Complex Lamellar Helical Solution for Cyclonically Driven Hybrid Rocket Engines

Abstract

This paper introduces a viscous approximation for the complex-lamellar helical solution of a cyclonically-driven, right-cylindrical chamber with sidewall mass injection. The flowfield may be used to describe the bulk gaseous motion in a bidirectional vortex-hybrid rocket engine, such as the one developed by NASA MSFC/ORBITEC. Our mathematical model is based on steady, rotational, axisymmetric, incompressible, and quasi-viscous flow conditions. Two distinctive perturbation parameters are used in this process: the ratio of sidewall-to-tangential 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 used as a basis to satisfy the fundamental conditions associated with the ensuing helical motion, including the ability to secure the sidewall mass injection requirement at the simulated burning surface. 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 viscous momentum equations are manipulated and expanded using perturbation tools. A uniformly valid, tripledeck approximation for the tangential velocity is then constructed using matched asymptotic expansions. Using a similar procedure, viscous corrections in the axial and radial directions are resolved. By relating fundamental variables to the bidirectional swirl number and wall regression rate, essential flow characteristics are captured throughout the chamber.