scholarly journals Transcritical flow of a stratified fluid: The forced extended Korteweg–de Vries model

2002 ◽  
Vol 14 (2) ◽  
pp. 755-774 ◽  
Author(s):  
R. H. J. Grimshaw ◽  
K. H. Chan ◽  
K. W. Chow
Keyword(s):  
Stratified Fluid ◽  
Transcritical Flow ◽  
De Vries

scholarly journals Transcritical flow of a stratified fluid over topography: analysis of the forced Gardner equation

2013 ◽  
Vol 736 ◽  
pp. 495-531 ◽  
Author(s):  
A. M. Kamchatnov ◽  
Y.-H. Kuo ◽  
T.-C. Lin ◽  
T.-L. Horng ◽  
S.-C. Gou ◽  
...  
Keyword(s):  
Single Layer ◽  
Stratified Fluid ◽  
Flow Speed ◽  
Flow State ◽  
Rarefaction Waves ◽  
Gardner Equation ◽  
Transcritical Flow ◽  
De Vries ◽  
Input Parameters ◽  
Nonlinear Modulation

AbstractTranscritical 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.

scholarly journals TRANSCRITICAL FLOW PAST AN OBSTACLE

2010 ◽  
Vol 52 (1) ◽  
pp. 2-26 ◽  
Author(s):  
R. GRIMSHAW
Keyword(s):  
Numerical Simulations ◽  
Recent Work ◽  
Vries Equation ◽  
Steady Solution ◽  
Underlying Theory ◽  
Flow Past ◽  
Transcritical Flow ◽  
De Vries ◽  
Large Width ◽  
Asymptotic Analyses

AbstractIt is well known that transcritical flow past an obstacle may generate undular bores propagating away from the obstacle. This flow has been successfully modelled in the framework of the forced Korteweg–de Vries equation, where numerical simulations and asymptotic analyses have shown that the unsteady undular bores are connected by a locally steady solution over the obstacle. In this paper we present an overview of the underlying theory, together with some recent work on the case where the obstacle has a large width.

Transcritical flow over obstacles and holes: forced Korteweg–de Vries framework

2019 ◽  
Vol 881 ◽  
pp. 660-678 ◽  
Author(s):  
Roger H. J. Grimshaw ◽  
Montri Maleewong
Keyword(s):  
Free Surface ◽  
Solitary Wave ◽  
Vries Equation ◽  
Solitary Wave Solution ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Main Concern ◽  
Oncoming Flow ◽  
Transcritical Flow ◽  
De Vries

This paper extends a previous study of free-surface flow over two localised obstacles using the framework of the forced Korteweg–de Vries equation, to an analogous study of flow over two localised holes, or a combination of an obstacle and a hole. Importantly the terminology obstacle or hole can be reversed for a stratified fluid and refers more precisely to the relative polarity of the forcing and the solitary wave solution of the unforced Korteweg–de Vries equation. As in the previous study, our main concern is with the transcritical regime when the oncoming flow has a Froude number close to unity. In the transcritical regime at early times, undular bores are produced upstream and downstream of each forcing site. We then describe the interaction of these undular bores between the forcing sites, and the outcome at very large times.

scholarly journals Steady transcritical flow over an obstacle: Parametric map of solutions of the forced extended Korteweg–de Vries equation

2011 ◽  
Vol 23 (4) ◽  
pp. 046602 ◽  
Author(s):  
Bernard K. Ee ◽  
R. H. J. Grimshaw ◽  
K. W. Chow ◽  
D-H. Zhang
Keyword(s):  
Vries Equation ◽  
Transcritical Flow ◽  
De Vries

Steady transcritical flow over a hole: Parametric map of solutions of the forced Korteweg–de Vries equation

2010 ◽  
Vol 22 (5) ◽  
pp. 056602 ◽  
Author(s):  
Bernard K. Ee ◽  
R. H. J. Grimshaw ◽  
D.-H. Zhang ◽  
K. W. Chow
Keyword(s):  
Vries Equation ◽  
Transcritical Flow ◽  
De Vries

