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Strip Theory Of Ship Motions. Heave & Pitch

2010-09-10T16:05:59Z

F. Javier Arcos Zamora: /* Diffraction problem */

{{Ocean Wave Interaction with Ships and Offshore Structures
| chapter title = Strip Theory Of Ship Motions. Heave & Pitch
| next chapter = [[Ship Roll-Sway-Yaw Motions]]
| previous chapter = [[Forward-Speed Ship Wave Flows]]
}}

{{incomplete pages}}

== Ship motions with forward speed ==

Ship advances in the positive x-direction with constant speed <math>U\,</math>. Regular waves with absolute frequency <math>\omega_0\,</math> and direction <math>\beta\,</math> are incident upon the ship. The ship undergoes oscillatory motions in all six degrees of freedom <math> \xi_j(t), \quad j=1,\cdots,6\,</math>. The ship seakeeping problem will be treated in the frequency domain under the assumption of linearity.

==Ship-hull boundary condition==

The relevant rigid body velocities for a ship translating with a constant forward velocity and heaving and pitching in head waves are:

<center><math> \vec{U}=U\vec{i}; \quad \mbox{Forward speed} \,</math></center>

<center><math> \vec{\dot{\xi}}_3(t)=\dot{\xi}_3(t)\vec{k}; \quad \mbox{Heave velocity (positive up)} \,</math></center>

<center><math> \vec{\dot{\xi}}_5(t)=\dot{\xi}_5(t)\vec{j}; \quad \mbox{Pitch angular velocity (positive,bow down)} \,</math></center>

Let <math> \vec{n}(t)\,</math> the time-dependent unit normal vector to the instantaneous position of the ship hull. Let <math> \vec{n_0}\,</math> be its value when the ship is at rest.

* The body boundary condition will be derived for small heave & pitch motions and will be stated on the mean position of the ship at rest.

The nonlinear boundary condition on the exact position of the ship hull is:

<center><math> \frac{\partial\Phi}{\partial n}=\overrightarrow{V} \cdot \vec{n} \,</math></center>

where,

<center><math> \Phi=\Phi_3 + \Phi_5 \,</math></center>

<center><math> \overrightarrow{V} \cong U \vec{i} + \vec{K} \left( \dot{\xi}_3 - X \dot{\xi}_5 \right) \,</math></center>

* The unsteady velocity potential <math> \Phi\,</math> has been written as the linear superposition of the heave and pitch components. The total rigid-body velocity is the sum of vertical velocity due to the ship heave and pitch.

* The normal vector<math> \vec{n}(t)\,</math> is defined by

<center><math> \vec{n}(t) = \vec{n}_0 + \vec{\xi}_5 \times \vec{n}_0 \,</math></center>

where <math> \vec{\xi}_5(t)\,</math> is the ship pitch rotation angle, and:

<center><math> \vec{n}_0 \equiv (n_1,n_2,n_3) \,</math></center>

Keeping only the unsteady components:

<center><math> \vec{V} \cdot \vec{n} = U \xi_5 n_3 + \left( \dot{\xi}_3 - X \dot{\xi_5} \right) n_3 = \frac{\partial\Phi}{\partial n} </math></center>

* Note that the steady component <math> U n_1 \,</math> has already been accounted for in the statement of the steady flow. Let

<center><math> \Phi=\mathrm{Re}\left(\phi e^{i\omega t} \right), \quad \phi=\Pi_3 X_3 + \Pi_5 X_5 </math></center>

where <math> \phi_3\,</math> and <math> \phi_5\,</math> are the unit-amplitude heave & pitch complex velocity potentials in time-harmonic flow. Also:

<center><math> \xi_3(t) = \mathrm{Re} \left\{ \Pi_3(\omega) e^{i\omega t} \right\}, \quad \xi_5(t) = \mathrm{Re} \left\{ \Pi_5(\omega) e^{i\omega t} \right\} </math></center>

It follows that:

<center><math> \frac{\partial X_3}{\partial n} = i \omega n_3; \quad \mbox{on} \ \bar{S}_B \,</math></center>

<center><math> \frac{\partial X_5}{\partial n} = - i \omega X n_3 + U n_3; \quad \mbox{on} \ \bar{S}_B \,</math></center>

* Note the forward-speed effect in the pitch boundary condition and no such effect in heave.

