Large eddy simulation of boundary layer flow under cnoidal waves

Water waves in coastal areas are generally non- linear, exhibiting asymmetric velocity profiles with different amplitudes of crest and trough. The behaviors of the bound- ary layer under asymmetric waves are of great significance for sediment transport in natural circumstances. While pre- vious studies have mainly focused on linear or symmetric waves, asymmetric wave-induced flows remain unclear, par- ticularly in the flow regime with high Reynolds numbers. Taking cnoidal wave as a typical example of asymmetric waves, we propose to use an infinite immersed plate oscillat- ing cnoidally in its own plane in quiescent water to simulate asymmetric wave boundary layer. A large eddy simulation approach with Smagorinsky subgrid model is adopted to investigate the flow characteristics of the boundary layer. It is verified that the model well reproduces experimental and theoretical results. Then a series of numerical experiments are carried out to study the boundary layer beneath cnoidal waves from laminar to fully developed turbulent regimes at high Reynolds numbers, larger than ever studied before. Results of velocity profile, wall shear stress, friction coeffi- cient, phase lead between velocity and wall shear stress, and the boundary layer thickness are obtained. The dependencies of these boundary layer properties on the asymmetric degree and Reynolds number are discussed in detail.

Numerical Model of Cnoidal Wave Flume

The volume of fluid (VOF) method is used to set up a wave flume with an absorbing wave maker of cnoidal waves. Based on the transfer function between wave surface and paddle velocity obtained by the shallow water wave theory, the velocity boundary condition of an absorbing wave maker is introduced to absorb reflected waves that reach the numerical wave maker. For Hid ranging from 0.1 to 0.59 and T root g/d from 7.9 to 18.3, the parametric studies have been carried out and compared with experiments.

An experimental investigation of the velocity field under cnoidal waves over the asymmetric rippled bed

The evolution of an initially flat sandy bed is studied in a laboratory wave flume under enoidal waves and acoustic Doppler velocimeter （ADV） was utilized in the detailed velocity measurements at different positions. The ripple formation and evolution have been analyzed by CCD images and the asymmetric rippled bed is induced by the nonlinear wave flow. The flow structure and a complete process of vortex formation, evolvement and disappearance were observed on the asymmetric rippled bed under cnoidal waves. With the increasing nonlinearity of waves, which is an important factor in the sand ripple formation, the vortex intensity becomes stronger and shows different characteristics on both sides of the ripple crest. The vorticity and wave velocity reach their maximum values at different phase angles. The vortex value reaches the maximum value at a small phase angle with the increasing Ursell number. The near bed flow patterns are mainly determined by the ripple forms and the averaged longitudinal velocity over a wave period above the ripple trough and crest are positive, which indicates the possibility of significant onshore sediment transport and a corresponding ripple drift. The phase averaged vertical velocity has noticeable positive values near the bottom of the ripple crest and trough. Sediments may be lifted from the ripple surface, picked up in suspension by the local velocity, and deposited over the crest and on the lee of the ripples.

Action of Cnoidal Waves on Vertical Walls

Based on the 2nd order cnoidal wave theory, the characters of shallow water standing waves and their action on vertical walls are studied in this paper. The theoretical expressions of the wave surface elevation in front of and the wave pressure on the vertical wall are obtained. In order to verify the theoretical results, model tests were made in the State Key Laboratory of Coastal and Offshore Engineering at DUT. For the wave surface elevation in front of the wall and the wave forces on the wall at the moment when the wave surface at the wall surface goes down to the bottom of the wave trough, the calculated results coincide quite well with the experimental results. For the wave forces on the wall at the moment when the wave surface at the wall surface goes up to the top of the wave crest, the theoretical expressions are modified by the experimental results. For the convenience of practical use, calculations are made for the wave conditions which usually occur in enginering practice by use of the investigated results obtained in this paper. Empirical formulas are fitted with these calculated results for designers to use.

