Wavelet-based methods to analyse, compress and compute turbulent flows

Transcription

1 Wavelet-based methods to analyse, compress and compute turbulent flows Marie Farge Laboratoire de Météorologie Dynamique Ecole Normale Supérieure Paris Mathematics of Planet Earth Imperial College, London 4 th November 2015

2 Outline 1. Choice of an adequate representation 2. The wavelet transform and its multiscale representation Continuous wavelet transform Orthogonal wavelet transform Wavelet-based filtering and denoising 3. Applications to 2D and 3D turbulent flows What is turbulence? Extraction of coherent structures New interpretation of the turbulent cascade Wavelet-based numerical simulation

3 Choice of an adequate representation 'A representation is a formal system for making explicit certain entities or types of information, together with a specification of how the system does this. For example, the Arabic, Roman and binary numerical systems are formal systems for representing numbers. [ ] A representation therefore is not a foreign idea at all, we all use representations all the time. However, the notion that we can capture some aspects of reality by making a description of it using a symbol, and that to do so can be useful, seems to me a fascinating and powerful idea... This issue is important, because how information is presented can greatly affect how easy it is to do different things with it. This is evident even from our number example : it is easy to add, to subtract and even to multiply if the Arabic or binary representation are used, but it is not at all easy to do these things, especially multiplication, with Roman numerals. This is a key reason why the Roman culture failed to develop mathematics in the way the Arabic culture had. David Marr! Vision! Freeman,1982!

4 An adequate representation for music Guido d Arezzo Micrologos half tones: 7 tones: ut re mi fa sol la si sa re ga ma pa da ni

5 Integral transforms Analysis : Synthesis : Phase-space (x,k) Discrete grid-point representation : A Discrete Fourier representation : A Uncertainty principle :

6 Continuous Fourier transform (1807) Function to analyze : Fourier coefficients : Reconstructed function : Analysis Synthesis Plancherel s identity Fourier modes as analyzing functions: Energy conservation

7 Fourier uncertainty principle Information plane (space x, wavenumber k) Uncertainty

8 Optimal tiling of phase-space Windowed Fourier representation (1946) : Wavelet representation (1984) : Space-wavenumber representation information atom Space-scale representation

9 The Wavelet Transform and its Multiscale Representation

10 Choice of a mother wavelet The mother wavelet should verify an admissibility condition : Jean Morlet By translating and dilating it, one generates a Alex Grossmann family of analyzing wavelets : Grossmann and Morlet, Decomposition of Hardy functions into square integrable wavelets of constant shape, SIAM J. math. Anal., 15(4), , 1984

11 Generation of the family of wavelets Δx In physical space : In spectral space : j = 7 j = 6 j = 5 j = 4 ψ ji j = 3 i Δk ˆ ψ ji Farge, Wavelet transforms and their applications to turbulence Ann. Rev. Fluid Mech., 92, 1992 Farge and Schneider, Wavelets: application to turbulence, Encyclopedia of Mathematical Physics, Springer, , 2006

12 Continuous wavelet transform (1984) Wavelet coefficients : Analysis Wavelet modes : Reconstructed function : Synthesis Plancherel s identity if Energy conservation Haar measure :

13 2D continuous wavelet transform 2D Morlet mother wavelet The wavelet family is generated by translating, dilating and rotating 2D mother wavelet

14 Analyzing wavelet Field to analyze Modulus of the wavelet coefficients Small scale Large scale Logarithm

15 2D real-valued Marr wavelet Bandpass filter with Δk/k constant

16 2D complex-valued Morlet wavelet The CWT acts as a local filter in k-space and as a polarizor

17 Discrete multiscale wavelet representation We can then select a finite number of wavelets restricted to a discrete grid optimally chosen, such that the wavelet family associated to this grid constitutes a quasi-orthogonal basis a wavelet frame For example for Marr wavelet we need : a 0 =2 1/2 b 0 =1/2

18 Orthogonal wavelet transform Wavelet analysis : with Wavelet synthesis : A signal sampled on N points is wavelet analyzed and synthetized in CN operations if one uses compactly-supported wavelets computed from a quadrature mirror filter of length C.

