Dynamic Free-surface Conditions Research Articles

Hamiltonian Variational Formulation of Three-Dimensional, Rotational Free-Surface Flows, with a Moving Seabed, in the Eulerian Description

Hamiltonian variational principles have provided, since the 1960s, the means of developing very successful wave theories for nonlinear free-surface flows, under the assumption of irrotationality. This success, in conjunction with the recognition that almost all flows in the sea are not irrotational, raises the question of extending Hamilton’s principle to rotational free-surface flows. The Euler equations governing the bulk fluid motion have been derived by means of Hamilton’s principle since the late 1950s. Nevertheless, a complete variational formulation of the rotational water-wave problem, including the derivation of the free-surface boundary conditions, seems to be lacking until now. The purpose of the present work is to construct such a missing variational formulation. The appropriate functional is the usual Hamilton’s action, constrained by the conservation of mass and the conservation of fluid parcels’ identity. The differential equations governing the bulk fluid motion are derived as usually, applying standard methods of the calculus of variations. However, the standard methodology does not provide enough structure to obtain the free-surface boundary conditions. To overcome this difficulty, differential-variational forms of the aforementioned constraints are introduced and applied to the boundary variations of the Eulerian fields. Under this transformation, both kinematic and dynamic free-surface conditions are naturally derived, ensuring the Hamiltonian variational formulation of the complete problem. An interesting feature, appearing in the present variational derivation, is a dual possibility concerning the tangential velocity on the boundary; it may be either the same as in irrotational flow (no condition) or zero, corresponding to the small-viscosity limit. The deeper meaning and the significance of these findings seem to deserve further analysis.

Modelling of depth-induced wave breaking in a fully nonlinear free-surface potential flow model

Two methods to treat wave breaking in the framework of the Hamiltonian formulation of free-surface potential flow are presented, tested, and validated. The first is an extension of Kennedy et al. (2000)’s eddy-viscosity approach originally developed for Boussinesq-type wave models. In this approach, an extra term, constructed to conserve the horizontal momentum for waves propagating over a flat bottom, is added in the dynamic free-surface condition. In the second method, a pressure distribution is introduced at the free surface that dissipates wave energy by analogy to a hydraulic jump (Guignard and Grilli, 2001). The modified Hamiltonian systems are implemented using the Hamiltonian Coupled-Mode Theory, in which the velocity potential is represented by a rapidly convergent vertical series expansion. Wave energy dissipation and conservation of horizontal momentum are verified numerically. Comparisons with experimental measurements are presented for the propagation of a breaking dispersive shock wave following a dam break, and then incident regular waves breaking on a mildly sloping beach and over a submerged bar.

A new multi-layer irrotational Boussinesq-type model for highly nonlinear and dispersive surface waves over a mildly sloping seabed

A new multi-layer irrotational Boussinesq-type model is proposed for both linear and nonlinear surface water waves over mildly sloping seabeds. The model is formulated in terms of computational horizontal and vertical velocity components within each layer and satisfies exact kinematic and dynamic free-surface conditions as well as kinematic seabed conditions. Using a Stokes-type expansion, a theoretical analysis of the new multi-layer model is carried out to examine both linear and nonlinear properties, including wave celerity, velocity profiles, shoaling amplitude, second- and third-order transfer functions and amplitude dispersion. The dispersive coefficients in the governing equations are determined by optimizing the linear celerity or linear velocity profiles. For example, the four-layer model shows extremely high accuracy and is applicable up to $kh=667$–800 (where $k$ is the wavenumber and $h$ is a typical water depth) with a 1 % error in wave phase celerity, and up to $kh=352$–423 with a 1 % error in the linear velocity components. The super- and subharmonic transfer functions are extremely accurate up to $kh=300$ (1 % error), the third-order harmonics and amplitude dispersion are accurate up to $kh=477$ (1 % error), and the shoaling property is optimized to cover the range of $0<kh<300$, which presents a 0.06 % tolerance error in shoaling amplitude. The high-accuracy nature of the model increases its suitability for simulating random wave propagation from extremely deep to shallow waters over mildly sloping topographies. The model is implemented numerically on a non-staggered grid via a composite fourth-order Adams–Bashforth–Moulton time integration. The numerical results show good agreement with both the analytical solutions and experimental data.

