of the functions ip{t), ip{t\) is (a, h) while th a t of c{t—ti ) is (—T, T) where T = b — a.

In order to represent the extended functions as Fourier series, i.e., we need to take the period to be

2T. However, if c{t — t i) has a compact support, or if it is different from zero only for \t — t i \ < A


123

for some finite A, then in cases where A <^T, the relative error in using the period being T rather

than 2T is of the order of A /T . In the literature, this approximation is sometime refered to as

spectral approximation [98, 99]. W ith both and c{t) being periodic functions of period T, we

represent them in their Fourier series as follows:

ipit)2Ti7T Stitt

tpcn COS — {t - a) + Ip sn Axi — {t - o) (58)

n = l

2mr 2wkCcn cos - j : r t + Csn Sm (59)

with

1 , / \ 2n7T, , , 1 / N . 2n7T, ,/ dtijj{t) c o s - ^ { t - a ) , = - d t i p { t ) s m ~ { t - a )

rTI2/ dtc{t) (

J-T/2

2 l-T/2 2mr I cos

2 2n7TCsn = - / dtc{t) sin t

J-T /2

(60)

(61)

In most applications c{t) is an even function then Cs„ = 0. Substituting Equations (60-61) into

(55), we obtain

Tn = l

StTTT OiTtTT0 c n COS -Y " {t - a) + tpsn sin (t - a) = 0 (62)

From this we find the set of eigenvalues and eigenfunctions as:

1. A — Aq — 2 Cco; 0cO 0 , 'ipcn — ‘4^sn — 0 f o r TI ^ Q.

0(0) =V T

(63)

2. A — Xfi — Wen 7 0, and 'Ipsn 7 — '4 sm — 0 ^ 7

, . / 2 2 ? T '7 r / . , f t j A , V / 2 , 2 ? 7 ^ 7 r , > / ^ < \

n C O S - j r i h - a ) , 0^ )(ti) = y - sm — (ti - a). (64)


124

The Karhunen-Loeve representation of the periodic random process v{t, is;

( n = l )

, 2n7T, , , 2 m r , .Ken c o s - j r i t - a) + Ksn sin ~ (65)

where

2mr 2nn/ c{t) cos—— t dt = 2 / c(t) cos—— t dt. J —T /2 «/ 0

(66)

Given a covariance function c{t), one can solve these integrals using appropriate numerical

quadrature methods. The corresponding covariance function which can be obtained directly from

Equation (65)^ S2, OriTT

(67)r“ ^ 71'jT

{v{t, = - f + Y l Ccn^— - t l ) = c(lt - t i) ,n = l

is just the Fourier series representation of the covariance function c.

4.1 O ne-D im ensional Kernels

In the following, the derivation of the second-order autoregressive processes and the analogy with

the corresponding covariance kernels were derived and kindly provided by Pr. C.-H. Su

First-O rder A utoregressive P rocess

One of the simplest form of in Equation (48), other than a diagonal one, happens to be

when Vi are velocities of a Brownian particle. In this case starting with the initial velocity Vo the

subsequent velocities at a set of discrete times t i , t 2 , . ■ - tn are given by

uq == aCo

Vi = bvi-i + af^ i for i = 1 ,2 ,. . . n

(68)



125

where Co,?i5 •• - are normal random variable with variance equal to one. This is a unilateral

type of scheme where the dependence is extended only in one direction, and it is the simplest

realistic time series. In Equation (68) a, h and / are all constants with /^ = 1 — 6 so as to ensure

th a t (u |) = a^, i.e., the average kinetic energy of the Brownian particle remains constant in time.

We can solve Vi in Equations (68) easily in terms of as:

Vi = afe=i

(69)

Comparing to Equation (48) we have;

CXi

for j = 0

a f V for 0 < j < z

0 for j > i

From Equation (69) one finds (vf) = 5 * + p il p = 1 — as we have

asserted before. Also from Equations (68)

(70)

Following our discretization scheme in Equation (68), the above expression represents the covari

ance of the velocity at ti and tj. If we fix these two points and make A t tend to zero, it is

reasonable to assume th a t b as used in Equation (48) will become closer and closer to unity.

6 = 1A t \ f A t \ a

(71)


126

where A is a scale factor of the same dimension of A t and a is any positive constant. Prom

Equation (70), it is seen tha t (viVj) will be independent of At , depending only on \ t i~ t j \ if a = 1.

In terms of the continuous variable t, we have from Equation (70);

{v{ti)v{tj)) = a^e ^ (72)

(73)

This is the most commonly used correlation function for stochastic processes and it corresponds

to a first-order Markov process and takes an exponential form where A > 0 is the correlation

length.

Second-O rder A utoregressive P rocess

For space series, a bilateral autoregression of the form:

Vi = 7Ui_i + Svi+i + Ci or

Vi = + U i + i ) + a / C i ( 7 4 )

where it is intuitively clear tha t 7 and b cannot be too large.

The spatial length of the periodic domain is L. We always take T = 1 in our numerical

examples. We discretize the domain with 2m + l equidistant points, so th a t we have; m — L j2 A x .

The expression for Vi is given by Equation (48) but because of the homogeneity of the system due

to the periodicity, it reduces to;2 m —1

j=0


127

The coefficients are such that:

a,' =

a/1-1

1 for j = 0

for 0 < j < j

for j > i

with

for A; = 1

Dk = <

2 — b‘ /D k - 1 for 1 < k < m

Because we take ao = 1, we conserve the variance of the process for each point in the domain and

we have: a / = 1 — }P/Dm. The covariance {viVj) between two points of the grid is obtained from

the dynamical system:2 m —1

“ (76)2 m —1

{viVj) ~ "y ( otjaj^i.-k 1=0

We define the mode number n th a t represents the number of waves th a t one could fit within

the domain L. The corresponding wave number is k. We have:

k =

a

27rn“ T ’

rnr.

(77)

(78)

There exists an analogy between a second-order regressive dynamical system of type (74) and the

discrete version of a non-homogeneous second-order ordinary equation:

Aa;2± fe Ui = F{xi). (79)


128

The system (74) can be transformed into Equation (79) by setting b = a-nd af^ i =

— 2+(fcAx)^^(^»)' d’dis gives in the limit of Ax going to zero:

b = ^ (go)

The expression of b for this case is similar to the expression of b for the unilateral case (71). This

time the wave number k has the dimension of the inverse of a correlation length A. Similarly to the

unilateral case, the coefficient b tends to unity as Ax (grid size) tends to zero. The plus or minus

signs in Equation (80) indicate th a t b can approach unity from above or below. This distinction

gives totally different results for the covariance kernel shape as we will see in the following. One

could argue tha t a value of 5 > 1 does not seem to make physical sense (74).

If 6 = Q-^(kAx)^ ^ which corresponds to Equation (79) with a negative sign, it can be

shown th a t the covariance kernel becomes;

1 ^+oo ^+ooc { x , y ) r = — J J (81)

Under the assumption th a t the variance is conserved for each point of the domain, i.e. {v{x)v{y)) =

a^S{x — y), it becomes:

=(*.») = = (1 + S|.r - (82)

Finally, if we assume periodicity over the domain, we have

c{x,y) = ———1- \2a + sinh (2a) + a |x - t/|(cosh (2a) - 1)1 cosh (o;|x - y\)2a + smh [2a)

+ [l — cosh (2a) — a |x — y\ sinh (2a)] sinh (a |x — y\) (83)

In the following, we will refer to the covariance kernel given by Equation (83) as K e rn e ll . Figures



129

3-(a) and 3-(b) give two examples of such a covariance kernel. The kernel is periodic, as expected,

and the decay rate is controlled by the correlation length value. The kernel decays faster for

smaller correlation lengths.

Similarly, if 6 = ^ which corresponds to Equation (79) with a positive sign, it can

be shown th a t the covariance kernel is very similar to K e rn e l l and becomes

c{x,y) = ------- :—r——Y [2a + sin (2a) + a |x ^ yl(cos (2q:) — 1)] cos (a|a; — y|)2iCX "|~ sin (2t(y.j

+ [l — cos (2a) + a |x — y\ sin (2a)] sin {a\x — y\)- (84)

In the following, we will refer to the covariance kernel given by Equation (84) as K ernel2. Figure

3-(c) shows an example of this kernel. Again, the kernel decays faster for smaller correlation

lengths. Interestingly, if n is an integer and b > 1, the covariance kernel given by Equation (84)

simply becomes:

c(x,y) = c o s ( a |x - j / | ) , (85)

and an example of such a kernel is shown in Figure 3-(d).

All those kernels being periodic, it is possible to use the Fourier decomposition method of

the integral equation described in the previous paragraph, in order to build a Karhunen-Loeve

representation of these processes. In this case, the time dimension becomes a spatial dimension

without loss of generality. We obtain the eigenvalues and eigenfunctions of the kernels from

Equations (63-64) and (60-61). The integrals of Equations (61) for K ern ell and K ernel2 are

computed numerically using numerical quadratures, except for the case of K ernel2 with n being

an integer. In this case, we can derive exact expressions for the eigenvalues and eigenfunctions of

the kernel. We only have one double eigenvalue corresponding to two eigenfunctions:

Ai = ^ (86)

®Thanks to Pr. C.-H. Su, Division of Applied Mathematics, Brown University


130

Figure 3: Covariance kernels c(x, y) with \x\ < L /2 and \y\ < L/2. (a): K e rn e ll with n = l \ k 2 t x \ a — 1/fc; (b): K ern ell with n = 2.5; k = Stt; A = 1/fc; (c): K ernel2 with n = 2.5; k 5w; A — \ j k \ (d): K ernel2 with n = 3; fc = Gtt; A = l / k . L = 1 for all cases.

= ^J~^cos{kx), = 'y ^ s in ( /c x ) (87)

Figure 4 shows the eigenvalues for K ern e ll (left) and for K ernel2 (right) for different corre

lation lengths A. Note that the smaller the value of A, the more contribution should be expected

from terms associated with smaller eigenvalues.

Figure 5 shows the first seven eigenfunctions for K ern ell (left) and K ernel2 (right) for

n = 2.5. As seen in Equation (64), each eigenvalue A„ is associated with a pair of eigenfunctions

and -^2” corresponding to a cosine and sine contributions respectively.

Figures 6 and 7 show the 5-term and 21-term approximations of K ern e ll and K ernel2 for

n = 2.5 and the associated errors.

It is shown tha t the bilateral type of scheme is not necessary in one dimension as it can be

effectively reduced to a unilateral one [100]. Thus the exponential correlation function can be

considered as the ‘elementary’ correlation in one dimension. It has been used extensively in the


131

ss

Figure 4: Eigenvalues A* for various values of the correlation length, here given by n, with n = and L = 1. Left: K ern ell. Right; KerneI2.

Bgenfunction _a_ 2^ 13!^ Eigenfunctions

4* /5* Bgenfunctlons -A - Eigenfunctions

1“ EigenfunctionEigenfuncSons

BgenfunctionsBgenhjnctions

-O.S -0,4 -0 .3 -0 .2 -0.1 0 0.1 0.2 0.3 0.4 O.S -0 .5 -0.4 -0 .3 -0 .2 -0.1 0 0.1 0.2 0.3 0.4 0.6

Figure 5; First 4-pairs eigenfunctions f i{x) with \x\ < L /2 and L = 1. Left: K ern e ll with n = 2.5. Right: K ernel2 with n = 2.5.


132

Figure 6: Approximation of Covariance kernels of c{x,y), see Figure (3-(b)). (a): 5-term approximation of K ern e ll with n = 2.5; k = 2tc;A = 1/fc; (b): 5-term absolute error of K ern e ll approximation with n = 2.5; k = 2tt: A = 1/fc; (c): 21-term approximation of K ern ell with n = 2.5; k = A = 1/fe; (d): 21-term absolute error of K ern e ll approximation with n = 2.5; k = 2tt; A = 1/fc.


133

Figure 7: Approximation of Covariance kernels of c{x,y), see Figure (3-(c)). (a): 5-term approximation of K e rn e l2 with n = 2.5; fc = 27t;A = 1/fc; (b): 5-term absolute error of K e r- nel2 approximation with n = 2.5; fc = 27t; A = 1/fc; (c): 21-term approximation of K ernel2 with n = 2.5; k = Stt; A = 1/fc; (d): 21-term absolute error of K ernel2 approximation with n = 2.5; fc = 27t; A = 1/fc.


134

literature and its Karhunen-Loeve decomposition can be solved analytically, see Section (2).

4.2 Two-Dim ensional K ernels

In two dimensions, the exponential correlation function can be written as C{r) = where

r is the distance between two spatial points. This function has been also used in the literature.

However, as W hittle pointed out in [100], it is necessary to introduce autoregression schemes with

dependence in all directions for more realistic models of random series in space. The simplest

such model is

= f l ( S s + l , t + C s - l , t + i s , t + l +

where is random field at grid (s ,t) and Cat is independent identically distributed random field.

This model corresponds to a stochastic Laplace equation in the continuous case:

d x J^{x,y) = e{x,y), (89)

where ^ = l / o — 4. The ‘elementary’ correlation function in two dimensions can be solved from

the above equation:

C (.) = [ k , ( t ) . (90)

where K i is the modified Bessel function of the second kind with order 1, b scales as the correlation

length and r is the distance between two points. On the other hand, the exponential correlation

function C{r) = in two dimensions corresponds to a rather artificial system

Adx

Aydy J 62 C(x,y) = e{x,y). (91)

It is difficult to visualize a physical mechanism which would lead to such a relation. For a detailed

discussion on this subject, see [100].

Since no analytical solution is available for the eigenvalue problem (9) of the Karhunen-Loeve


135

decomposition for the correlation function (90) of k and / , a numerical eigenvalue solver can be

employed. The following numerical results were kindly provided by Xiu et al. [101]:

Figure 8 shows the distribution of the first twenty eigenvalues. Here the parameter 6 is set

to b = 20. In Figures 9 and 10 the eigenfunctions corresponding to the first four eigenvalues are

plotted.

> 10 '

10“'

10“'

Index

Figure 8: Eigenvalues of KL decomposition with Bessel correlation function (90), b = 20.

0 .39 8 7 1 20 .398227

0 .39 7 2 5 7

0 .00633106

-0 .0189932-0 .031$553

Figure 9: Eigenfunctions of the KL decomposition with the Bessel correlation function (90), b = 20; Left: first eigenfunction. Right: second eigenfunction. (Dashed lines denote negative values.)


136

Level v3 6 0 .0316553

Level v4 6 0 ,0076245

Figure 10: Eigenfunctions of the KL decomposition with the Bessel correlation function (90), b = 20; Left: third eigenfunction, Right: fourth eigenfunction. (Dashed lines denote negative values.)

4.3 Three-D im ensional Kernels

The analysis presented in this section was derived and kindly provided by Pr. C.-H. Su In

three dimensions, the exponential correlation function can be w ritten as:

i u { X )u { Y ) ) r

{ u ^ X ) ) ’(92)

where r = |X — T | is the distance between two points located by X and Y position vectors in

the three-dimensional space.

If we consider the Helmoltz equation in three dimensions, we have:

(93)

The solution can be written as:

i k \ X ~ X ^ )


(94)


137

If we have:

{ f i X ) f { Y ) ) = S { X - Y ) , (95)

then:

( u ( X H Y ) ) = J (96)

We can integrate Equation (96) over X i by setting up a spherical polar coordinate system around

its origin X and take Y — X as the polar axis, we then have:

1 p ' ~ ^ P ~ ^ y ^ p ^ + r ^ ~ 2 p r m s e( u ( X ) u ( Y ) ) = ■. - ■■ ■ 27t / p^dp / sinffdd— ........................ (97) (47t)2 J o ^ P Jo y^p^ + ~ 2prcos9 ^

and

18ti 0

{ u \ X ) ) = ^ (98)

which give after simplification, the form of the covariance kernel of Equation (92). The exponential

kernel in the three-dimensional space is therefore directly related to the Helmoltz equation in three

dimensions. It can be shown that there exists a direct analogy between the dynamical system

associated with the exponential kernel and the discrete version of the Helmoltz equation in three

dimensions. The dynamical system in this case, involves the 6 adjacent neighbor points.


Chapter 6

Linear Oscillators

N om enclature

A: correlation length

c = 2Cwq: clamping factor

Wiener-Askey Polynomial Chaos coefficients

/ : external forcing

Hn{C)- Hermite polynomials

In{C) = ^ j i O - Wiener - Askey Polynomial Chaos basis

k = Uq: spring factor

Ln{C)' Legendre polynomials

n: number of random dimensions

p: highest polynomial order of the expansion

P + 1: number of terms in the Wiener-Askey Polynomial Chaos expansion

Pn^'^\C)' Jacobi polynomials

138


139

R x x { t i i t 2 )- covariance function

T: length of the time domain

var {X{9, t)) = ctx = {{X{6,t) — X {9, variance of X{9, t )

X{9), t): second-order random process

X{9, t) \ mean value of X{9 , t )

e: relative error in variance

(i{6): random variable

C = iCi i id) , --- , Ci„ (^)): random vector

9: random event

Aj; eigenvalues of R x x { t i , t 2 )

(j>i. eigenfunctions of R x x { i i R 2 )

ensemble average

1 Introduction

In the present work, we are interested in the non-stationary stochastic response of linear systems

{with and without random coefficients) subject to non-stationary Gaussian or non-Gaussian ran

dom external excitation. In particular, we consider random input processes tha t exhibit some de

gree of correlation in time and study the response of the single-degree-of-freedom (SDF) oscillator.

The response of more complex multi-degree-of-freedom linear systems is found by superimposing

modal responses, each obtained from the study of an SDF system, see [102]. Clearly, linear sys

tems with random coefficients are similar in some ways to non-linear systems and exhibit behavior

between constant coefficient linear systems and non-linear systems, see [103].


140

The output process of a linear system with deterministic coefficients subject to a Gaussian

input is Gaussian, and the knowledge of its mean and correlation function m atrix fully characterize

the solution response. Therefore, it may be tem pting to state, using arguments involving the

Central Limit theorem, th a t responses of linear systems are approximately Gaussian even for

non-Gaussian excitations. However, this is not generally true because physical systems have a

finite memory or relaxation time so tha t contributions of the input to the response are significant

only over a relatively short period.

Closed-form solutions for the statistical moments of the non-stationary response to a generally

defined external non-stationary input exist only in integral form. Only a few explicit closed-form

solutions in term s of elementary functions exist for particular cases of non-stationary inputs,

see [104] for an exhaustive list. Another approach is to first apply a Karhunen-Loeve spectral

decomposition to the excitation process covariance m atrix and then use orthogonal polynomials

(such as Chebyshev polynomials) in order to obtain a compact analytical description of the data,

see [105]. This compact excitation data expression allows the close-form solution of the problem.

An exact closed-form solution for the transient mean square response of a linear SDF oscillator

subject to unit step modulated white noise, which is a process with constant power spectral

density over the whole spectrum, was first derived in [106]. The steady-state covariance matrix

of the response of the SDF oscillator to a first-order Markov process has been reported in [107].

