nonmodal amplification of disturbances in inertialess...

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Nonmodal amplification of disturbances ininertialess flows of viscoelastic fluids

Mihailo Jovanovicwww.umn.edu/∼mihailo

Shaqfeh Research Group Meeting; July 5, 2010

http://www.ece.umn.edu/users/mihailo/

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M. JOVANOVIC, U OF M 1

Turbulence without inertiaNEWTONIAN: inertial turbulence VISCOELASTIC: elastic turbulence

Groisman & Steinberg, Nature ’00

NEWTONIAN: VISCOELASTIC:

+ FLOW RESISTANCE: increased 20 times!

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Good for mixing . . .

39 DEUTSCHE PHYSIKALISCHE GESELLSCHAFT

Figure 22. Photographs of the flow taken with the laser sheet visualization(figure 21) at different N. The field of view is 3.07 mm × 2.06 mm, andcorresponds to the region shown in figure 21 (rotated 90◦ counterclockwise).Bright regions correspond to high concentration of the fluorescent dye. (a) Flowof the pure solvent at N = 29; (b–d) flow of the polymer solution at Wi = 6.7and at N = 8, 29, 54, respectively.

The channel was illuminated from a side by an argon-ion laser beam converted by two cylindricallenses to a broad sheet of light with a thickness of about 40 µm in the region of observation. Itproduced a thin cut in the three-dimensional mixing pattern, parallel to the top and bottom of thechannel at a half of the channel depth.

Fluorescent light emitted by the liquid in the direction perpendicular to the beam and tothe channel plane was projected on to a CCD array by a camera lens and digitized by an 8-bit,512 × 512 frame grabber. Using homogeneous solutions with different amounts of the dye, wefound intensity of the fluorescent light captured by the camera to be linearly proportional tothe dye concentration. Therefore, concentration of the dye was evaluated from the fluorescentlight intensity. (The liquid without the dye appeared completely dark (figure 22) and signalfrom it was below the noise level of the camera.)

The flow was always observed near the middle of a half-ring on the side from whichthe laser beam was coming. So, the number, N, of a unit (figure 21) counted from the inletwas a natural linear co-ordinate along the channel. Since the total rate of liquid discharge,Q, was constant, N was also proportional to the average time of mixing. In order to observefurther stages of mixing (corresponding to N > 30), we carried out a series of experiments,

New Journal of Physics 6 (2004) 29 (http://www.njp.org/)

Groisman & Steinberg, Nature ’01

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. . . bad for processingDISTORTION OF A POLYMER MELT EMERGING FROM A CAPILLARY TUBE

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t in

lan

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■

Chr

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ger

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the

Hum

an O

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Gro

up, D

epar

tmen

t of P

alae

onto

logy

, Nat

ural

His

tory

Mus

eum

, Cro

mw

ell R

oad,

Lon

don

SW7

5BD

, UK

.1.

Kle

in, R

. The

Hum

an C

aree

r(U

niv

. Ch

icag

o P

ress

, 199

9).

2.La

hr,

M. &

Fol

ey, R

. Yb.

Phy

s. A

nthr

opol

. 41,

137–

176

(199

8).

3.R

elet

hfo

rd, J

. & J

orde

, L. A

m. J

. Phy

s. A

nthr

opol

. 108

,251

–260

(199

9).

4.W

alte

r, R

. C. e

t al.

Nat

ure

405,

65–6

9 (2

000)

.5.

Troe

ng,

J. A

cta

Arc

haeo

l. Lu

nden

sia

Ser.

8, N

o. 2

1 (1

993)

.6.

Bar

ton

, R. N

. et a

l.A

ntiq

uity

73,1

3–23

(19

99).

7.K

ingd

on, J

. Sel

f-M

ade

Man

and

his

Und

oing

(Sim

on &

Sch

ust

er,

Lon

don

, 199

3).

8.St

rin

ger,

C. A

ntiq

uity

73,8

76–8

79 (

1999

).9.

Th

orn

e, A

. et a

l. J.

Hum

. Evo

l.36

,591

–612

(19

99).

10.

Nat

iona

l Geo

grap

hic

(map

, Feb

. 199

7).

visc

osit

ies

that

can

be

hun

dred

s to

mill

ion

sof

tim

es h

igh

er t

han

wat

er.

Un

til

the

wor

k of

Gro

ism

an a

nd

Stei

nbe

rg2 ,

quan

tita

tive

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sure

men

ts o

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last

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nce

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han

din

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m l

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r ‘e

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e ba

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ility

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ech

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lve

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’. Nor

mal

stre

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prod

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the

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of p

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ol-

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to a

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last

ic f

orce

lik

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at i

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ubb

er b

and.

