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Stability of an oscillated fluid with a uniform density gradient

Published online by Cambridge University Press:  26 April 2006

David Jacqmin
David Jacqmin
Affiliation:
NASA Lewis Research Center, Cleveland, OH 44135, USA
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Abstract

We consider instabilities in a fluid with a constant density gradient that is subject to arbitrarily oriented oscillatory accelerations. With the Boussinesq approximation and for the case of an unbounded fluid, transformation to Lagrangian coordinates allows the reduction of the problem to an ordinary differential equation for each three-dimensional wavenumber. The problem has three parameters: the non-dimensional amplitude R of the base-state oscillation, the non-dimensional level of background steady acceleration, which for some cases can be represented in terms of a local (in time) Richardson number Ri, and the Prandtl number Pr. Some general bounds on stability are derived. For Pr = 1 closed-form solutions are found for impulse (delta function) accelerations and a general asymptotic solution is constructed for large R and general imposed accelerations. The asymptotic solution takes advantage of the fact that at large R wave growth is concentrated at ‘zero points’. These are times when the effective vertical wavenumber passes through zero. Kelvin–Helmholtz instabilities are found to dominate at low R while Rayleigh–Taylor instabilities dominate at high. At high R, the uniform shear of the Kelvin–Helmholtz case tends to distort and weaken instability waves. With unsteady flows, Ri = ¼ is no longer an instability limit. Significant instabilities have been found for sinusoidal forcing for Ri up to 0.6.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 219 , October 1990 , pp. 449 - 468
Copyright
© 1990 Cambridge University Press

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