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Numerical simulations of stratified inviscid flow over a smooth obstacle

Published online by Cambridge University Press:  26 April 2006

Kevin G. Lamb
Kevin G. Lamb
Affiliation:
Department of Physics, Memorial University of Newfoundland, St. John's, Newfoundland, Canada A1B 3X7
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Abstract

Results of numerical simulations of the flow of a non-rotating, inviscid, Boussinesq fluid over smooth two-dimensional obstacles are described. The fluid has finite depth and a rigid lid. Far upstream of the obstacle the horizontal velocity and buoyancy frequency are uniform. Comparisons with linear theory for small-amplitude obstacles are made and the long-time behaviour is compared with steady-state Long's model solutions. Comparisons with the time-dependent results of Baines (1979) are done. For Froude numbers between ½ and 1 the obstacle amplitude is varied in order to determine the amplitudes needed to initiate wave breaking. These results are compared with the predictions of Long's model and with the experimental results of Baines (1977) showing good agreement with the latter. It is found that wave breaking occurs for amplitudes significantly lower than Long's model predicts for a large range of Froude numbers. This is shown to be the result of the generation of large-amplitude lee waves with wavelengths longer than that of stationary lee waves, but not long enough to propagate upstream. The behaviour of these waves is coupled to the generation of both longer mode-one waves which do propagate upstream from the obstacle and to mode-two waves which propagate against the flow as they are advected downstream. It is also coupled to oscillations in the wave drag. The periods of the wave drag oscillations are compared to experimental results showing good agreement with cases for which oscillations have been observed. The behaviour of these large transient lee waves is compared with the theoretical results contained in Grimshaw & Yi (1991), showing some similarities. As the Froude number approaches 0.5 the breaking behaviour is no longer due to these large waves, with the result that wave breaking occurs much later.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 260 , 10 February 1994 , pp. 1 - 22
Copyright
© 1994 Cambridge University Press

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References

Baines, P. G. 1977 Upstream influence and Long's model in stratified flows. J. Fluid Mech. 82, 147–159.Google Scholar
Baines, P. G. 1979 Observations of stratified flow over two-dimensional obstacles in fluid of finite depth. Tellus 31, 351–371.Google Scholar
Baines, P. G. 1994 Topographic Effects in Stratified Flows. Cambridge University Press (to appear).
Bell, J. B., Colella, P. & Glaz, H. M. 1989a A second-order projection method for the incompressible Navier-Stokes equations. J. Comput. Phys. 85, 257–283.Google Scholar
Bell, J. B. & Marcus, D. L. 1992 A second-order projection method for variable-density flows. J. Comput. Phys. 101, 334–348.Google Scholar
Bell, J. B., Solomon, J. M. & Szymczak, W. G. 1989b A second-order projection method for the incompressible Navier Stokes equations on quadrilateral grids. AIAA 9th Computational Fluids Dynamics Conf., Buffalo, NY, June 14–16, 1989..
Benjamin, T. B. 1970 Upstream influence. J. Fluid Mech. 40, 49–79.Google Scholar
Boyer, D. L. & Tao, L. 1987 Impulsively started, linearly stratified flow over long ridges. J. Atmos. Sci. 44, 23–42.Google Scholar
Castro, I. P. 1994 Effects of stratification on separated wakes: Part I. Weak static stability. Proc. 3rd IMA Meeting on Stably Stratified Flows (Leeds, 1989) (to appear).
Castro, I. P. & Snyder, W. H. 1988 Upstream motions in stratified flow. J. Fluid Mech. 187, 487–506.Google Scholar
Castro, I. P. & Snyder, W. H. 1990 Obstacle drag in stratified flow. Proc. R. Soc. Lond. A 429, 119–140.Google Scholar
Davis, R. E. 1969 The two-dimensional flow of a stratified fluid over an obstacle. J. Fluid Mech. 36, 127–143.Google Scholar
Golub, G. H. & Loan, Van C. F. 1989 Matrix Computations, 2nd edn, pp. 170–171. The Johns Hopkins University Press.
Grimshaw, F. & Smyth, N. 1986 Resonant flow of a stratified fluid over topography. J. Fluid Mech. 169, 429–464.Google Scholar
Grimshaw, R. & Yi, Z. 1991 Resonant generation of finite-amplitude waves by the flow of a uniformly stratified fluid over topography. J. Fluid Mech. 229, 603–628 (referred to herein as GY).Google Scholar
Hanazaki, H. 1989 Upstream advancing columnar disturbances in two-dimensional stratified flow of finite depth. Phys. Fluids A 1, 1976–1987.Google Scholar
Hanazaki, H. 1992 A numerical study of nonlinear waves in a transcritical flow of stratified fluid past an obstacle. Phys. Fluids A 4, 2230–2243.Google Scholar
Hanazaki, H. 1993 On the nonlinear internal waves excited in the flow of a linearly stratified Boussinesq fluid. Phys. Fluids A 5, 1201–1205.Google Scholar
Janowitz, G. S. 1981 Stratified flow over a bounded obstacle in a channel of finite height. J. Fluid Mech. 110, 161–170.Google Scholar
Long, R. R. 1953 Some aspects of the flow of stratified fluids. I. A theoretical investigation. Tellus 5, 42–58.Google Scholar
Long, R. R. 1955 Some aspects of the flow of stratified fluids. III. Continuous density gradients. Tellus 7, 341–357.Google Scholar
Long, R. R. 1972 Finite amplitude disturbances in the flow of inviscid rotating and stratified fluids over obstacles. Ann. Rev. Fluid Mech. 4, 69–92.Google Scholar
Mcintyre, M. E. 1972 On Long's hypothesis of no upstream influence in uniformly stratified or rotating flow. J. Fluid Mech. 52, 209–243.Google Scholar
Wei, S. N., Kao, T. W. & Pao, H.-P. 1975 Experimental study of upstream influence in the two-dimensional flow of a stratified fluid over an obstacle. Geophys. Fluid Dyn. 6, 315–336.Google Scholar