A perturbation theory for unsteady cavity flows

Abstract

This investigation deals with a perturbation theory for unsteady cavity flows in which the time-dependent part of the flow may be considered as a small perturbation superimposed on an established steady cavity flow of an ideal fluid, the gravity effect being neglected in this study. In order to make a comparison between the various existing steady-cavity-flow models when applied to unsteady motions, some of these models have been employed to evaluate the small time behavior of, and the initial reaction to an unsteady disturbance. Furthermore, the mechanism by which the cavity volume may be changed with time is studied and the initial hydrodynamic force resulting from such change is calculated. The second kind of unsteady cavity flow problems treated here is characterized by the fact that the disturbances are limited to be small for all time instants. Based on a systematic linearization with respect to the steady basic flow, a general perturbation theory for unsteady cavity flows is formulated. From this perturbation theory the generation of surface waves along the cavity boundary is revealed, much in the same way as the classical gravity waves in water, except with the centrifugal acceleration due to the curvature of the free-streamlines in the basic flow playing the role of an equivalent gravity effect.