CVS filtering to study turbulent mixing

Transcription

1 CVS filtering to study turbulent mixing Marie Farge, LMD-CNRS, ENS, Paris Kai Schneider, CMI, Université de Provence, Marseille Carsten Beta, LMD-CNRS, ENS, Paris Jori Ruppert-Felsot, LMD-CNRS, ENS, Paris Inertial-Range Dynamics and Mixing Newton Institute for Mathematical Sciences, 30 th September 2008

2 Turbulent fluctuations At the 5th International Congress of Applied Mechanics in 1938 Tollmien and Prandtl suggested that turbulent fluctuations might consist of two components, a diffusive and a non-diffusive component [ ] Considerable masses of fluid move as more or less coherent units. The process cannot be smoothed by averaging over a small volume because it is not possible to choose dimensions small compared with a single element [ ]. Shall the flow be regarded as a mean flow that merely transports and distorts large eddies superposed on the flow, these eddies being of varying size and intensity. [ ] The ideas of Tollmien and Prandtl that measured fluctuations include both random and non random elements are correct, but as yet there is no known procedure, either experimental or theoretical, for separating them. Hugh Dryden, Adv. Appl. Mech. 1, 1948

3 CVS filtering In the spirit of Hugh Dryden s remark, we have proposed CVS filtering, based on the wavelet representation, to split each flow realization into two orthogonal components: coherent fluctuations incoherent fluctuations We will show that coherent fluctuations are non diffusive and drive the nonlinear dynamics, while incoherent fluctuations correspond to turbulent dissipation and have a purely diffusive effect.

4 2D turbulent flow in a cylindrical container Numerical experiment DNS N=10242 Random initial conditions No-slip boundary conditions using volume penalization Turbulent fluctuations with mean=0 Schneider & Farge Phys. Rev. Lett., December 2005

5 2D turbulent flow in a cylindrical container Numerical experiment Palinstrophy Enstrophy Enstrophy production at the wall Palinstrophy corresponds to dissipation Kai Schneider s talk on Thursday, 4 p.m. Energy Schneider & Farge Phys. Rev. Lett., December 2005

6 Introduction Extraction of coherent fluctuations using wavelets Application to PIV measures from rotating tank experiment Application to DNS results from homogeneous isotropic turbulence Conclusion

7 Orthogonal wavelet representation Orthogonal wavelet basis of N=512 = 2 9 functions ij small scales large scales Dyadic grid space

8 How to define coherent structures? Since there is not yet a universal definition of coherent structures observed in turbulent flows (from laboratory and numerical experiments), we adopt an apophetic method : instead of defining what they are, we define what they are not. We propose the minimal statement: Coherent structures are different from noise " Extracting coherent structures becomes a denoising problem, not requiring any hypotheses on the coherent structures but only on the noise to be eliminated. Choosing the simplest hypothesis as a first guess, we suppose we want to eliminate an additive Gaussian white noise, and use nonlinear wavelet filtering for this. Farge, Schneider and Kevlahan Phys. Fluids, 11 (8), 1999

9 Wavelet-based denoising 1. Goal: Extraction of coherent structures from a background noise. 2. Apophatic principle: - no hypothesis on the structures, - only hypothesis on the noise, - simplest hypothesis as our first choice. 3. Hypothesis on the noise: f B = f + n n σ 2 N : Gaussian white noise, : variance of the noise, : number of coefficients. f f B 4. Computation of the threshold: " D = 2# 2 ln(n) 5. Denoised signal: % f D = f ~ "# " ": f ~ " <$ Azzalini, Farge and Schneider ACHA, 18 (2), 2005 f D

10 Denoising Bose-Einstein Condensate Measured signal PDF Denoised signal Noise

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12 Introduction Extraction of coherent fluctuations using wavelets Application to PIV measures from rotating tank experiment Application to DNS results from homogeneous isotropic turbulence Conclusion

13 Rotating tank experiment with particle image velocimetry (PIV) Center for Nonlinear Dynamics, University of Texas, Austin ω min ω max Vorticity measured by PIV

14 CVS filtering of the flow evolution Total flow Coherent flow Incoherent flow

15 PIV N= D vortex extraction using CVS filter in a rotating tank experiment with PIV (Austin) 2% N 98% N Total vorticity 100% E 100% Z Coherent vorticity 99% E 80% Z ω min Incoherent vorticity 1% E 3% Z ω max

16 PDF of vorticity in laboratory experiment log p(ω) PIV N=128 2 Total Coherent Incoherent ω

17 Enstrophy spectrum in laboratory experiment log Z PIV N=128 2 Incoherent k +1 scaling, i.e. enstrophy equipartition Total Coherent k -3 scaling, i.e. long-range correlation log k

18 A posteriori proof of coherence in laboratory experiment ω PIV N=128 2 Total Coherent structures are regions where nonlinearity is depleted, thus, for 2D flows: Arnold, 1965, Joyce & Montgomery, 1973 Robert & Sommeria, 1991 ω Coherent ω ω = f(ψ) Incoherent ψ ψ ψ

19 Passive scalar advection in laboratory experiment PIV N=128 2 Transport by the coherent vortices : Diffusion by the incoherent background :

20 Lagrangian particle advection in laboratory experiment PIV N=128 2 Brownian-like Motion due to the incoherent background : Transport by the coherent vortices