19 Orthogonal wavelet representation Wavelets ψ ji ~ f Wavelet coefficients ji = ψ ji f scale j N = 512 = 2 9 position i The dyadic grid of the orthogonal wavelet space Mallat, 1998 A wavelet tour of signal processing, 3rd edition, Academic Press

20 2D orthogonal wavelets scaling function low-pass filter The father : The mother : wavelet, band-pass filter Coarse approximation Horizontal details Vertical details Diagonal details

21 2D orthogonal wavelet representation Image sampled on N=512 2 =(2 9 ) 2 pixels N=512 2 wavelet coefficients

22 Wavelet-based Filtering and Denoising

23 Linear compression Image reconstructed up to scale =0.1% N wavelet coefficients

24 Linear compression Image reconstructed up to scale =0.4% N wavelet coefficients

25 Linear compression Image reconstructed up to scale =1.6% N wavelet coefficients

26 Linear compression Image reconstructed up to scale =6.2% N wavelet coefficients

27 Linear compression Image reconstructed up to scale =25% N wavelet coefficients

28 Linear compression Image reconstructed up to scale =100% N wavelet coefficients

29 Local linear compression Image locally reconstructed up to scale 2 5

30 Local linear compression Image reconstructed up to scale 2 6

31 Local linear compression Image reconstructed up to scale 2 7

32 Local linear compression Image reconstructed up to scale 2 8

33 Local linear compression Image loccaly reconstructed up to scale 2 9

34 Nonlinear compression Image sampled on N=(2 9 ) 2 pixels N=512 2 wavelet coefficients

35 Nonlinear compression Image reconstructed from 3.3% N 3.3% N wavelet coefficients

36 Nonlinear compression Image reconstructed from 10% N 10% N wavelet coefficients

37 Linear compression / Nonlinear compression 2% N wavelet coefficients

38 Wavelet-based denoising Gaussian white noise is by definition equidistributed among all modes and the amplitude of the coefficients is given by its r.m.s. whatever the functional basis one considers to represent it. Therefore the coefficients of a noisy signal whose amplitudes are larger than the r.m.s. of the noise belong to the denoised signal. This procedure corresponds to wavelet-based denoising. The advantage of such a nonlinear filtering using the wavelet representation is that the wavelet coefficients preserve the space-scale locality, since wavelets are functions localized in both physical and spectral space. Since we do not know a priori the r.m.s. of the noise, we have proposed an iterative procedure which takes as first guess the r.m.s. of the noisy signal. Donoho, Johnstone, Biometrika, 81, 1994 Azzalini, M. F., Schneider, 2005 Appl. Comput. Harmonic Analysis, 18 (2)

39 Wavelet-based denoising algorithm Apophatic method : - no hypothesis on the structures, - only hypothesis on the noise, - simplest hypothesis as our first choice. f Hypothesis on the noise : f n = f d + n n Gaussian white noise, <f n2 > variance of the noisy signal, N number of coefficients of f n. Wavelet decomposition : f ji =< f ψ ji > Estimation of the threshold : ε n = 2 < f n 2 > ln(n) Wavelet reconstruction : f d = ji: f ~ ji >ε n f ~ ji ψ ji j scale, i position Donoho, Johnstone, Biometrika, 81, 1994 Azzalini, M. F., Schneider, ACHA, 18 (2), 2005 f n f d

40 Application to turbulent flows

41 What is turbulence? Turbulence is a state that fluid, gas or plasma flows reach when they become unstable and highly fluctuating. Etymology of the word turbulence : turba-ae, crowd, mob turbo-inis, vortex A mob of vortices interacting together on a wide range of temporal and spatial scales. Incompressible flows are governed by 3D Navier-Stokes equations The turbulent regime corresponds to the limit where the nonlinear terms dominate the linear terms. The flow is then highly unstable, chaotic and mixing.

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44 DNS of 2D homogeneous turbulence DNS N=512 2 Farge, 1987, Advances In turbulence Negative vorticity No vorticity Positive vorticity

45 DNS N=512 2 DNS of 2D homogeneous isotropic turbulence

46 Extraction of coherent structures Since there is not yet a universal definition of coherent structures which emerge out of turbulent fluctuations due to the nonlinear interactions, we adopt an apophatic method : instead of defining what they are, we define what they are not. For this we propose the minimal statement : Coherent structures are not noise Extracting coherent structures becomes a denoising problem, not requiring any hypotheses on the structures themselves but only on the noise to be eliminated. Choosing the simplest hypothesis as a first guess, i.e., Occam s razor principle, we suppose we want to eliminate an additive Gaussian white noise, and for this we use a nonlinear wavelet-based filtering. Farge, Schneider et al., Phys. Fluids, 15 (10), 2003 Azzalini, Farge, Schneider, ACHA, 18 (2), 2005

