Nonmodal amplification of disturbances in inertialess flows of viscoelastic fluids

Transcription

1 Nonmodal amplification of disturbances in inertialess flows of viscoelastic fluids Mihailo Jovanović mihailo Shaqfeh Research Group Meeting; July 5, 2010

2 M. JOVANOVIĆ, U OF M 1 Turbulence without inertia NEWTONIAN: inertial turbulence VISCOELASTIC: elastic turbulence Groisman & Steinberg, Nature 00 NEWTONIAN: VISCOELASTIC: FLOW RESISTANCE: increased 20 times!

3 M. JOVANOVIĆ, U OF M 2 Good for mixing 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, and corresponds to the region shown in figure 21 (rotated 90 counterclockwise). Bright regions correspond to high concentration of the fluorescent dye. (a) Flow Groisman & Steinberg, Nature 01

4 M. JOVANOVIĆ, U OF M 3... bad for processing D DISTORTION OF A POLYMER MELT EMERGING FROM A CAPILLARY TUBE description of hard turbulence, a power-law distribution in length scales is produced by this handing off of energy from larger to smaller eddies 4. For elastic turbulence, the basic mechanisms of instability in the base flow have only recently been discovered. These mechanisms involve normal stresses. Normal stresses are produced by the stretching of polymer molecules in a flow, leading to an elastic force like that in a stretched rubber band. If the streamlines are circular, the polymer rubber bands press inward from all directions, generating a radial pressure gradient. This (1999). 4. Walter, R. C. et al. Nature 405, (2000). 5. Troeng, J. Acta Archaeol. Lundensia Ser. 8, No. 21 (1993). 6. Barton, R. N. et al. Antiquity 73, (1999). 7. Kingdon, J. Self-Made Man and his Undoing (Simon & Schuster, London, 1993). 8. Stringer, C. Antiquity 73, (1999). 9. Thorne, A. et al. J. Hum. Evol. 36, (1999). 10. National Geographic (map, Feb. 1997). ants first t from the t it must go. Such rough the l foraging ithout inertia elastic turbulence 3. One such flow is that of a polymer melt emerging from the end of a capillary tube. Above a critical flow rate, the polymer jet becomes irregularly distorted (Fig. 1). Distortions in this and other polymer flows can mar products made from polymers, such as films and extruded parts. If the polymer is made more viscous by increasing the length of its polymer molecules, the minimum flow rate producing such irregular flow decreases. This is surprising because an increase in viscocity has the opposite effect on inertially driven turbulence. In fact, for fluids containing long polymer molecules, the Reynolds number at the onset of the instabilities can be minuscule, as low as Turbulent flow of water in a pipe, by contrast, has a much higher Reynolds number, around It is clear, then, that inertia has nothing whatsoever to do with this kind of polymeric turbulence. For viscous polymers, instabilities and irregular flows set in at a critical value of the so-called Weissenberg number, the ratio of elastic to viscous forces in the fluid, which for polymeric fluids plays the role of the Reynolds number in creating the nonlinearities that lead to unstable flow. Because elastic forces increase with polymer length more rapidly than do viscous forces, fluids containing long polymers are especially prone to such phenomena, despite having CURVILINEAR FLOWS: purely elastic instabilities RECTILINEAR FLOWS: no modal instabilities stablished mentum) served in e, plumes ies in the ny other s turbugh values nsionless Over the r characaccompag, but also temporal Figure 1 Is this jet turbulent? Turbulent flow is normally associated with a high Reynolds number. A polymer melt of high viscosity (5,000 pascal seconds) and low Reynolds number is greatly distorted after emerging from a capillary tube 12. Groisman and Steinberg 2 now confirm that the defining features of turbulence can appear in the flow of a polymer solution with high elasticity but low Reynolds number. sman and sed resisof turbuardly any is instead t in soluflow studsimple: a o parallel ne of the increased generate turbulent ong been tic fluids, dubbed com 2000 Macmillan Magazines Ltd 27 Kalika & Denn, J. Rheol. 87 Larson, Shaqfeh, Muller, J. Fluid Mech. 90

