Stokes Expansion Research Articles

Steady Boussinesq convection: Parametric analyticity and computation

AbstractSteady solutions to the Navier–Stokes equations with internal temperature forcing are considered. The equations are solved in two dimensions using the Boussinesq approximation to couple temperature and density fluctuations. A perturbative Stokes expansion is used to prove that that steady flow variables are parametrically analytic in the size of the forcing. The Stokes expansion is complemented with analytic continuation, via functional Padé approximation. The zeros of the denominator polynomials in the Padé approximants are observed to agree with a numerical prediction for the location of singularities of the steady flow solutions. The Padé representations not only prove to be good approximations to the true flow solutions for moderate intensity forcing, but are also used to initialize a Newton solver to compute large amplitude solutions. The composite procedure is used to compute steady flow solutions with forcing several orders of magnitude larger than the fixed‐point method developed in previous work.

Mass transport at the bottom of propagating surface waves over a rippled bottom

The sea surface can be described by means of the superposition of many sinusoidal functions. However, quite often the amplitude of each component turns out to be much smaller than its wavelength, and any component evolves independently of the others. Hence, it is common to investigate the dynamics of a simple monochromatic surface wave. Hereinafter, the flow generated by a monochromatic surface wave within the bottom boundary layer over a rippled sea bed is determined by means of the numerical integration of vorticity and continuity equations. The forcing term that drives the fluid motion within the boundary layer is evaluated assuming that the steepness of the monochromatic surface wave is much smaller than one and considering the first term of the Stokes expansion. Even though the irrotational flow that forces the viscous rotational flow near the sea bottom is symmetric with respect to the ripple crests, Blondeaux and Vittori [“A route to chaos in an oscillatory flow: Feigenbaum scenario,” Phys. Fluids A 3(11), 2492–2495 (1991a)] showed that the symmetry of the flow field is broken when the Reynolds number becomes larger than a threshold value Rδ,t1 that depends on the geometrical characteristics of the ripples. The results of Blondeaux and Vittori [“A route to chaos in an oscillatory flow: Feigenbaum scenario,” Phys. Fluids A 3(11), 2492–2495 (1991a)] suggest that, when the Reynolds number is larger than Rδ,t1 but not too far from it, a steady current is also generated. Hereinafter, the steady velocity component is determined as a function of the ripple characteristics.

Weakly nonlinear broadband and multi-directional surface waves on an arbitrary depth: A framework, Stokes drift, and particle trajectories

Surface gravity waves in coastal waters are broadband and multi-directional, whose quadratic properties are of considerable engineering and scientific interest. Based on a Stokes expansion and an envelope-type framework, a new semi-analytical approach is proposed in this paper for the description of weakly nonlinear broadband and multi-directional surface waves. This approach proposes solving for the second-order wave fields through the separation of harmonics, by using a Fast Fourier transform and a time integration method. Different from some other methods, e.g., the High-Order Spectral method, the approach introduces a spectral shift for the superharmonic waves, leading to computationally efficient and accurate spectral predictions. The approach has been validated through comparisons with the results based on Dalzell [“A note on finite depth second-order wave–wave interactions,” Appl. Ocean Res. 21, 105–111 (1999)]. An envelope-type framework for the fast prediction of particle trajectories and Stokes drifts up to the second order in wave steepness is also derived in this paper, based on the semi-analytical approach. This paper shows that the results based on a narrowband assumption lead to underestimates of Stokes drift velocities driven by broadband unidirectional focused wave groups. The cases, examined for particle trajectories below broadband unidirectional focused wave groups, show that a larger bandwidth and water depth can enhance the differences in the net mean horizontal displacement of particles at water surface relative to these at seabed.

