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1.
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. 相似文献
2.
A note on linear dispersion and shoaling properties in extended Boussinesq equations 总被引：1，自引：0，他引：1
A set of optimum parameter α is obtained to evaluate the linear dispersion and shoaling properties in the extended Boussinesq equations of [Madsen and Sorensen, 1992 and Nwogu, 1993], and [Chen and Liu, 1995]. Optimum α values are determined to produce minimal errors in each wave property of phase velocity, group velocity, or shoaling coefficient relative to the analytical one given by the Stokes wave theory. Comparisons are made of the percent errors in phase velocity, group velocity, and shoaling coefficient produced by the Boussinesq equations with a different set of optimum α values. The case with a fixed value of α = −0.4 is also presented in the comparison. The comparisons reveal that the optimum α value tuned for a particular wave property gives in general poor results for other properties. Considering all the properties simultaneously, the fixed value of α = −0.4 may give overall accuracies in phase velocity and shoaling coefficient for all the types of Boussinesq equations selected in this study. 相似文献
3.
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. 相似文献
4.
5.
A new form of higher order Boussinesq equations 总被引：3，自引：0，他引：3
On the basis of the higher order Boussinesq equations derived by the author (1999), a new form of higher order Boussinesq equations is developed through replacing the depth-averaged velocity vector by a new velocity vector in the equations in order to increase the accuracy of the linear dispersion, shoaling property and nonlinear characteristics of the equations. The dispersion of the new equations is accurate to a [4/4] Pade expansion in kh. Compared to the previous higher order Boussinesq equations, the accuracy of quadratic transfer functions is improved and the shoaling property of the equations have higher accuracy from shallow water to deep water. 相似文献
6.
One-dimensional numerical models of higher-order Boussinesq equations with high dispersion accuracy 总被引：3，自引：0，他引：3
Abstract-Nonlinear water wave propagation passing a submerged shelf is studied experimentally andnumerically. The applicability of the wave propagation model of higher-order Boussinesq equations de-rived by Zou(2000, Ocean Engneering, 27, 557～575) is investigated. Physical experiments areconducted; three different front slopes (1:10, 1:5 and 1:2) of the shelf are set-up in the experimentand their effects on the wave propagation are investigated. Comparisons of the numerical results withtest data are made and the higher-order Boussinesq equations agree well with the measurements since thedispersion of the model is of high accuracy. The numerical results show that the good results can also beobtained for the steep-slope cases although the mild-slope assumption is employed in the derivation of thehigher-order terms in the higher-order Boussinesq equations. 相似文献
7.
A formal derivation and numerical modelling of the improved Boussinesq equations for varying depth 总被引：1，自引：0，他引：1
A formal derivation of the improved Boussinesq equations of Madsen and Sørens (1992) is presented to provide the correct forms of the depth-gradient related terms. Linear shoaling characteristics of the new equations are investigated by the method of Madsen and Sørensen (1992) and by the energy flux concept separately and found to agree perfectly, whereas these approaches give conflicting results for the equations derived by Madsen and Sørensen (1992). Furthermore, Nwogu's (1993) modified Boussinesq model is found to produce a linear shoaling-gradient identical with the present work. Numerical modelling of the derived equations for directional waves is carried out by three-time-level finite-difference approximations. A higher-order radiation condition is implemented for effective absorption of the outgoing waves. Several test cases are included to demonstrate the performance of the model. 相似文献
8.
通过改进二阶全非线性 Boussinesq 波浪方程中的色散项,得到了一组没有改变原方程的数学形式但适用于更大变化水深的新方程,其色散性能和变浅性能都比原方程有了很大改进,所适用的水深范围更大,能更好地描述从深水到近岸浅水处的波浪传播;并基于新方程建立了波浪数值模型,通过模拟波浪从浅水到深水的传播变形来验证新方程的有效性. 相似文献
9.
In this study the evolution of internal solitary waves shoaling onto a shelf is considered. The results of high resolution two-dimensional numerical simulations of the incompressible Euler equations are compared with the predictions of several weakly-nonlinear shoaling models of the Korteweg–de Vries family including the Gardner equation and the cubic regularized long wave (or Benjamin–Bona–Mahoney) equation. Wave models in both physical x–t space and in s–x space are considered where s is a commonly used characteristic time variable. The effects of rotation, background currents and damping are ignored. The Boussinesq and rigid lid approximations are also used. The shoaling internal solitary waves generally fission into several waves. Reflected waves are negligible in the cases considered here. Several hyperbolic tangent stratifications are considered with and without a critical point. Among the equations in x–t space the cubic regularized long wave equation gives the best predictions. The Gardner equation in s–x space gives the best predictions of the shape of the leading waves on the shelf, but for many stratifications it predicts a propagation speed that is too large. 相似文献
10.
