Nonlinear Wave Force on Pier Group in Shallow Water

Based on the 1st order cnoidal wave theory, the wave diffraction around the pier group inshallow water is studied in this paper. The formulas for calculating the nonlinear wave forces are also presented here. In order to verify the theoretical results, model tests are conducted in the wave flume in The State Key Laboratory of Coastal and Offshore Engineering located in Dalian University of Technology. The range of the wave parameters in the experiments is characteristic wave period T g/d~(1/2) = 8.08- 22.86, characteristic wave height H/ d= 0.1 ~ 0.45. The results obtained from the experiments agree with the theoretical results quite well. It is shown that, in shallow water the nonlinear wave forces acting on a pier group are greater than those calculated by linear wave theory, the value of increment in wave force increases with the increases of the nonlinearity of the wave. In the wave range studied in this paper, the nonlinear wave force can reach over 4 times the force calculatecd by linear wave theory. Thus, it is suggested that, when Tg / d~(1/2)> 8, the wave force on the piers in the pier group in shallow water should be calculated by using the cnoidal wave theory.     

海啸波传播的线性和非线性特征及近海陆架效应影响的数值研究

浅水方程被广泛应用于海啸预警报业务及研究,而针对线性浅水方程与非线性浅水方程在不同海区水深地形条件下的适用范围、计算效率问题是海啸研究人员急需了解的。本文应用基于浅水方程的海啸数值预报模型就海啸波在南海、东海传播的线性、非线性特征以及陆架对其传播之影响进行了数值分析研究。海啸波在深水的传播表征为强线性特征,此时线性系统对海啸波幅的模拟计算具有较高的精度和效率,而弱的非线性特征及弱的色散特征对海啸波幅的预报影响甚微,可以忽略不计。海啸波传播至浅水大陆架后受海底坡度变化、海底粗糙度等因素影响,波动的非线性效应迅速传播、积累,与线性浅水方程计算的海啸波相比表现出较大差异,主要表现为:在南海区,水深小于100m时,海啸波首波以后的系列波动非线性特征比较明显,两者波幅差别较大,但首波波幅的区别不大,因此对于该区域在不考虑海啸爬高的情况下,应用线性系统计算得到的海啸波幅也可满足海啸预警报的要求;在东海区由于陆架影响,海啸波非线性特征明显增强,水深小于100m区域,首波及其后系列波波幅均差异较大,故在该区域必须考虑海啸波非线性作用。本文就底摩擦项对海啸波首波波幅的影响进行了数值对比分析,结果表明:底摩擦作用对海啸波首波波幅影响仅作用于小于100m水深。最后,该文通过敏感性试验,初步分析了陆架宽度及陆架边缘深度对海啸波波幅的影响,得出海啸波经陆架传播共振、变形后,海啸波幅的放大或减小与陆架的宽度及陆架边缘水深有关。     

The Nonlinear Wave Force on a Circular Cylinder in Shallow Water

Based on the 1st order cnoidal wave theory, the nonlinear wave diffraction around a circular cylinder in shallow water is studied in this paper. The equation of the wave surface around the cylinder is formulated and by using this formula the wave surface elevation on the cylinder surface can be obtained. In this paper, the formula for calculating the cnoidal wave force on a circular cylinder is also derived. For the wave conditions which are often encountered in practical engineering designs, the ratios of the nonlinear wave forces to the linear wave forces are calculated, and the results are plotted in this paper for design purposes. In order to verify the theoretical results, model tests are conducted. After comparing the test results with the theoretical ones, it is concluded that, in shallow water, for the case of T g / d~(1/2) > 8-10 and H / d > 0.3, the cnoidal wave theory should be used to calculate the wave action on a cylindrical pier.     

Reflection of long internal waves of small amplitudes from an underwater slope

The dynamics of long waves in the vicinity of a transition point of a two-layer flow into a single-layer one is studied within the linear theory of shallow water. The analogy between this problem and the classical problem of surface wave runup on the shore is shown. Conditions for breaking internal waves on a slope are discussed.     

Wave force on vertically submerged circular thin plate in shallow water

The paper presented is a solution of shallow water wave force, using small amplitude linear wave theory on two-dimensional vertically submerged circular thin plates under three different configurations: (1) a surface-piercing circular thin plate, (2) a submerged circular thin plate, and (3) a bottom-standing circular thin plate. Finally Morison's equation is used for the determination of wave force which is based on the linear wave theory. The plate is submerged in water near the shore on uniformly sloping bottom. The solution method is confined in a finite domain, which contains both the region of different depth of water and the plate. Laplace's equation and boundary value problems are solved in a finite domain, by the method of separation of variables and the small amplitude linear wave theory. The variation of horizontal force by single particle, total horizontal force and moment with respect to the wave amplitude are obtained at different depth of water and at different wave period. It is observed that the force and moment are converging with the increase of wave period and the gradients of force and moment with respect to the wave amplitude are extremely high for lower wave period.     

