Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-24T08:56:24.087Z Has data issue: false hasContentIssue false
  • English
  • Français

The structure of low-Froude-number lee waves over an isolated obstacle

Published online by Cambridge University Press:  08 November 2011

Stuart B. Dalziel ,
Michael D. Patterson ,
C. P. Caulfield  and
Stéphane Le Brun
Stuart B. Dalziel*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
Michael D. Patterson
Affiliation:
Department of Architecture and Civil Engineering, University of Bath, Bath BA2 7AY, UK
C. P. Caulfield
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK BP Institute, University of Cambridge, Madingley Road, Cambridge CB3 0EZ, UK
Stéphane Le Brun
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK École Polytechnique, Route de Saclay, 91120 Palaiseau, France
*
†Email address for correspondence: s.dalziel@damtp.cam.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We present new insight into the classical problem of a uniform flow, linearly stratified in density, past an isolated three-dimensional obstacle. We demonstrate how, for a low-Froude-number obstacle, simple linear theory with a linearized boundary condition is capable of providing excellent quantitative agreement with laboratory measurements of the perturbation to the density field. It has long been known that such a flow may be divided into two regions, an essentially horizontal flow around the base of the obstacle and a wave-generating flow over the top of the obstacle, but until now the experimental diagnostics have not been available to test quantitatively the predicted features. We show that recognition of a small slope that develops across the obstacle in the surface separating these two regions is vital to rationalize experimental measurements with theoretical predictions. Utilizing the principle of stationary phase and causality arguments to modify the relationship between wavenumbers in the lee waves, linearized theory provides a detailed match in both the wave amplitude and structure to our experimental observations. Our results demonstrate that the structure of the lee waves is extremely sensitive to departures from horizontal flow, a detail that is likely to be important for a broad range of geophysical manifestations of these waves.

JFM classification

Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Topographic effects
Type
Papers
Information
Journal of Fluid Mechanics , Volume 689 , 25 December 2011 , pp. 3 - 31
Copyright
Copyright © Cambridge University Press 2011

