Numerical Simulation of Unsteady Flow around a Sphere by a Vortex Method for Re Number from 300 to 1000

Abstract

Recendy, the present authors have developed an advanced vortex method for two and three-dimensional analysis of a viscous unsteady flow, aiming at the application of it toward various engineering problems. In order to investigate validity and apphcabUity to the analysis of the flow around three-dimensional bodies, numerical simulation of unsteady flow around a sphere in the range of Reynolds number from 300 to 1000 is carried out with the vortex method based on a new scheme of introduction of nascent vortex elements. The boundary layer separation and the development of vortical wake from the body surface were reasonably simulated for a starting flow. L INTRODUCTION Although the recent progress of computational fluid dynamics is quite rapid, the numerical analysis of high Reynolds number flow is not very easy in terms of engineering applications. The vortex methods have been developed and applied to analysis of complicated, unsteady and vortical flows related with problems in a wide range of industries, because they consist of simple algorithm based on physics of flow. Leonard' summarized the basic algorithm and examples of its applications. Sarpkaya'' presented a comprehensive review of various vortex methods based on Lagrangian or mixed Lagrangian-Eulerian schemes and the Biot-Savart law or the vortex in cell methods. Recently, Kamemoto summarized the mathematical basis of the Biot-Savart law methods. In considering the expression of viscous diffusion of vorticity in the Biot-Savart law methods, the core spreading method by Leonard', the random walk method by Chorin'' and the integrated vorticity equation method by Wincklemans and Leonard'' were proposed. Although Greengard' criticized the core spreading method, Nakajima and Kida' mathematically confirmed the validity of the core spreading method. Nakanishi and Kamemoto'' developed a Biot-Savart law method combined with the threedimensional core spreading method. Many investigations, related to two-dimensional analysis with vortex methods, have been done. However, only a few studies related to three-dimensional analysis with vortex methods have been reported. For examples, Ojima & Kamemoto' analyzed three-dimensional unsteady flow around a sphere and a prolate spheroid with a threedimensional vortex method based on a new scheme of introduction of nascent vortex elements. Kiya et al. applied a three-dimensional vortex blob method to the simulation of an impulsively started round jet. Dimas et al.' calculated the flow past a prolate spheroid by a threedimensional vortex method. Gharakhani et al.' applied a three-dimensional vortex-boundary method to the simulation of the compression stroke in combustion engines. The developments of three-dimensional analysis are necessary to understand the complicated vortical structures of the wake behind three-dimensional bodies. A lot of