Analytic Solution of Optimal Aspect Ratio of Bionic Transverse V-Groove for Drag Reduction Based on Vorticity Kinetics

Abstract

Previous studies have implied that the AR (aspect ratio) of the transverse groove significantly affects the stability of the boundary vortex within the groove and thus drives the variation in the drag-reduction rate. However, there is no theoretical model describing the relationship between the AR and the stability of the boundary vortex, resulting in difficulty in developing a forward method to obtain the optimum AR. In this paper, the velocity potential of the groove sidewalls to the boundary vortex is innovatively described by an image vortex model, thus establishing the relationship between the AR and the induced velocity. Secondly, the velocity profile of the migration flow is obtained by decomposing the total velocity inside the groove, by which the relationship between the AR and the migration velocity is established. Finally, the analytical solution of the optimal AR (ARopt = 2.15) is obtained based on the kinematic condition for boundary vortex stability, i.e., the induced velocity equals the migration velocity, and the forms of boundary vortex motion at other ARs are discussed. Furthermore, the stability of the boundary vortex at the optimal AR and the corresponding optimal drag-reduction rate are verified by the large eddy simulations method. At other ARs, the motion forms of the boundary vortex are characterized by “vortex shedding” and “vortex sloshing,” respectively, and the corresponding drag-reduction rates are smaller than those for vortex stability.