Dynamics of Elasto-Inertial Turbulence in Flows with Polymer Additives
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(2) Acknowledgements. Collaborators. • Yves Dubief • Julio Soria. MTFC RESEARCH GROUP. University of Vermont, USA. Monash University, Australia . King Abdulaziz University, Kingdom of Saudi Arabia. Financial support. • Marie Curie FP7 CIG. • Vermont Advanced Computing Center. • US National Institutes of Health. • Australian Research Council. • Center for Turbulence Research Summer Program. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 2.
(3) Outline. MTFC RESEARCH GROUP. • Context. • Models and numerical implementation. • Polymer drag reduction. • Elasto-inertial turbulence. • Conclusions and future work. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 3.
(4) Polymers and turbulence. MTFC RESEARCH GROUP. Turbulent drag reduction. Additives. Water. Fire hoses with and without polymer additives. • Up to 80% friction drag reduction, even at low concentration . Turbulent drag reduction. • No significant effect on drag in laminar flows . • Bounded by Maximum Drag Reduction (MDR) asymptote. • Pipeline. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 4.
(5) Polymers and turbulence. Elastic turbulence. MTFC RESEARCH GROUP. Elastic turbulence. Chaotic motion of a polymer solution in micro-channel . (Groisman & Steinberg, 2000). • Existence of elastic turbulence in flows with curved streamlines. Turbulent drag reduction. • Observed at low Reynolds number. • Strong increase in mixing properties. • Blood flow. • Micro-channel flow. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 5.
(6) MTFC. Polymers and turbulence. Early turbulence. RESEARCH GROUP. Elastic turbulence. Water. Early turbulence. Transition. Additives. Turbulent drag reduction. Transition to turbulence around an ogival head with ventilated cavity. (Hoyt, 1977). • Transition to turbulence promoted by polymers. • Biofluids. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 6.
(7) MTFC. Polymers and turbulence. Newtonian. 101. Viscoelastic. Elastic turbulence. Elastic turbulence. Laminar. Early turbulence. 103. Transition. Turbulent. RESEARCH GROUP. 105. Early turbulence. Drag reduction. Turbulent drag reduction. Re Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 7.
(8) Polymers and turbulence. Newtonian. 101. MTFC RESEARCH GROUP. Viscoelastic. Elastic turbulence. Laminar. Elasto-Inertial Turbulence (EIT). 103. Transition. Turbulent. 105. Early turbulence. Drag reduction. • State of small-scale turbulence. • Contributions from both elastic and inertial instabilities. • Observed over a wide range of Reynolds numbers. • Link between different phenomena. Re Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 8.
(9) Polymers and turbulence. Newtonian. 101. Transition. Turbulent. 105. RESEARCH GROUP. Viscoelastic. Elastic turbulence. Laminar. 103. MTFC. Early turbulence. Drag reduction. Key questions. • Is drag reduction. - a viscous and large-scale effect (Lumley). - an elastic and small-scale effect (de Gennes). • What is the nature of EIT?. - Relative contributions of elastic and inertial instabilities?. - Characteristics of MDR?. - Dynamical interactions between flow and polymers? . Re Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 9.
(10) Outline. MTFC RESEARCH GROUP. • Context. • Models and numerical implementation. • Polymer drag reduction. • Elasto-inertial turbulence. • Conclusions and future work. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 10.
(11) MTFC. Polymer dynamics in flows. RESEARCH GROUP. Coil-stretch dynamics. q. q. No flow: coiled. Flow: stretched. Non-Newtonian rheology. • Long and flexible macromolecules (PAM, PEO). • Flow deforms molecules. • Polymers exert stress onto flow. • Viscoelastic behavior (memory effect). • Extended polymers relax over a time λ. • Typically large increase in extensional viscosity. λ-DNA unraveling in extensional flow (Perkins et al., 1997). Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 11.
(12) MTFC. Modeling polymer dynamics. RESEARCH GROUP. Coarse graining. q. Real molecule. q. Bead-rod chain. q. Bead-spring chain. q. Bead-spring dumbbell. • Only dynamics of end-to-end vector q modeled. • Two beads connected by (nonlinear) spring. • Hydrodynamic and Brownian forces concentrated on beads. • Entropic restoring force modeled by spring. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 12.
