2. computational solutions based on hybrid spectral …computer methods and experimental...

Computational solutions of unsteady viscous flows

with oscillating boundaries

Dan Mateescu

Department of Mechanical Engineering, McGill University,

Montreal, Quebec, H3A 2K6, Canada

Abstract

This paper presents recent computational solutions for two- and three-dimensional unsteadyviscous flows with oscillating boundaries based on: (i) an enhanced hybrid-spectral method,and (ii) time-integration methods using artificial compressibility. A comparison is madebetween the results obtained with a mean-position analysis and with a formulation based on atime-dependent coordinate transformation, better suited for large amplitude oscillations.

1. Introduction

The analysis of unsteady confined flows with oscillating boundaries recentlyreceived an increasing research interest for its applications in many engineeringproblems. Thus, the unsteady annular flows between cylindrical structuresexecuting transverse oscillations, which are of particular interest for the flowinduced vibration problems encountered in many engineering systems, havebeen studied theoretically (based on simplified theoretical models) andexperimentally by numerous scientists, such as Chen et al.\ Inada & Hayama ,Mateescu and his coworkers " and others. These simplified theoretical modelswere proven to be in good agreement with experimental results, at least forsimple cylindrical geometries (Mateescu et al*). However, there is a need formore accurate solutions based on the time-accurate integration of the Navier-Stokes equations and applicable to more realistic and complex geometricconfigurations. The computational methods developed in this respect shoulddisplay, in addition to a good accuracy, a very good computational efficiency inorder to eventually permit the simultaneous integration of the Navier-Stokesequations and the structural equations of motion required in the study of thedynamics of structures subjected to fluid flows. Several such computationalmethods have recently been developed based on a pseudo-time-integration withartificial compressibility ( Mateescu et al™ ) and on spectral-collocation orhybrid-spectral formulations ( Mateescu et al. ™~" ), for unsteady viscous flowswith oscillating boundaries.

This paper presents several computational solutions recently obtained by thisauthor and his coworkers based on the time-accurate integration of the Navier-Stokes equations for two- and three-dimensional unsteady flows with oscillatingboundaries.

Transactions on Modelling and Simulation vol 16, © 1997 WIT Press, www.witpress.com, ISSN 1743-355X

Computer Methods and Experimental Measurements

2. Computational solutions based on hybrid spectral methods

The enhanced hybrid spectral method is illustrated here for the unsteady viscousflow in the annular space between two eccentric cylinders of radii a and bashown in Fig. 1, where ea represents the eccentricity of the cylinder axes. Theflow is referred to the cylindrical coordinates ax , ar and 9 , where x and rare nondimensional. The central portion of length al of the inner (or outer)cylinder executes transverse oscillations in the longitudinal plane of symmetry,0 = 0, or normal to it ( 0 = 90° ), defined as a e (x, t ) = a EQ E (x) cos oof =& [a 60 E (x) exp (/ cof )] , where GQ is the nondimensional amplitude, co theradian frequency and E (x) is the specified axial mode of oscillations. Twofixed cylindrical extensions of the same radii and lengths, al$ and al\ , aresituated upstream and downstream of the oscillating central portion. At its fixedinlet ( x = - /o ), the annular passage is subjected to a fully-developed laminarflow defined by the steady axial velocity UQ U (r, 0 ) , where UQ represents themean axial flow velocity, and the function £/ (r, 0 ) defines the knownvariation of the steady axial velocity in the eccentric annulus

10-11

Section A- A

al

&-T"

alt

Fig. 1. Geometry of the annular space between two eccentric cylinders, with thecentral portion of the inner cylinder executing transverse oscillations in thelongitudinal plane 0 = 0.

The boundary conditions on the moving portion of the cylinder can beexpressed, for oscillations in the plane of symmetry, in the form w* = 0, v* =a(3e/d/)cos0, w* = -0(3e/df)sin6, where w* , v* and M>* are theaxial, radial and circumferential velocity components, and d e / d1 = / co e (x, t).