If

the

stre

amlin

es a

re c

ircu

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the

poly

mer

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bber

ban

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s in

war

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all

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his

© 2

000

Mac

mill

an M

agaz

ines

Ltd

Kalika & Denn, J. Rheol. ’87

CURVILINEAR FLOWS: purely elastic instabilitiesLarson, Shaqfeh, Muller, J. Fluid Mech. ’90

RECTILINEAR FLOWS: no modal instabilities

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Transition in Newtonian fluids

• LINEAR HYDRODYNAMIC STABILITY: unstable normal modes

? successful in: Benard Convection, Taylor-Couette flow, etc.

? fails in: wall-bounded shear flows (channels, pipes, boundary layers)

• DIFFICULTY #1Inability to predict: Reynolds number for the onset of turbulence (Rec)

Experimental onset of turbulence:

{much before instabilityno sharp value for Rec

• DIFFICULTY #2Inability to predict: flow structures observed at transition

(except in carefully controlled experiments)

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LINEAR STABILITY:

? For Re ≥ Rec ⇒ exp. growing normal modescorresponding e-functions

(TS-waves)

}:= exp. growing flow structures

-x

��z

EXPERIMENTS: streaky boundary layers and turbulent spots

-x6z

Failure of classical linear stability analysis for wall-bounded shear flows

•

Classical theory assumes 2D disturbances

•

Experiments suggest transition to more complex flow is inherently 3D: streamwise streaks

Flow type Classical prediction Experiment

Plane Poiseuille 5772 ~ 1000

Plane Couette ∞ ~ 350Pipe flow ∞ ~ 2200-100000

•

Critical Reynolds number for instability

M. Matsubara and P. H. Alfredsson, J. Fluid Mech. 430 (2001) 149

Boundary layer flow with free-stream turbulence

Flow-x6z

Matsubara & Alfredsson, J. Fluid Mech. ’01

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• FAILURE OF LINEAR HYDRODYNAMIC STABILITY

caused by high flow sensitivity

? large transient responses

? large noise amplification

? small stability margins

TO COUNTER THIS SENSITIVITY: must account for modeling imperfections

TRANSITION ≈ STABILITY + RECEPTIVITY + ROBUSTNESS

←−

←−

flowdisturbances

unmodeleddynamics

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Tools for quantifying sensitivity

• INPUT-OUTPUT ANALYSIS: spatio-temporal frequency responses

-

d

Free-stream turbulenceSurface roughnessNeglected nonlinearities

Linearized Dynamics -

Fluctuatingvelocity field

}v

d1d2d3

︸︷︷︸

d

amplification−−−−−−−−−−−→

uvw

︸︷︷︸

v

IMPLICATIONS FOR:

transition: insight into mechanisms

control: control-oriented modeling

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Transient growth analysis

• STUDY TRANSIENT BEHAVIOR OF FLUCTUATIONS’ ENERGY

Farrell, Butler, Gustavsson, Henningson, Reddy, Trefethen, etc.

Re < 5772

ψt = Aψ

kinetic energy:

+ E-VALUES: misleading measure of transient response

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A toy example[ψ1

ψ2

]=

[−λ1 0R −λ2

] [ψ1

ψ2

], λ1, λ2 > 0

λ1 = 1, λ2 = 2:ψ1(t

),ψ2(t

)

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Non-modal amplification of disturbances[ψ1

ψ2

]=

[−λ1 0R −λ2

] [ψ1

ψ2

]+

[10

]d

-d 1

s + λ1-

ψ1

R -1

s + λ2-

ψ2

WORST CASE AMPLIFICATION VARIANCE AMPLIFICATION

maxenergy of ψ2

energy of d= max

ω|H(iω)|2 =

R2

(λ1λ2)21

2π

∫ ∞−∞|H(iω)|2 dω =

R2

λ1λ2 (λ1 + λ2)

ROBUSTNESS

-d 1

s + λ1-

ψ1

R -1

s + λ2s -ψ2

�Γ

modeling uncertainty(can be nonlinear or time-varying)

small-gain theorem:

stability for all Γ withmaxω |Γ(iω)| ≤ γ

mγ < λ1λ2/R

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Ensemble average energy density

Re = 2000

• Dominance of streamwise elongated structuresstreamwise streaks!

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Influence of Re: streamwise-constant model

[ψ1t

ψ2t

]=

[Aos 0

ReAcp Asq

] [ψ1

ψ2

]+

[0 B2 B3

B1 0 0

] d1d2d3

uvw

=

0 Cu

Cv 0Cw 0

[ ψ1

ψ2

]

-d2

B2- n - (iωI − Aos)

−1

Orr-Sommerfelds -ReAcp

coupling

- n - (iωI − Asq)−1

Squire

-ψ2

Cu-

u

-d1

B1

?

