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

Jamshidi, Sean; Trinh, Philippe H.

doi:10.1017/jfm.2020.95

Cited by 4 publications

(4 citation statements)

References 18 publications

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“…In this case, the existence of both upstream and downstream waves on the free boundary mean that it is difficult to verify the results computationally. Progress on studying these systems has been made in the context of gravity-capillary waves [26], but this remains a challenging numerical problem. We will briefly discuss the asymptotics of nonlinear geometries in Section 2.5.…”

Section: Linearizationmentioning

confidence: 99%

“…Any conclusions reached for the nonlinear system would require validation against numerical simulations, but the presence of surface waves in both directions far from the step makes it challenging to obtain sensible boundary conditions for the flow behaviour. For a detailed description of the numerical challenges involved in studying these systems, and substantial progress in overcoming these obstacles, see the numerical analysis of two dimensional gravity-capillary waves in [26]. A full analysis of the nonlinear problem is therefore beyond the scope of the present study.…”

Section: Nonlinear Problemmentioning

confidence: 99%

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Elastic and elastic-gravity waves on flow past submerged obstacles with small bending stiffness

Lustri¹

2022

Preprint

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Linearized flow past a submerged obstacle with an elastic sheet resting on the flow surface are studied in the limit of small bending stiffness, in two and three dimensions. Gravitational effects are included in the two-dimensional geometry, but absent in the three-dimensional geometry. In each of these problems, the waves are exponentially small in the asymptotic limit, and can be computed using exponential asymptotic methods. In the two-dimensional problem, flow past a submerged step is considered. It is found that the relative strength of the gravitational and elastic restoring forces produce two distinct classes of elastic sheet behaviour. In one parameter regime, constant-amplitude elastic waves and gravity waves extend indefinitely upstream and downstream from the obstacle. In the other parameter regime, all waves decay exponentially away from the obstacle. The equivalent nonlinear two-dimensional geometry is then studied; this asymptotic analysis predicts the existence of a third intermediate regime in which waves persist indefinitely only in one direction, depending on whether the submerged step rises or falls. In the three-dimensional geometry, it is predicted that the elastic waves extend ahead of the submerged source, decaying algebraically in space. The form of these elastic waves is computed, and validated by comparison with numerical computations of the elastic sheet behaviour.

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Section: Linearizationmentioning

confidence: 99%

Section: Nonlinear Problemmentioning

confidence: 99%

Elastic and elastic-gravity waves on flow past submerged obstacles with small bending stiffness

Lustri¹

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…In this case, the existence of both upstream and downstream waves on the free boundary mean that it is difficult to verify the results computationally. Progress on studying these systems has been made in the context of gravity-capillary waves in Jamshidi & Trinh (2020), but this remains a challenging numerical problem. We will discuss briefly the asymptotics of nonlinear geometries in § 2.5.…”

Section: Two-dimensional Elastic-gravity Wavesmentioning

confidence: 99%

“…Any conclusions reached for the nonlinear system would require validation against numerical simulations, but the presence of surface waves in both directions far from the step makes it challenging to obtain sensible boundary conditions for the flow behaviour. For a detailed description of the numerical challenges involved in studying these systems, and substantial progress in overcoming these obstacles, see the numerical analysis of two-dimensional gravity-capillary waves in Jamshidi & Trinh (2020). A full analysis of the nonlinear problem is therefore beyond the scope of the present study.…”

Section: Two-dimensional Elastic-gravity Wavesmentioning

confidence: 99%

Exponential asymptotics for elastic and elastic-gravity waves on flow past submerged obstacles

Lustri

2022

J. Fluid Mech.

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Linearized flow past a submerged obstacle with an elastic sheet resting on the flow surface are studied in the limit that the bending length is small compared to the obstacle depth, in two and three dimensions. Gravitational effects are included in the two-dimensional geometry, but absent in the three-dimensional geometry; the Froude number is chosen so that gravitational and elastic restoring forces are comparable in size. In each of these problems, the waves are exponentially small in the asymptotic limit, and can be computed using exponential asymptotic methods. In the two-dimensional problem, flow past a submerged step is considered. It is found that the relative strengths of the gravitational and elastic restoring forces produce two distinct classes of elastic sheet behaviour. In one parameter regime, constant-amplitude elastic waves and gravity waves extend indefinitely upstream and downstream from the obstacle. In the other parameter regime, all waves decay exponentially away from the obstacle. The equivalent nonlinear two-dimensional geometry is then studied; this asymptotic analysis predicts the existence of a third intermediate regime in which waves persist indefinitely in only one direction, depending on whether the submerged step rises or falls. In the three-dimensional geometry, it is predicted that the elastic waves extend ahead of the submerged source, decaying algebraically in space. The form of these elastic waves is computed, and validated by comparison with numerical computations of the elastic sheet behaviour.

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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

Cited by 4 publications

References 18 publications

Elastic and elastic-gravity waves on flow past submerged obstacles with small bending stiffness

Elastic and elastic-gravity waves on flow past submerged obstacles with small bending stiffness

Exponential asymptotics for elastic and elastic-gravity waves on flow past submerged obstacles

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

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