This work considers the wall-injected swirling motions evolving inside a right-cylindrical solid rocket motor (SRM) with and without headwall injection and, alternatively, a hybrid rocket engine (HRE) with an axisymmetric oxidizer showerhead at its forward closure. The bulk gaseous motion is modeled as a non-reactive, inviscid flow with a swirl velocity component that increases linearly along the axis of the chamber. Our approach is initiated from the compressible Bragg-Hawthorne equation, which is systematically solved using Rayleigh-Janzen perturbation expansions, to the extent of producing closed-form semi-analytical approximations for the stream function of a Trkalian motion from which all other flow attributes may be readily inferred. In the process, the case of a similarity-conforming headwall injection profile is considered, where the axial flow entering the chamber at the forward end enables us to simulate conditions associated with an idealized solid rocket motor with a reactive fore-end closure, or a simplified hybrid rocket engine with an oxidizer flux along its chamber's head-end section. Results are then compared to their counterparts obtained using a strictly incompressible Trkalian profile (Majdalani, J., and Fist, A., Improved Mean Flow Solution for Solid Rocket Motors with a Naturally Developing Swirling Motion, AIAA Paper 2014-4016, July 2014). They are also benchmarked against available compressible solutions in an effort to characterize the dilatational effects that are precipitated by flow acceleration in long rocket chambers or chambers with sufficiently large sidewafl injection. Besides the stream function, the velocity, pressure, temperature, and density are evaluated over a range of physical parameters corresponding to both SRM and HRE flows. Finally, the distortions affecting the velocity profiles are characterized and shown to result in a blunter motion near the center and a steeper curvature near the sidewall as a consequence of high speed flow acceleration. In comparison to non-swirling complex-lamellar solutions, we find the Trkalian solution to be generally faster and therefore able to reach sonic conditions in a shorter distance from the headwall.
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