Ideal shocks in 2‐layer flow Part I: Under a rigid lid

Previous work on the classical problem of shocks in a 2‐layer density‐stratified fluid used either a parameterized momentum exchange or an assumed Bernoulli loss. We propose a new theory based on a set of viscous model equations. We define an idealized shock in two‐layer density stratified flow under a rigid lid as a jump or drop of the interface in which (1) the force balance remains nearly hydrostatic in the shock, (2) there is no exchange of momentum between the two layers except by pressure forces on the sloping interface, and (3) dissipative processes can be treated with a constant viscosity. We proceed in two steps. First, we derive a necessary condition for shock existence based on a requirement for wave steepening. Second, we formulate and solve a set of viscous model equations. Some results are the following: Shocks require strong layer asymmetry; one layer must be much faster and/or shallower than the other layer. The linearized equations describing the shock tails provide boundary conditions and a proof of shock uniqueness. It is possible to derive an analytical solution for weak shocks if the steepening condition is met. The weak shock solutions provide closed form expressions for the Bernoulli loss in each layer. Bernoulli losses are strongly concentrated in the expanding layer as the relative layer depth change is much larger in that layer. Bernoulli losses are independent of layer viscosity. A sudden cessation of shock existence is found for strong shocks when the possible end state migrates into the supercritical regime. Surprisingly, the new ideal shock theory compares well with a 2‐D, time‐dependent shallow water model (SWM) with a flux formulation, but with no viscous formulation. Both the Bernoulli drop and shock cessation condition agree quantitatively.     

High order Hamiltonian water wave models with wave-breaking mechanism

Based on the Hamiltonian formulation of water waves, using Hamiltonian consistent modelling methods, we derive higher order Hamiltonian equations by Taylor expansions of the potential and the vertical velocity around the still water level. The polynomial expansion in wave height is mixed with pseudo-differential operators that preserve the exact dispersion relation. The consistent approximate equations have inherited the Hamiltonian structure and give exact conservation of the approximate energy. In order to deal with breaking waves, we extend the eddy-viscosity model of Kennedy et al. (2000) to be applicable for fully dispersive equations. As breaking trigger mechanism we use a kinematic criterion based on the quotient of horizontal fluid velocity at the crest and the crest speed. The performance is illustrated by comparing simulations with experimental data for an irregular breaking wave with a peak period of 12 s above deep water and for a bathymetry induced periodic wave plunging breaker over a trapezoidal bar. The comparisons show that the higher order models perform quite well; the extension with the breaking wave mechanism improves the simulations significantly.     

Numerical simulation of wave overtopping of coastal structures using the non-linear shallow water equations

A one-dimensional high-resolution finite volume model capable of simulating storm waves propagating in the coastal surf zone and overtopping a sea wall is presented. The model (AMAZON) is based on solving the non-linear shallow water (NLSW) equations. A modern upwind scheme of the Godunov-type using an HLL approximate Riemann solver is described which captures bore waves in both transcritical and supercritical flows. By employing a finite volume formulation, the method can be implemented on an irregular, structured, boundary-fitted computational mesh. The use of the NLSW equations to model wave overtopping is computationally efficient and practically flexible, though the detailed structure of wave breaking is of course ignored. It is shown that wave overtopping at a vertical wall may also be approximately modelled by representing the wall as a steep bed slope. The AMAZON model solutions have been compared with analytical solutions and laboratory data for wave overtopping at sloping and vertical seawalls and good agreement has been found. The model requires more verification tests for irregular waves before its application as a generic design tool.     

A weakly dispersive edge wave model

We derive a general linear, weakly dispersive, Boussinesq-type equation that can be used to study edge waves on beaches with slow cross-shore variation of the depth and the alongshore current. The equation is more accurate than the non-dispersive shallow water equations and simpler than the fully dispersive elliptic mild slope equation (especially for a non-zero alongshore current). The improved performance of the new Boussinesq-type model is demonstrated using analytic solutions for edge waves on a plane beach with zero alongshore current.     

Numerical solution of the equations for spatial nonlinear steady-state traveling waves on the free surfaces of homogeneous and two-layer bodies of water

This study is devoted to the construction of some numerical solutions to the previously proposed model equations for three-dimensional disturbances of a small but finite amplitude in liquids of a constant depth. The forms of progressive steady-state waves (both periodic in two horizontal coordinates and solitary) are demonstrated. In a homogeneous liquid, these forms depend on the magnitudes and directions of the wave vectors, whereas, in bodies of water with a small density jump, they also depend on the ratio of layer depths.     

