of the cavity induced pressure impulses on the hull.
Efforts to improve the prediction of the flow around the leading edge followed, among which the work of Kinnas should be noted. He applied the method of Lighthill's correction to the streamwise velocity component obtained in linear theory and was successful in reducing the cavity volume from the original work of Lee. The linear cavity theory is however found later to give a wrong trend by Uhlman and also by Lee et al, who show in their nonlinear theory that the predicted cavity extent decreases when the thickness of the blade increases, contrary to the trend in linear theory. The need to treat the finite blade thickness effect is evident.
Recently a surface panel method based on Morino's formulation has been applied successfully to the propeller problem with an improvement on the leading edge solution. Application to the cavitating flow problem is a natural extension. The method was first applied to a two-dimensional hydrofoil problem by Lee, Lee et al and also by Kinnas and Fine . They solved the exact nonlinear steady problem, in either partially or super-cavitating steady flow condition.
Kim et al then extended the method for the solution of the 3-D steady and unsteady cavitating hydrofoil problem, where the cavity surface was linearized on the suction side of the blade. By avoiding the linearization in blade thickness as in the lifting surface theory they could maintain the resolution of the flow prediction, especially in the area of the blade leading edge, and hence the cavity behavior could be predicted without loosing accuracy.
The unsteady flow problem around a marine propeller was solved subsequently by Fine and Kim. In the present paper we present the potential-based formulation of Kim to solve the steady or unsteady cavitating propeller flow.
The same formulation is applicable to 2- and 3-dimensional cavity flow problems. To verify the numerical procedure, we first show that the method is applicable to the analysis of the 2-dimensional unsteady hydrofoil in either heave or gust mode for both the partially and super-cavitating flow conditions. We then show the same method is applicable to the 3-dimensional hydrofoil with rectangular plan-form. Finally we predict the cavity behavior around the blades of two marine propeller operating either in screen-generated wake or in wake generated behind a ship model. Our predictions compares fairly well with the cavitation patterns observed at cavitation tunnels.
Let's consider a cavitating propeller operating in a nonuniform ship wake with a constant rotational speed n at a constant advance speed Vs. Boundaries such as the rudder and free-surface in contact with the atmosphere are ignored, and the presence of the hull is recognized only by the effective wake, which is assumed to be known. We assume the viscous effect on the blades and the trailing vortex sheets is confined within the infinitesimally thin boundary layer. The cavity is assumed to be a constant pressure surface which grows on the suction side of the blades. We assume that the cavity is a thin sheet cavity and the detachment point of the sheet cavity is known. The ship wake is nonuniform and is assumed undeformable under the influence of the propeller action, and the propeller is operating in an inviscid, incompressible and irrotational fluid field.
A Cartesian coordinate is chosen as shown in Figure 1; the x-axis coincides with the shaft centerline, defined positive downstream, the positive y-axis points upward, and the z-axis completes a right hand coordinate system. The cylindrical coordinate system is defined for convenience; r being the radius and θ the angle from the positive y-axis positive counter-clockwise when looking downstream.
The perturbation velocity may be expressed by the perturbation velocity potential Then conservation of the mass applied to the potential flow gives the Laplace equation as a governing equation in the fluid region, that is,
Motion of the flow satisfying the Laplace equation (1) can be uniquely defined by imposing the following boundary conditions on the boundary surfaces.