Non-stationary response of general multi-degree-of-freedorn linear systems subject to completely

general data-based non-stationary additive excitations

Several studies have treated the case of parametric random vibration of the SDF oscillator

with random tim e coefficients and stationary external forcing, see [108, 109]. The stability of the

response is essentially governed by the random parametric excitations regardless of the external

random excitation. The stability condition for the homogeneous case with random damping and

deterministic restoring force has been derived in [110]. It requires th a t the fraction of critical

damping be larger than a threshold value for the system to be stable. This value depends on the

magnitude of the spectral intensity of the random coefficient a t a frequency which is twice the


141

natural frequency of the system. In another study, a simply supported column under randomly

varying axial and transverse loads is modeled as a second-order oscillator with multiplicative

random coefficient restoring force, deterministic damping and additive random excitation, see

[111]. The system is solved by stochastic averaging and the stationary probability density solution

is derived with an appropriate condition on the fraction of critical damping. A stochastic averaging

m ethod is also used in [112] to find the stationary probability density for the case of a SDF

oscillator with param etric and additive random excitations. In this case, the random damping

and restoring force coefficients and the external forcing were taken to be mutually uncorrelated,

colored (which can be considered as the output of m ultidim ensional linear filters to white Gaussian

noise), and Gaussian noise excitations.

Conditions of moment stability were derived in [113] for the case with random time damping

and restoring force as well as random external forcing. The moment equation method is used in

[108] for solving the case of the SDF oscillator w ith random time damping, restoring force and

random external forcing, which are independent Gaussian random processes. In other work, the

stationary and non-stationary moment responses of a deterministic oscillator subjected to periodic

excitation with random amplitude and random phase disturbances were modeled as uncorrelated

stationary or non-stationary white noise processes, see [114, 115]. Here by white noise, we refer

to processes with very short correlation length.

In this chapter, we use the Generalized Polynomial Chaos approach to solve two different

classes of problems. We first consider the case of stochastic forcing represented by a first-order

Markov process with deterministic coefficients. In section 3, we derive an exact expression for

the variance of the solution to th a t problem over the entire time domain, i.e., including the

solution initial transient. Knowing the correlation kernel of the external forcing, we represent

the random input process using a Karhunen-Loeve (K-L) decomposition and solve the problem

using the Generalized Polynomial Chaos approach. We also develop a sharp error bound for K-L

representations and analyze the convergence rate of the Generalized Polynomial Chaos expansion

for different types of probability density distributions.


142

We then consider the mixed case of the random response of a SDF oscillator subject to both

external and parametric mutually uncorrelated random excitations with fully correlated distur

bances in time, which represents a random variable case. The additive noise in the system presents

a deterministic time-dependent periodic function multiplied by a stationary random variable. The

random input variables are represented by an appropriate Generalized Polynomial Chaos decom

position, and the problem is then integrated in time using effectively a coupled set of deterministic

equations derived by the Generalized Polynomial Chaos approach. This approach is an exten

sion of the Polynomial Chaos method, previously used to treat similar linear as well as nonlinear

random vibration problems [90]. However, it allows for different probability density functions for

the random disturbance. The novelty of this work resides in solving non-stationary stochastic

vibration problems with parametric excitation. It also provides careful theoretical validation of

the numerical results and accurate second-order moment convergence rates for different types of

probability density distributions.

2 Governing Equations

In this section, we determine the response of single-degree-of-freedom mechanical systems subject

to random excitations with possible randomness present in their mechanical properties. We are

particularly interested in the determination of the mean and covariance function of the response.

We consider the following linear oscillator subject to an external forcing f{t ,d):

x{t,8) + cx{t,0) + kx{t,6) = x{0,6) = xo Biid x{0,d) — Xo, t € [Q,T] (1)

The equation has been normalized with respect to the mass, so the forcing f {t , 8) has units of

acceleration. The damping factor c and spring factor k are defined as follows:

c = 2C,loq and k = lOq, (2)


143

where ( and loq are respectively, the damping ratio and the natural frequency of the system. This

system can become stochastic if the external forcing or the input parameters or both are some

random quantities. These random quantities can be evolving in time (i.e. random process) or not

evolving in time (i.e. random variable).

Let us consider the case where the damping factor c and the spring constant k are random

processes with unknown correlation functions and the external forcing is a random process with a

given correlation function. Complementary cases with different random param etric and/or forcing

inputs can be extrapolated from this case. We decompose the random process representing the

forcing term in its truncated Karhunen-Loeve expansion up to the n*^ random dimension to

obtain:

6) = f { i) + ^ ^ /i(i)?i- (3)i~Q

Because the correlation functions for the coefficients c and k are not known, we decompose the

random input parameters in terms of their Polynomial Chaos expansion:

p p= and fc(t, 61) = ^ %(t)«>j(4(6')). (4)

j=0 j=0

Finally, the solution of the problem is sought in the form given by its truncated Wiener-Askey

Polynomial Chaos expansion:

px{t, = (5)

i= 0

We substitute all expansions in the governing equation (see Equation (1)) to obtain:

p P P P P n

?)—0 i=0 j=0 i —0 i —0

We now project the above equation onto the random space spanned by our orthogonal polynomial

basis To this end, we take the inner product with each basis and average, then we use the


144

orthogonality relation (see Equation (5) in Chapter 5). We obtain a set of coupled deterministic

differential equations;

^ p p I ^ ^I 2 ^ "y "y y ^ kj(t)Xi{t)eijm — fmifyi m — 0,1,2, . . . P

(7)

where e^m = < >. These coefficients as well as < > can be determined analytically

or numerically using multi-dimensionnal numerical quadratures [21]. This system of equations

consists of (P + 1) linear equations, each equation corresponding to one random mode. More

details about the numerical implementation and the temporal discretization used to solve the

linear problem are given in the Appendix C.

In the next section, we study different combinations and different types of random inputs.

F irst, we consider the case of the random response of the system to a Gaussian or Uniform

random forcing with correlated disturbances in time (i.e. random process case). Next, we study

the case of bo th external and parametric Gaussian or Uniform random excitations with fully

correlated disturbances in time (i.e. random variable case).

3 Random Forcing Processes

The stochastic forcing is assumed to be a weakly stationary Gaussian or Uniform random process,

with zero mean and correlation function R f f { r ) , applied over a time interval [0, Tj. Equation (1)

becomes:

x{t) + 2C,LOox{t) + = f{t , 0), a;(0) = xq and i;(0) = xq, t e [0, T] (8)

In this case, we choose the random input process to be a first-order Markov process, specified by

its correlation function:

P / / ( r ) = cr^e” '^ , A > 0 (9)


145

where A is the correlation length and a f denotes the standard deviation of the process. It can be

checked tha t f { t , 9) is the stationary solution of the differential equation;

= (10)

in which W{t) is the zero-mean stationary white noise with covariance function 5{t), see [107].

Knowing the initial state of the system, there exists a theoretical solution for the asymptotic

state of the response covariance m atrix T = lim t-^oor(l) with initial condition F(0) = 0. The

time asymptotic value of the variance of the solution has been derived in [107]. Here, assuming

the same initial conditions, we derive a theoretical expression for the variance of the solution

over the entire time domain, i.e., including the solution initial transient. This expression

is independent of the type of probability distribution of the input but does depend on the type of

covariance kernel of the input. The final result is ^:

T x x { t ).g^(l + 27)(/3^ + ( l - - 7 )^ ) , 7 - 1 -2Co „t

a 2((l - 7)2 + /32) 1 4j(/3^ + + (1 + 7 )^) 47

+ sin(2at) + (/3 + 7(1 - 7 ) ) cos(2at))

/5 2 g - ( C w o + i : ) t 1 I

where

OL= w o \/l - c^; (3 = aA- j = (ujqA. (12)

Knowing the correlation function of the input in the time domain, we use the Karhunen-Loeve ex

pansion (see Equation (8) in Chapter 5) to decompose the random input process. The correspond

ing integral equation (see Equation (9) in Chapter 5) is equivalent to the following boundary-value



146

problem:

Ip" i t ) + = 0

Axp'iO) = rp{0)

Apy{T) = - ^ ( T )

and can be solved analytically. The eigenvalues and eigenfunctions are as follows:

(13)

= 77T F :T ^ cos{uit) + - — sin(wit) ) , j = 1 ,2 , . . . , n (14)( i ) + ^ i ^

where

21

Au)i^ s in (2ujiT) _ J_

2uJi \ V Auii Aw?( c o s ( 2 w iT ) — 1)

- 1 / 2

(15)

The normalization coefficient rj ensures th a t /g pi{t)dt — 1. Here, A is the correlation length,

[0, T] is the size of the time domain, and are determined numerically by solving:

w"Uli

— j j tan(w iT ) - 2 -y = 0, i = l,2 , (16)

For a given correlation length A and a standard deviation u / of the random process f { t ,9) , we

decompose the input in its truncated Karhunen-Loeve expansion up to the random dimension.

The number of random dimensions n needs to be large enough in order to resolve the scale

associated with the correlation length A. The solution of the problem is sought in the form given

by its truncated Wiener-Askey Polynomial Chaos expansion (Equation (18) in Chapter 5) where

n is the number of random dimensions and p is the highest polynomial order of the Polynomial

Chaos expansion. We expand the right-hand side of Equation (8) in its Karhunen-Loeve series and

we expand the response process x{t) in its Wiener-Askey Polynomial Chaos series. The system

does not exhibit any non-linearity in random space. This implies th a t quadratic or higher order


147

terms in the Polynomial Chaos expansion will not improve the accuracy of the solution. Therefore,

only linear terms are used in the expansion.

The general solution of the system is expressed as an integral form of the external forcing f{t):

x{t) — — [ f ( t )e sin (a{t — t i ) )d t i = — [ — sin{ati)dti , (17)Oi J o ' a J q

where a is given by Equation (12). We decompose the random forcing in its Karhunen-Loeve

representation;

1 7 * / - °° \x{t,9) = - J ^ / ( t - t i ) s i n ( a t i ) d t i . (18)

If we assume th a t the mean of the forcing is zero and we use the orthogonality property of the

decomposition, we can express the variance of the solution as-

oc 2< x ^ { t , e ) > = j - t i ) d t ^ =Txx{t)- (19)

i = i

Then, we introduce the truncated representation of the solution a;„(t, 9) and we compute the error

between the exact and approximate solutions as follows:

oo /•*< {x{ t,9)~Xn{t,0)Y > = < x ‘ {t,9) > - < xl^{t,9) >= (— )^ ^ A iM s in (a ;ti)0 i(t-ti)d fi j

i = n + l

(20)

We have already established the simplified form of the eigenvalues, eigenvectors and normalization

coefficient for large i, which are:

2 / 'in \ 2Ai ^ [ f * ) T '


148

Therefore, the absolute error becomes:

< f ; (22)

where we need to evaluate the following integral:

This integral is computed for large i and takes the value:

/j(f) = | e (a cos a t ~ Cuiosm at) - sin ( ^ t ) + a c o s ( ^ f ) | . (24)

For large time t, it simplifies to:

/ *7"' \ 2Ii{t) ^ ( -—J Ct cos (25)

and therefore, the error can be bounded by a function of the number of random dimensions n.

We obtain:2 “ cos‘ (m ) 4 < 7 ? T 5 j

< { x { t , 6 ) - X n { t , d ) ) > ~ ^ ^ 6 ^6 SAtT®

Then, we normalize the error by the asymptotic value of the variance of the solution T^xit —> oo).

If we call e the relative error, we have

where0(2 ((1 ^ )2 + ^ (-^2 + (1 + ^ )2 ) j ^ 5

(/32( l + 27) (/32 + (1 - 7 )2) ) A3’(28)



149

where the coefficients a, (3 and 7 are given in Equation (12). This error bound is particularly

sharp at the final time t = T and does not depend on the variance <7/ of the stochastic input.

n=50Asymptotic Exact Solution

n=10

i 'output

o n=5

10" “

n=2

-a- S to c h a s tic O u tp u t (PC ) S to c h a s tic O u tpu t E s tim a te S to ch a s tic Input E s tim a te

20cOot

Figure 1: Time evolution of second-order moment V^x (Case I) for different number of random dimensions n (a). Convergence rate of second-order moment of the output (Case I and II) and second-order moment F^f of the input (Case I and II) versus the number of random dimensions n a t uot = 20 (b).

We compute the second-order moment of the solution for different values of the random di

mensions n and for a correlation length A = 1.0. We present results for Gaussian (Case I) and

Uniform (Case II) random inputs. In Figure 1 (left), we present the evolution of the variance of

the solution for Case I versus non-dimensional time r = coot. As the number n increases, we see

tha t the numerical solution converges asymptotically to the exact solution for our particular set

of parameters. This is also the case for the Uniform input.

For Case I and Case II, we then compare the relative Too error at the final time versus the

exact solution (see Figure 1 (right)) and we examine the convergence rate of this error versus the

number of terms n in the Karhunen-Loeve expansion. The numerical results confirm the fact tha t

the theoretical estimate does not depend on the input distribution but on the input covariance

kernel. Indeed, we obtain exactly the same convergence rate for both distributions and the two

curves overlap to form the stochastic output. The particular convergence rate of 1/n® for this

specific kernel and equation is perfectly validated by the numerical results for large n and t. As

one would expect from the derivation, it remains an upper bound to the Polynomial Chaos solu-


150

— ■ n=10

n=50

O 1 0 '

206 12 16 180 4 8 142

Figure 2: Time evolution of the relative pointwise Too error of the second-order moment response Txx for different number of random dimensions n {Uniform distribution).

tion. We also plot the convergence rate of the corresponding stochastic input f {t , 0) based on the

same criteria developed in section 2.

In Figure 2, we plot the time evolution of the relative pointwise Too error between numerical

and theoretical solution of the variance of the solution. The error is relative to the asymptotic

expected value of the variance of the solution over the entire time domain. Again as n increases,

the accuracy of the stochastic input and output response improves, but a t a different rate. It is

also worth mentioning th a t we do not obtain a uniform convergence rate of the solution over the

time domain.

4 Random Parametric and Forcing Variables

Here, we consider a linear oscillator subject to both random parametric and external forcing

excitations. In this case, both non-stationary random forcing and random parameters are treated

as random variables. We consider two types of random inputs; Gaussian input (Case I) or

Uniform input (Case II). Equation (1) becomes:

x{t)+2(u)Qx{t)+UQx{t) — f{t ,6) = F{6) co8{ujt+(p), a;(0) = xq and i;(0) = To, t € [ 0 , T ] (29)


151

In this case, the random quantities do not vary in time, i.e. are fully correlated in time. We

assume that the probabilistic models of each of these random variables are given by:

2 C w o = C = C + CTcCl

CJq = k = k + CTfc 2

F = F + ap^s,

where ^i, ^2 and ^3 are three independent random variables with zero mean; Oc, (7k and ap are

the standard deviations of c, k and F , respectively.

The random inputs as well as the forcing function and the solution of the problem are represented

by their Wiener-Askey Polynomial Chaos expansion. The number of random dimensions n = 3

is equal to the number of independent random variables (^1, ^2 and ^3). The system does exhibit

non-linearity in random space. Therefore, quadratic or higher order term s in the Polynomial

Chaos expansion should improve the accuracy of the solution. We use the Newmark integration

scheme for this problem as described in the Appendix C.

For both Case I and II, the random parameters are set to; (c,Uc) = (0.1,0.01); (k,(7k) =

(1.05,0.105) and (F ,ap) ~ (0.1,0.01) with a frequency uj = 1.05 and a phase <(1 = 0 for the

forcing. The initial conditions xq and ±0 are set to 0. We notice tha t there is a non-zero prob

ability tha t the oscillator has a natural frequency uq = V k matching the forcing frequency w.

The time evolution of the dominant modes of the solution for Case I is represented in Figure

3 (a). We use a polynomial order p = 5 in the expansion (P + 1 = 5 6 terms). Only the first

four modes (mean plus Gaussian contribution to the solution) are presented in the plot. The

first mode (corresponding to i = 0 in Equation (7)) is the mean solution. As expected, due to

random diffusion, the amplitude of the mean solution is smaller than the deterministic solution

(not presented here). The maximum amplitude of the mean solution is about 30% lower than

the maximum amplitude of the deterministic solution. The higher modes, which describe the

stochastic part of the solution, all start from zero and then gradually grow as the interaction


152

0.2

- 0.6

1003 0

— G a u s s i a n in p u t U n ifo rm in p u t

10020t i m e

Figure 3: (a): Fifth-order (p = 5) Wiener-Askey Polynomial Chaos (P + 1 = 56 terms). Solution of the dominant random modes versus time for Case I. (b); Time evolution of the variance of the solution for Case I and II.


153

of the random modes, (through the non-linearity in random space), takes place. A stationary

periodic state is eventually reached by all modes (around t = 85 for the highest modes). The

dominant modes of the solution for Case II follow a similar pattern.

We then compute the second-order moment of the solution for Case I and II for different values

of the Polynomial Chaos order p (see Figure 3 (b) for p = 5) .

The numerical integration is performed up to T = 100 when the solution reaches the asymptotic

periodic state. We know the exact deterministic solution of the system and the probability

distribution functions of the random inputs in the system. We then integrate the solution over

the support defined by the corresponding distribution to obtain the exact mean and variance of

the solution. These integrations are performed numerically using a Gauss-Legendre quadrature.

A sufficient number of quadrature points is used to ensure convergence to converged values. We

also consider the convergence ra te of the relative Too error at the final time versus the order p of

the expansion (see Figure 4). For both cases, we see in this semi-log plot tha t the error of the

mean and variance decreases exponentially fast due to the spectral decomposition of the solution

in the random space. We notice th a t the rates of convergence are not identical and th a t the

mean decreases faster than the variance of the solution for the Gaussian case. The errors for the

Uniform case are lower than errors for the Gaussian case.

5 Summary

We have presented a convergence analysis and corresponding results of the Generalized Polynomial

Chaos approach, first presented in [92].

An efficient way to represent stochastic input processes for differential equations is through

the Kai'hunen-Loeve approach. To this end, we first developed a sharp error bound for K-L

representations, which is proportional to the interval of simulation T normalized by the correlation

length A and inversely proportional to the number of terms in the expansion n.

We then analyzed the convergence rate of the Generalized Polynomial Chaos expansion for


154

- s - Mean -A - Variance

10'

§W

10"*

p

-B~ M e a n -A - Variance

10’

10"“*

m

P

Figure 4: Convergence rate Lqo error of the mean and variance of the solution versus the order of the Wiener-Askey Polynomial Chaos expansion p. (a): Case I: Gaussian input, (b): Case II: Uniform input.


155

three different types of probability density distributions (Gamma, Beta and Gaussian). We found

th a t the most efficient representation is accomplished with the Wiener-Legendre chaos for the

Uniform distribution.

Finally, we applied the Generalized Polynomial Chaos to linear random oscillators with sta

tionary forcing as well as with parametric and non-stationary forcing. We derived a new analyti

cal solution for the time-dependent covariance and demonstrated exponential convergence of the

method with respect to the polynomial degree.

S p eed -u p fac to rs

^meanMonte-Carlo:

NGeneralized Polynomial Chaos:

P - f l S2% 350 56 6.25

Gaussian 0.8% 2,150 120 180.2% 33,200 220 1510.2% 13,000 10 13,000

Uniform 0.018% 1,580,000 20 79,0000.001% 610,000,000 35 17,430,000

Table 6.1; Speed-up factors S based on relative mean error tmean of Generalized Polynomial Chaos {{P -f 1) terms) versus Monte-Carlo simulations {N events) for Gaussian and Uniform distributions.

