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 $| using a Rayleigh-Janzen sequence in powers of |${\bf M}_{\rm w}^{\rm 2} $|, where Mw is the wall Mach number. At leading order, we readily recover the original Hart-McClure profile and, at |${\bf M}_{\rm w}^{\rm 2} $|, 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) https://doi.org/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.
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September 2012
Research Article|
September 06 2012
On the compressible Hart-McClure and Sellars mean flow motions
Brian A. Maicke;
Brian A. Maicke
a)
Department of Mechanical, Aerospace and Biomedical Engineering,
University of Tennessee (UTSI)
, Tullahoma, Tennessee 37388-9700, USA
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Tony Saad;
Tony Saad
b)
Department of Mechanical, Aerospace and Biomedical Engineering,
University of Tennessee (UTSI)
, Tullahoma, Tennessee 37388-9700, USA
Search for other works by this author on:
Joseph Majdalani
Joseph Majdalani
c)
Department of Mechanical, Aerospace and Biomedical Engineering,
University of Tennessee (UTSI)
, Tullahoma, Tennessee 37388-9700, USA
Search for other works by this author on:
Brian A. Maicke
a)
Tony Saad
b)
Joseph Majdalani
c)
Department of Mechanical, Aerospace and Biomedical Engineering,
University of Tennessee (UTSI)
, Tullahoma, Tennessee 37388-9700, USA
a)
Currently, Assistant Professor of Mechanical Engineering, The Pennsylvania State University, Middletown, PA 17057.
b)
Currently, Research Associate, Institute for Clean and Secure Energy, University of Utah, UT 84112.
c)
Author to whom correspondence should be addressed. Electronic mail: [email protected]. Telephone: (931) 393-7280. Fax: (931) 393-7444. Web: http://majdalani.net.
Physics of Fluids 24, 096101 (2012)
Article history
Received:
October 24 2011
Accepted:
August 03 2012
Citation
Brian A. Maicke, Tony Saad, Joseph Majdalani; On the compressible Hart-McClure and Sellars mean flow motions. Physics of Fluids 1 September 2012; 24 (9): 096101. https://doi.org/10.1063/1.4748349
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