47 Extraction of coherent structures in 2D flows DNS N= % of coefficients 99.8 % of kinetic energy 93.6 % of enstrophy Coherent flow Total flow 99.8 % of coefficients 0.2 % of kinetic energy 6.4 % of enstrophy Incoherent flow + +ω min +ω max

48 1D cut of the vorticity field DNS N=512 2 Total Coherent 0.2 % N 99.8 % E 93.6 % Z ω t =ω c +ω i Z t =Z c +Z i Incoherent 99.8 % N 0.2 % E 6.4 % Z

49 PDF of vorticity DNS N=512 2 Total log p(ω) Coherent 0.2 % N 99.8 % E 93.6 % Z Incoherent 99.8 % N 0.2 % E 6.4 % Z ω min ω max

50 Enstrophy spectrum DNS N=512 2 log Z(k) Coherent 0.2 % N 99.8 % E 93.6 % Z Incoherent k -1 scaling, i.e. enstrophy equipartition since E=k -2 Z Coherent Total k -5 scaling, i.e. long-range correlation Incoherent 99.8 % N 0.2 % E 6.4 % Z log k

51 A posteriori proof of coherence DNS N=512 2 Coherent structures are locally (in space and time) steady solutions of Euler equation, thus, for 2D flows : Arnold, 1965, Joyce & Montgomery, 1973 Robert & Sommeria, 1991 ω = sinh(ψ) Total Coherent ω ω ω Incoherent ψ ψ

52 DNS of 2D homogeneous isotropic turbulence

53 DNS of 2D homogeneous isotropic turbulence

54 2D turbulent flow observed in a rotating tank Rotate up to 1.0 Hz Mechanical pumping of fluid through hexagonal array of sources and sinks 100 mm seed particles PIV to measure velocity fields and calculate vorticity fields

55 Extraction of coherent structures in 2D flows PIV N= % N 98% N Total vorticity 100% E 100% Z Coherent vorticity 99% E 80% Z -ω min -ω max Incoherent vorticity 1% E 3% Z

56 Passive scalar advection DNS N= % N 98% N Total vorticity 100% E 100% Z Coherent vorticity 99% E 80% Z -ω min -ω max Incoherent vorticity 1% E 3% Z

57 Advection of tracer particles 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

58 3D fully-developed homogeneous turbulent flow Dissipation rate α Kaneda et al., 2003 Phys. Fluids, 12, Transition Fully-developed turbulence R λ 1200 When the fully-developed turbulent regime is reached the dissipation rate becomes independent of the Reynolds number

59 Resolution N= Modulus of the vorticity field ( ) 2π L, integral scale Computed in 2002 on ES1 14 Tflops 10 Tbytes Kaneda, et al., 2003, Phys. Fluids, 12, L

60 Zoom (sub-cube ) Resolution N= L, Integral scale L λ Kaneda et al., 2003, Phys. Fluids, 12, λ, Taylor microscale

61 Zoom (sub-cube ) Resolution N= L L, integral scale λ, Taylor macroscale Kaneda et al., 2003, Phys. Fluids, 12, λ

62 Zoom (sub-cube ) Resolution N= λ, Taylor macroscale η, Kolmogorov dissipative scale λ η Kaneda et al., 2003, Phys. Fluids, 12, 21-24

63 Zoom (sub-cube ) DNS N= Kaneda et al., 2003, Phys. Fluids, 12, 21-24

64 Zoom (sub-cube 64 3 ) DNS N= Kaneda, et al., 2003, Phys. Fluids, 12, 21-24

65 Extraction of coherent structures 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 Total vorticity + ω =5σ Okamoto, Yoshimatsu, Schneider, Farge, Kaneda, 2007, Phys. Fluids, 19, 1159 ω =5/3σ

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68 Energy spectrum DNS N= k -5/3 log E(k) k +2 Okamoto, Yoshimatsu, Farge, Schneider, Kaneda, 2007 Phys. Fluids, 19(11) 2.6 % 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

69 PDF of velocity DNS N= log p(v) Okamoto, Yoshimatsu, Farge, Schneider, Kaneda, 2007 Phys. Fluids, 19(11) 2.6 % N coefficients 99% energy v The total and coherent flows have the same extrema. The incoherent flow has a Gaussian PDF, therefore its effect should be easy to model

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

71 Turbulence modeling is the art of averaging Reynolds averaging (1883) : Field = Mean + Fluctuations with but nonlinearity is hard to handle since there is no scale separation : New way of averaging (1992): Fluctuations = coherent fluctuations + incoherent fluctuations = intermittent fluctuations + non-intermittent fluctuations