W ith regards to computational efficiency of this approach, we performed a sj^stematic compar

ison with the standard Monte-Carlo approach, i.e. with no special acceleration procedures. The

case we selected was the second-order random oscillator with random variables (random damping,

spring and forcing), random dimensions (n = 3), and long time integration {Tfinal = 100). The

results for Gaussian and Uniform distributions are summarized in Table 6.1. The worst speed-up

factor (<S fs 6) is for a Wiener-Hermite representation with relatively large error in the mean,

while the best speed-up (5 w 17 millions) is for a Wiener-Legendre representation with small

error in the mean. This is not very surprising given the poor resolution properties of the Hermite

polynomials in solving even deterministic problems, e.g. see [116]. However, we are not aware

of any previous efforts to compare the effectiveness of Wiener-Hermite expansions against other

representations as we have done here.

The advantages of Generalized Polynomial Chaos are clear in the cases of correlated input as


156

we demonstrated. However, in the limit of very small correlation length (e.g., T / A ^ oo) there

is a requirement for a large number of dimensions, n to represent accurately the stochastic input

process; this increases substantially the computational complexity of this approach. This is still

an unresolved problem, and in this case Monte-Carlo based approaches should be employed to

deal with the high dimensionality.


Chapter 7

Nonlinear Oscillators

1 Introduction

Fully nonlinear oscillators subject to mild or extreme noisy forces are of great interest for multiple

disciplinary engineering communities (e.g., ocean structures [117]). Many mechanical systems in

volving fiow-structure interaction can be modeled by the Duffing oscill&tor equation, see [118,119].

In the present work, we determine the response of nonlinear single-degree-of-freedom mechanical

systems subject to random excitations (Gaussian or non-Gaussian). We are particularly inter

ested in the determination of second-moment characteristics of the response of stochastic Duffing

oscillators.

The method we adopt in this work is an extension of the classical polynomial chaos ap

proach [26]. This representation is an infinite sum of multi-dimensional orthogonal polynomials

of standard random variables with deterministic coefficients. Practically, only a finite number

of term s in the expansion can be retained as the sum has to be truncated. Consequently, the

multi-dimensional random space has a finite number of dimensions n and the highest order of

the orthogonal polynomial is finite, denoted here by p. The Hermite-chaos expansion, which is

the basis of the classical polynomial chaos, is effective in solving stochastic differential equations

with Gaussian inputs as well as certain types of non-Gaussian inputs [21, 18, 19]. Its theoretical

157


158

justification is based on the Cameron-Martin theorem [91]. However, it has been found th a t for

general non-Gaussian random inputs, optimal exponential convergence rate is not achieved, and in

some cases the convergence rate is in fact severely deteriorated, see [92, 120). Another issue with

the polynomial chaos decomposition is the fast growth of the dimensionality of the problem with

respect to the number of random dimensions and the highest order of the retained polynomial,

see Table 7.1. This issue becomes critical if one deals with a very noisy input (white noise) or a

strongly nonlinear problem or both. Indeed, an accurate representation of a noisy input requires

using a large number of random dimensions while strong nonlinear dynamics can only be captured

accurately with the use of a high polynomial order.

In this chapter, we consider the case of the random response of a Duffing oscillator subject to

non-stationary additive noise, where the forcing is represented by a deterministic time-dependent

periodic function multiplied by a random variable with different distributions. We also study

the case of the random response of a Duffing oscillator subject to a stationary additive noise

represented by a random process with different distributions. The objective is twofold. First,

to investigate what type of stochastic solutions we obtain in comparison with the well-studied

deterministic Duffing oscillator. Second, to obtain the stochastic solutions at reduced cost using

adaptive procedures first pioneered by Ghanem in [121].

p=3 p=5 p=7 p=9n=2 10 21 36 55n=4 35 126 330 715n=8 165 1,287 6,435 24,310

n=16 969 20,349 245,157 2,042,975

Table 7.1: Number of unknown deterministic coefficients in the polynomial chaos representation, as a function of the number of random dimensions n and the highest polynomial order p.


159

2 Duffing Oscillator

2.1 Generalized Polynom ial Chaos R epresentation

We consider the Duffing oscillator subject to external forcing, i.e.

x{t, 9) + cx{t, 9) + k [x{t, 9) + ex^{t, 9)] = f{ t , 9). (1)

This equation has been normalized with respect to the mass, so the forcing f{ t) has units of

acceleration. The damping factor c and spring factor k are defined as follows:

c = 2(uio and k = ljq, (2)

where ( and tuo are respectively the damping ratio and the natural frequency of the system. This

system can become stochastic if the external forcing or the input parameters or both are some

random quantities. Those random quantities can evolve in time (random process) or not (random

variable).

Nonconservative restoring forces tend to correspond to hysteretic materials whose structural

properties change in time when subjected to cyclic stresses. A popular restoring force model used

in random vibration analysis consists of the superposition of a linear force ax{t) and a hysteretic

force (1 — a)Q{t), see [107].

x{t) + cx{t) + k {ax{t) + (1 - a)Q{t)) = f{ t) (3)

Q{t) = ax{t) - /3±(t)|Q(t)j” - p\x{t)\Q(t)\Q(t)\ '^^\ (4)

The coefficients a, /?, p and n control the shape of the hysteretic loop; some of these coefficients

may vary in time.

Here, we focus on the case of the Duffing oscillator, while other cases can be deducted from this


160

one. Let us consider the stochastic differential Equation (1) where the damping factor c and

the spring constant k are random processes with unknown correlation functions and the external

forcing is a random process with a given correlation function. We decompose the random process

representing the forcing term in its truncated Karhunen-Loeve expansion up to the n-th random

dimension, see [95]. We have:

/ ( t , 0) = f( t) + a t f ^ - E (5)i=l i=0

Assuming th a t the correlation functions for the coefficients c and k are not known, we can de

compose the random input parameters in their GPC expansion, [21, 18, 19] as follows:

p pc{t,0) = and k{ t ,0 ) = Y ^ k j { t ) ^ j { ^ { 0 )). (6)

j=0 j=0

Finally, the solution of the problem is sought in the form given by its truncated GPC expansion:

px{t,9) = (7)

i = 0

where n is the number of random dimensions and p is the highest polynomial order of the expan

sion.

By substituting all expansions in the governing equation (see Equation (1)), we obtain:

p p p

P / P / P P P

\ I = E / i ( 0 C i -j=0 \i=0 \i=0 fe=0 1=0 J / i=0

We project the above equation onto the random space spanned by our orthogonal polynomial

basis i-e. we take the inner product with each basis, then we use the orthogonality relation.


161

We obtain a set of coupled deterministic nonlinear differential equations:

p p p p

< 'Pm > < 'Pm > ^^0 i= 0

(P P P P \

5 > 5 5 '^ ' 5 1 j + f m{ ^ ) i (9)

i=0 j= 0 fe=0 i=0 j

where m = 0 ,1 ,2 , . . . P , d jm = < > and eijkim = < >; here < ■, • > denotes

an ensemble average. These coefficients as well as < > can be determined analytically or

numerically using multi-dimensionnal numerical quadratures. This system of equations consists

of (P + 1) nonlinear deterministic equations with each equation corresponding to one random

mode. Standard solvers can be employed to obtain the numerical solutions.

2.2 Periodic E xcitation w ith Random A m plitude

We consider a viscously damped nonlinear Duffing oscillator subject to random external forcing

excitations.

x(t, 9) + cx{t, 9) + k \x{t, 9) + ex^{t, 6)] = f { t , 9)(10)

a:(0,9) = xo and x{0,9) = xq, t & [0, T ] .

In this case, the random forcing is treated as a non-stationary random variable and has the form:

f i t , 9) = f i t ) + a f i t ) m + i f i t ) e [ o ) + h m H o ) , ( n )

where ^ is a random variable of known distribution and the coefficients are given by:

( 12)f = A { a + A^0] , o-/ = (a + 3A^/3) , 7/ = 3Aa\/3, Sf = cr\/3

a = ik — LO) cos iujt) ~ cuJ sin (w t), /? = fee cos® (w t).

An analytical solution can be obtained for this forcing of the form:

x(f) = (A + ctaC) cosiujt 4- <f>) (13)


162

with 4> = 0 and A, a a and w being some fixed constants. The random variable ^ can have

different distributions. In this section, we focus on a Gaussian (Case I) and a Uniform distribution

(Case l i b

If is a Gaussian random variable, the forcing can be represented exactly by the GPC basis using

Hermite-chaos, and has the following form:

/(C 6) = ( f i t ) + J f i t ) ) + { af { t ) + 35/(t)) C + 7 /(0 ~ 1) + <5/(0 - 3^) . (14)

If ^ is a random variable with a Uniform distribution (particular case of a Beta distribution),

the forcing can be represented exactly by the GPC basis using Legendre-chaos (particular case of

Jacobi polynomials), and has the following form:

f i t , 9) = f i t ) + ^ + K ( 0 + ^<5/(0]?+ 57 /(0 [^ (3C " - 1)] + l S f i t ) [ ^ i 5 e - 3 6 ]. (15)

We decompose the random forcing and the sought solution in its GPC expansion. After substi

tuting in the equation and projecting onto the random space, we obtain a set of coupled equations

similar to (9). This nonlinear system is simplified if we write it as a state equation. We obtain the

following discrete system which consists of a set of simultaneaous nonlinear first-order differential

equations:

X i i t ) ^ x l i t )

n A t ) + c E l o X f i t ) = - k E l o X H t ) (16)<

where Sikim = < >• These coefficients as well as < > can be determined analyti

cally or numerically very efficiently using multi-dimensional Gauss-Legendre quadratures.

Obviously, when e 0, we need at least a third-order GPC expansion to represent the forcing

exactly. Because of the form of the solution, we expect the energy injected in the system through


163

the forcing to concentrate mainly in the mean and the first mode of the solution. The energy

present in the other random modes should be zero.

Since the resulting ODEs are deterministic, we use standard exphcit schemes (Euler-forward,

Runge-Kutta of second-order and fourth order) to check the convergence rate of the solution in

time. The following results are obtained using the standard fourth-order Runge-Kutta scheme.

The structural parameters in the system (see Equation (10) and Equation (12)) and initial con

ditions are set to:

c = 0.05, fc ^ l.0 5 , (A, (Ta ) = (0.6,0.06), cu = 1.05, <)> = 0

x o ( t = 0 ) = R , x i { t = 0 ) = a A ,

xo(t = 0) = x i ( t = 0) = 0, Xj>i(f = 0) = ± i> i(t = 0) == 0.

(17)

0.6

0. 4

^ 0-2 I 03^ - 0.2

- 0 . 4

II 'I II II I I II

II III I

■I iiII ,1M ,1

i'

i\ ) I, I" n |l 11 ' I 'I |i I

‘bi i i ) .

. X (mean)

I 1 p ! II ,1

I II I !II i J 11

i;

I ' ' ' 11 " II

K “ !)

1 ‘ ' I ' > 1 1 ' , I ' ‘

' 1! I' " I ' ' I 'i 1

. 1 i ! i i i ! I

" I I ' '|I || II II I

ii W li >1 ||5 0

time

■' b d ; b y t i o yv-'vC;

time

Figure 1: Time evolution of the random modes solution for Case I (Gaussian) using a GPC expansion of 6 terms (p = 5); e = 1.0.

Figure 1 shows the time evolution of the random modes solution for Case I ( Gaussian) with

e = 1.0. A fifth-order polynomial is used to solve the problem. The top plot shows the mean and

the first mode of the solution. We notice th a t they have the proper amplitude and frequency tha t

we imposed by assuming the form of the solution. The lower plot represents the higher modes,


164

which should be indentically zero. They are very small and completely controlled by the temporal

discritization error. In this case, for fixed At , an increase of the polynomial order p does not

improve the solution error. In fact, we obtain the same results with a cubic order polynomial

as we know th a t it is enough to represent exactly the forcing term. We notice in the lower plot

th a t there exists a transient state with a burst of energy in the high modes which interact in a

nonlinear manner. At longer times the amplitude of the high modes remains bounded and the

system is stable. Similar observations and conclusions can be made for the case of the Uniform

input (Case II) for the same values of the parameters.

2 5 0 3 0 0

Figure 2: Time evolution of the random modes solution (Case II). Case of a GPC expansion of 4 terms (p = 3); e = 5.

Different values of the nonlinear param eter e were investigated for fixed values of the other

parameters. The magnitude and duration of the observed transient of the high modes mentioned

above depends on the value of e (and a^)- As e increases, the transient state takes place earlier in

time with an increased magnitude. Next, we choose e = 5 with the same set of parameters and an

input with Uniform distribution (Case II). We perform a long time integration for different values

of the polynomial order (from p = 3 to p = 11). Figure 2 shows results for p = 3. We present the

time evolution of the four random modes {xq (mean), x i , X2 and x^, see Figure 2). In this case,

we notice tha t both the mean and the first mode eventually deviate from the expected solution.


165

Higher modes also deviate toward another solution and their magnitude becomes non-negligible.

The temporal location of the onset of the bifurcation varies as a function of the temporal error

introduced by the scheme. However, the bifurcation always exists, even if the temporal error

introduced is slightly above machine precision. Moreover, we observe very similar asymptotic

behavior for higher values of p even though the transient states are somewhat different.

The critical value of e for Case II is around e 4.8. No bifurcation of the solution is obtained

for an e below this threshold value. The critical value of e for Case I is around e ss 3.7. Slightly

above this value, a long term instability develops th a t brings the initially regular (expected)

solution to a chaotic state. For both distributions, for a fixed value of e, a change in the standard

deviation of the input noise can change the regularity of the solution and bring it to another state.

For instance, for e = 5 for Case II, the transition in the solution to another state takes place if

a a ! A > 4%.

Because of the way the forcing term is defined, increasing values of the nonlinear param eter can

be seen as increasing forcing magnitudes in the equivalent normalized form of the Duffing equation

[122], Moreover, multi-frequencies are introduced in the forcing for e within some critical range.

For instance, for small values of e, the forcing is very close to a perfectly single-frequency harmonic

signal. However, in our case, the multi-frequencies forcing brings the oscillator’s mean value to

two limit cycles of different stability which coexist for certain values of the control parameters, see

plot (c-1) in Figure 3. For a limited param eter range, two stable closed orbits coexist. This kind

of jump phenomenon is observed for Duffing oscillator for which we change slightly the forcing

frequency [122]. We verified th a t once the oscillator jumps to the new solution, it does not switch

back to the original one. Concerning the first mode x i, a flip bifurcation-like occurs [122] where

the initial limit cycle loses its stability, while another closed orbit takes place whose period is half

the period of the original cycle, see plot (c-2) in Figure 3.

One fundamental question is whether the bifurcation is intrisic to the deterministic system or

whether it is in fact triggered by the uncertainty of the random input. Deterministic computations

for this case are done using the extreme values of the random input for the deterministic forcing.


166

This investigates the response of the deterministic oscillator subject to deterministic forcing whose

ampUtude is evaluated at the boundary of the density probability support (here Uniform distribu

tion). This is equivalent to setting the parameters {A, a a ) — (0 .6±0.06,0.0). While one case gives

a single hmit cycle solution (see plot (a) in Figure 3), the other case ((A, cr^) = (0.6 + 0.06,0.0),

see plot (b) in Figure 3), exhibits two hmit cycles with different amplitudes but same frequency.

this case the bifurcation is intrisic to the system.

Determlnisiic Deterministic

0.5(a) 1

0.5

> r 0 I )0

-0 .5-0 .5

-0 .5 0 0.5y

-1 -0 .5 0 0.5 1y

Stochastic; Mean Stochashc; First mode

0.6

0.40-2

0.2

0 f 1 >s- °-0 .2 V J -0.1

-0 .4 -0 .2

-0 .6 -0 .3-0 .5 0 0.5

y-0 .2 0 0.2

y

Figure 3: Phase projections of deterministic solutions and stochastic (uniform distribution) solutions.

In summary, what we have studied in this section shows complex and different dynamics for

the stochastic Duffing oscillator. A straightforward implementation of GPC is possible since

for the problem considered the relatively simple forcing does not require very high order in the

GPC expansion. However, in the general case of arbitrary stochastic forcing, the computational

complexity increases tremendously as shown in Table 7.1. To this end, we need to implement

adaptive procedures to lower the computaional complexity of stochastic nonlinear oscillators.


167

2.3 Random Forcing Processes

Solutions v ia A d ap tive G eneralized P olyn om ial C haos

We consider a nonlinear Dujfing oscillator subject to a random process excitation f {t , 9) applied

over a time interval. The equation governing the motion is given by:

x{t ,6)+2(u!ox{t ,9) + uJq (x{t,9) + ^x^{t ,9)) = f{ t ,9), a:(0) — i(0 ) = 0. (18)

We assume th a t the input process f {t , 6) is a weakly stationary random process, with zero mean

and correlation function R f f { t i , t 2 ), given by:

R f f i h M ) = o '/e" ' ' A > 0 (19)

where A is the correlation length and <t/ denotes the standard deviation of the process. If we

normalize the equation using non-dimensional time r = ixot and non-dimensional displacement

y = x / cx , where represents the standard deviation of the linear system {y = 0) with a

stationary excitation of infinite duration (T ^ co), we have:

y(t, 9) + 2Cy{t, 9) + {y{t, 9) + 6)) = y(0) = y(0) = 0, (20)

where e = /icr|. Using the above non-dimensional time, the autocorrelation function takes the

form:

R f f { A r ) = a j e ~ ' ^ ^ ^ , A > 0 (21)

where ax is given by:

(22 )


168

We also use Equation (5) to represent the stochastic forcing, i.e.

M M

f i t , = = / + ^ / E W- (23)i—0 i=l

and represent the solution y{t, 9) of the problem by its GPC expansion

py{t,9) = J 2 y i m i i m ) - (24)

i=0

The number (P + 1) of terms required in the expansion grows very rapidly with the number of

( M + 1) terms in the expansion for the input process f { t ,9) increases, see Table 7.1. However,

some of the terms in the expansion for y{t, 6) do not contribute significantly to its value. An

adaptive procedure, first introduced by Ghanem [121], can be designed in order to only keep the

terms which have the greatest contribution to the solution.

The expansion for the excitation (23) is decomposed into two summations;

/ K M \f i t , 9 ) = ! + a f [ Y , f i { t ) m + E m m \ - (25)

\ i= i /

The first summation contains the terms whose higher-order (nonlinear) contributions to the solu

tion y{t,9) will be kept at a given step of the iterative process. The second summation contains

the terms whose higher-order (nonlinear) contributions will be neglected in the computation.

