Finite-amplitude Surface Waves Research Articles

Development and Validation of Quasi-Eulerian Mean Three-Dimensional Equations of Motion Using the Generalized Lagrangian Mean Method

This study aims at developing a new set of equations of mean motion in the presence of surface waves, which is practically applicable from deep water to the coastal zone, estuaries, and outflow areas. The generalized Lagrangian mean (GLM) method is employed to derive a set of quasi-Eulerian mean three-dimensional equations of motion, where effects of the waves are included through source terms. The obtained equations are expressed to the second-order of wave amplitude. Whereas the classical Eulerian-mean equations of motion are only applicable below the wave trough, the new equations are valid until the mean water surface even in the presence of finite-amplitude surface waves. A two-dimensional numerical model (2DV model) is developed to validate the new set of equations of motion. The 2DV model passes the test of steady monochromatic waves propagating over a slope without dissipation (adiabatic condition). This is a primary test for equations of mean motion with a known analytical solution. In addition to this, experimental data for the interaction between random waves and a mean current in both non-breaking and breaking waves are employed to validate the 2DV model. As shown by this successful implementation and validation, the implementation of these equations in any 3D model code is straightforward and may be expected to provide consistent results from deep water to the surf zone, under both weak and strong ambient currents.

Unsteady free surface flow above a moving circular cylinder

A problem on non-stationary free surface flow of infinitely deep ideal fluid generated by the motion of a submerged body is considered. The water wave problem is reduced to the integral–differential system of equations for the functions defining free surface shape, normal, and tangential velocity components on the free boundary. Small-time asymptotic solution is constructed for the case of circular cylinder that moves with constant acceleration from rest. The role of non-linearity is clarified by analysis of this approximate solution which describes the formation of added mass layers, splash jets, and finite amplitude surface waves.

Large-Eddy Simulation of Marine Atmospheric Boundary Layers above a Spectrum of Moving Waves

Abstract Momentum and scalar transport in the marine atmospheric boundary layer (MABL) is driven by a turbulent mix of winds, buoyancy, and surface gravity waves. To investigate the interaction between these processes, a large-eddy simulation (LES) model is developed with the capability to impose a broadband spectrum of time-varying finite-amplitude surface waves at its lower boundary. The LES model adopts a Boussinesq flow model and integrates the governing equations on a time-varying, surface-fitted, nonorthogonal mesh using cell-centered variables with special attention paid to the solution of the pressure Poisson equation near the wavy boundary. Weakly unstable MABLs are simulated with geostrophic winds increasing from 5 to 25 m s−1 and wave age varying from swell-dominated to wind-wave equilibrium. The simulations illustrate cross-scale coupling as wave-impacted near-surface turbulence transitions into shear-convective rolls with increasing distance from the water. In a regime with swell, low winds, and weak heating, wave-induced vertical velocity and pressure signals are readily observed well above the standard reference height ζa = 10 m. At wind-wave equilibrium, the small-scale wave-induced signals are detectable only near the water surface. Below ζa, a nearly-constant-flux layer is observed where the momentum flux carried by turbulence, form stress, and subgrid-scale motions shifts with varying wave age and distance above the water. The spectral content of the surface form stress is wave-age dependent, especially at low wavenumbers. The LES wind profiles deviate from Monin–Obukhov similarity theory in nonequilibrium wind-wave conditions, and entrainment is greatly enhanced by shear-induced engulfment events.

Finite-amplitude surface waves on a thin film flow subject to a unipolar-charge injection

The interaction of an electric field with a dielectric liquid film is investigated as it drains under gravity down an inclined plane electrode emitting uniform positive ions into the liquid region. By applying long-wave approximation to the governing equations, the evolution equation for the free surface is derived up to the first order of a thin film parameter ξ To investigate the space charge effect on the development of a finite-amplitude surface wave, a neutral stability condition is obtained as a critical Reynolds number through a linear stability analysis, and the amplitude and velocity of a periodic disturbance are also calculated within a supercritically stable flow region. The presence of a unipolar space charge in the fluid makes a steady surface wave take on even higher amplitude and faster wave speed compared with the case of no space charge.