Numerical model of sloshing in rectangular tank based on Boussinesq-type equations

The highly accurate Boussinesq-type equations in terms of velocity potential are adopted for the simulation of sloshing phenomena in a two-dimensional rectangular tank. The fully nonlinear free surface conditions are used and linearized energy dissipation term is added into the dynamic free surface condition. The total velocity potential is divided into two parts: the particular solution and the rest which is calculated by the Boussinesq-type model. Different filling levels of liquid in the tank are considered for the validation of the numerical model, including shallow water, intermediate water depth and finite water depth cases. The linear theory based on the superposition of eigen-modes corresponding to natural sloshing frequencies is introduced. Compared with results calculated by linear theory and results from literatures, the numerical model based on Boussinesq-type equations presents good performances in predicting the sloshing motions in rectangular tank with small excitation amplitude.

On damping of two-dimensional piston-mode sloshing in a rectangular moonpool under forced heave motions

Based on a similarity between piston-mode sloshing in a two-dimensional rectangular moonpool and a small-amplitude oscillatory flow through a slot in a slatted screen, a pressure discharge in the moonpool opening is theoretically quantified. This makes it possible, by changing the dynamic free-surface condition inside the moonpool, to account for the vortex-induced damping and thereby to modify the inviscid potential flow statement. The present paper is an important addition to Faltinsenet al.(J. Fluid Mech., vol. 575, 2007, pp. 359–397), where a discrepancy between the potential flow theory and model tests was reported. New calculations are supported by the earlier model tests of the authors as well as numerical data obtained by means of viscous solvers.

Subelement Division Scheme in a Numerical Wave Tank that Uses Higher-Order Boundary Element Method

In this note, partially discontinuous elements are employed to solve the singular integrals in the simulation of waves in a numerical wave tank. A higher-order boundary element method (HOBEM) is employed in combination with a time-marching method to solve the system of partial differential equations. By using a predictor–corrector time integration scheme, we solve an integral equation at each time step by uncoupling the kinematic and dynamic free-surface conditions. Different progressive wave types are used at the inflow boundary, a special form of the Sommerfeld/Orlanski type open boundary condition is employed at the outflow boundary, and linear regular and irregular waves have been simulated. The results show that the adopted HOBEM, along with the open boundary condition, is accurate and robust, and furthermore, results have been obtained for a nonlinear Stokes wave.

Large-wave simulation of microscale breaking waves induced by a free-surface drift layer

The unsteady evolution of microscale breaking waves induced by a free-surface drift layer is studied by performing a large-wave simulation (LWS) of the corresponding free-surface flow. The initial flow condition is the superposition of a gravity wave and the surface layer. Their interaction models the propagation and breaking of gravity waves in the ocean under the influence of wind forcing. The governing Euler equations are filtered according to the LWS formulation and solved numerically by a spectral method for the spatial discretization and a fractional time-step scheme for the time advancement. Lengths are rendered dimensionless by the drift-layer thickness and velocities by the gravity-wave celerity. The flow depends on the Froude (gravity), Fr, and Weber (surface tension), We, numbers, through the dynamic free-surface condition, the initial surface drift velocity, q, and the initial wave height, H ∘, which sets the breaker intensity. Several cases are considered for Fr = 3.6, We = ∞, 10 and 5, q = 0.3, and H ∘ = 5.2, 6.5 and 7.8. The results, associated with microscale breakers, focus on the free-surface profile evolution and the subsurface unsteady flow characteristics. Decreasing the Weber number results in the reduction of the breaking crest elevation, the growth of the breaker bulge size and the lowering of the breaker toe elevation, while increasing the initial wave height results in stronger breakers and the increase of the surface tension influence. The free-surface acquires its steepest slope at the spilling face of the breaker, upstream of the toe, at start of breaking; this value is not affected by surface tension. The fluid streamwise velocity acquires its maximum magnitude at the crest of the breaker during breaking and after the occurrence of maximum surface slope. The time interval between the two events is equal to about three quarters of the wave period for the weaker breaker ( H ∘ = 5.2) and decreases to zero for the stronger breaker ( H ∘ = 7.8). The maximum crest fluid velocity becomes equal to the initial wave celerity C for We = ∞, but acquires smaller values for finite Weber numbers, which decrease with decreasing Weber number and decreasing initial wave height, i.e., it becomes equal to 0.92 C for We = 5 and H ∘ = 5.2. The speed of the breaker crest, on the other hand, acquires values that are larger than C and increase with increasing initial wave height but not affected by surface tension. In all cases, during post breaking, the crest fluid velocity is always much smaller than the wave celerity and in agreement with experimental measurements. The wave energy dissipation increases during breaking and becomes maximum at about the same time the crest fluid velocity becomes maximum. Finally, a vortex, which disrupts the vorticity field of the surface layer, is formed in the spilling region of the breaker, and its peak strength increases with decreasing Weber number. The disruption of the surface layer is stronger for the weak breakers and weaker for the strong ones.

