Irrotational Water Waves Research Articles

Sigma mapping for drainage problems with a time-dependent water table

Regional groundwater flow forecasting is frequently accomplished using model equations based on Darcy's law, the continuity equation, and kinematic and dynamic boundary conditions for a time-dependent water table, to predict hydraulic heads and fluxes. For homogeneous and isotropic aquifers, the model reduces to Laplace's equation for the hydraulic head and a transient non-linear free surface boundary condition. These equations are solved in the literature and in computer packages by a variety of numerical methods, including finite difference, finite element, and boundary element methods. The solution of this model is not simple and subject to instability problems. Computations in a vertical plane using finite-difference methods are not easy due to the irregular mesh needed near a highly curved water table, frequently involving smoothing of the numerical results to ensure stability. A technique to avoid the complications related to mesh generation near a curved boundary and its tracking entails using mappings to produce transformed domains. However, this is not so frequent to compute unsteady seepages. In particular, the so-called sigma mapping is widely used to model irrotational water waves; it applies to any flow involving a time-dependent free surface boundary. This mapping transforms the domain into a rectangle, so that the free surface becomes static in the transformed plane. There is close similarity between the mathematical model describing irrotational water waves and that of unsteady seepage flow, e.g., both are governed by Laplace's equation involving relevant boundary conditions at the moving boundary. Both moving free surfaces are different in character, e.g., the seepage is dissipative whereas the water wave is dispersive. However, it is possible to transfer mathematical approximations between both fields of research. This transfer is exploited in this work and the sigma mapping is newly and successfully applied to seepage flows involving both a steady and unsteady water table either receiving, or not, a recharge from rainfall or an artificial source.

Energy considerations for nonlinear equatorial water waves

<p style='text-indent:20px;'>In this article we consider the excess kinetic and potential energies for exact nonlinear equatorial water waves. An investigation of linear waves establishes that the excess kinetic energy density is always negative, whereas the excess potential energy density is always positive, for periodic travelling irrotational water waves in the steady reference frame. For negative wavespeeds, we prove that similar inequalities must also hold for nonlinear wave solutions. Characterisations of the various excess energy densities as integrals along the wave surface profile are also derived.</p>

On the energy of nonlinear water waves.

This article presents results concerning the excess kinetic and potential energies for exact nonlinear water waves. In particular, it is proven, for periodic travelling irrotational water waves, that the excess kinetic energy density is always negative, whereas the excess potential energy density is always positive, in the steady reference frame. A characterization of the total excess energy density as a weighted mean of the kinetic energy along the wave surface profile is also presented.

Global bifurcation of irrotational water waves using a flow force formulation

Using a novel formulation based on flow force, we prove the existence of a global continuum of periodic traveling wave solutions to the irrotational water wave problem in finite depth where the flow force is held fixed, instead of fixing the mass flux. We further establish the limiting behaviour of solutions in this continuum where the flow force remains fixed throughout the continuum of the solutions. The global solutions, with a fixed flow force, show the possibility that a weak stagnation point can be approached at the surface, which is a characteristic of large amplitude irrotational water waves. The formulation is suitable for applications to numerical studies on water waves.

An alternative approach to study irrotational periodic gravity water waves

We are concerned here with an analysis of the nonlinear irrotational gravity water wave problem with a free surface over a water flow bounded below by a flat bed. We employ a new formulation involving an expression (called flow force) which contains pressure terms, thus having the potential to handle intricate surface dynamic boundary conditions. The proposed formulation neither requires the graph assumption of the free surface nor does require the absence of stagnation points. By way of this alternative approach we prove the existence of a local curve of solutions to the water wave problem with fixed flow force and more relaxed assumptions.

On the decrease of velocity with depth in irrotational periodic water waves

We give an alternative proof for a classical result (due to Longuet-Higgins) that provides an estimate for the decay rate with depth of the velocity beneath two-dimensional, spatially periodic, irrotational water waves over a flat bed. Furthermore, an improvement to the same estimate is presented.

A flow force reformulation for steady irrotational water waves

The classical irrotational water wave problem described by Euler equations with a nonlinear free surface boundary condition and influenced by gravity over a flat bed is considered. Exploiting the monotonicity property of the flow force lines with depth, the unknown boundary problem is transformed into a problem with fixed domain. A flow force function formulation of the irrotational water wave problem is developed. Variational approach is used to prove the existence of small amplitude irrotational traveling wave solutions. Local bifurcation results have been proved relying on Crandall-Rabinowitz bifurcation theorem.