Wave dynamics features in a sea ice area

On a base of developed hydrodynamic models the reflecting capacities of edges, cracks and ice field breaks are studied when the surface waves pass through them. It was investigated the ice flexural rigidity and compression effects on the spatial distribution of wave disturbance amplitudes and bend stresses of the ice cover on both sides of the examined horizontal non-uniformity. Assessment of the ice breaking possibility within the edge-side zones was made. The surface wave propagation from the basin deep-water area through the bottom’s step into the area of the finite depth is investigated. The spatial and temporal features of the formed dynamic frontal zones on both sides of the step are revealed. The dependence of the basin surface displacement amplitudes and wave current velocities on the ice thickness, incident wave periods, and depth of the step occurrence, is studied. The three-dimensional bending vibrations of floating ice cover induced by a moving vortex formation are studied. Dependence of the amplitude-phase vibration characteristics on the ice rigidity in bending, angular velocity and forward speed of the vortex is investigated. Spatial distribution of the flexural-gravity disturbances both in front of the moving pressure region and in the back wake is analyzed. A mathematical model of the non-linear dynamics of surface waves of finite amplitude in the ice floes seas was developed involving the method of multiscale expansions. Using the model, it was studiedthe dependence of the amplitude-phase structure of disturbances, formed by the propagating non-linear periodic waves, on the basin depth, ice thickness and non-linearity of its vertical acceleration, frequency and steepness of initial wave harmonics. The ice Stokes drift velocity was estimated as well as the ice induced non-linear mass transport analyzed.&nbsp

On Simulation of Turbulent Nonlinear Free-Surface Flows

A method for numerical simulation of the unsteady, three-dimensional, viscous Navier–Stokes equations for turbulent nonlinear free-surface flows is presented and applied to simulations of a laminar standing wave and turbulent open-channel flow with a finite-amplitude surface wave. The solution domain is discretized with a boundary-orthogonal curvilinear grid that moves with the free surface, allowing surface deformations to be smoothly resolved down to the numerical grid scale. The nonlinear kinematic and dynamic boundary conditions for boundary-orthogonal curvilinear coordinates are developed and discussed with a novel approach for advancing the free surface in curvilinear space. Dynamic large-eddy-simulation techniques are used to model subgrid scale turbulence effects. The method is shown to correctly produce the shape of a nonlinear free-surface wave and its decay due to viscosity. Application to finite-amplitude waves moving over a turbulent channel flow allows demonstration of the clear differences between a channel flow with and without waves, particularly the instantaneous turbulence structure. An interesting sidelight is the appearance of short-crested cross-channel surface waves caused by natural resonance.

Dissipation of energy by finite-amplitude surface waves

Direct numerical simulations of the Navier–Stokes equations are used to study energy dissipation for deep-water finite-amplitude traveling waves. The effects of the dimensionless surface tension coefficient, the Reynolds number, the density ratio and the kinematic viscosity ratio on energy dissipation and wave profiles are investigated. It is found that the dimensionless surface tension plays an important role for the wave profiles and energy dissipation rates; energy dissipation occurs mostly near the crest for gravity waves, but mostly near the trough for capillary waves. Waves of lower Reynolds numbers have smoother profiles than those at higher Reynolds numbers. When the Reynolds number is low enough, the energy dissipation rate follows closely the viscous linear theory, even if the initial wave amplitude is large.

Nonlinear surface wave propagation in a piezoelectric material

Model equations were derived from first principles for nonlinear surface wave propagation in a piezoelectric material. There is no dependence on ad hoc or empirical parameters. The present work extends an earlier analysis of surface waves in crystals, described by the authors at the previous meeting [J. Acoust. Soc. Am. 99, 2538(A) (1996)]. As in the earlier work, the model equations account for crystals having arbitrary symmetry, and for surface wave propagation in arbitrary directions in planes having arbitrary orientations with respect to the crystalographic axes. Here, elastic, piezoelectric, dielectric, and electrostrictive properties of the material are taken into account. The model is formulated as spectral evolution equations that are integrated numerically to illustrate the distortion of a finite amplitude surface wave that is sinusoidal at the source. The material we considered is LiNbO3, for which measured values of all required second- and third-order elastic constants are available in the literature [e.g., the latter are reported by Cho and Yamanouchi, J. Appl. Phys. 61, 875 (1987)]. Analysis of the nonlinearity matrix permits identification of which physical effects contribute most to the distortion of nonlinear surface waves. [Work supported by ONR, NSF, and Schlumberger Foundation.]