Oblique propagation of nonlinear ion-acoustic cnoidal waves in magnetized electron–positron–ion plasmas with nonextensive electrons

Theoretical investigation of nonlinear electrostatic ion-acoustic cnoidal waves(IACWs) is presented in magnetized electron–positron–ion plasma with nonextensive electrons and Maxwellian positrons. Using reductive perturbation technique, Korteweg–de Vries equation is derived and its cnoidal wave solution is analyzed. For given plasma parameters, our model supports only positive potential(compressive) IACW structures. The effect of relevant plasma parameters(viz., nonextensive parameter q, positron concentration p, temperature ratio σ,obliqueness l3) on the characteristics of IACWs is discussed in detail.

NONLINEAR CNOIDAL WAVES AND SOLITARY WAVES IN ATMOSPHERE

In this paper, starting with the equations describing the atmospheric motion and by arelatively simple method, we find that, nearby the mechanical equilibrium point, all thefinite amplitude nonlinear inertia waves, internal gravity waves and Rossby waves in thedispersive atmosphere satisfy the KdV (Korteweg-de Vries) equation, its solution being thecnoidal waves and solitary waves. For the finite amplitude Rossby waves, we find the newdispersive relation which is different from the Rossby formula and contains the amplitudeparameter. It is shown that the larger the amplitude and width, the faster are the wavesfor the finite amplitude inertia waves and internal gravity waves, and the slower are thewaves for the Rossby solitary waves, to which perhaps the polar vortex and the blocking orcut-off systems belong. This treatise gives the nonlinear waves a new way and inspires usto study the nonlinear adjustment process and evolution process and the turbulence structure.

THE GENERATION OF CNOIDAL WAVE IN A WAVE FLUME

The method of cnoidal wave generation in a wave flume is studied in this paper.According to the wave equa- tion in shallow water,the wave paddle motion equation for given wave parameters and water depth is derived.The controlling signal for driving wavemaker can be converted from its solution,and high wave height shallow water waves can be generated.The experiment results show that,the parameters of the wave generated are very close to the required ones in a certain distance in front of the paddle of the wavemaker.

非线性发展方程的“Cnoidalwave”

分析了非线性方程的“Ｃｔｉｏｉｄａｌｗａｖｅ”，通过对雅可毕椭圆函数的研究，得到了“Ｃｎｏｉｄａｌｗａｖｅ”的传播速度、振幅及相邻振幅之间距与参量的依赖关系。

Numerical Analysis on Cnoidal Wave Induced Response of Porous Seabed with Definite Thickness

Severe water waves can induce seabed liquefaction and do harm to marine structures. Dynamic response of seabed with definite thickness induced by cnoidal water waves is investigated numerically. Biot's consolidation equations are employed to model the seabed response. Parametric studies are carried out to examine the influence of the air content in the pore water and the soil hydraulic conductivity. It has been shown that the air content and soil hydraulic conductivity can significantly afect the pore pressure in seabed. An increase of air content and/or a decrease of soil hydraulic conductivity can change the pore pressure gradient sharply.

Construction of Soliton-Cnoidal Wave Interaction Solution for the (2+1)-Dimensional Breaking Soliton Equation

In this paper, the truncated Painlev′e analysis and the consistent tanh expansion(CTE) method are developed for the(2+1)-dimensional breaking soliton equation. As a result, the soliton-cnoidal wave interaction solution of the equation is explicitly given, which is difficult to be found by other traditional methods. When the value of the Jacobi elliptic function modulus m = 1, the soliton-cnoidal wave interaction solution reduces back to the two-soliton solution. The method can also be extended to other types of nonlinear evolution equations in mathematical physics.

Flow Separation and Vortex Dynamics in Waves Propagating over A Submerged Quartercircular Breakwater

The interactions of cnoidal waves with a submerged quartercircular breakwater are investigated by a ReynoldsAveraged Navier–Stokes(RANS) flow solver with a Volume of Fluid(VOF) surface capturing scheme(RANSVOF) model. The vertical variation of the instantaneous velocity indicates that flow separation occurs at the boundary layer near the breakwater. The temporal evolution of the velocity and vorticity fields demonstrates vortex generation and shedding around the submerged quartercircular breakwater due to the flow separation. An empirical relationship between the vortex intensity and a few hydrodynamic parameters is proposed based on parametric analysis. In addition, the instantaneous and time-averaged vorticity fields reveal a pair of vortices of opposite signs at the breakwater which are expected to have significant effect on sediment entrainment, suspension, and transportation,therefore, scour on the leeside of the breakwater.

