Topics in Viscous Shear Flow Dynamics

Abstract

Improved upper bounds on viscous energy dissipation rates of wall-driven shear flow subject to uniform injection and suction rates are computationally determined. The so-called `background' variational formulation is implemented via a time-stepping numerical scheme to determine optimal estimates. Shear flow Reynolds numbers range from 50 to 40 000 with injection angles up to 2 degrees The computed upper bounds for pre-selected angles of injection at high Reynolds numbers significantly improve the rigorously estimated ones. Our results suggest that the steady laminar flow is nonlinearly stable for angles of injection greater than 2 degrees.

A viscous extension of Arnold's inviscid theory for non-inflectional plane shear flows is developed and viscous Arnold's identities are obtained. Special forms of our viscous Arnold's identities have been revealed that are closely related to the perturbation's enstrophy identity derived by Synge. As an application of our enstrophy identities, we quantitatively investigate mechanisms of linear stability/instability within the normal modal framework. The investigation reveals a subtle interaction between a critical layer and its adjacent boundary layer, which governs stability/instability of disturbances. As an implementation of relaxed wall boundary conditions imposed for our enstrophy identity, a control scheme is proposed that transitions wall settings from the no-slip condition to the free-slip condition, through which a flow is stabilized quickly within an early stage of the transition.