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{{Ocean Wave Interaction with Ships and Offshore Structures
 | chapter title = Solution of Wave-Body Flows, Green's Theorem
 | next chapter = [[Rankine Intergral Equations For Ship Flows]]
 | previous chapter = [[Seakeeping In Random Waves]]
}}

{{incomplete pages}}

==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 <math>\phi\,</math> over the body boundary <math>S_B\,</math> for the proper choice of the Green function:

<center><math> \iint_S \left( \phi_1 \frac{\partial\phi_2}{\partial n} - \phi_2 \frac{\partial\phi_1}{\partial n} \right) \mathrm{d}S = 0 \, </math></center>

For any <math>\phi_1, \ \phi_2\,</math> that solve the Laplace equation in a closed volume <math>V\,</math>.

Define the volume <math>V\,</math> and <math>S\,</math> as follows:

The fluid volume <math>V\,</math> is enclosed by the union of several surfaces

<center><math> S \equiv S_B + S_F + S_\infty + S_H + S_E </math></center>

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

Define two velocity potentials <math>\phi_i(\vec{x})\,</math>:

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

Two types of Green functions will be used:

<u>Rankine source</u>: <math> \nabla_x^2 G = 0 \, </math>

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

Note that the flux of fluid emitted from <math>\vec{\xi}\,</math> is equal to <math>1\,</math>.

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

===Havelock's wave source potential===

...Also known as the <math>U=0\,</math> wave Green function in the frequency domain.

Satisfies the free surface condition and near <math>\vec\xi=0\,</math> behaves like a Rankine source:

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

<center><math> 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) </math></center>

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

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

<center><math> \frac{\partial\phi_2}{\partial Z} - K \phi_2 = 0, \quad Z=0 </math></center>

<center><math> \phi_2 \sim - \frac{1}{4\pi} r^{-1}, \quad \vec{X} \to \vec{\xi} </math></center>

As <math> KR\to\infty\,</math>:

<center><math> G \sim - \frac{i}{2} K e^{K(Z+\zeta)} H_0^{(2)}(KR) </math></center>

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

At <math> KR\to\infty\,</math>:

<center><math> 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) </math></center>

Therefore the real velocity potential

<center><math> \mathbb{G} = \mathrm{Re} \left\{ G e^{i\omega t} \right\} </math></center>

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

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

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

It follows that on <math>S_\infty\,</math>:

<center><math> \frac{\partial\phi_1}{\partial n} = \frac{\partial\phi_1}{\partial R} = - i K \phi_1 \, </math></center>
<center><math> \frac{\partial\phi_2}{\partial n} = \frac{\partial\phi_2}{\partial R} = - i K \phi_2 \, </math></center>

Therefore:

<center><math> \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 </math></center>

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

On <math>S_F(Z=0)\,</math>:

<center><math> \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} </math></center>
<center><math> \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 </math></center>

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

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

There remains to interpret and evaluate the integral over <math> S_\epsilon\,</math> and <math>S_B\,</math>. Start with <math> S_\epsilon\,</math>:

<center><math> 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 </math></center>

or:

<center><math> I_\epsilon = \iint_{S_\epsilon} \left( \phi \frac{\partial G}{\partial n} - G \frac{\partial\phi}{\partial n} \right) \mathrm{d}S_X </math></center>

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

Near <math>\vec{\xi}\,</math>:

<center><math> 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} </math></center>
<center><math> \phi \to \phi(\vec{\xi}) = \phi(\vec{X}) \ \mbox{as} \ \epsilon \to 0, \vec{X} \to \vec{\xi} </math></center>

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

<center><math> 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}) </math></center>

In summary:

<center><math> \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 </math></center>

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

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

<center><math> \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 </math></center>

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

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

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

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

<center><math> \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 </math></center>

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

<center><math> \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}) </math></center>

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

==Infinite domain potential flow solutions==

In the absence of the free surface, the derivation of the Green integral equation remains almost unchanged using <math>G\,</math>:

<center><math> \phi_2(\vec{X}) = - \frac{1}{4\pi} \left| \vec{X} - \vec{\xi} \right|^{-1} \equiv G(\vec{X}; \vec{\xi}) </math></center>

The Rankine source as the Green function and using the property that as <math> R\to \infty\,</math>

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

<center><math> \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 </math></center>

with

<center><math> V(\vec{X}) = \frac{\partial\phi}{\partial n}, \quad \mbox{on} \ S_B \, </math></center>

===Example===
Uniform flow past <math> S_B\,</math>:
<center><math> \Phi = U X + \phi, \ \frac{\partial\Phi}{\partial n} = 0, \ \mbox{on} \ S_B \, </math></center>
<center><math> \Longrightarrow \frac{\partial\phi}{\partial n} = -\frac{\partial}{\partial n} \left(U X\right) = - n_1 U \equiv V(\vec{X}) </math></center>

So the RHS of the Green equation becomes:

<center><math> RHS = \iint_{S_B} G(\vec{X};\vec{\xi}) \left(-Un_1\right) \mathrm{d} S_X \, </math></center>.


[[Ocean Wave Interaction with Ships and Offshore Energy Systems]]