# 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 $\displaystyle{ \phi\, }$ over the body boundary $\displaystyle{ S_B\, }$ for the proper choice of the Green function:

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

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

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

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

$\displaystyle{ S \equiv S_B + S_F + S_\infty + S_H + S_E }$

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

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

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

Two types of Green functions will be used:

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

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

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

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

### Havelock's wave source potential

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

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

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

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

where:

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

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

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

As $\displaystyle{ KR\to\infty\, }$:

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

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

At $\displaystyle{ KR\to\infty\, }$:

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

Therefore the real velocity potential

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

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

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

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

It follows that on $\displaystyle{ S_\infty\, }$:

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

Therefore:

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

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

On $\displaystyle{ S_F(Z=0)\, }$:

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

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

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

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

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

or:

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

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

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

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

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

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

In summary:

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

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

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

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

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

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

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

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

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

where now both $\displaystyle{ \vec{X}\, }$ and $\displaystyle{ \vec{\xi}\, }$ lie no the body surface. This becomes an integral equation for $\displaystyle{ \phi(\vec{X})\, }$ over a surface $\displaystyle{ S_B\, }$ 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:

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

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

## Infinite domain potential flow solutions

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

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

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

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

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

with

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

### Example

Uniform flow past $\displaystyle{ S_B\, }$:

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

So the RHS of the Green equation becomes:

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

.