On the stability of slowly varying flow: the divergent channel

Abstract

The linear stability of a slowly varying flow, the flow in a diverging straight-walled channel, is studied using a modification of the ‘WKB’ or ‘ray’ method. It is shown that ‘quasi-parallel’ theory, the usual method for handling such flows, gives the formally correct lowest-order growth rate; however, this growth rate can be substantially in error if its magnitude is comparable to that of the rate of change of the basic state. The method used clearly demonstrates the dependence of the growth rate, wavenumber, neutral curves, etc., on the cross-stream variable and on the flow quantity under consideration. When applied to the divergent channel, the method yields a much wider ‘unstable’ region and a much lower ‘critical’ Reynolds number (depending on the flow quantity used) than those predicted by quasi-parallel theory. The determination of the downstream development of waves of constant frequency shows that waves of all frequencies eventually decay.