(13) MTFC. Modeling polymer dynamics. FENE model. RESEARCH GROUP. Evolution of end-to-end vector. q. dq 1 q 1 dW = ∇u ⋅ q − − dt Wi 2(1− q 2 L2 ) Wi dt Drag. Spring. Brownian. Polymer stress. F Drag + FSpring + F Brownian = 0. Parameters. • Wi Weissenberg number . (polymer relaxation time / flow time). • L maximum extensibility of polymer. 1 qq T= −I 2 2 Wi 1− q L. • Stochastic. • Lagrangian. • No closure model . Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 13.
(14) MTFC. Modeling polymer dynamics. FENE-P model. Conformation tensor. RESEARCH GROUP. Evolution of conformation tensor. DC = ∇u ⋅ C + C ⋅ ∇u T − T Dt. C = qq. Stretching. Restoring. Polymer stress. % 1 " C T= − I' $ 2 Wi # 1− trC L &. Parameters. • Wi Weissenberg number . (polymer relaxation time / flow time). • L maximum extensibility of polymer. • Deterministic (continuum) . • Eulerian. • Closure approximation. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 14.
(15) Viscoelastic NSE. MTFC RESEARCH GROUP. Continuity. ∇⋅u = 0. Momentum. ∂u β 2 1− β dP + (u ⋅ ∇)u = −∇p + ∇ u+ ∇⋅T+ ex ∂t Re Re dx. Polymer stress. % 1 " C T= − I' $ 2 Wi # 1− trC L &. Conformation tensor. ∂C + (u ⋅ ∇)C = ∇u ⋅ C + C ⋅ ∇u T − T ∂t. β. Ratio of solvent viscosity to zero-shear viscosity of solution. Ub H Reynolds number. ν λU Wi= b Weissenberg number. h. Re=. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 15.
(16) MTFC. Hyperbolic C equation. Equation for C is hyperbolic. ∂C + (u ⋅ ∇)C = ∇u ⋅ C + C ⋅ ∇u T − T ∂t • No spatial diffusion (Sc ~ • Sub-Kolmogorov scales of polymer stress. • Slow decay of energy spectrum. • Numerical instabilities . 105 –. 106). RESEARCH GROUP. Analogy with passive scalar (Batchelor, 1959). At Sc > 1, scalar energy spectrum decreases as. E ~ k -1. from Kolmogorov scale ηK to Batchelor scale. ηB ~ ηK / Sc1/2. k -1. LAD. k -3. E(Tyy). GAD. 2 Δz kz. Trace of C for different grid resolutions (Dubief, 2002). Spectrum of polymer stress Tyy at y+=15 (Dubief et al., 2005). Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 16.
(17) MTFC. Numerical approach. Space. • Structured grid. • 2nd order FD for velocity. • Non-dissipative 4th order compact scheme for polymer stress. • Compact upwind scheme for advection terms of conformation tensor. Time. • Semi-implicit fractional step. • 2nd order Crank-Nicolson/3rd order Runge-Kutta . • Implicit scheme for trace of C to ensure bounded trace. Artificial dissipation. • Local artificial dissipation (LAD). • Only used when determinant of tensor C becomes negative. RESEARCH GROUP. • Important to rely on accurate numerical method. • Global dissipation (Sceff ~ 1) damps all small scales. • Capturing small polymer scales is critical to represent the correct physics. Min et al. (2001),Vaithianathan & Collins (2003), Dubief et al. (2005), Dallas et al. (2010). Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 17.
(18) MTFC. Configuration. RESEARCH GROUP. Periodic channel flow. y. Ub. z. x. H=2h. • Mean pressure gradient in x. • Periodic in x and z. • Wall (no-slip) at y=±h. Typical parameters. • Reynolds number Reb = UbH/ν = 1’000 – 10’000. • Size:. 10h × 2h × 5h . • Grid:. 256 × 151 × 256. • Polymers:. L = 50 – 200. Wi = 3 – 40 . β = 0.9. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 18.
