Bounds on Surface Stress Driven Shear Flow

TL;DR

This paper uses the background method to establish rigorous limits on surface shear flow speeds and energy dissipation, providing theoretical bounds that support previous numerical estimates in shear-driven channel flows.

Contribution

It adapts the background method to derive rigorous bounds on surface speeds and energy dissipation, and establishes nonlinear energy stability thresholds for plane Couette flow with shear stress boundary conditions.

Findings

Critical Grashoff number for energy stability: 139.5 (2D), 51.73 (3D)

Upper bound on friction coefficient: 1/32 (2D), 1/8 (3D)

Results justify previous numerical estimates.

Abstract

The background method is adapted to derive rigorous limits on surface speeds and bulk energy dissipation for shear stress driven flow in two and three dimensional channels. By-products of the analysis are nonlinear energy stability results for plane Couette flow with a shear stress boundary condition: when the applied stress is gauged by a dimensionless Grashoff number $Gr$, the critical $Gr$ for energy stability is 139.5 in two dimensions, and 51.73 in three dimensions. We derive upper bounds on the friction (a.k.a. dissipation) coefficient $C_f = \tau/\bar{u}^2$, where $\tau$ is the applied shear stress and $\bar{u}$ is the mean velocity of the fluid at the surface, for flows at higher $Gr$ including developed turbulence: $C_f le 1/32$ in two dimensions and $C_f \le 1/8$ in three dimensions. This analysis rigorously justifies previously computed numerical estimates.