Numerical Study of Random Kelvin-Helmholtz Instability

TL;DR

This paper employs stochastic numerical methods and statistical analysis to study the complex, chaotic behavior of Kelvin-Helmholtz instabilities in compressible flows, revealing consistent features across random realizations.

Contribution

It introduces a novel statistical and reduced-order modeling approach to analyze random KH instabilities, enhancing understanding of their chaotic dynamics.

Findings

Random KH instabilities can be systematically characterized using statistical tools.

Averaging over multiple realizations reveals stable solution features.

Reduced-order models effectively capture the essential dynamics.

Abstract

In this paper, we study random dissipative weak solutions of the compressible Euler equations in the Kelvin-Helmholtz (KH) instability. Motivated by the fact that weak entropy solutions are not unique and can be viewed as inviscid limits of Navier-Stokes flows, we take a statistical approach following ideas from turbulence theory. Our aim is to identify solution features that remain consistent across different realizations and mesh resolutions. For this purpose, we compute stable numerical solutions using a stochastic collocation method implemented with the help of a fifth-order alternative weighted essentially non-oscillatory (A-WENO) scheme and seventh-order central weighted essentially non-oscillatory (CWENO) interpolation in the random space. The obtained solutions are averaged over several embedded uniform grids, resulting in Ces\'aro averages, which are studied using stochastic tools. The analysis includes Reynolds stress and energy defects, probability density functions of averaged quantities, and reduced-order representations using proper orthogonal decomposition. The presented numerical experiments illustrate that random KH instabilities can be systematically described using statistical methods, averaging, and reduced-order modeling, providing a robust methodology for capturing the complex and chaotic dynamics of inviscid compressible flows.