(19) MTFC. Transition to turbulence. RESEARCH GROUP. Simulated by-pass transition. Forced homogeneous isotropic turbulence. h. H=2h. Ub. • t < 0 :. - Free-slip. • t = 0 : . - No-slip. Simulated roughness-induced transition. Weak initial wall perturbation. (blowing). ' * ! 8π $ ! 8π $ v(x, y = ±h, z, t) = H (t)) Asin # x & sin # z & + ε (t), )( ,+ " L x % " Lz % Designed to trigger transition in Newtonian flow at Re=6000. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. Random noise. 19.
(20) Realism of the DNS. MTFC RESEARCH GROUP. • Hyperbolicity of C transport equation respected as best as numerically possible (Dubief et al., 2005) . • Polymer parameters (for low polymer concentrations). - shear thinning effect small . - extensional viscosity large (increasing with Wi). • Reproduce evolution of friction factor as function of Reynolds numbers observed in pipe flow experiments (Samanta, Dubief, Holzner, Shäfer, Morozov,Wagner & Hof, submitted) . Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 20.
(21) Outline. MTFC RESEARCH GROUP. • Context. • Models and numerical implementation. • Polymer drag reduction. • Elasto-inertial turbulence. • Conclusions and future work. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 21.
(22) Changes in mean velocity. MTFC RESEARCH GROUP. Mean velocity profile. 60. MDR. 50. %DR. 40. U+ 30. 20. Newtonian. 10. 0. 1. Modified log law. • Newtonian. U+ = 0.4 ln(y+) + 5.5. • Parallel shift at low drag reduction (LDR). • Change of slope at high drag reduction (HDR). • From buffer layer to edge of boundary layer. 10. y+. 100. MDR asymptote. • Universal. • Not laminar. • Virk’s log-law U+ = 11.7 ln(y+) - 17. 1000. Experimental measurements (Warholic et al., 1999). Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 22.
(23) Changes in velocity fluctuations. MTFC RESEARCH GROUP. Streamwise velocity fluctuations. 3.5. 3.0. %DR. • First, increase in peak value . 2.5. • Damping of peak value at higher DR. • Maximum value further away from the wall. 2.0. • Thicker buffer layer. u’+. 1.5. • Decrease of maximum streamwise velocity fluctuations not always observed. 1.0. 0.5. 0. 0. 0.2. 0.4. 0.6. 0.8. 1.0. y/H. Experimental measurements (Warholic et al., 1999). Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 23.
(24) Changes in velocity fluctuations. MTFC RESEARCH GROUP. Wall-normal velocity fluctuations. 1.2. 1.0. 0.8. • Strong damping with increasing DR. v’+. 0.6. • Maximum value further away from the wall. %DR. 0.4. 0.2. 0. 0. 0.2. 0.4. 0.6. 0.8. 1.0. y/H. Experimental measurements (Warholic et al., 1999). Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 24.
(25) MTFC. Changes in Reynolds stress. Reynolds stress and polymer stress. RESEARCH GROUP. • Strong damping of Reynolds stress . • Maximum value further away from wall. • Increasing contribution of polymer stress. + dU + y −u'v' + β + + (1− β )Txy+ = 1− + dy h +. %DR. %DR. Simulations (Dubief et al., 2004). Reynolds Viscous Polymer stress . stress. shear stress (“stress deficit”). • Does Reynolds stress vanishes at MDR or is it finite?. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 25.
(26) Changes in coherent structures. 0% DR. MTFC RESEARCH GROUP. 50% DR. 0% DR. Polymer stretch (contours on bottom and side walls) and vortices (isosurfaces of Qa) from DNS. 47% DR. PIV measurements in viscoelastic boundary layer (White et al., 2004) . • Strong damping of quasi-streamwise vortices. • Coarsening and stabilization of streaks. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 26.
(27) MTFC. Mechanism. RESEARCH GROUP. y+. Eα < 0. Polymer – turbulence interactions. • Energy transfer from flow to polymers. - Kinetic to elastic energy. - In up- and downwashes. - Around near-wall vortices. - Stretching caused by bi-axial extensional flow. • Energy transfer from polymers to flow. - Into high-speed streaks. - Very close to the wall. Ex > 0. 5. 0. u’ > 0. z+. Mean shear. y+ ~ 5. Streaks. y+ ≥ 20. Polymers. Ex >. 0. Vortices. Eα <. 0. Dubief et al. (2004), Terrapon et al. (2004). Nonlinear interactions. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 27.