Modeling of propagation and transformation of transient nonlinear waves on a current

A novel theoretical approach is applied to predict the propagation and transformation of transient nonlinear waves on a current. The problem was solved by applying an eigenfunction expansion method and the derived semi-analytical solution was employed to study the transformation of wave profile and the evolution of wave spectrum arising from the nonlinear interactions of wave components in a wave train which may lead to the formation of very large waves. The results show that the propagation of wave trains is significantly affected by a current. A relatively small current may substantially affect wave train components and the wave train shape. This is observed for both opposing and following current. The results demonstrate that the application of the nonlinear model has a substantial effect on the shape of a wave spectrum. A train of originally linear and very narrow-banded waves changes its one-peak spectrum to a multi-peak one in a fairly short distance from an initial position. The discrepancies between the wave trains predicted by applying the linear and nonlinear models increase with the increasing wavelength and become significant in shallow water even for waves with low steepness. Laboratory experiments were conducted in a wave flume to verify theoretical results. The free-surface elevations recorded by a system of wave gauges are compared with the results provided by the nonlinear model. Additional verification was achieved by applying a Fourier analysis and comparing wave amplitude spectra obtained from theoretical results with experimental data. A reasonable agreement between theoretical results and experimental data is observed for both amplitudes and phases. The model predicts fairly well multi-peak spectra, including wave spectra with significant nonlinear wave components.     

基于二阶斯托克斯波理论的辐射应力垂向分布

基于二阶斯托克斯波理论推导了辐射应力的垂向分布表达式,通过算例讨论了辐射应力在深水和有限水深条件下的垂向分布规律,并与基于微幅波理论的辐射应力进行了比较.结果表明,在波浪非线性不强时,基于二阶斯托克斯波理论的辐射应力与基于微幅波理论的辐射应力表达式计算结果接近;而当水深较浅波浪非线性较强时,基于二阶斯托克斯波理论的辐射应力在近表面处明显大于基于微幅波理论的辐射应力.采用二阶斯托克斯波理论推导的波浪辐射应力更为合理地反映了波浪非线性效应.     

A General Linear Wave Theory for Water Waves Propagating over Uneven Porous Bottoms

Starting from the widespread phenomena of porous bottoms in the near shore region, considering fully the diversity of bottom topography and wave number variation, and including the effect of evanescent modes, a general linear wave theory for water waves propagating over uneven porous bottoms in the near shore region is established by use of Green‘s scond identity. This theory can be reduced to a number of the most typical mild-slope equations curreutly in use and provide a reliable research basis for follow-up development of nonlinear water wave theory involving porous bottoms.     

Wave parameter tuning for the application of the mild-slope equation on steep beaches and in shallow water

The linear mild-slope equation (MSE) is examined in the limit of very shallow water. This is done by means of a series comparison with the more ‘exact’ linear classical theory (E) valid over arbitrary uniform slopes and known to have a “minimum norm” solution basis pair, respectively, regular and logarithmically singular at the shore line. It is shown that the agreement between E and MSE is exact for the first three terms for the regular wave and the first two for the singular wave. It is further demonstrated, by application of this example, that the MSE represents a better approximation than does the classical linearised shallow water equation (SWE) in the case of extremely small depth. In particular, if solutions to each are tuned to the same finite wave height at the shoreline, then MSE predicts the correct curvature of wave height there whereas SWE does not.The work of Booij (Booij, N.A., 1983. A note on the accuracy of the Mild-Slope Equation. Coastal Engineering 7, 191–203.) is supported and varied to allow performance on very steep beds to be tested against exact values rather than those of numerical simulation. Those tests are carried out both as Boundary Value Problems, BVP (Scheme A) and Initial Value Problems, IVP (Scheme B) with matching results on global error. Methods are found of specifying phase and group velocity, which are consistent with linear wave beach theory and lead to improvements in solving the MSE over steep flat beaches. The improvements are found generally superior, in the case considered, to those of some recently developed ‘modified’ and ‘extended’ MSEs. Finally, it is demonstrated, and confirmed by both asymptotic theory and calculation, that the addition of evanescent modes constitutes improvement only in intermediate depths and is not recommended in depths of the order of only a wavelength on a steep (e.g. 45°) beach.     

