Accurate fast computation of steady two-dimensional surface gravity waves in arbitrary depth

This paper describes an efficient algorithm for computing steady two-dimensional surface gravity waves in irrotational motion. The algorithm complexity is $O(N\log N)$, $N$ being the number of Fourier modes. This feature allows the arbitrary precision computation of waves in arbitrary depth, i.e. it works efficiently for Stokes, cnoidal and solitary waves, even for quite large steepnesses, up to approximately 99 % of the maximum steepness for all wavelengths. In particular, the possibility to compute very long (cnoidal) waves accurately is a feature not shared by other algorithms and asymptotic expansions. The method is based on conformal mapping, the Babenko equation rewritten in a suitable way, the pseudo-spectral method and Petviashvili iterations. The efficiency of the algorithm is illustrated via some relevant numerical examples. The code is open source, so interested readers can easily check the claims, use and modify the algorithm.

New exact relations for steady irrotational two-dimensional gravity and capillary surface waves

Steady two-dimensional surface capillary–gravity waves in irrotational motion are considered on constant depth. By exploiting the holomorphic properties in the physical plane and introducing some transformations of the boundary conditions at the free surface, new exact relations and equations for the free surface only are derived. In particular, a physical plane counterpart of the Babenko equation is obtained. This article is part of the theme issue ‘Nonlinear water waves’.

Nonlinear Hydroelastic Waves Generated due to a Floating Elastic Plate in a Current

Effects of underlying uniform current on the nonlinear hydroelastic waves generated due to an infinite floating plate are studied analytically, under the hypotheses that the fluid is homogeneous, incompressible, and inviscid. For the case of irrotational motion, the Laplace equation is the governing equation, with the boundary conditions expressing a balance among the hydrodynamics, the uniform current, and elastic force. It is found that the convergent series solutions, obtained by the homotopy analysis method (HAM), consist of the nonlinear hydroelastic wave profile and the velocity potential. The impacts of important physical parameters are discussed in detail. With the increment of the following current intensity, we find that the amplitudes of the hydroelastic waves decrease very slightly, while the opposing current produces the opposite effect on the hydroelastic waves. Furthermore, the amplitudes of waves increase very obviously for higher opposing current speed but reduce very slightly for higher following current speed. A larger amplitude of the incident wave increases the hydroelastic wave deflections for both opposing and following current, while for Young’s modulus of the plate there is the opposite effect.

The Resolution of d'Alembert's Paradox and Thermodynamic Admissibility

The d'Alembert paradox, annunciated in 1752, was established after it was shown that the result of a net zero drag, obtained by applying potential theory to the incompressible flow past a sphere, was in contradiction with experimental results which showed a positive drag. Interpreting the result as a flaw in the theory, resulted in the declaration of the paradox. Following d'Alembert, we assume a potential motion, and proceed to analyze the consequences of this assumption using the global principles of continuum mechanics. We show that if the fluid is inviscid, the potential motion is thermodynamically admissible, the drag is zero, and the motion can persist indefinitely. Although no conventional fluid is available to either falsify (or validate), this result experimentally, in principle, the theory could be tested by using a superfluid, such as liquid Helium. If the fluid is viscous, we show that the potential irrotational motion is thermodynamically inadmissible, it is in violation of the second law of thermodynamics, and hence such a motion cannot persist. Indeed, observations show that when a rigid body is impulsively set into motion, an irrotational motion may exist initially but does not persist. Any flow property which is derived from a thermodynamically inadmissible motion cannot be expected to be in agreement with experimental data. As an illustration we show that the drag, calculated from the viscous potential solution of the flow past a sphere, is zero. We submit that since the theory of continuum mechanics predicts that no agreement between results obtained from viscous potential theory and experimental data can be expected, there is no room for a paradox once a contradiction is in fact observed. In nature, or under experimental conditions, the nonpersistence of the thermodynamically inadmissible motion proceeds in a breakup of the irrotational motion which transforms into a rotational and obviously admissible motion. We show that by selecting boundary conditions, required in the solution of the differential equations of motion, such that they satisfy the Clausius–Duhem jump conditions inequality, the thermodynamic admissibility of the solution is a priori assured. We also show the vorticity distribution at the wall associated with the particular choice of boundary condition.

Note on the velocity and related fields of steady irrotational two-dimensional surface gravity waves

The velocity and other fields of steady two-dimensional surface gravity waves in irrotational motion are investigated numerically. Only symmetric waves with one crest per wavelength are considered, i.e. Stokes waves of finite amplitude, but not the highest waves, nor subharmonic and superharmonic bifurcations of Stokes waves. The numerical results are analysed, and several conjectures are made about the velocity and acceleration fields.

ELEMENTS OF WAVE THEORY

The first known mathematical solution for finite height, periodic waves of stable form was developed by Gerstner (1802). From equations that were developed, Gerstner (1802) arrived at the conclusion that the surface curve was trochoidal in form. Froude (1862) and Rankine (1863) developed the theory but in the opposite manner, i.e., they started with the assumption of a trochoidal form and then developed their equations from this curve. The theory was developed for waves in water of infinite depth with the orbits of the water particles being circular, decreasing in geometrical progression as the distance below the water surface increased in arithmetical progression. Recent experiments (Wiegel, 1950) have shown that the surface profile, represented by the trochoidal equations (as well as the first few terms of Stokes' theory), closely approximates the actual profiles for waves traveling over a horizontal bottom. However the theory necessitates molecular rotation of the particles, while the manner in which waves are formed by conservative forces necessitates irrotational motion.

Nonlinear Vibrations of Partially Fluid-Filled Cylindrical Shells

The present work investigates the nonlinear dynamic behavior and instabilities of partially fluid-filled cylindrical shell subjected to lateral pressure. Donnell shallow shell theory is employed to model the shell. The fluid is modeled as non-viscous and incompressible and its irrotational motion is described by a velocity potential which satisfies the Laplace equation. A discrete low-dimensional model for the nonlinear vibration analysis of thin cylindrical shells is derived to study the shell vibrations. First, a general expression for the nonlinear vibration modes that satisfy all the relevant boundary, continuity and symmetry conditions is derived using a perturbation procedure validated in previous studies and then the Galerkin method is used to discretize the equations of motion. The same modal solution is used to derive the hydrodynamic pressure on the shell wall. The influence played by the height of the internal fluid on the natural frequencies, nonlinear shell response and bifurcations is examined.

A note on the unsteady motion under gravity of a corner point on a free surface: a generalization of Stokes' theory

In this paper, we develop a theory based on local asymptotic coordinate expansions for the unsteady propagation of a corner point on the constant-pressure free surface bounding an incompressible inviscid fluid in irrotational motion under the action of gravity. This generalizes the result of Stokes and Michell relating to the horizontal propagation of a corner at constant speed.