Practical modified scheme of linear shallow-water equations for distant propagation of tsunamis

A simple but practical numerical model describing a distant propagation of tsunamis is newly proposed by introducing an additional term to the existing modified scheme. The numerical dispersion of the proposed model is manipulated to replace the physical dispersion of the linear Boussinesq equations without any limitation. The new model developed in this study is applied to propagation of a Gaussian hump over a constant water depth and the predicted free surface displacements are compared with available analytical solutions. A very reasonable agreement is observed.     

The bilinear oscillator: The response of an articulated mooring tower driven by irregular seas

A large oil tanker moored to an articulated mooring tower can be modelled as a bilinear oscillator, that is, a linear oscillator with different stiffnesses for positive and negative deflections due to the slackening of mooring lines. The response of the bilinear oscillator driven by irregular seas was studied using the qualitative concepts of dynamical systems. A normalized form of the semi-empirical Pierson-Moskowitz spectrum was used to describe the irregular forcing and features typical of nonlinear differential equations were found. Subharmonic resonances, multiple solutions dependent on the initial conditions as well as possible chaotic solutions were calculated. For irregular seas it was also found that large deflections of the tower may be generated when the peak frequency of the Pierson-Moskowitz spectrum is less than the bilinear frequency of the oscillator.     

Swash overtopping and sediment overwash on a truncated beach

New experimental laboratory data are presented on swash overtopping and sediment overwash on a truncated beach, approximating the conditions at the crest of a beach berm or inter-tidal ridge-runnel. The experiments provide a measure of the uprush sediment transport rate in the swash zone that is unaffected by the difficulties inherent in deploying instrumentation or sediment trapping techniques at laboratory scale. Overtopping flow volumes are compared with an analytical solution for swash flows as well as a simple numerical model, both of which are restricted to individual swash events. The analytical solution underestimates the overtopping volume by an order of magnitude while the model provides good overall agreement with the data and the reason for this difference is discussed. Modelled flow velocities are input to simple sediment transport formulae appropriate to the swash zone in order to predict the overwash sediment transport rates. Calculations performed with traditional expressions for the wave friction factor tend to underestimate the measured transport. Additional sediment transport calculations using standard total load equations are used to derive an optimum constant wave friction factor of f_w = 0.024. This is in good agreement with a broad range of published field and laboratory data. However, the influence of long waves and irregular wave run-up on the overtopping and overwash remains to be assessed. The good agreement between modelled and measured sediment transport rates suggests that the model provides accurate predictions of the uprush sediment transport rates in the swash zone, which has application in predicting the growth and height of beach berms.     

Numerical simulation of wave-induced currents combined with parabolic mild-slope equation in curvilinear coordinates

Researches on breaking-induced currents by waves are summarized firstly in this paper. Then, a combined numerical model in orthogonal curvilinear coordinates is presented to simulate wave-induced current in areas with curved boundary or irregular coastline. The proposed wave-induced current model includes a nearshore current module established through orthogonal curvilinear transformation form of shallow water equations and a wave module based on the curvilinear parabolic approximation wave equation. The wave module actually serves as the driving force to provide the current module with required radiation stresses. The Crank-Nicolson finite difference scheme and the alternating directions implicit method are used to solve the wave and current module, respectively. The established surf zone currents model is validated by two numerical experiments about longshore currents and rip currents in basins with rip channel and breakwater. The numerical results are compared with the measured data and published numerical results.     

Boussinesq–Green–Naghdi rotational water wave theory

Using Boussinesq scaling for water waves while imposing no constraints on rotationality, we derive and test model equations for nonlinear water wave transformation over varying depth. These use polynomial basis functions to create velocity profiles which are inserted into the basic equations of motion keeping terms up to the desired Boussinesq scaling order, and solved in a weighted residual sense. The models show rapid convergence to exact solutions for linear dispersion, shoaling, and orbital velocities; however, properties may be substantially improved for a given order of approximation using asymptotic rearrangements. This improvement is accomplished using the large numbers of degrees of freedom inherent in the definitions of the polynomial basis functions either to match additional terms in a Taylor series, or to minimize errors over a range. Explicit coefficients are given at O(μ²) and O(μ⁴), while more generalized basis functions are given at higher order. Nonlinear performance is somewhat more limited as, for reasons of complexity, we only provide explicitly lower order nonlinear terms. Still, second order harmonics may remain good to kh ≈ 10 for O(μ⁴) equations. Numerical tests for wave transformation over a shoal show good agreement with experiments. Future work will harness the full rotational performance of these systems by incorporating turbulent and viscous stresses into the equations, making them into surf zone models.     

