Resolvent Analysis of laminar and turbulent duct flows

TL;DR

This paper uses resolvent analysis to study energy amplification mechanisms in laminar and turbulent duct flows, revealing how flow structures and secondary flows influence flow stability across different aspect ratios.

Contribution

It extends resolvent analysis to finite-spanwise duct flows, analyzing the impact of secondary flow and aspect ratio on energy amplification mechanisms.

Findings

Secondary flow can enhance or suppress energy amplification.

Aspect ratio significantly affects flow response and mode shapes.

Secondary flow alters the structures leading to maximal amplification.

Abstract

This work applies resolvent analysis to incompressible flow through a rectangular duct, in order to identify dominant linear energy-amplification mechanisms present in such flows. In particular, we formulate the resolvent operator from linearizing the Navier--Stokes equations about a two-dimensional base/mean flow. The laminar base flow only has a nonzero streamwise velocity component, while the turbulent case exhibits a secondary mean flow (Prandtl's secondary flow of the second kind). A singular value decomposition of the resolvent operator allows for the identification of structures corresponding to maximal energy amplification, for specified streamwise wavenumbers and temporal frequencies. Resolvent analysis has been fruitful for analysis of wall-bounded flows with spanwise homogeneity, and here we aim to explore how such methods and findings can extend for a flow in spatial domains of finite spanwise extent. We investigate how linear energy-amplification mechanisms (in particular the response to harmonic forcing) change in magnitude and structure as the aspect ratio (defined as the duct width divided by its height) varies between one (a square duct) and ten. We additionally study the effect that secondary flow has on linear energy-amplification mechanisms, finding that in different regimes it can either enhance or suppress amplification. We further investigate how the secondary flow alters the forcing and response mode shapes leading to maximal linear energy amplification.