Two sets of higher-order Boussinesq-type equations for water waves

Based on the classical Boussinesq model by Peregrine [Peregrine, D.H., 1967. Long waves on a beach. J. Fluid Mech. 27 (4), 815–827], two parameters are introduced to improve dispersion and linear shoaling characteristics. The higher order non-linear terms are added to the modified Boussinesq equations. The non-linearity of the Boussinesq model is analyzed. A parameter related to h/L₀ is used to improve the quadratic transfer function in relatively deep water. Since the dispersion characteristic of the modified Boussinesq equations with two parameters is only equal to the second-order Padé expansion of the linear dispersion relation, further improvement is done by introducing a new velocity vector to replace the depth-averaged one in the modified Boussinesq equations. The dispersion characteristic of the further modified Boussinesq equations is accurate to the fourth-order Padé approximation of the linear dispersion relation. Compared to the modified Boussinesq equations, the accuracy of quadratic transfer functions is improved and the shoaling characteristic of the equations has higher accuracy from shallow water to deep water.     

On nondimensional correlation between roughness length and wind-friction velocity

Physical bases for nondimensional parameters,z ₀/(u ² _*/g) andz ₀/(u _*/), characterizing wind-wave interaction are discussed; data selected to support the latter are critically reviewed. Both parameters are herewith unified, with the former describing the primary growth of roughness length with wind and the latter the secondary effects due to waves.     

Nonlinear tidal waves in a kind of estuary with gradually varying cross-section

-Nonlinear tidal waves in a kind of estuary are studied in the paper using one-dimensional nonlinear hydrody-namic equations with friction. The estuary has exponentially varying width B=B0 e-bx and uniform depth h. The one-dimensional hydrodynamic equations are solved by perturbation method. It was found that our solution included two special cases, Pelisenpeki's solution and Airy's solution. The former can be got by letting b=0 in our solutions, and the latter by setting 6 = 0 and f= 0 (f is linear frictional coefficient). In terms of the second-order solution, the physical mechanism of nonlinear tidal waves in estuaries with gradually varying cross-section is explored. It is shown that, under the assumption of linear friction coefficient, shallow water constituent waves consist of two parts, one is produced by shallow water nonlinear effect outside the estuary, the other is generated by shallow water nonlinear effect inside estuary. In addition, the physical mechanism of the residual tidal current and     

A two-mode numerical model with applications to coastal upwelling

A continuously stratified, linear two mode numerical model has been developed. The model incorporates a free surface and finite amplitude topography.The vertical dependence in the equations is removed by applying a Galerkin procedure which uses the normal modes as test functions. The vertical structure is therefore determined by the normal modes.In order to find a suitable efficient numerical scheme to solve the equations a fairly general phase and stability analysis is carried out for the one dimensional gravity wave equations. The A.D.I. scheme was found to be the most suitable scheme.The model is applied to coastal upwelling. A number of two dimensional (x, z) experiments have been carried out. The advantage of the two mode model above the two layer models is that considerable detail of the vertical structure is readily obtained and that no difficulties with the intersection of interfaces with the topography or the seasurface are present. A three dimensional (x, y, z) test run was done for a region along the south western coast of Africa. The results of this experiment are discussed.     

A 3-D time-domain coupled model for nonlinear waves acting on a box-shaped ship fixed in a harbor

A 3-D time-domain numerical coupled model is developed to obtain an efficient method for nonlinear waves acting on a box-shaped ship fixed in a harbor. The domain is divided into the inner domain and the outer domain. The inner domain is the area beneath the ship and the flow is described by the simplified Euler equations. The remaining area is the outer domain and the flow is defined by the higher-order Boussinesq equations in order to consider the nonlinearity of the wave motions. Along the interface boundaries between the inner domain and the outer domain, the volume flux is assumed to be continuous and the wave pressures are equal. Relevant physical experiment is conducted to validate the present model and it is shown that the numerical results agree with the experimental data. Compared the coupled model with the flow in the inner domain governed by the Laplace equation, the present coupled model is more efficient and its solution procedure is simpler, which is particularly useful for the study on the effect of the nonlinear waves acting on a fixed box-shaped ship in a large harbor.     

