Stability of nonparallel developing flow in an annulus

The linear spatial stability of the developing flow in a concentric annulus, incorporating all nonparallel effects, is discussed. It is assumed that the disturbance is axisymmetric. The velocity profile in the developing flow region is arrived at using an implicit finite-difference scheme. The method of multiple scales is used to explain the nonparallel effects. The fourth-order Runge-Kutta method is used for integration together with a selective application of the Gram-Schmidt orthonormalization technique for circumventing the parasitic error-growth problem. The growth rate of the disturbance stream function is found to yield the minimum critical Reynolds number. The critical Reynolds number versus the axial location curves exhibit a minimum and the location of this minimum shifts downstream as the diameter ratio decreases. In general, at a given axial location, the critical Reynolds number decreases with the diameter ratio. The difference between the critical Reynolds number evaluated from the parallel flow theory and that based on the growth rate of the disturbance stream function is found to vary with the diameter ratio and to decrease as the flow develops.