scholarly journals New conformal mapping for adaptive resolving of the complex singularities of Stokes wave

2017 ◽  
Vol 473 (2202) ◽  
pp. 20170198 ◽  
Author(s):  
Pavel M. Lushnikov ◽  
Sergey A. Dyachenko ◽  
Denis A. Silantyev
Keyword(s):  
Free Surface ◽  
Conformal Mapping ◽  
Numerical Approximation ◽  
Travelling Wave ◽  
Stokes Wave ◽  
Numerical Convergence ◽  
Complex Singularities ◽  
The Family ◽  
Stokes Waves ◽  
Numerical Grid

A new highly efficient method is developed for computation of travelling periodic waves (Stokes waves) on the free surface of deep water. A convergence of numerical approximation is determined by the complex singularities above the free surface for the analytical continuation of the travelling wave into the complex plane. An auxiliary conformal mapping is introduced which moves singularities away from the free surface thus dramatically speeding up numerical convergence by adapting the numerical grid for resolving singularities while being consistent with the fluid dynamics. The efficiency of that conformal mapping is demonstrated for the Stokes wave approaching the limiting Stokes wave (the wave of the greatest height) which significantly expands the family of numerically accessible solutions. It allows us to provide a detailed study of the oscillatory approach of these solutions to the limiting wave. Generalizations of the conformal mapping to resolve multiple singularities are also introduced.

Oscillatory Third-Order Perturbation Solutions for Sums of Interacting Long-Crested Stokes Waves on Deep Water

1993 ◽  
Vol 37 (04) ◽  
pp. 354-383
Author(s):  
Willard J. Pierson
Keyword(s):  
Free Surface ◽  
Deep Water ◽  
Perturbation Expansion ◽  
The United States ◽  
Order Perturbation ◽  
Naval Academy ◽  
Third Order ◽  
First Order ◽  
Stokes Waves ◽  
Perturbation Solutions

Oscillatory third-order perturbation solutions for sums of interacting long-crested Stokes waves on deep water are obtained. A third-order perturbation expansion of the nonlinear free boundary value problem, defined by the coupled Bernoulli equation and kinematic boundary condition evaluated at the free surface, is solved by replacing the exponential term in the potential function by its series expansion and substituting the equation for the free surface into it. There are second-order changes in the frequencies of the first-order terms at third order. The waves have a Stokes-like form when they are high. The phase speeds are a function of the amplitudes and wave numbers of all of the first-order terms. The solutions are illustrated. A preliminary experiment at the United States Naval Academy is described. Some applications to sea keeping are bow submergence and slamming, capsizing in following seas and bending moments.

Three-dimensional coating flow of nematic liquid crystal on an inclined substrate

2015 ◽  
Vol 26 (5) ◽  
pp. 647-669 ◽  
Author(s):  
M. A. LAM ◽  
L. J. CUMMINGS ◽  
T.-S. LIN ◽  
L. KONDIC
Keyword(s):  
Free Surface ◽  
Liquid Crystal ◽  
Nematic Liquid Crystal ◽  
Nonlinear Partial Differential Equation ◽  
Three Dimensional ◽  
Travelling Wave ◽  
Linear Stability Theory ◽  
Newtonian Flow ◽  
Surface Evolution ◽  
Coating Flow

We consider a coating flow of nematic liquid crystal (NLC) fluid film on an inclined substrate. Exploiting the small aspect ratio in the geometry of interest, a fourth-order nonlinear partial differential equation is used to model the free surface evolution. Particular attention is paid to the interplay between the bulk elasticity and the anchoring conditions at the substrate and free surface. Previous results have shown that there exist two-dimensional travelling wave solutions that translate down the substrate. In contrast to the analogous Newtonian flow, such solutions may be unstable to streamwise perturbations. Extending well-known results for Newtonian flow, we analyse the stability of the front with respect to transverse perturbations. Using full numerical simulations, we validate the linear stability theory and present examples of downslope flow of nematic liquid crystal in the presence of both transverse and streamwise instabilities.

scholarly journals Dynamic-pressure distributions under Stokes waves with and without a current

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170103 ◽  
Author(s):  
Motohiko Umeyama
Keyword(s):  
Water Waves ◽  
Dynamic Pressure ◽  
Water Pressure ◽  
Travelling Waves ◽  
Stokes Wave ◽  
Front Face ◽  
Nonlinear Water Waves ◽  
Near Surface ◽  
Stokes Waves ◽  
A Current

To investigate changes in the instability of Stokes waves prior to wave breaking in shallow water, pressure data were recorded vertically over the entire water depth, except in the near-surface layer (from 0 cm to −3 cm), in a recirculating channel. In addition, we checked the pressure asymmetry under several conditions. The phase-averaged dynamic-pressure values for the wave–current motion appear to increase compared with those for the wave-alone motion; however, they scatter in the experimental range. The measured vertical distributions of the dynamic pressure were plotted over one wave cycle and compared to the corresponding predictions on the basis of third-order Stokes wave theory. The dynamic-pressure pattern was not the same during the acceleration and deceleration periods. Spatially, the dynamic pressure varies according to the faces of the wave, i.e. the pressure on the front face is lower than that on the rear face. The direction of wave propagation with respect to the current directly influences the essential features of the resulting dynamic pressure. The results demonstrate that interactions between travelling waves and a current lead more quickly to asymmetry. This article is part of the theme issue ‘Nonlinear water waves’.

