Two-Dimensional Supercavitating Plate Oscillating Under a Free-Surface

Abstract

The problem of a super cavitating flat plate at non-zero cavitation number oscillating under a free surface is analyzed by a linearized method using the acceleration potential. The flow is assumed two-dimensional and incompressible. The flow field is made simply connected by using a cut along the wake. The flow field is then mapped on to an upper half plane and the solution is expressed in an integral form by using Cheng and Rott's method. Equations for the cavity length, total force coefficient, moment coefficient and the frequency response function are expressed in closed form. Numerical results for some special cases are also obtained and presented graphically. When the flow is steady, the present theory agrees with experimental data and other existing theories. For the special case of infinite fluid and infinite cavity the present theory agrees with Parkins' original work. For the special case of zero submergence, the present theory indicates that the total force coefficient is one half that of the value for fully wetted flow in an infinite fluid for both steady and unsteady cases. An alternate analysis is also carried out for the infinite fluid case and the result shows that the effect of the wake assumption is of order of the square of the cavitation number when the cavitation number is small. The effect of the gravity field is also discussed qualitatively. It is also concluded that the effect of the free-surface is to shorten the cavity and to increase the total force coefficient. The steady part of the force coefficient at an arbitrary submergence is obtained by multiplying the value at infinite submergence by a correction factor, whereas the unsteady part is given by a more complicated function. Even with the presence of a free-surface and oscillation of the foil, the total force coefficient at small cavitation number is approximately equal to the corresponding value at zero cavitation number multiplied by a factor (1 + σ).