Correspondingly, the expansion of the solution becomes:

K M Nyi t,9) = y + Y ^ y , i t ) i i { e ) + E y*(OC*( ') + E yf(*)^i(^»(^) i £ i ) ’ N < P (26)

i=l i=K+l j = M+l

The first two summations represent the linear contributions. The third summation represents

higher-order terms, i.e. at least quadratic polynomials in the random variables {Ci}iE=i- Another

way to understand the method is to consider th a t we enrich the space of random variables by

adding L = {M — K ) linear terms to the standard GPC expansion (see Equation (7)). W ith the


169

expansions for y{t,9) and f { t ,6) , we now solve the system for the random modes yi{t) over the

time domain. Once the current computation is completed, we then evaluate the L 2 norm of each

function yi{t) over the time interval. The K linear components yi{t), (among i < M), with the

largest norm, are sorted and reordered and then used to produce the higher-order components in

the next iteration. The iterative process is repeated. Convergence is reached and the iterative

process stops when the ordering of the K largest contributors to the solution does not change.

We present numerical results for both Case I (Gaussian) and II ( Uniform) for different values

of the nonlinear param eter e and different K, L and polynomial order p combinations. The values

for the structural parameters are:

o;o = 1.0, ( / > ; ) = (0.0,1.0), A = 1.0. (27)

The time domain extends over th irty nondimensional units (T = 30). Values of the damping

coefficient ^ will be specified for the different cases as we will see th a t it plays a key role in the

efficiency of the adaptive method.

Because of the mean forcing / being zero, we have an asymptotic value of the mean of the

solution that tends to zero and only the random modes associated with polynomials of odd order

are excited due to the form of the nonlinearity. Therefore, we compare the second-order moment

reponses obtained by GPC with and without the use of the adaptive method and also by Monte-

Carlo simulation. The variance of the solution includes the square of aU random modes (except

mode zero), so we expect a truncated representation of the solution (without reordering) to always

underpredict the exact variance of the solution.

Figure 4 shows results for Case I (Gaussian case) with e = 1.0. The damping coefficient

C, = 0.1 is quite large so the solution converges quickly within the imposed time domain. Because

the problem has been normalized, the asymptotic value of the variance of the solution for the

linear case (e = 0.0) has to be one. The linear case is run first to estimate how many term s are

needed to capture the scale associated with the correlation length A of / . We found th a t twenty


170

0.9

- - I: GPC (K=20, L=0, p=1);£=0 II: WIG (M=30; 500,000 events) Ill: GPC(K=20, L=0, p=1)

IV: GPC (K=10, UO, p=3)- - V: AGPC (K=10, L=10, p=3)

5 0,7

iO.6

■ao 0.44=8 0.3

!Z1

0.2

IV0.1

Figure 4: Case I {Gaussian)-. Comparison of second-order moment response obtained by adaptive GPC (AGPC) and Monte-Carlo simulation (MC) (500,000 events), cjq = 1-0; C = 0-1; A = 1.0; e = 1.0.

terms {K = 20) are enough for the variance of the solution to reach its asymptotic value. Monte

Carlo simulation (MC) with 500,000 realizations for the nonhnear case (e = 1.0) was performed

with th irty random dimensions to keep a safety margin. We notice th a t cubic order {p = 3) GPC

with only ten random dimensions {K = 10) is far from converging to the Monte Carlo simulation.

Linear chaos with twenty random dimensions ( i f = 20) still underestimates the Monte Carlo

simulation. The adaptive GPC of cubic order with the addition of ten more random dimensions

{L = 10) shows very clear improvement to the standard GPC and it also improves the phasing of

the solution, but it still underestimates the value of the variance. In this case, cubic polynomials

are not enough to capture the strong nonlinear behavior of the oscillator. It is worth mentioning

tha t the use of the incomplete, adaptive third-order GPC (p = 3, i f = 10, L = 10) versus the

complete standard GPC expansion (p = 3, i f = 20, L = 0) lowers significantly the number of

unknown random coelEcients from 1,771 to 296.

Figure 5 shows very similar results for Case II. Structural parameters, correlation length and

nonlinear parameter are set to the same values and only the type of distribution of the input


171

0.9

- - I: GPC(K=20, L=0, p = 1);e =0 II: MC (IV!=30; 500,000 events) Ill: GPC (K=20, L=0, p=1)- IV:GPC(K=10, L=0, p=3)

- - V: AGPC (K=10, L=10, p=3)

O 0.7 o,

S0.6

g 0.4

0.3

0.2

Figure 5: Case II: Comparison of second-order moment response obtained by adaptive GPC and Monte-Carlo simulation (500,000 events), ujq = 1-0; C = 0.1; A = 1.0; e = 1.0.

is changed to the Uniform distribution. Here again, the nonlinearity is too large for a cubic

polynomial order even with reordering of the modes.

Figures 6 and 7 show results for Case I and II with an e = 0.1 smaller than the previous cases.

The damping coefficient = 0.02 is kept low. Consequently, the solution does not converge to its

asymptotic value within the imposed time domain. However, low damping implies sharper peaks

in the energy spectrum of the oscillator. Therefore, a finite number of random dimensions is more

likely to capture most of the energy in the system. Reordering in this case also helps by sorting

out the most significant random modes corresponding to the resonant frequencies and keeping the

associated nonlinear components.

Figure 6 shows the time evolution of variance of the solution for Case I. We see th a t the

adaptive GPC with reordering is very close to the Monte-Carlo simulation. In Figure 7, we

present differently the same type of results for Case II by showing the pointwise error of the

adaptive GPC solution against the Monte-Carlo simulation.

Figure 8 shows the energy distribution among the random modes for Case I before and after

reordering. Region I in the figure represents the linear terms or random modes associated with


172

0 .9

0 ,7

0 ,5

S 0 .3

0.2

0.1

3 020

Figure 6: Comparison of second-order moment response obtained by adaptive GPC and Monte- Carlo simulation (1,000,000 events), ujq = 1.0; C — 0.02; A — 1.0; e = 0.1, (Case I: Gaussian).

- - lll:APC(K=10, L=0, p=1) IV: APC (K=10, L=10, p=3), NO reordering V: APC (K=10, U1Q, p=3), WITH reordering0.02

m

- 0,01

2 5 3 0205 100

Figure 7: Comparison of second-order moment response obtained by adaptive GPC and Monte- Carlo simulation (1,000,000 events), wq = 1-0; C = 0.02; A = 1.0; e = 0.1, (Case II; Uniform).


173

NO reordering -O- iast reordering

Figure 8; L 2-norm of the random adaptive GPC modes with no reordering and with reordering. uiQ = 1-0; C = 0.02; A = 1.0; e = 0.1. Relates to V: adaptive GPC (K=10, M=10, p=3).

the linear polynomials. Similarly, region II represents the quadratic terms (which are zero as

explained previously) and finally the cubic terms are all grouped in the region III. The linear

terms distribution before reordering clearly illustrate the concentration of energy in the system

around the peak of resonance. We also represent the last reordering, after the iterative process

has converged. We notice tha t the most energetic frequencies have been placed first and the

corresponding cubic terms have increased by as much as four orders of magnitude and about two

orders of magnitude on average.

A N ew A daptive Approach to G eneralized Polynom ial C h a o s

The concept of truncated representation of the solution in the framework of adaptive GPC method

can be extended further. This time, the solution is again expanded as in Equation (26) but

with the distinction tha t not all of the nonlinear terms from the third summation based on the

K first random dimensions are kept in the decomposition. In our case, we observe th a t the

modal energy is always large for the nonlinear terms corresponding to cross products between

random dimensions (see Figure 8). Accordingly, we only keep the coefficients corresponding to


174

the nonlinear polynomials of the form:

i £ i ) = 1 , 2 . . . i f (28)

where the operator represents the product of the combination of the K possible linear

polynomials ^i{9) taken I a t a time.

S 0.2 -

I: MC(K=40). e=0.1- - II: A G PC(K = 7,L=13,p=7),e=0.1 , NO reordering Ill: A G P C (K = 7,U 13 .p= 7),£= 0 .1 ,WITH reordering

20 25 30

Figure 9: Comparison of second-order moment response obtained by adaptive GPC and Monte- Carlo simulation (1,000,000 events), wq — 1-0; C = 0.02; A = 1.0; e = 0.1 (Case I; Gaussian).

The case of Figure 4 is repeated using the aforementioned method w ith adaptive seventh-order

GPC (p=7, i f = 7, T = 13) which represents a to tal number of 141 random modes instead of

888,030 for a standard complete GPC expansion (p=7, i f = 20, L = 0). Results axe shown

in Figui'es 9 and 10. In this case, the adaptive GPC solution does not approach uniformly the

Monte-Carlo solution over the entire tim e domain, but it is locally very accurate. The error is very

small in some places and this is an example of local non-uniform convergence of the method. A

finite number of modes might be enough to capture the behavior of the oscillator at some instants

of time but insufficient at others.


175

0 .0 3

- - II: AG PC(K=7,L=13,p=7),e =0.1, NO reordering Ill: AGPC(K=7,L=13 ,p= 7),e =0.1,WITH reordering

20

Figure 10: Absolute value of second-order moment pointwise error obtained by adaptive GPC and Monte-Carlo simulation (1,000,000 events), wq = 1-0; C = 0.02; A = 1.0; e = 0.1 (Case II: Uniform)

3 Summary

High-order polynomial chaos solutions are prohibitively expensive for strongly nonlinear systems

when the number of dimensions of the stochastic input is large. Progress can be made, however, by

careful adaptive procedures and selectively incorporating the nonlinear expansion terms. In this

chapter, we demonstrated such a procedure, proposed previously by Ghanem [121], in the context

of the stochastic DulRng oscillator. The adaptive scheme improves the accuracy of this method

by reordering the random modes according to their magnification by the system. An extension

of the originally proposed adaptive procedure was presented that uses primarily contributions

corresponding to cross products between random dimensions.


Part III

Stochastic Flow-Structure

Interactions

176


Chapter 8

Generalized Polynom ial Chaos

Formulation

1 Incompressible Navier-Stokes Equations

1.1 Governing Equations

In this section we present the solution procedure for solving the stochastic Navier-Stokes equations

by use of a generalized Polynomial Chaos expansion. Moreover, we assume th a t the flow has one

homogeneous direction. In this direction, a Fourier expansion is used providing a natural parallel

paradigm. The randomness in the solution can be introduced through boundary conditions, initial

conditions, forcing, etc..

We employ the incompressible Navier-Stokes equations:

V - u = 0, (1)

^ + (u ■ V )u = - V n + E e-^V ^u, (2)

where u is the velocity field, II is the pressure and Re the Reynolds number. All flow quantities,

177


178

i.e., velocity and pressure are considered stochastic processes. A random dimension, denoted by

the parameter 9, is introduced in addition to the spatial-temporal dimensions {X , t ) , thus

u ^ u { x , t , e y , u = n { x , t , 9 ) . (3)

We apply the generalized polynomial chaos expansion to these random quantities and obtain:

p p

n {X, t - , e ) ^J2n j {X, t ) ^ j imy , = (4)j = o i = o

with:

S!s = l r = 0

If we assume th a t the deterministic coefficients Uj (X , t) and Ilj (X , t) of the expansion are periodic

in the ^-direction, we may use a Fourier expansion:

M - l

Uj(X, t ) = U j { x , y , z , t ) =

m=0M - l

I l j { X , t ) = U j { x , y , z , t ) = ^ (6)

th a t we combine with the generalized Polynomial Chaos expansion:

p M - l

n { X , t ; 0 ) =j™0 m=0

P M - l

n { x , f , 9 ) = (7)j=0 m~0

Substituting equations (7) into Navier-Stokes equations we obtain the following equations

P M - l

j = 0 m~0P M - l P P _ M - l M - l

E E + E E [(E Ejf= 0 m = 0 ji==0/c=0 m—Q l—O


179

P M -l P M -l= - E E E E (9)

j = 0 m=0 j~0 m—0

We now take the Fourier transform of equations 8 and 9 to get the coefhcient equation for

each Fourier mode m of the expansion

E V - u , „ # , = 0 , (10)1=0

E % ^ ^ i + E E [ ' ' F T m ( N ( u ) ) ] $ i $ f c (11)j~ 0 j —0 fe~0

p P= m = 0 - . - M - l , (12)

1=0 1=0

where F FT m is the component of the Fourier transform of the non-linear terms and,

M-l M-l

N (u) == ( ^ • V) (13)m=0 1=0

~ , d d .^

+ (I-*)

To maintain computational efficiency the non-linear product is calculated in physical space

while the rest of the algorithm may be calculated in transformed space. The non-linear term is

computed using a dealiasing 3/2 rule.

Similarly to what was done in Part II for the stochastic ODEs, we then project the above

equations onto the random space spanned by the basis polynomials {$ j} by taking the inner

product of above equation with each basis. By taking < •, $ „ > and utilizing the orthogonality

condition (5), we obtain the following set of equations:

For each n = 0 , . . . P and each m = 0 , . . . M — 1,

V • u „ „ = 0, (15)


180

a p Pdt + < ^2 > ' ^2 ^ [f F T ^ (N (u )) = - V n „ „ + Re ^Lm (u„„), (16)

” J = 0 k=0

where Cjkn = < >• The set of equations consists of (P + 1) system of deterministic

‘Navier-Stokes-like’ equations for each random mode coupled through the convective terms.

Discretization in space and time can be carried out by any conventional method. Here we

employ the spectral/hp element method in space in order to have better control of the numerical

error [28]. The high-order splitting scheme together with properly defined consistent pressure

boundarj'- conditions are employed in time [123]. In particular, the spatial discretization is based

on Jacobi polynomials on triangles or quadrilaterals in two-dimensions, and tetrahedra, hexahedra

or prisms in three-dimensions.

1.2 Post-Processing

After solving for the deterministic expansion coefficients, we obtain the analjdical form (in random

space) of the solution process. It is possible to perform a number of analytical operations on the

stochastic solution in order to carry out other analysis such as the sensitivity analysis. The mean

solution is contained in the expansion term with index of zero. The second-moment, i.e., the

covariance function is given by

Puu(Ali,fi;X2,<2) = < u(Xi,ti) — u(Xi,fi),u(X2,t2) ~ u(^2jf2) >p

= J 2 [ u j { X i , t i ) u j { X 2 , t 2 ) < ^ ^ ^ > ] . (17)i= i

The variance of the solution is obtained as:

Var {u{X , t ) ) = < ( u { X , t ) - u { X , t ) y >

J 2 [ ^ i i X , t ) < ^ ] > ] (18).7=1


181

and the root-mean-square (rms) is the square root of the variance. Note th a t the summation

starts from index j = 1 instead of 0 to exclude the mean, and th a t the orthogonality of the chaos

basis {$j} has been used in deriving the above equation. Similar expressions can be obtained for

the pressure field.

2 The coupled Flow-Structure Problem

2.1 Transformed Navier-Stokes Equations

For a deterministic problem, in order to couple the flow with a structure with moving bound

aries, one could employ A rbitrary Lagrangian-Eulerian (ALE) method at a moderate cost, [10].

However, for stochastic cases, due to the increase in dimensionality of the problem, this method

becomes prohibitively expensive. Therefore, we consider a stochastic boundary-fitted coordinate

approach based on the one already described in P art I. Moreover, we only consider the stochastic

rigid motion of the cylinder, which means th a t we can use identical structural solvers as the ones

we used for stochastic ODEs, see P art II.

In a stationary, Cartesian coordinate system (x', y', z') the non-dimensionalized incompressible

Navier-Stokes equations (in convective form) takes the form given in equation 13. This time,

velocity and pressure fields are considered as stochastic processes.

We now consider a coordinate system attached to a randomly moving rigid cylinder. This

maps the time-dependent and deforming random problem domain to a stationary and spatially

non-deforming one. It is worth noting tha t this mapping is stochastic when the cylinder motion

is random and needs to be represented by the chaos expansion as well. A convenient mapping is

described by the following transformations:

X = x ' —x(f' ,0), (19a)

y = (19b)

2: = 2 ', (19c)


182

t = t'. (19d)

where we can expand the cylinder motion in its polynomial chaos expansion:

p p

j=0 j=0

The partial derivative operators axe changed as followed:

A - Adx' dx

A - Ady' dyA _dz' dzd _ d dx{t ' ,0) d drj(t',0) d

dt' dt dt dx dt dy

(21a)

(21b)

(21c)

(21d)

In the case of a flow past a rigid cylinder (or a 2D flow), this transformation amounts to an

adjustment of u and v by the cylinder velocity:

u { X , t , e ) = u ' { X ' , t \ 9 ) ^ ^ ^ ^ (22a)

v { X X B ) = v \ X ' , t \ e ) - ^ ^ ^ ^ (22b)

p ( X, t , 0 ) = p ' { X ' , t ' , e ) . (22c)

The Navier-Stokes equation and continuity equation are transformed to:

11— + ( u - V ) u = - V p + i?e"i V ^u -b A{Re, u, p, ${t, 9)), (23a)

V - u = 0, (23b)

where the forcing term A(Re , u ,p, ^{t, 9)) is the extra acceleration term introduced by the trans

formation, consisting of both inviscid and viscous contributions. In 2D flow, A{Re ,u ,p ,^ { t ,9 ) )


has a very simple form:

183

(24,

which is not x ox y dependent.

j=0 j=0

This coupled system between the flow and the moving body precludes any of the two sub-

s3 stems from remaining deterministic as soon as the other one is treated stochastically. For

instance, even if we assume tha t only the damping coefficient and/or the spring factor of the

structure are random quantities (random variables or random processes), the entire coupled system

will eventually be driven to stochasticity. Indeed, the free structure, excited by the vortex shedding

of the flow initially deterministic, produces a random response. Therefore, the position of the

boundary of the cylinder becomes uncertain. This random boundary affects the flow domain

(through the random mapping) and the flow itself becomes a stochastic process. The fluid forces

on the cylinder derived from random flow velocity field and random pressure field are random

processes as well. The damped oscillator is then subject to random param etric (random variables)

and external forcing (random process) excitations.

The transformed Navier-Stokes/structure dynamics equations are discretized in space using an

spectral/hp element based method [28], th a t employs a hybrid grid in the x —y plane. The method

uses Fourier complex exponentials along the spanwise 2—direction and is covered in Appendix

A. The time integration algorithm uses the three-step time-splitting strategy for advancing the

Navier-Stokes equations in time using a stiffly stable time integration scheme.

Similarly to a deterministic case, when dealing with a stochastic coupled flow-structure prob

lem, one ideally wants to advance both the flow and the structure in time concurrently. This

is, however, impractical as the stochastic coupled system is prohibitively expensive and even a

predictor-corrector cycle is too costly because of the high operation count of the flow solver.

Therefore we choose to lag the flow solver.


184

Algorithm

1. We begin with both the flow solver and structure solver states at timestep n. We have

already calculated drag and lift at this timestep.

2. First the contributions of the non-linear term s and A{R e ,u ,p , ^ { t ,0 ) ) are calculated using

the same time integration scheme.

3. We use the structure’s state (velocity, acceleration) to calculate the adjustments to the

time-accurate pressure boundary conditions before solving for the pressure.

4. Then pressure is calculated by solving a Poisson equation th a t enforces the continuity con

straint; the gradient of the pressure is added to the non-linear terms.

5. We use the hydrodynamic forces at timestep n to advance the structure’s state to timestep

n + 1.

6. We use the structure’s velocity to calculate the adjustments to the velocity Dirichlet bound

ary conditions before the viscous correction calculation.

