Free-surface flow over a semicircular obstruction

Forbes, Lawrence K.; Schwartz, Leonard W.

doi:10.1017/s0022112082000160

Cited by 164 publications

(199 citation statements)

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“…Solutions to the full nonlinear problem were obtained numerically by Forbes [8,9], who considered a semi-elliptical obstruction on the stream bed, and Forbes and Schwartz [12] and Vanden-Broeck [21], who computed nonlinear solutions for flow over a semicircular obstruction. Forbes [11] also computed hydraulic fall flows over a semi-circular obstruction, which were verified by comparison with the results of a laboratory experiment.…”

Section: Introductionmentioning

confidence: 99%

Waveless subcritical flow past symmetric bottom topography

Holmes¹,

Hocking²,

Forbes³

et al. 2012

Eur. J. Appl. Math

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The subcritical flow of a stream over a bottom obstruction or depression is considered with particular interest in obtaining solutions with no downstream waves. In the linearised problem this can always be achieved by superposition of multiple obstructions, but it is not clear whether this is possible in a full nonlinear problem. Solutions computed here indicate that there is an effective nonlinear superposition principle at work as no special shape modifications were required to obtain wave-cancelling solutions. Waveless solutions corresponding to one or more trapped waves are computed at a range of different Froude numbers and are shown to provide a rather elaborate mosaic of solution curves in parameter space when both negative and positive obstruction heights are included.

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Section: Introductionmentioning

confidence: 99%

Waveless subcritical flow past symmetric bottom topography

Holmes¹,

Hocking²,

Forbes³

et al. 2012

Eur. J. Appl. Math

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“…While the important point is that these are singularities of the complexified free surface, it happens that they are also singularities on the boundary of the physical flow field. For flows past curved bottom obstructions, such as flow past a semi-circle [18], there are no in-fluid angles greater than 2π/3, but we would still expect there to be exponentially small waves in the limit that the Froude number F → 0, suggesting there are Stokes lines that intersect the free surface. For these flows the relevant singularities of the complexified free surface may be off the real ξ-axis, although for some geometries it may not be obvious where they are located, or whether they are of the form treated in [12].…”

Section: Discussionmentioning

confidence: 97%

“…Within the linearised framework, the former is defined as having F > 1 and F * > 1, while the latter has either F < 1 and F * > 1 or F > 1 and F * < 1. For a further discussion on the three flow regimes, see [17,18], for example. )…”

Section: Introductionmentioning

confidence: 99%

“…Similar analysis was undertaken by Wehausen & Laitone [41] for the linearised problem of two-dimensional flow past a submerged cylinder. However, as noted by Zhang and Zhu [42], if either stagnation points or singularities occur on the boundary of the flow (at corners, for example), then for a linear solution to be valid throughout the flow field, the linearisation must take place in an appropriate conformally mapped plane (as in [18,22], for example).…”

Section: Introductionmentioning

confidence: 99%

“…In particular, Forbes & Schwartz [18] treated the flow past a semi-circular obstacle, while King & Bloor [22] solved for the flow over a step. Further early studies are given by [16,23,42,43], for example.…”

Section: Introductionmentioning

confidence: 99%

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Free surface flow past topography: A beyond-all-orders approach

2012

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The problem of steady subcritical free surface flow past a submerged inclined step is considered. The asymptotic limit of small Froude number is treated, with particular emphasis on the effect that changing the angle of the step face has on the surface waves. As demonstrated by Chapman & Vanden-Broeck, (2006) Exponential asymptotics and gravity waves. J. Fluid Mech. 567, 299-326, the divergence of a power series expansion in powers of the square of the Froude number is caused by singularities in the analytic continuation of the free surface; for an inclined step, these singularities may correspond to either the corners or stagnation points of the step, or both, depending on the angle of inclination. Stokes lines emanate from these singularities, and exponentially small waves are switched on at the point the Stokes lines intersect with the free surface. Our results suggest that for a certain range of step angles, two wavetrains are switched on, but the exponentially subdominant one is switched on first, leading to an intermediate wavetrain not previously noted. We extend these ideas to the problem of flow over a submerged bump or trench, again with inclined sides. This time there may be two, three or four active Stokes lines, depending on the inclination angles. We demonstrate how to construct a base topography such that wave contributions from separate Stokes lines are of equal magnitude but opposite phase, thus cancelling out. Our asymptotic results are complemented by numerical solutions to the fully nonlinear equations.

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