Improved explicit approximation of linear dispersion relationship for gravity waves: A discussion

Recently, an accurate explicit approximation to linear dispersion relationship is proposed based on Eckart's explicit relationship (Beji, 2013). The author has nicely improved Eckart's explicit dispersion relationship by introducing an empirical correction function. The resulting expression is valid for the entire range of relative water depths and accurate to within 0.044%.In this discussion, the proposed expression by the author is simplified and improved to an accuracy of 0.019%. Moreover, a near exact solution with 0.001% accuracy is also given.     

Improved explicit approximation of linear dispersion relationship for gravity waves: Comment on another discussion

Recently, a simple explicit approximation to linear dispersion relationship with an accuracy of 0.044% has been proposed (Beji, 2013). Then, this solution was simplified and improved to an accuracy of 0.019% (Vatankhah and Aghashariatmadari, 2013). Moreover, by considering Beji's approximation as a seed, Newton's method was used (Simarro and Orfila, 2013) to obtain an accurate and explicit two-step solution to linear dispersion relationship with percentage error less than 0.0000082%.Newton's method works very well, if a good seed is given. In this discussion, Beji's expression is simplified and improved as a seed for Newton's method. Using this new expression (initial guess), the solution is improved to an accuracy of 0.00000028% which is 30 times smaller than the solution proposed by Simarro and Orfila (2013).     

水波在软淤泥底床上的衰减

本文研究了二层流体系统中波浪的衰减。上层为理想流体，下层为粘弹性Ｖｏｉｇｔ体。导出了色散关系，计算了波浪衰减系数。对于粘性或弹性很大或很小的情况，导出了各种水深情况下近似的显式的衰减系数表示式。与精确的数值结果比较，近似程度很好。可供工程设计参考、使用。     

非线性弥散效应及其对波浪变形的影响

针对Hedges,Kirby和Dalrymple提出的非线性弥散关系的修正式在浅水区存在的较大偏差的问题，给出了一个在整个水深范围内具有单值性的非线性弥散关系。比较可知，它具有在深水与中等水深逼近二阶Stokes波的弥散关系式，在浅水较Hedges,Kirby和Dalymple的修正表达式与Hedges的关系更加吻合的优点，且形式简练，用近似该非线性弥散关系的显式表达式，结合弱非线性效应的缓坡方程，得到考虑非线性弥散影响的波浪变形模型。数值模拟结果表明，用新的非线性弥散关系得到的模型对复杂地形进行模拟的结果和实测结果吻合很好。     

具有精确色散性的非线性波浪水平二维数学模型

建立具有色散性的水平二维非线性波浪方程,方程的非线性近似到了三阶。方程以波面升高和自由表面速度势表达的微分-积分型数学方程,给出方程的数值求解方法和算例,对方程积分项的处理给出了计算方法。计算结果与Boussinesq方程模型和缓坡方程模型的对应计算结果进行了对比。     

Nonlinear Dispersion Effect on Wave Transformation

—A new nonlinear dispersion relation is given in this paper.which can overcome the limitationof the intermediate minimum value in the dispersion relation proposed by Kirby and Dalrymple(1986).and which has a better approximation to Hedges'empirical relation than the modified relations by Hedges(1987).Kirby and Dalrymple(1987)for shallow waters.The new dispersion relation is simple in form.thusit can be used easily in practice.Meanwhile,a general explicit approximation to the new dispersion rela-tion and other nonlinear dispersion relations is given.By use of the explicit approximation to the newdispersion relation along with the mild slope equation taking into account weakly nonlinear effect.amathematical model is obtained,and it is applied to laboratory data.The results show that the model de-veloped with the new dispersion relation predicts wave transformation over complicated topography quitewell.     

A higher-order non-hydrostatic σ model for simulating non-linear refraction–diffraction of water waves

A higher-order non-hydrostatic σ model is developed to simulate non-linear refraction–diffraction of water waves. To capture non-linear (or steep) waves, a 4th-order spatial discretization is utilized to approximate the large horizontal pressure gradient. A higher-order top-layer pressure treatment is further implemented to resolve wave propagation. The model's characteristics including linear wave dispersion and non-linearity are carefully examined. The accuracy of the present model using only two vertical layers is validated by laboratory data and the available results predicted by the non-linear Schrödinger equation, Boussinesq-type equations, the non-linear mild slope equation, and the Laplace equation. Features of harmonic generation as well as the influences of dispersion and non-linearity on wave energy transfer processes are discussed.     

