# Forward-Speed Ship Wave Flows

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## Linear free surface conditions

The linear free surface conditions are as follows,

$\displaystyle{ \frac{\mathrm{d}^2\Phi}{\mathrm{d}t^2} + g \frac{\mathrm{d}\Phi}{\mathrm{d}z} = 0, \quad z=0 }$

We can calculate the second derivative above as

$\displaystyle{ \frac{\mathrm{d}^2\Phi}{\mathrm{d}t^2} = \frac{\partial^2 \bar{\Phi}}{\partial t^2} - 2 U \frac{\partial^2\bar{\Phi}}{\partial x \partial t} + U^2 \frac{\partial \bar{\Phi}}{\partial x^2} }$

When no ambient waves are present $\displaystyle{ \frac{\partial\bar{\Phi}}{\partial t} = 0 \, }$ and we obtain the free surface condition for the steady Kelvin ship wave problem

$\displaystyle{ U^2 \frac{\partial^2 \bar{\Phi}}{\partial x^2} + g \frac{\partial\bar{\Phi}}{\partial z} = 0, \qquad z=0 }$

This is the famous Neumann-Kelvin free-surface condition governing the linear steady wave pattern generated by a translating ship.

Bernoulli equation:

$\displaystyle{ P = - \rho \left( \frac{\mathrm{d}\Phi}{\mathrm{d}t} + \frac{1}{2} \nabla\Phi \cdot \nabla\Phi + gz \right) = - \rho \left[ \left( \frac{\partial\bar{\Phi}}{\partial t} - U \frac{\partial\bar{\Phi}}{\partial x} \right) + \frac{1}{2} \nabla \phi \cdot \nabla \phi + g z \right] }$

Free surface elevation:

$\displaystyle{ \zeta = - \left. \frac{1}{g} \frac{\mathrm{d}\Phi}{\mathrm{d}t} \right|_{z=0} = -\frac{1}{g} \left. \left( \frac{\partial\bar{\Phi}}{\partial t} - U \frac{\partial\bar{\Phi}}{\partial x} \right)\right|_{z=0} }$

In the Kelvin ship wave problem

$\displaystyle{ \frac{\partial\phi}{\partial t} = 0 }$

Hence

$\displaystyle{ \zeta = \frac{U}{g} \frac{\partial\bar{\Phi}}{\partial x}, \qquad z=0 }$

## Neumann-Kelvin linearization of $\displaystyle{ U\gt 0 }$ ship seakeeping problem

Let $\displaystyle{ \Phi(X_0,Y_0,Z_0,t) \, }$be the total potential relative to the inertial coordinate system

$\displaystyle{ X_0 = x + U t \, }$

Let $\displaystyle{ \Phi( X,Y,Z,t) \, }$ be the same potential expressed relative to the translating frame. It was shown before that

$\displaystyle{ \frac{\mathrm{d}\Phi}{\mathrm{d}t} = \left( \frac{\partial}{\partial t} - U \frac{\partial}{\partial X} \right) \phi (\vec{X}, t) \, }$

where now the time dependence of $\displaystyle{ \phi\, }$ w.r.t. time will be the $\displaystyle{ e^{i\omega t} \, }$ form in the ship seakeeping problem.

The total potential $\displaystyle{ \Phi\, }$ consists of the sum of two components in a linearized setting

$\displaystyle{ \mathbf{\Phi}_{TOT} = \mathbf{\bar{\Phi}} + \mathbf{\Phi} \, }$

where $\displaystyle{ \mathbf{\bar{\Phi}}\, }$ is the velocity potential due to the vessel forward translation with constant speed $\displaystyle{ U\, }$ and $\displaystyle{ \Phi\, }$ is the seakeeping component due to vessel motions in waves.

Relative to the ship frame:

$\displaystyle{ \mathbf{\bar{\Phi}} = \bar{\Phi} (X,Y,Z) \, }$
$\displaystyle{ \mathbf{\Phi} = \Phi(X,Y,Z,t) = \mathbf{Re} \left\{ \phi(X,Y,Z)e^{i\omega t} \right\} \, }$

where $\displaystyle{ \omega\, }$ is the encounter frequency and

$\displaystyle{ \phi = \phi_I + \sum_{j=1}^7 \phi_j \, }$

with $\displaystyle{ \phi_j, \ j=1,\cdots,6\, }$ being the radiation and $\displaystyle{ \phi_7\, }$ being the diffraction potentials.

## Boundary-value problem for $\displaystyle{ \bar{\Phi} }$

Free surface condition:

$\displaystyle{ U^2 \bar{\Phi}_{XX} + g \bar{\Phi}_Z = 0, \quad Z=0 }$

Ship-hull condition

$\displaystyle{ \vec{n} \cdot \nabla\vec{\Phi} = \vec{n} \cdot \vec{U} = U n_1 \, }$

where $\displaystyle{ \vec{n} = ( n_1, n_2, n_3 ) \, }$ is the unit vector pointing inside the ship hull.

