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The Quarterly Journal of Mechanics and Applied Mathematics 1989 42(1):115-130; doi:10.1093/qjmam/42.1.115
© 1989 by Oxford University Press
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ON THE INTERACTION OF TOLLMIEN-SCHLICHTING WAVES IN AXISYMMETRIC SUPERSONIC FLOWS

P. W. DUCK1 and P. HALL2

( 1Department of Mathematics, University of Manchester Oxford Road, Manchester M13 9PL
2Department of Mathematics, University of Exeter North Park Road, Exeter EX4 4QE )

It is known that two-dimensional lower-branch Tollmien-Schlichting waves described by triple-deck theory are always stable for planar supersonic flows. Here the possible occurrence of axisymmetric unstable modes in the supersonic flow around an axisymmetric body is investigated. In particular flows around bodies with typical radii comparable with the thickness of the upper deck are considered. It is shown that such unstable modes exist below a critical non-dimensional radius of the body {alpha}0. At values of the radius above {alpha}0 all the modes are stable whilst if unstable modes exist they are found to occur in pairs. The interaction of these modes in the nonlinear regime is investigated using a weakly nonlinear approach and it is found that, dependent on the frequencies of the imposed Tollmien—Schlichting waves, either of the modes can be set up.


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