Applications of cnoidal and snoidal wave solutions via optimal system of subalgebras for a generalized extended (2+1)-D quantum Zakharov-Kuznetsov equation with power-law nonlinearity in oceanography and ocean engineering

The nonlinear evolution equations have a wide range of applications,more precisely in physics,biology,chemistry and engineering fields.This domain serves as a point of interest to a large extent in the world’s mathematical community.Thus,this paper purveys an analytical study of a generalized extended(2+1)-dimensional quantum Zakharov-Kuznetsov equation with power-law nonlinearity in oceanography and ocean engineering.The Lie group theory of differential equations is utilized to compute an optimal system of one dimension for the Lie algebra of the model.We further reduce the equation using the subalgebras obtained.Besides,more general solutions of the underlying equation are secured for some special cases of n with the use of extended Jacobi function expansion technique.Consequently,we secure new bounded and unbounded solutions of interest for the equation in various solitonic structures including bright,dark,periodic(cnoidal and snoidal),compact-type as well as singular solitons.The applications of cnoidal and snoidal waves of the model in oceanography and ocean engineering for the first time,are outlined with suitable diagrams.This can be of interest to oceanographers and ocean engineers for future analysis.Furthermore,to visualize the dynamics of the results found,we present the graphic display of each of the solutions.Conclusively,we construct conservation laws of the understudy equation via the application of Noether’s theorem.

An available formula of the sandy beach state induced by plunging waves

Laboratory experiments are performed to explore the response rule of a sandy beach profile under plunging wave on a non-uniform sediment-bed slope. The initial beach slope of combination of 1/10 and 1/20 is exposed to regular waves and cnoidal waves respectively. The free surface elevation, process of wave propagation, wave breaking, uprush and backwash and the change of a cross-shore beach profile are measured and recorded. The beach profile under the regular waves action exhibits two parts： a sandbar profile and a beach berm profile, and only one typical profile transformation under the cnoidal waves action is obtained, which is the beach berm profile. In the laboratory experiments, it is found that the beach states under wave action related to the previous factors. In addition, they are related to the characteristic of breaking waves such as the breaking intensity of the plunging wave. A concept about the characteristic angle of the plunging wave has been put forward through the observation and analysis of the phenomenon of the laboratory experiment. A qualitative analysis about the sediment transport carrying by currents generated from the plunging wave and the state of beach profile under the wave action has been done. The quantitative analysis about the relationship between the characteristic angle and Irribarren number has been done. An available formula of equilibrium states for the sandy beach induced by the plunging wave has been established based on the relationship between Irribarren number and the beach profile. By fitting these experimental results and others＇ experimental results to three lines, the three fitting coefficients can be calculated in their formula respectively. The recommended empirical formulas can divide three states of a beach morphology profile obviously, which include a depositive beach, an erosive beach and an intermediate beach.

Nonlinear effect on inertia component of wave forces on a cylinder

A series of experiments on wave forces on a cylinder have been carried out when inertia component isdominant for a small she cylinder. The influence of nonlinear effect on the inertia component of wave forces on a cylinder is analyzed. The applicable range of nonlinear wave theories, such as Stokes and cnoidal wave theories, in calculating wave forces on a cylinder is discussed. A correction method is suggested for linear wave theory in calculated waveforces on a cylinder under the nonlinear condition.

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.

