On reduced models for gravity waves generated by moving bodies

Trinh, Philippe H.

doi:10.1017/jfm.2016.818

Cited by 6 publications

(47 citation statements)

References 38 publications

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“…If is decreased to 0.5, the relative error decreases to ≈ 3.6%. In reference [8], a careful check of numerical versus asymptotic results verifies these claims properly.…”

Section: The Reduced Water-wave Equationsmentioning

confidence: 68%

“…The derivation of this framework is the topic of a companion paper [8]. There it is shown that, by extending earlier ideas by Tulin [12] and Tuck [13], solutions can be expressed in the form of a (nearly explicit) integral.…”

Section: Introductionmentioning

confidence: 89%

“…Once done, an asymptotic expansion of the integral is derived by consideration of dominant contributions from the endpoints, as well as any relevant saddle or critical points of the integrand. In reference [8], it is shown that the endpoint contributions lead to a reexpansion of higher-order algebraic terms from (2.4), whereas the exponentially small water waves are caused by contributions away from the positive real axis. Therefore,Ĥ [θ], which only depends on the free surface values ofθ , can be ignored at leading order.…”

Section: The Reduced Water-wave Equationsmentioning

confidence: 99%

“…This is why we may think of the surface waves as being generated by the ship's corner. The analytical form of (4.4) is given in [8].…”

Section: (Ii) Crossing the Stokes Linementioning

confidence: 99%

“…The inability to detect the surface waves using standardized methods is known as Ogilvie's low-Froude paradox [6,7]. In the 50 years following Ogilvie's work, the low-Froude paradox was eventually resolved through the combined efforts of many researchers, and a review of the historical progression can be found in [7,8]. Chapman & Vanden-Broeck [9] were the first, to the best of our knowledge, to derive the correct leading-order wave solution of the low-Froude problem, and their approach uses exponential asymptotic techniques of matched asymptotics and Stokes line smoothing [10,11].…”

Section: Introductionmentioning

confidence: 99%

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A topological study of gravity free-surface waves generated by bluff bodies using the method of steepest descents

Trinh¹

2016

Proc. R. Soc. A.

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The standard analytical approach for studying steady gravity free-surface waves generated by a moving body often relies upon a linearization of the physical geometry, where the body is considered asymptotically small in one or several of its dimensions. In this paper, a methodology that avoids any such geometrical simplification is presented for the case of steady-state flows at low speeds. The approach is made possible through a reduction of the water-wave equations to a complex-valued integral equation that can be studied using the method of steepest descents. The main result is a theory that establishes a correspondence between different bluff-bodied free-surface flow configurations, with the topology of the Riemann surface formed by the steepest descent paths. Then, when a geometrical feature of the body is modified, a corresponding change to the Riemann surface is observed, and the resultant effects to the water waves can be derived. This visual procedure is demonstrated for the case of two-dimensional free-surface flow past a surface-piercing ship and over an angled step in a channel.

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“…If is decreased to 0.5, the relative error decreases to ≈ 3.6%. In reference [8], a careful check of numerical versus asymptotic results verifies these claims properly.…”

Section: The Reduced Water-wave Equationsmentioning

confidence: 68%

Section: Introductionmentioning

confidence: 89%

Section: The Reduced Water-wave Equationsmentioning

confidence: 99%

“…This is why we may think of the surface waves as being generated by the ship's corner. The analytical form of (4.4) is given in [8].…”

Section: (Ii) Crossing the Stokes Linementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A topological study of gravity free-surface waves generated by bluff bodies using the method of steepest descents

Trinh¹

2016

Proc. R. Soc. A.

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Pathologies in the Asymptotics of low-Froude Free-surface Waves Over Smooth Bodies

Johnson-Llambias,

Trinh

2024

Water Waves

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In the study of low-speed or low-Froude flows of a potential gravity-driven fluid past a wave-generating object, the traditional asymptotic expansion in powers of the Froude number predicts a waveless free-surface at every order. This is due to the fact that the waves are, in fact, exponentially small and beyond-all-orders of the naive expansion. The theory of exponential asymptotics indicates that such exponentially-small water waves are switched-on across so-called Stokes lines—these curves partition the fluid-domain into wave-free regions and regions with waves. In prior studies, Stokes lines are associated with singularities in the flow field, such as stagnation points, or corners of submerged objects or rough beds. In this work, we present a smoothed geometry that was recently highlighted by Pethiyagoda et al. [Int. J. Numer. Meth. Fluids. 2018; 86:607–624] as capable of producing waves, yet paradoxically exhibiting no obvious Stokes line according to conventional exponential asymptotics theory. In this work, we demonstrate that the Stokes line for this smooth geometry originates from an essential singularity at infinity in the analytic continuation of free-surface quantities. We discuss some of the difficulties in extending the typical methodology of exponential asymptotics to general wave-structure interaction problems with smooth geometries.

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Gravity–capillary waves in reduced models for wave–structure interactions

Jamshidi

Trinh

2020

J. Fluid Mech.

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In order to determine the steady-state subcritical gravity-capillary waves that are produced by potential flow past a wave-making body, it is typically necessary to impose a radiation condition that allows for capillary waves upstream, but disallows those corresponding to gravity. However, this radiation condition is not known a priori and consequently, the computation of accurate numerical solutions to the steady-state problem remains problematic. Although the physical model might be modified (e.g. with viscosity), recovery of the original problem is not always possible.In this work, we discuss the above radiation problem, and show how, in the low-speed limit, the steady gravity-capillary waves can be resolved using a Sommerfeld-type boundary condition applied to an asymptotically reduced set of water-wave equations. These results allow us to validate the specialized classes of low-speed waves theoretically predicted by Trinh & Chapman (2013b) using methods in exponential asymptotics [J. Fluid Mech. 724,. The issues of numerically solving the full set of gravity-capillary equations for a potential flow are discussed, and the sensitivity to errors in the boundary conditions is clearly demonstrated.

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On reduced models for gravity waves generated by moving bodies

Cited by 6 publications

References 38 publications

A topological study of gravity free-surface waves generated by bluff bodies using the method of steepest descents

A topological study of gravity free-surface waves generated by bluff bodies using the method of steepest descents

Pathologies in the Asymptotics of low-Froude Free-surface Waves Over Smooth Bodies

Gravity–capillary waves in reduced models for wave–structure interactions

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