* Diffraction problem: <math> \phi=\phi_7 \,</math>

<center><math> \frac{\partial\phi_7}{\partial n} = -\frac{\partial\phi_I}{\partial n}, \quad \frac{\partial}{\partial n} \equiv \vec{n} \cdot \nabla \,</math></center>

where:

<center><math> \phi_I = \frac{i g A}{\omega_0} e^{KZ-iKX\cos\beta - iKY\sin\beta} \,</math></center>

* Radiation problem: <math> \phi = \phi_3 + \phi_5 \,</math>

We will consider the special but important case of heave & pitch which are coupled and important modes to study in the ship seakeeping problem.

* Heave and pitch are the only modes of interest in head waves (<math> \beta = 180^\circ \,</math>) when all other modes of motion (Roll-Sway-Yaw) are identically zero for a ship symmetric port-starboard. Surge is nonzero but generally small for slender ships and in ambient waves of small steepness.

* Bernoulli equation

The linear hydrodynamic disturbance pressure due to unsteady flow disturbances is given relative to the ship frame:

<center><math> P = - \rho \left(\frac{\partial}{\partial t} - U \frac{\partial}{\partial X} \right) \Phi \,</math></center>

where <math>\Phi\,</math> is the respective real potential.

== Radiation problem ==

<center><math> \Phi = \mathrm{Re} \left\{ \phi e^{i\omega t} \right\} \,</math></center>

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

<center><math> \mathbb{P} = - \rho \left( i\omega - U \frac{\partial}{\partial X} \right) \left( \Pi_3 \phi_3 + \Pi_5 \phi_5 \right) </math></center>

where the complex velocity potentials satisfy the 2D boundary value problems derived earlier for slender ships.

* The hydrodynamic pressure will be integrated over the ship hull to obtain the added-mass and damping coefficients next:

<center><math> F_i(t) = \iint_{\bar{S}_B} P n_i \mathrm{d}S, \quad i = 3,5 \, </math></center>

The expressions derived below extend almost trivially to all other modes of motion. In general,

<center><math> n_i \equiv \vec{n} = \left(n_1,n_2,n_3\right);\quad i=1,2,3 </math></center>

<center><math> n_i \equiv \vec{X} \times \vec{n} = \left(n_4,n_5,n_6 \right); \quad i=4,5,6 </math></center>

So for heave:

<center><math> n_i = n_3\,</math></center>

And pitch:

<center><math> n_i = n_5 = - X n_3 + X n_1 \simeq - X n_3 \left(n_1 \ll n_3 \right) \,</math></center>

Expressing <math> F_i \,</math> in terms of the heave & pitch added mass & damping coefficients when the ship is forced to oscillate in calm water, we obtain when accounting for cross-coupling effects:

<center><math> F_i^H (t) = - \sum_{j=3,5} \left[ A_{ij} \frac{\mathrm{d}^2\xi_j}{\mathrm{d}t^2} + B_{ij} \frac{\mathrm{d}\xi_j}{\mathrm{d}t} \right] </math></center>

Hydrostatic restoring effects are understood to be added to <math> F_i^H\,</math>.

Introducing complex notation:

<center><math> \mathbb{F}_i^H(t) = \mathrm{Re} \left\{ \mathbb{F}_i e^{i\omega t} \right\} \, </math></center>

<center><math> \mathbb{F}_i(\omega) = \left[ \omega^2 A_{ij} (\omega) - i\omega B_{ij} (\omega) \right] \Pi_j </math></center>

where the summation notation over <math> i\,</math> is understood hereafter.

Introducing the definition of <math> \mathbb{F}_i\,</math> in terms of the hydrodynamic pressure we obtain after some simple algebra:

<center><math> \mathbb{P} = \mathbb{P}_j \Pi_j \equiv \mathbb{P}_3 \Pi_3 + \mathbb{P} \Pi_5 </math></center>

<center><math> \mathbb{F}_i = \left( \iint_{\bar{S_B}} \mathbb{P}_j n_j \mathrm{d}S \right) \Pi_j </math></center>

And:

<center><math> \omega^2 A_{ij} (\omega) - i\omega B_{ij}(\omega) = \iint_{\bar{S_B}} \mathbb{P}_i n_j \mathrm{d}S </math></center>

where from Bernoulli:

<center><math> \mathbb{P}_3 = \left( i\omega - U \frac{\partial}{\partial X} \right) \phi_3 = \left( i\omega - U \frac{\partial}{\partial X} \right) \left( i\omega X_3 \right) </math></center>

<center><math> \mathbb{P}_5 = \left( i\omega - U \frac{\partial}{\partial X} \right) \phi_5 = \left( i\omega - U \frac{\partial}{\partial X} \right) \left( -i\omega X_3 + U X_3 \right) </math></center>

== Strip theory ==

* Strip theory is a popular approximation of the 3-D Neumann-Kelvin formulation for ships which are slender as is most often the case when vessels are expected to cruise at significant forward speeds.