Comparison of four numerical models for simulating seepage from salt marsh sediments

A boundary integral equation method (BIEM) model and three differently formulated finite element method (FEM) models were implemented to explore the spatial and temporal patterns in marsh pore water seepage that each generated. The BIEM model is based on the Laplace equation coupled to a dynamic free-surface condition that assumes that, as the water-table changes, the aquifer instantaneously loses or gains an amount of water equal to the change in head times the specific yield. The FEM models all implement a simplified Richards equation that allows gradual desaturation or resaturation and thus flow in both the saturated and unsaturated zones of the aquifer. Two of the FEM models are based on the governing equation for the USGS model SUTRA and thus take into account fluid and aquifer compressibility. One of these was modified to take into account the effect of tidal loading on the total stress, which is assumed to be constant in the derivation of the original version of SUTRA. The third FEM model assumes that neither the fluid or aquifer matrix is compressible so that changes in storage are due solely to changes in saturation. The unmodified SUTRA model generated instantaneous boundary fluxes that were up to two orders of magnitude greater, and spatially more uniform, than those of the other models. The FEM model without compressibility generated spatial and temporal patterns of the boundary fluxes very similar to those produced by the BIEM model. The SUTRA model with the tidal stress modification gave fluxes similar in magnitude to the BIEM and no compressibility models but with distinctly different distributions in space and time. These results indicate that accurate simulation of seepage from marsh soils is highly sensitive to aquifer compressibility and to proper formulation of the effect of tidal loading on the total stress in the aquifer. They also suggest that accurate simulation may require total stress correction not only for tidal loading but for changes in the water table as well. Finally, to aid the development of methods for the measurement of compressibility, we present a schematic, pore-scale model to illustrate the factors that may govern the compressibility of marsh soils.

A Generalized Wagner Method for Three-Dimensional Slamming

Impact between the water and ship, that is, slamming, can cause important global and local effects. A numerical method has been applied to predict water entry loads on three-dimensional bodies. The problem is solved as an initial value problem using the boundary element method. The Green second identity is used to represent the velocity potential as a distribution of Rankine sources and dipoles over the body surface and free surface. The problem is stepped up in time using the information from the boundary conditions. The kinematic free-surface condition is used to determine the intersection between the body surface and free surface at each time step. The exact body boundary condition is used, whereas the dynamic free-surface condition, φ = 0, is approximated on to a horizontal line and not on the exact free-surface profile. The approach presented by Zhao et al (1996) for two-dimensional water entry problems was extended to arbitrary three-dimensional bodies in this presented work. An idealized shape, which consists of cylindrical mid-body and hemispherical ends, was studied. The wetted body surface is calculated with great detail and is considered to be more important than the free-surface elevation away from the body. Drop tests have been carried out to verify and validate the numerical simulation. The effect of the angle between the free surface and the body surface has also been studied. The agreement between theory and experiments is good, and the effect of three-dimensionality is documented. The presented computational method is found to be robust for engineering use and computationally less demanding. The experimental results for vertical force have a strong oscillatory nature, and this has been analyzed using a simplified hydroelastic model. The hydroelastic model gives reasonable representation of the dynamic oscillations found in the vertical force. Reasons for the observed deviations between the numerical and the experimental results are documented. Recommendations for conducting drop tests with minimal dynamic effects are also presented.

