New Interpretation for error propagation of data-driven Reynolds stress closures via global stability analysis

TL;DR

This paper investigates the causes of convergence issues in data-driven Reynolds stress closures for RANS equations, revealing that numerical instability, not just ill-conditioning, can lead to error propagation, with stability depending on flow type and eddy viscosity distribution.

Contribution

It introduces a global stability analysis approach to distinguish between ill-conditioning and numerical instability in RANS closures, highlighting their different impacts on error propagation.

Findings

High Reynolds number turbulent channel flow is ill-conditioned but stable.

Separated flow over hills shows instability without ill-conditioning.

Eddy viscosity distribution affects numerical stability and error propagation.

Abstract

In light of the challenges surrounding convergence and error propagation encountered in Reynolds-averaged Navier-Stokes (RANS) equations with data-driven Reynolds stress closures, researchers commonly attribute these issues to ill-conditioning through conditional number analysis. This paper delves into an additional factor, numerical instability, contributing to these challenges. We conduct global stability analysis for the RANS equations, closed by the Reynolds stress of direct numerical simulation (DNS), with the time-averaged solution of DNS as the base flow. Our findings reveal that, for turbulent channel flow at high Reynolds numbers, significant ill-conditioning exists, yet the system remains stable. Conversely, for separated flow over periodic hills, notable ill-conditioning is absent, but unstable eigenvalues are present, indicating that error propagation arises from the mechanism of numerical instability. Furthermore, the effectiveness of the decomposition method employing eddy viscosity is compared, results show that the spatial distribution and amplitude of eddy viscosity influences the numerical stability.