Experimental study of particle trajectories below deep-water surface gravity wave groups

Owing to the interplay between the forward Stokes drift and the backward wave-induced Eulerian return flow, Lagrangian particles underneath surface gravity wave groups can follow different trajectories depending on their initial depth below the surface. The motion of particles near the free surface is dominated by the waves and their Stokes drift, whereas particles at large depths follow horseshoe-shaped trajectories dominated by the Eulerian return flow. For unidirectional wave groups, a small net displacement in the direction of travel of the group results near the surface, and is accompanied by a net particle displacement in the opposite direction at depth. For deep-water waves, we study these trajectories experimentally by means of particle tracking velocimetry in a two-dimensional flume. In doing so, we provide visual illustration of Lagrangian trajectories under groups, including the contributions of both the Stokes drift and the Eulerian return flow to both the horizontal and the vertical Lagrangian displacements. We compare our experimental results to leading-order solutions of the irrotational water wave equations, finding good agreement.

Numerical investigations of two-dimensional irrotational water waves over finite depth with uniform current

We investigate large amplitude steady water waves in the presence of underlying uniform current with using a numerical continuation method. The two branches of traveling wave solutions, whose existence has been proved recently by Basu [On the existence of two-dimensional irrotational water waves over finite depth with uniform current. Appl Anal. 2018;97(14):2523–2532.] for a parameter or are computed numerically. The numerical results for the case have not been previously reported in the literature. The properties of the waves and flows in this solution branch are discussed as comparison to those in the case of .

On the Motion of a Compressible Gravity Water Wave with Vorticity

We prove a priori estimates for the compressible Euler equations modeling the motion of a liquid with moving physical vacuum boundary with unbounded initial domain. The liquid is under influence of gravity but without surface tension. Our fluid is not assumed to be irrotational. But the physical sign condition needs to be assumed on the free boundary. We generalize the method used in Lindblad and Luo (Commun Pure Appl Math, 2008) to prove the energy estimates in an unbounded domain up to arbitrary order. In addition to that, the a priori energy estimates are in fact uniform in the sound speed $$\kappa $$ . As a consequence, we obtain the convergence of solutions of compressible Euler equations with a free boundary to solutions of the incompressible equations, generalizing the result of Lindblad and Luo (2008) to when you have an unbounded domain. On the other hand, we prove that there are initial data satisfying the compatibility condition in some weighted Sobolev spaces, and this will propagate within a short time interval, which is essential for proving long time existence for slightly compressible irrotational water waves.

Global Regularity for 2d Water Waves with Surface Tension

We consider the full irrotational water waves system with surface tension and no gravity in dimension two (the capillary waves system), and prove global regularity and modified scattering for suitably small and localized perturbations of a flat interface. An important point of our analysis is to develop a sufficiently robust method (the “quasilinear I-method”) which allows us to deal with strong singularities arising from time resonances in the applications of the normal form method (the so-called “division problem”). As a result, we are able to consider a suitable class of perturbations with finite energy, but no other momentum conditions. Part of our analysis relies on a new treatment of the Dirichlet-Neumann operator in dimension two which is of independent interest. As a consequence, the results in this paper are self-contained.

Existence and amplitude bounds for irrotational water waves in finite depth.

We prove the existence of solutions to the irrotational water-wave problem in finite depth and derive an explicit upper bound on the amplitude of the nonlinear solutions in terms of the wavenumber, the total hydraulic head, the wave speed and the relative mass flux. Our approach relies upon a reformulation of the water-wave problem as a one-dimensional pseudo-differential equation and the Newton-Kantorovich iteration for Banach spaces.This article is part of the theme issue 'Nonlinear water waves'.

Recent advances on the global regularity for irrotational water waves.

We review recent progress on the long-time regularity of solutions of the Cauchy problem for the water waves equations, in two and three dimensions. We begin by introducing the free boundary Euler equations and discussing the local existence of solutions using the paradifferential approach. We then describe in a unified framework, using the Eulerian formulation, global existence results for three- and two-dimensional gravity waves, and our joint result (with Deng and Pausader) on global regularity for the gravity-capillary model in three dimensions. We conclude this review with a short discussion about the formation of singularities and give a few additional references to other interesting topics in the theory.This article is part of the theme issue 'Nonlinear water waves'.

Shallow-water models for a vibrating fluid

We consider a layer of an inviscid fluid with a free surface which is subject to vertical high-frequency vibrations. We derive three asymptotic systems of equations that describe slowly evolving (in comparison with the vibration frequency) free-surface waves. The first set of equations is obtained without assuming that the waves are long. These equations are as difficult to solve as the exact equations for irrotational water waves in a non-vibrating fluid. The other two models describe long waves. These models are obtained under two different assumptions about the amplitude of the vibration. Surprisingly, the governing equations have exactly the same form in both cases (up to the interpretation of some constants). These equations reduce to the standard dispersionless shallow-water equations if the vibration is absent, and the vibration manifests itself via an additional term which makes the equations dispersive and, for small-amplitude waves, is similar to the term that would appear if surface tension were taken into account. We show that our dispersive shallow-water equations have both solitary and periodic travelling wave solutions and discuss an analogy between these solutions and travelling capillary–gravity waves in a non-vibrating fluid.