(28) Outline. MTFC RESEARCH GROUP. • Context. • Models and numerical implementation. • Polymer drag reduction. • Elasto-inertial turbulence. - MDR. - Early turbulence. - Transition. • Conclusions and future work. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 28.
(29) MTFC. Transition in Newtonian BL. Reθ = 250-450. Skin friction coefficient (ZPGBL). ×10-3. Reθ=200 Reθ=300. RESEARCH GROUP. Reθ=900. 5. Cf. 4. Reθ = 1050-1250. 3. 2. 0. 200. 400. Reθ. 600. 800. Nonlinear breakdown of instabilities. DNS of a ZPGBL (Wu & Moin, 2009). Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 29.
(30) MTFC. Transition in Newtonian BL. U+. Skin friction coefficient (ZPGBL). ×10-3. Reθ=200 Reθ=300. Mean velocity. Reθ=900. RESEARCH GROUP. Virk. 30. Reθ=300. Reθ=200. 20. Reθ=900. 10. 5. Cf. 0. 1. 4. 10. 100. -<u’v’>+. 0.8. 3. 2. 0. 200. 400. 600. Reθ. 800. Reynolds stress. Virk’s asymptote very similar to velocity profiles around nonlinear breakdown of instabilities. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. Reθ=900. 0.6. 0.4. 0.2. 0. 0. Reθ=300. Reθ=200. 50. 100. y+. 150. Wu & Moin (2009). 30.
(31) MDR and Newtonian transitional flow. Mean velocity. Virk. Viscoel.. Reθ=300. 30. Reθ=200. U+. 20. 0.4. 0.3. Reθ=300. 0.2. -<u’v’>+. 0.1. 10. 1. RESEARCH GROUP. Reynolds stress. 40. 0. MTFC. Reθ=200. 0. 10. y+. 100. • Velocity profile at MDR (Virk) similar to Newtonian transitional flow . • Time fluctuations at MDR span both pre- and post-breakdown velocity profiles. -0.1. 1. Viscoel.. 10. 100. y+. • Average MDR Reynolds stress negligible. • Time fluctuations at MDR larger than prebreakdown Newtonian case. • Correlation with dP/dx. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 31.
(32) MTFC. MDR oscillating behavior. Vortical activity vs. wall-shear. RESEARCH GROUP. Elastic energy vs. vortical activity. ×10-2. 3. 0.8. NQ>0.1. [%]. Ep. 2. 0.6. 1. 0.4. 0. 5. 5.5. 6. Swall(t). 6.5. 7. 0. 1. 2. 3. NQ>0.1 [%]. 4. 5. • Correlation between increasing vortical activity and increasing shear stress. • Elastic energy increases when vortical activity grows. • When vortices are damped, shear stress decreases. • Elastic energy peaks during decaying portion of vortical activity. • Rapid drop when vortical activity vanishes. • Phase lag depends on elasticity. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 32.
(33) MTFC. Is MDR logarithmic?. RESEARCH GROUP. Mean velocity profile. V: h+=300. 40. Virk. V: h+=180. Exp.*. U+. N: h+=590. 20. Log-law. 0. 1. 10. y+. • MDR experiment, BL transitional flow, and MDR simulations straddle Virk’s asymptote. • But is it a logarithmic scaling?. 100. * Escudier et al.(2009). Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 33.
(34) MTFC. Is MDR logarithmic?. RESEARCH GROUP. Indicator function ζ = y+(dU+/dy+). V: h+=300. Reθ=200. 15. • If logarithmic behavior, indicator function should be constant. V: h+=180. Virk. 10. ς. Reθ=300. Exp.*. 5. N: h+=590. Log-law. 0. 1. 10. y+. 100. • No apparent logarithmic behavior . - MDR simulations. - MDR experiment. - BL transitional flow. • Possibly due to low Re but other experiments at higher Re show similar velocity profile. • Need more data at higher Re . White et al.(2012). * Escudier et al.(2009). Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 34.