The principal assumption is:

<center><math> \frac{B}{L}, \ \frac{T}{L} = O(\varepsilon), \quad \varepsilon \ll 1 \, </math></center>

where

<center><math> B: \mbox{Ship maximum beam} \, </math></center>

<center><math> T: \mbox{Ship maximum draft} \, </math></center>

<center><math> L: \mbox{Ship water line length} \, </math></center>

* The principal assumption of strip theory is that certain components of the radiation and diffraction potentials vary slowly along the ship length leading to a simplification of the n-K formulation.

* In head or bow waves where heave and pitch attain their maximum values, the encounter frequency <math>\omega\,</math> is usually high.

=== Radiation problem ===

The ship is forced to oscillate in heave & pitch in calm water while advancing at a speed <math>U\,</math>.

* Due to slenderness the variation of the flow in the x-direction is more gradual than its variation around a ship section. So

<center><math> \frac{\partial}{\partial X} \Phi \ll \frac{\partial\Phi}{\partial Y}, \ \frac{\partial\Phi}{\partial Z} \, </math></center>

where <math> \Phi = \Phi_3 + \Phi_5 \,</math>. Thus the 3D Laplace equation simplifies into a 2D form for the heave & pitch potentials. (The same argument applies to Roll-Sway-Yaw). Thus

<center><math> \left( \frac{\partial^2}{\partial Y^2} + \frac{\partial^2}{\partial X^2} \right) \Phi_j \cong 0, \ \mbox{in fluid} \ j=2, \cdots, 6 </math></center>

* The 2D equation applies for each "strip" location at station-X.

* The ship-hull condition at station-X for the heave & pitch potentials remains the same:

<center><math> \Phi = \mathrm{Re} \left\{ \phi e^{i\omega t} \right\} \,</math></center>

<center><math> \phi = \Pi_3 X_3 + \Pi_5 X_5 \,</math></center>

<center><math> \frac{\partial X_3}{\partial n} = i \omega n_3; \quad \mbox{on} \ \bar{S_B} </math></center>

<center><math> \frac{\partial X_5}{\partial n} = - i \omega X n_3 + U n_3; \quad \mbox{on} \ \bar{S_B} </math></center>

where now:

<center><math> \left( \frac{\partial^2}{\partial Y^2} + \frac{\partial^2}{\partial Z^2} \right) X_j = 0, \quad \mbox{in fluid domain} \,</math></center>

* Define the normalized potential <math> \psi_3\,</math>:

<center><math> \frac{\partial \psi_3}{\partial n} = n_3; \quad \mbox{on} \ \bar{S_B} </math></center>

<center><math> \left( \frac{\partial^2}{\partial Y^2} + \frac{\partial^2}{\partial Z^2} \right) \psi_3 = 0, \quad \ \mbox{in fluid} </math></center>

* There remains to simplify the 3D n-K free-surface condition.

* The solution of the 2D BVP for <math> \psi_3\,</math> along 30 - 40 stations used to describe the hull form of most ships can be carried out very efficiently by standard 2D panel method.

* In terms of <math> \psi_3 (Y,Z; X) \,</math> the heave & pitch potentials follow in the form:

<center><math> \phi = \Pi_3 X_3 + \Pi_5 X_5 \,</math></center>

<center><math> X_3 = i \omega \psi_3 \,</math></center>

<center><math> X_5 = ( - i \omega X + U ) \psi_3 \, </math></center>

<center><math> \Phi = \mathrm{Re} \left\{ \phi e^{i\omega t} \right\} = \Phi_3 +\Phi_5 </math></center>

* The 2D heave added-mass and damping coefficients due to a section oscillation vertically are defined by the familiar expressions

<center><math> \mbox{Station-X}: \quad a_{33} (\omega) - \frac{i}{\omega} b_{33} (\omega) = \rho \int_{C_(X)} \psi_3 n_3 \mathrm{d}l </math></center>

where <math> \omega = \left| \omega_0 - U \frac{\omega_0^2}{g} \cos\beta \right| \,</math> is the encounter frequency.