5 M. JOVANOVIĆ, U OF M 4 Transition in Newtonian fluids LINEAR HYDRODYNAMIC STABILITY: unstable normal modes D successful in: Benard Convection, Taylor-Couette flow, etc. fails in: wall-bounded shear flows (channels, pipes, boundary layers) DIFFICULTY #1 Inability to predict: Reynolds number for the onset of turbulence (Re c ) Experimental onset of turbulence: { much before instability no sharp value for Re c DIFFICULTY #2 Inability to predict: flow structures observed at transition (except in carefully controlled experiments)

6 linear stability analysis ded shear flows M. JOVANOVIĆ, U OF M 5 LINEAR STABILITY: r instability For Re Re c exp. growing normal modes } corresponding e-functions l prediction Experiment := exp. growing flow structures (TS-waves) ~ 1000 disturbances z x EXPERIMENTS: streaky boundary layers and turbulent spots on to more complex flow is ise streaks am turbulence ~ 350 ~ z x Flow z x Matsubara & Alfredsson, J. Fluid Mech. 01

7 M. JOVANOVIĆ, U OF M 6 D 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 flow disturbances unmodeled dynamics

8 M. JOVANOVIĆ, U OF M 7 Tools for quantifying sensitivity D INPUT-OUTPUT ANALYSIS: spatio-temporal frequency responses d Free-stream turbulence Surface roughness Neglected nonlinearities Linearized Dynamics Fluctuating velocity field } v d 1 d 2 d 3 }{{} d amplification u v w }{{} v IMPLICATIONS FOR: transition: insight into mechanisms control: control-oriented modeling

9 M. JOVANOVIĆ, U OF M 8 Transient growth analysis D STUDY TRANSIENT BEHAVIOR OF FLUCTUATIONS ENERGY Farrell, Butler, Gustavsson, Henningson, Reddy, Trefethen, etc. kinetic energy: Re < 5772 ψ t = A ψ E-VALUES: misleading measure of transient response

10 M. JOVANOVIĆ, U OF M 9 [ ψ 1 ψ 2 ] = A toy example [ ] [ ] λ1 0 ψ1, λ R λ 2 ψ 1, λ 2 > 0 2 λ 1 = 1, λ 2 = 2: ψ1(t), ψ2(t)

11 M. JOVANOVIĆ, U OF M 10 Non-modal amplification of disturbances [ ψ 1 ψ 2 ] = [ ] [ ] λ1 0 ψ1 R λ 2 ψ 2 + [ 1 0 ] d d 1 s + λ 1 ψ 1 R 1 s + λ 2 ψ 2 WORST CASE AMPLIFICATION VARIANCE AMPLIFICATION max energy of ψ 2 energy of d = max ω H(iω) 2 = R2 (λ 1 λ 2 ) 2 1 2π H(iω) 2 dω = R 2 λ 1 λ 2 (λ 1 + λ 2 ) ROBUSTNESS small-gain theorem: d 1 s + λ 1 ψ 1 R 1 s + λ 2 ψ 2 stability for all Γ with max ω Γ(iω) γ Γ modeling uncertainty (can be nonlinear or time-varying) γ < λ 1 λ 2 /R

12 M. JOVANOVIĆ, U OF M 11 Ensemble average energy density Re = 2000 Dominance of streamwise elongated structures streamwise streaks!