Surface wavepackets subject to an abrupt depth change. Part 1. Second-order theory

Abstract

Extreme runup events around a ship-shaped floating production, storage and offloading vessel in transient wave groups

Abstract

Progressive flexural–gravity waves with constant vorticity

Abstract

Rigorous Asymptotic Models of Water Waves

We develop a rigorous asymptotic derivation of two mathematical models of water waves that capture the full nonlinearity of the Euler equations up to quadratic and cubic interactions, respectively. Specifically, letting $$ \epsilon $$ denote an asymptotic parameter denoting the steepness of the water wave, we use a Stokes expansion in $$ \epsilon $$ to derive a set of linear recursion relations for the tangential component of velocity, the stream function, and the water wave parameterization. The solution of the water wave system is obtained as an infinite sum of solutions to linear problems at each $$O( \epsilon ^k)$$ level, and truncation of this series leads to our two asymptotic models, which we call the quadratic and cubic h-models. These models are well posed in spaces of analytic functions. We prove error bounds for the difference between solutions of the h-models and the water wave system. We also show that the Craig–Sulem models of water waves can be obtained from our asymptotic procedure. We then develop a novel numerical algorithm to solve the quadratic and cubic h-models as well as the full water wave system. For three very different examples, we show that the agreement between the model equations and the water wave solution is excellent, even when the wave steepness is quite large. We also present a numerical example of corner formation for water waves.

Gravity-capillary waves in finite depth on flows of constant vorticity.

This paper considers two-dimensional periodic gravity-capillary waves propagating steadily in finite depth on a linear shear current (constant vorticity). A perturbation series solution for steady periodic waves, accurate up to the third order, is derived using a classical Stokes expansion procedure, which allows us to include surface tension effects in the analysis of wave-current interactions in the presence of constant vorticity. The analytical results are then compared with numerical computations with the full equations. The main results are (i) the phase velocity is strongly dependent on the value of the vorticity; (ii) the singularities (Wilton singularities) in the Stokes expansion in powers of wave amplitude that correspond to a Bond number of 1/2 and 1/3, which are the consequences of the non-uniformity in the ordering of the Fourier coefficients, are found to be influenced by vorticity; (iii) different surface profiles of capillary-gravity waves are computed and the effect of vorticity on those profiles is shown to be important, in particular that the solutions exhibit type-2-like wave features, characterized by a secondary maximum on the surface profile with a trough between the two maxima.

Numerical study of the generalised Klein–Gordon equations

In this study, we discuss an approximate set of equations describing water wave propagating in deep water. These generalised Klein–Gordon (gKG) equations possess a variational formulation, as well as a canonical Hamiltonian and multi-symplectic structures. Periodic travelling wave solutions are constructed numerically to high accuracy and compared to a seventh-order Stokes expansion of the full Euler equations. Then, we propose an efficient pseudo-spectral discretisation, which allows to assess the stability of travelling waves and localised wave packets.

A generalized approach to the mechanics of three-dimensional nonlinear ocean waves

The paper suggests a generalized approach to the analysis of second-order three-dimensional random wave fields that can represent either ocean waves in an undisturbed field or standing ocean waves. A nonlinear stochastic family is introduced that has a linear stationary Gaussian random contribution and a second-order correction exact in a Stokes expansion. This family can describe the nonlinear mechanics of a random wave field. Subsequently, in the considered random ocean field, the nonlinear mechanics of wave groups can be investigated conditional to the occurrence of a very high wave. The free-surface displacement and velocity potential may be obtained, as well as the nonlinear wave velocity and acceleration, for both the stochastic family and the very high wave groups. The results are of interest for the analysis of nonlinear wave kinematics when extreme surface wave events occur and thus for Morison′s wave forces on slender structures.

