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On the stability and the numerical solution of the unsteady interactive boundary-layer equation

Published online by Cambridge University Press:  21 April 2006

O. R. Tutty  and
S. J. Cowley
O. R. Tutty
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge CB3 9EW Present address: Department of Aeronautics and Astronautics, The University, Highfield, Southampton SO9 5NH.
S. J. Cowley
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT Present address: Department of Mathematics, Imperial College of Science and Technology, Huxley Building, 180 Queen's Gate, London SW7 2BZ.
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Abstract

By means of a high-frequency analysis it is shown that a Rayleigh instability is possible within the interactive boundary-layer formulation. This instability reflects the tendency of large-Reynolds-number flows to be unstable. For most, but not all, pressure-displacement relations both Rayleigh's and Fjørtoft's theorems hold, although Fjørtoft's criterion is not a sufficient condition for instability. However, for two pressure-displacement relations neither theorem could be proved, and for one of these, unstable flows exist which are free of inflexion points. Analytically, the existence of this instability may result in a finite-time singularity, while numerically the presence of Rayleigh modes often leads to accuracy problems which cannot be overcome by simple grid refinement. A test integration resulted in the generation of small grid-dependent eddies. It is suggested that the instability may be a possible cause of the eddy splitting observed in experiments on unsteady flows through distorted channels. This Rayleigh instability is also possible within the ‘inverse’ boundary-layer formulation, but is absent from classical boundary-layer problems.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 168 , July 1986 , pp. 431 - 456
Copyright
© 1986 Cambridge University Press

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