Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-19T05:33:00.932Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical simulations of uniformly stratified fluid flow over topography

Published online by Cambridge University Press:  26 April 2006

James W. Rottman ,
Dave Broutman  and
Roger Grimshaw
James W. Rottman
Affiliation:
Praxis, Inc., 6080 Franconia Road, Alexandria, VA 22310, USA and E. O. Hulburt Center for Space Sciences, Naval Research Laboratory, Washington DC 20375, USA
Dave Broutman
Affiliation:
Department of Applied Mathematics, University of New South Wales, Sydney, NSW 2052, Australia
Roger Grimshaw
Affiliation:
Department of Mathematics, Monash University, Clayton, VIC 3168, Australia
Get access Rights & Permissions [Opens in a new window]

Abstract

We use a high-resolution spectral numerical scheme to solve the two-dimensional equations of motion for the flow of a uniformly stratified Boussinesq fluid over isolated bottom topography in a channel of finite depth. The focus is on topography of small to moderate amplitude and slope and for conditions such that the flow is near linear resonance of either of the first two internal wave modes. The results are compared with existing inviscid theories: the steady hydrostatic analysis of Long (1955), time-dependent linear long-wave theory, and the fully nonlinear, weakly dispersive resonant theory of Grimshaw & Yi (1991). For the latter, we use a spectral numerical technique, with improved accuracy over previously used methods, to solve the approximate evolution equation for the amplitude of the resonant mode. Also, we present some new results on the modal similarity of the solutions of Long and of Grimshaw & Yi. For flow conditions close to linear resonance, solutions of Grimshaw & Yi's evolution equation compare very well with our fully nonlinear numerical solutions, except for very steep topography. For flow conditions between the first two resonances, Long's steady solution is approached asymptotically in time when the slope of the topography is sufficiently small. For steeper topography, the flow remains unsteady. This unsteadiness is manifested very clearly as periodic oscillations in the drag, which have been observed in previous numerical simulations and tow-tank experiments. We explain these oscillations as mainly due to the internal waves that according to linear theory persist longest in the neighbourhood of the topography.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 306 , 10 January 1996 , pp. 1 - 30
Copyright
© 1996 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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. & Guest, F. 1988 The nature of upstream blocking in uniformly stratified flow over long obstables. J. Fluid Mech. 188, 23–45.Google Scholar
Boyd, J. P. 1989 Chebyshev and Fourier Spectral Methods. Springer.
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1988 Spectral Methods in Fluid Dynamics. Springer.
Castro, I. P., Snyder, W. H. & Baines, P. G. 1990 Obstacle drag in stratified flow. Proc. R. Soc. Lond. A 429, 119–140.Google Scholar
Grimshaw, R. H. J. & Smyth, N. 1986 Resonant flow of a stratified fluid over topography. J. Fluid Mech. 169, 429–464.Google Scholar
Grimshaw, R. H. J. & Yi Zengxin. 1991 Resonant generation of finite-amplitude waves by the flow of a uniformly stratified fluid over topography. J. Fluid Mech. 229, 603–628.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
Lamb, K. G. 1994 Numerical simulations of stratified inviscid flow over a smooth obstable. J. Fluid Mech. 260, 1–22.Google Scholar
Long, R. R. 1955 Some aspects of the flow of stratified fluid. III. Continuous density gradients. Tellus 7, 341–357.Google Scholar
Paisley, M. F., Castro, I. P. & Rockliff, N. J. 1994 Steady and unsteady computations of strongly stratified flows over a vertical barrier. In Stably Stratified Flows: Flow and Dispersion over Topography (ed. I. P. Castro & N. J. Rockliff), pp. 39–59. Clarendon.
Yi Zengxin. & Warn, T. 1987 A numerical method for solving the evolution equation of solitary Rossby waves on a weak shear. Adv. Atmos. Sci. 4, 43–54.Google Scholar