Statistical characteristics of long waves nearshore

The random long wave runup on a beach of constant slope is studied in the framework of the rigorous solutions of the nonlinear shallow water theory. These solutions are used for calculation of the statistical characteristics of the vertical displacement of the moving shoreline and its horizontal velocity. It is shown that probability characteristics of the runup heights and extreme values of the shoreline velocity coincide in the linear and nonlinear theory. If the incident wave is represented by a narrow-band Gaussian process, the runup height is described by a Rayleigh distribution. The significant runup height can also be found within the linear theory of long wave shoaling and runup. Wave nonlinearity nearshore does not affect the Gaussian probability distribution of the velocity of the moving shoreline. However the vertical displacement of the moving shoreline becomes non-Gaussian due to the wave nonlinearity. Its statistical moments are calculated analytically. It is shown that the mean water level increases (setup), the skewness is always positive and kurtosis is positive for weak amplitude waves and negative for strongly nonlinear waves. The probability of the wave breaking is also calculated and conditions of validity of the analytical theory are discussed. The spectral and statistical characteristics of the moving shoreline are studied in detail. It is shown that the probability of coastal floods grows with an increase in the nonlinearity. Randomness of the wave field nearshore leads to an increase in the wave spectrum width.     

Run-Up of a Nonlinear Monochromatic Wave on a Plane Beach in Presence of a Tide

Oceanology - Abstract—The nonlinear problem of run-up of a long wave on a plane beach in presence of a tide is solved within nonlinear shallow water theory using the Carrier–Greenspan...     

Numerical study on water waves and wave-induced longshore currents in Obaky coastal water

In this paper, the water waves and wave-induced longshore currents in Obaky coastal water which is located at the Mediterranean coast of Turkey were numerically studied. The numerical model is based on the parabolic mild-slope equation for coastal water waves and the nonlinear shallow water equation for the wave-induced currents. The wave transformation under the effects of shoaling, refraction, diffraction and breaking is considered, and the wave provides radiation stresses for driving currents in the model. The numerical results for the water wave-induced longshore currents were validated by the measured data to demonstrate the efficiency of the numerical model. Then the water waves and longshore currents induced by the waves from main directions were numerically simulated and analyzed based on the numerical results. The numerical results show that the movement of the longshore currents was different while the wave propagated to a coastal zone from different directions.     

An Experimental Investigation on Wave Loading on a Platform Mat

Overall wave loading and local hydrodynamic pressure distributions on a platform mat in reg-ular waves for shallow waters are experimentally investigated in order to examine the severity of thenonlinear effects and shallow water effects.Wave focusing phenomenon is observed in the tests.The meas-ured results may also provide a comparison basis for the theoretical development to consider the nonlinearinteraction between waves and viscous flow by introducing viscosity into wave theories.     

数值波浪水槽中完全非线性浅水波仿真特性研究

应用基于势流理论的时域高阶边界元方法,建立一个完全非线性的三维数值波浪水槽,通过实时模拟推板造波运动的方式产生波浪。通过混合欧拉-拉格朗日方法和四阶Runge-Kutta方法更新自由水面和造波板的瞬时位置。利用所建模型分别模拟了有限水深波和浅水波,与试验结果、相关文献结果和浅水理论结果吻合较好,且波浪能够稳定传播。系统地讨论造波板的运动圆频率、振幅和水深等对波浪传播和波浪特性的影响,并对波浪的非线性特性进行分析,研究发现造波板运动频率、运动振幅以及水深均将对波浪形态和波浪非线性产生显著影响。结果为真实水槽造波机的运动控制以及波浪生成试验提供了依据,便于实验室设置更合理的参数来准确模拟不同条件下的波浪。     

Analytical and experimental investigation on the hydrodynamic performance of onshore wave-power devices

The hydrodynamic performance of the oscillating water column type shoreline-mounted wave-power device is numerically studied within linear wave theory by using a boundary element method based on the Wehausen and Laitone 3D shallow water Green's function. In order to verify the numerical model, a 1:12 physical model with different bottom slopes was constructed and tested in a wave basin under regular wave conditions. The effects of the bottom slope on the hydrodynamic performance are investigated by both analytical and experimental methods.     

