Transcritical flow of a stratified fluid over topography: Analysis of the forced Gardner equation

A. M. Kamchatnov, Y. H. Kuo, T. C. Lin, T. L. Horng, S. C. Gou, R. Clift, G. A. El, R. H.J. Grimshaw

Research output: Contribution to journalArticle

8 Citations (Scopus)

Abstract

Transcritical flow of a stratified fluid past a broad localised topographic obstacle is studied analytically in the framework of the forced extended Korteweg-de Vries, or Gardner, equation. We consider both possible signs for the cubic nonlinear term in the Gardner equation corresponding to different fluid density stratification profiles. We identify the range of the input parameters: the oncoming flow speed (the Froude number) and the topographic amplitude, for which the obstacle supports a stationary localised hydraulic transition from the subcritical flow upstream to the supercritical flow downstream. Such a localised transcritical flow is resolved back into the equilibrium flow state away from the obstacle with the aid of unsteady coherent nonlinear wave structures propagating upstream and downstream. Along with the regular, cnoidal undular bores occurring in the analogous problem for the single-layer flow modelled by the forced Korteweg-de Vries equation, the transcritical internal wave flows support a diverse family of upstream and downstream wave structures, including kinks, rarefaction waves, classical undular bores, reversed and trigonometric undular bores, which we describe using the recent development of the nonlinear modulation theory for the (unforced) Gardner equation. The predictions of the developed analytic construction are confirmed by direct numerical simulations of the forced Gardner equation for a broad range of input parameters.

Original languageEnglish
Pages (from-to)495-531
Number of pages37
JournalJournal of Fluid Mechanics
Volume736
DOIs
Publication statusPublished - 2013 Dec 10

Fingerprint

Topography
topography
Fluids
fluids
upstream
cavities
Korteweg-de Vries equation
subcritical flow
Froude number
supercritical flow
Direct numerical simulation
equilibrium flow
internal waves
stratification
direct numerical simulation
elastic waves
hydraulics
Hydraulics
Modulation
modulation

All Science Journal Classification (ASJC) codes

  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering

Cite this

Kamchatnov, A. M., Kuo, Y. H., Lin, T. C., Horng, T. L., Gou, S. C., Clift, R., ... Grimshaw, R. H. J. (2013). Transcritical flow of a stratified fluid over topography: Analysis of the forced Gardner equation. Journal of Fluid Mechanics, 736, 495-531. https://doi.org/10.1017/jfm.2013.556
Kamchatnov, A. M. ; Kuo, Y. H. ; Lin, T. C. ; Horng, T. L. ; Gou, S. C. ; Clift, R. ; El, G. A. ; Grimshaw, R. H.J. / Transcritical flow of a stratified fluid over topography : Analysis of the forced Gardner equation. In: Journal of Fluid Mechanics. 2013 ; Vol. 736. pp. 495-531.
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Kamchatnov, AM, Kuo, YH, Lin, TC, Horng, TL, Gou, SC, Clift, R, El, GA & Grimshaw, RHJ 2013, 'Transcritical flow of a stratified fluid over topography: Analysis of the forced Gardner equation', Journal of Fluid Mechanics, vol. 736, pp. 495-531. https://doi.org/10.1017/jfm.2013.556

Transcritical flow of a stratified fluid over topography : Analysis of the forced Gardner equation. / Kamchatnov, A. M.; Kuo, Y. H.; Lin, T. C.; Horng, T. L.; Gou, S. C.; Clift, R.; El, G. A.; Grimshaw, R. H.J.

In: Journal of Fluid Mechanics, Vol. 736, 10.12.2013, p. 495-531.

Research output: Contribution to journalArticle

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AU - Kamchatnov, A. M.

AU - Kuo, Y. H.

AU - Lin, T. C.

AU - Horng, T. L.

AU - Gou, S. C.

AU - Clift, R.

AU - El, G. A.

AU - Grimshaw, R. H.J.

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