Far from the ship $\displaystyle{ \bar{\Phi}\, }$ represents outgoing waves which are known as the Kelvin ship wake studied earlier.

The solution for $\displaystyle{ \bar{\Phi}\, }$ by the above formulation known as the Neumann-Kelvin problem and its generalizations discussed in the literature is carried out by panel methods.

The principal output quantities of interest in practice are:

• Free-surface elevation
$\displaystyle{ \zeta = - \frac{1}{g} \left. \frac{\mathrm{d}\bar{\Phi}}{\mathrm{d}t} \right|_{\mbox{Inertial frame}} = \frac{U}{g} \left. \bar{\Phi}_X \right|_{\mbox{Ship frame}}, \quad Z=0 }$
• Hydrodynamic pressure (linear)
$\displaystyle{ P = - \rho \left. \frac{\mathrm{d}\bar{\mathbf{\Phi}}}{\mathrm{d}t} \right|_{\mbox{Inertial frame}} = \rho U \left. \bar{\Phi}_X \right|_{\mbox{Ship frame}} }$
• Hydrodynamic pressure (total)
$\displaystyle{ P_T = \rho \left( \frac{\mathrm{d}\bar{\mathbf{\Phi}}}{\mathrm{d}t} = \frac{1}{2} \nabla \bar{\mathbf{\Phi}} \cdot \nabla \bar{\mathbf{\Phi}} + g Z \right)_{\mbox{Inertial frame}} }$
$\displaystyle{ = - \rho \left( - U \bar{\Phi_X} + \frac{1}{2} \nabla \bar{\Phi} \cdot \nabla \bar{\Phi} + g Z \right)_{\mbox{Ship frame}} \, }$

If $\displaystyle{ \bar{S_w}\, }$ is the ship wetted surface due to its steady forward translation on a free surface and $\displaystyle{ \bar{n}\, }$ is the unit normal vector pointing out of the fluid domain the ship ideal-fluid force is given by

$\displaystyle{ \overrightarrow{F} = \iint_{\overline{S_W}} P_T \vec{n} \mathrm{d}S \, }$

The wave resistance is: $\displaystyle{ R_W = \vec{i} \cdot \vec{F} \, }$.

We will derive boundary value problems for the potentials $\displaystyle{ \bar{\Phi}\, }$ and $\displaystyle{ \Phi\, }$ relative to the ship fixed frame.

The principal assumption underlying the ensuing derivation is that the ship is slender, thin or flat or in general streamlined in the longitudinal direction. More explicitly, if $\displaystyle{ B\, }$ is the ship beam, $\displaystyle{ T\, }$ its draft and $\displaystyle{ L\, }$ its length we will assume that:

$\displaystyle{ \frac{B}{L}, \quad \frac{T}{L} = O (\varepsilon), \quad \varepsilon \ll 1 \, }$

where $\displaystyle{ \varepsilon\, }$ is the slenderness parameter assumed to be small compared to $\displaystyle{ 1 \, }$.

The ship slenderness justifies the use of the linear free-surface condition in the forward-speed problem for a broad range of speeds and hull shapes.

## Boundary-value problem for time-Harmonic velocity potential

From the Galilean transformation:

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

Relative to the ship frame. In terms of the complex potential:

$\displaystyle{ \Phi = \mathbf{Re} \left\{ \phi e^{i\omega t} \right\} \, }$
$\displaystyle{ \left( i\omega - U \frac{\partial}{\partial X} \right)^2 \phi + g \phi_Z = 0 , \quad Z=0 \, }$

where $\displaystyle{ \omega\, }$ is the encounter frequency and $\displaystyle{ \phi\, }$ is any of the $\displaystyle{ \phi_j\, }$ potentials.

• The above time harmonic Neumann-Kelvin free surface condition is being treated by state-of-the-art panel methods. An important simplification for slender ships and large values of $\displaystyle{ \omega\, }$ will lead to the popular strip theory.
• The solution for $\displaystyle{ \bar{\Phi}\, }$ is far from simple numerically. A lot of research has been devoted to this effort, in particular towards the evaluation of the ship Kelvin wake and the ship wave resistance.
• The linearization of the pressure and vessel wetted surface $\displaystyle{ \overline{S_W}\, }$ about its static value in calm water must be carried out carefully! Nonlinear effects are known to contribute appreciably to the wave resistance.
• If available, a fully nonlinear solution of the forward-speed steady ship wave problem is preferable. Numerical issues must be carefully treated and are the subject of state-of-the-art research.
• Coupling with viscous effects is often strong and important for predicting the total resistance of the ship.

Relative to the ship-fixed coordinate system the ambient wave elevation oscillates with frequency $\displaystyle{ \omega\, }$.