An efficient model for three-dimensional surface wave simulations. Part II: Generation and absorption

Water wave generation procedures and efficient numerical beaches are crucial components of a fully non-linear numerical tank for water wave simulations. Linear formulae for pneumatic wave makers are optimized for efficient fully non-linear wave generation in three dimensions. Analytical integration of the (linear) applied free surface pressure provides formulae valid for all times of the simulation. The purely non-linear part of the wave making procedure becomes integrated in the fully non-linear formulation. Novel numerical beaches are introduced, damping the (scaled) tangential velocity at the free surface. More specifically, an additional term is introduced in the Bernoulli equation at the free surface, namely ∇ - 1 · ( γ ∇ ϕ ˜ ) , where γ is a non-zero (smooth) function in regions where damping is required and zero in the wave propagation domain, ∇ ϕ ˜ is the scaled tangential velocity at the free surface, and ∇ −1 the inverse horizontal gradient operator. The new term results in a modified dynamic free surface condition which is integrated in time in the fully non-linear formulation. Extensive numerical tests show that the energy of the outgoing waves is completely absorbed by the new damper. Neither wave reflection nor emission are observed. A steep solitary wave is completely absorbed at the numerical beach. Damping of waves due to advancing pressure distributions are efficient as well. The implementation of the absorber in any existing numerical tank is rather trivial.

Simulating nonlinear waves on the free surface in surf zones with two-dimensional sloping beach

Unsteady nonlinear wave motions on the free surface in shallow water and over slopes of various geometries are numerically simulated using a finite difference method in rectangular grid system. Two-dimensional Navier–Stokes equations and the continuity equation are used for the computations. Irregular leg lengths and stars are employed near the boundaries of body and free surface to satisfy the boundary conditions. Also, the free surface which consists of markers or segments is determined every time step with the satisfaction of kinematic and dynamic free surface conditions. Moreover, marker-density method is also adopted to allow plunging jets impinging on the free surface. Either linear or Stokes wave theory is employed for the generation of waves on the inflow boundary. For the simulation of wave breaking phenomena, the computations are carried out with various wave periods and sea bottom slopes in surf zone. The results are compared with other existing computational and experimental results. Agreement between the experimental data and the computation results is good.

On energy arguments applied to the hydrodynamic impact force

The hydrodynamic impact problem is investigated within the framework of potential-flow theory. The vertical load acting on the rigid body is derived based on either momentum or energy conservation, and using the concept of added mass together with a homogeneous Dirichlet condition for the potential on the free surface as usually done to model an impact problem. It is demonstrated that the use of this simplified dynamic free-surface condition, instead of the fully nonlinear one, has a direct influence on the computation of the loads. In particular, the equivalence of momentum and energy analysis is in general not recovered. The situation is then highlighted by performing an asymptotic analysis of the two-dimensional blunt-body asymmetric impact problem. The asymptotic solution is given explicitly and validated through comparisons with experimental results. The energy distribution is then studied. It is shown that the contradiction between momentum and energy analysis can be removed, provided that the flux of energy through the jets is taken into account in the energy balance. If the simplified free-surface condition is indeed valid in the far-field, nonlinear terms must be retained near the body, in the spray-root domains. To leading order, the energy distribution during the gravity-free inertia stage does not depend on the blunt-body shape. The general analysis based on momentum or energy conservation suggests that this result also applies for arbitrary body shape as soon as a homogeneous Dirichlet condition can be applied as a dynamical free-surface boundary condition. In this case, and for a constant vertical impact velocity, half the work performed by the body would seem to be transferred to the fluid as kinetic energy within the spray.

