The stability of flows in channels with small wall curvature

Abstract

The ‘stability’ of flows in symmetric curved-walled channels is investigated by essentially combining Fraenkel's ‘small’ wall-curvature theory with the multiple-scaling (or WKB) method. The basic flow is characterized by the steady-state stream function Ω, which varies ‘slowly’ in the streamwise direction. An asymptotic scheme is posed for Ω in such a way that at lowest order Ω represents a class of Jeffery–Hamel solutions. An infinitesimal disturbance is superimposed on the basic flow through a time-dependent stream function Φ, and the resulting linearized disturbance equation suggests that fixed-frequency disturbances with ‘slowly’ varying wavenumber are appropriate. The asymptotic scheme for Φ yields the Orr–Sommerfeld equation at lowest order. Two classes of channels are considered. In the first class the curvature is constant in sign, and under certain conditions they reduce to symmetric divergent straight-walled channels. In the second class of channels the curvature varies in sign, and these may be more suitable for experimentation. A spatially dependent growth rate of the disturbance relative to the basic flow is defined; this forms the basis of the ‘stability’ analysis. Critical Reynolds numbers are deduced, below which the disturbance decays as it travels downstream, and above which the disturbance grows for a limited range in the streamwise direction. For the first class of channels the ‘stability’ analysis is carried out locally, and the dependence of the critical Reynolds numbers on curvature and higher-order terms is investigated. For the second class of channels the ‘stability’ analysis is carried out at various positions downstream, and an overall minimum critical Reynolds number is predicted for a range of channels and flows.