7. Finally the viscous correction is calculated.

8. We calculate the new forces at timestep n -|-1.

The flow problem and the structure problem interact with each other through the boundary

conditions and the hydrodynamic forces. Specifically:

® The structure’s motion is affected by the calculated lift and drag forces on the cylinder. For

these calculations we include both the pressure and the viscous force contribution.

® As the structure moves, the flow is affected by the change in the shape and speed of its

boundary. Because of the mapping th a t keeps the mesh constant, this translates to:

- Adjustments in the velocity Dirichlet boundary conditions at the farfield. The non

slip velocity boundary conditions a t the cylinder’s surface remain zero because of the

mapping.


185

- Adjustments in the time accurate pressure boundary conditions employed (see [123])

for the acceleration term (which is zero for simulations with fixed boundaries).

— Addition of the extra term s to the Navier-Stokes equations A{Re, u,p,^).

In this chapter we formulated our stochastic flow-induced vibration problem using the stochas

tic incompressible Navier-Stokes equations and the one degree-of-freedom stochastic linear second-

order oscillator equation. We proposed a stochastic coordinate transformation th a t mapped our

moving boundary flow problem to a spatially stationary boundary flow problem, with time depen

dent boundary conditions. We derived the form of the stochastic Navier-Stokes equations under

this transformation, and described the tem poral and spatial discretizations used to solve for the

deterministic part of the flow problem. Numerical methods used to solve for the linear random

oscillators were described in P art II.


Chapter 9

Stationary Cylinders Simulations

1 Two-Dimensional Simulations with Random Inflow

1.1 Random Force R esponse

The method described in the previous chapter is applied to a random two-dimensional flow past a

stationary cylinder. We study the case of an unsteady flow at averaged Reynolds number Re = 50

and Re = 100. The uncertainty is introduced a t the inflow boundary condition and the averaged

Reynolds number is based on the averaged inflow velocity u. The uncertainty at the inflow takes

the form of a stationary random variable. The inflow velocity is {u{9) = u + (T„^(0); u = 0) where

C s a random variable of zero mean and unit variance. We have the choice of the pdf of C While a

Gaussian distribution seems to be the natural choice, it has been shown for some stochastic partial

differential equations th a t the Hermite Polynomial Chaos expansion is not necessarily well-posed

for some values of the polynomial order p [124]. This is consistent with our observation and we

have noticed th a t our simulations would become unstable for large polynomial chaos order p in

the case of a Gaussian pdf. Because of the Gaussian tails, one can see th a t there is a non-zero

probability of a negative streamwise velocity at the inflow. This possibility is unphysical and

would make the computation unstable. We therefore decide to adopt a uniform distribution. This

186


187

choice lets us have control of the bounds of the random inflow so th a t we insure positive incoming

streamwise velocity. Moreover, the Legendre-Chaos seems to be the most robust and requires the

least number of modes for the same integration time, see Section 3 in Chapter 5 of Part II.

It is possible to modify the onset of the primary instability for a laminar flow past a cylinder by

adding some noise at the inflow, as seen in [125]. In this case, the flow bifurcates from steady

state to periodic vortex shedding at a lower averaged Reynolds number than the critical Reynolds

number Rc ~ 40 given by the deterministic computation. The idea, in the present work, is to

choose an averaged Reynolds number Re = 50 above but close to the bifurcation threshold. Our

experience from deterministic simulations in tha t Reynolds number range indicates th a t the wake

flow is governed by a purely harmonic vortex shedding with small amplitudes and tha t we can fully

resolve it in space. This implies a low dimensionality of the deterministic problem at this value

of the Reynolds number tha t could be benificial to the convergence of the stochastic simulations.

Later on, comparisons will be made with the case of Re = 100.

The computational domain consists of 412 triangular elements with periodic conditions spec

ified in the crossflow direction, see Figure 1. The boundary condition at the inflow is projected

onto the polynomial chaos basis. Fifth-order Jacobi polynomials in each element are used for

the spatial decomposition of the solution. Legendre polynomials of order p are used for the de

composition of the solution in the random space. The initial condition for the mean flow is the

fully developed time-dependent deterministic simulation. Initial conditions for the higher random

modes of the flow are set to zero. Solutions with different polynomial chaos order from p = 2 to

p = 15 are computed for different values of the variance cr of the inflow.

Figure 2 shows the time evolution of the mean lift (left) and drag (right) coefficients for different

resolutions in random space. The mean values are compared to the deterministic coefficients.

Because, the initial condition for the flow is deterministic, mean and deterministic values are

identical a t the initial time of the computation. The mean lift and drag exhibit oscillatory behavior

similar to the deterministic solution. The mean lift decays to an almost zero value after about 8

shedding periods for any resolution p. The mean drag is larger than the deterministic one and


188

Figure 1: MeshS: 412 elements triangular grid in the x — y plane; [x x y] = [(—15D;25H) x (^9D ;9D )].

tends to an almost constant value in time as p increases. Figure 3 shows the time evolution of the

p=2

Figure 2: Time history of deterministic and mean lift (left) and drag (right) coefficients for different values of p. Inflow velocity: u = u + cXu ', v = 0; u = 1.0 and <iu = 0.05; Re = 50.

variance of the lift (left) and drag (right) coefficients for different resolutions in random space.

Mean lift and drag exhibit oscillatory behavior similar to the deterministic solution. The mean lift

decays to an almost zero value after about 8 shedding periods for any resolution p. The mean drag

is larger than the deterministic one and tends to an almost constant value in time as p increases.

Because both the initial condition and inflow boundary condition for the crossflow component of

the flow V are deterministic, the variance of the lift coefficient is zero at the initial time of the

computation. However, energy is initially injected in the streamwise component of the flow u

through the random inflow boundary condition which explained the positive value of the variance

of the drag coefficient a t the initial time. The variance of the lift coefficient increases to a certain

level after about 8 shedding periods for any resolution p. After this time, the behavior depends


189

on the order p. The variance of the drag coefficient initially decreases to a minimum value after

about 2 shedding periods and then increases to a certain level after about 6 shedding periods for

any resolution p. After this time, the behavior depends on the order p.

1020 1040 1080 1080 1100

Figure 3: Time history of the variance of lift (left) and drag (right) coefficients for different values of p. Inflow velocity; u = u + cr«^; u = 0; u = 1.0 and o-„ = 0.05; Re = 50.

v w w w v v w w w v w v w w w v

Figure 4: Time history of deterministic and mean lift (left) and drag (right) coefficients for different values of <r„ at p = 15. Inflow velocity: u = u + v = 0; u = 1.0: Re ~ 50.

Prom the plots of the mean and the variance of the forces on Figures 2 and 3, it seems th a t

the force signals converge to a unique soution as the polynomial order p is increased. This is

particularly obvious for the mean quantities. For instance, there is a very slight difference for

the mean drag and lift forces for p = 8 and p = 15, especially at early times. In Figure 6, we

plot the integral of the absolute value of the difference between the solutions with polynomial


190

Figure 5: Time history of the variance of lift (left) and drag (right) coefficients for different values of (T„ at p = 15. Inflow velocity: u = u + auC, v = 0; u = 1.0; Re = 50.

Figure 6: Convergence rate of lift (square symbols) and drag (circle symbols) coefficients at different times {ti = 1021; t 2 = 1046; is = 1071). Inflow velocity: u = u + cr„^; u = 0; u = 1.0 and cr„ = 0.05; Re = 50.


191

- Deterministic- P=2- p=6- P=8

» Deterministic« p=2

1000 1020 1040 1060 1080

Figure 7: Time history of deterministic and mean lift (left) and drag (right) _coefhcients for different values of p. Inflow velocity: u = u + v = 0] u = 1.0 and = 0.05; Re = 100.

1070 1C«0 1090 1000 1010 1020 1030 1040 1 050 1060 1070 1080 1090

Figure 8: Time history of the variance of lift (left) and drag (right)_coefficients for different values of p. Inflow velocity: u = u + cr„^; v = 0; u = 1.0 and = 0.05; Re — 100.


192

(a)

_ _ _ P = 1 5 ; R e ^ e a n = 5 0

p=15; =100^ m ean0.5

-0 .5

1030 1060 1070 1080 10901000 1010 1020 1040 1050

(b)

___ p=15; Re^neanp=15; Re =100

0.4

0.2

1030 1070 1080 10901000 1010 1020 1040 1050 1060

Figure 9: Comparison of normalized mean (a) and variance (b) lift coefficients for different values of Re with same <r„ = 0.05 and p — 15.

order p (p = 2 . . . 10) and the solution with polynomial order p = 15. This integral is computed

(using trapezoidal rule) from the initial time of the computation up to three different times

{ti = 1021; t 2 = 1046; ta = 1071). It is worth pointing out th a t this is not a rigorous computation

of the convergence rate as we do not compare the numerical solution to an exact solution bu t to

the best candidate (highest order p) available to us. We see from Figure 6, th a t we obtain spectral

convergence for both the drag and the lift coefficients. The rate of convergence is the same for

both coefficients but worsens (the slopes of the profiles become flatter) for increasing time. This

indicates th a t there exists a dependence between the accuracy of the method and time. Roughly

speaking, the polynomial order p should increase in time in order to keep the same accuracy. This

is consistent with what we found in Section 3, Chapter 5, P art II.

We then perform identical simulations where we only change the noise level by increasing

from (Tu = 0.05 to = 0.1. Figures 4 and 5 compare mean and variance values of the lift and

drag coefficients for the two different cr„ with p = 15. As is increased , transient responses are

shortened for both mean and variance. Mean lift forces tend faster to zero. Mean drag forces are

increased. Both drag and lift variance are increased too. Mean drag forces do not remain close to

a constant value for large (t„. This observation hints to a dependence between p and cr„. As we


193

will see later, the polynomial chaos order p needs to be increased as cr„ increases in order to keep

the same accuracy. Next, we test the influence of the Reynolds number. We perform simulations

at Re = 100 with cr„ = 0.05 for different p. We obtain similar mean and variance signals of lift

and drag coefficients as in the case of Re = 50 with the same level of noise. The main differences

are shorter transients, higher frequency of oscillations and slower convergence, see Figures 7 and

8. Moreover, the mean drag signal is always within the deterministic envelope, see Figure 7-right

compared to the Figure 2-right. Shorter transients in the case of Re = 100 are easier to spot

when we compare the normalized mean response (Figure 9-(a)) and normalized variance response

(Figure 9-(b)). The signals have been normalized by their maximum values.

For a deterministic simulation, if we use a linear model for the vortex shedding transverse

force imposed on the cylinder, we obtain an harmonic signal at the shedding frequency /„ [46]:

FL{t) = ^pU^DCLCQs{27rfJ) with S t = f ^ D / U (1)

th a t we can also write;

Cl (t) = cos(27r/«t). (2)

If we use the same kind of model for the stochastic case, the uncertainty in the incoming inflow

velocity u{6) will most likely introduce an uncertainty into the force frequency as well as into

the force amplitude. Similarly to the random inflow velocity we can assume th a t the random

frequency takes an expression of the form:

/ ; = / « ( 3 )

where ^ follows a uniform distribution. This is a strong assumption th a t neglects nonlinear effects

so th a t the same distribution is used for the force frequency and for the random inflow. Moreover,

we assume that the forcing amplitude is not strongly affected by the noise at the inflow, and


194

remains quasi constant, we have:

9) = A fl cos{2ttfyt) = cos (27r(o-/„tC + fvt)) (4)

The previous assumptions simplify considerably the model and its analysis. Our goal is to identify

and describe the difhculties of solving such a simplified version of the problem. The mean value

of the forcing can be computed and is in this case:

{Cl) = j + f 4 ) ) d i

= (5)

The variance of the transverse forcing can also be computed and is in this case:

( (O l - (C i))^ ) = j - ^ ( ^ A f „ cos {27T{afJ^ + %t))'^ { C i f

= sin(4\/37rcr/„f) cos(47rfyt) j - { C L f . (6)

We see from Equations (5) and (6) th a t mean and variance tend asymptotically to stationary

values. We have:

(C i)t_oo = 0 (7)a2

(8)

It can be shown th a t those asymptotic limits are independent of the type of probability dis

tribution functions of the random variable fy From the numerical simulations using truncated

GPC representation, it seems tha t the lift and drag coefficients asymptotically tend to stationary

processes as well.

In order to validate our simple model against the full GPC simulation, we need to estimate

the standard deviation (std) of the shedding frequency cr/ by relating it to the std of the random


195

inflow (T„ which is given. A tempting conceptual approach to the problem is to consider the

polynomial chaos simulation as equivalent to a collection of numerous deterministic simulations

(Monte Carlo simulation). In the case of a random inflow u{9) = u + au^{6) with random variable

each deterministic Monte Carlo event would represent a different inflow velocity u{9) with a

corresponding Reynolds number Re{9) = u{9)D/u and associated Strouhal frequency St{9).

If ^ obeys a uniform distribution with zero mean and unit variance, the bounds of the ensemble

described by the inflow velocity are such that: u{9) G [u — \ /3au;u + V3(t„]. This is equivalent

to a Reynolds number in the interval: Re(9) £ [{u — \/3au)D/u; {u + \/3(Tm)£>/V]. For small std

the range of Reynolds number remains close to the averaged Re. Experimental laws give the

Strouhal-Reynolds number dependence for three-dimensional laminar flows past circular cylinders

[86]. For instance, in the case of laminar parallel shedding for 47 < Re < 180, the expression of

the Strouhal number is given with good accuracy by:

= 0.2684 - (9)v R e

We see th a t there is, in this case, a mild nonlinearity between the Strouhal number and the

Reynolds number. For example, at Re = 50, if we choose (t„ = 0.035, we obtain a Reynolds

number 47 < Re < 53 which gives 0.1173 < St < 0.1262 {St = 0.1217) according to Equation (9).

At Re = 100, if we choose cr„ = 0.035, we obtain a Reynolds number 94 < Re < 106 which gives

0.1616 < S t < 0.1678 {St = 0.1647).

In Figure 10-left, we show the correspondence between Reynolds number and Strouhal number.

The same uncertainty (T„ at the inflow corresponds to a wider Reynolds number region for Re =

100 than Re = 50, but corresponds to a narrower Strouhal number range for Re = 100 than

Re = 50 due to the change in the slope of the profile, see Figure 10-left.

In terms of the shedding frequency fv{9), we have:

f ,{9) = St{0)u{9)/D (10)


196

For a uniform distribution, this means that:

/ „ ( 0 ) G [ S t { e ) m i n U { 9 ) m i n / D ' , S t { 9 ) m a x U { 0 ) m a x / D ] - ( H )

Based on this estimate, our previous numerical example gives a range of shedding frequency

in the interval [0.1102; 0.1338] for Re = 50 and /d(0) G [0.1518; 0.1780] for Re = 100.

Assuming th a t the frequency obeys Equation (4), one can derive the corresponding value of the

std (7/ appropriately so th a t the frequency remains within the imposed range. This is equivalent to

computing the std of a uniform distribution with imposed bounds in order to insure normalization

of the pdf.

The model is now closed as we have an expression for In Figure 10-right, we plot mean

lift coefficients (a) according to Equation (5) and variance of the lift coefficients (b) according to

Equation (6) for the two frequency ranges described herebefore. In this case, the amplitude of

the signals is not relevant and is taken to be the amplitude of the deterministic solution.

Those results show good qualitative agreement with the stochastic simulations, see Figure 9.

In particular, the effect of the increase of the Reynolds number is properly captured by the model.

However, we notice quantitative differences. For instance, the transients are shorter than in the

case of the firll stochastic simulations. This discrepancy is related to our simplifying assumptions.

Our Monte-Carlo-like approach is based on an empirical Strouhal-Reynolds number model for an

ensemble of fully-developed independent three-dimensional flows. However, our computation is

two-dimensional. Moreover, we use a unique deterministic initial condition, which means th a t our

solution is fully correlated at the initial time. Long time integration is therefore needed for the

uncertainty to fully propagate through the system which explains why the transients are longer

in the full simulation. Another reason for the discrepancy is the fact th a t we ignore the nonlinear

effect, between the Strouhal number and the Rejmolds number, on the distribution of the shedding

frequency. The nonlinearity affects the computation of the bounds of the frequency range but it

should also affect the form of the pdf of the shedding frequency. Instead, we recast the problem


197

by assuming tha t the frequency obeys a uniform distribution which allows us to compute the std

based on the given interval range.

(b)

Figure 10: Left: Strouhal-Reynolds number dependence for Re G [47; 180]. Right: models of mean (a) and variance (b) of lift coefScients for (T„ = 0.035 for two different cases: {Re = 50; St — 0.1217) and (Re = 100; S i = 0.1647).

Figure 11: Time evolution of the pdf of Cl given by Equation (4) for = 0.06, cr/„ = 0.05 and fv — 0.13. Ty is the deterministic shedding period.

Figure 11 shows the time evolution of the pdf of the signal Cp given by Equation (4). The pdfs

are obtained by Monte Carlo simulations using 500,00 events at each tim e instant and we choose

the amplitude Ap^ » 0.06 and the frequency fy = 0.13 from the deterministic lift coefficient

signal. A constant value of Ap^ implies th a t the support of the pdf oi C p is always included


198

within the \ interval (see Figure 11). At t = 0, the pdf takes the form of a delta

function. Later on, the distribution flattens out and moves from side to side, spreading out untils

it fills up the entire interval. It then takes a form similar to a uniform distribution with larger

values at the boundaries of the support. Step function-like discontinuities move in time along the

distributions and eventually vanish as t —> oo. As t oo, the distribution eventually stabilizes

to a stationary profile looking like a uniform distribution with elevated sides.

Figure 12: Time evolution of the pdf of Cp given by the full scale simulation. Inflow velocity: u = u + au^; u = 0; u = 1.0 and cr« = 0.05; Re = 50 and p = 7. is the deterministic shedding period.

Figure 12 shows the tim e evolution of the pdf of Cp given by the full scale GPC simulation, in

this case using p = 7 polynomial order. We obtain qualitatively comparable results. The bounds

of the pdfs oscillate in time from side to side. The pdf flattens and spreads out as time increases

in a more complicated manner than in the theoretical case. This is probably due to the fact th a t

the amplitude of the random signal also varies in time. Small discontinuities “crawl” along the

profiles similarly to the theoretical case.


199

1.2 Random -Flow Visualizations

All instantaneous random flow visualizations we present in the following correspond to the final

time of the computation tU /D = 1096. Figure 13 shows isocontours of the deterministic solution

of the vorticity. Figures 14, 15 and 16 show isocontours of the mean solution of the vorticity

in the wake of the cylinder. When we compare those figures with Figure 13, we see that, as we

increase p, the classic features of an unsteady Von-Karman deterministic wake, disappear. The

wake becomes almost anti-symmetric with respect to the y = 0 axis and large values of vorticity

only remain present in the direct vicinity of the cylinder for x /D < 4. The mean values become

close to zero along the y = 0 axis, even close to the body.

Figure 13: Instantaneous spatial distribution of deterministic value of vorticity at tU /D = 1096.

Figure 14; Instantaneous spatial distribution of mean value of vorticity at tU /D = 1096; p = 2.

