The Bidirectional Vortex with Sidewall Injection

Abstract

The purpose of this paper is to derive an approximate solution for the bidirectional vortex in a right-cylindrical chamber with sidewall injection. The flowfield may be used to describe the bulk gas motion in a vortex-hybrid rocket chamber as well as in other cyclonic devices that combine circulatory motion with mass transfer. Our mathematical model is based on steady, rotational, axisymmetric, incompressible, and quasi-viscous flow conditions. Two distinctive perturbation parameters are used: the ratio of sidewall-totangential injection velocities and the reciprocal of the vortex Reynolds number, which combines the swirl number, chamber aspect ratio, and viscous Reynolds number. First, an Euler-type solution is obtained using variation of parameters and suitable boundary conditions that secure the sidewall mass injection requirement. This enables us to reproduce the two-cell, bipolar motion observed in vortex-hybrid thrust chambers. Second, to capture the viscosity-dominated forced vortex and sidewall boundary layers, the regularized tangential momentum equation is expanded in the reciprocal of the vortex Reynolds number. A uniformly valid, triple-deck approximation for the tangential velocity is then constructed using matched asymptotics. Viscous corrections in the axial and radial directions are also resolved. Additionally, we calculate pressure distributions, axial and radial velocity extrema, vorticity formation, roll torques, and the dynamic mantle location that separates inner and outer vortices. Finally, by relating fundamental variables to the bidirectional swirl number and wall regression rate, essential flow characteristics are captured throughout the chamber. As a windfall, an explicit relation is obtained linking the mantle location to the wall injection rate.