On the swirling Trkalian mean flow field in solid rocket motors

Abstract

In this work, an exact Euler solution is derived under the fundamental contingencies of axisymmetric, steady, rotational, incompressible, single-phase, non-reactive and inviscid fluid, which also stand behind the ubiquitously used mean flow profile named ‘Taylor–Culick.’ In comparison with the latter, which proves to be complex lamellar, the present model is derived in the context of a Trkalian flow field, and hence is capable of generating a non-zero swirl component that increases linearly in the streamwise direction. This enables us to provide an essential mathematical representation that is appropriate for flow configurations where the bulk gaseous motion is driven to swirl. From a procedural standpoint, the new Trkalian solution is deduced directly from the Bragg–Hawthorne equation, which has been repeatedly shown to possess sufficient latitude to reproduce several existing profiles such as Taylor–Culick’s as special cases. Throughout this study, the fundamental properties of the present model are considered and discussed in the light of existing flow approximations. 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. Furthermore, the 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 motion, where the velocity and vorticity vectors remain orthogonal. By way of verification, a numerical simulation is carried out using a finite-volume solver, thus leading to a favourable agreement between theoretical and numerical predictions.