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

Ee, Bernard K.; Grimshaw, Roger; Chow, K. W.; Zhang, D.-H.

doi:10.1063/1.3582523

Cited by 8 publications

(8 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The first of these is when the waves are very steep and are approaching the Stokes limiting configuration of an included angle of 120 • at the wave crest(s) [49,69]. The second regime is when the flow is intrinsically unsteady, and is often referred to as transcritical flow [25,32,34,52,55,61,70], which is characterised by solitons being periodically emitted upstream of the disturbance with a uniform depression and wake downstream (e.g., see the waterfall plots in Figure 6 [34,49,71]). However, when within the range of validity of the weakly nonlinear theory, analysis in the phase plane of the problem provides a systematic way to classify the different types of solutions.…”

Section: Methodsmentioning

confidence: 99%

Section: Flow Over a Step (Up Or Down)mentioning

confidence: 99%

“…Note that Equations (31) and (32) relax to those of Equations (10) and (11) when h = 0. The comparison between weakly nonlinear, Equations (27) and (30), and fully nonlinear, Equations (31) and (32), results for h = ±0.05 is presented in Figure 8, and qualitatively similar results are also found for other non-zero values of the step height, h = 0. In the case of when h > 0, we see that there is a gap in the solution space around the value of F = 1 in Figure 8c,d, and the turning points are given by Equations (29) and (33), which correspond to the maximum height of the step.…”

Section: Flow Over a Step (Up Or Down)mentioning

confidence: 99%

“…The study of steady two-dimensional flow in a finite depth channel has been a classical problem in hydrodynamics with a rich and long history [8,9,[20][21][22][23][24][25][26]. More recently, the flow problem has continued to attract attention through both the computation of numerical solutions to the full problem and analysis of the weakly nonlinear problem [27][28][29][30][31][32][33][34][35][36]. The primary purpose of this paper is to review the relatively recent advances made in this field of research through the examination of the steady solution space in the weakly nonlinear phase plane of the problem.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Steady Two-Dimensional Free-Surface Flow Past Disturbances in an Open Channel: Solutions of the Korteweg–De Vries Equation and Analysis of the Weakly Nonlinear Phase Space

Binder¹

2019

Fluids

View full text Add to dashboard Cite

Two-dimensional free-surface flow past disturbances in an open channel is a classical problem in hydrodynamics—a problem that has received considerable attention over the last two centuries (e.g., see Lamb’s Treatise, 1879). With traces back to Russell’s experimental observations of the Great Wave of Translation in 1834, Korteweg and de Vries (1895), and others, derived an unforced equation to describe the balance between nonlinearity and dispersion required to model the solitary wave. More recently, Akylas (1984) derived a forced KdV equation to model a pressure distribution on the free-surface (e.g., a ship). Since then, the forced KdV equation has been shown to be a useful model approximation for two-dimensional flow past disturbances in an open channel. In this paper, we review the stationary solutions of the forced KdV equation for four types of localised disturbances: (i) a flat plate separating two free surfaces; (ii) a compact bump, or dip in the channel bottom topography; (iii) a compact distribution of pressure on the free surface and (iv) a step-wise change in the otherwise constant horizontal level of the channel bottom topography. Moreover, Dias and Vanden-Broeck (2002) developed a phase plane method to analyse flow over a bump, and this general approach has also been applied to the three other types of forcing (see Binder et al., 2005–2015, and others). In this study, we use eleven basic flow types to classify the steady solutions of the forced KdV equation using the phase plane method. Additionally, considering solutions that are wave-free both far upstream and far downstream, we compare KdV model approximations of the uniform flow conditions in the far-field with exact solutions of the full problem. In particular, we derive a new KdV model approximation for the upstream dimensionless flow-rate which is conveniently given in terms of the known downstream dimensionless flow-rate.

show abstract

Section: Methodsmentioning

confidence: 99%

Section: Flow Over a Step (Up Or Down)mentioning

confidence: 99%

Section: Flow Over a Step (Up Or Down)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Steady Two-Dimensional Free-Surface Flow Past Disturbances in an Open Channel: Solutions of the Korteweg–De Vries Equation and Analysis of the Weakly Nonlinear Phase Space