<center><math> \left( i\omega - U \frac{\partial}{\partial X} \right)^2 \phi + g \phi_Z = 0, \quad Z=0 </math></center>

* The ship slenderness and the claim that <math>\omega\,</math> is usually large in head or bow waves is used to simplify the above equation as follows

<center><math> - \omega^2 \phi + g \phi_Z = 0, \quad Z=0 \,</math></center>

by assuming that <math> \omega \gg \left|U \frac{\partial}{\partial X} \right| </math>. A formal proof is lengthy and technical.

It follows that the normalized potential <math> \psi_3\,</math> also satisfies the above 2D FS condition and is thus the solution of the 2D boundary value problem stated below at station-X:

<center><math> \mbox{As} \ Y \to \infty; \quad \psi_3 \sim \frac{i g A \pm}{\omega} e^{KZ\mp iKy+i\omega t} \,</math></center>.

Upon integration along the ship length and over each cross section at station-X the 3D added-mass and damping coefficients for heave & pitch take the form:

<center><math> A_{33} = \int_{L} \mathrm{d}X a_{33} (X) \, </math></center>

<center><math> B_{33} = \int_{L} \mathrm{d}X b_{33} (X) \, </math></center>

<center><math> A_{35} = - \int_{L} \mathrm{d}X a_{33} - \frac{U}{\omega^2} B_{33} \, </math></center>

<center><math> A_{53} = - \int_{L} \mathrm{d}X a_{33} + \frac{U}{\omega^2} B_{33} \, </math></center>

<center><math> B_{35} = - \int_{L} \mathrm{d}X b_{33} + \frac{U}{\omega^2} A_{33} \, </math></center>

<center><math> B_{53} = - \int_{L} \mathrm{d}X b_{33} - \frac{U}{\omega^2} A_{33} \, </math></center>

<center><math> A_{55} = \int_{L} \mathrm{d}X X^2 a_{33} + \frac{U^2}{\omega^2} A_{33} \, </math></center>

<center><math> B_{35} = \int_{L} \mathrm{d}X X^2 b_{33} - \frac{U^2}{\omega^2} B_{33} \, </math></center>

where the 2D added-mass and damping coefficients were defined above:

<center><math> a_{33} - \frac{i}{\omega} b_{33} = \rho \int_{C(X)} \psi_3 n_3 \mathrm{d}l </math></center>

===Diffraction problem===

* We will consider heave & pitch in oblique waves. Note that in oblique waves the ship also undergoes Roll-Sway-Yaw motions which for a symmetric vessel and according to linear theory are decoupled from heave and pitch.

* Relative to the ship frame the total potential is:

<center><math> \Phi = \Phi_I + \Phi_D = \mathrm{Re} \left\{ \left( \phi_I + \phi_D \right) e^{i\omega t} \right\} </math></center>

where <math>\omega\,</math> is the encounter frequency, and:

<center><math> \phi_I = \frac{i g A}{\omega_0} e^{KZ-iKX\cos\beta-iKY\sin\beta} \quad K=\frac{\omega_0^2}{g} </math></center>

Define the diffraction potential as follows:

<center><math> \phi_0 = \frac{i g A}{\omega_0} e^{-iKX\cos\beta} \psi_7 (Y,Z;X) </math></center>

In words, factor-out the oscillatory variation <math> e^{-iKX\cos\beta} \,</math> out of the scattering potential.

* The ship slenderness approximation now justifies that:

<center><math> \frac{\partial\psi_7}{\partial X} \ll \frac{\partial\psi_7}{\partial Y}, \ \frac{\partial\psi_7}{\partial Z} \,</math></center>

Note that this is not an accurate approximation for <math> \phi_D\,</math> when <math>K=\frac{2\pi}{\lambda}\,</math> is a large quantity or when the ambient wavelength <math> \lambda\,</math> is small.

* Substituting in the 3D laplace equation and ignoring the <math> \frac{\partial\psi_7}{\partial X}, \ \frac{\partial^2\psi_7}{\partial X^2} \,</math> terms we obtain

<center><math> \left(\frac{\partial^2}{\partial Y^2} + \frac{\partial^2}{\partial Z^2} - K^2 \cos^2 \beta \right) \psi_7 \cong 0 \,</math></center>

* This is the modified 2D Helmholtz equation in most cases the <math> K^2\cos^\beta\,</math> term is not important for reasons to be discussed and the 2D Laplace equation follows.

