Theory: Dipole panels

On all other help pages, the fundamental governing equations and their solution are discussed with the assumption that the body surface $\SB$ and interior surface $\SI$ together form a closed surface, as shown in the geometry definitions. In particular, at every point on $\SB$ one side is wet (i.e. on the exterior of the body) and the other side is dry (i.e. in the interior of the body). On this page we describe an extension to the theory, allowing some (or all) of the body surface to be wet on both sides – for example, an idealised (zero thickness) keel on the bottom of a boat.

Keels, heave plates and other appendages of course have finite thickness in reality, so could be represented with a panel mesh in the usual way. However, the integral equations used in diffraction analysis are ill-conditioned and give unreliable results if the thickness, $t$, of an appendage is small. One way to resolve the problem would be to use panels of size less than $t$; however, that is often impractical because it implies a large number of mesh panels and a prohibitively expensive calculation. A better alternative is to use the well-conditioned integral equations which apply in the idealised limit (thickness $t=0$).

The geometry notation is extended to include a surface $\SE$, defined as portions of the body surface which are infinitely thin, wet on both sides, and rigidly connected to the rest of the body. The thin surface $\SE$, together with $\SB$ & $\SI$, is part of the body mesh in an OrcaWave model. The mesh panels that constitute $\SE$ are referred to as dipole panels.

Integral equations

Integral equations to solve the general boundary value problem in the presence of thin surfaces $\SE$ are obtained by considering the limit $t \rightarrow 0$ of the integral equations given on other help pages (the basic equations, and extended equations for removing irregular frequency effects and modelling a damping lid).

Integral equations for the potential formulation are as follows \begin{equation} \begin{aligned} 2\pi\phi(\vec{X}) + \mathcal{D}[\phi, G](\vec{X}) & = \mathcal{B}[G](\vec{X}) & & \vec{X}\in \SB \\ -4\pi\phi(\vec{X}) + \mathcal{D}[\phi, G](\vec{X}) & = \mathcal{B}[G](\vec{X}) & & \vec{X}\in \SI \\ 4\pi\phi(\vec{X}) + \mathcal{D}[\phi, G](\vec{X}) & = \mathcal{B}[G](\vec{X}) & & \vec{X}\in \SF \\ \mathcal{D}\left[\phi, \PD{G}{n_x}\right](\vec{X}) & = \mathcal{B}\left[\PD{G}{n_x}\right](\vec{X}) - 4\pi q_B(\vec{X}) & & \vec{X}\in \SE \end{aligned} \label{eqBIE-Pot-Dipoles} \end{equation} where the operators $\mathcal{B}$ & $\mathcal{D}$ are defined to condense the notation \begin{equation} \begin{aligned} \mathcal{B}[\tau](\vec{X}) = & \int_{\SB} q_B(\vec{\xi}) \tau(\vec{X}, \vec{\xi}) \ud S_{\xi} + \int_{S_F}\frac{q_F(\vec{\xi})}{g} \tau(\vec{X}, \vec{\xi}) \ud S_{\xi} \\ \mathcal{D}[\mu, \tau](\vec{X}) = & \CPVint{\SB}{} \mu(\vec{\xi}) \frac{\partial \tau(\vec{X}, \vec{\xi})}{\partial n_{\xi}} \ud S_{\xi} - \nu \int_{\SI} \mu(\vec{\xi}) \tau(\vec{X}, \vec{\xi}) \ud S_{\xi} \nonumber \\ & + \textrm{i}\nu\int_{\SF} \epsilon(\vec{\xi}) \mu(\vec{\xi}) \tau(\vec{X}, \vec{\xi}) \ud S_{\xi} + \CPVint{\SE}{} \Delta \mu(\vec{\xi})\frac{\partial \tau(\vec{X}, \vec{\xi})}{\partial n_{\xi}}\ud S_{\xi} \end{aligned} \end{equation}

In the integral equations (\ref{eqBIE-Pot-Dipoles}) the unknown quantities are:

• the potential $\phi(\vec{X})$ for $\vec{X} \in \SB$, $\SI$ & $\SF$
• the jump in the potential, $\Delta\phi(\vec{X})$, between opposite sides of the thin surface for $\vec{X} \in \SE$

The quantity $\Delta\phi$ corresponds physically to a discontinuity in the fluid pressure, $\Delta p$, across $\SE$. The orientation of the normal vector $\vec{n}$ on $\SE$ is arbitrary, and without loss of generality we define \begin{equation} \begin{aligned} \Delta\phi(\vec{X}) = & \lim_{e \rightarrow 0+} \phi(\vec{X} - e\vec{n}) - \phi(\vec{X} + e\vec{n}) \end{aligned} \end{equation} so that a positive pressure jump $\Delta p$ gives a local force in the direction of $\vec{n}$.

OrcaWave obtains the unknown values of $\phi$ & $\Delta\phi$ by solving a matrix equation derived from (\ref{eqBIE-Pot-Dipoles}), the thin surface $\SE$ being described by the dipole panels contained in the body mesh. From a computational perspective, the significant new feature of (\ref{eqBIE-Pot-Dipoles}) is that the equation for $\vec{X}\in \SE$ involves integrals of the second derivative of the Green's function. This higher derivative is more challenging to evaluate, but the approach to computing panel integrals remains the same – the singularities of the Green's function are subtracted and integrated analytically, and the remaining nonsingular portion is integrated numerically.

Restrictions

The solve type must be potential formulation only.

