Dynamics of inertialess flows of viscoelastic fluids: the role of uncertainty

Transcription

1 Dynamics of inertialess flows of viscoelastic fluids: the role of uncertainty Mihailo Jovanović mihailo joint work with: Satish Kumar, Binh Lieu Workshop on Complex Fluids and Flows in Industry and Nature

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

3 39 DEUTSCHE PHYSIKALISCHE GESELLSCHAFT Turbulence: good for mixing... 2 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 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 bad for processing 3

5 HOOKEAN SPRING: raft (Re/We) t v = Re (v ) v p + β v + (1 β) τ + d VISCOSITY RATIO: 0 = v t τ = τ + v + ( v) T + We ( τ v + ( v) T τ (v )τ ) β := solvent viscosity total viscosity Oldroyd-B fluids 4 WEISSENBERG NUMBER: We := fluid relaxation time characteristic flow time REYNOLDS NUMBER: Re := inertial forces viscous forces

6 kinetic energy: Transient growth analysis 5 STUDY TRANSIENT BEHAVIOR OF FLUCTUATIONS ENERGY SPECTRUM: misleading measure of transient response Sureshkumar et al., JNNFM 99; Atalik & Keunings, JNNFM 02; Kupferman, JNNFM 05; Doering et al., JNNFM 06; Renardy, JNNFM 09; Jovanović & Kumar, Phys. Fluids 10

7 HIGH FLOW SENSITIVITY even in the absence of inertia large transient responses large noise amplification small stability margins TO COUNTER THIS SENSITIVITY: must account for modeling imperfections 6 STABILITY + RECEPTIVITY + ROBUSTNESS flow disturbances unmodeled dynamics

8 INPUT-OUTPUT ANALYSIS: spatio-temporal frequency responses body forcing fluctuations polymer stress fluctuations velocity fluctuations Equations of motion Constitutive equations Tools for quantifying sensitivity 7 d 1 d 2 d 3 }{{} d amplification u v w }{{} v

9 d { body forcing fluctuations raft Linearized Dynamics INSIGHT INTO AMPLIFICATION MECHANISMS importance of streamwise elongated structures velocity fluctuations } v 8 Hoda, Jovanović, Kumar, J. Fluid Mech. 08, 09 Jovanović & Kumar, JNNFM 11

10 FREQUENCY RESPONSE raft harmonic forcing: d(x, y, z, t) = ˆd(k x, y, k z, ω) e i(k xx + k z z + ωt) steady-state response v(x, y, z, t) = ˆv(k x, y, k z, ω) e i(k xx + k z z + ωt) Non-modal amplification of disturbances 9 ˆd(k x, y, k z, ω) H(k x, k z, ω) ˆv(k x, y, k z, ω) H(k x, k z, ω): operator in y ˆv(k x, y, k z, ω) = 1 1 H ker (k x, k z, ω; y, η) ˆd(k x, η, k z, ω) dη

11 G(k x, k z ) = max σ 2 max (H(ω)) worst case amplification: output energy input energy = max ω σ2 max (H(k x, k z, ω)) Input-output gains Determined by singular values of H(k x, k z, ω) 10

12 ˆd Nominal Linearized Dynamics Γ modeling uncertainty (can be nonlinear or time-varying) small-gain theorem: ˆv Robustness interpretation 11 stability for all Γ with max ω σ2 max (Γ(ω)) γ γ < 1 G LARGE worst case amplification small stability margins

13 We = 10, β = 0.5, Re = 0 raft G(k x, k z ): Inertialess channel flow: worst case amplification 12

14 We = 50, β = 0.5, Re = 0 raft G(k x, k z ): 13

15 We = 100, β = 0.5, Re = 0 raft G(k x, k z ): 14 Dominance of streamwise elongated structures streamwise streaks!

16 Inertial Newtonian t η = η + Re C p1 v source Inertialess viscoelastic t η = (1/β) η + We (1 1/β) C p2 ϑ t ϑ = ϑ + v source Streamwise-constant flows LINEARIZED DYNAMICS OF NORMAL VORTICITY η 15 SOURCE TERM IN INERTIALESS VISCOELASTIC FLOWS C p2 ϑ = yz (U (y)τ 22 ) + zz (U (y)τ 23 )

17 t η = (1/β) η + We (1 1/β) C p2 ϑ = (1/β) η + We (1 1/β) ( yz (U (y)τ 22 ) + zz (U (y)τ 23 )) Inertialess lift-up mechanism 16

18 INERTIALESS VISCOELASTIC: glorified normal/spanwise diffusion forcing 1 iωβ + 1 A 1 os raft polymer stretching We Cp2 viscous dissipation (1 β) iωβ streamwise velocity Highest amplification: (d 2, d 3 ) u Amplification mechanism 17 INERTIAL NEWTONIAN: normal/spanwise forcing glorified diffusion (iωi A os ) 1 vortex tilting ReCp1 viscous dissipation (iωi ) 1 streamwise velocity

19 (d 2, d 3 ) amplification u INERTIAL NEWTONIAN: G(k z ; Re) = Re 2 f(k z ) INERTIALESS VISCOELASTIC: G(k z ; We, β) = We 2 g(k z ) (1 β) 2 vortex tilting: f(k z ) polymer stretching: g(k z ) Spatial frequency responses 18

20 19 Dominant flow patterns raf t F REQUENCY RESPONSE PEAKS + streamwise vortices and streaks Inertial Newtonian: Inertialess viscoelastic: ( C HANNEL CROSS - SECTION VIEW : color plots: streamwise velocity contour lines: stream-function

21 ONGOING EFFORT raft direct numerical simulations of inertialess flows track linear and nonlinear stages of disturbance development secondary stability/receptivity analysis study influence of streamwise-varying disturbances on streaks Challenge: relative roles of flow sensitivity and nonlinearity Outlook 20 self-sustaining process for transition to elastic turbulence?

22 TEAM: SUPPORT: raft Satish NSF CAREER Award CMMI U of M Digital Technology Center Seed Grant Binh Acknowledgments 21 COMPUTING RESOURCES: Minnesota Supercomputing Institute