建立基于四阶完全非线性Boussinesq水波方程的二维波浪传播数值模型。采用Kennedy等提出的涡粘方法模拟波浪破碎。在矩形网格上对控制方程进行离散,采用高精度的数值格式对离散方程进行数值求解。对规则波在具有三维特征地形上的传播过程进行了数值模拟,通过数值模拟结果与实验结果的对比,对所建立的波浪传播模型进行了验证。同时,为了考察非线性对波浪传播的影响,给出和上述模型具有同阶色散性、变浅作用性能但仅具有二阶完全非线性特征的波浪模型的数值结果。通过对比两个模型的数值结果以及实验数据,讨论非线性在波浪传播过程中的作用。研究结果表明,所建立的Boussinesq水波方程在深水范围内不但具有较精确的色散性和变浅作用性能,而且具有四阶完全非线性特征,适合模拟波浪在近岸水域的非线性运动。 相似文献
11.
《Coastal Engineering》2006,53(4):311-318
The extended mild-slope equations of Suh et al. [Suh, K.D., Lee, C., Park, W.S., 1997. Time-dependent equations for wave propagation on rapidly varying topography. Coastal Eng., 32, 91–117] and Lee et al. [Lee, C., Kim, G., Suh, K.D., 2003. Extended mild-slope equation for random waves. Coastal Eng., 48, 277–287] are compared analytically and numerically to determine their applicability to random wave transformation. The geometric optics approach is used to compare the two models analytically. In the model of Suh et al., the wave number of the component wave with a local angular frequency ω is approximated with an accuracy of O(ω − ω¯) at a constant water depth, where ω¯ is the carrier frequency of random waves. In the model of Suh et al., however, the diffraction effects and higher-order bottom effects are considered only for monochromatic waves, and the shoaling coefficient of random waves is not accurately approximated. This inaccuracy arises because the model of Suh et al. was derived for regular waves. In the model of Lee et al., all the parameters of random waves such as wave number, shoaling coefficient, diffraction effects, and higher-order bottom effects are approximated with an accuracy of O(ω − ω¯). This approximation is because the model of Lee et al. was developed using the Taylor series expansion technique for random waves. The result of dispersion relation analysis suggests the use of the peak and weighted-average frequencies as a carrier frequency for Suh et al. and Lee et al. models, respectively. All the analytical results are verified by numerical experiments of shoaling of random waves over a slightly inclined bed and diffraction of random waves through a breakwater gap on a flat bottom. 相似文献
12.
Nonlinear water wave propagation passing a submerged shelf is studied experimentally and numerically. The applicability of two different wave propagation models has been investigated. One is higher-order Boussinesq equations derived by Zou (1999) and the other is the classic Boussinesq equations. Physical experiments are conducted, three different front slopes (1:10, 1:5 and 1:2) of the shelf are set up in the experiment and their effects on wave propagation are investigated. Comparisons of numerical results with test data are made, the model of higher-order Boussinesq equations agrees much better with the measurements than the model of the classical Boussinesq equations. The results show that the higher-order Boussinesq equations can also be applied to the steeper slope case although the mild slope assumption is employed in the derivation of the higher order terms of higher order Boussinesq equations. 相似文献
13.
This work focuses on linear shoaling performance of low order Boussinesq-type equations. It is shown that the linear shoaling errors can be important in well known equations in the literature. New sets of coefficients are presented for three well known sets of equations. The sets are found so as to minimize a global linear error that includes celerity and shoaling errors. Finally, a new set of enhanced bilayer low order equations is presented, with much improved linear behavior (errors in wave celerity and wave amplitude below 1% up to kh = 20). For completeness, the equations are written in their fully nonlinear version, and the nonlinear coefficients are also given. 相似文献
14.
From the phase-resolving improved Boussinesq equations (Beji and Nadaoka, Ocean Engineering 23 (1996) 691), a phase-averaged Boussinesq model for water waves is derived by more effectively describing carrier wave groups and accompanying long wave evolution with less CPU time. Linear shoaling characteristics of carrier wave equations are investigated and found to agree exactly with the analytical expression obtained from the constancy of energy flux for the improved Boussinesq equations themselves, showing that the present model equations are the results of a consistent derivation procedure regarding energy considerations. Numerical simulations of the derived equations for the single wave group and narrow-banded random waves show the validity of the present model and its high performance, especially on the CPU time. 相似文献
15.
Alternative forms of the higher-order Boussinesq equations: Derivations and validations 总被引：2，自引：0，他引：2
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(μ2/3) (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. 相似文献
16.
