On the theory of unsteady shearing flow over a slot

Abstract

A linearized theory is proposed of the unstable motion of a thin shear layer formed in a slot in a thin, rigid plate by unequal, parallel mean flows on opposite sides of the plate. Liepmann’s asymptotic displacement thickness representation of the influence of an unsteady boundary layer is used to model the quasi-periodic ejection of vorticity from the slot at its trailing edge. Discrete intervals in frequency are found within which energy is extracted from the mean flow by the unsteady motion in the slot. When the mean velocities on opposite sides of the plate approach a common, non-zero value, the number of these intervals increases without limit, indicating that perturbation energy is supplied by the mean flow even when the Kelvin-Helmholtz instabilities of a mean shear layer are absent. A detailed analysis of the motion in the general case reveals that the net flux of fluid through the slot is composed of a component q caused by the to-and-fro motion of the shear layer, together with a component qh that accompanies the ejection of vorticity at the trailing edge. Except at very small Strouhal numbers, q and qh are of almost equal magnitude but of opposite sign, so that the net flux arises from a delicate imbalance between these opposing flows, and amounts to a small fraction of either one. Application of the theory is made to determine the influence of the ejected vorticity on the excitation of self-sustaining wall cavity oscillations, and on the diffraction of sound by a perforated screen. Such screens are used to attenuate aerodynamic sound