Solution of Wave-Body Flows, Green's Theorem

Wave and Wave Body Interactions
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Introduction

Two types of wave body interaction problems are encountered frequently in applications and solved by the methods described in this section: zero-speed linear wave body interactions in the frequency domain in 2D and 3D, forward-speed seakeeping problems in the frequency or time domain in three dimensions (linear and nonlinear).

A consensus has been reached over the past two decades that the most efficient and robust solution methods are based on Green's Theorem using either a wave-source potential or the Rankine source as the Green function. The numerical solution of the resulting integral equations in practice is in almost all cases carried out by panel methods.

Frequency-domain radiation-diffraction. U = 0

Boundary-value problem

Green's Theorem generates a boundary integral equation for the complex potential $\phi\,$ over the body boundary $S_B\,$ for the proper choice of the Green function:

$\iint_S \left( \phi_1 \frac{\partial\phi_2}{\partial n} - \phi_2 \frac{\partial\phi_1}{\partial n} \right) \mathrm{d}S = 0 \,$

For any $\phi_1, \ \phi_2\,$ that solve the Laplace equation in a closed volume $V\,$.

Define the volume $V\,$ and $S\,$ as follows:

The fluid volume $V\,$ is enclosed by the union of several surfaces

$S \equiv S_B + S_F + S_\infty + S_H + S_E$

$S_B \,$: mean position of body surface
$S_F \,$: mean position of the free surface
$S_\infty \,$: Bounding cylindrical surface with radius $R = \left( x^2 +y^2 \right)^{1/2} \,$. Will be allowed to expand after the statement of Green's Theorem
$S_H \,$: Seafloor (assumed flat) of a surface which will be allowed to approach $Z=-\infty\,$
$S_E \,$: Spherical surface with radius $V = \epsilon \ ,$ centered at point $\vec\xi \,$ in the fluid domain
$\vec{n}\,$: Unit normal vector on $S\,$, at point $\vec{x}\,$ on $S\,$

Define two velocity potentials $\phi_i(\vec{x})\,$:

$\phi_1(\vec{x}) = \phi(\vec{x}) \equiv \,$ Unknown complex radiation or diffraction potential
$\phi_2(\vec{x}) = G(\vec{x};\vec{\xi}) \equiv \,$ Green function value at point $\vec{x}\,$ due to a singularity centered at point $\vec{\xi}\,$.

Two types of Green functions will be used:

Rankine source: $\nabla_x^2 G = 0 \,$

$G(\vec{x};\vec{\xi}) = - \frac{1}{4\pi} \left|\vec{x}-\vec{\xi}\right|^{-1} = - \frac{1}{4\pi r} \,$ $= - \frac{1}{4\pi} \left\{ (x-\xi)^2 + (Y-n)^2 + (Z-\zeta)^2 \right\}^{-1/2} \,$

Note that the flux of fluid emitted from $\vec{\xi}\,$ is equal to $1\,$.

This Rankine source and its gradient with respect to $\vec\xi\,$ (dipoles) is the Green function that will be used in the ship seakeeping problem.

Havelock's wave source potential

...Also known as the $U=0\,$ wave Green function in the frequency domain.

Satisfies the free surface condition and near $\vec\xi=0\,$ behaves like a Rankine source:

The following choice for $G(\vec{X};\vec{\xi})\,$ satisfies the Laplace equation and the free-surface condition:

$F(\vec{X},\vec{\xi}) = - \frac{1}{4\pi} \left( \frac{1}{r} + \frac{1}{r_1} \right) - \frac{K}{2\pi} \int_0^\infty \frac{\mathrm{d}u}{u-K} e^{u(Z+\zeta)} J_0(uR)$

where:

$K = \frac{\omega^2}{g} \,$ $R^2 = (X-\xi)^2 + (Y-n)^2 \,$ $J_0(uR) = \mbox{Bessel Function of order zero} \,$ $\mbox{Contour indented above pole} n = K \,$

Verify that with respect to the argument $\vec{X}\,$, the velocity potential $\phi_2(\vec{X}) \equiv G(\vec{X};\vec{\xi})\,$ satisfies the free surface condition:

$\frac{\partial\phi_2}{\partial Z} - K \phi_2 = 0, \quad Z=0$ $\phi_2 \sim - \frac{1}{4\pi} r^{-1}, \quad \vec{X} \to \vec{\xi}$

As $KR\to\infty\,$:

$G \sim - \frac{i}{2} K e^{K(Z+\zeta)} H_0^{(2)}(KR)$

where $H_0^{(2)} (KR)\,$ is the Hankel function of the second kind and order zero.

