Flutter-Like Oscillationas of a Planing Plate

Abstract

In the two-dimensional steady nonlinear problem of a planing plate, two distinct limiting flow regimes occur: In one, the usual planing problem, a jet (or spray) is thrown forward approximately parallel to the plate, whereas in the other there is a flow over the plate in the downstream direction and no fluid is thrown forward. The latter is equivalent to the case of a hydrofoil under the free surface with an infinitely long cavity at atmospheric pressure. If the plate has a large fixed angle of attack, there is a gradual transition from the hydrofoil flow to the true planing flow as speed is increased and/or load is decreased. However, if the angle of attack is small, the transition is relatively abrupt. Furthermore, for small angle of attack, either limiting type of flow can be analyzed on the basis of a linearized mathematical model, although the transition is intrinsically nonlinear. &&In order to study the instability, the planing plate is assumed to oscillate sinusoidally in heave at a prescribed frequency, and the pertinent linearized problem is solved. Only the lightly-loaded condition is considered, with the jet thrown forward. It is assumed that the effective location of the leading edge of the wetted portion oscillates fore and aft; the problem is equivalent to that of an airfoil with variable chord length. It is necessary to solve for the location of the leading edge by finding the time-dependent elevation of the water ahead of the plate. Then the lift on the planing plate is computed; it depends only on the reduced frequency parameter, and the damping is negative whenever this parameter has a value less than a critical value. This suggests that the flow is always unstable. However, it is found that the value of the frequency parameter for zero damping agrees fairly well with the value observed in experiments under actual conditions of spontaneous oscillation. &&It is concluded that oscillation cannot occur until the speed is high enough that the flow has changed from a hydrofoil_like flow to a planing type of flow. After this transition has occurred, the flow is invariably unstable, and oscillation occurs at that frequency at which the damping coefficient just becomes negative. Unfortunately, the speed at which transition occurs cannot yet be predicted, because the steady-motion problem is here treated in the classical manner, in which gravity is not considered; the steady solution is not unique. Also, no convincing arguments have yet been found to explain the observation that oscillation apparently occurs at the frequency at which the damping just becomes negative, rather than at some higher frequency at which damping would be more negative and the flow more unstable.