The problem is solved in a computational domain (x, Z, 0) defined by thecoordinate transformation

Z = l-(r-1)7/2(0) , x = x , 0 =0 , 6(0) = b* - e* sin* 0 -ecos0 -1 (1)In this computational domain, the unsteady velocity components ( u =

w*-t/ot/(r,0), v = v* , w = w*can be expressed in the form

u = aco 80 / exp(/co t) u (x,Z,0) ,

and the unsteady pressure (p = p* -

v = v(x,Z,0) , v(*,Z,0)= ZE%/*)7:(Z)fX0), (3)7=0 k =0

= aa>e,iexp(ia>0 w(jc,Z,9)= (4)



w-2 np = paW So exp(zo) /) p(x,Z,Q) , p(x,Z,Q) = H ? (*) 7 (Z) F,(6) (5)

j=0*=0where 7} (Z) represents the Chebyshev polynomials of order j , F*( 8 ) and7* (6) are orthogonal Fourier functions (e. g. cos &6 and sin kQ ) , whilst%%*(*), */*(*), %%*(%) and Pjk(x) are a priori unspecified complexfunctions, and where w(jc,Z,6) , v(*,Z,9), 6(%,Z,G) and (%,Z,6) representthe nondimensional reduced unsteady velocity components and pressure.

The Navier-Stokes equations and the boundary conditions are transformed inthe computational domain, via the spectral expansions (2) - (5), into a system ofdifferential equations for the unknown functions *£*(*), *J k ( * ) > # * ( x )and ?,&(% ). These equations are then discretized using an efficient hybrid approachbased on :

(a) An azimuthal Fourier identification with respect to 6 , based on anefficient Fourier expansion of the relative annular clearance h(Q).

(b) A collocation approach in the quasi-radial direction Z , imposed atcollocation points conveniently distributed within the computational domain(solution accuracy varies exponentially with the number of collocation points**).

(c) A mixed central-upwind finite-difference scheme for the x direction,based on a discretization performed at N axial stations, */ (with / = 1,2,...JV).The central or upwind schemes are used in function of the local mesh Reynoldsnumber .

p=500

6 = 0

— O —— A —

-D-— • —— A—

Ere,0000.40.4

Re0

125025000

1250

6=1

—V—

-•-

Erel0.40.4

Re12502500

-02 12

Fig. 2. Hybrid spectral method. Typical axial variations of the real andimaginary components of the reduced unsteady pressure, /?(x,0 ), on theoscillating cylinder at 0 = 8 x 90° for (3=500, three values of the hydraulicReynolds number, Re» , and two relative eccentricities, e,ei = 0 and 0.4 .



This efficient hybrid discretization approach leads to the solution of asystem of algebraic equations for the a priori unknown complex grid values

can be expressed in the general matrix form

where A is a sparse block-tridiagonal matrix ( 2% non-zero elements or less).After validation, this hybrid spectral method has been used to obtain for the

first time the solution of the three-dimensional unsteady flow between eccentriccylinders, when the inner cylinder executes transverse oscillations following thefirst flexural beam mode*\ The solutions are illustrated in Figures 2 and 3 forboth cases of oscillations in the plane of symmetry (8 = 0) and normal to thesymmetry plane (5=1).

The typical axial variations of the real and imaginary components of thereduced unsteady pressure, p(x,Q ), on the oscillating inner cylinder (Z=l), areshown in Fig. 2 for an oscillatory Reynolds number p = ooa^/v = 500, threevalues of the hydraulic Reynolds number, Re^ = 2 (ft-1) a UQ I v , and twovalues of the relative eccentricity, e^\ = e I ( b -1) = 0 and 0.4 .

Ere, = 0.4

6 = 0

— # —— A—— T—•

P5005005000500

Re0

125012502500

6=1

—A—P500

Re1250

-02

Fig. 3. Hybrid spectral method. Influence of the hydraulic and oscillatoryReynolds numbers, Re*, and p, on the typical axial variations of the real and

Aimaginary components of the nondimensional reduced unsteady force, F(x),acting on the oscillating cylinder for a relative eccentricity £/•£/•= 0.4 .



Typical axial variations of the real and imaginary components of the reducednondimensional unsteady force acting on the oscillating cylinder, defined as

F(x) - F(x,t) /[% pa^Eo exp(/co /)] and obtained by integrating the unsteady

pressure and skin friction, are shown in Fig. 3 for a relative eccentricity e i =0.4.It was found that the imaginary components are strongly influenced by the

increase in the hydraulic Reynolds number Re^ = 2(6-1) aU^ I v , due to theCorriolis effects which become increasingly important for larger values of Re% .

