Numerical computations of two-dimensional solitary waves generated by moving disturbances

Two-dimensional solitary waves generated by disturbances moving near the critical speed in shallow water are computed by a time-stepping procedure combined with a desingularized boundary integral method for irrotational flow. The fully non-linear kinematic and dynamic free-surface boundary conditions and the exact rigid body surface condition are employed. Three types of moving disturbances are considered: a pressure on the free surface, a change in bottom topography and a submerged cylinder. The results for the free surface pressure are compared to the results computed using a lower-dimensional model, i.e. the forced Korteweg–de Vries (fKdV) equation. The fully non-linear model predicts the upstream runaway solitons for all three types of disturbances moving near the critical speed. The predictions agree with those by the fKdV equation for a weak pressure disturbance. For a strong disturbance, the fully non-linear model predicts larger solitons than the fKdV equation. The fully non-linear calculations show that a free surface pressure generates significantly larger waves than that for a bottom bump with an identical non-dimensional forcing function in the fKdV equation. These waves can be very steep and break either upstream or downstream of the disturbance.     

Weakly nonlocal envelope solitary waves: numerical calculations for the Klein-Gordon (φ) equation

“Weakly nonlocal” solitary waves differ from ordinary solitary waves by possessing small amplitude, oscillatory “wings” that extend indefinitely from the large amplitude “core”. Such generalized solitary waves have been discovered in capillarygravity water waves, particle physics models, and geophysical Rossby waves. In this work, we present explicit calculations of weakly nonlocal envelope solitary waves. Each is a sine wave modulated by a slowly-varying “envelope” that itself propagates at the group velocity. Our example is the cubically nonlinear Klein-Gordon equation, which is a model in particle physics (φ⁴ theory) and in electrical engineering (with a different sign). Both cases have weakly nonlocal“breather” solitons. Via the Lorentz invariance, each breather generates a one-parameter family of nonlocal envelope solitary waves. The φ⁴ breather was described and calculated in earlier work. This generates envelope solitons which have “wings” that are (mostly) proportional to the second harmonic of the sinusoidal factor. In this article, we calculate breathers and envelope solitary waves for the second, electrical engineering case. Since these, unlike the φ⁴ waves, contain only odd harmonics, the envelope solitary waves are nonlocal only via the third harmonic.     

Uni-directional waves over slowly varying bottom. Part I: Derivation of a KdV-type of equation

The exact equations for surface waves over an uneven bottom can be formulated as a Hamiltonian system, with the total energy of the fluid as Hamiltonian. If the bottom variations are over a length scale L that is longer than the characteristic wavelength ℓ, approximating the kinetic energy for the case of “rather long and rather low” waves gives Boussinesq type of equations. If in the case of an even bottom one restricts further to uni-directional waves, the Korteweg-de Vries (KdV) is obtained. For slowly varying bottom this uni-directionalization will be studied in detail in this part I, in a very direct way which is simpler than other derivations found in the literature. The surface elevation is shown to be described by a forced KdV-type of equation. The modification of the obtained KdV-equation shares the property of the standard KdV-equation that it has a Hamiltonian structure, but now the structure map depends explicitly on the spatial variable through the bottom topography. The forcing is derived explicitly, and the order of the forcing, compared to the first order contributions of dispersion and nonlinearity in KdV, is shown to depend on the ratio between ℓ and L; for very mild bottom variations, the forcing is negligible. For localized topography the effect of this forcing is investigated. In part II the distortion of solitary waves will be studied.     

