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

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.

scholarly journals On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal feedback

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Julie Valein
Keyword(s):  
Asymptotic Stability ◽  
Exponential Stability ◽  
Numerical Simulations ◽  
Vries Equation ◽  
Stability Result ◽  
Internal Feedback ◽  
De Vries

<p style='text-indent:20px;'>The aim of this work is to study the asymptotic stability of the nonlinear Korteweg-de Vries equation in the presence of a delayed term in the internal feedback. We first consider the case where the weight of the term with delay is smaller than the weight of the term without delay and we prove a semiglobal stability result for any lengths. Secondly we study the case where the support of the term without delay is not included in the support of the term with delay. In that case, we give a local exponential stability result if the weight of the delayed term is small enough. We illustrate these results by some numerical simulations.</p>

Darboux transformation and soliton solutions of the (2+1)-dimensional Schwarz–Korteweg–de Vries equation

2020 ◽  
Vol 34 (25) ◽  
pp. 2050270
Author(s):  
Xuemei Li ◽  
Mingxiao Zhang
Keyword(s):  
Darboux Transformation ◽  
Trivial Solution ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Lax Pairs ◽  
De Vries ◽  
Asymptotic Analyses

In this paper, we deduce the (2+1)-dimensional Schwarz–Korteweg–de Vries equation from two (1+1)-dimensional equations. Based on the resulting Lax pairs, we present its [Formula: see text]-fold Darboux transformation. From a trivial solution, we get [Formula: see text]-soliton and [Formula: see text]-soliton solutions of the (2+1)-dimensional Schwarz–Korteweg–de Vries equation. The asymptotic analyses of the [Formula: see text]-soliton, [Formula: see text]-soliton and [Formula: see text]-soliton solutions are presented theoretically and graphically.

Integration of the nonlinear modified Korteweg-de Vries equation with a loaded term

2020 ◽  
Vol 2020 (2) ◽  
pp. 85-98
Author(s):  
A.B. Khasanov ◽  
T.J. Allanazarova
Keyword(s):  
Vries Equation ◽  
De Vries ◽  
Loaded Term

Convergence of the Rosenau-Korteweg-de Vries Equation to the Korteweg-de Vries One

2020 ◽  
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
A Priori Estimates ◽  
Vries Equation ◽  
A Priori ◽  
Diffusion Parameter ◽  
Priori Estimates ◽  
De Vries ◽  
Wall Interactions

The Rosenau-Korteweg-de Vries equation describes the wave-wave and wave-wall interactions. In this paper, we prove that, as the diffusion parameter is near zero, it coincides with the Korteweg-de Vries equation. The proof relies on deriving suitable a priori estimates together with an application of the Aubin-Lions Lemma.

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