On the existence of two-dimensional irrotational water waves over finite depth with uniform current

ABSTRACTThe existence of large amplitude two-dimensional water waves with wave speed c, propagating on the surface of an irrotational flow over a flat bed with underlying uniform current, is investigated by means of a bifurcation approach. Two branches of travelling wave solutions exist for a parameter or 1, respectively (where is the flow velocity at the wave crest). The existence of the global bifurcating branch of solution for has not been reported in the literature and can exist in the presence of uniform underlying current, while the other branch of solution for can exist without any underlying current as has been shown by previously.

Particle motions beneath irrotational water waves

Neutral and buoyant particle motions in an irrotational flow are investigated under the passage of linear, nonlinear gravity, and weakly nonlinear solitary waves at a constant water depth. The developed numerical models for the particle trajectories in a non-turbulent flow incorporate particle momentum, size, and mass (i.e., inertial particles) under the influence of various surface waves such as Korteweg-de Vries waves which admit a three parameter family of periodic cnoidal wave solutions. We then formulate expressions of mass-transport velocities for the neutral and buoyant particles. A series of test cases suggests that the inertial particles possess a combined horizontal and vertical drifts from the locations of their release, with a fall velocity as a function of particle material properties, ambient flow, and wave parameters. The estimated solutions exhibit good agreement with previously explained particle behavior beneath progressive surface gravity waves. We further investigate the response of a neutrally buoyant water parcel trajectories in a rotating fluid when subjected to a series of wind and wave events. The results confirm the importance of the wave-induced Coriolis-Stokes force effect in both amplifying (destroying) the pre-existing inertial oscillations and in modulating the direction of the flow particles. Although this work has mainly focused on wave-current-particle interaction in the absence of turbulence stochastic forcing effects, the exercise of the suggested numerical models provides additional insights into the mechanisms of wave effects on the passive trajectories for both living and nonliving particles such as swimming trajectories of plankton in non-turbulent flows.

Symmetric irrotational water waves are traveling waves

We show that a spatially periodic solution to the irrotational two-dimensional gravity water wave problem, with the property that the horizontal velocity component at the surface, as well as the wave profile is symmetric, necessarily defines a traveling wave. The proof makes use of maximum principles for harmonic functions and structural properties of the governing equations for nonlinear water waves.

Recovery of the wave profile for irrotational periodic water waves from pressure measurements

Using a conformal hodograph transform and the governing equations for periodic traveling irrotational water waves, we obtain a relation between the bottom pressure and the free surface, expressed in terms of Fourier coefficients. This permits us to perform a formal asymptotic analysis that yields an approximate solution for the free surface provided the pressure at the flat bed is known.

Variational derivation of two-component Camassa–Holm shallow water system

By a variational approach in the Lagrangian formalism, we derive the nonlinear integrable two-component Camassa–Holm system (1). We show that the two-component Camassa–Holm system (1) with the plus sign arises as an approximation to the Euler equations of hydrodynamics for propagation of irrotational shallow water waves over a flat bed. The Lagrangian used in the variational derivation is not a metric.

Lagrangian Solution for an Irrotational Progressive Water Wave Propagating on a Uniform Current

Abstract Experiments are conducted to measure the motion properties of water particle for the progressive water wave propagation in the presence of following and adverse uniform currents. The experimental data are used to validate the fifth-order Lagrangian solution from Chen and Chen. The experimental results show that the measured data of the particle motion properties such as the b line (denoted as the line connecting the positions of consecutive particles of the same b label), the particle velocity, the particle transport velocity (drift velocity), the particle trajectory, the particle motion period, and the Lagrangian mean level are in close agreement with those of the fifth-order Lagrangian solution. The study also shows that the particle label could adopt the position coordinates of the particle as if it were in still water. The motion of the b line oscillates like wave motion: its wavelength is equal to the progressive wavelength and its wave velocity obeys the Doppler effect so the sum of the velocities of the progressive wave and current, the particle motion period, the Lagrangian mean level, and the particle transport velocity less current velocity are the same as for the case of pure progressive waves. For following currents, the shape of particle trajectory depends on the horizontal particle velocity at the trajectory trough. For adverse currents, the shape of particle trajectory depends on the horizontal particle velocity at the trajectory crest. For a description of the flow motion, the Lagrangian solution could be more effective and precise than the Eulerian solution.