Solution of Wave-Body Flows, Green's Theorem

Solution of wave-body interaction problems

• 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 & 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]\displaystyle{ \phi\, }[/math] over the body boundary [math]\displaystyle{ S_B\, }[/math] for the proper choice of the Green function:

[math]\displaystyle{ \iint_S \left( \phi_1 \frac{\partial\phi_2}{\partial n} - \phi_2 \frac{\partial\phi_1}{\partial n} \right) dS = 0 \, }[/math]

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

Define the volume [math]\displaystyle{ V\, }[/math] and [math]\displaystyle{ S\, }[/math] as follows:

The fluid volume [math]\displaystyle{ V\, }[/math] is enclosed by the union of several surfaces

[math]\displaystyle{ S \equiv S_B + S_F + S_\infty + S_H + S_E }[/math]

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

Define two velocity potentials [math]\displaystyle{ \phi_i(\vec{X})\, }[/math]:

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

Two types of Green functions will be used:

Rankine source: [math]\displaystyle{ \nabla_X^2 G = 0 \, }[/math]

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

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

• This Rankine source and its gradient with respect to [math]\displaystyle{ \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]\displaystyle{ U=0\, }[/math] wave Green function in the frequency domain.

• Satisfies the free surface condition and near [math]\displaystyle{ \vec\xi=0\, }[/math] behaves like a Rankine source:

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

[math]\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{du}{u-K} e^{u(Z+\zeta)} J_0(uR) }[/math]

where:

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

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

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

As [math]\displaystyle{ KR\to\infty\, }[/math]:

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

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

At [math]\displaystyle{ KR\to\infty\, }[/math]:

[math]\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) }[/math]

• Therefore the real velocity potential

[math]\displaystyle{ \mathbb{G} = \mathbb{R}\mathbf{e} \left\{ G e^{i\omega t} \right\} }[/math]

Represents outgoing ring waves of the form [math]\displaystyle{ \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]\displaystyle{ \phi_1(\vec{X})\equiv\phi(\vec{X})\, }[/math]

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

It follows that on [math]\displaystyle{ S_\infty\, }[/math]:

[math]\displaystyle{ \frac{\partial\phi_1}{\partial n} = \frac{\partial\phi_1}{\partial R} = - i K \phi_1 \, }[/math] [math]\displaystyle{ \frac{\partial\phi_2}{\partial n} = \frac{\partial\phi_2}{\partial R} = - i K \phi_2 \, }[/math]

Therefore:

[math]\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 }[/math]

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

On [math]\displaystyle{ S_F(Z=0)\, }[/math]:

[math]\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} }[/math] [math]\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 }[/math]

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

• Over [math]\displaystyle{ S_H\, }[/math], either [math]\displaystyle{ \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]\displaystyle{ \frac{\partial\phi_1}{\partial Z} \to 0, \ \frac{\partial\phi_2}{\partial Z}\to 0 \, }[/math] as [math]\displaystyle{ 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]\displaystyle{ S_\epsilon\, }[/math] and [math]\displaystyle{ S_B\, }[/math]. Start with [math]\displaystyle{ S_\epsilon\, }[/math]:

[math]\displaystyle{ I_\epsilon = \iint_{S_\epsilon} \left( \phi_1 \frac{\partial\phi_2}{\partial n} - \phi_2 \frac{\partial\phi_1}{\partial n} \right) dS }[/math]

or:

[math]\displaystyle{ I_\epsilon = \iint_{S_\epsilon} \left( \phi \frac{\partial G}{\partial n} - G \frac{\partial\phi}{\partial n} \right) dS_X }[/math]

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

Near [math]\displaystyle{ \vec{\xi}\, }[/math]:

[math]\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} }[/math] [math]\displaystyle{ \phi \to \phi(\vec{\xi}) = \phi(\vec{X}) \ \mbox{as} \ \epsilon \to 0, \vec{X} \to \vec{\xi} }[/math]

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

[math]\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}) }[/math]

In summary:

[math]\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] d S_n = 0 }[/math]

on [math]\displaystyle{ 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]\displaystyle{ \phi(\vec{\xi})\, }[/math] at some point in the fluid domain and its values [math]\displaystyle{ \phi_(\vec{X})\, }[/math] and normal derivatives over the body boundary:

[math]\displaystyle{ \phi(\vec{\xi}) + \iint_{S_B} \phi(\vec{X}) \frac{\partial G ( \vec{X}; \vec{\xi} )} {\partial n_X} d S_X = \iint_{S_B} G (\vec{X};\vec{\xi}) B(\vec{X}) d S_X }[/math]

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

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

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

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

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

where now both [math]\displaystyle{ \vec{X}\, }[/math] and [math]\displaystyle{ \vec{\xi}\, }[/math] lie no the body surface. This becomes an integral equation for [math]\displaystyle{ \phi(\vec{X})\, }[/math] over a surface [math]\displaystyle{ 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 as follows:

[math]\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}) }[/math]

where derivatives are taken w.r.t. the first argument for a point source centered at point [math]\displaystyle{ \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]\displaystyle{ G\, }[/math]:

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

The Rankine source as the Green function and using the property that as [math]\displaystyle{ R\to \infty\, }[/math]

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

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

with

[math]\displaystyle{ B(\vec{X} = \frac{\partial\phi}{\partial n}, \quad \mbox{on} \ S_B \, }[/math]

Example: uniform flow past [math]\displaystyle{ S_B\, }[/math]

[math]\displaystyle{ \Phi = U X + \phi_1, \ \frac{\partial\Phi}{\partial n} = 0, \ \mbox{on} \ S_B \, }[/math] [math]\displaystyle{ \Longrightarrow \frac{\partial\phi}{\partial n} = -\frac{\partial}{\partial n} \left(U_X\right) = - n_1 U \equiv V(\vec{X}) }[/math]

So the RHS of the Green equation becomes:

[math]\displaystyle{ RHS = \iint_{S_B} G(\vec{X};\vec{\xi}) \left(-Un_1X\right) d S_X \, }[/math]

.

Ocean Wave Interaction with Ships and Offshore Energy Systems