Theory: First-order equations

Substituting the expansion in wave steepness into the governing equations and retaining only first-order terms gives a linear boundary value problem for the first-order complex potential, $\phi\fo$. All quantities on this page are understood to be first order and we omit the superscript from $\phi\fo$ for visual clarity.

The linearity of the problem allows a decomposition of $\phi$ as \begin{equation} \phi = \phi_I + \phi_S + \phi_R \label{eqPotentialDecomposition} \end{equation} where

$\phi_I$ is the potential of the incident wave

$\phi_S$ is the scattered potential, due to the presence of a fixed obstructing body

$\phi_R$ is the radiation potential, caused by first-order oscillatory motion of the body in the fluid

It is also convenient to define two combinations of these potentials

$\phi_D \equiv \phi_I + \phi_S$ is known as the diffraction potential, thus $\phi = \phi_D + \phi_R$

$\phi_B \equiv \phi_S+\phi_R$ is known as the body potential, thus $\phi = \phi_I + \phi_B$

The potential of the incident wave, $\phi_I$, can be written down in closed form whereas the other components of the potential must be obtained by solving integral equations as described below.

Boundary value problems

The different definitions of the components of $\phi$ in (\ref{eqPotentialDecomposition}) imply slightly different boundary conditions for each. However, each component satisfies a boundary value problem of the following general form \begin{equation} \begin{aligned}%implicit alignment is {rlrlrl...} \nabla^2\phi & = 0 & & \vec{X} \in \fV \\ \PD{\phi}{n} & = q_B(\vec{X}) & & \vec{X} \in \SB \\ g\PD{\phi}{Z} - \omega^2 \phi & = q_F(\vec{X}) & & \vec{X} \in \SF \\ \PD{\phi}{Z} & = 0 & & Z\rightarrow - \infty \textrm{ (or on seabed)} \end{aligned} \label{eqGeneralBVP} \end{equation} where the forcing functions $q_B$ and $q_F$ are different for each component in (\ref{eqPotentialDecomposition}). To simplify the subsequent equations, it will be convenient to define $\nu = \omega^2/g$.

The incident wave potential

The incident wave potential, $\phi_I$, is the solution to the BVP (\ref{eqGeneralBVP}) in the absence of any body and with no forcing ($q_B = q_F = 0$). The solution is known analytically. For an incident wave with complex amplitude $A$ ($A$ is a height with dimensions of length), frequency $\omega$, wavenumber $k$ and wave heading $\beta$, the potential is given by \begin{equation} \label{eqFirstOrderIncidentPotential} \phi_I(\vec{X}) = \frac{\textrm{i} g A}{\omega} f(k Z) \textrm{e}^{-\textrm{i} k(X \cos \beta + Y \sin \beta)} \end{equation}

In water of infinite depth \begin{equation} \begin{aligned} f(k Z) & = \textrm{e}^{k Z} \\ k & = \nu \end{aligned} \end{equation}

In water of finite depth, $h$ \begin{equation} \begin{aligned} f(k Z) & = \frac{\cosh(k(Z + h))}{\cosh(kh)} \\ k \tanh k h & = \nu \end{aligned} \end{equation}

Integral equations

Potential formulation

OrcaWave solves the general boundary value problem (\ref{eqGeneralBVP}) using Green's theorem and the established boundary integral equation method. The integral equations are all formulated and solved in global coordinates $\GXYZ$. Application of Green's theorem gives the following integral equation for the potential on the body surface \begin{equation} \begin{aligned} 2\pi\phi(\vec{X}) + \CPVint{\SB}{} \phi(\vec{\xi})\PD{G}{n_{\xi}}\ud S_{\xi} & = \int_{\SB}{} q_B(\vec{\xi}) G \ud S_{\xi} + \int_{\SF} \frac{q_F(\vec{\xi})}{g} G \ud S_{\xi} & & \vec{X}\in \SB \end{aligned} \label{eqBIE-Pot-Classical} \end{equation} where $G(\vec X, \vec \xi)$ is the classical Green's function for the problem. We omit the arguments of $G$ to avoid further complicating the appearance of the equation, but every occurrence of $G$ should be understood as shorthand for $G(\vec X, \vec \xi)$. The notation $\ud S_{\xi}$ is employed to emphasise that $\vec \xi$ is the dummy integration variable in each of the surface integrals in (\ref{eqBIE-Pot-Classical}). The notation $\PD{}{n_{\xi}}$ stands for the normal derivative with respect to the $\vec \xi$ variable, i.e. $\vec n(\vec \xi) \cdot \nabla_{\xi}$.

OrcaWave solves (\ref{eqBIE-Pot-Classical}) by assuming that $\phi$ is constant on each mesh panel of $\SB$. This transforms (\ref{eqBIE-Pot-Classical}) into a matrix equation for the value of the potential on each panel \begin{equation} \begin{aligned} \frac{1}{2}\phi_i + \sum_{P_j \in \SB} D_{ij}\phi_j & = \sum_{P_j \in \SB} S_{ij}q_{B,j} + \frac{1}{g}\sum_{P_j \in \SF}S_{ij}q_{F,j} \\ \phi_i & = \phi(\vec{C}_i) \\ q_{B,i} & = q_B(\vec{C}_i) \\ q_{F,i} & = q_F(\vec{C}_i) \\ S_{ij} & = \frac{1}{4\pi}\int_{P_j} G(\vec{C}_i, \vec{\xi}) \ud S_{\xi} \\ D_{ij} & = \frac{1}{4\pi}\CPVint{P_j}{} \PD{G}{n_{\xi}}(\vec{C}_i, \vec{\xi}) \ud S_{\xi} \end{aligned} \label{eqBIE-Pot-ClassicalMatrixForm} \end{equation} where $P_i$ is the $i^{\textrm{th}}$ mesh panel and $\vec{C}_i$ is its centroid. $S_{ij}$ and $D_{ij}$ are known as influence matrices. Equation (\ref{eqBIE-Pot-ClassicalMatrixForm}) is a complex matrix equation which can be solved by standard numerical methods.