Numerical simulation of three-dimensional tidal current in the Bobal Sea

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Wave evolution over submerged sills: tests of a high-order Boussinesq model

A Boussinesq model accurate to O(μ)⁴, μ=k₀h₀ in dispersion and retaining all nonlinear effects is derived for the case of variable water depth. A numerical implementation of the model in one horizontal direction is described. An algorithm for wave generation using a grid-interior source function is derived. The model is tested in its complete form, in a weakly nonlinear form corresponding to the approximation δ=O(μ²), δ=a/h₀, and in a fully nonlinear form accurate to O(μ²) in dispersion [Wei, G., Kirby, J.T., Grilli, S.T., Subramanya R. (1995). A fully nonlinear Boussinesq model for surface waves: Part 1. Highly nonlinear unsteady waves. J. Fluid Mech., 294, 71–92]. Test cases are taken from the experiments described by Dingemans [Dingemans, M.W. (1994). Comparison of computations with Boussinesq-like models and laboratory measurements. Report H-1684.12, Delft Hydraulics, 32 pp.] and Ohyama et al. [Ohyama, T., Kiota, W., Tada, A. (1994). Applicability of numerical models to nonlinear dispersive waves. Coastal Engineering, 24, 297–313.] and consider the shoaling and disintegration of monochromatic wave trains propagating over an elevated bar feature in an otherwise constant depth tank. Results clearly demonstrate the importance of the retention of fully-nonlinear effects in correct prediction of the evolved wave fields.     

Investigation on Effects of Mooring Line Fractures and Connector Failures for A Hybrid Modular Floating Structure System

The present work reports a Hybrid Modular Floating Structure (HMFS) system with typical malfunction conditions. The effects of both fractured mooring lines and failed connectors on main hydrodynamic responses (mooring line tensions, module motions, connector loads and wave power production) of the HMFS system under typical sea conditions are comparatively investigated. The results indicate that the mooring tension distribution, certain module motions (surge, sway and yaw) and connector loads (M_z) are significantly influenced by mooring line fractures. The adjacent mooring line of the fractured line on the upstream side suffers the largest tension among the remaining mooring lines, and the case with two fractured mooring lines in the same group on the upstream side is the most dangerous among all cases of two-line failures in view of mooring line tensions, module motions and connector loads. Therefore, one emergency strategy with appropriate relaxation of a proper mooring line has been proposed and proved effective to reduce the risk of more progressive mooring line fractures. In addition, connector failures substantially affect certain module motions (heave and pitch), certain connector loads (F_z and M_y) and wave power production. The present work can be helpful and instructive for studies on malfunction conditions of modular floating structure (MFS) systems.

     

A new kind of nonlinear mild-slope equation for combined refraction-diffraction of multifrequency waves

This paper presents the numerical solution of a new nonlinear mild-slope equation governing waves with different frequency components propagating in a region of varying water depth. There are two new nonlinear equations. The linear part of the equations is the mild-slope equation, and one of the models has the same non-linearity as the Boussinesq equations. The new equations are directly applicable to the problems of nonlinear wave-wave interactions over variable depth. The equations are first simplified with the parabolic approximation, and then solved numerically with a finite difference method. The Crank-Nicolson method is used to discretize the models. The numerical models are applied to a set of published experimental cases, which are nonlinear combined refraction-diffraction with generation of higher harmonic waves. Comparison of the results shows that the present models generally predict the measurements better than other nonlinear numerical models which have been applied to the data set.     

Sensitive study of the long and short surface wave-induced vertical mixing in a wave-circulation coupled model

The previous studies by the MASNUM research team have shown the effectiveness of the wave-induced mixing (Bv) in improving the simulation of upper-ocean thermal structure. The mechanisms of Bv are further investigated by incorporating different Bv products into the MASNUM wave-circulation coupled model. First, experiments were designed to explore the effects of Bv, which contain the contributions at different wave lengths (l). The results of three experiments, the non-Bv case, the short-wave case (l <300 m), and the long-wave case (l >300 m) are compared, and it is found that the long waves are the most important component for Bv to generate mixing in the upper ocean. As the swell plays dominant role in mixing, the parameterization of Bv into wind may be not a proper way. Second, Bv effects at different time-scales, including daily and monthly, were examined. The results show that the monthly averaged Bv has larger impact than the daily averaged Bv, especially in summer.     