Over-reflection and instabilities in shear flows of shallow water with bottom friction

In a two-dimensional shear flow of shallow water, the bottom friction relates uniquely the spanwise profile of the depth-averaged velocity to the bottom topography. If the basic flow varies weakly in the spanwise direction, the local analysis of stability at every spanwise position gives the region of the flow parameters for which the classic hydraulic instability due to the bottom friction cannot occur. In this region, the linear analyses of the waves scattering and instability due to the lateral shear can be performed effectively by means of the frictionless linearized equations if both the bottom slope and friction are equally small.The energy of the total perturbed flow can be split into three main parts that correspond to the basic flow, small amplitude wave motion and induced mean flow. The waves can be either amplified or damped near the critical layers, where their streamwise phase velocity equals the velocity of the basic flow. Two physical mechanisms of this amplification exist. The first one is similar to that suggested by Takehiro and Hayashi for a linear frictionless shallow water flow. The incident and transmitted waves carry energy of opposite signs, which results in an increase in the amplitude of the reflected wave compared to that of the incident one. This mechanism of over-reflection operates for any combination of the flow parameters. The other mechanism is similar to Landau damping in plasma flows; it is related to the energy exchange between the waves and fluid particles at the critical layers due to the velocity synchronism. It may lead to either additional amplification or damping of the waves for different flow conditions. In particular, its significance can be reduced by stronger bottom friction. If the basic flow has uniform potential vorticity, Landau damping is negligible, and over-reflection always occurs. If the feed-back is provided by another critical layer, the net over-reflection results in the formation of trapped modes.     

Solitary waves on the surface of a viscoelastic fluid running down an inclined plane

In this paper, we study the existence and the role of solitary waves in the finite amplitude instability of a layer of a second-order fluid flowing down an inclined plane. The layer becomes unstable for disturbances of large wavelength for a critical value of Reynolds number which decreases with increase in the viscoelastic parameter M. The long-term evolution of a disturbance with an initial cosinusoidal profile as a result of this instability reveals the existence of a train of solitary waves propagating on the free surface. A novel result of this study is that the number of solitary waves decreases with in crease in M. When surface tension is large, we use dynamical system theory to describe solitary waves in a moving frame by homoclinic trajectories of an associated ordinary differential equation.     

Internal wave computations using the ghost fluid method on unstructured grids

Two‐layer incompressible flows are analysed using the ghost fluid method on unstructured grids. Discontinuities in dynamic pressure along interfaces are captured in one cell without oscillations. Because of data reconstructions based on gradients, the ghost fluid method can be adopted without additional storages for the ghost nodes at the expense of modification in gradient calculations due to the discontinuity. The code is validated through comparisons with experimental and other numerical results. Good agreements are achieved for internal waves generated by a body moving at transcritical speeds including a case where upstream solitary internal waves propagate. The developed code is applied to analyse internal waves generated by a NACA0012 section moving near interfaces. Variations of the lift acting on the body and configurations of the interfaces are compared for various distances between the wing and the interface. The effects of the interface are compared with the effects of a solid wall. Copyright © 2004 John Wiley & Sons, Ltd.     

Generation and evolution of oblique solitary waves in supercritical flows

It is considered that a thin strut sits in a supercritical shallow water flow sheet over a homogeneous or very mildly varying topography. This stationary 3-D problem can be reduced from a Boussinesq-type equation into a KdV equation with a forcing term due to uneven topography, in which the transverse coordinate Y plays a same role as the time in original KdV equation. As the first example a multi-soliton wave pattern is shown by means of N-soliton solution. The second example deals with the generation of solitary wave-train by a wedge-shaped strut on an even bottom. Whitham's average method is applied to show that the shock wave jump at the wedge vertex develops to a cnoidal wave train and eventually to a solitary wavetrain. The third example is the evolution of a single oblique soliton over a periodically varying topography. The adiabatic perturbation result due to Karpman & Maslov (1978) is applied. Two coupled ordinary differential equations with periodic disturbance are obtained for the soliton amplitude and phase. Numerical solutions of these equations show chaotic patterns of this perturbed soliton.     

