ON THE UNSTEADY KUTTA CONDITION

In their study of the time-harmonic instability of the vortex sheet shed from the trailing edge of a plate dividing a uniform stream from a stagnant fluid, Orszag and Crow (1) describe three possible inviscid solutions of the problem which correspond to different forms of the vortex sheet downstream of the trailing edge. The present paper considers the matching of these inviscid solutions to a consistent viscous flow structure at the trailing edge. For amplitudes of oscillation O(R~h) and frequencies O(R^), where R » 1 is a Reynolds number, the application of the Kutta condition of smooth flow at the trailing edge in the inviscid problem is shown to lead to a consistent viscous flow and is also consistent with the oscillation of the dividing streamline in the shape of a parabola near the trailing edge as found experimentally by Bechert and Pfizenmaier (2). Larger amplitudes of oscillation lead to the separa-tion of the flow from the plate. The inviscid solutions which do not satisfy the Kutta condition are not consistent for amplitudes of the same order, but for smaller amplitudes O(R~ty their consistency appears to depend upon the existence of a solution of the full Navier-Stokes equations in a region of dimension O(R^) at the trailing edge. 1