Addition of dispersive terms to the method of averaged Lagrangian

Whitham's method of averaged Lagrangian is applied to the problem of Stokes waves on the surface of a layer of ideal fluid. We derive a Lagrangian which contains additional terms with \documentclass[12pt]{minimal}\begin{document}$a_ {x} ^ {2}$\end{document}ax2 and aaxx besides nonlinear terms of Whitham with a2 and a4, a being the wave amplitude. These terms with derivatives appear after taking into account (1) additional terms in the Stokes expansions with the same approximation as the one was taken into consideration by Whitham; (2) supplementary slow (in comparison to the rapid phase oscillations) dependence of the amplitudes of harmonics on coordinates and time. We demonstrate the need for the account of such terms in the Lagrangian for obtaining the correct coefficients of dispersive terms of evolution equations from the corresponding variational equations.

Internal Waves at an Interface between Two Layers of Differing Stability

Abstract Internal waves in a two-layer fluid are considered. The layers have different values of the buoyancy frequency, assumed to be constant in each layer. The density profile is chosen to be continuous across the interface and the flow is Boussinesq. The solution is an expansion in the wave amplitude, similar to a Stokes expansion for free surface waves. The results show that the nonlinear terms in the interfacial boundary conditions require higher harmonics and result in nonlinear wave steepening at the interface. The first few harmonics are scattered by the interface, whereas the higher harmonics are evanescent in the vertical. The second-order correction to the wave speed is negative, similar to previous results with a rigid upper boundary.

The exact solution of a non-linear boundary-value problem of the theory of waves on the surface of a liquid of finite depth

A non-linear boundary-value problem of the theory of waves on the surface of a heavy ideal incompressible liquid, which arises as a result of the expansion of the required functions in amplitude, taking quadratic terms into account, is investigated. A solution is constructed, on the one hand, suitable for describing long waves, and on the other, matched to the Stokes expansion (i.e., with the expansion in amplitude of the first order of infinitesimals). A function is sought which conformally maps a strip into the plane of the complex potential in the flow region. An exact solution is obtained for this problem, defined by fairly simply formulae. This solution, in the limit of long and short waves, gives linear sinusoidal waves and cnoidal waves respectively.

Further on stokes expansions for the finite amplitude water waves

An analytic method is applied to study the higher order approximate solution of Stokes waves. For comparison and effective analysis, solutions of first to fifth order approximations are studied. From the various expressions derived, the numerical computations and the graphical solutions, there is a gradual increase in the wave height from the first to fifth order and they follow one another very closely. However, there is a significant difference in that of fourth and fifth order compared with those of first to third. This analysis suggests that the wave height of fifth order is twice that of third order but with slight increase with respect to the fourth order solution. Fifth order Stokes waves may be peculiar to waves with unusual characters in view of the significant difference in the wave height when compared with those of lower order solution. However, dispersive phenomena among wave modes often limit the wave growth in practice. Journal of the Nigerian Association of Mathematical Physics Vol. 8 2004: pp. 227-236

On the leading nonlinear correction to gravity-wave dynamics

The principal nonlinear correction to the dynamics of gravity waves on an irrotational fluid is traditionally derived as a non-resonant perturbation solution to the Stokes expansion. When the problem is reformulated in the Hamiltonian description and limited to moderately collimated random waves over infinite depth, the perturbation term assumes a very simple and descriptive form. The sum-frequency component for the surface height is just a bilinear product of the height with the associated scalar strain, and the accompanying term in the potential is half the time derivative of the squared linear height. This solution is exact in one surface dimension and remains quite accurate for long-crested waves in two dimensions, with an error small to second order in the angular spread of constituent wave vectors. It is a natural generalization for random, disordered wave ensembles of the second-order Stokes solution, and its effect is to sharpen the random crests and to flatten the troughs. For wave sets of narrow relative bandwidth the difference-frequency component consists of a negligible elevation term and a non-negligible potential term whose gradient is the surface value of the volume return flow balancing the quadratic wave transport of fluid.