Figures 17, 18 and 19 show isocontours of the rms solution of the vorticity in the wake. Again,

as p increases, the wake is drastically changed as it becomes, this time, symmetric with respect


200

■isg

Figure 15: Instantaneous spatial distribution of mean value of vorticity at tU /D — 1096; p = 8.

'Mu.:..

Figure 16: Instantaneous spatial distribution of mean value of vorticity at tU /D = 1096; p = 15.


201

to the y = 0 axis. The largest rms values concentrate in a bubble looking region in the near

wake, but away from the cylinder. The bubble takes an arrowhead-like shape (pointing toward

the cylinder) as p is increased.

Figure 17; Instantaneous spatial distribution of rms value of vorticity at tU / D = 1096; p = 2.

Figure 18; Instantaneous spatial distribution of rms value of vorticity at t U / D = 1096; p = 8.

Figure 20 shows 4 instantaneous images of the rms value of vorticity. The images are taken

at 4 different instants within one shedding period. We see th a t the wake features do not change

much in time. This is consistent with our hypothesis of stationarity of the solution as time tends

to infinity.

Figures 21, 22, 23, 24, 25 and 26 show mean and rms values of streamwise and crossflow velocity

components for p = 15. Similarly to the mean vorticity, the mean of the crossflow component of

velocity in the far wake is totally removed compared to its deterministic counterpart, see Figures

24 and 25.


202

Figure 19; Instantaneous spatial distribution of rms value of vorticity at tU /D = 1096; p ~ 15.

Figure 20: Instantaneous spatial distribution of rms value of vorticity at 4 different instants within one period; p = 15.

Figure 21: Instantaneous spatial distribution of deterministic value of streamwise velocity at tU /D = 1096.


203

Figure 22: Instantaneous spatial distribution of mean value of streamwise velocity at tU /D 1096; p = 15.

Figure 23: Instantaneous spatial distribution of rms value of streamwise velocity at tUJD = 1096; p = 15.

Figure 24: Instantaneous spatial distribution of deterministic value of crossflow velocity &ttU/D 1096.


204

JPM

y p l

Figure 25: Instantaneous spatial distribution of mean value of crossflow velocity at tU /D = 1096; p = 15.

Figure 26: Instantaneous spatial distribution of mean value of crossflow velocity a t tU /D = 1096; p — 15.


205

2 Three-Dim ensional Simulations with Random Inflow

The method described in Chapter 8 is applied to a random three-dimensional flow past a stationary

cylinder. We study the case of an unsteady flow above the onset of three-dimensionality. The

uncertainty is injected a t the inflow and the averaged Reynolds number based on the averaged

inflow velocity is Re = 300. The uncertainty at the inflow takes the form of a stationary uniform

random variable. The inflow velocity is (u{d) — u + cr„^(0); u = 0; w = 0) where ^ is a random

variable of zero mean and unit variance and cr„ = 0.05. The computational domain consists of 708

triangular elements with periodic conditions specified in the crossflow direction, see Figure 27.

The aspect ratio of the cylinder is taken to be L = 47t and we use 8 Fourier modes (corresponding

to 16 Fourier planes) along the spanwise direction. The boundary condition at the inflow is

projected onto the polynomial chaos basis. Fifth-order Jacobi polynomials in each element are

used for the spatial decomposition of the solution. Legendre polynomials of order p are used for

the decomposition of the solution in the random space. The initial condition for the mean flow is a

three-dimensional time-dependent deterministic simulation a t Re = 300. Initial conditions for the

higher random modes of the flow are set to zero. Solutions with different polynomial chaos order

from p = 2 to p = 6. I t is worth mentioning tha t the three-dimensional deterministic solution

is not perfectly harmonic but it is subject to some modulation in the time domain. Moreover,

it exhibits a symmetry with respect to the {x , y ,L/2 ) plane which explains for instance that

the span-averaged values of the spanwise forces acting on the cylinder are very small. This could

become an issue in the route to three-dimensionality of the stochastic solution as the deterministic

solution is used as the initial solution for the stochastic solution.

2.1 Random-Force R esponse

Figures 28, 29 and 30 shows the time evolution of the span-averaged deterministic and mean lift,

drag and spanwise force coefficient solutions for different resolutions. Those figures are to be

related to the isocontours of Figures 31, 32. Temporal and spatial evolution of the variance of the


206

Figure 27: Mesh4: 708 elements triangular grid in the x — y plane; [x x y] ~ [(—22D;55.D) x (-22D ; 22D)].

spanwise force coefficients is presented in Figure 33.

0.6

0.4

0.2

-0.2

-0 .4

-0.( —- Deterministic p=2

-O.i p ^ 4

p=5 P=6

230 240 250210 220180 190 200

Figure 28: Time evolution of span-averaged deterministic and mean lift coefficient Cl solutions for different p. Re — 300; au ~ 0.05.

Figures 35 and 36 show the time evolution of the span-averaged values of the variance of the

lift and drag coefficients for different resolutions in random space. Those Figures relate to Figures

37, 38. and 39.

Results for the random lift forces are qualitatively similar to the two-dimensional simulations.

The mean solutions amplitude are lower than the deterministic solution and tend to small values

after about 6 periods. The variance grows from a zero value and reaches some kind of intermediate


207

— Deterministic— p=2

p=4p=5P =61.35

1.25

1.2

1.15240 250220 230210190 200180

tU/D

Figure 29: Time evolution of span-averaged deterministic and mean drag coefficient Cd solutions for different p. Re = 300; cTu ~ 0.05.

xio

2

1

0

•1

-2

-3

— Deterministic— p=2

p=3 P=4 p=5

P=6

-4

-5

250220 230 240210190 200180

Figure 30: Time evolution of span-averaged deterministic and mean spanwise force coefficient Cz solutions for different p. Re — 300; = 0.05.


208

° 6 4

2

0

■

190 2 0 0 2 1 0 22 0 23 0 2 4 0 250

0.6

0.4

0.2

0

- 0.2

- 0 .4

- 0.6

- 0.8

(b)

10

8

e 64

2

0

10

ft ■lii 1 ) : ■ • 1

B

1

.... ......... ....... p I p

i l ln h

_190 200 210 220

tU/D

(c )

23 0 2 4 0 250

190 2 0 0 210 22 0 23 0 240 250

0.6

0.4

0.2

0- 0.2

- 0 .4

- 0.6

0,4

0.2

0

- 0.2

- 0 .4

(d)

10

8

% 6 4

2 0

190 2 0 0 210 220 tU/D

2 3 0 2 4 0 250

Figure 31: Comparison between deterministic (a) and mean (b-c-d) isocontours of lift coefficient Cl solutions for different p; (b): p — 2; (c): p = 4; (d): p — 6.


209

(a)

§ 6

11 H I ) y^ I J a i i 11 ki I i H

190 2 0 0 21 0 2 2 0 2 3 0 2 4 0 250

1.35

1.3

1.25

1.2

1.15

(b)

190 2 0 0 21 0 220 2 3 0 2 4 0 250tU/D

1.25

1.15

I I l f m W w

u.ul

1.35

1.3

1.25

1.2

1.15

190 2 0 0 21 0 22 0 230 24 0 25 0

(b )

...

..

190 2 0 0 21 0 22 0 23 0 2 4 0 250tU/D

1.25

1.15

Figure 32: Comparison between deterministic (a) and mean (b-c-d) isocontours of drag coefficient Cd solutions for different p; (b): p = 2; (c): p — 4; (d): p = 6.


210

(a)I B j8o ,

iH li iiillMiilWiiI B i •# iBStf-BHIlllillllilij

n ililMtiasSilB IliB I llBiii'liBi*190 200 210 220 230 240

(b)

10

8

§ 6 4

"IS0

10

§ 6 4

2- L

190 200 210 220 230 240tU/D

(C)

n■

0.015

G.Q1

0 005

C

-0.005

- 0.01

-0.015

250

0.015

0.01

^0.005

io1-0.005

- 0.01

0.015

- 0.02

0.005

0.0 5

210 220 230 240 250

(d )

10

4'

2

0190 200 210 220

tU/D230

n0150.01

0.005

0

-0.005- 0.01

-0.015- 0.02

250

Figure 33: Comparison between deterministic (a) and mean (b-c-d) isocontours of spanwise force coefficient Cz solutions for different p; (b): p ~ 2; (c): p — 4; (d): p ~ 6.


211

Determ inistic p=2p -3

p=5p=6

210 220 tU/D

Figure 34; Time evolution of spanwise standard deviation deterministic and mean lift coefRcient C l solutions for different p. i?e = 300; Uu =0.05.

stage before to grow again. Results for the random drag forces are more difficult to interpret as

they do not exhibit a definite trend. Mean values are sometimes above (at early time) or below (at

later time) the deterministic solution. The variance grows and momentarily reaches a plateau but

low resolutions overestimate its value at later time. Random spanwise forces are smoother than

the deterministic ones. High temporal oscillation frequencies are damped out and lower temporal

frequencies with same spanwise wavelength appear as the resolution increases.

The quantitative difference with the two-dimensional results {Re = 100) at the same level of

noise is maily due to the difference in Reynolds number. The variance levels for lift and drag forces

are much larger than the two-dimensional case. R is about 5 times larger for the lift and 10 times

larger for the drag. The mean values for the lift also decay faster. These trends are consistent

with our two-dimensional results where we compared 2 different Reynolds number simulations

with the same level of noise. Consequently, the spectral convergence of the solution as we increase

the polynomial order p deteriorates much faster as time increases and p = 6 is obviously not

enough to accurately resolve the wake for long time integration.

An interesting aspect of the noise is its smoothing action on the mean solution. In Figure 34,

we plot the time evolution of the standard deviation (std) taken along the spanwise direction of

the deterministic and mean lift forces . At any given time, a zero value of the standard deviation


212

0.5P=2p=4P =60.45

0.4

0,35

0.3

0.25

0.2

0.15

0.05

190 200 210 220 230 240 250tU/D

Figure 35; Span-averaged variance of lift force coefficient solutions for different p. = 300; — 0.05. ^

0.03p=2

p=6

0.025

0.02

0.015

0.01

0.005

190 200 210 220 230 240 250tU/D

Figure 36: Span-averaged variance of drag force coefficient Co solutions for different p. W ■cr„ = 0.05. 300;


213

J0.4

10.3

0.2

0.1

200 210 220 (b)

250

v i i l l l .220 230

S i l i a *240 250

0.02

0.015

0.01

1 0.005

xio"'

190 200 210 220 tU/D

230 240 250

Figure 37: Isocontours of variance of lift (a), drag (b) and spanwise (c) force coefficient solutions for p — 2.

230 240 250220B

0.5

0.4

io.3

0.2

0.1

Wiw

220 230 240I

250

0.025

0.02

0.015

0.01

0.005

I j

X 10"' 15

190 200 210 220 tU/D

230 240 250

Figure 38: Isocountours of variance of lift (a), drag (b) and spanwise (c) force coefficient solutions for p — 4.


214

U

Figure 39; Isocontours of variance of lift (a), drag (b) and spanwise (c) force coefficient solutions for p = 6.

taken along the span would indicate a perfectly two-dimensional force with no z-variations. It is

therefore a measure of the three-dimensionality of the wake. Results from Figure 34 show tha t

the standard deviation amplitude is about constant in time for the deterministic case. However,

values drop in time as we increase the resolution p. More tests with different noise levels would

indicate the dependence of the three-dimensionality of the mean wake to the noise level. We can

then predict th a t the correlation lengths of the mean wake would be most likely larger than the

correlation lengths of the deterministic wake.

2.2 Random -Flow Visualizations

In this section, we present some deterministic and random instanteneous flow visualizations cor

responding to the problem described in the previous section. All visualizations of the flow are

taken at the final time of the computation. The polynomial order in the GPC method used to

solve this case is p = 6.


215

Figure 40: Isocontour of the deterministic pressure Pdetfield in the near wake pdet -

Figure 41: Isocontour of the mean pressure po cr,, =0 .05 .

field in the near wake po -= -0.15. Re = 300;

Reproduced with petntlssioh of the copyhght ownerFurther reproductioh prohibited without permission.

216

Figure 42: Isocontour of deterministic spanwise vorticity in the near wake ± 1.

I

Figure 43: Isocontour of mean spanwise vorticity in the near wake ± 1- -Re = 300; = 0.05.

Figure 44: Isocontour of rms of crossflow velocity in the near wake Vrms = 0.5. Re = 300; <T„ = 0.05.


Chapter 10

M oving Cylinders Simulations

1 Two-Dimensional Simulations with Random Inflow and

Forced Structural M otion

Simple random inflows, represented by random variables, past stationary cylinders constitute

severe test cases for the GPC method, especially for distributions with long tails. For finite

order expansions, resolution problems in the random space arise as time increases during the

computation. This is due to the increasing dimensionality of the random fluid-structure interaction

problem with time. We have seen in Chapter 9 th a t the spectral convergence deteriorates in time

whenever the uncertainty seeps into the frequency response of the system. As no definite time

or spatial (in the case of a random variable inflow) scales are imposed to the random problem,

the time-depency is being removed from the solution as time tends to infinity and the solution

becomes stationary! Accurate spectral representation of the time-correlations between different

realizations requires the size of the polynomial chaos basis to grow in time.

One idea is then to impose dominant and finite frequency and spatial scales to the random

wake in order to be able to resolve it for sufficiently high resolution in random space. This is

achieved by forcing the cylinder in a deterministic purely harmonic motion th a t imposes a leading

217


218

forcing frequency to the system. This setup is realistic as we could face a situation where the

inflow is random, bu t where we force the cylinder in a perfectly controlled deterministic manner.

We investigate the effect of the inflow noise level on the vortex dynamics of the wake.

1.1 Random -Flow V isualizations

The inflow takes the form of a stationary uniform random variable as described in Section 1 of

Chapter 9. Convergence study of the polynomial order p th a t we need in order to resolve the

random wake indicates th a t a value of p = 10 is sufficient. We choose Re — 140 and test different

levels of noise. The cylinder is forced to oscillate in a purely harmonic deterministic motion of

amplitude A /D = 1.0 and reduced velocity based on the forcing frequency = 7.5. This choice

of parameters should give a P+S type of vortex shedding mode with one pair of vortices plus one

single vortex shed per cycle [126].

Deterministic vorticity results a t different instants within one cycle (see top-left images from

Figures 1, 2 and 3 prove th a t we do obtain a P-fS shedding mode with one pair of vortices shed

from the cylinder upper side and one single vortex shed from the lower side. The other images are

instanteneous mean vorticity representation at the corresponding times for different level of noise.

The effect of the noise is striking and the vortical topology of the wake is drastically re-organized

as the level of noise increases, see Figures 1, 2 and 3).

Figure 4 presents a similar comparison for the case of a P+S shedding mode of a forced cylinder

at Re = 400. In this case, due to the higher value of the Reynolds number, a small noise level

(7„ = 0.1 is enough to completely deteriorate the dynamics of the wake.

These results seem to indicate th a t the noise especially affects and deteriorates the formation

of the second vortex of each pair from the P+S mode.

1.2 R an d o m Force R esponse

Next, we look at the effect of the noise on the force response. We present results in both the time

and frequency domains for the case of the forced cylinder at Re = 140. A polynomial order p = 10


219

^ 0 7S6B42 >6316

0.315788 15263 15S83

>^-031576.8 >931S

•0.1O6K315789

-0626316^ -0.735842

m l

- 17388 3S842

1 (1628316 16769

S 4 07S5B42 J 062B316

0315789I 0 35293

38263 15789

M -0.528316

§, 0

” ' , - -

t 3105263 M-i 03157S9 I -> 0 526316 j~-« 0738812

m s r

11 -2|i -5

‘•-j \ J

Figure 1: Comparison between deterministic (Top-left) and mean instanteneous vorticity fields for different levels of noise (t„ at identical time to; Cu = 0.1 (Top-right); cr„ = 0.2 (Bottom-left); (T„ = 0.25 (Bottom-right). Re = 140; p = 10.

Voflioly

I 1.36612

P’ ' 0917368 0736642

0315789 0105263

•0105263 -0.315739 -Q 526316 -0736842

•u 316769 26316 '36842

IFigure 2; Comparison between deterministic (Top-left) and mean instanteneous vorticity fields for different levels of noise it„ at identical time to + == 0.1 (Top-right); (t„ = 0.2 (Bottom-left);(j„ = 0.25 (Bottom-right). Re = 140; p = 10.


220

|Si i:=i— 09*17368 h?rj 0.7515842

' 0 5i6316 ^ D.31S789

" d -1106263

26316

i , =

094.7360 0739642 0526316 QS15788

■yJ 010S2B3 0 10B283

1 0 315769

• 0 736642

I«2.\.38S42ii™” IP

T::; I'SC 0 3 4 ^ — 0.W7368

-j 0526318 4 5789

0106263 03293

-0.738842m .0.947396

Figure 3; Comparison between deterministic (Top-left) and mean instanteneous vorticity fields for different levels of noise at identical tim e to + = 0.1 (Top-right); (t„ = 0.2 (Bottom-left);au = 0.25 (Bottom-right). Re = 140; p = 10.

r (®)

w ^

°

Figure 4: Comparison between deterministic (bottom) and mean (top) instanteneous vorticity field at identical time; (7„ = 0.1; Re = 400; p = 10.


221

for the Legendre basis is used for all computations. Figure 5 compares the time evolution of the

deterministic and mean lift coefficient solutions for different levels of noise while Figure 6 displays

the same for the drag coefficients. Figures 7 and 8 show the spectrum of the deterministic and

mean lift and base pressure coefEcients respectively.

—— Deterministic a^=0.1

a , =0.25

0.5

-0 .5

840825 835805 815 820 830800 810

Figure 5: Time evolution of deterministic and mean lift coefficients for different noise level Re = 140; p — 10.

3.2

2.6

2.4

2.2

840825 835815 820tU/D

830800 810

Figure 6: Time evolution of deterministic and mean drag coefficients for different noise level (T„; Re = 140; p = 10.

We see from Figures 5 and 6 tha t the mean amplitude values of the forces are lower than


222

the deterministic case. This is consistent with what we found for free and stationary cases. A

threshold in the noise level seems to exist above which the force mean values are noticeable

different from the deterministic forces. Indeed, a value of (7„ = 0.1 does not affect significantly

the mean solution but cr„ > 0.2 does. Interestingly, the mean solutions for sufBciently large noise

level are somewhat smoother than the deterministic solution. The additional weaker oscillation

of the deterministic solution (see for instance Figure 5 in the [—0.5; 0] range), symptomatic of the

shedding of the pair of vortices of the P+S mode, is almost completely removed from the mean

solutions.

S !0'

10"’

1 2 3 4C forcing

6 7 85

Figure 7: Comparison between deterministic and mean spectrums of lift coefficientfor = 0.25; l e = 140; p = 10.

