Transverse Waves in Simulated Liquid Rocket Engines

Abstract

This work seeks to provide a closed-form analytical solution for the transverse vorticoacoustic wave in a circular cylinder with headwall injection. This particular configuration mimics the conditions leading to the onset of traveling radial and tangential waves in an idealized liquid rocket engine. Assuming a short cylindrical chamber with two injection showerhead models (a top hat, uniform flow, and a bell-shaped sinusoidal profile), regular perturbations are used to linearize the problem’s conservation equations. Flow decomposition is subsequently applied to the first-order disturbance equations, thus giving rise to a compressible, inviscid, acoustic set that is responsible for driving the unsteady motion, and to an incompressible, viscous, vortical set that is driven by virtue of coupling with the acoustic mode along solid boundaries. While the acoustic mode is readily recovered from the wave equation, the induced vortical mode is resolved using boundary-layer theory and a judicious expansion of the rotational equations with respect to a small viscous parameter, . After some effort, an explicit formulation is arrived at for each of the uniform and bell-shaped injection profiles. The two solutions are then presented, verified numerically, and compared at fixed spatial locations within the chamber. The penetration depth of the unsteady boundary layer is also characterized. Unlike the solution based on uniform headwall injection, the vorticoacoustic wave based on the bell-shaped mean flow is found to be more realistic; being capable of securing the no-slip requirement at both headwall and sidewall boundaries.