The Linear Stability of Finite Amplitude Surface Waves on a Linear Shearing Flow

We give some instability characteristics of two-dimensional finite amplitude surface waves on a linear shearing flow to three-dimensional infinitesimal rotational disturbances using a numerical method. We have found three kinds of instabilities: two of them are related with four- and five-wave resonances and the rest is intrinsically caused by a shearing flow. The growth rates of instabilities associated with four- and five-wave resonances are larger than those without a shearing flow for the same wave steepness.

The Linear Stability of Finite Amplitude Surface Waves on a Linear Shearing Flow

We give some instability characteristics of two-dimensional finite amplitude surface waves on a linear shearing flow to three-dimensional infinitesimal rotational disturbances using a numerical method. We have found three kinds of instabilities: two of them are related with four- and five-wave resonances and the rest is intrinsically caused by a shearing flow. The growth rates of instabilities associated with four- and five-wave resonances are larger than those without a shearing flow for the same wave steepness.

Nonlinear properties of Rayleigh and Stoneley waves in solids

Bulk wave reflection formulas are extended to the surface-wave case, considering Rayleigh and Stoneley waves in terms of refracted inhomogeneous longitudinal and transversal waves, linked by the surface coupling. In this manner, a physical mechanism of second-harmonic generation process is clearly shown. The analytical expressions and physical pictures of Rayleigh- and Stoneley-wave harmonic generation have been derived in a second-order approximation. The solution shows that the second-harmonic surface waves have a field structure different from the fundamental ones and varying with the propagation distance. Some parameters were defined to characterize the elastic surface-wave second-harmonic generation efficiency. Numerical data for some isotropic materials are present. The steady-state form of finite amplitude surface wave is described and explained by a clear physical model.

Effects of surfactants on wave-induced drainage of foam and emulsion films

It is well established that the plane-parallel models of foam and emulsion films underestimate the velocity of film thinning by up to several orders of magnitude and show an incorrect dependence of thinning velocity on film radius. A new theory of film thinning has been previously formulated for tangentially immobile films [12, 13], and shows that the reason for this discrepancy is the neglect of experimentally observed finite amplitude surface waves. For thin films of relatively large radii (> 1o−2 cm), the pumping of the fluid generated by oscillations of the surface waves, provides the dominant contribution to film thinning velocity. The present hydrodynamic model includes the effects of surfactants (Marangoni-Gibbs-effect, surface viscosity and surface diffusion) and surface waves on thinning velocity. As in the case of a tangentially immobile film, it is concluded that the thinning velocity varies inversely with less than the first power of the film radius, and not with the square of the film radius, as predicted by the plane-parallel models of thin film. Also, the velocity of thinning is found to be up to several orders of magnitudes larger than that evaluated from the plane-parallel models. The influence of waves in enhancing the thinning velocity is found to be most significant for a tangentially immobile film and this effect decreases by a factor of up to 3, with a decrease in surface elasticity and surface viscosity.

Critical thickness and lifetimes of foams and emulsions: Role of surface wave-induced thinning

A theory is developed for predicting simultaneously the velocity of thinning, the critical thickness of rupture, and the lifetimes of the tangentially immobile foam and emulsion films. The theory accounts for the fact that thin free films are not plane-parallel, but are periodically nonhomogeneous in thickness. Based on the equations of motion, it is shown that the pumping of fluid generated by the oscillations of the finite amplitude surface waves enhances the velocity of thinning predicted by Reynolds' theory by up to several orders of magnitudes. In addition, for very thin films (thickness <500 Å), the growth of these “hydrodynamic” surface waves due to van der Waals dispersion forces increases even further the velocity of film thinning. The velocity of thinning obtained from the present theory varies inversely with about 8 10 power of the film radius, which is in agreement with the experimental observations. In contrast, Reynolds' velocity of thinning is inversely proportional to the square of the film radius. The thermal perturbations superimposed on the finite amplitude hydrodynamic surface waves also start to grow for very thin films due to the dispersion interactions. The growth rate of the thermal perturbations being much larger than the growth rate of the “hydrodynamic” surface waves, the former lead to the film rupture. The theory leads to an explicit correlation for the critical thickness as a function of the velocity of thinning, initial amplitude of thermal perturbations, viscosity, surface tension, and Hamaker constant. Finally, the velocities of thinning, critical thicknesses, and the lifetimes calculated from the present theory are all in qualitative and quantitative agreement with the hitherto unexplained experimental results.

