A model for the core of a slender viscous vortex

A model for the core of a viscous vortex is presented. It is based on a s i d a d t y solution to the Naner-Stokes equations in which the following assumptions are made: the flow is incompressible; . the flow is axisymmetric; ., the flow is conical; 2 the Reynolds number is high; the vortex is slender. Under them assumptions, the Navier-Stokes equations and the continuity equation take the form of four coupled, nonlinear, ordinary differential equations. These equations are discretized using centered differences, and solved by a Newton procedure, after imposing appropriate boundary conditions. The model is found to agree fairly well with experimeutal data of Earnshaw [l], and very well with the matched asymptotic solution for a vortex core developed by Hall [2]. The level of total pressure loss pxedicted in the core of the vortex is found to be independent of Reynolds number, depending only on the edge circumferential velocity. Some implications of this result on the prediction of the onset of vortex bursting are given. W Introduction Due to the critical role that vortices play in aeronautical engineering, naval engineering and metmrology, a number of modela for vortex cores have been developed through the years. These have ranged from very simple nd hoc models to complicated mnltiple-scale analyses. In this paper, the vortex core wiU be modeled as a region of distributed vorticity. Similar models have been proposed by Long [3], Hall [2], Luckring [4] and Brown (51, among others. While a vortex core of distributed vorticity can occur only in a viscous flow, Euler calculations of vortex flows by the authors [6,7] and others [8,9,10] indicate that the dis Crete Euler equations model the flow inside a vortex core surprisingly well. In particular, the level of total pressure loss inside the core of a given vortex computed by the discrete Euler method is independent of computational parameters such as mesh spacing and artificial viscosity level [Ill. and agrees well with losses measured in experiment W 'Asastanc Pmfeuor, AlAA Member tPmfesessor, AlAA Fellow 1 Copyright @ 1988 by the American Institute of Aeronautics and Astronautics, Inc. AU rights reserved. Figure 1: Cylindrical Coordinate System [12]. Navier-Stokes c d d a t i o l u [9,13,14] have comparable l o w . This numerical evidence has driven the authors to determine whsther there is a vortex core model which accounts for viscosity, but has a total presanre loss level that is independent of the level of viscosity. Burgers' Vortex Burgers developed a core model [I51 that, it will be shown, has this characteristic. It is an exact solution to the axisymmetric, incomprnvlible Navier-Stokes equations. For the cylindrical (r,6,2) system, shown in Figure 1, the velocity field is given by u = -AI u = u(r) w = 2Az where u(r) is to be determined from the 8-momentum equation. The velocities are non-dimensionalired by a reference velocity W , and r and I are nondimensionalized by a reference length L. This velocity field satisfies the continuity eauation. aw 0, I S 3 --(In)+ = r a? a2 identically. The r-momentum equation (with pressure nondimensiondized by pW2) reduces, for the above choices of u. u and w. to The z-momentum equation redncea to