3. Time-integration method with artificial compressibility

3. 1. Two-dimensional solutions for unsteady viscous flowsover a backward step with an oscillating floor

The time-integration method with artificial compressibility is illustrated firstfor the two-dimensional unsteady flows in ducts with a backward-facing stepand with an oscillating floor behind the step (configuration shown in Fig. 4).The flow is referred to Cartesian coordinates centered at the step corner, Hxand Hy , where x and y are nondimensional coordinates with respect to thedownstream height, //, which is twice the heights of the step and of theupstream channel (H/2 each). The upstream inlet of the channel is subjected toa fully-developed laminar flow defined by the steady axial velocity U (y) =24 UQ y ( 0.5 -y ) , where (/<, represents the mean axial flow velocity. Justbehind the backstep, a portion of the floor of length HI executes transversaloscillations defined as He(t)sm(nx /1), where e (t) = e^ cos ( CD /) , inwhich t = UQ /* / H and co =o* HI UQ represents the nondimensional timeand reduced frequency of the oscillations.

TH/2 iHy

#/H/2

Hx

HI,

H

Fig. 4. Geometry of the unsteady channel flow with a backward step followedby an oscillating floor over the length H 1 .

The incompressible time-dependent Navier-Stokes and continuity equationscan be expressed in nondimensional conservation law form as

o tV.V = (7)


336 Computer Methods and Experimental Measurements

where V, denoting the nondimensional fluid velocity, and G ( V, p ), whichincludes the convective derivative, pressure and viscous terms, can be expressedin 2-D Cartesian coordinates in the form

V = { %, v }\ G ( V,f ) = { G,(%, v,f), G/%, v,p) }? (8)

dx 8y dx Re\d x d y

ap l|"-^-—

(9)

(10),dx dy dy Re\dx dy

in which Re = H UQ I v represents the Reynolds number.In the present approach, the momentum equation is discretized in real

time based on a second order three-point-backward implicit scheme:3 V / 3 f = (3 V "*' - 4V " + V "~* ) / ( 2 AO , where the superscripts n-l , n and«+l indicate three consecutive time levels, A/ represents the time step andG "** = G ( V "% p"** ) . Thus, equations (7) can be expressed at the time levelf"^=(w+l)Af in the form

V^+aG^ = F", V*V"**=0, (11)where a = 2Af/3, F"= (4V" + V^)/3 .

An iterative pseudo-time relaxation procedure using artificial compressibilityis then used in order to advance the solution of the semi-discretized equationsfrom the real time level /" to t"^ in the form

V + aG = F" , 6 (d p / di) + V •V = 0 , (12)

where V (T) and p (T) denote the pseudo-functions corresponding to the variablevelocity and pressure at pseudo-time T , between the real time levels t " andf" , and 6 represents an artificially-added compressibility (the optimum valueof 8 is determined based on the theory of characteristics ) . An implicit Eulerscheme is then used to discretize equations (12) between the pseudo-time levelsT * and T = T * + AT , and the resulting equations are expressed in function ofthe pseudo-time variations A u = iT** - u* , A v = v""^ - v " , A p — p^^ - p* ,in the matrix form

[ I + a AT ( D, + D^ ) ] Af = AT S (13)

where Af = [Aw, Av, Ap]\ a = 2 Af / 3 , I is the identity matrix, and

D =

o a /a%

M O

o o

D =

# 0 0

o

o (i/a8)(a/a)/) o

where (14)

,a% Rea%" a;/ Rea/

A factored Alternate Direction Implicit (ADI) scheme is used to separateequation (13) into two successive sweeps in x and y defined by the equations

[ I + a AT D^ ) ] Af * = AT S , [ I + a AT D^ ) ] Af = Af * , (16)

where Af * is a convenient intermediate variable vector. These equations arethen spatially discretized by central differencing on a stretched staggered grid.A special decoupling procedure, based on the utilization of the continuity



equation, is then used for each sweep to reduce the resulting systems ofdiscretized equations to two sets of decoupled scalar tridiagonal equations ' .

The results obtained with this method for the two-dimensional unsteadyflows in ducts with a backward-facing step and with an oscillating floor behindthe step are illustrated in Fig. 5. Computations have been performed for a rangeof values of the oscillatory Reynolds number, P =CD*//^/V , between 5 and 50,and of the Reynolds number, Re = H UQ I v , between 100 and 1 000 . Thenondimensional length of the oscillating floor portion was / = 0.5 H, and theupstream and downstream lengths of the computational domain are /<,'= 1 and/i =30. The real-time step used was A/ = 2 n I (20 co), where co = P / Re.

