Hyperbolicity, shadowing directions and sensitivity analysis of a turbulent three-dimensional flow

TL;DR

This study investigates the hyperbolic nature, shadowing directions, and sensitivities in a turbulent 3D cylinder flow using CLVs and shadowing algorithms, revealing complex hyperbolic behavior and potential for sensitivity analysis in chaotic fluid flows.

Contribution

It provides the first detailed analysis of hyperbolicity and shadowing directions in a turbulent 3D flow, demonstrating the applicability of shadowing methods for chaotic fluid dynamics.

Findings

Unstable CLVs active in near-wake, stable CLVs in far-wake

System not uniformly hyperbolic due to neutral CLV and tangencies

Shadowing methods may be effective for chaotic fluid problems

Abstract

This paper uses compressible flow simulation to analyze the hyperbolicity, shadowing directions, and sensitivities of a weakly turbulent three dimensional cylinder flow at Reynolds number 525 and Mach number 0.1. By computing the first 40 Covariant Lyapunov Vectors (CLVs), we find that unstable CLVs are active in the near-wake region, whereas stable CLVs are active in the far-wake region. This phenomenon is related to hyperbolicity since it shows that CLVs point to different directions; it also suggests that for open flows there is a large fraction of CLVs that are stable. However, due to the extra neutral CLV and the occasional tangencies between CLVs, our system is not uniform hyperbolic. By the Non-intrusive least-squares shadowing (NILSS) algorithm, we compute shadowing directions and sensitivities of long-time-averaged objectives. Our results suggest that shadowing methods may be valid for general chaotic fluid problems.