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Stability of stratified flow of large depth over finite-amplitude topography

Published online by Cambridge University Press:  26 April 2006

Dilip Prasad ,
Jaime Ramirez  and
T. R. Akylas
Dilip Prasad
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology Cambridge, MA 02139, USA
Jaime Ramirez
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology Cambridge, MA 02139, USA
T. R. Akylas
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology Cambridge, MA 02139, USA
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Abstract

The flow of a Boussinesq density-stratified fluid of large depth past the algebraic mountain (‘Witch of Agnesi’) is studied in the hydrostatic limit using the asymptotic theory of Kantzios & Akylas (1993). The upstream conditions are those of constant velocity and Brunt–Väisälä frequency. On the further assumptions that the flow is steady and there is no permanent alteration of the upstream flow conditions (no upstream influence), Long's model (Long 1953) predicts a critical amplitude of the mountain (ε = 0.85) above which local density inversions occur, leading to convective overturning. Linear stability analysis demonstrates that Long's steady flow is in fact unstable to infinitesimal modulations at topography amplitudes below this critical value, 0.65 [lsim ] ε < 0.85. This instability grows at the expense of the mean flow and may be attributed to a discrete spectrum of modes that become trapped over the mountain in the streamwise direction. The transient problem is also solved numerically, mimicking impulsive startup conditions. In the absence of instability, Long's steady flow is reached. For topography amplitudes in the unstable range 0.65 [lsim ] ε < 0.85, however, the flow fluctuates about Long's steady state over a long timescale; there is no significant upstream influence and no evidence of transient wave breaking is found for ε [les ] 0.75.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 320 , 10 August 1996 , pp. 369 - 394
Copyright
© 1996 Cambridge University Press

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