Numerical study of combined overflow and wave overtopping over a smooth impermeable seawall

A numerical wave flume is used to investigate the discharge characteristics of combined overflow and wave overtopping of impermeable seawalls. The numerical procedure computes solutions to the Reynolds-averaged Navier–Stokes equations and includes the generation of an irregular train of waves, the simulation of wave breaking and interaction with a sloping, impermeable wall. The numerical model is first tested against published experimental observations, approximate analytical solutions and empirical design formulae for the cases of pure overflow and pure overtopping. A sequence of numerical experiments simulating combined overflow and overtopping are described. The results are used to determine empirical discharge formulae of the form used in current practice.     

An Efficient Hydrodynamic Model for Surface Waves

In the present study,a semi-implicit finite difference model for non-bydrostatic,free-surface flows is analyzed and discussed.The governing equations are the three-dimensional free-surface Reynolds-averaged Navier-Stokes equations defined on a general,irregular domain of arbitrary scale.At outflow,a combination of a sponge layer technique and a radiation boundary condition is applied to minimize wave reflection.The equations are solved with the fractional step method where the hydrostatic pressure component is determined first,while the non-hydrostatic component of the pressure is computed from the pressure Poisson equation in which the coefficient matrix is positive definite and symmetric.The advectiou and horizontal viscosity terms are discretized by use of a semi-Lagrangian approach.The resulting model is computationally efficient and unrestricted to the CFL condition.The developed model is verified against analytical solutions and experimental data,with excellent agreement.     

The calculation of zig-zag maneuver in regular waves with use of the impulse response functions

The traditional governing equations for sway–yaw maneuvering motion are a set of ordinary differential equations with constant coefficients. But, as is well known, integro–differential equations with impulse response functions are more strict governing equations that can handle the frequency dependence of hydrodynamic forces.In this paper, the two types of equation are compared and used to calculate the 10°–10° zig-zag maneuver in waves. Differences between the solutions are discussed.     

A Unique Solvable Higher Order BEM for Wave Diffraction and Radiation

- For the discretization of higher order elements, this paper presents a modified integral domain method to remove the irregular frequencies inherited in the integral equation of wave diffraction and radiation from a surface-piercing body. The set of over-determined linear equations obtained from the method is modified into a normal set of linear equations by superposing a set of linear equations with zero solutions. Numerical experiments have also been carried out to find the optimum choice of the size of the auxiliary domain and the discretization on it.     

Accurate determination of internal kinematics from numerical wave model results

The internal kinematics for surface waves propagating over a locally constant depth are expressed as convolution integrals. Given the wave kinematics at the still water level (SWL), this provides explicit and exact potential flow expressions for the internal kinematics as convolutions in space with appropriate impulse response functions. These functions are derived in closed form and they are shown to decay exponentially. This effectively reduces the limits of the convolution integral to a horizontal distance of approximately three water depths from the water column of interest. The SWL kinematics must be provided within this region. The source of SWL kinematics may, e.g. be one of the recently developed highly accurate Boussinesq-type formulations. The method is valid for multidirectional, irregular waves of arbitrary nonlinearity at any constant water depth.     