(35) Our understanding of MDR. MTFC RESEARCH GROUP. • MDR is a transitional state corresponding to the onset of the nonlinear breakdown of instabilities . • MDR flow oscillates between pre- and post-breakdown state. - Polymers disrupt the near-wall autonomous cycle of wall turbulence. - Vortices are damped out (or sufficiently weakened as to “turn-off” their ability to stretch polymers). - Stretching mechanism for polymers is limited to wall-shear and spanwise shear layers between streaks (both modest source of polymer stretching compared to biaxialextensional flow). - Polymers recoil, allowing instabilities to grow and new vortices to form. • Oscillation “amplitude” depends on polymer elasticity. • There is no apparent logarithmic scaling of velocity at MDR or in transitional flows. • There are indications of elastic instabilities at small scales that contribute to the regeneration of vortices following the low vortical activity. White et al. (2012), Xi & Graham (2010), L’vov et al. (2004). Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 35.
(36) Outline. MTFC RESEARCH GROUP. • Context. • Models and numerical implementation. • Polymer drag reduction. • Elasto-inertial turbulence. - MDR. - Early turbulence. - Transition. • Conclusions and future work. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 36.
(37) Transitional viscoelastic flows. MTFC RESEARCH GROUP. Pipe flow experiment with PAAm solution . Friction factor . Turbulent. 10-2. f MDR. 500 ppm unperturbed. 500 ppm perturbed. • Departure from laminar state at Re~800 . • Smooth transition from laminar to MDR state. • Flow dynamics controlled by elastic and inertial instabilities. Laminar. 10-3. 103. ReD. 104. Samanta et al. (2012, submitted). Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 37.
(38) Transitional viscoelastic flows. Channel flow simulations. Friction factor . 10-2. MTFC RESEARCH GROUP. Re = 1000 . • Not laminar. • Elastic contributions. Turbulent. Isosurface of Qa invariant. f Laminar. MDR. Newtonian. Viscoelastic. 10-3. 103. ReH. Re = 6000. • Inertial & elastic contributions. • Turbulent?. • New state?. 104. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 38.
(39) Qualitative flow behavior. MTFC RESEARCH GROUP. Re = 1000 . Wi = 8. Turbulent. 10-2. f Laminar. 10-3. Newtonian. Viscoelastic. 103. Re. MDR. 104. Second invariant of the velocity gradient tensor: Qa = ½ (Ω2-S2). Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 39.
(40) Qualitative flow behavior. MTFC RESEARCH GROUP. Re = 1000 . Wi = 8. Turbulent. 10-2. f Laminar. 10-3. Newtonian. Viscoelastic. 103. Re. MDR. 104. Polymer extension (Cii / L2)1/2. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 40.
(41) Qualitative flow behavior. MTFC RESEARCH GROUP. Re = 6000 . Wi = 8. Turbulent. 10-2. f Laminar. 10-3. Newtonian. Viscoelastic. 103. Re. MDR. 104. Second invariant of the velocity gradient tensor: Qa = ½ (Ω2-S2). Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 41.
(42) Qualitative flow behavior. MTFC RESEARCH GROUP. Re = 6000 . Wi = 8. Turbulent. 10-2. f Laminar. 10-3. Newtonian. Viscoelastic. 103. Re. MDR. 104. Polymer extension (Cii / L2)1/2. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 42.
(43) MTFC. Flow topology. RESEARCH GROUP. Local flow pattern at fluid particle. Qa. Invariants:. Pa = −Aii = 0 Velocity gradient tensor:. Aij = (∇u)ij. 1 Qa = − Aij A ji 2 1 Ra = − Aij A jk Aki 3. Ra. Da = 0. Discriminant:. 27 2 Da = Ra + Qa3 4 Chong et al. (1990), Soria et al. (1994). Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 43.
(44) MTFC. Flow topology. RESEARCH GROUP. EIT flow (JPDF). 0.02. Re=1000. 0.01. Re=6000. Qa 0. • At low Re, symmetric distribution around 2D flow (Ra=0). • At higher Re, “teardrop” shape similar to Newtonian turbulence. -0.01. -0.02. -0.001. -0.0005. 0. 0.0005. 0.001. Ra. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 44.
(45) MTFC. Flow topology. RESEARCH GROUP. Newtonian turbulent flow (JPDF). Velocity gradient tensor:. Aij = (∇u)ij = Sij + Ωij. -QS. Irrotational dissipation. Vortex sheets. 2nd invariant. Qa = QS + QΩ 1 QS = − Sij S ji 2 1 QΩ = − Ωij Ω ji 2. Vortex tubes. QΩ. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 45.