波浪的非线性显式弥散关系

波浪的非线性弥散关系在应用于求解波浪的变形问题时很不方便，需要与含非线性效应的缓坡方程一起进行迭代运算，往往导致数值计算的计算量太大，计算过于复杂。采用显式形式表达非线性弥散关系，可以克服上述缺点，大为简化波浪变形数值计算的计算量。本文通过将现有的非线性弥散关系进行分析比较，给出了一个更为一般的非线性弥散关系及其显式表达式，经比较可知，该显式弥散关系与相对应非线性弥散关系吻合的很好。本文最后用该显式结合含弱非线性效应的缓坡方程，对复式浅滩地形上的波浪折射绕射进行了计算。结果表明，考虑弱非线性可以得出与实验数据更为相符的结果，而采用显式弥散关系可以有效提高计算效率,在波浪的非线性计算中不失为一种切实有效的方法。     

Alternative forms of the higher-order Boussinesq equations: Derivations and validations

An alternative form of the Boussinesq equations is developed, creating a model which is fully nonlinear up to O(μ⁴) (μ is the ratio of water depth to wavelength) and has dispersion accurate to the Padé [4,4] approximation. No limitation is imposed on the bottom slope; the variable distance between free surface and sea bottom is accounted for by a σ-transformation. Two reduced forms of the model are also presented, which simplify O(μ⁴) terms using the assumption ε = O(μ^2/3) (ε is the ratio of wave height to water depth). These can be seen as extensions of Serre's equations, with dispersions given by the Padé [2,2] and Padé [4,4] approximations. The third-order nonlinear characteristics of these three models are discussed using Fourier analysis, and compared to other high-order formulations of the Boussinesq equations. The models are validated against experimental measurements of wave propagation over a submerged breakwater. Finally, the nonlinear evolution of wave groups along a horizontal flume is simulated and compared to experimental data in order to investigate the effects of the amplitude dispersion and the four-wave resonant interaction.     

An analytic solution to the mild slope equation for waves propagating over an axi-symmetric pit

An analytic solution to the mild slope equation is derived for waves propagating over an axi-symmetric pit located in an otherwise constant depth region. The water depth inside the pit decreases in proportion to an integer power of radial distance from the pit center. The mild slope equation in cylindrical coordinates is transformed into ordinary differential equations by using the method of separation of variables, and the coefficients of the equation in radial direction are transformed into explicit forms by using the direct solution for the wave dispersion equation by Hunt (Hunt, J.N., 1979. Direct solution of wave dispersion equation. J. Waterw., Port, Coast., Ocean Div., Proc. ASCE, 105, 457–459). Finally, the Frobenius series is used to obtain the analytic solution. Due to the feature of the Hunt's solution, the present analytic solution is accurate in shallow and deep waters, while it is less accurate in intermediate depth waters. The validity of the analytic solution is demonstrated by comparison with numerical solutions of the hyperbolic mild slope equations. The analytic solution is also used to examine the effects of the pit geometry and relative depth on wave transformation. Finally, wave attenuation in the region over the pit is discussed.     

加强的适合复杂地形的水波方程及其一维数值模型验证

在他人给出的方程的基础上,通过在其动量方程中引入含4个参数的公式,推导出了加强的适合复杂地形的水波方程,新方程的色散、变浅作用以及非线性均比原来适合复杂地形的方程有了改善:色散关系式与斯托克斯线性波的Padé(4,4)阶展开式一致;变浅作用在相对水深(波数乘水深)不大于6时与解析解符合较好;非线性在相对水深不大于1.05时保持在5%的误差之内.基于该方程,在非交错网格下建立的时间差分格式为混合4阶Adams-Bashforth-Moulton的一维数值模型,并在数值计算中利用了五对角宽带解法.数值模拟了潜堤上波浪传播变形,并将数值计算结果与实验结果进行了对比,验证了该数值模型是合理的.     

A new efficient 3D non-hydrostatic free-surface flow model for simulating water wave motions

A new three-dimensional, non-hydrostatic free surface flow model is presented. For simulating water wave motions over uneven bottoms, the model employs an explicit project method on a Cartesian the staggered gird system to solve the complete three-dimensional Navier–Stokes equations. A bi-conjugated gradient method with a pre-conditioning procedure is used to solve the resulting matrix system. The model is capable of resolving non-hydrostatic pressure by incorporating the integral method of the top-layer pressure treatment, and predicting wave propagation and interaction over irregular bottom by including a partial bottom-cell treatment. Four examples of surface wave propagation are used to demonstrate the capability of the model. Using a small of vertical layers (e.g. 2–3 layers), it is shown that the model could effectively and accurately resolve wave shoaling, non-linearity, dispersion, fission, refraction, and diffraction phenomena.     

An explicit approximation to the wavelength of nonlinear waves

An explicit and concise approximation to the wavelength in which the effect of nonlinearity is involved and presented in terms of wave height, wave period, water depth and gravitational acceleration. The present approximation is in a rational form of which Fenton and Mckee's (1990, Coastal Engng 14, 499–513) approximation is reserved in the numerator and the wave steepness is involved in the denominator. The rational form of this approximation can be converted to an alternative form of a power-series polynomial which indicates that the wavelength increases with wave height and decreases with water depth. If the determined coefficients in the present approximation are fixed, the approximating formula can provide a good agreement with the wavelengths numerically obtained by Rienecker and Fenton's (1981, J. Fluid Mech. 104, 119–137) Fourier series method, but has large deviations when waves of small amplitude are in deep water or all waves are in shallow water. The present approximation with variable coefficients can provide excellent predictions of the wavelengths for both long and short waves even, for high waves.     