Computations of self-propulsion free to sink and trim and of motions in head waves of the KRISO Container Ship (KCS) model

Two computations of the KCS model with motions are presented. Self-propulsion in model scale free to sink and trim are studied with the rotating discretized propeller from the Hamburg Model Basin (HSVA) at Fr = 0.26. This case is particularly complex to simulate due to the close proximity of the propeller to the rudder. The second case involves pitch and heave in regular head waves. Computations were performed with CFDShip-Iowa version 4.5, a RANS/DES CFD code designed for ship hydrodynamics. The self-propulsion computations were carried out following the procedure described in Carrica et al. [1], in which a speed controller is used to find the propeller rotational speed that results in the specified ship velocity. The rate of revolutions n, sinkage, trim, thrust and torque coefficients K_T, K_Q and resistance coefficient CT_(SP) are thus obtained. Comparisons between CFD and EFD show that the rate of revolutions n, thrust and torque coefficients K_T and K_Q have higher prediction accuracies than sinkage and trim. For the simulation of pitch and heave in head waves, the geometry includes KCS hull and rudder under three conditions with two Froude numbers and three wave length and amplitude combinations. 0th and 1st harmonic amplitudes and 1st harmonic phase are computed for total resistance coefficient C_T, heave motion z and pitch angle θ. Comparisons between CFD and EFD show that pitch and heave are much better predicted than the resistance. In both cases comparisons with simulations by other authors presented at the G2010 CFD Workshop [2] using different CFD methodologies are included.     

非线性波浪时域计算的三维耦合模型

将计算区域Ω划分为内域Ω₁和外域Ω₂(Ω₂=Ω-Ω₁),外域控制方程采用改进线性频散特性的二维Boussinesq方程,用预报一校正法数值求解;结构物附近的内域控制方程为三维Navier-Stokes方程,由VOF方法数值求解。通过在外域和内域相匹配的交界面上设置合适的速度和波面边界条件,建立了三维非线性波浪时域计算的耦合模型。模拟试验表明:(1)耦合模型数值波浪水池可以产生稳定的、重复性较好的波动过程;(2)用耦合模型数值波浪水池求解较大浅水区域上的非线性波浪数值计算问题可以取得较高的计算效率,同时又能得出结构物附近的复杂流场。     

Surf zone dynamics simulated by a Boussinesq type model. Part I. Model description and cross-shore motion of regular waves

This is the first of three papers on the modelling of various types of surf zone phenomena. In this first paper, part I, the model is presented and its basic features are studied for the case of regular waves. The model is based on two-dimensional equations of the Boussinesq type and it features improved linear dispersion characteristics, possibility of wave breaking, and a moving boundary at the shoreline. The moving shoreline is treated numerically by replacing the solid beach by a permeable beach characterized by an extremely small porosity. Run-up of nonbreaking waves is verified against the analytical solution for nonlinear shallow water waves. The inclusion of wave breaking is based on the surface roller concept for spilling breakers using a geometrical determination of the instantaneous roller thickness at each point and modelling the effect of wave breaking by an additional convective momentum term. This is a function of the local wave celerity, which is determined interactively. The model is applied to cross-shore motions of regular waves including various types of breaking on plane sloping beaches and over submerged bars. Model results comprise time series of surface elevations and the spatial variation of phase-averaged quantities such as the wave height, the crest and trough elevations, the mean water level, and the depth-averaged undertow. Comparisons with physical experiments are presented. The phaseaveraged balance of the individual terms in the momentum and energy equation is determined by time-integration and quantities such as the cross-sectional roller area, the radiation stress, the energy flux and the energy dissipation are studied and discussed with reference to conventional phase-averaged wave models. The companion papers present cross-shore motions of breaking irregular waves, swash oscillations and surf beats (part II) and nearshore circulations induced by breaking of unidirectional and multidirectional waves (part III).     

On the wave dependence of sea-surface wind stress

Analysis is made of wind and wave data, which were obtained during the passage of Typhoon 8013 at an Ocean Data Buoy Station south of Honshu operated by the Japan Meteorological Agency, in order to investigate the wave dependence of sea-surface roughness parameter in the situation where wind waves are dominant with less significant swells. The data fit better the wave-dependent expression of the wind stress,z _{0 p}/u*=, than to Charnock's formula,gz ₀/u*²=, wherez ₀ is the roughness length, _p the angular frequency of the spectral peak of wind waves,u* the friction velocity of air,g the acceleration of gravity, and are non-dimensional constants. The results are very similar to those of our previous study using data from an oil producing platform in the Bass Strait, Australia, although the type of observation system and the synoptic situation of the winds and wind waves were totally different.     