(46) MTFC. Flow topology. RESEARCH GROUP. EIT flow (JPDF). Velocity gradient tensor:. 3. Aij = (∇u)ij = Sij + Ωij. 2. 2nd invariant. Qa = QS + QΩ 1 QS = − Sij S ji 2 1 QΩ = − Ωij Ω ji 2. Re=10000. -QS / <QΩ>. 1. Dominated by sheet topology!. 0 . 0. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 1. -QΩ / <QΩ>. 2. 3. 46.
(47) Observations. MTFC RESEARCH GROUP. • Flow is perfectly laminar in the absence of polymers . • Polymer addition creates a self-sustained chaotic flow consisting of trains of cylindrical weakly rotational regions (positive Qa) and weakly extensional regions (negative Qa) . • There is a hierarchy of cylindrical structures, the smallest one being of the order of the Kolmogorov scale . • The polymer extension field is organized in sheets . • Polymers cause the flow to evolve from pure shear flow to mix extensionalshear flow . • The cylindrical Qa structures are attached to sheets of large polymer extension. • Elasto-inertial turbulence results from the combination of the hyperbolic transport equation of C and the elliptic equation of p . • Pressure redistributes energy with trains of cylindrical structures to attenuate the anisotropy caused by sheets of extensional viscosity . Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 47.
(48) MTFC. Proposed mechanism. RESEARCH GROUP. Channel flow. Polymer extension. Turbulent. 10-2. f. • Formation of sheets. Laminar. 10-3. 103. • Excitation of extensional sheet flow. • Elliptical pressure redistribution of energy. MDR. Newtonian. Viscoelastic. 104. Re. • Increase of extensional viscosity (anisotropic). Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 48.
(49) Outline. MTFC RESEARCH GROUP. • Context. • Models and numerical implementation. • Polymer drag reduction. • Elasto-inertial turbulence. - MDR. - Early turbulence. - Transition. • Conclusions and future work. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 49.
(50) MTFC. By-pass transition. RESEARCH GROUP. Re = 10000 L = 200. Time evolution of drag. 4 ×10-3. Newtonian. u’/Ub=0.015. u’/Ub=0.025. 2. u’/Ub=0.020. • Viscoelastic flow transitions with weaker perturbations. 3. dP/dx. Wi = 40. β = 0.9. • Same perturbation level would lead to laminar Newtonian flow. Viscoelastic. 1. • Viscoelastic flow transitions to MDR and stays at MDR. u’/Ub=0.006. 0. 500. 1000. tUb/h. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 50.
(51) MTFC. By-pass transition. RESEARCH GROUP. Time evolution of drag. 4 ×10-3. Newtonian. dP/dx. 3. 2. Viscoelastic. 1. 0. 500. 1000. tUb/h. 2nd invariant Qa of the velocity gradient tensor. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 51.
(52) MTFC. By-pass transition. RESEARCH GROUP. Time evolution of drag. 4 ×10-3. Newtonian. dP/dx. 3. 2. Viscoelastic. 1. 0. 500. 1000. tUb/h. 2nd invariant Qa of the velocity gradient tensor. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 52.
(53) MTFC. By-pass transition. RESEARCH GROUP. Time evolution of drag. 4 ×10-3. Newtonian. dP/dx. 3. 2. Viscoelastic. 1. 0. 500. 1000. tUb/h. 2nd invariant Qa of the velocity gradient tensor. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 53.
(54) MTFC. Spatial statistics. RESEARCH GROUP. Mean velocity profile. Time evolution of drag. 4 ×10-3. 1.5 Newtonian. 1. U. U. dP/dx. 3. 2. Viscoelastic. 0.5. 1. 0. 500. tUb/h. 1000. 0 -1. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. -0.5. 0. 0.5. y y. 54. 1.
(55) MTFC. Spatial statistics . RESEARCH GROUP. Mean polymer extension. 1. Time evolution of drag. 4 ×10-3. Newtonian. 0.75. CCiiii/L / L2 2. dP/dx. 3. 2. 0.5. Viscoelastic. 0.25. 1. 0. 500. tUb/h. 1000. 0 -1. -0.5. 0. 0.5. yy. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 55. 1.