The unsteady flow over the backward-facing step, generated by thetransversal oscillations of the floor portion behind the step, displays aninteresting unsteady separated flow pattern . Two separated flow regionsappear on the lower and upper walls, and the position of their separation andreattachment edges substantially vary in time, due to the floor oscillations. Thisis shown in Fig. 5 a & b for two harmonic cycles defined by: (a) Re = 1000and p = 50 , and (b) Re = 500 and (3 = 25. Interestingly, it was found thatin the case (b) the separation bubble simply disappears for a certain portion ofthe oscillatory cycle.

(») Re = 1000

215 220 225Number of real time steps

215 220 225Number of real time steps

Fig. 5. Unsteady flow over a backstep with an oscillating floor. Variation intime of the separation and reattachment points positions on the upper and lowerwalls: (a) Re=1000, p=50, and (b) Re =500, p = 25.



3. 2. Two- and three-dimensional solutions for unsteady annular flowsbased on time-dependent coordinate-transformation

The time-integration method with artificial compressibility based on time-dependent coordinate transformation is illustrated here for the 2-D unsteadyviscous flows between two initially concentric cylinders (e = 0 ), when the outercylinder executes transverse translational oscillations defined by a s(x, t ) =a BO E (jc) exp (/ co/ ) , where E (x) = 1. The time-dependent coordinatetransformation used in this case is

,f), x = x, 8=6, (17)

c,f)f sin^G -e(jc,/)cos9 -1 , (18)

where A(0,x,f) defines the time-dependent azimuthal variation of the relativeannular clearance.

In the fixed-grid computational domain obtained by this transformation, theboundary conditions on the oscillating cylinder can be rigorously implemented,which permit to obtain time-accurate solutions even in the case of largeamplitude oscillations. In this computational domain, the Navier-Stokes andcontinuity equations can be expressed in the form

where t is the nondimensional time, A - dZ/dr , B = ( dZ/dQ ) / Z ,C= ( dZ/de ) ( ddd t) , r = 1 + Zh (6, f) , V = [ v, w ] is the nondimensionalvelocity vector with the radial and circumferential components v = v*/ f/ f andw = w* / £/ref, where U^f = a& in this case, /? = (p*-pQ*)/(pU^ ) is thenondimensional unsteady pressure, and where the vector Q ( V, p ) =[ 2 v (X w, p), Q „ (v, M>, p) f includes the convective derivatives, pressure andviscous terms.

Equations (19), representing the Navier-Stokes and continuity equationstransposed in the fixed-grid computational domain, are further discretized inreal time based on a second order three-point-backward implicit scheme, and apseudo-time relaxation procedure with artificial compressibility is used in asimilar manner to that presented in the previous section. Similarly, a factoredADI scheme is then used in conjunction with spatial differencing on a staggeredgrid and with a special decoupling procedure in order to reduce the problem tothe solution of several sets of decoupled scalar tridiagonal systems procedure .

Two-dimensional unsteady annular flow solutions. The solutions obtainedfor the 2-D unsteady annular flow problem defined above are illustrated in Fig.5, which shows the typical variations of the real and imaginary components ofthe reduced unsteady pressure, p- p I exp(/co /) , with the relative amplitude ofthe transverse oscillations, SQ (relative to the annular gap, ba-a ). The presentsolution based on a time-dependent coordinate transformation is compared inFig. 5 with the solution based on a mean-position analysis (without coordinatetransformation). The imaginary components of these solutions (determining thephase lag of the unsteady pressure) were found increasingly different for largeramplitude of oscillations. This shows the importance of the time-dependentcoordinate transformation for the analysis of unsteady flows with largeamplitude oscillations of the moving boundaries.



0.04 0.06 0.12 0.16Amplitude,

0.05 0.1Amplitude, e$

0.15

Fig. 6. 2-D unsteady annular flows. Typical influence of the relative amplitudeof oscillations on the real and imaginary components of the reduced unsteadypressure, p on the oscillating cylinder at 9=7.5 °. Comparison between:

I , solution based on a time-dependent coordinate transformation, and—Q—, solution based on mean-position analysis.

Fig. 7. 3-D unsteady annular flows. Typical axial variation of the reducedunsteady pressure amplitude at Z =0.127 and 9=7.5° for Re =2900, co=0.416and BO =0.054 . Comparison between:

, solution based on time-dependent coordinate transformation;, solution based on mean-position analysis; • , experimental results.

