# Rankine Intergral Equations For Ship Flows

Wave and Wave Body Interactions
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Rankine integral equations for ship flow problems with forward speed

• The Green integral equation extends easily to flows past ships in calm water and in waves when the free surface condition is more complex than that of the $\displaystyle{ U=0\, }$ frequency domain problem.
• Neumann-Kelvin problem in time domain

Consider a vessel which starts from rest at $\displaystyle{ t=0\, }$ and translates forward with constant velocity $\displaystyle{ U\, }$ and also possibly oscillating with amplitudes $\displaystyle{ \xi_i(t)\, }$ if ambient waves are present.

It was shown earlier that the simplest forward speed free surface condition for the forward problem takes the form:

$\displaystyle{ \begin{cases} \left( \frac{\partial}{\partial t} - U \frac{\partial}{\partial X} \right)^2 \phi + g \frac{\partial\phi}{\partial Z} = 0, \quad Z=0 \\ \frac{\partial\phi}{\partial n} = V, \quad \mbox{on} \ \bar{S}_B \end{cases} }$

Relative to the ship frame. The normal velocity $\displaystyle{ V\, }$ on $\displaystyle{ \bar{S}_B\, }$ can be of three forms:

$\displaystyle{ V(\vec{X}) = \begin{cases} U n_1, \quad t\gt 0: \ \mbox{forward translation} \\ n_i \xi_i (t), \quad t\gt 0: \ \mbox{radiation} \\ - \frac{\partial\phi_I}{\partial n}, \quad t\gt 0: \ \mbox{diffraction} \end{cases} }$
• More general free-surface conditions with space dependent coefficients arising from gradients of the double body flow exist and are described in the literature the steps in deriving the relevant integral equations are very similar to the ones that follows:
• Wave Green functions that satisfy analytically the time-domain free surface condition stated above exist and are derived in W&L. Their evaluation is how ever time-consuming and they apply only to the Neumann-Kelvin formulation.
• Proceeding with the derivation of the Green integral equation as above and using the Rankine source as the Green function:
$\displaystyle{ \phi_2(\vec{X}) = - \frac{1}{4\pi} \left| \vec{X} - \vec{\xi} \right|^{-1} \equiv G(\vec{X};\vec{\xi}) \, }$

We obtain:

$\displaystyle{ \frac{1}{2} \phi (\vec{\xi}) + \iint_{S_B} \phi(\vec{X}) \frac{\partial G(\vec{X};\vec{\xi})}{\partial n_X} \mathrm{d}S_X + \iint_{S_F(Z=0)} \left[ \phi( \vec{X} \frac{\partial G}{\partial Z} (\vec{X}; \vec{\xi}) - G(\vec{X};\vec{\xi}) \frac{\partial\phi}{\partial Z} \right] \mathrm{d}X \mathrm{d}Y }$
$\displaystyle{ = \iint_{S_B} G(\vec{X};\vec{\xi}) B(\vec{X}) \mathrm{d} S_X \, }$
• Note that the integral over the free surface (Z=0) does not vanish since we have not use the relevant wave Green function.
• Otherwise the remaining integral over $\displaystyle{ S_B\, }$ retains its form. The integral over $\displaystyle{ S_\infty\, }$ can be shown to vanish. The proof if non-trivial and may be found in references.

Over $\displaystyle{ Z=0\, }$, it follows from the free-surface condition:

$\displaystyle{ \frac{\partial\phi}{\partial Z} = - \frac{1}{g} \left( \frac{\partial}{\partial t} - U \frac{\partial}{\partial X} \right) ^2 \phi, \quad Z=0 }$

Upon substitution, the second integral over $\displaystyle{ S_F\, }$ becomes:

$\displaystyle{ I_F = \iint_{Z=0} \left[ \phi(\vec{X}) \frac{\partial G(\vec{G};\vec{\xi})}{\partial Z} + \frac{1}{g} G(\vec{X};\vec{\xi}) \left( \frac{\partial}{\partial t} - U \frac{\partial}{\partial X} \right)^2 \phi(\vec{X}) \right] \mathrm{d}S }$

It follows that over $\displaystyle{ Z=0\, }$, only values and tangential gradients of $\displaystyle{ \phi(\vec{X})\, }$ are now present leading to an integro-differential equation:

$\displaystyle{ \frac{1}{2} \phi(\vec{\xi}) + \iint_{S_B} \phi(\vec{X}) \frac{\partial G(\vec{X};\vec{\xi})}{\partial n_X} \mathrm{d}S_X }$
$\displaystyle{ + \iint_{S_F(Z=0)} \left[ \phi(\vec{X}) \frac{\partial G(\vec{X};{\vec{\xi}})}{\partial Z} + \frac{1}{g} G(\vec{X};\vec{\xi}) \left( \frac{\partial}{\partial t}- U \frac{\partial}{\partial X} \right)^2 \phi(\vec{X}) \right] \mathrm{d}X \mathrm{d}Y = \iint_{S_B} V(\vec{X}) G(\vec{X};\vec{\xi}) \mathrm{d}S_X \, }$
• Unknown is $\displaystyle{ \phi(\vec{X})\, }$ over $\displaystyle{ S_B\, }$ & $\displaystyle{ S_F\, }$. Its X-derivatives may be approximated by carefully selected numerical differentiation schemes forming a core part of Rankine panel methods, discussed below.
• The integral over the infinite free surface $\displaystyle{ S_F (Z=0) \, }$ is truncated at some finite distance from the ship as drawn below
• A domain denoted by the shaded area is also introduced defined as the "beach". This is located as the outer boundary of $\displaystyle{ S_F\, }$ and selected so that over its surface the following free surface condition is enforced:
$\displaystyle{ \left( \frac{\partial}{\partial t} - U \frac{\partial}{\partial X} \right)^2 \phi + g \frac{\partial\phi}{\partial Z} + 2 V \left( \frac{\partial}{\partial t} - U \frac{\partial}{\partial X} \right) \phi + V^2 \phi = 0, \quad Z=0 }$
• This condition differs from the Neumann-Kelvin condition by the addition of the terms that are multiplied by the dissipative parameter $\displaystyle{ V(\vec{X})\, }$ which varies from $\displaystyle{ V=0\, }$ at the inner boundary to a finite value at the outer boundary of the beach.
• It can be shown that converting from the time to the frequency domain via $\displaystyle{ \frac{\partial}{\partial t} \to i\omega, V \, }$ is the familiar Rayleigh viscosity that plays a key role in the enforcement of the radiation conditions (See W&L).