Generation and propagation of transient nonlinear waves in a wave flume

A semi-analytical nonlinear wavemaker model is derived to predict the generation and propagation of transient nonlinear waves in a wave flume. The solution is very efficient and is achieved by applying eigenfunction expansions and FFT. The model is applied to study the effect of the wavemaker and its motion on the generation and propagation of nonlinear waves. The results indicate that the linear wavemaker theory may be applied to predict only the generation of waves of low steepness for which the nonlinear terms in the kinematic wavemaker boundary condition and free-surface boundary conditions are of secondary importance. For waves of moderate steepness and steep waves these nonlinear terms have substantial effects on wave profile and wave spectrum just after the wavemaker. A wave spectrum corresponding to a sinusoidally moving wavemaker possesses a multi-peak form with substantial nonlinear components, which disturbs or may even exclude physical modeling in wave flumes. The analysis shows that the widely recognized weakly nonlinear wavemaker theory may only be applied to describe the generation and propagation of waves of low steepness. This is subject to further restrictions in shallow and deep waters because the kinematic wavemaker boundary condition as well as the nonlinear interaction of wave components and the evolution of wave energy spectrum is not properly described by weakly nonlinear wavemaker theory. Laboratory experiments were conducted in a wave flume to verify the nonlinear wavemaker model. The comparisons show a reasonable agreement between predicted and measured free-surface elevation and the corresponding amplitudes of Fourier series. A reasonable agreement between theoretical results and experimental data is observed even for fairly steep waves.     

A comparison of weakly and fully non-linear models of the shoaling of a solitary internal wave

This study investigates the behaviour of internal solitary waves crossing a continental slope in the presence of a seasonal thermocline. Comparisons are made between a fully non-linear computational fluid dynamics (CFD) model, and weakly non-linear theory. Previous observations suggested that the amplitudes of solitary waves are capped as they pass across the continental slope, which may be due to laminar dynamics, or due to the effect of turbulence. Across the continental slope, CFD and second order variable depth KdV (vEKdV) predictions agree well with observations of a limited change in solitary wave amplitude. First order variable depth KdV theory overpredicts the final amplitude significantly. In terms of the wave shape, the CFD modeled wave changes from a KdV shape in deep water towards an EkdV solution in shallow water, as observations suggest. The phase speed of the CFD and vEKdV waves are similar to that observed in waters of 400–500 m deep, but are slightly lower than observed in 140 m depth. CFD predictions using a standard k, turbulence model showed that turbulence had little effect on the amplitude. These preliminary results indicate that in this situation wave capping is due to laminar, large amplitude solitary wave dynamics and is independent of turbulent mixing.     

Static wave setup with emphasis on damping effects by vegetation and bottom friction

Wave setup can contribute significantly to elevated water levels during severe storms. In Florida we have found that wave setup can be 30% to 60% of the total 100-year storm surge. In areas with relatively narrow continental shelves, such as many locations along the Pacific Coast of the United States, wave setup can be an even larger proportionate contributor of anomalous water levels during major storms. Wave setup can be considered as comprising two components, with the first being the well-known static wave setup resulting from the transfer of breaking wave momentum to the water column. The second, oscillating component, is a result of nonlinear transfer of energy and momentum from the primary (linear) spectrum to waves with length and time scales on the order of the wave groups.Static wave setup is the focus of this paper with emphasis on effects due to internal or surface forces that act on the wave system and cause both dissipation of wave energy and transfer of momentum. In particular, the effects of wave damping by vegetation and bottom friction are considered. Linear wave theory is applied to illustrate these effects and, for shallow water waves, the setup is reduced by two-thirds the amount that would occur if the same amount of energy dissipation occurred in the absence of forces. Effects of nonlinear waves are then considered and it is found, for a shallow water wave of approximately one-half breaking height, that a wave setdown rather than setup occurs due to damping by vegetation and bottom friction.The problem of wave setup as waves propagate through vegetation was stimulated by studies to establish hazard zones associated with 100-year storm events along the shorelines of the United States. These storms can generate elevated water levels exceeding 4 to 6 m and can result in overland wave propagation. As these waves propagate through vegetation and damp, the question arose as to the contribution of this process to elevated mean water levels through additional wave setup.     

Oblique wave interaction with a floating structure near a wall with stepped bottom

Diffraction of obliquely incident waves by a floating structure near a wall with step-type bottom topography is investigated under the three-dimensional small amplitude wave theory. Full solution of the problem under the potential flow approach is obtained by the matched eigenfunction expansion method. The wave-induced forces on the structure and on the wall, the reflection and transmission characteristics and the wave elevations in the free surface regions are studied for different incident wave angles, water depth ratios and dimension of the structure and the distance of the wall from the center of the structure. The problem is reformulated under shallow water approximations and results are compared with the finite depth results.     

强非线性和色散性Boussinesq方程数值模型检验

采用同位网格有限差分法,建立了强非线性和色散性Boussinesq方程数值计算模型。以稳恒波Fourier近似解给定入射波边界条件,对均匀水深深水和浅水域不同非线性的行进波、缓坡地形上深水至浅水域的浅水变形波、以及缓坡和陡坡地形上的波浪水槽实验进行了数值计算,并将计算结果与解析解、解析数值解以及实验值进行了较为详细的比较,从而检验了模型的色散性、非线性以及不同底坡下非线性波的浅水变形性能。