# Numerical Simulation of the Kelvin Wake Patterns
## 1 Introduction  
When a vessel or a body steadily translates into calm water, free surface waves are generated and these profiles are referred to as \"Kelvin ship waves\" [1]. The larger the Kelvin wake generated by a ship, the larger the drag force will be on the ship [2]. Meanwhile, the ship waves contain energy, which impacts the coastline and riverbank [3]. Calculating how much the wave drag and knowing how the energy distribution of a ship wave changes with respect to certain properties (e.g., ship speed, ship size and ship shape) can help design the ship hull and inform environmentally friendly shipping policies. Inferring characteristics of wake waveform can help identify ships that have been detected by monitoring instruments [4; 5; 6]. Initially, [PERSON] [7] was fascinated by the V\(-\)shaped wake created by a ship advancing through water; one of his most remarkable results concerned the well\(-\)known constant wake angle 19.47\({}^{\circ}\), which is known in the community as the Kelvin angle. [PERSON] [8] suggested a simplified stationary phase analysis of the dispersion relation, showing that the open dispersion curve is divided by an inflexion into two branches corresponding to transverse waves and divergent waves, respectively. [PERSON] [9] noted that for sufficiently fast\(-\)moving ships, contrary to commonly held views, the wake angle that is observed behind a steadily moving ship is less than the well\(-\)known Kelvin angle, according to simulation results. This finding aroused the interest of many academics. [PERSON] [10] utilized a steadily moving pressure disturbance to analyze the variation in wake angle for large Froude numbers. [PERSON] [11] provided an explanation for this phenomenon, namely that the apparent wake angle is naturally provided by lining up the points of maximum wave amplitude along the divergent waves. Furthermore, this is similar fornon\(-\)axisymmetric simplified ship models and interference effects [12; 13; 14; 15; 16], the presence of shear current, submergence depth and finite depth [17; 18; 19; 20], effects of surface tension and the bottom topography [21; 22], as well as the effects of viscosity [23]. [PERSON] [24] analyzed the wake angles on the low Froude number regime, which arose when the disturbance was traveling sufficiently slowly. Previously, analytical algorithms were usually utilized to explore ship waves. [PERSON] [25] firstly presented the linear solution for flow due to a moving point source. [PERSON] and [PERSON] [26] solved the solutions for infinitely deep fluid, as well as fluid of arbitrary finite depth. With the development of numerical methods and computer technology, numerical simulation methods became popular. Meanwhile, the study of ship waves had been extended from two\(-\)dimensional linear problems to three\(-\)dimensional nonlinear problems. [PERSON] [27] used the finite\(-\)difference method to solve the three\(-\)dimensional free surface hydrodynamical problem. [PERSON] [28] utilized the finite\(-\)element method to explore the flow over a submerged weir. [PERSON] [29] overcame the difficulty about the singularities and stagnation points of the boundary integral equation, using the boundary integral method in the plane of the physical variables. [PERSON] and [PERSON] et al. [30] used the high\(-\)order boundary element method to solve the steady wave problem of high\(-\)speed ships. There are three methods for the discretization of the entire fluid domain: finite\(-\)difference method, finite\(-\)element method and boundary integral method [31; 32; 33]. Generally, significant simplifications can be made if the problem is transformed into one involving a boundary integral. By reducing a three\(-\)dimensional problem to a two\(-\)dimensional problem, the boundary integral method is appealing to solve the three\(-\)dimensional nonlinear free surface hydrodynamical problems [34]. Numerical solutions to nonlinear free surface hydrodynamical problems in three dimensions are considerably rare, because of the obvious computational difficulty.