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The Quarterly Journal of Mechanics and Applied Mathematics Advance Access published online on March 4, 2006
The Quarterly Journal of Mechanics and Applied Mathematics, doi:10.1093/qjmam/hbl001
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© The author 2006. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org
Received August 31, 2005
Revised January 25, 2006

Article

INSTABILITIES OF A FLEXIBLE SURFACE IN SUPERSONIC FLOW

J. D. A. Walker 1, A. Fletcher 1, and A. I. Ruban 2 *

( 1 Department of Mechanical Engineering and Mechanics, Lehigh University, 19 Memorial Drive West, Bethlehem, PA 18015, USA
2 Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL
)

* To whom correspondence should be addressed.
A. I. Ruban, E-mail: ruban{at}ma.man.ac.uk


  Abstract

The stability of a supersonic boundary layer above a flexible surface is considered in the limit of large Reynolds number and for Mach numbers O(1). Asymptotic theory of viscous-inviscid interaction has been used for this purpose. We found that for a simple elastic surface, for which deflections are proportional to local pressure differences, the boundary-layer flow remains stable as it is for a rigid wall. However, when either damping or surface inertia is included the flow becomes unstable. Moreover, in a certain range of wave numbers the boundary layer develops more then one unstable mode. It is interesting that these modes are connected to one another via saddle points in the complex-frequency plane. A more complex Kramer-type surface is also analysed and in some parameter ranges is found to permit the evolution of unstable Tollmien-Schlichting waves. The neutral curves are found for a variety of situations related to the parameters associated with the flexible surface.


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