13 M. JOVANOVIĆ, U OF M 12 Influence of Re: streamwise-constant model [ ψ1t ] ψ 2t u v w = = [ A os 0 Re A cp 0 C u C v 0 C w 0 A sq ] [ ψ1 [ ψ1 ψ 2 ] ψ 2 ] + [ 0 B2 B 3 B ] d 1 d 2 d 3 d 1 B 1 Orr-Sommerfeld d 2 B 2 (iωi A os ) 1 ψ 1 coupling ReAcp Squire (iωi A sq) 1 ψ 2 u C u d 3 B 3 C v v C w w Jovanović & Bamieh, J. Fluid Mech. 05

14 M. JOVANOVIĆ, U OF M 13 Amplification mechanism in flows with high Re HIGHEST AMPLIFICATION: (d 2, d 3 ) u glorified diffusion d 2 B 2 (iωi 1 2 ) 1 vortex tilting ψ 1 ReAcp viscous dissipation (iωi ) 1 ψ 2 u C u d 3 B 3 AMPLIFICATION MECHANISM: vortex tilting or lift-up Stream-wise vortices and streaks wall-normal direction spanwise direction

15 Simpler than full 3D models M. JOVANOVIĆ, U OF M 14 Linear analyses: Input-output vs. Stability Has much of the phenomenology of boundary la ut analysis of Linearized Navier-Stokes (Cont.) AMPLIFICATION: 1 y. z x 1 STABILITY: Transitional v = (bypass) H d and fully turbulent ψ t = A ψ cohe.. singular values of H e-values of A w structures ortices and streaks very elongated in streamwise direction. v u w typical structures cross sectional dynamics sta w typical structures cross-sectional dynamics 2D models 30

16 M. JOVANOVIĆ, U OF M 15 Oldroyd-B fluids HOOKEAN SPRING: D (Re/We) u t = Re (u ) u p + β u + (1 β) τ + d 0 = u τ 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

17 M. JOVANOVIĆ, U OF M 16 D 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 v Equations of motion τ Polymer equations importance of streamwise elongated structures Hoda, Jovanović, Kumar, J. Fluid Mech. 08 Hoda, Jovanović, Kumar, J. Fluid Mech. 09

18 M. JOVANOVIĆ, U OF M 17 Inertialess channel flow: streamwise-constant model 0 = p + β u + (1 β) τ + d 0 = u τ t = τ + u + ( u) T + We ( τ u + ( u) T τ (u )τ )

19 M. JOVANOVIĆ, U OF M 18 Inertialess Oldroyd-B vs. Inertial Newtonian d 1 B1 D φ 1 polymer stretching d 2 B2 (iωi Ā11) 1 WeĀcp φ (iωi Ā22) 2 u 1 Cu d 3 B3 v Cv w Cw OLDROYD-B W/O INERTIA: Weissenberg number & Polymer Stretching NEWTONIAN WITH INERTIA: Reynolds number & Vortex Tilting Jovanović & Kumar, Phys. Fluids 10 Jovanović & Kumar 10, (submitted)

20 M. JOVANOVIĆ, U OF M 19 Spatial frequency responses (d 2, d 3 ) amplification u INERTIAL NEWTONIAN: E(k z ; Re) = Re 2 f(k z ) INERTIALESS OLDROYD-B: E(k z ; We, β) = We 2 g(k z ) (1 β) 2 /β vortex tilting: f(k z ) polymer stretching: g(k z )

21 M. J OVANOVI C, U OF M 20 Dominant flow patterns F REQUENCY RESPONSE PEAKS + streamwise vortices and streaks Inertial Newtonian: Inertialess Oldroyd-B: ( C HANNEL CROSS - SECTION VIEW : color plots: streamwise velocity contour lines: stream-function

22 M. JOVANOVIĆ, U OF M 21 Outlook WISH LIST D direct numerical simulations of stochastically forced flows track linear and nonlinear stages of disturbance development secondary sensitivity analysis study influence of streamwise-varying disturbances on streaks Challenge: relative roles of flow sensitivity and nonlinearity

23 M. JOVANOVIĆ, U OF M 22 Acknowledgments COLLABORATORS: D Satish Kumar (CEMS, U of M) Nazish Hoda (ExxonMobil) Binh Lieu (ECE, U of M) SUPPORT: NSF CAREER Award CMMI COMPUTING RESOURCES: Minnesota Supercomputing Institute