Formation of solitary waves on gas-sheared liquid layers

The origin of solitary waves on gas-liquid sheared layers is studied by comparing the behavior of the wave field at sufficiently low liquid Reynolds number, R_L, where solitary waves are observed to form, to measurements at higher R_L where solitary waves do not occur. Observations of the wave field with high-speed video imaging suggest that solitary waves, which appear as a secondary transition of the stratified gas-liquid interface, emanate from existing dominant waves, but that not all dominant waves are transformed. From measurements of interface tracings it is found that for low R_L, waves which have amplitude/substrate depth (a/h) ratios of 0.5–1 occur while for higher R_L, no such waves are observed. A comparison of amplitude/wavelength ratios shows no distinction for different R_L. Consequently, it is conjectured that solitary waves originate from waves with sufficiently large a/h ratios; this change of form being similar to wave breaking. The dimensionless wavenumber is found to be smaller at low R_L, where solitary waves are observed. This suggests that perhaps, larger precursor (to solitary wave) waves are possible because the degree of dispersion, which acts to break waves into separate modes, is lower.     

Solitary waves and conjugate flows in a three-layer fluid

Interfacial symmetric solitary waves propagating horizontally in a three-layer fluid with constant density of each layer are investigated. A fully nonlinear numerical scheme based on integral equations is presented. The method allows for steep and overhanging waves. Equations for three-layer conjugate flows and integral properties like mass, momentum and kinetic energy are derived in parallel. In three-layer fluids the wave amplitude becomes larger than in corresponding two-layer fluids where the thickness of a pycnocline is neglected, while the opposite is true for the propagation velocity. Waves of limiting form are particularly investigated. Extreme overhanging solitary waves of elevation are found in three-layer fluids with large density differences and a thick upper layer. Surprisingly we find that the limiting waves of depression are always broad and flat, satisfying the conjugate flow equations. Mode-two waves, obtained with a periodic version of the numerical method, are accompanied by a train of small mode-one waves. Large amplitude mode-two waves, obtained with the full method, are close to one of the conjugate flow solutions.     

Modeling of long nonlinear waves on the interface in a horizontal two-layer viscous channel flow

The dynamics of two-dimensional waves of small but finite amplitude are theoretically studied for the case of a two-layer system bounded by a horizontal top and bottom. It is shown that for relatively large steady-state flow velocities and at certain fluid depth ratios the vertical velocity profile is nonlinear. An evolutionary equation governing the fluid interface disturbances and allowing for the long-wave contributions of the layer inertia and surface tension, the weak nonlinearity of the waves, and the unsteady friction on all the boundaries of the system is derived. Steady-state solutions of the cnoidal and solitary wave type for the disturbed flow are determined without regard for dissipation losses. It is found that the magnitude and the direction of the flow can alter not only the lengths of the waves but also their polarity.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, 2005, pp. 143–158. Original Russian Text Copyright © 2005 by Arkhipov and Khabakhpashev.     

Numerical analysis on flow dynamics and heat transfer of falling liquid films with interfacial waves

Flow dynamics and heat transfer of falling liquid films with interfacial waves flowing on a vertical plate have been studied with originally proposed numerical simulation method. To discretize basic equations a staggered grid fixed on a physical space is employed. A small amplitude disturbance generated at inflow boundary develops to a solitary wave which consists of a large amplitude roll wave and small amplitude capillary waves. Instantaneous streamwise velocity profiles at the wave crest and trough are very different from a laminar flow. A circulation flow occurs in the roll wave and it affects temperature distributions, especially the strong effect is observed for high Prandtl number liquids. The interfacial wave enhances the heat transfer by two kinds of effects which are a film thinning effect and a convection effect. The dominating effect depends on the Prandtl number. Received on 23 December 1998     

Propagation and Evolution of Wave Fronts in Two-Phase Porous Media

An investigation is made into the propagation and evolution of wave fronts in a porous medium which is intended to contain two phases: the porous solid, referred to as the skeleton, and the fluid within the interconnected pores formed by the skeleton. In particular, the microscopic density of each real material is assumed to be unchangeable, while the macroscopic density of each phase may change, associated with the volume fractions. A two-phase porous medium model is concisely introduced based on the work by de Boer. Propagation conditions and amplitude evolution of the discontinuity waves are presented by use of the idea of surfaces of discontinuity, where the wave front is treated as a surface of discontinuity. It is demonstrated that the saturation condition entails certain restrictions between the amplitudes of the longitudinal waves in the solid and fluid phases. Two propagation velocities are attained upon examining the existence of the discontinuity waves. It is found that a completely coupled longitudinal wave and a pure transverse wave are realizable in the two-phase porous medium. The discontinuity strength of the pore-pressure may be determined by the amplitude of the coupled longitudinal wave. In the case of homogeneous weak discontinuities, explicit evolution equations of the amplitudes for two types of discontinuity waves are derived.     

