Improved Mean Flow Solution for Solid Rocket Motors

Abstract

In this work, a new exact Euler solution is presented as a potential flow candidate for describing the internal gaseous motion in simulated solid rocket motors both with and without headwall injection. For this reason, the solution may be simultaneously employed to capture the streamtube motion in conventional hybrid rocket engines with circular crosssections and large oxidizer injection velocities at their forward closure. As usual, the motor is idealized as a right-cylindrical chamber with a permeable sidewall and either finite or infinite headwall impedance. Furthermore, the solution is derived under the same fundamental contingencies of axisymmetric, steady, rotational, incompressible, single phase, non-reactive, and inviscid fluid, which stand behind the ubiquitously used mean flow known as the Taylor-Culick profile. In comparison to the latter, which proves to be complex lamellar, the present model is shown to be of the Beltramian type, hence capable of generating a nonzero swirl velocity that increases linearly in the streamwise direction. This enables us to provide an essential mathematical representation that is appropriate of flow configurations where the bulk gaseous motion is permitted to swirl. Examples abound and one may cite the classic experiments of Dunlap and co-workers, which focus on porous chambers with circular cross-sections, and where the inevitable presence of swirl as a natural flow companion is clearly demonstrated (Dunlap, R., Blackner, A., Waugh, R., Brown, R., and Willoughby, P., “Internal Flow Field Studies in a Simulated Cylindrical Port Rocket Chamber,” Journal of Propulsion and Power, Vol. 6, No. 6, 1990, pp. 690-704. doi: 10.2514/3.23274). Besides its relevance to the modeling of solid and hybrid rocket motor flow fields, the basic solution that we obtain can therefore be applied to problems involving paper manufacture, drainage of watery suspensions, isotope separation, boundary layer control, water irrigation, surface ablation, and sweat cooling. From a procedural standpoint, the new Beltramian solution is deduced directly from the Bragg-Hawthorne equation, a reduced form of Euler’s momentum equation, which has been repeatedly shown to possess sufficient generality to reproduce several existing profiles, such as TaylorCulick’s, as special cases. Throughout this study, the fundamental properties and benefits of the present model are highlighted and discussed in the light of existing flow approximations and experiments. Consistently with the original Taylor-Culick mean flow motion, the Beltramian velocity is seen to exhibit an axial component that increases linearly with the distance from the headwall, and a radial component that remains axially invariant.