Wavelet-based regularization of the Galerkin truncated three-dimensional incompressible Euler flows

Marie Farge, Naoya Okamoto, Kai Schneider, and Katsunori Yoshimatsu
Phys. Rev. E 96, 063119 – Published 29 December 2017

Abstract

We present numerical simulations of the three-dimensional Galerkin truncated incompressible Euler equations that we integrate in time while regularizing the solution by applying a wavelet-based denoising. For this, at each time step, the vorticity field is decomposed into wavelet coefficients, which are split into strong and weak coefficients, before reconstructing them in physical space to obtain the corresponding coherent and incoherent vorticities. Both components are multiscale and orthogonal to each other. Then, by using the Biot-Savart kernel, one obtains the coherent and incoherent velocities. Advancing the coherent flow in time, while filtering out the noiselike incoherent flow, models turbulent dissipation and corresponds to an adaptive regularization. To track the flow evolution in both space and scale, a safety zone is added in wavelet coefficient space to the coherent wavelet coefficients. It is shown that the coherent flow indeed exhibits an intermittent nonlinear dynamics and a k5/3 energy spectrum, where k is the wave number, characteristic of three-dimensional homogeneous isotropic turbulence. Finally, we compare the dynamical and statistical properties of Euler flows subjected to four kinds of regularizations: dissipative (Navier-Stokes), hyperdissipative (iterated Laplacian), dispersive (Euler-Voigt), and wavelet-based regularizations.

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  • Received 17 July 2017

DOI:https://doi.org/10.1103/PhysRevE.96.063119

©2017 American Physical Society

Physics Subject Headings (PhySH)

  1. Research Areas
Fluid Dynamics

Authors & Affiliations

Marie Farge

  • CNRS–INSMI, LMD–IPSL, Ecole Normale Supérieure–PSL, 24 rue Lhomond, 75231 Paris Cedex 05, France

Naoya Okamoto

  • Center for Computational Science, Nagoya University, Nagoya, 464-8603, Japan

Kai Schneider

  • I2M-CNRS, Centre de Mathématiques et d'Informatique, Aix-Marseille Université, 39 rue F. Joliot-Curie, 13453 Marseille Cedex 13, France

Katsunori Yoshimatsu

  • Institute of Materials and Systems for Sustainability, Nagoya University, Nagoya, 464-8603, Japan

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Issue

Vol. 96, Iss. 6 — December 2017

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