Existence and uniqueness of a solution to a problem of steady composite waves on the surface of a liquid flowing over a wavy bottom

Composite waves on the surface of the stationary flow of a heavy ideal incompressible liquid are steady forced waves of finite amplitude which do not disappear when the pressure on the free surface becomes constant but rather are transformed into free nonlinear waves [1]. It will be shown that such waves correspond to the case of nonlinear resonance, and mathematically to the bifurcation of the solution of the fundamental integral equation describing these waves. In [2], a study is made of the problem of composite waves in a flow of finite depth generated by a variable pressure with periodic distribution along the surface of the flow. In [3], such waves are considered for a flow with a wavy bottom. In this case, composite waves are defined as steady forced waves of finite amplitude that, when the pressure becomes constant and the bottom is straightened, do not disappear but are transformed into free nonlinear waves over a flat horizontal bottom. However, an existence and uniqueness theorem was not proved for this case. The aim of the present paper is to fill this gap and investigate the conditions under which such waves can arise.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 88–98, July–August, 1980.     

Liquid sloshing in a horizontally forced vessel with bottom topography

This paper presents a numerical study of the free-surface evolution for inviscid, incompressible, irrotational, horizontally forced sloshing in a two-dimensional rectangular vessel with an inhomogeneous bottom topography. The numerical scheme uses a time-dependent conformal mapping to map the physical fluid domain to a rectangle in the computational domain with a time-dependent aspect ratio Q(t), known as the conformal modulus. The advantage of this approach over conventional potential flow solvers is the solution automatically satisfies Laplace's equation for all time, hence only the integration of the two free-surface boundary conditions is required. This makes the scheme computationally fast, and as grid points are required only along the free-surface, high resolution simulations can be performed which allows for simulations for mean fluid depths close to the shallow water water regime. The scheme is robust and can simulate both resonate and non-resonate cases, where in the former, the large amplitude waves are well predicted.Results of nonlinear simulations are presented in the case of non-breaking waves for both an asymmetrical ‘step’ and a symmetric ‘hump’ bottom topography. The natural free-sloshing mode frequencies are compared with the small topography asymptotic results of Faltinsen and Timokha (2009) (Sloshing, Cambridge University Press (Cambridge)), and are found to be lower than this asymptotic prediction for moderate and large topography magnitudes. For forced periodic oscillations it is shown that the hump profile is the most effective topography for minimizing the nonlinear response of the fluid, and hence this topography would reduce the stresses on the vessel walls generated by the fluid. Results also show that varying the width of the step or hump has a less significant effect than varying its magnitude.     

Problem of Breakdown of Solitary Waves and Discontinuities

The evolution of initial data of the solitary-wave type with time is investigated numerically. The solitary wave amplitude decreases due to the generation of short-wave radiation. This solution is interpreted as the solution with a discontinuity qualitatively analogous to the solution of the problem of the breakdown of an arbitrary discontinuity in dissipationless systems. The solitary wave amplitude reduction rate is estimated, first for a generalized Korteweg-de Vries equation and then for plasma waves. Features of the investigation are analyzed for cold and hot-electron plasmas.     

The splitting of solitary waves running over a shallower water

The Korteweg-de Vries type of equation (called KdV-top) for uni-directional waves over a slowly varying bottom that has been derived by Van Groesen and Pudjaprasetya [E. van Groesen, S.R. Pudjaprasetya, Uni-directional waves over slowly varying bottom. Part I. Derivation of a KdV-type of equation, Wave Motion 18 (1993) 345–370.] is used to describe the splitting of solitary waves, running over shallower water, into two (or more) waves. Results of numerical computations with KdV-top are presented; qualitative and quantitative comparisons between the analytical and numerical results show a good agreement.     

