Steady free-surface flow over spatially periodic topography

Binder, Benjamin J.; Blyth, Mark; Balasuriya, Sanjeeva

doi:10.1017/jfm.2015.507

Cited by 3 publications

(12 citation statements)

References 23 publications

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“…3(e) there is more than one solitary wave part. These results are broadly similar to those found by Binder et al [12] for an infinite corrugation, and the nonuniqueness can be explained as follows. For flat topography there are an uncountable number of solitary waves which can be obtained by taking arbitrary shifts, x s , of the well-known free solitary wave solution For corrugated topography the nonautonomous theory of Balasuriya et al [29] can be adapted to characterise the solitary waves.…”

Section: Numerical Resultssupporting

confidence: 91%

“…In this work we investigate steady potential flow past semi-infinite and finite-length corrugations with a particular focus on computing the shape of the free surface. This extends the work of Binder et al [12], who determined the free-surface response in the case of an infinite corrugation. We assume that there is uniform flow far upstream where the topography becomes flat (see Fig.…”

Section: Introductionsupporting

confidence: 86%

“…As x → ∞ this matches to the small amplitude periodic solution obtained for an infinite sinusoidal corrugation by Ref. [12]. The comparison between the linearized and weakly nonlinear results will be examined in the next section.…”

Section: Linearized Analysissupporting

confidence: 80%

“…As we require that η remains bounded as x → ∞ we truncate the problem on a domain sufficiently wide and demand that the free surface reaches a local maximum at the downstream truncation point, which itself is found as part of the solution. In practice by taking the downstream truncation point at a sufficiently large value of x we find that the local maximum on the free surface almost coincides with a local maximum on the topography in line with the linearized solution (12). We compute both weakly nonlinear and fully nonlinear solutions for flow past semi-infinite and finite-length corrugations.…”

Section: Numerical Resultsmentioning

confidence: 64%

“…For flat topography there are an uncountable number of solitary waves which can be obtained by taking arbitrary shifts, x s , of the well-known free solitary wave solution For corrugated topography the nonautonomous theory of Balasuriya et al [29] can be adapted to characterise the solitary waves. As shown in Binder et al [12], there are exactly two possible types of solitary waves on the corrugated portion of the topography: (1) those centered around crests and (2) those centered around troughs of the topography. Solitary waves within each of these two families are simple shifts of each other by multiples of kπ, and therefore there are only a countable number of such solitary wave solutions.…”

Section: Numerical Resultsmentioning

confidence: 92%

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Steady two-dimensional free-surface flow over semi-infinite and finite-length corrugations in an open channel

2018

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Free-surface flow past a semi-infinite or a finite-length corrugation in an otherwise flat and horizontal open channel is considered. Numerical solutions for the steady flow problem are computed using both a weakly nonlinear and fully nonlinear model. The new solutions are classified in terms of a depth-based Froude number and the four classical flow types (supercritical, subcritical, generalized hydraulic rise, and hydraulic rise) for flow over a small bump. While there is no hydraulic fall solution for semi-infinite topography, we provide strong numerical evidence that such a solution does exist in the case of a finite-length corrugation. Numerical solutions are also found for the other flow types for either semi-infinite or finite-length corrugation. For subcritical flow over a semi-infinite corrugation, the free-surface profile is found to be quasiperiodic downstream.

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Section: Numerical Resultssupporting

confidence: 91%

Section: Introductionsupporting

confidence: 86%

Section: Linearized Analysissupporting

confidence: 80%

Section: Numerical Resultsmentioning

confidence: 64%

Section: Numerical Resultsmentioning

confidence: 92%

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Steady two-dimensional free-surface flow over semi-infinite and finite-length corrugations in an open channel

2018

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Near-critical turbulent free-surface flow over a wavy bottom

Schneider

Murschenhofer

2022

Acta Mech

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Steady plane turbulent free-surface flow over a slightly wavy bottom is considered for very large Reynolds numbers, very small bottom slopes, and Froude numbers close to the critical value 1. As in previous works, the slope and the deviation from the critical Froude number are assumed to be coupled such that turbulence modeling is not required. The amplitudes of the periodic bottom elevations, however, are assumed to be half an order of magnitude larger than in the previous case of bumps or ramps of finite length. Asymptotic expansions give a steady-state version of an extended Korteweg–deVries (KdV) equation for the surface elevation. The extension consists of a forcing term due to the unevenness of the bottom and a damping term due to friction at the bottom. Other flow quantities, such as pressure, flow velocity components, local Froude number and bottom friction force, can be expressed in terms of the surface elevation. Exact solutions of the extended KdV equation, describing stationary cnoidal waves, are obtained for bottoms of particular periodic shapes. As a limiting case, the solitary waves over a bottom ramp are re-obtained in accord with previous results.

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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

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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.

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Steady free-surface flow over spatially periodic topography

Cited by 3 publications

References 23 publications

Steady two-dimensional free-surface flow over semi-infinite and finite-length corrugations in an open channel

Steady two-dimensional free-surface flow over semi-infinite and finite-length corrugations in an open channel

Near-critical turbulent free-surface flow over a wavy bottom

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

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