Unsteady ship waves in shallow water of varying depth based on Green–Naghdi equation

Based on Green–Naghdi equation this work studies unsteady ship waves in shallow water of varying depth. A moving ship is regarded as a moving pressure disturbance on free surface. The moving pressure is incorporated into the Green–Naghdi equation to formulate forcing of ship waves in shallow water. The frequency dispersion term of the Green–Naghdi equation accounts for the effects of finite water depth on ship waves. A wave equation model and the finite element method (WE/FEM) are adopted to solve the Green–Naghdi equation. The numerical examples of a Series 60 (C_B=0.6) ship moving in shallow water are presented. Three-dimensional ship wave profiles and wave resistance are given when the ship moves in shallow water with a bed bump (or a trench). The numerical results indicate that the wave resistance increases first, then decreases, and finally returns to normal value as the ship passes a bed bump. A comparison between the numerical results predicted by the Green–Naghdi equation and the shallow water equations is made. It is found that the wave resistance predicted by the Green–Naghdi equation is larger than that predicted by the shallow water equations in subcritical flow , and the Green–Naghdi equation and the shallow water equations predict almost the same wave resistance when , the frequency dispersion can be neglected in supercritical flows.     

Internal wave generation in an improved two-dimensional Boussinesq model

A set of Boussinesq-type equations with improved linear frequency dispersion in deeper water is solved numerically using a fourth order accurate predictor-corrector method. The model can be used to simulate the evolution of relatively long, weakly nonlinear waves in water of constant or variable depth provided the bed slope is of the same order of magnitude as the frequency dispersion parameter. By performing a linearized stability analysis, the phase and amplitude portraits of the numerical schemes are quantified, providing important information on practical grid resolutions in time and space. In contrast to previous models of the same kind, the incident wave field is generated inside the fluid domain by considering the scattered wave field in one part of the fluid domain and the total wave field in the other. Consequently, waves leaving the fluid domain are absorbed almost perfectly in the boundary regions by employment of damping terms in the mass and momentum equations. Additionally, the form of the incident regular wave field is computed by a Fourier approximation method which satisfies the governing equations accurately in water of constant depth. Since the Fourier approximation method requires an Eulerian mean current below wave trough level or a net mass transport velocity to be specified, the method can be used to study the interaction of waves and currents in closed as well as open basins. Several computational examples are given. These illustrate the potential of the wave generation method and the capability of the developed model.     

Internal wave generation in an improved two-dimensional Boussinesq model

A set of Boussinesq-type equations with improved linear frequency dispersion in deeper water is solved numerically using a fourth order accurate predictor-corrector method. The model can be used to simulate the evolution of relatively long, weakly nonlinear waves in water of constant or variable depth provided the bed slope is of the same order of magnitude as the frequency dispersion parameter. By performing a linearized stability analysis, the phase and amplitude portraits of the numerical schemes are quantified, providing important information on practical grid resolutions in time and space. In contrast to previous models of the same kind, the incident wave field is generated inside the fluid domain by considering the scattered wave field in one part of the fluid domain and the total wave field in the other. Consequently, waves leaving the fluid domain are absorbed almost perfectly in the boundary regions by employment of damping terms in the mass and momentum equations. Additionally, the form of the incident regular wave field is computed by a Fourier approximation method which satisfies the governing equations accurately in water of constant depth. Since the Fourier approximation method requires an Eulerian mean current below wave trough level or a net mass transport velocity to be specified, the method can be used to study the interaction of waves and currents in closed as well as open basins. Several computational examples are given. These illustrate the potential of the wave generation method and the capability of the developed model.     

考虑非线性弥散影响的波浪变形数学模型

提出了逼近Kirby和Dalrymple的非线性弥散关系的显式非线性弥散关系的表达式,该显式表达式与他们的非线性弥散关系的精度几乎完全相同.采用显式非线性弥散关系,结合含弱非线性效应的缓坡方程,得到考虑非线性弥散影响的波浪变形数学模型,并对该数学模型进行了数值验证.结果表明,考虑非线性弥散影响的波浪变形数学模型更为精确.     

Efficient computation of surf zone waves using the nonlinear shallow water equations with non-hydrostatic pressure

A numerical method for non-hydrostatic, free-surface, irrotational flow governed by the nonlinear shallow water equations including the effects of vertical acceleration is presented at the aim of studying surf zone phenomena. A vertical boundary-fitted grid is used with the water depth divided into a number of layers. A compact finite difference scheme is employed for accurate computation of frequency dispersion requiring a limited vertical resolution and hence, capable of predicting the onset of wave breaking. A novel wet–dry algorithm is applied for a proper handling of moving shoreline. Mass and momentum are strictly conserved at discrete level while the method only dissipates energy in the case of wave breaking. The numerical results are verified with a number of tests and show that the proposed model using two layers without ad-hoc assumptions enables to resolve propagating nonlinear shoaling, breaking waves and wave run-up within the surf and swash zones in an efficient manner.