A Nonlinear Method for Predicting Unsteady Sheet Cavitation on Marine Propellers

Abstract

A method is presented which provides a basis for predicting the nonlinear dynamic behavior of unsteady propeller sheet cavitation. The method separates the fluid velocity potential boundary-value problem into two parts, static and dynamic, which are solved sequentially in a forward time stepping procedure. The static potential problem is for the cavity fixed instantaneously relative to the propeller and the propeller translating through the nonuniform wake field. This problem can be solved by standard methods. The dynamic potential represents the instantaneous reaction of the cavity to the static potential field and thus predicts the cavity's deformation and motion relative to the blade. A solution is obtained for the dynamic potential by using the concepts of slender-body theory to define near-and far-field potentials which are matched to form the complete solution. In the far field, the cavity is represented by a three-dimensional spanwise line distribution of sources. In the near field, the cavity is approximated at each cross section as a semi-ellipse with unknown axes a(t), b(t), and position l(t) along the chord of the foil section. Conditions are derived that determine (a, b, l) by minimizing the square error in satisfying the dynamic boundary condition. These conditions yield the equations of motion of the cavity in the form of three coupled nonlinear second-order ordinary differential equations with time as the independent variable. The theory is presented for the general foil and not specifically for propellers. However, the method incorporates features in its formulation which facilitate its application to marine propellers. The method is demonstrated by using the steady noncavitating potential for the two-dimensional half-body as an approximation to the static potential. Both fixed and unsteady cavities are calculated. The unsteady cavities are calculated by varying the hydrostatic pressure in the half-body pressure field sinusoidally.