Proof:

$\displaystyle{ \zeta = - \frac{1}{g} \left( \frac{\partial}{\partial t} - U \frac{\partial}{\partial X} \right) \Phi_I }$

where:

$\displaystyle{ \Phi_I = \mathbf{Re} \left\{ \frac{i g A}{\omega_0} e^{KZ-iKX\cos\beta-iKY\sin\beta+i\omega t} \right\} }$
$\displaystyle{ \frac{\partial}{\partial t} = i \omega, \qquad \frac{\partial}{\partial X} = - i K \cos \beta }$
$\displaystyle{ \frac{\partial}{\partial t} - U \frac{\partial}{\partial X} = i \left( \omega + K U \cos \beta \right) = i \omega_0 \, }$

Thus:

$\displaystyle{ \zeta = \mathbf{Re} \left\{ A e^{-iKX\cos\beta-iKY\sin\beta+i\omega t} \right\} \, }$
$\displaystyle{ = \mathbf{Re} \left\{ A e^{i\omega t} \right\}_{X=Y=0} }$

where $\displaystyle{ \Pi_j(\omega) \, }$ is the complex amplitude of the vessel motion in mode $\displaystyle{ -j\, }$, a function of the frequency of encounter $\displaystyle{ \omega\, }$, known as the response amplitude operator (RAO).

The ship equations of motion follow as in the $\displaystyle{ U=0\, }$ case using linear system theory:

$\displaystyle{ \sum_{j=1}^6 \left[ - \omega^2 \left(M_{ij} + A_{ij} \right) + i\omega B_{ij} + C_{ij} \right] \Pi_j (\omega) = \mathbf{X}_i(\omega),\quad i=1,\cdots,6 }$

where the hydrodynamic coefficients $\displaystyle{ A_{ij}(\omega), \ B_{ij}(\omega) \, }$ and exciting forces are now functions of the encounter frequency $\displaystyle{ \omega\, }$ and other forward-speed effects.

• Inertia & hydrostatic matrices the same as in the zero-speed case
• Will derive BVP's governing the coefficients $\displaystyle{ A_{ij}(\omega), \ B_{ij}(\omega) \, }$ and exciting forces $\displaystyle{ \mathbf{X}_i (\omega) \, }$.

Explicitly:

$\displaystyle{ \omega = \omega_0 - U \frac{\omega_0^2}{g} \cos\beta \, }$
• $\displaystyle{ \omega \gt \ \mbox{or} \ \lt 0 \, }$: Both positive and negative values of $\displaystyle{ \omega\, }$ are possible. In practice will always deal with the absolute value of $\displaystyle{ \omega\, }$.
• Given the absolute wave frequency $\displaystyle{ \omega_0 \gt 0 \, }$ there exists a unique $\displaystyle{ \omega\, }$.
• Given a positive absolute encounter frequency $\displaystyle{ |\omega|\, }$, there exist possibly multiple $\displaystyle{ \omega_0\, }$'s satisfying the above relation. More discussion of this topic will follow.
• Assuming small amplitude motions the ship responses are modeled after linear system theory, input signal $\displaystyle{ \sim e^{i\omega t} \ \longrightarrow\, }$ output signal $\displaystyle{ \sim e^{i\omega t} \, }$.

Relative to the earth-fixed frame the ambient wave velocity potential takes the form:

$\displaystyle{ \Phi_I = \mathbf{Re} \left\{ \phi_I \right\} \, }$
$\displaystyle{ \phi_I = \frac{i g A}{\omega_0} e^{KZ_0 - iKX_0 \cos\beta - iKY_0 \sin\beta +i\omega_0 t} \, }$

where in deep water: $\displaystyle{ K = \frac{\omega_0^2}{g} \, }$

Let:

$\displaystyle{ X_0 = x+ Ut \qquad Y_0 = y \qquad Z_0 = z \, }$

It follows that:

$\displaystyle{ \phi_I = \frac{i g A}{\omega_0} e^{ Kz - iKx\cos\beta - iKy\sin\beta + i (\omega_0 -UK \cos\beta) t} }$

Let:

$\displaystyle{ \omega = \omega_0 - UK\cos\beta \, }$

Be defined to be the encounter frequency which accounts for the Doppler effect included in the second term in the RHS.

## Comments on N-K formulation

• The ship is assumed to be streamlined in order to justify the decomposition of the steady & time harmonic components.
• The vessel motions are assumed small and of the same order as the ambient wave amplitude. Terms omitted are of $\displaystyle{ O\left(A^2\right)\, }$.
• When Taylor expanding the free-surface and body-boundary condition about $\displaystyle{ Z=0\, }$ and $\displaystyle{ \overrightarrow{S_B}\, }$ respectively, the steady flow potential $\displaystyle{ \bar{\Phi}\simeq 0 \, }$.
• For ships with appreciable thickness a better approximation for $\displaystyle{ \bar\Phi\, }$ is that of the double-body flow disturbance such that $\displaystyle{ \overrightarrow{\Phi_Z} = 0\, }$ on $\displaystyle{ Z=0\, }$ and $\displaystyle{ \overrightarrow{\Phi_n} = U n\, }$, on $\displaystyle{ \overrightarrow{S_B}\, }$, This leads to the state-of-the-art linear 3D steady flow and seakeeping formulation discussed later in connection with panel methods.
• The N-K formulation is the staring point of strip theory.