Using numerical methods without grid in modeling nonlinear waves of water surface caused by wind

Master's thesis in the field

Road, building and environmental engineering - hydraulic structures

Abstract

In this research, nonlinear wave differential equations are solved by local RBF-DQ numerical method. These differential equations are in the form of Laplace's equation (as a governing equation) and nonlinear boundary conditions on the free surface; The basis of the mathematical model in this research. Using this mathematical model, it is possible to simulate the propagation and changes of the water level after the generation of waves. RBF-DQ numerical method is a new meshless numerical method; which has been used to solve problems such as Neurastokes equations, heat transfer problem modeling, non-permanent leakage simulation and so on. It has been used and has obtained acceptable results. In this method, in addition to using the features of the differential quadrature method in the direct estimation of the derivative, by using radial basis functions, the advantages of numerical methods without grid can also be used. In addition, the resulting method can be used in problems with irregular boundaries. One of the most important factors affecting the accuracy of this method is the parameter of the shape of the radial basis function, which in this research, its appropriate values ??are estimated using the analysis of the state number of the weight coefficients matrix. In this research, instead of the general form, the local form of the RBF-DQ method has been used. This method can expand its application range and reduce computational costs by maintaining the accuracy of RBF-DQ method. In order to simulate the free surface, which is the main part of the simulation; Eulerian and Lagrangian composite methods have been used. The validity and accuracy of the current model has been verified by analytical models, available numerical models and laboratory results. In this research, firstly, the wave propagation model in the numerical tank is investigated and then the wave propagation from the wave generator is studied. The results of this research showed that in a problem with a variable boundary condition, in terms of the amount of calculations, using a method without a grid is preferred over the grid-based methods.  Local RBF-DQ method is able to solve equations well and in some cases its accuracy is better than other analytical and numerical methods. Also, the investigation of the factors affecting the nonlinearity of the wave showed that the wave height is more effective than the water depth and wavelength. Keywords: Nonlinear wave model - Non-grid numerical methods Chapter 1 Introduction Introduction 1-1 Generalities Oceans And the seas are considered huge capitals of the universe and have important effects on people's livelihood, economy, tourism and transportation. In these infinite water environments, various phenomena occur; One of the most obvious phenomena that has an inseparable connection with the seas and oceans; The waves are caused by the wind. Knowing and predicting these waves is essential for the correct and safe exploitation of oceans and seas. In the current research, these waves have been investigated and a mathematical model for their simulation has been presented. This topic itself shows the importance of knowing and investigating the phenomena that occur in this vast part of the globe. Waves are one of the most important phenomena in aquatic environments. Therefore, their prediction and simulation play a significant role in serving and controlling the seas and oceans. For example, the construction of coastal structures for the safety of the coast and the control of the sea area, the design of offshore structures for the exploitation of oil and gas, environmental studies, the design of ships and their safe transportation, and the transfer of sediments all require accurate and complete information about water waves.

Getting information about waves and their characteristics is possible in two ways. The first method is to estimate waves by means of measurement tools, such as wave measuring vessels [1] or satellites. And the second method is wave modeling, which can be done by mathematical or physical models.  Since the measurements carried out by wave measurement floats; are points and satellite images are not accurate enough; Simulation by mathematical and physical models is very important. On the other hand, preparing physical models is difficult and requires a lot of time and money; Therefore, with the advancement of computers, mathematical models have found an important place in the simulations and modeling of engineering problems.In recent years, numerical models have been used to simulate waves.

Waves are created under the influence of various factors. Wind, seabed disturbances and the gravitational force of the sun and the moon are the three main factors of tidal production. Wind waves are short and have a smaller period. On the other hand, there are waves caused by bed disturbances (tsunami) and waves caused by gravity (seismic tides), which belong to the group of long waves. The classification of waves and energy of each type based on period is shown in figure (1-2).

In this research, short waves caused by wind have been investigated. After the waves are created by the wind, they start moving. During the movement, the waves are separated from each other and their height decreases, but their wavelength and period are maintained. This process is called wave separation. The waves that are located in the production area are irregular, short and sharp [2] (Reeve et al., 2004), but when they move away from this area, they find an almost regular and short form and finally become distant waves (Figure (1-3)). For example, the process of wave breaking in deep water (white cap[3]) is strongly nonlinear locally. But on average, the consumption of energy is weak compared to that on a large scale. Another example is structures exposed to waves. For example, in measuring the forces acting on a marine structure, in some cases the waves must be modeled non-linearly. In general, for modeling very sharp waves or waves in shallow waters or on small scales, linear models are not responsive and non-linear models should be used (Holthuijsen, 2007). The purpose of this research is to investigate and simulate nonlinear waves.

Until now, researchers have done many researches in the field of modeling nonlinear waves caused by wind. The primary theories are analytical theories. But new theories are based on partial differential equations[4] and their solution is possible with numerical methods (Holthuijsen, 2007)). Finite element [5] and finite difference [6] methods are used in this field. For example, Mei (1978) used the finite element method and Chan and Street (1970) used the finite difference method. One of the most widely used methods in solving nonlinear wave equations is the boundary element method [7], which has been used by many researchers such as Cokelet and Longuet-Higgings (1976). The mentioned methods require computational domain networking. This networking should be done according to specific criteria. Because the shape and manner of connecting the elements that control the quality of the network; They directly affect the accuracy of the results. Besides, in most of the problems, due to the deviation of the elements, the meshing must be done again in all time steps or some of them, and these meshings are as expensive and time-consuming as the initial mesh. For this reason, numerical methods without grid [8] in modeling nonlinear waves were also considered as in other engineering fields. One of the gridless numerical methods used by researchers in recent years is the RBF-DQ method. where the DQ method is used to estimate the derivative. With the help of the DQ network-based method [9], good results can be obtained despite the few nodes in the domain. But this method cannot be used in irregular domains (Hashemi and Hatam, 2011); Because the derivative of the function is expressed by DQ in each direction as a weighted linear sum of the function values ??in the same direction, and it is not possible to provide regular nodes in a specific direction in irregular domains. But by using radial basis functions [10] as a shape function in DQ, this problem can be avoided. Besides, the use of radial functions in the DQ method will turn it into a grid-free method that does not have the mentioned disadvantages of grid-based methods. Among the different types of radial functions, in this research, due to the good performance of the MQ function, this type of function is used in solving problems. This function has a parameter called the shape parameter [11], which greatly affects the accuracy of the results. So far, many researches have been presented to calculate the optimal value of this parameter. But none of them presented a theoretical and comprehensive method. For this reason, the research in this field continues.

The aim of this research is to simulate the nonlinear waves caused by the wind with the RBF-DQ method.