scholarly journals Third-order stokes wave solutions of the free surface capillary-gravity wave and the interfacial internal wave

2017 ◽  
Vol 31 (6) ◽  
pp. 781-787
Author(s):  
Rui-jun Meng ◽  
Ji-feng Cui ◽  
Xiao-gang Chen ◽  
Bao-le Zhang ◽  
Hong-bo Zhang
Keyword(s):  
Free Surface ◽  
Internal Wave ◽  
Gravity Wave ◽  
Stokes Wave ◽  
Third Order ◽  
Wave Solutions

A Monte Carlo method for backward stochastic differential equations with Hermite martingales

2019 ◽  
Vol 25 (1) ◽  
pp. 37-60
Author(s):  
Antoon Pelsser ◽  
Kossi Gnameho
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Approximation ◽  
Mathematical Finance ◽  
Backward Stochastic Differential Equations ◽  
Terminal Condition ◽  
Dimensional System ◽  
Infinite Dimensional ◽  
The Family ◽  
Infinite Dimensional System

Abstract Backward stochastic differential equations (BSDEs) appear in many problems in stochastic optimal control theory, mathematical finance, insurance and economics. This work deals with the numerical approximation of the class of Markovian BSDEs where the terminal condition is a functional of a Brownian motion. Using Hermite martingales, we show that the problem of solving a BSDE is identical to solving a countable infinite-dimensional system of ordinary differential equations (ODEs). The family of ODEs belongs to the class of stiff ODEs, where the associated functional is one-sided Lipschitz. On this basis, we derive a numerical scheme and provide numerical applications.

Lagrangian moments and mass transport in Stokes waves

1987 ◽  
Vol 179 ◽  
pp. 547-555 ◽  
Author(s):  
Michael Longuet-Higgins
Keyword(s):  
Mass Transport ◽  
Phase Speed ◽  
Simple Relation ◽  
Finite Amplitude ◽  
Stokes Wave ◽  
Random Waves ◽  
The Third ◽  
Stokes Waves ◽  
Mass Transport Velocity ◽  
High Degree

The orbital motions in surface gravity waves are of interest for analysing wave records made by accelerometer buoys. In this paper we derive some exact expressions for the first, second and third cumulants of the vertical orbital displacements in a regular Stokes wave of finite amplitude in terms of previously known integral quantities of the wave: the kinetic and potential energies, the phase speed c and the mass-transport velocity U at the free surface. These results generalize a remarkably simple relation found previously between the Lagrangian-mean surface level and the product Uc.Expansions are given in powers of the wave steepness parameter ak which show that the third Lagrangian cumulant is very small – of order (ak)6, indicating a high degree of vertical symmetry in the orbit. This contrasts with the situation in random waves, where the third cumulant is of order (ak)4 only. It is shown that the increased skewness in random waves is due mainly to an O(ak)2 shift in the Lagrangian mean level of individual waves. Such shifts in mean level may be too gradual to be fully detected by some accelerometer buoys. In that case the apparent skewness will be reduced.

Expansions for the shape of maximum amplitude Stokes waves

1974 ◽  
Vol 66 (2) ◽  
pp. 261-265 ◽  
Author(s):  
A. C. Norman
Keyword(s):  
Maximum Amplitude ◽  
Stokes Wave ◽  
Stokes Waves

Grant (1973) showed that the expansion giving the profile of a steady Stokes wave near a 120° corner was more complicated than had previously been assumed. This paper gives further terms in such an expansion, and shows that generating them cannot introduce transcendental quantities beyond those noted by Grant.

scholarly journals A BEM for the Propagation of Nonlinear Planar Free-surface Waves

2007 ◽  
Vol 5 (1) ◽  
Author(s):  
V. Vinayan ◽  
S. A. Kinnas
Keyword(s):  
Free Surface ◽  
Surface Waves ◽  
Mixed Boundary ◽  
Hydrodynamic Characteristics ◽  
Roll Motion ◽  
Surface Boundary ◽  
Free Surface Waves ◽  
Surface Boundary Conditions ◽  
Stokes Waves ◽  
Fifth Order

A Boundary Element Method (BEM) model for the propagation of non- linear free-surface waves is described and its application to the study of the hydrodynamic characteristics associated with the roll-motion of 2-D hull sec- tions is presented. The roll-motion of the hull section is modeled as a mixed boundary value problem and solved using a higher-order (linear strength dis- tribution) BEM coupled with a Mixed-Eulerian-Lagrangian (MEL) scheme for the time-dependent free-surface boundary conditions. Applications, that in- clude the propagation of fifth-order Stokes waves and waves generated by a piston wave-maker, used to validate the BEM scheme prior to its application to the hull roll-motion are also described.

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