• The source formulation is not available. The key benefit of the source formulation, that it yields velocity results without requiring double derivatives of the Green's function, no longer holds if the geometry contains thin surfaces $\SE$.
• Full QTFs are not available. The second-order theory implemented in OrcaWave is not valid if the geometry contains thin surfaces $\SE$.

Other modified equations

In addition to the integral equations themselves, the following equations are also modified by the presence of thin surfaces $\SE$.

Added mass and damping

The added mass and damping are \begin{equation} A_{ij} - \frac{\textrm{i}}{\omega} B_{ij} = \rho \left\{ \int_{\SB}(n_{\textrm{vel}})_i \phi_j \ud S + \int_{\SE}(n_{\textrm{vel}})_i \Delta\phi_j \ud S \right\} \end{equation} where $\phi_j$ are the components of the radiation potential.

Diffraction load RAOs

The diffraction load RAO is \begin{equation} F_i = -\textrm{i}\omega\rho \left\{ \int_{\SB} (n_{\textrm{vel}})_i \phi_D \ud S + \int_{\SE} (n_{\textrm{vel}})_i \Delta\phi_D \ud S \right\} \end{equation}

Haskind load RAOs

The Haskind formula for the load RAO is \begin{equation} F_i = -\textrm{i}\omega \rho \left[ \int_{\SB} \left\{(n_{\textrm{vel}})_i\phi_I - \phi_i\PD{\phi_I}{n}\right\} \ud S - \int_{\SE} \Delta\phi_i \PD{\phi_I}{n} \ud S -\textrm{i}\nu \int_{\SF} \epsilon \phi_i \phi_I \ud S \right] \end{equation}

Sea state RAOs

The body potential at a field point in the fluid domain, used to obtain sea state RAOs from the potential formulation, is \begin{equation} \begin{aligned} \phi_B(\vec X) & = \frac{1}{4\pi}\int_{\SB} \left\{\PD{\phi_B(\vec{\xi})}{n_{\xi}} G - \phi_B(\vec{\xi}) \PD{G}{n_{\xi}} \right\} \ud S_{\xi} -\frac{1}{4\pi}\int_{\SE} \Delta\phi_B(\vec{\xi}) \PD{G}{n_{\xi}} \ud S_{\xi} \\ & \phantom{=} -\frac{\textrm{i}\nu}{4\pi}\int_{\SF} \epsilon(\vec{\xi}) \phi(\vec{\xi}) G \ud S_{\xi} \\[7pt] \nabla\phi_B(\vec X) & = \frac{1}{4\pi}\int_{\SB} \left\{\PD{\phi_B(\vec{\xi})}{n_{\xi}} \nabla_x G - \phi_B(\vec{\xi}) \nabla_x \PD{G}{n_{\xi}} \right\} \ud S_{\xi} -\frac{1}{4\pi}\int_{\SE} \Delta\phi_B(\vec{\xi}) \nabla_x \PD{G}{n_{\xi}} \ud S_{\xi} \\ & \phantom{=} -\frac{\textrm{i}\nu}{4\pi}\int_{\SF} \epsilon(\vec{\xi}) \phi(\vec{\xi}) \nabla_x G \ud S_{\xi} \end{aligned} \end{equation}

Panel results

Field points located on $\SE$, or closer than a typical panel diameter, may give unreliable sea state RAOs. Therefore the panel results are extended to include results at the centroids of dipole panels.

The pressure jump on $\SE$ is given directly by the solution of the potential formulation \begin{equation}\label{eqPressureFromPotential} \Delta p(\vec X) = -\textrm{i}\omega\rho\Delta\phi(\vec X) \end{equation} Results for panel velocity are not available because the solve type must be potential formulation only for models with dipole panels.

Control surface mean drift loads

If all dipole panels are fully submerged, the usual formulae for the force, $\vec{f}_q$, and moment, $\vec{m}_q$, are unchanged. If there are dipole panels intersecting the free surface, the force has an additional contribution \begin{equation} \rho g \vec{e}_z \int_{\CDWL} \Delta\eta \left\{ -\vec{d}\cdot\vec{n}^h + \left(\frac{1}{2}\Sigma\eta - d_3\right) \frac{n_z}{\sqrt{1-n_z^2}} \right\} \ud l \end{equation} where $\CDWL$ is the dipole waterline, defined as the intersection of $\SE$ with the free surface. $\Sigma\eta$ and $\Delta\eta$ are the sum and difference, respectively, of the free surface elevation on opposite sides of $\CDWL$. All other quantities have the same definitions as in the usual formulae.

The moment also has a similar additional contribution \begin{equation} \rho g \int_{\CDWL} \Delta\eta \left\{ -\vec{d}\cdot\vec{n}^h + \left(\frac{1}{2}\Sigma\eta - d_3\right) \frac{n_z}{\sqrt{1-n_z^2}} \right\} \left(\vec{x}\times\vec{e}_z\right) \ud l \end{equation}

Momentum conservation mean drift loads

The Kochin function used to compute these mean drift loads is modified \begin{align} H_{\beta}(\theta) = &\; \int_{S_B} \left\{ \psi(\vec{\xi}, \theta) \PD{\phi}{n_\xi}(\vec{\xi}) - \phi(\vec{\xi})\PD{\psi}{n_\xi}(\vec{\xi}, \theta) \right\} \ud S_\xi - \int_{S_E} \Delta \phi(\vec{\xi}) \PD{\psi}{n_\xi}(\vec{\xi}, \theta) \ud S_\xi \end{align}