Fully non-linear free-surface simulations by a 3D viscous numerical wave tank

A finite difference scheme using a modified marker-and-cell (MAC) method is applied to investigate the characteristics of non-linear wave motions and their interactions with a stationary three-dimensional body inside a numerical wave tank (NWT). The Navier–Stokes (NS) equation is solved for two fluid layers, and the boundary values are updated at each time step by a finite difference time marching scheme in the frame of a rectangular co-ordinate system. The viscous stresses and surface tension are neglected in the dynamic free-surface condition, and the fully non-linear kinematic free-surface condition is satisfied by the density function method developed for two fluid layers. The incident waves are generated from the inflow boundary by prescribing a velocity profile resembling flexible flap wavemaker motions, and the outgoing waves are numerically dissipated inside an artificial damping zone located at the end of the tank. The present NS–MAC NWT simulations for a vertical truncated circular cylinder inside a rectangular wave tank are compared with the experimental results of Mercier and Niedzwecki, an independently developed potential-based fully non-linear NWT, and the second-order diffraction computation. Copyright © 1999 John Wiley & Sons, Ltd.

Viscous free surface effect on coastal embankment hydrodynamics

A finite-differnece method was used to calculate the nonlinear hydrodynamic pressures acting on the coastal embankment faces by seismic-wave actions. The nonlinearity of free surface flow, convective acceleration, viscosity and surface tension of fluid are included in the analysis. The kinematic and dynamic free surface boundary conditions are employed for calculating the horizontal fluid velocity, pressure at the free surface and the surface profile of the fluid. The time-dependent water surface is transformed to the horizontal plane, and the flow field is mapped onto a rectangular, making it convenient to model the complex sea bottom geometry and the wavy water surface by the finite-difference method. Fully nonlinear and weakly nonlinear dynamic free surface conditions are used and compared. The effects of surface tension of fluid are also discussed. The nonslip boundary condition is applied on the most part of the interface between fluid and solid face, except the region near the intersection between free surface and wall face. The numerical results are presented for various water depths and ground motion intensities, and their associate viscous effects on coastal embankment hydrodynamics are discussed.

Finite difference method of transinet nonlinear free surface wave problems

A finite difference method is developed for computing the two-dimensional transient potential flow generated by an impulse on the free surface. Both the dynamic and kinematic free surface conditions are considered in nonlinear version. the primary features of the present paper include the use of special coordinates transformations so that the geometry of the flow field is transformed into a time-invariant region, presents an iteration process, by which the velocity potential is computed as the solution of a Poisson equation, the application of fast Fourier transform (FFT) technique results in a tri-diagonal system of equations which can be readily solved by the Thomas algorithm, the computing time is significantly reduced. Thus an efficient technique for handling the transient potential problems is well justified. The feasibility of the present method has been verified by two examples including different initial disturbances respectively.

Numerical Analysis of Two-Dimensional Stern Waves by Finite-Difference Method

A numerical method for two-dimensional stern wave problems is developed by use of a time-dependent stream function. Fully nonlinear dynamic and kinematic free surface conditions are employed. The governing equations and the boundary conditions are expressed in finite-difference forms in a boundary-fitted curvilinear coordinate system. Present method has potentiality to explain viscous effects on the stern waves by including vorticity.Preliminary computational tests are carried out in order to refine the computational scheme. Some computed results of the potential flow past a semi-infinite flat-bottomed body with a transom stern are presented and the viscous effect is discussed by comparing the calculated inviscid results with the measured.

Free-surface flow over a semicircular obstruction

The two-dimensional steady flow of a fluid over a semicircular obstacle on the bottom of a stream is discussed. A linearized theory is presented, along with a numerical method for the solution of the fully nonlinear problem. The nonlinear free-surface profile is obtained after solution of an integrodifferential equation coupled with the dynamic free-surface condition. The wave resistance of the semicircle is calculated from knowledge of the solution at the free surface.