Pressure gradient effects on nonequilibrium far wakes

Abstract

To summarize, the following conclusions are noteworthy: 1) The azimuthal component of velocity is nonzero in contrast to nonrotating case in which v$> = 0.2) The rotating effect generates vorticity in that, for large distances, the flow pattern is irrotational in the nonrotating case. In the rotating case, the azimuthal component of velocity is responsible for vorticity for large R.3) The wave-like terms f n (hR) (corresponding to a boundary layer) appearing in the solution, effective in a small distance from the surface of the sphere, and behaving like exp[-(1 + i)(R e /2) lf2 (R -1)] are essentially similar in nature in rotating and nonrotating cases.4) The drag on the sphere remains unchanged (due to rotation) to the order 6, since v x , v r , and p remain unchanged to that order.In a forthcoming paper, the preceding discussion will be extended to a compressible fluid in the presence of a magnetic field.T HE purpose of this note is to present some results of recent calculations 1 pertaining to the far wake of re-entry vehicles. In these calculations, the effect of pressure gradient on a laminar-turbulent nonequilibrium mixture is assessed. The far wake problem has attracted many investigators, for example, Refs. 2-4; however, to the authors' knowledge, the quantitative effect of pressure gradient on a dissociatedionized fluid medium undergoing nonequilibrium chemical kinetics has not appeared in the literature. AnalysisAn integral method of solution is employed similar to that of Ref . 5. The chemical model consists of 22 chemical reactions involving the 8 species 0, N, 0 2 , N 2 , NO, NO+, e~, 0 2~. The O 2~ species appears in an attachment reaction and a mutual neutralization reaction. The back rate constants for these reactions are taken from Ref. 6.The Sutherland relation is used to compute the viscosity in the laminar region. In the turbulent region, the turbulent diffusivity model given bywas used. This expression is attributed to Lees and Hromas 2 and Ting and Libby. 7 Here 5 m is a transformed wake thickness, p e and u e refer to the density and velocity at the edge of the wake, and U Q refers to the velocity at the centerline of symmetry. The transition criteria given by Zeiberg 8 was employed in determining the transition point. The procedure given by Pallone, et al. 4 for calculating the transition point could have been used as an alternative method. The pressure gradient in the wake was determined (with a slight modification) from the blast-wave theory of Sakurai. 9