Binder¹

2019

Fluids

View full text Add to dashboard Cite

show abstract

“…In the context of this article the most relevant is the article by Pratt (1984) where a combination of steady hydraulic theory, numerical simulations using the nonlinear shallow water equations and laboratory experiments are used to infer that the formation of dispersive waves between the obstacles is needed to obtain a stable solution. More recently Dias and Vanden-Broeck (2004) Ee et al (2010Ee et al ( , 2011 have examined the possible presence of such waves for steady flows, while Grimshaw et al (2009) considered the related problem of unsteady flow over a wide hole.Thus a new feature of interest when considering two obstacles is that the waves generated by each obstacle may interact in the region between the two obstacles, and then the question is how this interaction might affect the long-time outcome. In this paper we examine this scenario using the nonlinear shallow water equations, so that although finite-amplitude effects are included, wave dispersion is neglected and the generated waves are represented as shock waves.…”

Section: Introductionmentioning

confidence: 99%

Critical control in transcritical shallow-water flow over two obstacles

Grimshaw

Maleewong

2015

J. Fluid Mech.

View full text Add to dashboard Cite

The nonlinear shallow-water equations are often used to model flow over topography In this paper we use these equations both analytically and numerically to study flow over two widely separated localised obstacles, and compare the outcome with the corresponding flow over a single localised obstacle. Initially we assume uniform flow with constant water depth, which is then perturbed by the obstacles. The upstream flow can be characterised as subcritical, supercritical, and transcritical respectively. We review the well-known theory for flow over a single localised obstacle, where in the transcritical regime the flow is characterised by a local hydraulic flow over the obstacle, contained between an elevation shock propagating upstream and a depression shock propagating downstream. Classical shock closure conditions are used to determine these shocks. Then we show that the same approach can be used to describe the flow over two widely spaced localised obstacles. The flow development can be characterized by two stages. The first stage is the generation of upstream elevation shock and downstream depression shock from each obstacle alone, isolated from the other obstacle. The second stage is the interaction of two shocks between the two obstacles, followed by an adjustment to a hydraulic flow over both obstacles, with criticality being controlled by the higher of the two obstacles, and by the second obstacle when they have equal heights. This hydraulic flow is terminated by an elevation shock propagating upstream of the first obstacle and a depression shock propagating downstream of the second obstacle. A weakly nonlinear model for sufficiently small obstacles is developed to describe this second stage. The theoretical results are compared with fully nonlinear simulations obtained using a well-balanced finite volume method. The analytical results agree quite well with the nonlinear simulations for sufficiently small obstacles.

show abstract

The effect of obstacle length and height in subcritical free-surface flow

Michalski,

Mattner,

Balasuriya

et al. 2024

Theor. Comput. Fluid Dyn.

View full text Add to dashboard Cite

Two-dimensional free-surface flow past a submerged rectangular disturbance in an open channel is considered. The forced Korteweg–de Vries model of Binder et al. (Theor Comput Fluid Dyn 20:125–144, 2006) is modified to examine the effect of varying obstacle length and height on the response of the free-surface. For a given obstacle height and flow rate in the subcritical flow regime an analysis of the steady solutions in the phase plane of the problem determines a countably infinite set of discrete obstacle lengths for which there are no waves downstream of the obstacle. A rich structure of nonlinear behaviour is also found as the height of the obstacle approaches critical values in the steady problem. The stability of the steady solutions is investigated numerically in the time-dependent problem with a pseudospectral method.

show abstract

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

Cited by 8 publications

References 21 publications

Steady Two-Dimensional Free-Surface Flow Past Disturbances in an Open Channel: Solutions of the Korteweg–De Vries Equation and Analysis of the Weakly Nonlinear Phase Space

Steady Two-Dimensional Free-Surface Flow Past Disturbances in an Open Channel: Solutions of the Korteweg–De Vries Equation and Analysis of the Weakly Nonlinear Phase Space

Critical control in transcritical shallow-water flow over two obstacles

The effect of obstacle length and height in subcritical free-surface flow

Contact Info

Product

Resources

About