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

TL;DR

This paper introduces a wavelet-based regularization method for 3D Euler flows, effectively modeling turbulence by filtering incoherent components and capturing key turbulent features like intermittency and the energy spectrum.

Contribution

The paper presents a novel wavelet-based denoising approach to regularize Galerkin truncated Euler equations, capturing turbulence characteristics and comparing it with other regularization methods.

Findings

Coherent flow exhibits intermittent nonlinear dynamics.

Energy spectrum follows a $k^{-5/3}$ law.

Wavelet regularization effectively models turbulence.

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 filed is decomposed into wavelet coefficients, that 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 noise-like incoherent flow, models turbulent dissipation and corresponds to an adaptive regularization. In order 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 $k^{-5/3}$ energy spectrum, where $k$ is the wavenumber, 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.