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Numerical Study of Random Kelvin-Helmholtz Instability

Alina Chertock, Michael Herty, Arsen S. Iskhakov, Anna Iskhakova, Alexander Kurganov, Mária Lukáčová-Medvid'ová

TL;DR

This work addresses nonuniqueness in the multidimensional compressible Euler equations by adopting a turbulence-inspired statistical framework for random KH instabilities. It combines a stochastic collocation method with a high-order A-WENO solver in space and CWENO interpolation in the random space to compute Cesàro averages across embedded meshes, enabling robust analysis of averaged fields, Reynolds-stress, and energy defects. The study demonstrates that DW solutions, Cesàro averages, and POD can coherently describe the chaotic, multi-scale dynamics of inviscid compressible flows, revealing turbulence-like statistics such as nontrivial PDFs and broad modal spectra. The approach offers a practically meaningful description of random KH evolution and lays groundwork for extensions to broader flow regimes and long-time behavior, with potential applications in turbulence modeling for compressible fluids.

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 Cesáro 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.

Numerical Study of Random Kelvin-Helmholtz Instability

TL;DR

This work addresses nonuniqueness in the multidimensional compressible Euler equations by adopting a turbulence-inspired statistical framework for random KH instabilities. It combines a stochastic collocation method with a high-order A-WENO solver in space and CWENO interpolation in the random space to compute Cesàro averages across embedded meshes, enabling robust analysis of averaged fields, Reynolds-stress, and energy defects. The study demonstrates that DW solutions, Cesàro averages, and POD can coherently describe the chaotic, multi-scale dynamics of inviscid compressible flows, revealing turbulence-like statistics such as nontrivial PDFs and broad modal spectra. The approach offers a practically meaningful description of random KH evolution and lays groundwork for extensions to broader flow regimes and long-time behavior, with potential applications in turbulence modeling for compressible fluids.

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 Cesáro 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.

Paper Structure

This paper contains 15 sections, 2 theorems, 38 equations, 9 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Dissipative Weak (DW) Solutions
  3. Deterministic Solutions of the Compressible Euler Equations
  4. Basic Properties of DW Solutions
  5. Random Solutions of the Compressible Euler Equations
  6. Methodology
  7. Numerical Study of KH Instabilities
  8. Cesàro averages.
  9. Reynolds stress and energy defects.
  10. Statistical analysis.
  11. Proper Orthogonal Decomposition (POD)
  12. POD for $\rho$
  13. POD for Cesàro Averages
  14. Conclusions
  15. Appendix A

Key Result

Theorem 2.1

Let the initial data $\{(\rho_{0,m},\bm m_{0,m},E_{0,m})\}_{m=1}^\infty$ satisfy where $\underline{\rho}$ is a constant independent of $m$, and let $\{(\rho_m,\bm m_m,S_m)\}_{m=1}^\infty$ be a consistent approximate solution of (1.1)--(1.3). Further, let where $\hbox{$E$}$ is another constant independent of $m$. Then, the sequence $\{(\rho_m,\bm m_m,S_m)\}_{m=1}^\infty$ is uniformly bounded and

Figures (9)

  • Figure 4.1: $\rho(x,y,0;\xi)$ for $\xi=-1$ (top row), $0$ (middle row), and $1$ (bottom row), and five embedded uniform meshes with $m=1,\dots,5$ (from left to right). Yellow and green squares in the top left panel show the regions $D_1=[0.46,0.54]\times[0.71,0.79]$ and $D_2=[0.76,0.84]\times[0.71,0.79]$, which will be used below.
  • Figure 4.2: $\rho(x,y,2;\xi)$ for $\xi=-1$ (top row), $0$ (middle row), and $1$ (bottom row) and five embedded uniform meshes with $m=1,\dots,5$ (from left to right).
  • Figure 4.3: Cesàro averages for the deterministic problem with $\xi=0$.
  • Figure 4.4: Means of the Cesáro averages for the stochastic problem.
  • Figure 4.5: Reynolds stress and energy defect, along with their ratio.
  • ...and 4 more figures

Theorems & Definitions (3)

  • Theorem 2.1: ${\cal K}$-convergence
  • Theorem 2.2: Convergence of the Monte Carlo method
  • Definition A.1: DW solution