Theory: Governing equations

The fluid flow is assumed to be incompressible, inviscid and irrotational. The fluid velocity is given by $\nabla \Phi$, where the velocity potential, $\Phi$, satisfies Laplace's equation in the fluid domain \begin{equation} \nabla^2 \Phi(\vec{X}, t) = 0 \end{equation} Substituting into the Navier-Stokes equation and integrating yields the Bernoulli equation for the pressure \begin{equation} p(\vec{X}, t) = -\rho \left( \PDt{\Phi} + \frac{1}{2} (\nabla \Phi)^2 + g Z \right) \end{equation}

Boundary conditions

• On the seabed, or as $Z\rightarrow -\infty$ in infinite-depth water, the velocity potential satisfies \begin{equation} \PD{\Phi}{Z}=0 \end{equation}

• On the surface of a rigid body moving with velocity $\vec{U}$ and angular velocity $\vec{\Omega}$ \begin{equation} \vec{n} \cdot \nabla \Phi = \vec{n} \cdot (\vec{U} + \vec{\Omega}\times\vec{X}) \end{equation} where $\vec{n}$ is a unit normal. This condition applies over the wet surface of the body at its instantaneous position.

• On the free surface the standard kinematic boundary condition of fluid dynamics becomes \begin{equation} \PDD{\Phi}{t} + g\PD{\Phi}{Z} + 2\nabla\Phi \cdot \nabla\PDt{\Phi} + \frac{1}{2}\nabla\Phi\cdot\nabla (\nabla\Phi)^2 = 0 \end{equation} This condition holds on the instantaneous free surface $Z=\eta(X,Y,t)$, which is itself determined by the Bernoulli equation \begin{equation} \eta(X,Y,t) = -\frac{1}{g}\left.\left(\PDt{\Phi} + \frac{1}{2}(\nabla\Phi)^2 \right)\right|_{Z=\eta} \end{equation}

• The radiation/causality condition that the disturbance due to the presence of a body consists of outgoing waves decaying at infinity.

Perturbation expansion

In full generality, the governing equations can only be solved by computational fluid dynamics approaches. This is because, whilst Laplace's equation is linear, the boundary conditions on the free surface and on moving body surfaces are nonlinear. In practical applications, wave steepness is often small and it is standard to proceed with a perturbation expansion in this parameter \begin{equation} \begin{aligned} \Phi(\vec{X}, t) & = \Phi\fo(\vec{X}, t) + \Phi\so(\vec{X}, t) + \ldots \\ p(\vec{X}, t) &= p\zo(\vec{X}, t) + p\fo(\vec{X}, t) + p\so(\vec{X}, t) + \ldots \end{aligned} \label{eqPerturbationExpansion} \end{equation}

The equations at first and second order in wave steepness are significantly more tractable. OrcaWave solves these equations using the standard boundary integral equation method. RAOs and added mass and damping result from solving the first-order problem for $\Phi\fo$. QTFs arise from continuing the expansion to second order and solving for $\Phi\so$.

Complex notation

At each order in wave steepness the equations governing the velocity potential are linear and have translational invariance with respect to time, permitting a standard frequency domain approach in which each wave frequency is considered individually. For a single wave frequency, $\omega$, we write the velocity potential, $\Phi(\vec{X}, t)$, in terms of a complex potential, $\phi(\vec{X})$, as \begin{equation} \begin{aligned} \Phi(\vec{X}, t) & = \Re \left\{\phi(\vec{X}) \mathrm{e}^{\mathrm{i}\omega t}\right\} \label{eqComplexPotential} \\ \omega & > 0 \;\;\textrm{(without loss of generality)} \end{aligned} \end{equation} Almost all subsequent equations will involve the complex potential $\phi$. The real part operator $\Re\{\;\}$ is understood but generally omitted for visual clarity.

Geometry notation

We define the following surfaces and volumes:

$\SB$ is the surface of a body. Specifically, it is the wet portion ($Z\le 0$) of the body surface when the body is in its mean position

$\SI$ is the free surface of the fluid displaced by a body. Specifically, it is the portion of the plane $Z=0$ interior to $\SB$

$\CWL$ is the waterline of a body, i.e. the line of intersection between $S_B$ and $S_I$

$\SF$ is the water free surface. Specifically, it is the portion of the plane $Z=0$ exterior to all body surfaces, $\SB$

$\fV$ is the volume occupied by the fluid. $\fV$ is the space exterior to all body surfaces, $\SB$, and beneath the free surface $Z\le 0$

$\dV$ is the boundary surface of $\fV$. $\dV$ includes $\SB$, $\SF$ and, if present, the seabed

$\VInt$ is the volume interior to a body, in other words the fluid displaced by $\SB$

$\dVInt$ is the boundary surface of $\VInt$. Therefore $\dVInt = \SB \cup \SI$

We adopt the following convention for the orientation of normal vectors, $\vec n$:

On $\SB$ the normal $\vec n$ points out of the fluid and into the body