Trapped bottom topographic waves over the inclined bottom

In the Boussinesq approximation, we study baroclinic topographic waves trapped by the flat meridional slope. The existence of these waves is explained by stratification, inclined bottom, and Earth's rotation. We deduce the evolutionary equation for the square of the envelope of a narrow-band wave packet of trapped waves. In the second order of smallness relative to the wave amplitude, we find the mean fields of velocity and density induced by the packet. It is shown that, in the limiting case of weakly nonlinear plane waves, the induced current is zonal. In the Northern hemisphere, depending on the slope of the bottom γ₁, the sign of the phase velocity σ/k (k is the zonal wave number) is either always positive (for γ₁>γ_1cr) or always negative (for γ₁<γ_1cr). If we neglect the vertical component of the Coriolis acceleration, then γ_1cr=0. Translated by Peter V. Malyshev and Dmitry V. Malyshev     

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.     

Internal generation of waves for extended Boussinesq equations

It is studied whether the mass transport or energy transport is the proper viewpoint for internally generating waves in the extended Boussinesq equations of Nwogu [J. Waterw., Port, Coastal Ocean Eng. 119 (1993) 618–638]. Numerical solutions of the Boussinesq equations with the internal generation of sinusoidal waves show that the energy transport approach yields the required wave amplitude properly while the mass transport approach yields wave amplitude different from the required one by the ratio of phase velocity to energy velocity. The waves which pass through the wave generation point do not cause any numerical distortion while the incident waves are generated. The technique of internal generation of waves shows its capability of generating nonlinear cnoidal waves as well as linear sinusoidal waves.     

浅水中浮体在波浪作用下运动的三维时域计算模型

港口中系泊船在波浪作用下运动问题的本质是浅水波浪与浮体的相互作用。与深水情况不同,浅水问题应当考虑水底、水域边界的影响及浅水波浪自身的特性,单一模型很难实现该模拟过程。为此,建立了Boussinesq方程计算入射波和Laplace方程计算散射波的全时域组合计算模型。有限元法求解的Boussinesq方程能使入射波充分考虑到水底、水域边界的影响和浅水波浪的特性;散射波被线性化,采用边界元法求解,并以浮体运动时的物面条件为入射波和散射波求解的匹配条件。该方法为完全的时域方法,计算网格不随时间变动,计算过程较为方便。通过与实验及其他数值方法的结果进行比较,验证了本模型对非线性波面、浮体的运动都有比较理想的计算结果,显示了本模型对非线性问题具有较好的计算能力。     

Velocity predictions for shoaling and breaking waves with a Boussinesq-type model

In this paper, the performance of a 1-D Boussinesq model is evaluated against laboratory data for its ability to predict surf zone velocity moments. Wave evolution over a plane beach and a complex bathymetry both extending into the surf-zone is examined for six cases. For the plane beach, these comprise two cases, a spilling and a plunging cnoidal wave. For the complex bathymetry, these comprise four cases of longer and short wavelengths (spilling and plunging breakers), with regular and irregular periodicity. The model evaluation places emphasis on parameters of the wave field that could be used for the prediction of sediment transport; orbital velocity, undertow, velocity skewness, kurtosis and asymmetry. It is found that, despite an overestimation of the depth-averaged horizontal velocity in the regular waves cases, the predicted higher order velocity moments and undertow are in good agreement with the laboratory data. A bispectral analysis demonstrates that the nonlinear transfers of energy amongst the low order harmonics are well reproduced, but energy exchanges with the higher harmonics are less well predicted. As a result, the model handles velocity moments better in the shorter wave tests than in the long wave cases where triad interactions are stronger. Of the four parameters describing wave breaking, the model behaviour is most sensitive to the critical wave front slope φ_B, especially with regard to velocity skewness and kurtosis predictions. It is also found that increasing the thickness of the surface roller for the case of plunging breakers improves the model's performance.