Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-16T15:46:15.309Z Has data issue: false hasContentIssue false
  • English
  • Français

Multiple asymptotic solutions for axially travelling waves in porous channels

Published online by Cambridge University Press:  25 September 2009

JOSEPH MAJDALANI
JOSEPH MAJDALANI*
Affiliation:
Mechanical, Aerospace and Biomedical Engineering Department, University of Tennessee Space Institute, 411 B. H. Goethert Parkway, Tullahoma, TN 37388-9700, USA
*
†Email address for correspondence: maji@utsi.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Travelling waves in confined enclosures, such as porous channels, develop boundary layers that evolve over varying spatial scales. The present analysis employs a technique that circumvents guessing of the inner coordinate transformations at the forefront of a multiple-scales expansion. The work extends a former study in which a two-dimensional oscillatory solution was derived for the rotational travelling wave in a porous channel. This asymptotic solution was based on a free coordinate that could be evaluated using Prandtl's principle of matching with supplementary expansions. Its derivation required matching the dominant term in the multiple-scales expansion to an available Wentzel-Kramers-Brillouin (WKB) solution. Presently, the principle of least singular behaviour is used. This approach leads to a multiple-scales approximation that can be obtained independently of supplementary expansions. Furthermore, a procedure that yields different types of WKB solutions is described and extended to arbitrary order in the viscous perturbation parameter. Among those, the WKB expansion of type I is shown to exhibit an alternating singularity at odd orders in the perturbation parameter. This singularity is identified and suppressed using matched asymptotic tools. In contrast, the WKB expansion of type II is found to be uniformly valid at any order. Additionally, matched asymptotic, WKB and multiple-scales expansions are developed for several test cases. These enable us to characterize the essential vortico-acoustic features of the axially travelling waves in a porous channel. All solutions are numerically verified, compared and discussed.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 636 , 10 October 2009 , pp. 59 - 89
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions. National Bureau of Standards.Google Scholar
Barron, J., Majdalani, J. & Van Moorhem, W. K. 2000 A novel investigation of the oscillatory field over a transpiring surface. J. Sound Vib. 235, 281–297.CrossRefGoogle Scholar
Berman, A. S. 1953 Laminar flow in channels with porous walls. J. Appl. Phys. 24, 1232–1235.CrossRefGoogle Scholar
Berman, A. S. 1958 Effects of porous boundaries on the flow of fluids in systems with various geometries. Proceedings of the Second United Nations Intl Conf. on the Peaceful Uses of Atomic Energy, Geneva, Switzerland. P/720 vol. 4, pp. 351–358.Google Scholar
Bosley, D. L. 1996 A technique for the numerical verification of asymptotic expansions. SIAM Rev. 38, 128–135.CrossRefGoogle Scholar
Brown, R. S., Blackner, A. M., Willoughby, P. G. & Dunlap, R. 1986 Coupling between acoustic velocity oscillations and solid propellant combustion. J. Propul. Power 2, 428–437.CrossRefGoogle Scholar
Casalis, G., Avalon, G. & Pineau, J.-P. 1998 Spatial instability of planar channel flow with fluid injection through porous walls. Phys. Fluids 10, 2558–2568.CrossRefGoogle Scholar
Dunlap, R., Blackner, A. M., Waugh, R. C., Brown, R. S. & Willoughby, P. G. 1990 Internal flow field studies in a simulated cylindrical port rocket chamber. J. Propul. Power 6, 690–704.CrossRefGoogle Scholar
Fabignon, Y., Dupays, J., Avalon, G., Vuillot, F., Lupoglazoff, N., Casalis, G. & Prévost, M. 2003 Instabilities and pressure oscillations in solid rocket motors. Aerosp. Sci. Technol. 7, 191–200.CrossRefGoogle Scholar
Griffond, J. 2002 Receptivity and aeroacoustic resonance in channels with blowing walls. Phys. Fluids 14, 3946–3962.CrossRefGoogle Scholar
Ma, Y., Van Moorhem, W. K. & Shorthill, R. W. 1991 Experimental investigation of velocity coupling in combustion instability. J. Propul. Power 7, 692–699.CrossRefGoogle Scholar
Majdalani, J. 1995 Improved Flowfield Models in Rocket Motors and the Stokes Layer with Sidewall Injection, Ph.D. Dissertation, University of Utah.Google Scholar
Majdalani, J. 1998 A hybrid multiple scale procedure for boundary layers involving several dissimilar scales. J. Appl. Math. Phys. (ZAMP) 49, 849–868.CrossRefGoogle Scholar
Majdalani, J. 1999 Vortical and acoustical mode coupling inside a two-dimensional cavity with transpiring walls. J. Acoust. Soc. Am. 106, 46–56.CrossRefGoogle Scholar
Majdalani, J. 2001 The oscillatory channel flow with arbitrary wall injection. J. Appl. Math. Phys. (ZAMP) 52, 33–61.CrossRefGoogle Scholar
Majdalani, J. & Flandro, G. A. 2002 The oscillatory pipe flow with arbitrary wall injection. Proc. R. Soc. Lond. A 458, 1621–1651.CrossRefGoogle Scholar
Majdalani, J. & Rienstra, S. W. 2002 Two asymptotic forms of the rotational solution for wave propagation inside viscous channels with transpiring walls. Q. J. Mech. Appl. Math. 55, 141–162.CrossRefGoogle Scholar
Majdalani, J. & Van Moorhem, W. K. 1998 Improved time-dependent flowfield solution for solid rocket motors. AIAA J. 36, 241–248.CrossRefGoogle Scholar
Munson, A. G. 1964 The vortical layer on an inclined cone. J. Fluid Mech. 20, 625–643.CrossRefGoogle Scholar
Richardson, E. G. & Tyler, E. 1929 The transverse velocity gradient near the mouths of pipes in which an alternating or continuous flow of air is established. Proc. R. Soc. Lond. A 42, 1–15.Google Scholar
Terrill, R. M. 1965 Laminar flow in a uniformly porous channel with large injection. Aeronaut. Q. 16, 323–332.CrossRefGoogle Scholar
Terrill, R. M. 1973 On some exponentially small terms arising in flow through a porous pipe. Q. J. Mech. Appl. Math. 26, 347–354.CrossRefGoogle Scholar
Ugurtas, B., Avalon, G., Lupoglazoff, N., Vuillot, F. & Casalis, G. 2000 Stability and acoustic resonance of internal flows generated by side injection. In Solid Propellant Chemistry, Combustion, and Motor Interior Ballistics (ed. V. Yang, T. B. Brill & W.-Z. Ren), vol. 185, pp. 823–836. AIAA Progress in Astronautics and Aeronautics.CrossRefGoogle Scholar
Van Dyke, M. 1975 Perturbation Methods in Fluid Mechanics. The Parabolic Press.Google Scholar
Yuan, S. W. 1956 Further investigation of laminar flow in channels with porous walls. J. Appl. Phys. 27, 267–269.CrossRefGoogle Scholar