Minimising wave drag for free surface flow past a two-dimensional stern

Ogilat, Osama; McCue, Scott W.; Turner, Ian; Belward, John A.; Binder, Benjamin J.

doi:10.1063/1.3609284

Cited by 12 publications

(12 citation statements)

References 16 publications

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“…For example, Binder et al [30] considered flow with a curved plate and in the phase plane this gives a curved jump between trajectories and fixed points instead of the horizontal jump that we have seen for a flat plate. A similar approach has also been used to examine the waves in subcritical flow at the stern of a ship [11,85]. In these problems, the stern consists of a curved portion of the hull that is connected to a semi-infinite horizontal plate, which is assumed to have a constant pressure, P, applied to the plate as x → ∞.…”

Section: Discussionmentioning

confidence: 99%

“…In these problems, the stern consists of a curved portion of the hull that is connected to a semi-infinite horizontal plate, which is assumed to have a constant pressure, P, applied to the plate as x → ∞. Under these assumptions, the equations for a step Equations (26-28) with h = P are the same as those for flow at the stern of a ship, and furthermore, Ogilat et al [85] showed that an exact solution to Equation (28) is given in terms of Jacobian elliptic functions.…”

Section: Discussionmentioning

confidence: 99%

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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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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

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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“…[40][41][42] The plate considered can either be horizontal for mathematical simplicity 13 or curved for a better representation of a ship's hull. 12,33,43 For the following work, we only consider a horizontal plate in an infinitely deep fluid. For our purposes, this flow configuration provides an interesting test case as there are semi-analytical results to compare with.…”

Section: Flow Under a Semi-infinite Platementioning

confidence: 99%

Efficient computation of two‐dimensional steady free‐surface flows

Pethiyagoda

Moroney

McCue

2017

Numerical Methods in Fluids

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Summary We consider a family of steady free‐surface flow problems in two dimensions, concentrating on the effect of nonlinearity on the train of gravity waves that appear downstream of a disturbance. By exploiting standard complex variable techniques, these problems are formulated in terms of a coupled system of Bernoulli equation and an integral equation. When applying a numerical collocation scheme, the Jacobian for the system is dense, as the integral equation forces each of the algebraic equations to depend on each of the unknowns. We present here a strategy for overcoming this challenge, which leads to a numerical scheme that is much more efficient than what is normally used for these types of problems, allowing for many more grid points over the free surface. In particular, we provide a simple recipe for constructing a sparse approximation to the Jacobian that is used as a preconditioner in a Jacobian‐free Newton‐Krylov method for solving the nonlinear system. We use this approach to compute numerical results for a variety of prototype problems including flows past pressure distributions, a surface‐piercing object and bottom topographies.

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“…Other early examples of similar observations are given for flows past submerged bodies and pressure distributions by Dagan [13] and Doctors & Dagan [15]. Note that these are in contrast to certain other configurations, such as flow past a semi-infinite plate [2,26,28,29,31,37], for which solutions with downstream waves do not exist for sufficiently small Froude numbers, and so there is no corresponding small Froude number nonuniformity.…”

Section: Introductionmentioning

confidence: 95%

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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Minimising wave drag for free surface flow past a two-dimensional stern

Cited by 12 publications

References 16 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

Efficient computation of two‐dimensional steady free‐surface flows

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

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