Coherent enstrophy dissipation in the inviscid limit of 2D - - PowerPoint PPT Presentation

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Coherent enstrophy dissipation in the inviscid limit of 2D - - PowerPoint PPT Presentation

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Coherent enstrophy dissipation in the inviscid limit of 2D turbulence Romain Nguyen van yen 1 Marie Farge 1 Kai Schneider 2 APS Meeting, Minneapolis, Nov. 22nd, 2009 1 Laboratoire de Mtorologie Dynamique-CNRS, ENS Paris, France 2 Laboratoire

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Coherent enstrophy dissipation in the inviscid limit of 2D turbulence

Romain Nguyen van yen1 Marie Farge1 Kai Schneider2

1 Laboratoire de Météorologie Dynamique-CNRS, ENS Paris, France

2 Laboratoire de Mécanique, Modélisation et Procédés Propres-CNRS, CMI-Université d’Aix-Marseille, France

APS Meeting, Minneapolis, Nov. 22nd, 2009

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Introduction

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How to decompose turbulent fluctuations?

‘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

turbulent fluctuations = non random + random = coherent structures + incoherent noise turbulent dynamics = chaotic non diffusive + stochastic diffusive = inviscid nonlinear dynamics + turbulent dissipation

Farge, Schneider, Kevlahan,

  • Phys. Fluids, 11 (8), 1999

Farge et al., Fluid Dyn. Res., 10, 229, 1992 Farge, Pellegrino, Schneider

  • Phys. Rev, Lett. 87 (5), 2001

Coherent Vorticity Simulation (CVS)

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SLIDE 4

Definition of coherent enstrophy

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1D Wavelet bases

  • Orthogonal wavelet bases on the real line are obtained by dilating and

translating a single, well chosen oscillating function.

  • They have good locality properties both in scale and space.

The « coiflet 12 » wavelets and their corresponding energy spectra. the construction can be generalized to any dimension using the multiresolution formalism.

  • S. Mallat, A wavelet tour of signal processing, Academic Press (1999)
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SLIDE 6

kx ky

2D Wavelet bases

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Scalewise and directionwise extraction

→ incoherent → coherent μ is the direction j is the scale i is the position (*) Azzalini et al., ACHA 18 (2004) As a first guess, we make the hypothesis that the incoherent part is an additive Gaussian noise. Hence we can separate them by thresholding: Gaussian contributions will correspond to the smallest wavelet coefficients at their respective scale. The scalewise and directionwise thresholds are determined from the field itself using a fixed-point iterative procedure (*).

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Results

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SLIDE 9

N=5122 Re=103 N=10242 Re=2.104 N=20482 Re=105 N=40962 Re=3.105 N=81922 Re=106

t=4 10 80 120 40 20

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Coherent Vorticity Extraction

Total Incoherent Coherent

t = 40

3% N 99.6% Z 2% N 98% Z 2% N 96% Z 2% N 92% Z 2% N 88% Z N = 512 Re = 4 103 N = 2048 Re = 6 104 N = 1024 Re = 1.5 104 N = 4096 Re = 2.5 105 N = 8192 Re = 106 97% N 0.4% Z 98% N 2% Z 98% N 4% Z 98% N 8% Z 98% N 12% Z

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Coherent Vorticity Extraction t = 40

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SLIDE 12

enstrophy spectra

wavenumber

vorticity PDF k-1

  • the incoherent part has a k-1

inertial range spectrum

Scalewise statistics: Extraction Results

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enstrophy spectra

wavenumber

k-1

  • the incoherent part has a k-1

inertial range spectrum

  • the coherent part dominates in the

dissipative range (!!)

Scalewise statistics: Extraction Results

vorticity PDF

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SLIDE 14

enstrophy spectra

wavenumber

k-1

  • the incoherent part is

close to marginally Gaussian

  • the incoherent part has a k-1

inertial range spectrum

  • the coherent part dominates in the

dissipative range (!!)

Scalewise statistics: Extraction Results

vorticity PDF

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SLIDE 15

Dissipation of coherent enstrophy

initial enstrophy enstrophy at t = 50 normalized by initial enstrophy enstrophy that has been dissipated between t = 0 and t = 50 (*) Dmitruk & Montgomery 2005, Tran & Dritschel 2006 total enstrophy does not dissipate in the inviscid limit (*)

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Dissipation of coherent enstrophy

initial enstrophy coherent enstrophy that has been dissipated between t = 0 and t = 50 ...but coherent enstrophy dissipates in the inviscid limit total enstrophy does not dissipate in the inviscid limit enstrophy at t = 50 normalized by initial enstrophy

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Dissipation of coherent enstrophy

...but coherent enstrophy dissipates in the inviscid limit due to the production

  • f incoherent enstrophy

total enstrophy does not dissipate in the inviscid limit enstrophy at t = 50 normalized by initial enstrophy

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Conclusion

  • Coherent enstrophy was defined using scalewise statistics of the vorticity field

that could be obtained thanks to a wavelet transform.

  • The Navier-Stokes equations at increasingly high Reynolds numbers were solved

using a classical pseudo-spectral method.

  • The analysis of the numerical solutions shows that coherent enstrophy is

dissipated in the inviscid limit, even though total enstrophy is conserved.

  • The remainder, incoherent enstrophy, gets spread between wavelet coefficients

that behave like a correlated Gaussian process with a spectral slope -1, like the total vorticity field.

  • We conjecture that only the coherent coefficients have to be solved for

deterministically using Coherent Vortex Simulation, while the incoherent ones could be modelled by a random process. More references: http://wavelets.ens.fr Numerical tools (incl. parallel wavelet transform): http://justpmf.com/romain/kicksey_winsey