Numerical convergence of the Lyapunov spectrum computed using low Mach number solvers

TL;DR

This paper develops and analyzes a robust numerical method to compute Lyapunov exponents and vectors in low Mach number turbulent flow simulations, revealing non-universal convergence behaviors.

Contribution

It introduces a new algorithm for Lyapunov spectrum computation in turbulent flows and investigates its convergence properties across different flow complexities.

Findings

Spatial convergence rates follow discretization order.

Temporal convergence is weakly dependent on time step.

Convergence depends on specific Lyapunov exponents.

Abstract

In the dynamical systems approach to describing turbulent or otherwise chaotic flows, an important quantity is the Lyapunov exponents and vectors that characterize the strange attractor of the flow. In particular, knowledge of the Lyapunov exponents and vectors will help identify perturbations that the system is most sensitive to, and quantify the dimension of the attractor. However, reliably computing these Lyapunov quantities requires robust nu- merical algorithms. While several perturbation-based techniques are available in literature, their application to commonly used turbulent flow solvers as well the numerical convergence properties have not been studied in detail. The goal of this work is two-fold: a) develop a robust algorithm for obtaining Lyapunov exponents and vectors for low-Mach based sim- ulation of turbulent flows, b) quantify the spatial and temporal convergence properties of this algorithm using a series of progressively complex flow problems. In particular, a manu- factured solutions approach is devised using the Orr-Sommerfeld (OS) perturbation theory to extract Lyapunov exponents and vectors from OS eigen solutions. While individual test cases show interesting results, overall the spatial convergence rates of Lyapunov exponents follow the truncation order for the discretization scheme. However, temporal convergence rates were found to be only a weak function of time step used. Additionally, for some config- urations, the convergence properties are found to depend on the Lyapunov exponent itself. These results indicate that convergence properties do not follow universal rates, and require careful analysis for specific configurations considered.