72 Decomposition of turbulent flows In 1938 Tollmien and Prandtl suggested that turbulent fluctuations might consist of two components, a diffusive and a non-diffusive. Their ideas that fluctuations include both random and non random elements are correct, but as yet there is no known procedure for separating them. Hugh Dryden, Adv. Appl. Mech., 1, 1948 mean + turbulent fluctuations = mean + non random + random = mean + coherent structures + incoherent noise Coherent Vorticity Extraction (CVE) turbulent dynamics = chaotic non diffusive + stochastic diffusive = inviscid nonlinear dynamics + turbulent dissipation Coherent Vorticity Simulation (CVS) Farge, Ann. Rev. Fluid Mech., 24,1992 Farge, Schneider, Kevlahan, Phys. Fluids, 11 (8), 1999 Farge, Pellegrino, Schneider Phys. Rev. Lett., 87 (5), 2001

73 New interpretation of the turbulence cascade Fourier space viewpoint No spectral gap between production and dissipation

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

75 New interpretation of turbulence cascade Wavelet space viewpoint Small scales Linear dissipation <η> Interface η Large scales Nonlinear interactions

76 Wavelet-based Numerical Simulation

77 Coherent Vorticity Simulation (CVS) 1. Selection of the wavelet coefficients whose modulus is larger than the threshold. 2. Construction of a graded-tree which defines the interface between the coherent and incoherent wavelet coefficients. 3. Addition of a security zone which corresponds to dealiasing. Schneider & Farge, 2000, Comp. Rend. Acad. Sci. Paris, 328 Schneider & Farge, 2002, Appl. Comput. Harmonic Anal., 12 Schneider, Farge et al., 2005, J. Fluid Mech., 534(5)

78 Vorticity dipole impinging on a wall

79 Adapted grid automatically generated by CVS

80 CVS of Kelvin-Helmholtz instability in 2D Schneider, Farge, Koster,Griebel, Numerical Flow Simulation, 75, Springer, , 2001

81 Comparison DNS / CVS for 3D mixing layer 4 eddy turnover times DNS CVS Schneider, Farge, Pellegrino, Rogers 2005, J. Fluid Mech., 534(5)

82 Comparison DNS / CVS for 3D mixing layer 8 eddy turnover times DNS CVS Schneider, Farge, Pellegrino, Rogers 2005, J. Fluid Mech., 534(5)

83 Comparison DNS / CVS for 3D mixing layer 12 eddy turnover times DNS CVS Schneider, Farge, Pellegrino, Rogers 2005, J. Fluid Mech., 534(5)

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85 Present General Simulation Models Clouds Computational grid Topography Oceans Vegetation Source: CEA

86 Future General Simulation Models Adaptative icosahedric grid generated using bi-orthogonal wavelets Thomas Dubos (LMD, France) et Nicholas Kevlahan (McMasters, Canada)

87 Turbulence is still an open problem! Hans Liepmann ( ) As long as we are not able to predict the drag on a sphere or the pressure drop in a pipe from continuous, incompressible and Newtonian assumptions without any other complications, namely from first principles, we would not have made it! Turbulence Workshop, UC Santa Barbara, 1997 Turbulence is still an open problem for physicists and mathematicians. Indeed, the Berlin 1750 Prize problem posed by Euler is not yet solved!

88 Review papers on wavelets Marie Farge, 1992 Wavelet Transforms and Their Applications to Turbulence Ann. Rev. Fluid Mech., 24, Marie Farge, Nicholas Kevlahan, Valerie Perrier and Eric Goirand, 1996 Wavelets and Turbulence IEEE Proceedings, 84, 4, 1996, Marie Farge, Nicholas Kevlahan, Valérie Perrier and Kai Schneider, 1999 Turbulence Analysis, Modelling and Computing using Wavelets Wavelets in Physics, ed. J. van den Berg, Cambridge University Press, Marie Farge and Kai Schneider, 2002 Analyzing and computing turbulent flows using wavelets Summer Course, Les Houches LXXIV, New trends in turbulence, Springer Kai Schneider and Marie Farge, 2006 Wavelets: Mathematical Theory Encyclopedia of Mathematical Physics, Elsevier, Marie Farge and Kai Schneider, 2015 Wavelets transforms and their applications to MHD and plasma turbulence Journal of Plasma Physics, Cambridge University Press, in press

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90 You can download movies from : Results You can download papers from : Publications You can download codes from : Codes