Zonal vortex method and impulsively started flow around a circular cylinder

Abstract

IN this paper, a zonal vortex method and its application to simulating the impulsively started flow of a circular cylinder are presented. This method treats the attached viscous flow region and the separated flow region separately. The attached flow region is computed through solving boundarylayer equations by the finite-difference method. Only the separated flow is computed through solving Navier-Stokes equations by the vortex method. Since the separated flow region has a length scale of 0(1), in the vortex method used for this region, the number of new vortices introduced at the surface of the body per time step is relatively insensitive to the Reynolds number of the flow. For simulation of highReynolds-number flows with massive separation, the total number of vortices in the flowfield, hence the computer storage and computer time, is greatly reduced. Contents The impulsively started flow around a circular cylinder is complex and all of the phenomena of fluid mechanics are presented. Because of its fundamental importance, this flow has become a typical problem in the study of separated flows. Bouard and Coutanceau1 have done careful experimental studies on this problem and provided valuable data for comparison with numerical solutions. Smith and Stansby2 have recently calculated this flow using a vortex method. In this method, the vorticity equation is solved using random walk for diffusion and the vortex-in-cell method for convection in a time-stepping procedure. Vortices are introduced at the surface at each time step to satisfy the no-slip condition. Their calculations revealed detailed flow features of the wake and were in good agreement with experimental observations in Ref. 1. Their numerical experiment showed that accurate simulation requires the introduction of a sufficiently large number of vortices at the surface per time step. This number increases as the Reynolds number increases. Therefore, for simulation of a high-Reynolds-number flow by the vortex method, the total number of vortices in the flow may become very large as time increases, requiring large computer storage and computer time. In this paper, the vortex method is improved by a zonal approach. For a high-Reynolds- number flow with massive separation, the vorticity in the upstream of the separation point is confined in a layer near the surface of the body. The length scale of the thickness of this layer is 0(Re~l/2). The length scale of the separated flow region is 0(1). It is difficult to accommodate these two scales when solving the NavierStokes equations in the whole flow region. The approach of