Non-modal stability of Jeffery-Hamel flow

Abstract

The destabilization of modal perturbations in the classical diverging Jeffery-Hamel (JH) flow has been long-known. The converging JH flow is far less-studied, but it is known that convergence suppresses modal instabilities. We make a parallel-flow approximation following previous studies, to examine its non-modal stability at small convergent and divergent angles and show that non-modal growth is extremely sensitive to the angle of convergence/divergence at high Reynolds numbers. The transient growth of energy is significantly suppressed at high Reynolds numbers as the wall angle is varied from divergence to convergence by just a few hundredths of a degree. This finding is especially relevant for convergent channels, where the flow is stable to linear modal perturbations up to the Reynolds numbers of the order of 105 or larger. In all the cases, streamwise-aligned rolls (which are a characteristic of the lift-up mechanism) are the initial perturbations that display the largest energy growth. The spanwise separation between the rolls decreases significantly with channel convergence. Our findings indicate that extremely small imperfections in the wall alignment in channel flows can drastically affect the experimental measurements of algebraic growth of the disturbance kinetic energy, as minute amounts of wall convergence can strongly reduce the maximum transient growth.