A number of solutions are obtained to ideal axisymmetric flow past cavitating disks for both cases of a free surface jet and flow confined in a constant radius conduit. Finite difference methods are utilized in obtaining the solutions from an inverse formulation which considers the velocity potential, and Stokes’ stream function, as the independent variables and the radial and axial dimensions, r and z, as the dependent variables. The resulting inverse boundary value problem for r, for which the basis solution is obtained, is nonlinear. The solution technique uses a Newton-Raphson iteration to evaluate the implicit finite difference operator at each grid point of the finite difference mesh under the assumption that the values of r at the surrounding point are known. The Gauss-Seidel iterative methods with an over-relaxation factor adjusts the values at the surrounding grid points until the system of nonlinear finite difference equations is solved. The solutions which have been obtained are for cavitation numbers between 0.1 and 0.4, and disk radii relative to the radius of the incoming jet of approximately .15 to .42. For the confined flow case the maximum relative disk radius is approximately .27, and most of the solutions for this case indicate blockage of the flow has occurred. Preliminary analyses of the results from the solutions have related parameters commonly used in describing cavity flows such as the coefficient of drag, cavitation number, relative size of the disk and maximum size of the cavity. These preliminary analyses indicate that the magnitudes of the drag coefficients fro the free jet case are less than those for the infinite fluid case as determined by Garabedian’s solution or rotation of the pressure distribution of a two-dimensional flat plate; whereas for the confined flow case the magnitudes of the drag coefficients are larger. A small increase in the magnitude of the coefficient of drag based on the velocity on the cavity surface occurs with increasing magnitude of the cavitation number for the free jet case whereas for the confined flow case this drag coefficient decreases with an increase of the cavitation number. This decrease in the magnitude of the drag coefficient for the confined flow case is strongly influenced by a decrease in the magnitude of blockage with an increase of the cavitation number. In the free jet case the drag coefficient shows practically no relationship to the relative disk radius (after deletion of the data from the smallest disk sizes whose solution results are in doubt), whereas an increase in the magnitude of the coefficient of drag occurs with an increase in disk size for the confined flow case. The maximum radius of the cavity for the confined flow case is considerably less than the corresponding radius for the free jet case. For both cases the maximum radius of the cavity divided by the relative disk radius increases rapidly with decreasing disk radius. For the free jet case the ratio of maximum cavity radius to disk radius increases very slightly with a decrease in the cavitation number. For the case of t confined cavity flow, the ratio of maximum cavity radius to disk radius increases with an increase in the cavitation number. A number of solutions were also obtained to two variations of the basic problems. The first variation consists of placing the cavitating disk in a partially confined flow in which the first portion of the jet is confined in a constant radius conduit and the latter portion consists of the free jet. The second variation consists of the cavity flow of a free jet past a truncated cone.
Jeppson, Roland W., "Finite Difference Solutions to Free Jet and Confined Cavity Flows Past Disks with Preliminary Analyses of the Results" (1969). Reports. Paper 311.