very elongated in streamwise direction

Transcription

1 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) T d and fully turbulent ψ t = A ψ cohe.. singular values of T e-values of A w structures ortices and streaks very elongated in streamwise direction. v u w Simpler than full 3D models EE 8235: Lecture 21 1 Lecture 21: Input-output analysis in fluid mechanics typical structures cross sectional dynamics sta w typical structures cross-sectional dynamics 2D models 30

2 EE 8235: Lecture 21 2 Transition in Newtonian fluids 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 ) LINEAR HYDRODYNAMIC STABILITY: unstable normal modes 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)

3 linear stability analysis EE 8235: Lecture 21 3 ded LINEAR shear STABILITY: flows For Re Re c exp. growing normal modes r instability } corresponding e-functions := exp. growing flow structures l prediction Experiment (TS-waves) disturbances ~ 1000 ~ 350 ~ z x EXPERIMENTS: streaky boundary layers and turbulent spots on to more complex flow is ise streaks am turbulence z x Flow z x Matsubara & Alfredsson, J. Fluid Mech. 01

4 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 EE 8235: Lecture 21 4 TRANSITION STABILITY + RECEPTIVITY + ROBUSTNESS flow disturbances unmodeled dynamics

5 EE 8235: Lecture 21 5 Tools for quantifying sensitivity INPUT-OUTPUT ANALYSIS: spatio-temporal frequency responses d Free-stream turbulence Surface roughness Acoustic waves Linearized Dynamics d 1 d 2 d 3 }{{} d Fluctuating velocity field amplification } v u v w }{{} v IMPLICATIONS FOR: transition: insight into mechanisms control: control-oriented modeling

6 EE 8235: Lecture 21 6 Ensemble average energy density Re = 2000 Dominance of streamwise elongated structures streamwise streaks!

7 EE 8235: Lecture 21 7 Influence of Re: streamwise-constant model d 1 B 1 [ 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 Orr-Sommerfeld d 2 B 2 (jωi A os ) 1 ψ 1 coupling ReAcp Squire (jωi A sq) 1 ψ 2 u C u d 3 B 3 C v v C w w Jovanović & Bamieh, J. Fluid Mech. 05

8 glorified diffusion d 2 B 2 (jωi 1 2 ) 1 d 3 B 3 ψ 1 vortex tilting ReAcp AMPLIFICATION MECHANISM: vortex tilting or lift-up viscous dissipation (jωi ) 1 ψ 2 u C u EE 8235: Lecture 21 8 Amplification mechanism in flows with high Re HIGHEST AMPLIFICATION: (d 2, d 3 ) u wall-normal direction spanwise direction

9 Groisman & Steinberg, Nature 00 EE 8235: Lecture 21 9 Turbulence without inertia NEWTONIAN: inertial turbulence VISCOELASTIC: elastic turbulence NEWTONIAN: VISCOELASTIC: FLOW RESISTANCE: increased 20 times!

10 EE 8235: Lecture Turbulence: 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

11 EE 8235: Lecture bad for processing 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

12 EE 8235: Lecture Oldroyd-B fluids HOOKEAN SPRING: (Re/We) v t = Re (v ) v p + β v + (1 β) τ + d 0 = v VISCOSITY RATIO: τ t = τ + v + ( v) T + We ( τ v + ( v) T τ (v )τ ) β := solvent viscosity total viscosity WEISSENBERG NUMBER: We := fluid relaxation time characteristic flow time REYNOLDS NUMBER: Re := inertial forces viscous forces

13 EE 8235: Lecture body forcing fluctuations polymer stress fluctuations Input-output analysis velocity fluctuations Equations of motion Constitutive equations d 1 d 2 d 3 }{{} d amplification u v w }{{} v INSIGHT INTO AMPLIFICATION MECHANISMS importance of streamwise elongated structures Hoda, Jovanović, Kumar, J. Fluid Mech. 08, 09 Jovanović & Kumar, JNNFM 11

14 EE 8235: Lecture Inertialess channel flow: worst case amplification No single constitutive equation can describe the range of phenomena important to quantify influence of modeling imperfections on dynamics We = 10, β = 0.5, Re = 0 G(k x, k z ) = sup σmax 2 (T (k x, k z, ω)): ω

15 We = 50, β = 0.5, Re = 0 G(k x, k z ): EE 8235: Lecture 21 15

16 We = 100, β = 0.5, Re = 0 G(k x, k z ): EE 8235: Lecture Dominance of streamwise elongated structures streamwise streaks!

17 INERTIALESS VISCOELASTIC: glorified normal/spanwise diffusion forcing 1 jωβ + 1 A 1 os polymer stretching We Acp2 viscous dissipation (1 β) jωβ streamwise velocity EE 8235: Lecture Amplification mechanism Highest amplification: (d 2, d 3 ) u INERTIAL NEWTONIAN: normal/spanwise forcing glorified diffusion (jωi A os ) 1 vortex tilting ReAcp1 viscous dissipation (jωi ) 1 streamwise velocity

18 EE 8235: Lecture Inertialess lift-up mechanism η t = (1/β) η + We (1 1/β) A cp2 ϑ ( ) = (1/β) η + We (1 1/β) yz (U (y) τ 22 ) + zz (U (y) τ 23 )

19 EE 8235: Lecture Spatial frequency responses (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 )

20 EE 8235: Lecture 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 Newtonian fluids: self-sustaining Self-Sustaining process Process for(ssp) transition to turbulence Waleffe, Phys. Fluids 97 EE 8235: Lecture Flow sensitivity vs. nonlinearity Challenge: relative roles of flow sensitivity and nonlinearity advection of mean shear Streaks O(1) instability of U(y,z) O(1/R) Streamwise Rolls nonlinear self interaction Streak wave mode (3D) O(1/R)