Viscous Instabilities in Transversely Strained Channel Flows

TL;DR

This paper analyzes linear stability in a three-dimensional boundary layer with spanwise pressure gradient, revealing how spanwise shear influences stability metrics and transient growth, with implications for flow control.

Contribution

It introduces a coordinate transformation simplifying stability analysis of spanwise pressured channel flows and derives new stability metrics.

Findings

Critical Reynolds number decreases with spanwise pressure gradient.

Stability characteristics approach those of standard channel flow after re-scaling.

Transient growth increases quadratically with spanwise pressure differential.

Abstract

We investigate here linear stability in a canonical three-dimensional boundary layer generated by the superposition of a spanwise pressure gradient upon an otherwise standard channel flow. As the main result, we introduce a simple coordinate transformation that enables the complete description of modal and non-modal stability using previous results on Poiseuille flow. We leverage this insight to derive closed forms for some relevant stability metrics. In particular, the critical Reynolds number for exponential-in-time growth is found to monotonically decrease with the strength of the cross-flow. A suitably chosen re-scaling, however, shows that the stability characteristics ultimately approach those of channel flow, despite the presence of a non-zero spanwise shear. Unstable eigenmodes akin to the Tollmien-Schlichting wave are found to propagate along the direction of the net flow. From a non-modal perspective, the maximal transient (algebraic) growth increases quadratically with the spanwise pressure differential and, similar to two-dimensional flows, is fueled by the lift-up effect. In this regard, the linear energy budget highlights a dramatic increase in energy production against the spanwise shear.