# Solution of Wave-Body Flows, Green's Theorem

Wave and Wave Body Interactions
Current Chapter Solution of Wave-Body Flows, Green's Theorem
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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:

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