New asymptotic description of nonlinear water waves in Lagrangian coordinates

A new description of two-dimensional continuous free-surface flows in Lagrangian coordinates is proposed. It is shown that the position of a fluid particle in such flows can be represented as a fixed point of a transformation in . Components of the transformation function satisfy the linear Euler-type continuity equation and can be expressed via a single function analogous to an Eulerian stream function. Fixed-point iterations lead to a simple recursive representation of a solution satisfying the Lagrangian continuity equation. Expanding the unknown function in a small-perturbation asymptotic expansion we obtain the complete asymptotic formulation of the problem in a fixed domain of Lagrangian labels. The method is then applied to the classical problem of a regular wave travelling in deep water, and the fifth-order Lagrangian asymptotic solution is constructed, which provides a much better approximation of steep waves than the corresponding Eulerian Stokes expansion. In contrast with early attempts at Lagrangian regular-wave expansions, the asymptotic solution presented is uniformly valid at large times.

Non linear highest sea wave groups in an undisturbed field and in front of a vertical wall

In this paper some non-linear effects for the mechanics of sea wave groups with large waves are investigated, either for waves in an undisturbed field or for waves in front of a vertical wall. To the first-order in a Stokes expansion, Boccotti s quasi-determinism theory enables us to foresee the mechanics of wave groups, either in undisturbed or in diffracted fields, when a large wave occurs. The first formulation of this theory shows the random group mechanics when a large crest height occurs (New wave); the second theory formulation gives the random group mechanics when a large crest-to-trough wave height occurs. The quasi-determinism theory in both formulations, for undisturbed fields, was extended recently to the second-order by the author. In this paper the procedure to derive the second-order solution is analyzed and is applied to random wave groups in front of a vertical wall. The non-linear effects are then investigated in space-time domain, and it is obtained a good agreement of analytical predictions with both field data and data from numerical simulation.

On non-linear very large sea wave groups

The paper deals with the non-linear effects for sea wave groups. Boccotti's quasi-determinism theory, which is exact to the first-order in a Stokes expansion, gives the mechanics of sea wave groups when either a very high crest (first formulation of the theory—‘New wave’), or a large crest-to-trough wave height (second formulation of the theory) occurs. In this paper, the quasi-determinism theory, in both formulations, is extended to the second-order, by obtaining the expressions of free surface displacement and velocity potential, as a function of wave spectrum. Finally it is shown that analytical predictions are in good agreement with both field data and data of Monte Carlo simulations of non-linear random waves.

Second-order sloshing over an arbitrary bed

The two-dimensional problem of the free sloshing of an inviscid fluid in a vertically walled tank with an arbitrary bed shape is solved at both first and second order in the Stokes expansion of the velocity potential. The approach employed at both orders uses Green's functions for a flat bed in conjunction with the Cauchy–Riemann equations to derive integral equations for the tangential flux along the varying bed. The first- and second-order potentials everywhere in the fluid may then be related to these fluxes. Significant analytic progress is made with the calculation of various contributions to the integral equations at second order. The equations at first and second order are ultimately solved using a variational principle equivalent to the Galerkin method, giving efficient and accurate results. In particular, the work involved in determining the second-order solution is no more intensive than in solving the first-order problem. The first-order solution is shown to reproduce known results for specific bed shapes. The method is applied to a range of bed shapes and the second-order correction to the free-surface elevation is illustrated.

Weakly nonlinear statistics of high random waves

It is well known that in a Gaussian sea state for an infinitely narrow spectrum the crest height and the trough depth follow the same Rayleigh distribution, because of linearity of the first order Stokes expansion solution. For spectra of finite bandwidth, Boccotti obtained, as a corollary of his first formulation of the theory of quasideterminism (which is exact to the first order in a Stokes expansion), that the crest height and the trough depth still follow asymptotically the Rayleigh law for high waves in Gaussian sea states. In this paper we extend the theory of quasideterminism of Boccotti to the second-order, deriving new wave crest and wave trough distributions that take into account nonlinear effects and are valid for finite bandwidth of the spectrum in deep water. Nonlinear Monte Carlo simulations validate our theoretical predictions and comparisons with experimental data and the recent model of Forristall are finally presented.