At $KR\to\infty\,$:

$H_0^{(2)}(KR) \sim \sqrt{\frac{2}{\pi K R}} e^{-i \left( KR-\frac{\pi}{4} \right)} + O \left( \frac{1}{R} \right)$

Therefore the real velocity potential

$\mathbb{G} = \mathrm{Re} \left\{ G e^{i\omega t} \right\}$

Represents outgoing ring waves of the form $\propto e^{i(\omega T -KR)}\,$ hence satisfying the radiation condition.

A similar far-field radiation condition is satisfied by the velocity potential $\phi_1(\vec{X})\equiv\phi(\vec{X})\,$

$\phi_1 \sim \frac{\mathbb{A}(\theta)}{(KR)^{1/2}} e^{KZ-iKR} + O\left(\frac{1}{R}\right)$

It follows that on $S_\infty\,$:

$\frac{\partial\phi_1}{\partial n} = \frac{\partial\phi_1}{\partial R} = - i K \phi_1 \,$ $\frac{\partial\phi_2}{\partial n} = \frac{\partial\phi_2}{\partial R} = - i K \phi_2 \,$

Therefore:

$\phi_1 \frac{\partial\phi_2}{\partial n} - \phi_2 \frac{\partial\phi_1}{\partial n} = - i K \left( \phi_1 \phi_2 - \phi_2 \phi_1 \right) = 0$

with errors that decay like $R^{-3/2}\,$, hence faster than $R\,$, which is the rate at which the surface $S_\infty\,$ grows as $R\to\infty\,$.

On $S_F(Z=0)\,$:

$\frac{\partial\phi_1}{\partial n} = \frac{\partial\phi_1}{\partial Z}, \quad \frac{\partial\phi_2}{\partial n} = \frac{\partial\phi_2}{\partial Z}$ $\phi_1 \frac{\partial\phi_2}{\partial n} - \phi_2 \frac{\partial\phi_1}{\partial n} = \phi_1 \frac{\partial\phi_2}{\partial Z} - \phi_2\frac{\partial\phi_1}{\partial Z} = Y \left( \phi_1\phi_2 - \phi_2\phi_1\right) = 0$

It follows that upon application of Green's Theorem on the unknown potential $\phi_1 \equiv \phi\,$ and the wave Green function $\phi_2 \equiv G\,$ only the integrals over $S_B\,$ and $S_\epsilon\,$ survive.

Over $S_H\,$, either $\frac{\partial\phi_1}{\partial n} = \frac{\partial \phi_2}{\partial n} = 0 \,$ by virtue of the boundary condition if the water depth is finite or $\frac{\partial\phi_1}{\partial Z} \to 0, \ \frac{\partial\phi_2}{\partial Z}\to 0 \,$ as $Z\to - \infty\,$ by virtue of the vanishing of the respective flow velocities at large depths.

There remains to interpret and evaluate the integral over $S_\epsilon\,$ and $S_B\,$. Start with $S_\epsilon\,$:

$I_\epsilon = \iint_{S_\epsilon} \left( \phi_1 \frac{\partial\phi_2}{\partial n} - \phi_2 \frac{\partial\phi_1}{\partial n} \right) \mathrm{d}S$

or:

$I_\epsilon = \iint_{S_\epsilon} \left( \phi \frac{\partial G}{\partial n} - G \frac{\partial\phi}{\partial n} \right) \mathrm{d}S_X$

Note that the integral over $S_\epsilon\,$ is over the $\vec{X}\,$ variable with $\vec{\xi}\,$ being the fixed point where the source is centered.

Near $\vec{\xi}\,$:

$G \sim - \frac{1}{4\pi r}, \quad \frac{\partial G}{\partial n} = - \frac{\partial G}{\partial r} \sim \frac{1}{4\pi r^2}$ $\phi \to \phi(\vec{\xi}) = \phi(\vec{X}) \ \mbox{as} \ \epsilon \to 0, \vec{X} \to \vec{\xi}$

In the limit as $r\to 0 \,$ the integrand over the sphere $S_\epsilon\,$ becomes spherically symmetric and with vanishing errors

$I_\epsilon \to 4 \pi r^2 \left[ \phi(\vec{\xi}) \frac{1}{4 \pi r^2} + G \frac{\partial\phi}{\partial r} \right] = \phi(\vec{\xi})$

In summary:

$\phi(\vec{\xi}) + \iint_{S_B} \left[ \phi(\vec{X}) \frac{\partial G(\vec{X};\vec{\xi})}{\partial n_X} - G(\vec{X}; \vec{\xi}) \frac{\partial\phi}{\partial n_X} \right] \mathrm{d} S_n = 0$

on $S_B: \quad \frac{\partial\phi}{\partial n_X} = V(X) = \,$ known from the boundary condition of the radiation and diffraction problems.

