On the compressible Hart-McClure and Sellars mean flow motions

Abstract

We consider the compressible flow analogue of the solution known colloquially as the Hart-McClure profile. This potential motion is used to describe the mean flow in the original energy-based combustion instability framework. In this study, we employ the axisymmetric compressible form of the potential equation for steady, inviscid, irrotational flow assuming uniform injection of a calorically perfect gas in a porous, right-cylindrical chamber. This equation is expanded to order M 4 w using a Rayleigh-Janzen sequence in powers of M 2 w , where M w is the wall Mach number. At leading order, we readily recover the original Hart-McClure profile and, at M 2 w , a closed-form representation of the compressible correction. By way of confirmation, the same solution is reconstructed using a novel application of the vorticity-streamfunction technique. In view of the favorable convergence properties of the Rayleigh-Janzen expansion, the resulting approximation can be relied upon from the headwall down to the sonic point and slightly beyond in a long porous tube or nozzleless chamber. As a windfall, the compressible Sellars motion that arises in the reverse flow problem driven by wall suction is deduced. Based on the simple closed-form expressions that prescribe this motion, the principal flow attributes are quantified parametrically and compared to existing incompressible and one-dimensional theories. In this effort, the local Mach number and pressure are calculated and shown to provide an improved formulation when gauged against one-dimensional theory. Our results are also compared to the two-dimensional axisymmetric solution obtained by Majdalani ["On steady rotational high speed flows: The compressible Taylor-Culick profile," Proc. R. Soc. London, Ser. A 463, 131-162 (2007)]. After rescaling the axial coordinate by the critical length L s , a parametrically-free form is obtained that is essentially independent of the Mach number. This behavior is verified analytically, thus confirming Majdalani's geometric similarity with respect to the critical distance. A secondary verification by computational fluid dynamics is also undertaken. When compared to existing rotational models, the compressible Hart-McClure plug-flow requires, as it should, a slightly longer distance to reach the speed of sound at the centerline. With this model, however, not only the centerline but the entire cross-section becomes fully choked.

Energy Steepened States of the Swirling Mean Flow in a Solid Rocket Motor

“…Additionally, it very interesting to note the "potential flow-like" mean flow for the Type I solution at 2 q = , which was previously identified with the Hart-McClure classic profile. 4,13 The present characteristics mirror those of the Hart-McClure mean flow in the radial and axial directions, thus predicting a uniform axial velocity profile and a linearly decreasing radial velocity as 0. r → However, the Trkalian swirl velocity is seen to decrease asymptotically as the distance to the wall is successively increased. These observations are quite similar to those of the Type I class of solutions described in Saad and Majdalani's work, although velocity magnitudes remain different here.…”