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

1. Baines, P. G. 1987 Upstream blocking and airflow over mountains. Annu. Rev. Fluid Mech. 19, 75–97.CrossRefGoogle Scholar
2. Baines, P. G. 1995 Topographic Effects in Stratified Flows, p. 482. Cambridge University Press.Google Scholar
3. Booker, J. R. & Bretherton, F. P. 1967 The critical layer for internal gravity waves in a shear flow. J. Fluid Mech. 27, 513–539.CrossRefGoogle Scholar
4. Dalziel, S. B., Hughes, G. O. & Sutherland, B. R. 1998 Synthetic Schlieren. In Proceedings of the 8th International Symposium on Flow Visualization, ed. Carlomagno & Grant, paper 062.Google Scholar
5. Dalziel, S. B., Hughes, G. O. & Sutherland, B. R. 2000 Whole field density measurements by ‘Synthetic Schlieren’. Exp. Fluids 28, 322–335.CrossRefGoogle Scholar
6. Dalziel, S. B., Carr, M., Sveen, K. J. & Davies, P. A. 2007 Simultaneous Synthetic Schlieren and PIV measurements for internal solitary waves. Meas. Sci. Technol. 18, 533–547.CrossRefGoogle Scholar
7. Drazin, P. G. 1961 On the steady flow of a fluid of variable density past an obstacle. Tellus 13, 239–251.CrossRefGoogle Scholar
8. Gill, A. E. 1982 Atmosphere-Ocean Dynamics, p. 662. Academic Press.Google Scholar
9. Greenslade, M. D. 1994 Strongly stratified airflow over and around mountains. In Stably Stratified Flows: Flow and Dispersion over Topography (ed. Castro, I.P. & Rockcliff, N.J. ). Oxford.Google Scholar
10. Greenslade, M. D. 2000 Drag on a sphere moving horizontally through a stratified fluid. J. Fluid Mech. 418, 339–350.CrossRefGoogle Scholar
11. Mercier, M. J., Garnier, N. B. & Dauxois, T. 2008 Reflection and diffraction of internal waves analysed with the Hilbert transform. Phys. Fluids 20, 086601.CrossRefGoogle Scholar
12. Hazewinkel, J., Grisouard, N. & Dalziel, S. B. 2010 Comparison of laboratory and numerically observed scalar fields of an internal wave attractor. Eur. J. Mech. (B/Fluids) 30, 51–56.CrossRefGoogle Scholar
13. Hunt, J. C. R & Snyder, W. H. 1980 Experiments on stably and neutrally stratified flow over a model three-dimensional hill. J. Fluid Mech. 9, 671–704.CrossRefGoogle Scholar
14. Hunt, J. C. R., Vilenski, G. G. & Johnson, E. R. 2006 Stratified separated flow around a mountain with an inversion layer below the mountain top. J. Fluid Mech. 556, 105–119.CrossRefGoogle Scholar
15. Huppert, H. E. & Miles, J. W. 1969 Lee waves in a stratified flow. Part 3. Semi-elliptical obstacle. J. Fluid Mech. 35, 481–496.CrossRefGoogle Scholar
16. Lighthill, M. J. 1978 Waves in Fluids, p. 504. Cambridge University Press.Google Scholar
17. Long, R. R. 1955 Some aspects of the flow of stratified fluids III. Continuous density gradients. Tellus 7, 341–357.Google Scholar
18. Miles, J. W. 1961 On the stability of heterogeneous shear flows. J. Fluid Mech. 10, 496–508.CrossRefGoogle Scholar
19. Miles, J. W. & Huppert, H. E. 1968 Lee waves in a stratified flow. Part 2. Semi-circular obstacles. J. Fluid Mech. 33, 803–814.CrossRefGoogle Scholar
20. Odell, G. M. & Kovasznay, L. S. G. 1971 A new type of water channel with density stratification. J. Fluid Mech. 50, 535–543.CrossRefGoogle Scholar
21. Riley, J. J. & Lelong, M.-P. 2000 Fluid motions in the presence of strong stable stratification. Annu. Rev. Fluid Mech. 32, 613–657.CrossRefGoogle Scholar
22. Scase, M. M. & Dalziel, S. B. 2006 Internal wave fields generated by a translating body in a stratified fluid: an experimental comparison. J. Fluid Mech. 564, 305–331.CrossRefGoogle Scholar
23. Scorer, R. S. 1949 Theory of waves in the lee of mountains. Q. J. R. Meteorol. Soc. 75, 41–56.CrossRefGoogle Scholar
24. Sutherland, B. R., Dalziel, S. B., Hughes, G. O. & Linden, P. F. 1999 Visualisation and measurement of internal waves by ‘Synthetic Schlieren’. Part 1. Vertically oscillating cylinder. J. Fluid Mech. 390, 93–126.CrossRefGoogle Scholar
25. Turner, J. S. 1973 Buoyancy Effects in Fluids, p. 367. Cambridge University Press.CrossRefGoogle Scholar
26. Voisin, B. 1994 Internal wave generation in uniformly stratified fluids. Part 2. Moving point sources. J. Fluid Mech. 261, 333–374.CrossRefGoogle Scholar
27. Voisin, B. 2007 Lee waves from a sphere in a stratified flow. J. Fluid Mech. 574, 273–315.CrossRefGoogle Scholar
28. Vosper, S. B., Castro, I. P., Snyder, W. H. & Mobbs, S. D. 1999 Experimental studies of strongly stratified flow past three-dimensional orography. J. Fluid Mech. 390, 223–249.CrossRefGoogle Scholar
29. Wurtele, M. G., Sharman, R. D. & Datta, A. 1996 Atmospheric lee waves. Annu. Rev. Fluid Mech. 28, 429–476.CrossRefGoogle Scholar