Three-dimensional annular flow solutions. The solutions obtained with thistime-integration method for the 3-D unsteady viscous flow problems withoscillating boundaries are illustrated in Fig. 7 for an annular passage betweentwo concentric cylinders subjected to a fully-developed laminar flow with themean axial velocity UQ , when the central portion (of length al ) of theouter cylinder executes transversal translational oscillations. Figure 7 shows thetypical axial variation of the reduced unsteady pressure amplitude, p(x), actingon the oscillating cylinder. In this figure the solution based on a time-dependentcoordinate transformation is validated by comparison with experimental resultsand also compared with a similar solution based on a mean-position analysis;the amplitude of oscillation used in these experiments (EQ = 0.054) was just atthe limit when the mean position analysis is still reasonably accurate.



4. Conclusions

Recent time-accurate computational solutions are presented for several two-and three-dimensional unsteady viscous flows with oscillating boundaries Thesesolutions have been obtained with computational methods recently developedby this author and his coworkers, such as an enhanced hybrid-spectral methodand time-integration methods using artificial compressibility and based eitheron a mean-position analysis or on a time-dependent coordinate transformation.These methods have been used to obtain solutions for unsteady viscous flows in2-D and 3-D concentric and eccentric annular configurations with oscillatinginner or outer walls, as well as for unsteady flows over a backward-facing stepwith an oscillating floor behind the step; some of these unsteady flow solutionswere obtained for the first time by using these methods. Very good accuracyand computational efficiency have been displayed by these methods in allproblems solved. The annular flow solutions have been validated by comparisonwith experimental results; a very good agreement was found with theexperimental results. The method based on a time-dependent coordinatetransformation was found better suited for the case of large amplitudeoscillations.

References

1. Chen, S. S., Wambsganss, M. W. W. & Jendrzejczyk, J. A. Added mass anddumping of a vibrating rod in a confined viscous fluid, J. of Applied Mechanics, 1976,43,325-329.2. Inada, F. and Hayama, S. A study on leakage-flow-induced vibrations, J.of Fluidsand Structures, 1990, 4, 395-412 and 413-428.3. Mateescu, D. & Paidoussis, M. P., The unsteady potential flow in an axially-variable annulus and its effect on the dynamics of the oscillating rigid center-bodyJ. of Fluids Engineering, 1985,. 107, 615-628.4. Mateescu, p. & Pai'doussis, M. P., Unsteady viscous effects on the annular-flow-induced instabilities of a rigid cylindrical body in a narrow annular duct, J. of Fluidsand Structures, 1987,. 1, 197-215.5. Mateescu, D., Paidoussis, M. P. and Belanger, F. Unsteady pressure measurementson an oscillating cylinder in narrow annular flow, J.of Fluids and Structures, 1988, 2,615-628.6. Mateescu, D., Paidoussis, M. P. and Belanger, F. A theoretical model comparedwith experiments for the unsteady pressure on an cylinder oscillating in turbulentannular flow, J. of Sound and Vibration, 1989, 135, 487-498.7. Mateescu, D., Paidoussis, M. P. and Belanger, F. A time-integration method usingartificial compressibility for unsteady viscous flows, J. of Sound and Vibration, 1994,.177, 197-2058. Mateescu, D., Paidoussis, M. P. and Belanger, F. Unsteady annular viscous flowsbetween oscillating cylinders. Part I: Computational solutions based on a timeintegration method. & Part II: A hybrid time-integration solution based on azimuthalFourier expansions for configurations with annular backsteps, J. Fluids & Structures1994, 8,. 489-507 and 509-527.9. Mateescu, D., Mekanik, A. and Paidoussis, M. P. Analysis of 2-D and 3-Dunsteady annular flows with oscillating boundaries based on a time-dependentcoordinate transformation, J. of Fluids & Structures, 1996, 10, pp. 57-77.10. Mateescu, D., Paidoussis, M. P. and Sim, W.-G. A spectral collocation methodfor confined unsteady flows with oscillating boundaries, J. Fluids & Structures, 1994,8,157-181.11. Mateescu, D., Pettier, T., Perotin, L. and Granger, S. Three-dimensional unsteadyflows between oscillating eccentric cylinders by an enhanced hybrid spectral method,J. of Fluids and Structures, 1995, 9, 671-695.12. Mateescu, D. and Venditti, D. Unsteady viscous flows over a backward step withan oscillating floor behind the step (1997-to be published).


2. computational solutions based on hybrid spectral …computer methods and experimental...

Documents