The Bidirectional Vortex. Part 2: Viscous Core Corrections

Abstract

This article focuses on the viscous core of the bidirectional flowfield arising in a swirldriven thrust chamber. By regularizing the momentum equation in the tangential direction, the boundary layer equation that controls the forced vortex near the chamber axis is obtained. After identifying the coordinate transformation needed to resolve the rapid changes near the core, an inner expansion is arrived at. This expansion is then matched with the outer solution associated with the free vortex; the latter is known to prevail in the outer region. By combining inner and outer expansions, uniformly valid approximations are obtained for the swirl velocity, vorticity, and pressure. These are shown to be strongly influenced by a dynamic similarity parameter that combines the mean flow Reynolds number and the chamber aspect ratio. Referred to as the vortex Reynolds number V, this dimensionless grouping enables us to quantify the characteristic features of the bidirectional vortex. Among them is the thickness of the viscous core which is found to decrease with the square root of V. The converse can be said of the maximum swirl velocity. In the same vein, the angular frequency of the rigid-body rotation of the forced vortex near the core is found to be linearly proportional to V. The form of the swirl velocity is reminiscent of the Burgers vortex; here, it is based on the aspect ratio of the thrust chamber. The resulting theoretical predictions compare favorably with experimental measurements and computational results over the length of the chamber.