21 Introduction Extraction of coherent fluctuations using wavelets Application to PIV measures from rotating tank experiment Application to DNS results from homogeneous isotropic turbulence Conclusion

22 Transport of passive tracer by turbulence DNS N=512 2 Beta, Schneider & Farge, Chem. Eng. Sci., 58, 2003 Vorticity field Tracer concentration

23 PDF of vorticity in numerical experiment DNS N=512 2 Coherent Non Gaussian Incoherent Gaussian P(ω) ω

24 Energy spectrum in laboratory experiment DNS N=512 2 E(k) Coherent k -5 scaling, i.e. long-range correlation Incoherent k -1 scaling, i.e. enstrophy equipartition k

25 A posteriori proof of coherence in numerical experiment DNS N=512 2 Coherent structures are regions where nonlinearity is depleted, thus, for 2D flows: ω = sinh(ψ) Arnold, 1965, Joyce & Montgomery, 1973 Robert & Sommeria, 1991 Total Coherent ω ω ω Incoherent ψ ψ ψ

26 Passive scalar advection in numerical experiment DNS N=10242 Beta,Schneider, Farge 2003, Chemical Eng. Sci., %N 99.8%E 93.6%Z 99.8%N 0.2%E 6.4%Z = + Beta,Schneider, Farge 2003, Nonlinear Sci. Num. Simul., 8 Total flow Coherent flow Incoherent flow

27 Evolution of the concentration variance DNS N= t 3/2 Coherent flow σ 2 (0) σ 2 (t) anomalous diffusion due to transport by vortices t 1/2 Incoherent flow classical diffusion t

28 Lagrangian particle advection in laboratory experiment DNS N= % of coefficients 99.8 % of kinetic energy 93.6 % of enstrophy 99.8 % of coefficients 0.2 % of kinetic energy 6.4 % of enstrophy by the total flow by the coherent flow by the incoherent flow = + Transport by vortices Beta,Schneider, Farge 2003, Nonlinear Sci. Num. Simul., 8 Diffusion by Brownian motion

29 Modulus of the 3D vorticity field DNS N= ω =5σ with σ=(2ζ) 1/2 Coherent vorticity 2.6 % N coefficients 80% enstrophy 99% energy Incoherent vorticity 97.4 % N coefficients 20 % enstrophy 1% energy ω =5σ Total vorticity R λ =732 N= Visualization at Naoya Okamoto et al., Phys. Fluids, 19(11), 2007 ω =5/3σ

30 Energy spectrum DNS N= k -5/3 log E(k) k % N coefficients 80% enstrophy 99% energy Multiscale Coherent k -5/3 scaling, i.e. long-range correlation Multiscale Incoherent k +2 scaling, i.e. energy equipartition log k

31 Nonlinear transfers and energy fluxes ccc coherent flux = total flux L Inertial range iic, iii incoherent flux = 0 η cci ttt icc, iic

32 New interpretation of the energy cascade Physical space viewpoint No vortex fission a la Richardson Vortex stretching and bursting

33 New interpretation of the energy cascade Fourier space viewpoint No spectral gap between production and dissipation

34 New interpretation of the energy cascade Wavelet space viewpoint Small scales Linear dissipation <η> Interface η Large scales Nonlinear interactions

35 Introduction Extraction of coherent fluctuations using wavelets Application to PIV measures from rotating tank experiment Application to DNS results from homogeneous isotropic turbulence Conclusion

36 Conclusion The description of turbulent flows in terms of mean value plus random fluctuations seems eroded since there is no scale separation in the fully-developed turbulent regime. We have developed a wavelet-based filter to separate the coherent fluctuations from the incoherent ones. The algorithm works in any time and space dimension and is fast, requiring only order N operations, N being the resolution. We split each flow realization into two orthogonal components which are both multiscale but exhibit different statistics: - non-random fluctuations, which are non-gaussian, long-range correlated and correspond to coherent vortices, - random fluctuations, which are Gaussian, decorrelated, and correspond to incoherent background flow.

37 Conclusion We have shown that the nonlinear wavelet filter disentangles two different dynamics : a nonlinear dynamics, which corresponds to the transport by coherent vortices which are responsible for mixing, a linear dynamics, which corresponds to the turbulent dissipation. Therefore discarding the incoherent flow may be sufficient to model turbulent dissipation Coherent Vortex Simulation (CVS). To download papers and information about the wavelet course: //wavelets.ens.fr

38 Interpretation of the turbulent cascade 'The terms "scale of motion" or "eddy of size l " appear repeatedly in the treatments of the inertial range. One gets an impression of little, randomly distributed whirls in the fluid, with the cascade process consisting of the fission of the whirls into smaller ones, after the fashion of Richardson's poem. This picture seems drastically in conflict with what can be inferred about the qualitative structures of high-reynolds number turbulence from laboratory vizualization techniques and from the application of the Kelvin's circulation theorem'. Robert Kraichnan, 1974 On Kolmogorov s inertial range theories, J. Fluid Mech., 62, We should find a new interpretation of the turbulent cascade, taking into account the nonlinear dynamics of Navier-Stokes equations and the formation of coherent vortices in regions of strong shear. In memory of Robert Kraichnan and Kunio Kuwahara