# Numerical Simulation of the Kelvin Wake Patterns
## 2 Problem Formulation  
Based on potential flow theory, this paper considers the irrotational flow of an inviscid, incompressible fluid of infinite depth and ignores the effects of surface tension. The fluid possesses an upper free boundary and is subject to the downward acceleration \(g\) of gravity. A Cartesian coordinate is defined such that the \(x-\) and \(y-\)axes in the plane of the free surface and \(z-\)axis point vertically with the origin at (0,0,0). The flow is directed along the positive \(x-\)axis with uniform speed \(U\). Meanwhile, a source singularity whose strength is \(m\) or a Rankine source whose strength is \(x\) is introduced at a distance \(L\) below the surface, as illustrated in Figure 1. The source strength represents ship size, a higher source strength means a larger ship size. The transient waves will be generated on the free surface, on account of the disturbance caused by a source. This is the steady\(-\)state problem, considering the wave pattern does not change over time. The free surface location is described by the equation \(z^{\prime}=~{}\zeta^{\prime}(x^{\prime},y^{\prime})\), with labelling the velocity potential \(\Phi^{\prime}(x^{\prime},y^{\prime},z^{\prime})\); here, the apostrophe means the dimensionless variable. Meanwhile, dimensionless variables can be defined by scaling all lengths with respect to \(L\) and all speeds with respect to \(U\). Assuming the fluid is inviscid and incompressible, [PERSON]'s equation can be satisfied, as follows:  
\[\
abla^{2}\;\Phi^{\prime}=\frac{\partial^{2}\;\Phi^{\prime}}{\partial x^{ \prime 2}}+\frac{\partial^{2}\;\Phi^{\prime}}{\partial y^{\prime 2}}+\frac{ \partial^{2}\;\Phi^{\prime}}{\partial z^{\prime 2}}=0\quad\text{for}\;z^{ \prime}<\zeta\big{(}x^{\prime},y^{\prime}\big{)}. \tag{1}\]  
By deploying a source point, the velocity potential can be described, as follows:  
\[\Phi^{\prime}\sim x^{\prime}-\frac{\epsilon^{\prime}}{4\pi\sqrt{x^{\prime 2 }+y^{\prime 2}+(z^{\prime}+1)^{2}}}\quad\text{as}\;(x^{\prime},y^{\prime},z^{ \prime})\rightarrow(0,0,-1). \tag{2}\]  
As for the neighbourhood of the Rankine source point, the appropriate limiting behaviour can be described, as follows:  
\[\Phi^{\prime}\sim x^{\prime}+\frac{\mu^{\prime}x^{\prime}}{4\pi({x^{\prime 2 }+y^{\prime 2}+(z^{\prime}+1)})^{\frac{3}{2}}}\quad\text{as}\;(x^{\prime},y^{ \prime},z^{\prime})\rightarrow(0,0,-1), \tag{3}\]  
where \(\epsilon^{\prime}\) and \(\mu^{\prime}\) are the Kelvin source strength and Rankine source strength in the form:  
\[\epsilon^{\prime}=\frac{m}{UL^{2}}, \tag{4}\]  
\[\mu^{\prime}=\frac{\kappa}{UL^{3}}. \tag{5}\]  
Since the kinematic and dynamic boundary conditions must be satisfied on the free surface, the nonlinear conditions are adopted, as follows:  
\[\[PERSON]^{\prime}{}_{x}\xi^{\prime}{}_{x}+\[PERSON]^{\prime}{}_{y}\xi^{\prime}{}_{y}= \[PERSON]^{\prime}{}_{z}\qquad\qquad\text{on}\;z^{\prime}=\zeta^{\prime}\big{(}x^{ \prime},y^{\prime}\big{)}, \tag{6}\]  
\[\frac{1}{2}\Big{(}\[PERSON]^{\prime}{}_{x}^{2}+\[PERSON]^{\prime}{}_{y}^{2}+\[PERSON]^{ \prime}{}_{z}^{2}\Big{)}+\frac{\zeta^{\prime}}{F^{\prime 2}}=\frac{1}{2}\quad\text{ on}\;z^{\prime}=\zeta^{\prime}\big{(}x^{ \prime},y^{\prime}\big{)}, \tag{7}\]  
Figure 1: Illustration of the disposition about the fluid.