Improved Mean Flow Solution for Solid Rocket Motors with a Naturally Developing Swirling Motion

Abstract

In this work, a new exact Euler 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 profile named “Taylor-Culick.” In comparison to the latter, which proves to be complex lamellar, the present model is shown to be of the Trkalian type, hence capable of generating a nonzero swirl component 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). From a procedural standpoint, the new Trkalian solution is deduced directly from the Bragg-Hawthorne equation, which has been repeatedly shown to possess sufficient platitude to reproduce several existing profiles such as Taylor-Culick’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. Consistent with the original Taylor-Culick mean flow motion, the Trkalian velocity is seen to exhibit both axial and tangential components that increase linearly with the distance from the headwall, and a radial component that remains axially invariant. However, unlike its axial and radial counterparts, which identically satisfy the no slip requirement at the sidewall (even in their purely inviscid form), the tangential velocity must be treated asymptotically so as to capture the evolving sidewall shear layer while resolving the core layer near the centerline. At the outset, a matched-asymptotic expression for the swirl velocity is obtained and discussed. Finally, our Trkalian model is shown to form a subset of the Beltramian class of solutions for which the velocity and vorticity vectors are not only parallel but also directly proportional. This characteristic feature is interesting as it stands in sharp contrast to the complex lamellar nature of the Taylor-Culick flow, where velocity and vorticity vectors remain orthogonal.