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Theory: Green's function |
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Theory: Green's function |
OrcaWave uses the classical Green's function, $G(\vec{X}, \vec{\xi})$, which is defined as the complex potential of the response to a point source at $\vec{\xi}$ in the absence of a body and in the absence of a damping lid; hence, $G$ satisfies the general boundary value problem with $\epsilon, q_F, q_B \equiv 0$ and no body surface $\SB$. This Green's function is well known and understood.
| Notes: | It is important to be clear about the precise definition of $G$ because alternative choices are available, e.g. a Rankine Green's function that satisfies Laplace's equation but does not satisfy the free surface condition. Such choices can be used for a valid analysis, but they would lead to different integral equations from those solved by OrcaWave because different cancellations would occur (or not) in the various surface integrals that arise from applying Green's theorem. |
The Green's Function, $G$, is given by\begin{equation} \begin{aligned} G(\vec X, \vec \xi) & = \left[R^2 + (Z-\zeta)^2 \right]^{-1/2} + \int_0^{\infty}\frac{k+\nu}{k-\nu} \textrm{e}^{k(Z+\zeta)}J_0(kR)\ud k \\ R & = \left[(X-\xi)^2 + (Y-\eta)^2 \right]^{1/2} \end{aligned} \label{eqGDefInfinite} \end{equation} where $J_0$ is the Bessel function of the first kind and the two vector arguments have coordinates $\vec{X}=(X,Y,Z)$ and $\vec{\xi}=(\xi,\eta,\zeta)$. $\nu \equiv \omega^2 / g$, where $\omega$ is the angular frequency of the wave and $g$ is the acceleration due to gravity. The integration contour is indented above the pole on the real axis. OrcaWave evaluates the Green's function (\ref{eqGDefInfinite}) using the methods of (Newman 1985) and (Newman 1992).
Allowance must be made for the singularities of the Green's function (\ref{eqGDefInfinite}):
A singularity when the two arguments are equal, $\vec X = \vec \xi$, is immediately visible in (\ref{eqGDefInfinite}) with \begin{equation} G \sim |\vec X - \vec \xi |^{-1} \end{equation} This singular term is called the true Rankine singularity.
A singularity also occurs if one argument equals the reflection of the other in the free surface, i.e. $\vec X = R_F(\vec\xi) = (\xi, \eta, -\zeta)$. In this case \begin{equation} G \sim |\vec X - R_F(\vec \xi) |^{-1} - 2\nu \log\left(|\vec X - R_F(\vec \xi) | + |Z+\zeta|\right) \end{equation} These two singular terms are called the image Rankine and logarithmic singularities respectively.
Close to these singularities, the Green's function takes large values and shows rapid variation over short length scales; therefore, they have the potential to harm the accuracy of the numerical integration of $G$ that is performed over mesh panels. OrcaWave subtracts each of these singularities and performs the integration of the singular terms analytically over each panel using the methods of (Newman 1986) for the Rankine terms and (Newman & Sclavounos 1988) for the logarithmic term. The remaining nonsingular portion of the Green's function is integrated numerically over each panel by evaluating the value at the panel centroid and multiplying this by the panel area.
In water of constant depth $h$, the Green's function $G$ is given by\begin{equation} \begin{aligned} G(\vec X, \vec \xi) & = \left[R^2 + (Z-\zeta)^2 \right]^{-1/2} + \left[R^2 + (Z+\zeta+2h)^2 \right]^{-1/2} \\ & \phantom{=\;\,} + 2\int_0^{\infty} \frac{(k+\nu)\cosh k(z+h) \cosh k(\zeta+h)}{k\sinh kh -\nu\cosh kh} \textrm{e}^{-kh}J_0(kR)\ud k \\ R & = \left[(X-\xi)^2 + (Y-\eta)^2 \right]^{1/2} \end{aligned} \label{eqGDefFinite} \end{equation} where the integration contour is again indented above the pole on the real axis. OrcaWave evaluates the Green's function (\ref{eqGDefFinite}) using the methods of (Newman 1985) and (Newman 1992).
The Green's function (\ref{eqGDefFinite}) has more singularities in addition to the true Rankine, image Rankine and logarithmic singularities displayed by the infinite-depth $G$.
A singularity if one argument equals the reflection of the other in the seabed, i.e. $\vec X = R_B(\vec\xi) = (\xi, \eta, -\zeta-2h)$. In this case \begin{equation} G \sim |\vec X - R_B(\vec \xi) |^{-1} \end{equation} This singular term is called the seabed Rankine singularity.
Further singularities also occur at $R=0$, $|Z - \zeta|= 2h$. These are far enough from the physical domain (i.e. $-h\le Z,\zeta\le 0$) to have a negligible effect.
OrcaWave analytically integrates the seabed Rankine singularity in the same manner as above and the remainder of the Green's function is integrated numerically over each panel.