(56) Excitation of instabilities. MTFC RESEARCH GROUP. Power spectrum of the elastic energy before nonlinear breakdown of instabilities. y+. Isotropic turbulence. No diffusion yet. • Instabilities in near-wall regions not caused by diffusion of turbulence from channel center. • What triggers the instability?. Elastic instabilities. log10 kx. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 56.
(57) Poisson equation for pressure. Extended Poisson equation. MTFC RESEARCH GROUP. Green function G(0, -0.9H, 0; x, y, z). f(x). + B.C.. Pressure kernel – Green function G. y. x. z. “Influence” function F(x, ξ) represents the contribution of point x to the pressure at point ξ. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 57.
(58) MTFC. Influence function. Before nonlinear breakdown of instabilities. RESEARCH GROUP. After nonlinear breakdown of instabilities. F(ξ,x). F(ξ,x). Point P=(0,-0.9h,0). Source of contribution to pressure at point P from relative “unorganized” free-stream turbulence in channel center. Source of contribution to pressure at point P from elastically induced more “organized” structures in near-wall region. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 58.
(59) Plane-averaged pressure fluctuations. MTFC RESEARCH GROUP. h. p 2 (η ) =. ∫ h(y;η )dy. −h. • h(y;η) represents the influence of plane y on the pressure fluctuations averaged over plane η. • Three contributions. - Qa alone. - T alone. - the interaction of Qa and T. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 59.
(60) Plane-averaged pressure fluctuations. MTFC RESEARCH GROUP. dP/dx. h 2. p (η ) =. ∫ h(y;η )dy. −h tUb/h. 1. 10-7. h(Q). h(T). h(Q,T). Total. h(z). 0.5. h(y;η) 0. Before nonlinear breakdown of instabilities. • Major contribution from . - free-stream turbulence at center of the channel. - polymer stress. -0.5. η=-0.9h. -1 -1. -0.5. 0 z. 0.5. 1. y. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 60.
(61) Plane-averaged pressure fluctuations. MTFC RESEARCH GROUP. dP/dx. h 2. p (η ) =. ∫ h(y;η )dy. −h tUb/h. 20. 10. -7. h(Q). h(T). h(Q,T). Total. 15. h(z). 10. h(y;η) 5. After nonlinear breakdown of instabilities. • Major contribution from . - near-wall region. - Q. • Contribution from free-stream turbulence negligible. 0 -5. η=-0.9h. -10 -1. -0.5. 0 z. 0.5. 1. y. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 61.
(62) Plane-averaged pressure fluctuations. MTFC RESEARCH GROUP. dP/dx. h 2. p (η ) =. ∫ h(y;η )dy. −h tUb/h. 3. 10. -4. h(Q). h(T). h(Q,T). Total. 2. h(z). 1. h(y;η). 0. • Major contribution from . - near-wall region. - the interaction between Q and T. • Free-stream turbulence fully decayed. -1. • Activity mostly at lower wall at this specific instant. η=-0.9h. -2 -1. Much after nonlinear breakdown of instabilities. -0.5. 0 z. 0.5. 1. y. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 62.
(63) dP/dx. Budget of turbulent kinetic energy. 4. Dissipation. Production of elastic energy . -5. 10. D. Before nonlinear breakdown of instabilities. 2. PT. -2. -4 -1. • Dissipation dominates. • Larger production of k from polymers. • Free-stream turbulence decays. PS. Budget k 0. RESEARCH GROUP. Transport. Production from mean shear. ∂k = T k +P S −PT −D ∂t. tUb/h. MTFC. -0.5. 0. y y. 0.5. 1. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 63.
(64) dP/dx. Budget of turbulent kinetic energy. 20. Dissipation. Production of elastic energy . -5. 10. D. 0. After nonlinear breakdown of instabilities. PS. • Production of k from polymers dominates. • Larger values at upper wall. Budget k. -20. -40 -1. RESEARCH GROUP. Transport. Production from mean shear. ∂k = T k +P S −PT −D ∂t. tUb/h. MTFC. PT. -0.5. 0. y y. 0.5. 1. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 64.
