A three-dimensional source-vorticity method for simulating incompressible potential flows around a deforming body without the Kutta condition

Abstract

Abstract For predicting three-dimensional incompressible potential flows around a body accompanied by a wake vortex, surface singularity methods (i.e., panel methods) have been employed extensively, owing to their ease of use and low solution times. In the case of lifting/vortical flow, the Kutta condition is applied, in order to insure smooth flow at the trailing edge. However, the Kutta condition is inapplicable in the case of blunt bodies. For this reason, a three-dimensional source-vorticity method for simulating incompressible potential flows around a deforming body without using the Kutta condition is presented. For lifting/vortical flows, three components of the surface vorticity vectors are placed on the panels instead of the doublet as the unknowns. In place of the Kutta condition, additional equations are employed for determining the total circulations for the three vorticity components about the body. To validate the proposed method, simple examples, such as a sphere in a uniform flow and a sphere in an accelerated flow, are treated as non-lifting/non-vortical cases. For lifting/vortical cases where the Kutta condition cannot be applied, a rotating sphere in a uniform flow and a sphere with a vortex ring are considered. To assess the accuracy of the proposed method, the numerical results are compared with the analytical solutions. Finally, to highlight the applicability of the method in the case of unsteady lifting/vortical flow and to show its versatility as well as suitability in treating deforming bodies, a swimming great white shark is simulated with and without the wake vortex. Based on the results obtained in the absence of the wake vortex, it was found that, even in an inviscid flow, a thrust force is produced by the movement of the shark. Further, the results obtained for the case where a wake vortex was shed from the tail fin suggested that the wake vortex sheets decrease the amplitude of the side force and increase that of the thrust force.