It follows that a relationship is obtained between the value of $\phi(\vec{\xi})\,$ at some point in the fluid domain and its values $\phi_(\vec{X})\,$ and normal derivatives over the body boundary:

$\phi(\vec{\xi}) + \iint_{S_B} \phi(\vec{X}) \frac{\partial G ( \vec{X}; \vec{\xi} )} {\partial n_X} \mathrm{d} S_X = \iint_{S_B} G (\vec{X};\vec{\xi}) V(\vec{X}) \mathrm{d} S_X$

Stated differently, knowledge of $\phi\,$ and $\frac{\partial\phi}{\partial n}\,$ over the body boundary allows the determination of $\phi\,$ and upon differentiation of $\nabla\phi\,$ in the fluid domain.

In order to determine $\phi(\vec{X})\,$ on the body boundary $S_B\,$, simply allow $\vec{\xi}\to S_B\,$ in which case the sphere $S_\epsilon\,$ becomes a $\frac{1}{2}\,$ sphere as $\epsilon\to 0 \,$:

Note that $\vec{\xi}\,$ is a fixed point where the point source is centered and $\vec{X}\,$ is a dummy integration variable moving over the body boundary $S_B\,$.

The reduction of Green's Theorem derived above survives almost identically with a factor of $\frac{1}{2}\,$ now multiplying the $I_\epsilon\,$ integral since only $\frac{1}{2}\,$ of the $S_\epsilon\,$ surface lies in the fluid domain in the limit as $\epsilon\to 0 \,$ and for a body surface which is smooth. It follows that:

$\frac{1}{2} \phi(\vec{\xi}) + \iint_{S_B}\phi(\vec{X}) \frac{\partial G (\vec{X}; \vec{\xi})}{\partial n_X} \mathrm{d} S_X = \iint_{S_B} G(\vec{X}; \vec{\xi}) V(\vec{X}) \mathrm{d} S_X$

where now both $\vec{X}\,$ and $\vec{\xi}\,$ lie no the body surface. This becomes an integral equation for $\phi(\vec{X})\,$ over a surface $S_B\,$ of boundary extent. Its solution is carried out with panel methods described below.

The interpretation of the derivative under the integral sign is as follows:

$\frac{\partial G}{\partial n_X} \equiv \vec{n}_X \cdot \nabla_X G(\vec{X};\vec{\xi}) \equiv \left( n_1 \frac{\partial}{\partial X} + n_2 \frac{\partial}{\partial Y} + n_3 \frac{\partial}{\partial Z} \right) G (\vec{X}; \vec{\xi})$

where derivatives are taken with respect to the first argument for a point source centered at point $\vec{\xi}\,$.

Infinite domain potential flow solutions

In the absence of the free surface, the derivation of the Green integral equation remains almost unchanged using $G\,$:

$\phi_2(\vec{X}) = - \frac{1}{4\pi} \left| \vec{X} - \vec{\xi} \right|^{-1} \equiv G(\vec{X}; \vec{\xi})$

The Rankine source as the Green function and using the property that as $R\to \infty\,$

For closed boundaries $S_B\,$ with no shed wakes responsible for lifting effects the resulting integral equation for $\phi(\vec{X})\,$ over the body boundary becomes:

$\frac{1}{2} \phi(\vec{\xi}) + \iint_{S_B} \phi(\vec{X}) \frac{\partial G(\vec{X};\vec{\xi})}{\partial n_X} \mathrm{d} S_X = \iint_{S_B} G(\vec{X};\vec{\xi}) V(\vec{X}) \mathrm{d} S_X$

with

$V(\vec{X}) = \frac{\partial\phi}{\partial n}, \quad \mbox{on} \ S_B \,$

Example

Uniform flow past $S_B\,$:

$\Phi = U X + \phi, \ \frac{\partial\Phi}{\partial n} = 0, \ \mbox{on} \ S_B \,$ $\Longrightarrow \frac{\partial\phi}{\partial n} = -\frac{\partial}{\partial n} \left(U X\right) = - n_1 U \equiv V(\vec{X})$

So the RHS of the Green equation becomes:

$RHS = \iint_{S_B} G(\vec{X};\vec{\xi}) \left(-Un_1\right) \mathrm{d} S_X \,$.

Ocean Wave Interaction with Ships and Offshore Energy Systems