Nonlinear Disturbances and Weak Discontinuities in a Supersonic Boundary Layer

The processes of wave disturbance propagation in a supersonic boundary layer with self-induced pressure [1–4] are analyzed. The application of a new mathematical apparatus, namely, the theory of characteristics for systems of differential equations with operator coefficients [5–8], makes it possible to obtain generalized characteristics of the discrete and continuous spectra of the governing system of equations. It is shown that the discontinuities in the derivatives of the solution of the boundary layer equations are concentrated on the generalized characteristics. It is established that in the process of flow evolution the amplitude of the weak discontinuity in the derivatives may increase without bound, which indicates the possibility of breaking of nonlinear waves traveling in the boundary layer.     

Frontal collision of internal solitary waves of first mode

The dynamics and energetics of a frontal collision of internal solitary waves (ISW) of first mode in a fluid with two homogeneous layers separated by a thin interfacial layer are studied numerically within the framework of the Navier–Stokes equations for stratified fluid. It was shown that the head-on collision of internal solitary waves of small and moderate amplitude results in a small phase shift and in the generation of dispersive wave train travelling behind the transmitted solitary wave. The phase shift grows as amplitudes of the interacting waves increase. The maximum run-up amplitude during the wave collision reaches a value larger than the sum of the amplitudes of the incident solitary waves. The excess of the maximum run-up amplitude over the sum of the amplitudes of the colliding waves grows with the increasing amplitude of interacting waves of small and moderate amplitudes whereas it decreases for colliding waves of large amplitude. Unlike the waves of small and moderate amplitudes collision of ISWs of large amplitude was accompanied by shear instability and the formation of Kelvin–Helmholtz (KH) vortices in the interface layer, however, subsequently waves again become stable. The loss of energy due to the KH instability does not exceed 5%–6%. An interaction of large amplitude ISW with even small amplitude ISW can trigger instability of larger wave and development of KH billows in larger wave. When smaller wave amplitude increases the wave interaction was accompanied by KH instability of both waves.     

A fully non‐linear model for three‐dimensional overturning waves over an arbitrary bottom

An accurate three‐dimensional numerical model, applicable to strongly non‐linear waves, is proposed. The model solves fully non‐linear potential flow equations with a free surface using a higher‐order three‐dimensional boundary element method (BEM) and a mixed Eulerian–Lagrangian time updating, based on second‐order explicit Taylor series expansions with adaptive time steps. The model is applicable to non‐linear wave transformations from deep to shallow water over complex bottom topography up to overturning and breaking. Arbitrary waves can be generated in the model, and reflective or absorbing boundary conditions specified on lateral boundaries. In the BEM, boundary geometry and field variables are represented by 16‐node cubic ‘sliding’ quadrilateral elements, providing local inter‐element continuity of the first and second derivatives. Accurate and efficient numerical integrations are developed for these elements. Discretized boundary conditions at intersections (corner/edges) between the free surface or the bottom and lateral boundaries are well‐posed in all cases of mixed boundary conditions. Higher‐order tangential derivatives, required for the time updating, are calculated in a local curvilinear co‐ordinate system, using 25‐node ‘sliding’ fourth‐order quadrilateral elements. Very high accuracy is achieved in the model for mass and energy conservation. No smoothing of the solution is required, but regridding to a higher resolution can be specified at any time over selected areas of the free surface. Applications are presented for the propagation of numerically exact solitary waves. Model properties of accuracy and convergence with a refined spatio‐temporal discretization are assessed by propagating such a wave over constant depth. The shoaling of solitary waves up to overturning is then calculated over a 1:15 plane slope, and results show good agreement with a two‐dimensional solution proposed earlier. Finally, three‐dimensional overturning waves are generated over a 1:15 sloping bottom having a ridge in the middle, thus focusing wave energy. The node regridding method is used to refine the discretization around the overturning wave. Convergence of the solution with grid size is also verified for this case. Copyright © 2001 John Wiley & Sons, Ltd.