A Method for Determining the Velocity Induced by Highly Anisotropic Vorticity Blobs

Abstract

Resolution of boundary layer flows at moderate or high Reynolds numbers with the vortex blob method requires a great many isotropic elements. In this paper, an approximate method for determination of the induced velocity from highly anisotropic vorticity blobs is presented, and issues related to use of anisotropic elements in calculations with vortex blob algorithms for high Reynolds number near-wall flows are examined. The method presented here can be used to determine the induced velocity from smooth blob functions of arbitrary form, provided that the vorticity length scale associated with the blob is much less in one direction than in orthogonal directions. The ratio of these length scales is called the blob aspect ratio, ϵ, and is used as a small parameter to construct an asymptotic approximation to the induced velocity field. This method is applied in the present paper to derive induced velocity expressions for anisotropic Gaussian blob functions in both two and three dimensions. It is argued, using test calculations for a Blasius boundary layer, that although direct calculation of the induced velocity requires about an order of magnitude more CPU time for anisotropic Gaussian elements than for isotropic elements, this difference is more than made up for by a reduction of several orders of magnitude in the number of elements needed to resolve boundary layer flows at moderate to high Reynolds numbers. It is also found that the standard vortex blob representation leads to errors in the calculation of wall slip velocity and wall shear stress due to smoothing of the discontinuity between the real and image vorticity fields at the wall, but that these errors can be avoided by placing doublet-type elements along the wall.