© 2003 by Oxford University Press
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On the Evolution of Time-Dependent Localized Disturbances to Dean Flow in a Channel with Slowly Varying Curvature and Gap-Width
( 1 Department of Mathematics, The City University, Northampton Square, London EC1V OHB )
The first approximation to steady flow in channels with slowly varying curvature and gap width is locally of the same form as in the classical Dean problem. Eagles (1992) considered small disturbances to this flow with a W. K. B. factor to take account of the slow streamwise variation of the base flow. These disturbances were sinusoidal in the z-direction (perpendicular to the main stream) and of a steady form. The idea was that if the disturbance grows streamwise in any section of the flow then the flow is unstable. In the present paper we allow a factor exp(i
0t) in the disturbance, where 0 is a real constant and t is the time variable appropriate here. Single modes with such a factor are combined together by integrals with respect to 0 to give initial-value problems for disturbances localized in the streamwise direction. The aim is to see whether or not the disturbances move downstream as t increases, and whether they grow or decay in size as they move, for various values of the parameters. The results are consistent with the above supposition of Eagles that if the steady state disturbance grows downstream for any distance the flow is unstable.
Received 11 July 2001. January and May 2002.