Higher order Boussinesq equations 总被引：2，自引：0，他引：2
Z. L. Zou 《Ocean Engineering》1999,26(8):239
A new form of Boussinesq-type equations accurate to the third order are derived in this paper to improve the linear dispersion and nonlinearity characteristics in deeper water. Fourth spatial derivatives in the third order terms of the equations are transformed into second derivatives and present no difficulty in numerical computations. With the increase in accuracy of the equations, the nonlinear and dispersion characteristics of the equations are of one order of magnitude higher accuracy than those of the classical Boussinesq equations. The equations can serve as a fully nonlinear model for shallow water waves. The shoaling property of the equations is also of high accuracy through shallow water to deep water by introducing an extra source term into the second order continuity equation. An approach to increase the accuracy of the nonlinear characteristics of the new equations is introduced. The expression for the vertical distribution of the horizontal velocities is a fourth order polynomial. 相似文献
17.
Hemming A. Schäffer 《Coastal Engineering》2009,56(5-6):517-533
The transformation of irrotational surface gravity waves in an inviscid fluid can be studied by time stepping the kinematic and dynamic surface boundary conditions. This requires a closure providing the normal surface particle velocity in terms of the surface velocity potential or its tangential derivative. A convolution integral giving this closure as an explicit expression is derived for linear 1D waves over a mildly sloping bottom. The model has exact linear dispersion and shoaling properties. A discrete numerical model is developed for a spatially staggered uniform grid. The model involves a spatial derivative which is discretized by an arbitrary-order finite-difference scheme. Error control is attained by solving the discrete dispersion relation a priori and model results make a perfect match to this prediction. A procedure is developed by which the computational effort is minimized for a specific physical problem while adapting the numerical parameters under the constraint of a predefined tolerance of damping and dispersion error. Two computational examples show that accurate irregular-wave transformation on the kilometre scale can be computed in seconds. Thus, the method makes up a highly efficient basis for a forthcoming extension that includes nonlinearity at arbitrary order. The relation to Boussinesq equations, mild-slope wave equations, boundary integral equations and spectral methods is briefly discussed. 相似文献
18.
Essential properties of Boussinesq equations for internal and surface waves in a two-fluid system 总被引：1，自引：0，他引：1
Boussinesq equations describing motions of internal waves in a two-fluid system with the presence of free surface are theoretically derived, and the associated essential properties are examined in this study. Eliminating the dependence on the vertical coordinate from all variables, four equations constitute the Boussinesq model with two flexible parameters, zu and zl, which indicate the specific elevations, respectively, in the upper and lower fluids. Similar to the Boussinesq model for a single-layer fluid, zu and zl are determined by matching the linear dispersion relation with Lamb's solution. This determines the optimal model. In the analysis stage, this problem is classified into two cases, the thicker-upper-layer case and the thicker-lower-case case, to avoid the possible divergence of wave properties as the thickness ratio grows. Since there exist two modes of motions that may be excited, cases of both modes are separately analyzed. Linear characteristics including the amplitude ratios and normalized particle velocities are analyzed. Second-order harmonic waves are examined to validate nonlinear behaviors of present model. Results of linear and nonlinear investigations show that the present model indeed extends the applicable range of traditional Boussinesq equations. 相似文献
19.
A fully nonlinear Boussinessq-type model with several free coefficients is considered as a departure point. The model is monolayer and low order so as to simplify numerical solvability. The coefficients of the model are here considered functions of the local water depth. In doing so, we allow to improve the dispersive and shoaling properties for narrow banded wave trains in very deep waters. In particular, for monochromatic waves the dispersion and shoaling errors are bounded by ~ 2.8% up to kh = 100, being k the wave number and h the water depth. The proposed model is fully nonlinear in weakly dispersive conditions, so that nonlinear wave decomposition in shallower waters is well reproduced. The model equations are numerically solved using a fourth order scheme and tested against analytical solutions and experimental data. 相似文献
20.
For simulating water wave propagation in coastal areas, various Boussinesq-type equations with improved properties in intermediate or deep water have been presented in the past several decades. How to choose proper Boussinesq-type equations has been a practical problem for engineers. In this paper, approaches of improving the characteristics of the equations, i.e. linear dispersion, shoaling gradient and nonlinearity, are reviewed and the advantages and disadvantages of several different Boussinesq-type equations are compared for the applications of these Boussinesq-type equations in coastal engineering with relatively large sea areas. Then for improving the properties of Boussinesq-type equations, a new set of fully nonlinear Boussinseq-type equations with modified representative velocity are derived, which can be used for better linear dispersion and nonlinearity. Based on the method of minimizing the overall error in different ranges of applications, sets of parameters are determined with optimized linear dispersion, linear shoaling and nonlinearity, respectively. Finally, a test example is given for validating the results of this study. Both results show that the equations with optimized parameters display better characteristics than the ones obtained by matching with padé approximation. 相似文献