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 ${\bf M}_{\rm w}^4 $Mw4 using a Rayleigh-Janzen sequence in powers of ${\bf M}_{\rm w}^{\rm 2} $Mw2, where Mw is the wall Mach number. At leading order, we readily recover the original Hart-McClure profile and, at ${\bf M}_{\rm w}^{\rm 2} $Mw2, a closed-form representation of the compressible correction. By way of confirmation, the same solution is re-constructed 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)10.1098/rspa.2006.1755]. After rescaling the axial coordinate by the critical length Ls, 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.