Numerical solution of a mathematical model for water wavesin large coastal areas

Based on the evolution equation for water waves, a mathematical model for wave propaga tion in large mild - slope areas is derived. The model is solved by the finite difference method with the staggered grid system. The computational results are in good agreement with experimental data and show that the model can obtain better results with relatively coarser grids. The model can be used to simulate water wave propagation in large coastal areas and can be efficiently solved without much programming effort.

Applications of sediment sudden deposition model based on the third-generation numerical model for shallow water wave

The existing numerical models for nearshore waves are briefly introduced, and the third-generation numerical model for shallow water wave, which makes use of the most advanced productions of wave research and has been adapted well to be used in the environment of seacoast, lake and estuary area, is particularly discussed. The applied model realizes the significant wave height distribution at different wind directions. To integrate the model into the coastal area sediment, sudden deposition mechanism, the distribution of average silt content and the change of sediment sudden deposition thickness over time in the nearshore area are simulated. The academic productions can give some theoretical guidance to the applications of sediment sudden deposition mechanism for stormy waves in the coastal area. And the advancing directions of sediment sudden deposition model are prospected.

A THIRD GENERATION SHALLOW WATER WAVE NUMERICAL MODEL-YE-WAM

This paper pnesents a third gneration shallow Whter disode spedtal wave nbotal medeIYE-WAM based on the spedtal action balance equation. The mode accounts for all edevan effectsof currents on waves, incuding tmpotally and spatialy varying depth and current inded refraction,sttalning and fequency shift and also explidtly takeS into aanunt all source terms, speclally adePth-limited breaking dheipation. In addition, an energy forcing scheme is propond and applied to themode’s open boundaries to areUn for the propagution of sedIs into the study spstem The upwinddiffeIenng scheme and a standard hybrid diffdrencing scheme for the propagaion terrn and a simpleEuler method for the source teme are employed.

Numerical study on water waves and wave-induced longshore currents in Obaky coastal water

In this paper, the water waves and wave-induced longshore currents in Obak6y 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 proDaRated to a coastal zone from different directions.

Dimension-Reduced Model for Deep-Water Waves

Starting from the 2D Euler equations for an incompressible potential flow, a dimension-reduced model describing deep-water surface waves is derived. Similar to the Shallow-Water case, the z-dependence of the dependent variables is found explicitly from the Laplace equation and a set of two one- dimensional equations in x for the surface velocity and the surface elevation remains. The model is nonlocal and can be formulated in conservative form, describing waves over an infinitely deep layer. Finally, numerical solutions are presented for several initial conditions. The side-band instability of Stokes waves and stable envelope solitons are obtained in agreement with other work. The conservation of the total energy is checked.

Analysis and Dynamics of Fractional Order Mathematical Model of COVID-19in Nigeria Using Atangana-Baleanu Operator

We propose a mathematical model of the coronavirus disease 2019(COVID-19)to investigate the transmission and control mechanism of the disease in the community of Nigeria.Using stability theory of differential equations,the qualitative behavior of model is studied.The pandemic indicator represented by basic reproductive number R0 is obtained from the largest eigenvalue of the next-generation matrix.Local as well as global asymptotic stability conditions for the disease-free and pandemic equilibrium are obtained which determines the conditions to stabilize the exponential spread of the disease.Further,we examined this model by using Atangana–Baleanu fractional derivative operator and existence criteria of solution for the operator is established.We consider the data of reported infection cases from April 1,2020,till April 30,2020,and parameterized the model.We have used one of the reliable and efficient method known as iterative Laplace transform to obtain numerical simulations.The impacts of various biological parameters on transmission dynamics of COVID-19 is examined.These results are based on different values of the fractional parameter and serve as a control parameter to identify the significant strategies for the control of the disease.In the end,the obtained results are demonstrated graphically to justify our theoretical findings.

A Mathematical Model for Redshift

We have used model scaling so that the propagation of light through space could be studied using the well-known nonlinear Schrödinger equation. We have developed a set of numerical procedures to obtain a stable propagating wave so that it could be used to find out how wavelength could increase with distance travelled. We have found that broadening of wavelength, expressed as redshift, is proportional to distance, a fact that is in agreement with many physical observations by astronomers. There are other reasons for redshifts that could be additional to the transmission redshift, resulting in the deviation from the linear relationship as often observed. Our model shows that redshift needs not be the result of an expanding space that is a long standing view held by many astrophysicists. Any theory about the universe, if bases on an expanding space as physical fact, is open to question.

