Some Considerations on the Laminar Stability of Time-Dependent Basic Flows

Abstract

As a stability criterion for infinitesimal disturbances in an incompressible, parallel but time-dependent basic flow, it is proposed to introduce the concept of stability, which is said to prevail at the instant if the kinetic energy of the disturbances, as a fraction of the kinetic energy of the basic flow, tends to decrease. The significance of such a criterion is briefly discussed. For special time-dependent basic flows which are described by similar velocity profiles at all times (except for changes in amplitude), in the inviscid limit only a change of the time scale is needed to reduce the solution essentially to that for the steady case. The disturbances may be of either the transverse-wave or the longitudinal-vortices type. The result indicates a very strong destabilizing influence of deceleration, which is likely to overshadow tha t of the velocity profile under normal circumstances. The observations of Fales (on flow over the flat plate) and Coles (on flow between rotating cylinders) are believed to be largely due to the deceleration. At finite Reynolds numbers, the usual procedure of calculating the stability solution on the basis of the instantaneous profile is further shown to be valid only for extremely slow acceleration or deceleration. Even when the solution is acceptable, the condition a = 0 for neutral stability may not be used without reservation. To calculate momentary stability properly, a procedure for a slowly varying but more general profile is also described.