SOLITARY WAVES, SOLITON BOUND STATES AND CHAOS IN A DISSIPATIVE KORTEWEG-DE VRIES EQUATION

1994 ◽  
Vol 04 (05) ◽  
pp. 1135-1146 ◽  
Author(s):  
VLADIMIR I. NEKORKIN ◽  
MANUEL G. VELARDE
Keyword(s):  
Internal Waves ◽  
Solitary Waves ◽  
Liquid Layer ◽  
Bound States ◽  
Vries Equation ◽  
Stratified Fluid ◽  
Localized Structures ◽  
Fluid Layers ◽  
De Vries ◽  
Wave Trains

Propagating dissipative (localized) structures like solitary waves, pulses or “solitons,” “bound solitons,” and “chaotic” wave trains are shown to be solutions of a dissipation-modified Korteweg-de Vries equation that in particular appears in Marangoni-Bénard convection when a liquid layer is heated from the air side and in the description of internal waves in sheared, stably stratified fluid layers.

scholarly journals Transcritical Flow Over a Bump using Forced Korteweg-de Vries Equation

MATEMATIKA ◽  
2018 ◽  
Vol 34 (3) ◽  
pp. 179-187
Author(s):  
Vincent Daniel David ◽  
Arifah Bahar ◽  
Zainal Abdul Aziz
Keyword(s):  
Vries Equation ◽  
Fundamental Problem ◽  
Bottom Topography ◽  
Suitable Model ◽  
Special Choice ◽  
Analysis Method ◽  
Water Flows ◽  
Transcritical Flow ◽  
De Vries ◽  
Homotopy Analysis

The flow of water over an obstacle is a fundamental problem in fluid mechanics. Transcritical flow means the wave phenomenon near the exact criticality. The transcritical flow cannot be handled by linear solutions as the energy is unable to propagate away from the obstacle. Thus, it is important to carry out a study to identify suitable model to analyse the transcritical flow. The aim of this study is to analyse the transcritical flow over a bump as localized obstacles where the bump consequently generates upstream and downstream flows. Nonlinear shallow water forced Korteweg-de Vries (fKdV) model is used to analyse the flow over the bump. This theoretical model, containing forcing functions represents bottom topography is considered as the simplified model to describe water flows over a bump. The effect of water dispersion over the forcing region is investigated using the fKdV model. Homotopy Analysis Method (HAM) is used to solve this theoretical fKdV model. The HAM solution which is chosen with a special choice of }-value describes the physical flow of waves and the significance of dispersion over abump is elaborated.

scholarly journals Generation of solitary waves by transcritical flow over a step

2007 ◽  
Vol 587 ◽  
pp. 235-254 ◽  
Author(s):  
R. H. J. GRIMSHAW ◽  
D.-H. ZHANG ◽  
K. W. CHOW
Keyword(s):  
Analytical Solutions ◽  
Water Waves ◽  
Numerical Solutions ◽  
Vries Equation ◽  
Undular Bore ◽  
Downstream Side ◽  
Upstream Side ◽  
Transcritical Flow ◽  
De Vries ◽  
Positive Step

It is well-known that transcritical flow over a localized obstacle generates upstream and downstream nonlinear wavetrains. The flow has been successfully modelled in the framework of the forced Korteweg–de Vries equation, where numerical and asymptotic analytical solutions have shown that the upstream and downstream nonlinear wavetrains have the structure of unsteady undular bores, connected by a locally steady solution over the obstacle, which is elevated on the upstream side and depressed on the downstream side. Inthispaper we consider the analogous transcritical flow over a step, primarily in the context of water waves. We use numerical and asymptotic analytical solutions of the forced Korteweg–de Vries equation, together with numerical solutions of the full Eulerequations, to demonstrate that a positive step generates only an upstream-propagating undular bore, and a negative step generates only a downstream-propagating undular bore.

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