NUMERICAL SIMULATION OF A MATHEMATICAL MODEL FOR WAVE PROPAGATION ON NON-UNIFORM CURRENT AND DEPTH

A water wave evolution equation is developed from the combinedrefraction-diffraction equation on non-uniform current in water of slowly varying topography byusing the perturbation method. A numerical model is presented with the governing equationdiscretized with an improved Alternating Direction Impicit (ADI) method involving a relaxationfactor which can improve convergent rate. The calculation results show that the model caneffectively reflect the effects of current on wave propagation.

Wave Numerical Model for Shallow Water

The history of forecasting wind waves by wave energy conservation equation Is briefly described. Several currently used wave numerical models for shallow water based on different wave theories are discussed. Wave energy conservation models for the simulation of shallow water waves are introduced, with emphasis placed on the SWAN model, which takes use of the most advanced wave research achievements and has been applied to several theoretical and field conditions. The characteristics and applicability of the model, the finite difference numerical scheme of the action balance equation and its source terms computing methods are described in detail. The model has been verified with the propagation refraction numerical experiments for waves propagating in following and opposing currents; finally, the model is applied to the Haian Gulf area to simulate the wave height and wave period field there, and the results are compared with observed data.

An Analytic-Numerical Solution of Wave Transformation

An analytic-numerical solution of wave transformation in shoaling water is presented in this paper. The analytical expression for wave heights along the wave rays is derived in consideration of the combined effect of water depth shoaling, the wave refraction and the sea bottom friction. The wave rays (orthogonals) are calculated by a fourth order Runge-Kutta algorithm and the wave crest lines are computed by an iteration procedure. The numerical results are compared with analytical solution for a special case of parallel- straight contour shore and field data, and comparisons show that the proposed mathematical model and computation method are very useful and convenient for engineering application.

Numerical Simulation of Nonlinear Surface Gravity Waves Transformation under Shallow-Water Conditions

This work considers the problems of numerical simulation of non-linear surface gravity waves transformation under shallow bay conditions. The discrete model is built from non-linear shallow-water equations. Are resulted boundary and initial conditions. The method of splitting into physical processes receives system from three equations. Then we define the approximation order and investigate stability conditions of the discrete model. The sweep method was used to calculate the system of equations. This work presents surface gravity wave profiles for different propagation phases.

WATER WAVE SIMULATION IN CURVILINEAR COORDINATES USING A TIME-DEPENDENT MILD SLOPE EQUATION

The purpose of this article is to model the detailed progress of wave propagation in curvilinear coordinates with an effective time-dependent mild slope equation. This was achieved in the following approach, firstly deriving the numerical model of the equation, i.e., Copeland＇s hyperbolic mild-slope equation, in orthogonal curvilinear coordinates based on principal of coordinate transformation, and then finding the numerical solution of the transformed model by use of the Alternative Directions Implicit （ADI） method with a space-staggered grid. To test the curvilinear model, two cases of a channel with varying cross section and a semi-circular channel were studied with corresponding analytical solutions. The model was further investigated through a numerical simulation in Ponce de Leon Inlet, USA. Good agreement is reached and therefore, the use of the present model is valid to calculate the progress of wave propagation in areas with curved shorelines, nearshore breakwaters and other complicated geometries.

Numerical Simulation of Wave Height and Wave Set-Up in Nearshore Regions

Based on the time dependent mild slope equation including the effect of wave energy dissipation, an expression for the energy dissipation factor is derived in conjunction with the wave energy balance equation, and then a practical method for the simulation of wave height and wave set- up in nearshore regions is presented. The variation of the complex wave amplitude is numerically simulated by use of the parabolic mild slope equation including the effect of wave energy dissipation due to wave breaking. The components of wave radiation stress are calculated subsequently by new expressions for them according to the obtained complex wave amplitude, and then the depth-averaged equation is applied to the calculation of wave set-up due to wave breaking. Numerical results are in good agreement with experimental data, showing that the expression for the energy dissipation factor is reasonable and that the new method is effective for the simulation of wave set-up due to wave breaking in nearshore regions.

Modeling the Infection Disease (Covid-19) and the Effect of Vaccination

In this paper we provide different types of approach in mathematical biology about infection disease and understanding the dynamic of epidemic mathematical models specially in COVID-19 disease which first outbroke in China and fast spread around the world. We work in the connection between the mathematical models and the solution analytically and numerically. At first, we emphasize the Susceptible-Infectious-Recovered (SIR) models’ extension for policy significance. Then, we found the improved SIER model done by research. In third section, we examine the improved model when an appropriate vaccine has been found, we introduce the model of SIR with vaccine term which ends up with discussion and conclusion about the effect of vaccinate. The comprehension of COVID-19 transmission methods, structures, and characteristics is greatly aided by these mathematical models analytically and numerically.