This observation is confirmed by the spectrum analysis of the forces and the base pressure

signal measured at the anti-stagnation point of the cylinder, see Figures 7 and 8. The 2 leading

frequencies of the deterministic signals are the Strouhal frequency corresponding to the 2S type

shedding mode and the first super-harmonic at twice this frequency corresponding to the 2P

type shedding mode. While the energy remains constant in the first peak, it is clear tha t there

is a decrease of the energy in the second peak for the random case. A purely 2S mode would

correspond to a case where all the energy would be removed from the second peak and would

only concentrate in the first peak {Jc l / fforcing = 1)- We notice th a t the noise affects the entire


223

— Deterministic a^=0.25

7 81 2 3 5 6

Figure 8: Comparison between deterministic and mean spectrums of base pressure coefficient for o-„ = 0.25; S e = 140; p = 10.

spectra and flattens it.

As we had suspected from the flow visualizations in the previous section, the noise first affects and

suppresses the formation of the second vortex of each pair of vortices from the P + S mode. The

vortical topology of the random wake then becomes closer to a classic von Karman 2S shedding

mode as we increase the noise level.

2 Two-Dimensional Simulations with D eterm inistic Inflow

and Random Structural Parameters

We consider the case of a two-dimensional flow past an elastically mounted circular cylinder with

random structural parameters, subject to vortex-induced vibrations. We study the case of an

unsteady flow in the subcritical regime with Re = U d /v = 100 {U is the free-stream velocity, d

the cylinder diameter and v is the kinematic viscosity). The flow is computed using the procedure

outlined in Chapter 6, Part II while the structural response of the moving cylinder is computed

using the procedure described in Chapter 8, P art III.

In this section, we assume the damping coefficient and the spring factor of the structure to be


224

two independent Gaussian random variables. The free structure, excited by the vortex shedding

of the flow which is initially deterministic, produces a random response. Therefore, the position of

the boundary of the cylinder becomes stochastic. This random boundary affects the flow domain,

and consequently the flow itself becomes a stochastic process. The fluid forces on the cylinder,

derived from the random flow velocity field and the random pressure field, are random processes

as well. The damped oscillator is then subject to random parametric (random variables) and

external forcing (random process) excitations.

Because the response of the cylinder to the vortex shedding is mainly in the cross-flow direction

and due to the increased complexity introduced by the random character of the system, we have

constrained the cylinder movement in one direction. Therefore, the cylinder is free to oscillate in

the cross-flow y-direction but it is forced to have no motion in the streamwise x-direction. If we

denote by 77 the non-dimensional response of the structure in the y-direction, then we have the

following governing equation for the structure response;

ri{t,9)+c{9)r](t,9) + k{0)rj{t,6) = F (t ,9 ) , 77(0) = 770 and 77(0) = yo (1)

The damping coeflicient c and spring factor k are both random variables with Gaussian dis

tributions. The forcing excitation term on the right-hand-side of Equation (1) is not known a

priori and has to be computed once the flow distribution has been obtained. This is done every

timestep, with the fluid forces acting on the cylinder surface computed as follows:

F{t, ^ ( ~ n ( t , 9)n -h + Vu(f, 9)'^) ■ n)ds, (2)

where n is the unit normal. For the tem poral discretization of equation (1) we use the implicit,

second-order stochastic Newmark scheme, as described in Appendix ??.


225

2.1 Random Structural R esponse and Forces D istribution

The Reynolds number is Re = 100, and the random parameters for the structure are set to:

® (c,<7e) - (0 .1, 0 .01), (fc,afe) = (1 .0 , 0 .2)

(see Equation (1)), while the initial conditions rjo and ?)o are set to 0. We note th a t there is a

non-zero probability th a t the oscillator has a natural frequency u>o = "Jk matching the vortex

shedding frequency of a fixed cyhnder at this Reynolds number. Also, the system has two random

dimensions n = 2 (one for each uncertainty in the structure), and we use third-order Hermite

Polynomial Chaos expansion {p = 3), which corresponds to a 10-term chaos expansion (i.e.,

P + 1 = 10), see Chapter ??, P art II. This is considered to be a quite low resolution as it

represents non-linearity up to cubic terms only.

0.5

-0.5

A !ii t I Ml III f

ii A - u F ^ A 'u '■a: ^fr AnP A?IU • V [

r l r M k k m M k' i j rl'T m mMM

\h lit I fW uV r y }f t U : (' 1i: t': T C -I

:■ I; 1 y 7: >■ y I-•' J it r e . .V. V- ;. •• •;

Deterministic Mean

First modeSecond mode

: I

n IV //Vi I!i\ I'jfs /iV> /-V /V. /Vjili /i' T 1/ -1 S;

i t (1; H ; l;i: i | : i t i t 1 ] : i j

r 'J; V- V- V "■

200 220 240 260 280 300 320 340 360 380 400tU/d

0.3■ ■ Deterministic Mean— First mode Second mode

0.2

0,1

- 0.1

- 0.2

340 380220 260 280 300 320 360 400200 240tU/d

Figure 9: Time evolution of the leading random modes for the cylinder cross-flow response p (top) and the lift coefflcient Cl (bottom).

In figure 9, we plot the time evolution of the leading random modes of the cylinder response. We

show the non-dimensional cross-flow displacement 77 of the cylinder (top) and the corresponding


226

lift coefficient Cl = F l/(0 .5p-D ^^) (bottom; with Fl the lift force), along with the deterministic

solution. As expected, the mean response, due to the diffusion effect of the randomness, has

a smaller amplitude than the deterministic solution. The first and second modes represent the

Gaussian part of the solution. In figure 10, we show the time evolution of the variances of the non-

dimensional cross-fiow displacement r) (top) and the corresponding lift coefficient Cl (bottom).

Both curves exhibit a similar trend. In both cases, after an initial peak in the response, the signal

eventually reaches a stationary periodic state with smaller oscillation amplitudes. The initial peak

value can be two times larger than tha t of the final periodic state.

Figure 11 shows instantaneous flood countours (gray scale) and countour lines (white color) of

rm s and mean of cross-fiow velocity, respectively. Figure 12 shows the same kind of plot for the

vorticity field. Both snapshots are taken at t = 600 (non-dimensional time units) corresponding to

more than 100 shedding cycles from the beginning of the simulation. Regions of the flow domain

with high uncertainty for this value of p are the shear layers and the near-wake of the cylinder.

0.12 -

0.1

0 .0 4

0.02

4 0 0 4 5 0200 2 5 0 3 0 0 3 5 0tU /d

0.1

0 .0 6

0 .0 4

0.02

200 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0tU /d

Figure 10: Time evolution of the variance of the structural solution. Variance of the cylinder cross-flow response rj (top) and variance of the lift coefficient C l (bottom).


227

15 20x / d I

0.30.250.20.150.10.05

Figure 11; Instantaneous spatial distribution of rm s (gray scale) and mean (white line) of crossflow velocity.

Figure 12: Instantaneous spatial distribution of rm s (gray scale) and mean (white line) of vorticity.


228

Figure 11 suggests th a t the rm s values of cross-flow velocity are not strongly spatially corre

lated to the mean values; for example the contours with the largest values are not aligned. In the

near-wake up to x /d = 7.0, large variance is obtained at the boundary between flow structures

of mean, but further downstream, as the system reaches equilibrium, large variance realigns with

large values of the mean. Figure 12 shows this more consistently along the wake, where there is

a strong correlation between variance and mean values of the vorticity. For this low resolution,

large variance of the vorticity is obtained in the regions of large shear or strong mean vorticity.

6 0 0 6 0 2 6 0 4tU /d

0 .4

/ ® \ ) 0 .2/

90 1 . 0

- 0 .2

- 0 .4

t ^ + T / 4 v y120 60

0 .4

/ \ ^ / -o / \ /^ 1 0

t +T/2 t +3T/4 V y° . . . . ______ . . . b . . 0 .5 - 0 .4 - ° ............6 0 2 5 0 4 6 0 6 6 0 6

lU /d

r r r o r - b a r reg ion— W ea n so lu tio n------D eterm in is tic so lu t io n

2 7 0

Figure 13: Polar plots of pressure distribution on the cylinder surface relative to the cylinder mean cross-flow position at four different instants within one shedding cycle. Deterministic pressure solution (dashed line); Stochastic pressure solution (solid line and shaded area).


229

Figure 13 presents the pressure distribution on the cylinder surface at different instants within

one shedding cycle. The figure shows four different instants, from non-dimensional time to = 602.4

(top left plot) to to + 3T /4 (bottom right plot) within one shedding cycle of period T. Each plot

shows an instantaneous polar view of the pressure distribution on the cylinder surface as well

as the mean cross-flow position of the cylinder r] at the corresponding time. The cylinder is

represented by a black disk. The flow orientation is from left to right in each plot. Therefore,

the angle 0 — 180° on the polar view corresponds to the front stagnation point and 0 = 0°

corresponds to the rear stagnation point. Successive dashed circles give pressure value levels (the

zero value is a solid circle). The deterministic pressure solution is represented by a dashed line

while the stochastic solution is represented by a solid line (mean pressure solution) and a shaded

area {^error — bar' region of the pressure solution). This region is centered around the mean curve

and its span is two standard deviations (i.e., one std above and one std below the mean value).

Pressure values are mainly in the range from —1.0 to 1.0, and both deterministic and stochastic

pressure solutions take positive values around the front stagnation point. Noticeable differences

exist between stochastic and deterministic solutions. In particular, tem poral as well as spatial

changes in the pressure variance (or ‘e rro r — bar' region) can be seen. However, the deterministic

signal remains, most of the time, inside the envelope of the stochastic solution. Small uncertainty

(in terms of Gaussian response) is obtained at the front stagnation point and at two points close

to separation on both sides of the cylinder. Values of the std a t those locations are small and

mean solutions are equivalent to the deterministic solutions. The polar angular position of these

two nodes is always in the range 6 € [90°, 120°] or in the range 9 € [240°, 270°]. It would be

interesting to investigate the relationship between these nodes and the separation points on the

cylinder.

Having obtained all the random modes, we can now “reconstruct” the solution and examine

its probability distribution function at different times within a shedding cycle. Figures 14 and

15 show the probability distribution functions of the pressure at the rear stagnation point and of

the crossflow amplitude of oscillation at different time instants. It is clear th a t the probability


230

e

- 0 .5Base pressure at rear stagnation point

Figure 14: Probability distribution function for the pressure at the rear stagnation point at different instants within a shedding cycle.

Figure 15: Probability distribution function for the amplitude of the crossflow oscillation r] at different instants within a shedding cycle.


231

distribution function of the amplitude of the cylinder oscillation is symmetric, as expected, but

the probability distribution function of the base pressure shows a strong bias towards one side.

-0,3

III-0.4

,0.5

- 0.1 Deterministic— Mean IV-0.7

589585 586 587 588 590584tU/d

10

8

6

I4

2

0- 0.2-0.7 - 0.6 -0.5 -0.4 -0.3 - 0.1-0.9 - 0.1

Figure 16: Upper: Time variation of the mean base pressure (with error bars) versus the deterministic solution. Lower: Probability distribution function of base pressure at five time instants marked in the top plot.

In Figures 16 and 17 we plot the time evolution of the base pressure and of the cylinder am

plitude of cross-flow oscillation. Also shown are error bars at five time instants and corresponding

probability distribution functions. There is a large difference of the deterministic versus the mean

stochastic solution as a function of time. Specifically, for the cylinder oscillation the amplitude

probability distribution function approaches a ‘lognormal’ form on the left side, it transitions

through a uniform-like distribution (between time IV and time V), and finally reverses to the

other side resembling a lognormal distribution again. It is also interesting to observe th a t the

probability distribution function with the sharp left side (time II) occurs when the cylinder has

maximum negative displacement in this case. Often, smaller variance occurs when the dispace-

ment is close to its maximum value as opposed to when it is closed to some “average” value. Here,

the left side of the probability distribution function of time II is very sharp, which means th a t

the cylinder has almost “no chance” to go any lower down (because it reaches its minimum value)


232

0.5

T3

-0.5589 590586 587 588584 585

tU/d

M.I l:tu/d=586 It: tU/d=587- - lll:tU/d=588— ■ IV:tU/d=589

• V: tU/d=590

0. 2

0.5-0.5y/d

Figure 17: Upper: Time variation of the mean amplitude of cylinder oscillation (with error bars) versus the deterministic solution. Lower: Probability distribution function of amplitude at five time instants marked in the top plot.

but it will most likely go back up.

The accuracy of those results for long time integration is questionable due to the low polynomial

order p = 3 th a t we used. In fact, stability issues would arise as soon as we would perform long

time integration for p > 4 for the same problem and the computations would eventually blow-up.

This could be due to the fact th a t Gaussian distributions do not prescribe the structure from

having a non-zero probability to have negative damping or spring factor values.


A ppendix A

Parallel A lgorithm s and

Im plem entation Issues

1 Spectral//ip D iscretizations on Unstructured and Hybrid

Grids

The parallel code M e n 'T a .r th a t we employ in the two aforementioned parallel paradigms is

based on a new spectral basis [127, 128, 129]. It is appropriate for unstructured meshes based on

triangles or tetrahedra in two- and three-dimensions, respectively. In many simulations, however,

involving complex-geometry domains or external flows it is more efiicient to employ hybrid dis

cretizations, i.e. discretizations using a combination of structured and unstructured subdomains.

Such an approach combines the simplicity and convenience of structured domains with the geomet

ric flexibihty of an unstructured discretization. In two-dimensions, hybrid discretization simply

implies the use of triangular and rectangular subdomains, however in three-dimensions the hybrid

strategy is more complex requiring the use of hexahedra, prisms, pyramids and tetrahedra.

Hexahedral domains have been used quite extensively in the hp finite element field [130, 131].

More recently an unstructured hp finite element approach, based upon theoretical work in two-

233


234

dimensions by Dubiner [132], has been developed for unsteady problems in fluid dynamics [133,

128, 134]. In the following, we will show how these expansions can be constructed using a unified

approach which incorporates all the hybrid subdomains.

This unified approach generates polynomial expansions which can be expressed in terms of

a generalized product of the form <ppgr{x,y,z) = (/}p{x)<j>pg{y)(l>pgp{z). Here we have used the

Cartesian coordinates x ,y and z but, in general, they can be any set of coordinates defining

a specified region. The standard tensor product is simply a degenerate case of this product

where the second and third functions are only dependent on one index. The primary motivation

in developing an expansion of this form is computational efficiency. Such expansions can be

evaluated in three-dimensions in 0 {N ^) operations as compared to 0 {N ^) operations necessary

with non-tensor products based expansions.

1.1 Local coordinate System s

We start by defining a convenient set of local coordinates upon which we can construct the

expansions. Moving away from the use of barycentric coordinates, which are typically applied

to unstructured domains, we define a set of collapsed Cartesian coordinates in non-rectangular

domains. These coordinates will form the foundation of the polynomial expansions. The advantage

of this system is th a t every domain can be bounded by constant limits of the new local coordinates;

accordingly operations such as integration and differentiation can be performed using standard

one-dimensional techniques.

V (-1.1) I (1.1)

(0,0)

—=----- (-i.f-i) (i.fi)T l ,= - ) t! ,= ( l T l,= l

Figure 1: Triangle to rectangle transformation

The new coordinate systems are based upon the transformation of a triangular region to a


235

rectangular domain (and vice versa) as shown in figure 1. The main effect of the transformation is

to map the vertical lines in the rectangular domain (i.e. lines of constant 771) onto lines radiating

out of the point (^1 = —1,^2 = 1) in the triangular domain. The triangular region can now be

described using the “ray” coordinate (771) and the standard horizontal coordinate (^2 = ''72) • The

triangular domain is therefore defined by (—1 < 771,772 < 1) rather than the Cartesian description

(—1 < ^1,^2; 5i + C2 < 0) where the upper bound couples the two coordinates. The “ray”

coordinate (771) is multi-valued at (^1 = —1,^2 = 1)- Nevertheless, we note th a t the use of

singular coordinate systems is very common arising in both cylindrical and spherical coordinate

systems.

T), = u±nij£izM- 1 2

^ 5 , = (l-H ll)(l-tl2) - 1f , = ( l+ 'la X l-n a l - 1

Figure 2: Hexahedron to tetrahedron transformation

As illustrated in figure 2 , the same transformation can be repeatedly applied to generate new

coordinate systems in three-dimensions. Here, we sta rt from the bi-unit hexahedral domain and

apply the triangle to rectangle transformation in the vertical plane to generate a prismatic region.

The transformation is then used in the second vertical plane to generate the pyramidal region.

Finally, the rectangle to triangle transformation is applied to every square cross section parallel

to the base of the pyramidal region to arrive at the tetrahedral domain.

By determining the hexahedral coordinates (771, 772, 773) in terms of the Cartesian coordinates

of the tetrahedral region (Ci, C2, Cs) we can generate a new coordinate system for the tetrahedron.

This new system and the planes described by fixing the local coordinates are shown in figure

3. Also shown are the new systems for the intermediate domains which are generated in the

same fashion. Here we have assumed th a t the local Cartesian coordinates for every domain are

(6 !? 2 ,6 )-


236

(i-y- 2= (1-y

w

/ // // // /

T13.

/ // /

1/1.

iwV

Figure 3: The local coordinate systems used in each of the hybrid elements and the planes described by fixing each local coordinate.

1.2 Spectral Hierarchical Expansions

For each of the hybrid domains we can develop a polynomial expansion based upon the local

coordinate system derived in section 1.1. These expansions will be polynomials in term s of the

local coordinates as well as the Cartesian coordinates This is a significant property as

primary operations such as integration and differentiation can be performed with respect to the

local coordinates but the expansion may still be considered as a polynomial expansion in terms

of the Cartesian system.

We shall initially consider expansions which are orthogonal in the Legendre inner product.

We define three principle functions (j}'}(z),4>^j{z) and ™ terms of the Jacobi polynomial,

Pp''^{z), as:

6t{z) = p r i z ) , =


237

Using these functions we can construct the orthogonal polynomial expansions:

Hexahedral expansion: Prismatic expansion:

Pyramidal expansion:

^ P 9 r(C l,6 ,6 ) = (l>p{m)(pg{mWpgr(.V3)

Tetrahedral expansion:

< /> p g r ( € l ,6 ,? 3 ) = < P p { m ) < P % { m ) 4 > p q r i m )

where,

2 (1 + gi)( - 6 - 6 ) ’

_ 2 (1 + e i ) , 2 ( 1 + 6 ) ,Vi = -71— r r ” 1’ V2 = ~ — t t - = 6 ,(1 - 6 ) (1 - 6 )

are the local coordinates illustrated in figure 3.

The hexahedral expansion is simply a standard tensor product of Legendre polynomials (since

pO>f’(2) = Lp{z}). In the other expansions the introduction of the degenerate local coordinate

systems is linked to the use of the more unusual functions and <t>ijk{z). These functions

both contain factors of the form which is necessary to keep the expansion as a polynomial

of the Cartesian coordinates (6 i 6 i 6 ) - For example, the coordinate 772 in the prismatic expansion

necessitates the use of the function (pgri^a) which introduces a factor of . The product

of this factor with <t>q{rj2 ) is a polynomial function in 6 nnd 6 - Since the remaining part of the

prismatic expansion, is already in term s of a Cartesian coordinate the whole expansion is

a polynomial in terms of the Cartesian system.