(65) dP/dx. Budget of turbulent kinetic energy. 40. Dissipation. Production of elastic energy . -5. 10. Much after nonlinear breakdown of instabilities. 20. D. Budget k 0. • Production of k from polymers dominates. • Larger values at lower wall. PS. PT. -20. -40 -1. RESEARCH GROUP. Transport. Production from mean shear. ∂k = T k +P S −PT −D ∂t. tUb/h. MTFC. -0.5. 0. y y. 0.5. 1. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 65.
(66) Outline. MTFC RESEARCH GROUP. • Context. • Models and numerical implementation. • Polymer drag reduction. • Elasto-inertial turbulence. • Conclusions and future work. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 66.
(67) Conclusion. MTFC RESEARCH GROUP. Viscoelasticity leads to new phenomena. • Common to many fluids (e.g., blood). • Viscoelasticity can dramatically reduce drag at higher Reynolds number. • Viscoelasticity can promote departure from laminar state at lower Reynolds number. • Elasto-Inertial Turbulence (EIT) identified as new regime explaining this seemingly contradictory behavior. Drag reduction. • Up to 80% drag reduction can be achieved with dilute polymer or surfactant solutions. • Maximum drag reduction achievable bounded by MDR asymptote. • Polymers stretched in bi-axial flow regions around near-wall vortices. • Polymers damp quasi-streamwise vortices in near-wall region. • Polymers can store kinetic energy from the flow as elastic energy. • Polymers release elastic energy to the flow in high-speed streaks. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 67.
(68) Conclusion. MTFC RESEARCH GROUP. Maximum drag reduction. • MDR is transitional state corresponding to onset of nonlinear breakdown of instabilities . • MDR flow oscillates between pre- and post-breakdown state. • No apparent logarithmic scaling of velocity at MDR or in transitional flows. • Elastic instabilities at small scales contribute to regeneration of vortices. Elasto-inertial turbulence. • EIT identified as new regime describing both MDR and early turbulence. • Hyperbolic transport equation leads to creation of thin sheets of high polymer extension and large extensional viscosity. • Self-sustained chaotic flow consisting of trains of cylindrical weakly rotational and extensional regions. • Pressure redistributes energy with trains of cylindrical structures to attenuate the anisotropy caused by sheets of extensional viscosity . • Transfer of elastic energy from polymers into turbulent kinetic energy of the flow. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 68.
(69) Conclusion. MTFC RESEARCH GROUP. Transition. • Long-range excitation of elastic instabilities through pressure. • Feeding of energy to elastic instabilities from mean flow. • Transfer of elastic energy from the polymers into turbulent kinetic energy of the flow. • Elastic instabilities self-sustained. Numerics. • Accurate numerical approach is required. • Small-scales must be captured to simulate EIT. • Use of global artificial dissipation prevents formation of elastic instabilities. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 69.
(70) Conclusion. MTFC RESEARCH GROUP. Overall. • Viscoelastic turbulent flows are a very exciting area. • EIT explains seemingly contradictory phenomena in viscoelastic turbulence. • EIT provides support to de Gennes’ theory. • Understanding EIT is relevant for biofluids, but also to better understand and control Newtonian turbulence. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 70.
(71) Future work. MTFC RESEARCH GROUP. • Further characterize EIT. • Understand the exact mechanisms during transition process. • Develop new control strategies for turbulent flows. • Identify potential role of EIT in biofluid flows. Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 71.
(72) Future work – DMD analysis. MTFC RESEARCH GROUP. Most amplified mode from DMD analysis . Qa invariant (streamwise – wall-normal plane) at Re=1000. y/h. 0.15-0.2. • Mostly two-dimensional structures. • Located in near-wall region. • Alternating pressure minima and maxima. • “Discontinuity” close to wall corresponding to location of maximum of mean polymer extension (critical layer?). Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 72.
(73) MTFC. Future work – DMD analysis. RESEARCH GROUP. Mode amplitude as a function of frequency from DMD analysis . Qa invariant (streamwise – wall-normal plane). 101. Re = 1000. Re = 6000. Inertial ?. A. Elastic ?. 100. -5. 0. fr. 5. -5. 0. 5. fr. • Shape change with increasing Re • At larger Re, two apparent contributions. • Hypothesis: elastic and inertial (to be verified). Von Karman Institute (VKI) for Fluid Dynamics - 7 December 2012 - V. Terrapon, Y. Dubief & J. Soria. 73.
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