Periodic Rayleigh waves of finite amplitude on an isotropic solid

Existence of a periodic progressive wave solution to the nonlinear boundary value problem for Rayleigh surface waves of finite amplitude is demonstrated using an extension of the method of strained coordinates. The solution, obtained as a second-order perturbation of the linearized monochromatic Rayleigh wave solution, contains harmonics of all orders of the fundamental frequency. It is shown that the higher harmonic content of the wave increases with amplitude, but the slope of the waveform remains finite so long as the amplitude is less than a critical value.

Numerical simulation of breaking waves

Recent accidents with smaller fishing vessels in breaking waves have focused the interest of naval architects on the dynamics of extreme waves. In the field of ocean engineering there is also a lack of knowledge about what velocities and accelerations to use when calculating the forces from breaking waves on structural members of offshore structures. A number of theories for steady, finite amplitude surface waves are available. In general they are, however, only valid for symmetric, progressive waves. LonguetHiggins and Cokelet 12 have developed a numerical technique for solving the periodic, 2-dimensional, deep water breaking wave problem. This method is based on potential theory and a conformal mapping of the physical plane inside a closed contour in the mapped plane, and the equations of motion are solved in this plane. Then the wave form in the physical plane is found by an inversion of the mapping. Details about the numerical techniques and the time integration procedures are found in the references. In this paper we present a similar method, but with the exceptions that the problem is solved in the physical plane and finite depth is introduced. Except for this, the problem is stated in the same way. The advantage of working in the physical plane is that certain other effects can be included without much modification of the program. Presently we are introducing a floating cylinder into the problem to try to solve the ship motion problem in 2-dimensional breaking waves and a submerged floating or fixed cylinder may quite easily be introduced.

Finite-amplitude surface waves in electrohydrodynamics

The stability of weakly nonlinear waves on the surface of a fluid layer in the presence of an applied electric field is investigated by using the derivative expansion method. A nonlinear Schrödinger equation for the complex amplitude of quasi-monochromatic traveling wave is derived. The wave train of constant amplitude is unstable against modulation. The equation governing the amplitude modulation of the standing wave is also obtained which yields the nonlinear cut-off wave number.

Shoaling of finite-amplitude surface waves on water of slowly-varying depth

Periodic wave trains propagating over water which varies in depth in the direction of wave propagation are studied by using accurate solutions for wave trains in constant depth of water. The accurate solutions are (i) Cokelet's (1977) extension of Stokes’ approximation and, for the longer waves, (ii) a solution for trains of solitary waves using the solitary-wave solution of Longuet-Higgins &amp; Fenton (1974).A representative selection of results is shown in diagrams. A feature which arises from the use of these accurate solutions is that near the highest wave two solutions are possible for a given incoming wave. Although the solutions cannot describe waves that break, it is shown that as depth is decreased a point is reached beyond which no solution can be found. This is taken to indicate the region in which waves break.The limitations of the theory are discussed and analysed. Comparisons with experimental measurements of Hansen &amp; Svendsen (1979) are included.

Shoaling of finite-amplitude surface waves on water of lowly-varying depth : Stiassnie, M. and D. H. Peregrine, 1980 J. Fluid Mech., 97(4): 783–805

Shoaling of finite-amplitude surface waves on water of lowly-varying depth : Stiassnie, M. and D. H. Peregrine, 1980 J. Fluid Mech., 97(4): 783–805

Standing waves of finite amplitude on surface of ideal liquid of finite depth

The nonlinear problem of plane gravitational standing waves of finite amplitude on the surface of an ideal incompressible liquid of infinite depth has been solved analytically [1], It was also solved in [2] using a new method, in which the dimensionless velocity potential, the profile of the free surface, and the frequency were expressed as power series in the parameter ɛ, equal to the ratio between the the amplitude and the wavelength. The results of these two papers agree. Below, the method of [2] is used to study plane standing surface waves of finite amplitude on the surface of a liquid of finite depth. The frequency, the profile of the free surface, and the velocity potential are expressed as power series in the small parameter ɛ. The solution is obtained in the third approximation. An expression for the amplitude dependence of the frequency is obtained.

Surface waves of finite amplitude in axisymmetric swirling flow

A novel perturbation technique originally proposed by Benjamin is applied to the study of surface waves in axisymmetric, swirling flow of an inviscid fluid. A first approximation to finite amplitude waves of large wavelength is obtained. The results are found to correspond to the classical solitary and conidial wave solutions given by Korteweg and deVries.