Source formulation

Another consequence of Green's theorem is that the potentials $\phi_R$ and $\phi_S$ can be written indirectly in terms of a source function $\sigma(\vec{X})$ \begin{equation} \begin{aligned} \phi(\vec{X}) & = \int_{\SB} \sigma(\vec{\xi}) G \ud S_{\xi} + \int_{\SF}\frac{q_F(\vec{\xi})}{4\pi g} G \ud S_{\xi} & & \vec{X}\in \fV \end{aligned} \label{eqClassicalSourceDef} \end{equation} A similar integral equation for the source function can also be derived from Green's theorem \begin{equation} \begin{aligned} 2\pi \sigma(\vec{X}) + \CPVint{\SB}{} \sigma(\vec{\xi}) \frac{\partial G}{\partial n_{x}} \ud S_{\xi} & = q_B(\vec{X}) - \int_{S_F}\frac{q_F(\vec{\xi})}{4\pi g} \frac{\partial G}{\partial n_{x}} \ud S_{\xi} & & \vec{X} \in \SB \end{aligned} \label{eqBIE-Sor-Classical} \end{equation} This equation can be discretised and numerically solved for $\sigma$ in an analogous fashion to (\ref{eqBIE-Pot-ClassicalMatrixForm}).

OrcaWave always solves the potential formulation, and can optionally solve the source formulation as well. The potential formulation gives the most accurate values for the basic results that are computed directly from the values of $\phi$: added mass and damping, load RAOs and displacement RAOs. However, the source formulation gives more accurate results for the fluid velocity, $\nabla \phi$. You must select to solve the source formulation if you wish to obtain results that depend on $\nabla \phi$ on $\SB$, such as panel velocity and mean drift loads using the pressure integration method.

The scattered and diffraction potentials

The scattered potential $\phi_S$ is the solution to the BVP (\ref{eqGeneralBVP}) with \begin{equation} \begin{aligned} q_B & = -\frac{\partial \phi_I}{\partial n} \\ q_F & = 0 \end{aligned} \end{equation} and the corresponding version of the integral equation (\ref{eqBIE-Pot-Classical}) applies.

OrcaWave in fact solves an integral equation for the diffraction potential, $\phi_D$, because this is marginally quicker computationally. The integral equation for $\phi_D$ has the same left-hand side as the equation for $\phi_S$, but a simpler right-hand side \begin{equation} \begin{aligned} 2\pi\phi(\vec{X}) + \CPVint{\SB}{} \phi(\vec{\xi})\PD{G}{n_{\xi}}\ud S_{\xi} & = 4\pi \phi_I(\vec X) & & \vec{X}\in \SB \end{aligned} \end{equation}

If the source formulation is solved as well, then the integral equation (\ref{eqBIE-Sor-Classical}) is solved for $\sigma_S$, the source function of the scattered potential (there is no source function representation of the diffraction potential).

The radiation potential

The radiation potential, $\phi_R$, is itself decomposed into six components, one for each rigid-body degree of freedom with respect to the body axes $\Bxyz$ (e.g. surge is in the $\mat{B}_\mathrm{x}$ direction). Letting $\xi_j$ represent the complex amplitude of oscillatory body motion in degree of freedom $j$, we can write \begin{equation}\label{eqRadiationPotentialDecomposition} \phi_R = \textrm{i} \omega \sum_{j} \xi_j \phi_j \end{equation}

Each radiation potential $\phi_j$ is the solution to the BVP (\ref{eqGeneralBVP}) with \begin{equation} \begin{aligned} q_B & = (n_{\textrm{vel}})_j \\ q_F & = 0 \end{aligned} \end{equation} where $n_{\textrm{vel}}$ is the normal component on the body surface of the oscillatory rigid body motion in body axes \begin{equation} \begin{aligned} (n_{\textrm{vel}})_j & = n_j & j=1,2,3 \\ (n_{\textrm{vel}})_j & = (\vec{x} \times \vec{n})_j & j=4,5,6 \end{aligned} \end{equation}

OrcaWave solves the corresponding version of the integral equation (\ref{eqBIE-Pot-Classical}) for the radiation potentials and, if applicable, the integral equation (\ref{eqBIE-Sor-Classical}) for the source functions of the radiation potentials.

Irregular frequencies

The mapping from the full boundary value problem (\ref{eqGeneralBVP}) to the integral equations (\ref{eqBIE-Pot-Classical}) and (\ref{eqBIE-Sor-Classical}) reduces a three-dimensional partial differential equation on the unbounded domain $\fV$ to the two-dimensional problem of finding the unknown $\phi$ on the surface $\SB$. The simplification resulting from this mapping is the key to the success of the boundary integral method: it the makes efficient numerical solution tractable.

However the mapping also introduces irregular frequencies. These are a set of wave frequencies at which the derived matrix equation (\ref{eqBIE-Pot-ClassicalMatrixForm}) is ill-conditioned and can result in unreliable numerical results. Irregular frequency effects generally become a problem at higher wave frequencies (i.e. shorter waves). OrcaWave will warn you if irregular frequency effects are likely to be encountered, in which case you are encouraged to extend your mesh to remove irregular frequency effects.