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.

On the Sidewall Boundary Layer of Transverse Waves in Simulated Liquid Rocket Engines

“…Because the bulk streamwise motion of the fluid remains axial, a contribution of the injection mechanism is expected to be significant (and is therefore captured) in the leading-order solution. 19 In contrast, the radial direction represents a secondary axis with respect to the mean flow motion. It then follows that the bulk development of the boundary layer at the sidewall Figure 3 illustrates the dependence of the wave's boundary layer thickness on the viscous parameter.…”

“…17,22 In both geometric configurations, the velocity adherence constraint is imposed at the injecting surfaces, and these correspond to either the headwall or the sidewall of the simulated LRE and SRM, respectively. However, since the vortical waves driven by the injecting surfaces are determined in previous studies, [17][18][19] slippage along the non-injecting surface (sidewall) must not be allowed in the improved formulation. Consistent with other boundary layer studies, attenuation of the unsteady vorticity component is expected away from the sidewall.…”

“…This change manifests itself in an overshoot taking place in the near-wall region. 19,23 As usual, the phase shift between the vortical and acoustic fields leads to a positive coupling of the tangential velocity. This behavior is illustrated in Figs.…”

“…In-depth formulations that focus on wave characterization are presented by Haddad and Majdalani,18,19 where detailed solutions for axisymmetric injection profiles are constructed. In both studies, the vortical wave is generated at the headwall and propagates downstream.…”

“…Assuming waves generated by uniform headwall injection, these researchers also analyze the vorticoacoustic boundary layer forming above the injector faceplate. Later studies by Haddad and Majdalani 18,19 introduce a framework for describing the transverse evolution of vorticoacoustic waves and their dependence on variable headwall injection patterns. However, despite the ability of their solutions to satisfy the no-slip requirement at the headwall, and partially at the sidewall, their tangential motion still permits slippage along the sidewall.…”