Wave Attenuation Properties of Double Trapezoidal Submerged Breakwaters on Flat-Bed

This paper investigates the wave attenuation properties of the double trapezoidal submerged breakwaters on the flat-bed by conducting physical experiments subjected to linear and cnoidal incident waves.The method of Goda's two points is used to separate the heights of incident,reflected and transmitted waves based on the experimental data.The possible factors affecting the wave attenuation properties of the double trapezoidal submerged breakwaters(i.e.,the relative submerged water depth,relative breakwater spacing,wave steepness and relative wave height) are investigated with respect to the reflection and transmission coefficients.The results show that there is a range,within which the breakwater spacing has little impact on the reflection coefficient,and the transmission coefficient tends to be a constant.The influence of the wave steepness is reduced while the breakwater spacing is too large or too small.Within the range of the relative wave height tested in this study,the reflection and transmission coefficients increase and decrease with the relative wave height,respectively.The double trapezoidal submerged breakwaters model indicates a good attenuation effect for larger wave steepness,big relative wave height and within the range of the relative breakwater spacing between 12.5 and 14 according to linear and cnoidal waves.The changes of wave energy spectra between the double submerged breakwaters on the flat-bed are investigated by the fast Fourier transform(FFT) method,showing that wave energy dissipation can be reached more effectively when the relative breakwater spacing is 12.5.

含变系数或强迫项的KdV方程的新解

Jacobi椭圆函数展开法被推广并用于求解另一种形式的KdV方程的新的精确解,所求解的这类KdV方程包括一种典型的变系数的KdV方程和具有强迫项(随机项)的KdV方程。用这种方法得到的新的类周期解在极限条件下可以退化为类孤立波解或类冲击波解。

作用于透空圆环墩柱的浅水波绕射波浪力与波浪渗流力

透空结构具有降低海浪作用的功效。该文基于椭圆余弦波一阶分量的浅水波理论和Biot渗流固结理论,分别推导了浅水区固立于可渗透弹性海床上透空圆环墩柱的绕射波浪压力以及由波浪所致海床内渗流压力的数学解,并据此对墩柱所受绕射波浪力与力矩以及渗流浮托力与倾覆力矩进行了实算。结果表明,海况条件和结构参数的变化对波浪所引起的各种载荷存在不同的影响效应,波浪渗流力与波浪直接作用可能具有相同量级。圆环柱外表面的可渗透性对浅水波的直接波浪作用具有明显的减弱效应,而对渗流倾覆力矩也具有一定的减弱效应。此外,圆环墩柱内外柱半径比存在优化取值,以使墩柱达到减载和稳定的双重效果。与Airy波浪理论计算结果相比,浅水波理论的结果能有效反映水波非线性因素的可能影响。

斜入射椭圆余弦波对直立防波堤的波浪渗流作用

基于椭圆余弦波的浅水波理论,结合对Biot波浪渗流固结理论的推广应用,给出了固立于可渗透海床上的直立防波堤在浅水波斜入射条件下的反射波问题的数学解,并据此通过特征函数展开法推导了浅水波引起的海床内渗流压力的解析解式,同时计算了作用于防波堤底部的浅水波波浪渗流浮托力和倾覆力矩且与水平波浪力和力矩进行了比较。计算结果表明,在一定条件下,椭圆余弦波的波浪渗流载荷与波浪的直接载荷可能具有相同量级。浅水波入射角、海水和渗流条件以及防波堤几何条件等因素的相对变化对波浪渗流载荷均存在一定的影响,其中对渗流倾覆力矩的影响更为明显。椭圆余弦波理论所给波浪渗流载荷的估值明显高于浅水条件下Airy波理论的对应估值,反映了水波非线性因素的影响效应。

ADV在波浪边界层流动特性研究中的应用

本文介绍新声学多普勒流速仪(Acoustic-Doppler Velocimeter)技术在波浪流动特性研究中的应用。通过波浪水槽试验,利用ADV测量椭圆余弦波作用下不同底床情况,垂线上各点的瞬时流速。将瞬时流速进行相位平均,计算了各点的紊动强度。试验结果显示:在平底床面,水平流速u沿水深变化曲线体现了椭圆余弦波流速的不对称性;分析紊动强度沿水深分布,两种床面上水平流速u的紊动强度明显大于另外两个方向,沙纹床面上由于受到分离涡的影响,紊动强度沿水深分布和平坦床面上的特性有较大不同。