Numerical Study of Random Kelvin-Helmholtz Instability

Communications in Computational Physics, vol. 40, pp. 1394–1415

Abstract

In this paper, we study random dissipative weak solutions of the compressible Euler equations for the Kelvin-Helmholtz (KH) instability problem. 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 using 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 Cesàro 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.

Authors 5

1. Alina E. Chertock

   North Carolina State University

   Affiliation as printed

   Department of Mathematics, North Carolina State University, Raleigh, NC, USA

2. Michaël Herty Aachen Department of Mathematics

   RWTH Aachen University

   Affiliation as printed

   Department of Mathematics, RWTH Aachen University, Aachen, Germany

3. Arsen S. Iskhakov

   Kansas State University

   Affiliation as printed

   Department of Mechanical and Nuclear Engineering, Kansas State University, Manhattan, KS, USA

4. Anna Iskhakova

   Kansas State University

   Affiliation as printed

   Department of Mechanical and Nuclear Engineering, Kansas State University, Manhattan, KS, USA

5. Mária Lukáčová-Medviďová

   Johannes Gutenberg University Mainz

   Affiliation as printed

   Institute of Mathematics, Johannes Gutenberg University Mainz, Mainz, Germany