-d3

B3

6

s - Cv-

v

- Cw-

w

ψ1

Jovanovic & Bamieh, J. Fluid Mech. ’05

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Amplification mechanism in flows with high Re

• HIGHEST AMPLIFICATION: (d2, d3)→ u

-d2

B2- n -(iωI − ∆−1∆2)−1

‘glorifieddiffusion’

-ψ1

ReAcp

vortextilting

- (iωI − ∆)−1

viscousdissipation

-ψ2

Cu-

u

-d3

B3

6

+ AMPLIFICATION MECHANISM: vortex tilting or lift-up

wal

l-nor

mal

dire

ctio

n

Stream-wise vortices and streaks

31

spanwise direction

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Linear analyses: Input-output vs. Stability

AMPLIFICATION:

v = H d

singular values of H

STABILITY:

ψt = Aψ

e-values of A

The cross sectional model

2 Dimensional, 3 Component (2D3C) model for parallel flowsDynamics of stream-wise constant pertrubations

• Simpler than full 3D models

• Has much of the phenomenology of boundary layer turbulence

– Transitional (bypass) and fully turbulent coherent structures arevery elongated in streamwise direction

typical structures cross sectional dynamics standard 2D models

8

Input-output analysis of Linearized Navier-Stokes (Cont.)

��

��

................................................................................

................................................................................

.................................................................

................................................................................

................................................................................

.................................................................

�

�

�� ..

�

.........................................................................................................................................................................................................

��

��

�

.........................................................................................................................................................................................................

��

��

x1

−1uw

vyz

“Most resonant flow structures”are stream-wise vortices and streaks

Cross sectional view

30

typical structures cross-sectional dynamics 2D models

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Oldroyd-B fluids

HOOKEAN SPRING:

(Re/We)ut = −Re (u · ∇)u − ∇p + β∆u + (1− β)∇ · τ + d

0 = ∇ · uτ t = − τ + ∇u + (∇u)T + We

(τ · ∇u + (∇u)T · τ − (u · ∇)τ

)

VISCOSITY RATIO: β :=solvent viscosity

total viscosity

WEISSENBERG NUMBER: We :=fluid relaxation time

characteristic flow time

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• TRANSIENT GROWTH ANALYSIS

Sureshkumar et al., JNNFM ’99; Atalik & Keunings, JNNFM ’02;Kupferman, JNNFM ’05; Doering et al., JNNFM ’06; Renardy, JNNFM ’09

• INPUT-OUTPUT ANALYSIS

-d

Equations of motion s -v

�Polymer equationss�

τ

-

importance of streamwise elongated structures

Hoda, Jovanovic, Kumar, J. Fluid Mech. ’08Hoda, Jovanovic, Kumar, J. Fluid Mech. ’09

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Inertialess channel flow: streamwise-constant model

0 = −∇p + β∆u + (1− β)∇ · τ + d

0 = ∇ · uτ t = − τ + ∇u + (∇u)T + We

(τ · ∇u + (∇u)T · τ − (u · ∇)τ

)

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Inertialess Oldroyd-B vs. Inertial Newtonian

-d2

B2- n - (iωI − A11)

−1 s -WeAcp

polymerstretching

- n - (iωI − A22)−1 -

φ2Cu

-u

-d1

B1

?

-d3

B3

6

s - Cv-

v

- Cw-

w

φ1

OLDROYD-B W/O INERTIA: Weissenberg number & Polymer Stretching≈

NEWTONIAN WITH INERTIA: Reynolds number & Vortex Tilting

Jovanovic & Kumar, Phys. Fluids ’10

Jovanovic & Kumar ’10, (submitted)

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Spatial frequency responses

(d2, d3)amplification−−−−−−−−−−−→ u

INERTIAL NEWTONIAN: E(kz;Re) = Re2 f(kz)

INERTIALESS OLDROYD-B: E(kz;We, β) = We2 g(kz) (1− β)2/β

vortex tilting: f(kz) polymer stretching: g(kz)

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Dominant flow patterns• FREQUENCY RESPONSE PEAKS

+ streamwise vortices and streaks

Inertial Newtonian: Inertialess Oldroyd-B:

• CHANNEL CROSS-SECTION VIEW:

{color plots: streamwise velocitycontour lines: stream-function

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Outlook

• WISH LIST

? direct numerical simulations of stochastically forced flowstrack ‘linear’ and ‘nonlinear’ stages of disturbance development

? secondary sensitivity analysisstudy influence of streamwise-varying disturbances on streaks

• Challenge: relative roles of flow sensitivity and nonlinearity

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Acknowledgments

COLLABORATORS:

Satish Kumar (CEMS, U of M)Nazish Hoda (ExxonMobil)Binh Lieu (ECE, U of M)

SUPPORT:

NSF CAREER Award CMMI-06-44793

COMPUTING RESOURCES:

Minnesota Supercomputing Institute

nonmodal amplification of disturbances in inertialess...

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