Transverse Waves in Simulated Liquid Rocket Engines with Variable Headwall injection

Abstract

*† In this study, the transverse vorticoacoustic wave in a circular cylinder is characterized for a variable velocity profile at the injector faceplate. This particular configuration mimics the conditions leading to the onset of traveling radial and tangential waves in a simple liquid rocket engine (LRE). To capture the unsteady behavior in this physical setting, we consider a short thrust chamber with an injecting headwall and combine the benefits of three techniques: regular perturbations, Helmholtz decomposition, and boundary layer theory. First, regular perturbations are leveraged to linearize the equations of motion and, in the process, help to identify the unsteady interaction equations. Second, a Helmholtz decomposition of the first-order disturbance equations gives rise to a compressible, inviscid, and acoustic set that is responsible for driving the unsteady motion. This is accomplished in conjunction with an essentially incompressible, viscous, and vortical set that materializes by virtue of coupling with the acoustic wave at the boundaries. After recovering the acoustic mode from the resulting wave equation, the last step is to solve for the vortical mode by applying boundary layer theory and a judicious expansion of the rotational set with respect to a small viscous parameter,  . After some effort, an explicit formulation for variable headwall injection is obtained and validated by means of a limiting process verification that is based on two previously investigated cases, the uniform and bell-shaped injection profiles. The solution is then illustrated using two new configurations corresponding to laminar and turbulent profiles. In the process of comparing the four representative cases, the characteristics of the vorticoacoustic wave, including its penetration depth, spatial wavelength, and overshoot factor, are systematically explored and discussed. Most characteristics are found to depend on the penetration and Strouhal numbers along with the distance from the centerline. Along the axis of the chamber, the waves attributed to different injection profiles behave similarly to the extent that behavioral deviations among them increase as the sidewall is approached. This work also accounts for the presence of a downstream boundary that stands to produce left-traveling reflections whose pairing with the right-traveling waves promotes the establishment of a standing wave environment. The combined waves are formulated analytically and shown to be appreciable in view of their amplitudes that twice exceed those associated with traveling waves.