施工期跨海桥梁矩形设置基础波流力研究

为给跨海桥梁矩形设置基础施工提供参考,研究跨海桥梁矩形设置基础定位下沉过程中墩位水深及结构吃水深度对其波流力的影响。基于垂向多层σ坐标变换模型和大涡模拟LES方法建立三维波浪-流与结构物相互作用的数值仿真模型,分析不同水深及结构吃水深度时,单独波浪作用和波流共同作用下矩形设置基础的波流力及其统计值和流场分布特征,并提出施工期矩形设置基础下沉过程波流力简化算法。结果表明:矩形设置基础在定位下沉过程中,垂直来流传播方向的结构阻水面积随之增大,流场的复杂绕射作用使得波流力峰值呈非线性增大趋势;由于浅水波在整个水深范围影响都比较大,矩形设置基础波流力对结构吃水深度的变化相对深水条件更为敏感;结构波流力峰值不对称系数随吃水深度增加逐渐减小,且波流力表现为正向峰值大于负向峰值的强烈不对称性特点;随着结构吃水深度增加,单独波浪作用、波流共同作用时,在流场复杂绕射作用下,矩形设置基础迎浪侧的波面爬高增大,涡脱现象愈加明显;提出的跨海桥梁矩形设置基础下沉着床过程波流力简化算法可用于估算设置基础施工期波流力。

STUDY ON THE INTERACTION BETWEEN WAVES AND MUD BOTTOM

An investigation is made on the wave motion in stratified fluids, the upper layer being water and the lower layer being fluid mud. Water is treated as viscous fluid and fluid mud as plastic fluid. The orbital velocities, dispersion relation, interfacial amplitude and wave height attenuation are given and checked by experiment. Theoretical and experimental results are in good agreement.

Calculation of Wave Radiation Stress in Combination with Parabolic Mild Slope Equation

A new method for the calculation of wave radiation stress is proposed by linking the expressions for wave radiation stress with the variables in the parabolic mild slope equation. The governing equations are solved numerically by the finite difference method. Numerical results show that the new method is accurate enough, can be efficiently solved with little programming effort, and can be applied to the calculation of wave radiation stress for large coastal areas.

一种半潜式商用6 MW级风力机载荷数值模拟研究

针对海上漂浮式风力机在其运行状态下的叶轮气动载荷与平台运动状态难以预测的问题,本文以一种商用6 MW级漂浮式风力机为研究对象,基于非稳态的雷诺平均湍流模型,进行了全耦合数值模拟和水池模型试验研究。结果显示:浮式风力机自由衰减周期的模拟值在纵摇、横揺和垂荡上与实验值吻合度较高,而在纵荡、横荡和艏摇上,衰减周期的模拟值与实验值存在一定误差,这主要是由数值模型中对系泊系统进行悬链线简化时造成的。在静水工况下,风力机气动载荷的波动主要受塔影效应影响,机舱加速度的波动与叶轮推力系数变化呈正相关,浮式风力机纵荡和纵摇运动主要受叶轮气动载荷波动的影响;在规则波和非规则波工况下,叶轮推力的波动受浮式基础纵荡和纵摇的影响较大,叶轮推力系数变化和机舱速度呈负相关。浮式风力机的运动主要受波浪载荷的影响,气动载荷对其影响相对较小。

Large-scale edge waves generated by a moving atmospheric pressure

Long waves generated by a moving atmospheric pressure distribution, associated with a storm, in coastal region are investigated numerically. For simplicity the moving atmospheric pressure is assumed to be moving only in the alongshore direction and the beach slope is assumed to be a constant in the on-offshore direction. By solving the linear shallow water equations we obtain numerical solutions for a wide range of physical parameters, including storm size （2a）, storm speed （U）, and beach slope （a）. Based on the numerical results, it is determined that edge wave packets are generated if the storm speed is equal to or greater than the critical velocity, Ucr, which is defined as the phase speed of the fundamental edge wave mode whose wavelength is scaled by the width of the storm size. The length and the location of the positively moving edge wave packet is roughly Ut/2 〈 y 〈 Ut, where y is in the alongshore direction and t is the time. Once the edge wave packet is generated, the wavelength is the same as that of the fundamental edge wave mode corresponding to the storm speed and is independent of the storm size, which can, however, affect the wave amplitude. When the storm speed is less than the critical velocity, the primary surface signature is a depression directly correlated to the atmospheric pressure distribution.