Quasi-periodic travelling gravity–capillary waves

Wilkening, Jon; Zhao, Xinyu

doi:10.1017/jfm.2021.28

Cited by 10 publications

(48 citation statements)

References 73 publications

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“…Note that the tangent velocity of the curve differs from that of the underlying fluid particles. This is similar in spirit to a method of Hou et al (1994Hou et al ( , 2001, who proposed a tangential velocity that maintains a uniform parametrization of the curve (rather than a conformal one); see also (Choi and Camassa 1999;Turner and Bridges 2016;Ambrose et al 2021;Wilkening and Zhao 2021). Combining (2.35) and (2.37), we obtain the kinematic boundary conditions in conformal space,…”

Section: Governing Equations In Conformal Spacementioning

confidence: 56%

“…The non-resonant case leads to a smooth two-parameter family of traveling waves with wave speed and surface tension depending nonlinearly on the amplitude parameters. This is the case most analogous to the spatially quasi-periodic traveling waves that we compute in (Wilkening and Zhao 2021).…”

Section: Introductionmentioning

confidence: 61%

“…They use a spatial Hamiltonian structure to construct weakly nonlinear approximations of spatially quasi-periodic traveling gravity-capillary waves for two special cases: deep water and shallow water. This inspired us to develop a conformal mapping framework for computing spatially quasi-periodic, fully nonlinear traveling gravitycapillary waves, which is the topic of the companion paper, namely Wilkening and Zhao (2021). We show in (Wilkening and Zhao 2021) that these spatially quasi-periodic traveling waves come in two-parameter families in which the amplitudes of the base modes with wave numbers k 1 and k 2 serve as bifurcation parameters.…”

Section: Introductionmentioning

confidence: 99%

“…This inspired us to develop a conformal mapping framework for computing spatially quasi-periodic, fully nonlinear traveling gravitycapillary waves, which is the topic of the companion paper, namely Wilkening and Zhao (2021). We show in (Wilkening and Zhao 2021) that these spatially quasi-periodic traveling waves come in two-parameter families in which the amplitudes of the base modes with wave numbers k 1 and k 2 serve as bifurcation parameters. The wave speed and surface tension depend nonlinearly on these parameters as well.…”

Section: Introductionmentioning

confidence: 99%

“…The present paper focuses on the more general spatially quasi-periodic initial value problem, which we use to validate the traveling wave computations of Wilkening and Zhao (2021) and explore new dynamic phenomena. In recent years, conformal mapping methods have proved useful for studying two-dimensional traveling water waves (Choi and Camassa 1999;Milewski et al 2010;Wang et al 2013;) and time-dependent water waves (Dyachenko et al 1996a;Dyachenko 2001;Dyachenko et al 1996b;Dyachenko and Newell 2016;Li et al 2004;Milewski et al 2010;Turner and Bridges 2016;Zakharov et al 2002) with periodic boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

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Spatially Quasi-Periodic Water Waves of Infinite Depth

Wilkening

Zhao

2021

J Nonlinear Sci

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We formulate the two-dimensional gravity-capillary water wave equations in a spatially quasi-periodic setting and present a numerical study of solutions of the initial value problem. We propose a Fourier pseudo-spectral discretization of the equations of motion in which one-dimensional quasi-periodic functions are represented by two-dimensional periodic functions on a torus. We adopt a conformal mapping formulation and employ a quasi-periodic version of the Hilbert transform to determine the normal velocity of the free surface. Two methods of time-stepping the initial value problem are proposed, an explicit Runge–Kutta (ERK) method and an exponential time-differencing (ETD) scheme. The ETD approach makes use of the small-scale decomposition to eliminate stiffness due to surface tension. We perform a convergence study to compare the accuracy and efficiency of the methods on a traveling wave test problem. We also present an example of a periodic wave profile containing vertical tangent lines that is set in motion with a quasi-periodic velocity potential. As time evolves, each wave peak evolves differently, and only some of them overturn. Beyond water waves, we argue that spatial quasi-periodicity is a natural setting to study the dynamics of linear and nonlinear waves, offering a third option to the usual modeling assumption that solutions either evolve on a periodic domain or decay at infinity.

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Section: Governing Equations In Conformal Spacementioning

confidence: 56%

Section: Introductionmentioning

confidence: 61%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Spatially Quasi-Periodic Water Waves of Infinite Depth

Wilkening

Zhao

2021

J Nonlinear Sci

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Traveling Quasi-periodic Water Waves with Constant Vorticity

Berti

Franzoi

Maspero

2021

Arch Rational Mech Anal

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We prove the first bifurcation result of time quasi-periodic traveling wave solutions for space periodic water waves with vorticity. In particular, we prove the existence of small amplitude time quasi-periodic solutions of the gravity-capillary water waves equations with constant vorticity, for a bidimensional fluid over a flat bottom delimited by a space-periodic free interface. These quasi-periodic solutions exist for all the values of depth, gravity and vorticity, and restrict the surface tension to a Borel set of asymptotically full Lebesgue measure.

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Numerical algorithms for water waves with background flow over obstacles and topography

et al. 2022

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We present two accurate and efficient algorithms for solving the incompressible, irrotational Euler equations with a free surface in two dimensions with background flow over a periodic, multiply connected fluid domain that includes stationary obstacles and variable bottom topography. One approach is formulated in terms of the surface velocity potential while the other evolves the vortex sheet strength. Both methods employ layer potentials in the form of periodized Cauchy integrals to compute the normal velocity of the free surface, are compatible with arbitrary parameterizations of the free surface and boundaries, and allow for circulation around each obstacle, which leads to multiple-valued velocity potentials but single-valued stream functions. We prove that the resulting second-kind Fredholm integral equations are invertible, possibly after a physically motivated finite-rank correction. In an angle-arclength setting, we show how to avoid curve reconstruction errors that are incompatible with spatial periodicity. We use the proposed methods to study gravity-capillary waves generated by flow around several elliptical obstacles above a flat or variable bottom boundary. In each case, the free surface eventually self-intersects in a splash singularity or collides with a boundary. We also show how to evaluate the velocity and pressure with spectral accuracy throughout the fluid, including near the free surface and solid boundaries. To assess the accuracy of the time evolution, we monitor energy conservation and the decay of Fourier modes and compare the numerical results of the two methods to each other. We implement several solvers for the discretized linear systems and compare their performance. The fastest approach employs a graphics processing unit (GPU) to construct the matrices and carry out iterations of the generalized minimal residual method (GMRES).

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Quasi-periodic travelling gravity–capillary waves

Abstract: Abstract

Cited by 10 publications

References 73 publications

Spatially Quasi-Periodic Water Waves of Infinite Depth

Spatially Quasi-Periodic Water Waves of Infinite Depth

Traveling Quasi-periodic Water Waves with Constant Vorticity

Numerical algorithms for water waves with background flow over obstacles and topography

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