The polynomial space, in Cartesian coordinates, for each expansion is:

V = Span{ef Cl 6 } (1 )


238

where pqr for each domain is

Hexahedron 0 < p < N i 0 < q < N 2 0 < r < N 3

Prism 0 < p < i V i 0 < q < N 2 0 < q + r < Ns

Pyram idal 0 < p < N i 0 < q < N 2 0 < p + q + r < N 3

Tetrahedron 0 < p < N i 0 < p + q < N 2 0 < p + q + r < N 3 .

(2 )

The range of the p, q and r indices indicate how the expansions should be expanded to generate

a complete polynomial space. We note tha t if N i = N 2 = N 3 then the tetrahedral and pyramidal

expansions span the same space and are in a subspace of the prismatic expansion which is in turn

a subspace of the hexahedral expansion.

To enforce C° continuity the orthogonal expansion is modified by decomposing the expansion

into an interior and boundary contribution [129, 127, ?]. The interior modes (or bubble functions)

are defined to be zero on the boundary of the local domain. The completeness of the expansion

is then ensured by adding boundary modes which consist of vertex, edge and face contributions.

The vertex modes have unit value at one vertex and decay to zero at all other vertices; edge

modes have local support along one edge and are zero on all other edges, and vertices and face

modes have local support on one face and are zero on all other faces, edges and vertices, figure 4

shows the decomposition of the domain into such elements, with the vertex, edge, face, and interior

modes marked for one element. continuity between elements can then be enforced by matching

similar shaped boundary modes. The local coordinate systems do impose some restrictions on

the orientation in which triangular faces may connect. However, it has been shown in [127, 135]

that a tetrahedral expansion can be constructed for any tetrahedral mesh. A similar strategy

could be applied to a hybrid discretization [136].

Finally, we note th a t the bases are all hierarchical, which means th a t increasing the poljmomial

order of any expansion simply adds extra modes to the existing basis. Hierarchical expansions

naturally lend themselves to p-type adaptivity where the polynomial order of the expansion can

differ within each elemental domain. This is a very attractive property as it permits the polynomial


239

Figure 4: In the spectral//ip method the solution domain is decomposed into elements of characteristic size h and then a polynomial expansion of order N is used within every element. On the left we see a cuboid decomposed into 3072 tetrahedral elements within which we use a polynomial expansion of order 4 as indicated by the mode shapes on the right. (Courtesy of S.J.S. Sherwin)

order of the expansion to be altered in order to capture the spatial characteristics of the solution.

Jacobi polynomials of mixed weights are used for the trial basis th a t form tensor products

of generalized type. The nonlinear products are handled using effectively a super-collocation

approach followed by a Galerkin projection [28].

This is accomplished by arranging the trial basis in terms of vertex, boundary and bubble

modes and ensuring matching of the boundary modes, thus satisfying the (7° continuity condition

required in the Galerkin formulation.

1,3 Tim e Integration Algorithm

A popular time-stepping algorithm for integrating the Navier-Stokes equations is the splitting or

fractional scheme. Although many different versions have been developed, here we describe a

particular implementation tha t can give high-order time accuracy [123].

W ithin a domain 0 , the fluid velocity u and the pressure p can be described by the incom-


240

pressible Navier Stokes equations,

du- V P + i^L(u) + N (u) in Q

V ■ u = 0 (3)

where

L(u) = V^u; w = V x u

N(u) = u X u;; P = p + ^ V ( u • u) (4)

The non-linear operator N (u) has been w ritten in rotational form to minimize the number of

derivative evaluations (6 vs. 9 for the convective form). A semi-implicit time integrator is used

to integrate the system (3), (4) by using a 3-substep splitting scheme [123]:

---- = X ] /3 ,N (u--«) (5)

^ = -V P "+ ^ (6)

(7)A t ^

The time-stepping algorithm can then be summarized in three steps:

1. Calculate the advective terms eqn. (4) and advance the solution in time using a stiffly-stable

multi-step integrator.

2. Solve a Poisson equation for the dynamic pressure P to satisfy the divergence-free condition

for the solution. Consistent pressure boundary conditions are used to ensure stability and

high order accuracy [123].

3. Implicitly solve the viscous terms, advancing the solution to the next timestep. This gives


241

rise to a Helmholtz equation for each of the velocity components.

2 The A fe n 'T c x r T code

The M e n T oltT code is appropriate for flows with one homogeneous direction. In this direction

a Fourier expansion is used providing a natural parallel paradigm.

2.1 Fourier D ecom position

If we assume th a t the problem is periodic in the z-direction, we may use a Fourier expansion to

describe the velocity and the pressure, i.e. for the velocity,

M -l

u ( a ; , y , z , t ) = ^ (8)m —0

where (3 is the z-direction wave number defined as /? = and is the length of the

computational domain in the ^-direction. We now take the Fourier transform of equation (3) to

get the coefficient equation for each mode m of the expansion,

^11 ^+ lyLminm) + P F T „ [N (u )] in m = 0...M - 1, (9)

where F F T ,„ is the m component of the Fourier transform of the non-linear term s and,

- , 5 3 . , ,

Tm(Um) = ~

The computational domain Qm is an x-y slice of the domain f2, implying th a t all are identical

copies. From equation (9) we see th a t the only couphng between modes is through the non

linear terms. Therefore the computation of each mode m can be treated independently of one

another. The obvious parallelization is to compute the Fourier mode on processor m for m =


242

0...M — 1. Therefore, the three-dimensional computation essentially becomes a set of = 2M

two-dimensional problems computed in parallel on P processors where M is a multiple of P . We

note th a t the factor of two comes from the real/im aginary part pairs for the Fourier modes.

To maintain computational efhciency the non-linear product is calculated in physical space

while the rest of the algorithm may be calculated in transformed space. The paradigm may

therefore be thought of as a two pass process as illustrated in figure 5. As mentioned previously

the spectral//ip representation in the x — y plane is hierarchical and so we may also consider this

representation as a set of elemental modes and corresponding coefficients. In the first pass of

the paradigm we need to obtain the physical data values at the quadrature points within each

elemental domain. The inverse Fourier transform and differentiation are then performed at these

points. For each timestep, Pass I can be summarized by the following substeps:

1. The velocity gets transformed to Quadrature space.

2. Calculation of the vorticity.

3. To form the non-linear terms:

(a) Global transpose of the velocity and vorticity components.

(b) Nxy ID inverse FFTs for each velocity and vorticity component,

(where N^y is the number of points in one x-y plane divided by the number of proces

sors).

(c) Computation of N (u) using a dealiasing 3/2 rule.

(d) Nxy ID FFTs for each non-linear term.

(e) Global transpose of non-linear terms.

In Pass II the explicit time-integration of the non-linear terms and then the Helmholtz solves

for pressure and velocity may be performed independently on each processor.


243

ProcO 1 Proc 1 ------- Proc N-1

I Computational Loop

Global Excliange

Inverse FFT I1

Inverae FFT ------- Inverse FFT

Nonlinear Nonlinear ------- Nonlinear

FFT FFT ------- FFT

Global Exchange

Advectiion Advecbon ------- Advection

Pressurerj Pressure Pressure

Viscous 1 Viscous ------- Viscous

Analysis 1 Analysis Analysistime-history point analysis

Figure 5: Solution process in M e n 'T o l t T


A ppendix B

Brief Overview of Algebra of

Random Variables

These useful notes are extracts from the work of Su [97] and were kindly provided to me.

Given n random variables ^1,^2 • ■ • Cn with the probability distribution function (PDF) / (^i ,C25 • • ■ :^n)

suppose a new random variable r] is defined in term s of by

?7 = 9 ' ( 6 ,6 , - - -^ n ) - (1)

The PDF of 77 is then given by

/oo ^ 0 0 pocd i l j d ^ 2 - - - d U f { ^ i , ^ 2 , - - - C n ) S [ v - g i i i , ^ 2 - - - x i n ) ] - ( 2 )

-oo J — oo J ~~oo

Here we take the probability space of each as well as r; to be (—00, 00) as indicated by the range

of the integrals (2). 5{x) is the 5-function.

One can use (2) to calculate expectation value of any function of say H(j]) as

< F(r?) > = p H{ri)Firj)drj - J d^i ■ ■ ■ J dCn/(Ci • ■ • Cn)i?[<?(Ci • ■ • Cn)j (3)

244


245

For example,

< 1 > =

< r}>

< r f >

J F{ji)dr] — 1

j r]F{ri)dri = j d ^ i . . . j d ^n fi^ i ■ ■ ■ ^n)g)^i

v ‘" F { v ) d r , = 1 d U i i i • ■■■Cn).

To obtain explicitly the PDF of ry, it is convenient to deal with the problem separately for the

case of a single random variable ^ and of multivariables Ci, ^2 ■ ■ ■ Cn-

lA . Single variable ^

F { r ] )= ^ f d^f{^)5[r] - g{^)].J — OO

To perform the integral in we need to find all real zeros of 77 — g{^) for a given value of g.

Then just pick up the contribution of the integral at each of these points as

(4)

where the summation i goes over all the real zeros o» and we assume th a t g'{^oi) 0.^ Examples

lAa. V = F{g) = -J {g lc )

lAb. V = C + n, F(ri) = f{g - a)

lAc. V = F iv) = 2 ^ [ / ( v ^ ) + / ( ^ % g > 0

lAd. V =

lAe. g = In^, F{g) = eV {e^)

lAf. g =

Jens if there axe double zeros?


246

where for the first case ^ = 77 a has double real roots ±r]i while there is only one real root r]a for

the second.

IB . More than one random variable.

One of the most frequent occurrence involving more than one random variables is the linear

combination of a group of variables as

T] = where a* are constants. (5)i~l

F{rf) = [ f ■ • ■ ^n)Slv - ak^k] (6)k=l

In this case, it is most convenient to use the following expression for the d-function

1 /■“

We use this in Equation (6) and we obtain after the interchange of integrals:

F(v) = ■ ■ ■ /

dA:e‘“ .

(7)

If ?i ■ • • are m utually independent, i.e..

/(Cl • • - Cz) = /l(Cl)/2(C2) • • • f n i i n ) -

then-j p O C p O D

^This is attributed to A.A. AlarkofF, see S. Chandrasekhar. Rev. of Modern Physics 15, January 1943. The latter was collected in “Selected Papers on Noise and Stochastic Processes” by N. Wax, Dover Pub. (1954).


247

We will use Equation (8) to obtain F{r]) for linear sum of two random variables in the remainder

of this section. Its extension to more than two variables can be done readily.

IBa. Two normal random variables, i.e ., 77 = ai^i + 0 2 ^ 2

f i iO = h i O = - i ey/2n (9)

f{k a

1 /'°° 1 F{n) = ^ / dkF^V{ka,)f{ka2) - --T- 3'-..

J — OO ■ \ / 2 7 r [ C l ^ ~ h Ci'

77 /2(01+02) (10)

IB b. Two uniform random variables, i.e .,, rj — ai^i + 0 2 ^ 2

/ i ( 0 = / 2(C) =

0 for l i > a/ 3

Here we choose to have the mean and the variance to be zero and 1 respectively.

( 11)

L sinVSfegjf(k a j)

F in)

W d .-vS \/3fcc

1 r°° dkc''^^ v^fcai sin \ / ik a 2

27t J_ o o \/dkai \/Ska2( 12 )

The integral on the right can be obtained by the following manipulation involving 5-function.

First differentiate the expression (12) twice with respect to 77 to have F"{rj) as

F"{r)) = Y2I a ^ \/3 (a i -f 02) -l- 5[?7 - V s{a i + 02)] - 5[t7 - V i{a i - 02)] - S[r} - V3 (a2 - a i ) ] |


248

Integrating this once to obtain F (r?) as a function of step functions, and twice to obtain

for \r]\ > \/3 (a i + 02)

F{v) = < — — [“ hi + \/3 (a i + a 2)] for - tt2) < |r/| < \/3 (a i + a 2)12ciiCi2

for h i < %/3(ai “ 02)2 v ^ a i

here we take a i > a2 > 0 .

IBc. Two exponential r a n d o m variables

g - ( € + l ) fQJ. ^

for C < “ 1

/O O ^ i k a j

, = T T 5 i -

i ^^-^{V+ai+a2) _ (»?+«!+02)1 for 77 + a i + tt2 > 0

1 r pjfe(ai+a2)

(1 + ika i){ l + ika 2 )

a\ — «2

0 for 77 + tti + G2 < 0 .

for ai = 02 = a

F(77) =

IB d. A normal random variable added to a uniform one 2 -

A (e .) =yZTT


249

2^/3for [ 1 < V s

/2(C2)

0 for 11 > V s

fV k a i) =

M ka 2 ) =Sin VSka2

VSktt2

1 r°°Fiv) = 7T / dke^>^ fi{kai)fika2)

J — O O

F'iv) 47T\/3a2

4 \/fea i0 2

- (7? + V a2) _ (Tj—\/3a2)1 — e

T7+%/3a2

IB e. N uniform random variables

We consider rj = Ylf=i fV i ) given by (10). Then

f ik)sin VSk

V sk

and

Sin VSkN

VSk


250

We consider —> oo it is easy to show

1F{r]) = — / d fce* ''" -" - = ~ ^ = = e - ^ .

J —oo

n -N \ _V 2 t "

As one would expect th a t sum of large independent random variables is a normal random variable.

IB f. F-distribution function

/(C; A, t) = —— for 0 < ^ < 00 and f > 0 and A > 0r( f)

where r( f ) = is a F function defined for f > 0. F irst, if 77 = a^, then F(r}) =

i f in f a; A, t) = f(r/; A/a, t). Now if rf = + ^2 , and / i ( 6 ) = f i^ i ',^ , i), /2 (6 ) = / ( 6 ; s), then

2 n J _ ^ \ \ + i k ) r{t + s)

Thus the sum of two F random variables with the same A is a F random variable of the same

param eter A. Note in the normal random variables th a t any linear combination of a set of normal

variables is a normal variable itself. In the case of F variable, the statem ent is true only for the

sum of random variables with the same A.

Finally we derive a formula for the sum of two random variables in the form of convolution

integral widely used in the regular text on probability.


251

Let r] = + ^ 2 w ith PDF for and ^2 as /(C i,^ 2) then

F{V) = J j dî,d^2f{î ,^2) S i v - ^ i - ^ 2)

- J d^2f d ^ i f {^1,^2)S{î - rj + ^2)

= J d^2 f i v ~ ^2,^2)

If / ( ? i , 6 ) = /i(C i) /2(6 ) then F{ri) = /rfC2/i(^? - 6 ) / 2(6 )-


Appendix C

Stochastic Newm ark Scheme

We formulate the problem of Equation (1) in Chapter 6 in m atrix form as follows:

X (t) = A X (t) + T ( 1)

with

X i t )X \ t ) x{t)

, A =0 1

and T =0

. .±{t) k{t) c(f) m

(2)

where X { t) is the state vector, A is the system m atrix , and T is the input distribution m a trix .

Revisiting Equation (9) in Chapter 6 and writing the system in compact form gives us:

x l i t ) = x i i t )

X i i t ) + ^ E f= o E f= o c jX f(t)e ,,m + ^ E l o E f= o kjX }{t)e i,m = fm (t)

with TO = 0,1, 2 , . . . P.

252

(3)


253

Tem poral D iscretization

By using the Wiener-Askey poljuiomial chaos expansion, the randomness of the system is trans

ferred into the basis polynomials. Therefore the deterministic coefficients are smooth in time and

any deterministic ODE solver can be employed such as explicit second-order and fourth-order

Runge-Kutta scheme. In this section, we focus on an implicit Newmark scheme for stochastic

systems. We describe in details the implementation of the Newmark scheme for single degree of

freedom stochastic differential equations with correlated random inputs in the context of Polyno

mial Chaos. Both fourth-order Runge-K utta scheme and Newmark scheme were used in Sections

3 and 4 of Chapter 6 and gave identical results.

It is worth mentioning th a t in the case of stochastic systems under white-noise inputs, the va

lidity and properties of the crude Newmark scheme are not necessarily conserved. This is because

the Wiener process even though continuous may have an unbounded variation over any given time

interval. Therefore, the acceleration vector does not exist mathematically [137]. If the accelera

tion is not smooth enough and the C'^-regularity of the forcing is not satisfied, a modification of

the usual Newmark scheme can be done in order to show almost sure convergence of the scheme

[138]. This modification is only in the stochastic part. However, in our case, the forcing term is

correlated in time and the aforementioned restrictions to white-noise inputs do not apply.

We consider Equation (1) in Chapter 6 where both the external forcing / and the input pa

rameters c and k are random. We call v{t) = x{t) {v has the dimension of a velocity) the first

temporal derivative of the Polynomial Chaos solution x{t) and a{t) = x{t) {a has the dimension

of an acceleration) the second temporal derivative of the solution x{t). x { t), v{t) and a{t) are

vectors which components are the modes of the Polynomial Chaos decomposition. To update the

state of the system S at timestep n, S'” = [a” (t),w” (t),a:” (t)]^, using the forcing at time level

we use the following

r s “+’- = A S " + A"+^ (4)


254

where F and A are matrices th a t can be build by blocks and is such as in Equation (10) in

Chapter 6 , see below. The matrices take the form:

/ \ F i i F i2 F i3

\

/

/ V

A ll A i 2 A i 3

A 21 A 2 2 A 23

A 31 A 3 2 A 33

(5)

/

T21 F22 T23

T 3 1 F 3 2 F 3 3

where each block submatrix Fy and A y is of size (P + 1) x (P + 1).

Concerning F, only F i 2 and F 13 receive a contribution from the random quantities of the left-

hand-side of Equation (1) in C hapter 6 , the rest of the matrix is determ inistic.

We have

^ * 2 .^ and F i 3 (y) ^1=0 ® 1=0 »

and F 22 and F ss are zero matrices, F n , F 23 and F 32 are identity matrices; the other blocks are

diagonal matrices with constant term s on the diagonal {dt is the time step used in the temporal

scheme).

r 2i(ii) — —fddt^, F 3i(jj) — —jd t (7)

The matrix A is fully determ inistic. A n , A 12, A 13 and A 33 are zero matrices, A 13 and A 32

are identity matrices, the other blocks are diagonal matrices with constant term s on the diagonal.

A2i(jj) = (1/2 — P)dt'^, A22(m) = dt, A3x(ii) = (1 “ l ) d t (8)

The Newmark method is unconditionally stable and second-order accurate if we choose 7 = 1/2

and /3 = 1/4. The vector A of size 3 (P + 1) is such that:


255

r + \ t )

0

0

(9)

jn+l it)

x^+\t) x^+Ht)

x^+\t),v^+ \t) =

x^+ \t), x ^ + \ t ) ^

x^+\t)

.x-+i(t) x T \ t ) _

(10)

Once the matrices and the forcing vector have been evaluated, the m atrix F is inverted numerically

and the solution of the system for the next time step is obtained by computing m a trix — m a trix

and m a trix — vector multiplications. We have

r - i A 5 " + r - ’-A"+^ (11)

This method is computationally very efhcient if the input param eters are random variables. In

this case, the m atrix F needs to be computed and inverted only once at the beginning of the

computation and only the forcing term is updated at every time step. If the input parameters

are random processes, only blocks Fxa and F 13 need to be computed every time step in addition

of the forcing term. The inversion of the matrix F at every time step should be simplified if one

takes into account its structure by blocks where only two blocks are time-dependent.


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