Interaction of water waves with vertical cylinders using null-field integral equations

The scattering of water waves by bottom-mounted vertical circular cylinders is solved by using the null-field integral equations in conjunction with degenerate kernels and Fourier series to avoid calculating the Cauchy and Hadamard principal values. In the implementation, the null-field point can be exactly located on the real boundary owing to the introduction of degenerate kernels for fundamental solutions. An adaptive observer system of polar coordinates is considered to fully employ the properties of degenerate kernels. For the hypersingular equation, vector decomposition for the radial and tangential gradients is carefully considered. This method can be seen as a semi-analytical approach since errors attribute from the truncation of Fourier series. Neither hypersingularity in the Burton and Miller approach nor the CHIEF concept was required to deal with the problem of irregular frequencies. Five advantages of free of calculating principal value, well-posed algebraic system, convergence rate of exponential order, meshfree and elimination of boundary-layer effect, are achieved by using the present approach. Numerical results are given for the forces and free-surface elevation around the circular boundaries. Also, the near-trapped behavior arisen from the physical resonance is detected. A general-purpose program for water wave impinging several circular cylinders with arbitrary number, radii, and positions was developed. Several examples of water wave structure interaction by vertical circular cylinders were demonstrated to see the validity of the present formulation.     

An analytic investigation of oscillations within a harbor of constant slope

Longitudinal and transverse oscillations within a harbor of constant slope are analyzed. Based on the linear shallow water approximation, longitudinal oscillations are described with Bessel equations. Ignoring friction, oscillations are forced using the period of the incident perpendicular wave field by the method of matched asymptotics. The analytic results show that the varying depth shifts the resonant wave numbers to lower values than those for the same geometric harbor with constant depth. Furthermore, we extend the shallow water equations to a linear, weakly dispersive, Boussinesq-type equation by modifying the offshore velocity component, and then use it to investigate possible existing transverse oscillations in the harbor of constant slope. These oscillations are types of standing edge waves. Their character is quite sensitive to the boundary condition at the backwall of the harbor.     

Generating–absorbing sponge layers for phase-resolving wave models

The accurate generation and absorption of water waves in phase-resolving models are critical issues in representing nearshore processes. Here, we present a source function method for combined wave generation and absorption using modified sponge layers. This technique can be easily adapted to a wide variety of systems, and does not require the solution of Green's functions but rather the simpler knowledge of solutions for free waves. These solutions may be linear or nonlinear, regular or irregular, and generated waves can be made arbitrarily accurate through simple selection of sponge layer coefficients. Generating–absorbing sponge layer systems are shown to have a close correspondence to relaxation zones for wave generation if relaxation coefficients are chosen appropriately.     

Numerical investigation of oscillations induced by submerged sliding masses within a harbor of constant slope

Oscillations within a rectangular harbor of constant slope induced by submerged sliding masses are investigated numerically based on Boussinesq-type equations and results are used to reveal the characteristics of the generated oscillations. The numerical result of each transverse eigenfrequency is very close to the theoretical prediction and the spatial structure of each mode of the oscillations may also be well captured by the existing analytical solutions based on shallow water equations. The investigation shows that relatively small-scale sliding masses whose width is small compared with the harbor width may induce obvious transverse oscillations. The predominant transverse components are those with small mode numbers when the solid slides start moving from the backwall. In comparing the oscillations induced by the slides of constant velocity and those accelerated by gravity force with bottom friction, it is observed that the movements accelerated by gravity force may facilitate the development of very low transverse oscillation modes while those with constant velocity may also be in favor of the higher ones. The augmentation of the sliding velocity along the constant slope may shift the amplitudes of the oscillation components to smaller values, which corresponds to the physical understandings of the waves generated by underwater sliding masses or landslides. While the sliding masses may not act on an isolated point of the bottom but follow a certain trajectory along the harbor, the transverse oscillations induced by them are sensitive to their position of departure in both the cross-harbor direction and the offshore direction. Longitudinal oscillations may be induced by relatively large sliding masses of harbor width on a constant slope within the harbor. Although the longitudinal oscillations may not reach a steady state without forcing terms at the entrance of the harbor, some patterns of several low-mode ones occur and wavelet spectra are used to analyze their evolutions and comparisons are made with theoretical predictions. It is revealed that the longitudinal oscillations are also sensitive to the moving velocity and initial location of the sliding masses.     

New Method for Predicting the Motion Responses of A Flexible Joint Multi-Body Floating System to Irregular Waves

A new hybrid method of frequency domain and time domain is developed in this paper to predict the motion responses of a flexibly joint multi-body floating system to irregular waves. The main idea of the method is that the three-dimensional frequency method is used to obtain the hydrodynamic coefficients and the response equations are solved in time domain step by step. All the forces can be obtained at the same time. The motions and nonlinear mooring forces of a box type six-body floating system are predicted. A comparison of the theoretical method-based solutions with experimental results has shown good agreement.