Linear stability of slowly varying unsteady flows in a curved channel

Abstract

The linear stability of viscous flows in a curved channel due to time dependent slowly varying pressure gradients is considered by an approach of the W.K.B. type. The asymptotic behaviour of small perturbation waves is determined, allowing their characteristics (amplitude, transverse structure, amplification rate) to be slowly varying with time. The evolution of such disturbances is followed and the instantaneous marginal states are determined according to a ‘momentary’ criterion for stability which is based on the definition of a ‘growth rate’ of the disturbance. An asymptotic representation for the growth rate is found which, at lowest order, reduces to the quasi-steady amplification rate of the disturbance velocity field associated with the lowest order component of the instantaneous basic flow. A quasi-steady correction term appears at the next order together with terms which arise from the slow variation with time of the actual disturbance structure. Slowly accelerated basic flows are also investigated. The effect of the quasi-steady correction is found to be stabilizing whereas part of the ‘slowly varying’ terms exhibit the opposite behaviour. However the influence of the correction terms appears to be small, as expected since the characteristic scale of the instability is much bigger than the ‘slow’ scale of the basic state. Under such circumstances the quasi-steady approximation may be considered a fairly accurate approach. Low frequency modulated basic flows are also investigated by using the periodicity criterion to define the marginal state. The modulation is always found to destabilize the mean flow and the ‘critical’ wavenumber is found to decrease from its unmodulated value when the amplitude of the oscillation increases. Some discussion follows.