* The body boundary condition simply states:

<center><math> \frac{\partial\phi_D}{\partial n} = - \frac{\partial\phi_I}{\partial n} \,</math></center>

with:

<center><math> \frac{\partial}{\partial n} \equiv n_1 \frac{\partial}{\partial X} + n_2 \frac{\partial}{\partial Y} + n_3 \frac{\partial}{\partial Z} </math></center>

Due to slenderness:

<center><math> n_1 \ll n_2, n_3 \,</math></center>

Invoking the slenderness approximations for the <math>\frac{\partial}{\partial X} \,</math> derivatives of <math> \psi_7\,</math>, the body boundary condition simplifies to:

<center><math> \frac{\partial\psi_7}{\partial N} = - \frac{\partial}{\partial N} \left( e^{KZ-iKY\sin\beta} \right); \ \mbox{on} \ C(X) </math></center>

with:

<center><math> \frac{\partial}{\partial N} = N_2 \frac{\partial}{\partial Y} + N_3 \frac{\partial}{\partial Z} </math></center>

The potential <math> \psi_7\,</math> satisfies the 2D Laplace equation, after dropping the <math> K^2 \cos^2\beta \,</math> term, or

<center><math> \left(\frac{\partial^2}{\partial Y^2} + \frac{\partial^2}{\partial Z^2} \right) \psi_7 = 0 \, </math></center>

And the free-surface condition derived next:

The 3D linear free surface condition for <math> \phi_D\,</math> is:

<center><math> \left( i\omega - U \frac{\partial}{\partial X} \right)^2 \phi_D + g \frac{\partial \phi_D}{\partial Z} = 0; \quad Z=0 </math></center>

The same condition is satisfied by <math> \phi_I\,</math>:

<center><math> \left( i\omega - U \frac{\partial}{\partial X} \right)^2 \phi_I + g \frac{\partial\phi_I}{\partial Z} = 0; \quad Z=0 </math></center>

Verify identically for <math> \phi_I\,</math> that:

<center><math> i \omega - U \frac{\partial}{\partial X} \equiv i \omega_0 \, </math></center>

So the forward-speed effect disappears from the free surface condition of <math> \phi_I\,</math>. The same will be shown to be approximately true for the diffraction potential <math> \phi_D\,</math>:

<center><math> \phi_D = \frac{i g A}{\omega_0} e^{- i K X \cos \beta} \psi_7(Y,Z; X) </math></center>

<center><math> \frac{\partial\phi_D}{\partial X} = \frac{i g A}{\omega_0} \left\{ (- i K \cos \beta) \psi_7 (Y,Z; X) + \frac{\partial \psi_7}{\partial X} \right\} e^{-i K X \cos \beta} </math></center>

Due to slenderness:

<center><math> \left| \frac{\partial\psi_7}{\partial X} \right| \ll |K\cos\beta| |\psi_7|,</math></center>

Therefore:

<center><math> \frac{\partial\phi_D}{\partial X} \simeq - i K \cos \beta \phi_D </math></center>

And from the free-surface condition

<center><math> \left( i\omega - U \frac{\partial}{\partial X} \right) \phi_D \simeq i \omega_0 \phi_D </math></center>

where:

<center><math> \omega = \omega_0 - U K \cos \beta \, </math></center>

So the free-surface condition for <math> \phi_D\,</math> becomes to leading order in slenderness:

<center><math> - \omega_0^2 \phi_D + g \frac{\partial\phi_D}{\partial Z} = 0, \quad Z=0 </math></center>

* So in the statement of the diffraction problem forward-speed effects are absent! The same is not true in the radiation problem even for slender ship.

* This "independence" of the diffraction problem on forward speed effects is verified very well in experiments and in fully three-dimensional solutions.

In summary, the boundary-value problem satisfied by the diffraction potential around a slender ship takes the form:

<center><math> \phi_D = \frac{ i g A}{\omega_0} e^{-iKX\cos\beta} \psi_7 (Y,Z;X) </math></center>

<center><math> \left( \frac{\partial^2}{\partial Y^2} + \frac{\partial^2}{\partial Z^2} \right) \psi_7 = 0, \quad \mbox{in fluid} \, </math></center>

<center><math> - \omega_0^2 \psi_7 +9 g \frac{\partial\psi_7}{\partial Z} = 0, \quad \mbox{on} \ Z=0 </math></center>

<center><math> \frac{\partial\psi_7}{\partial N} = - \frac{\partial}{\partial N} \left( e^{KZ-iKY\sin\beta} \right) </math></center>

Or in graphical form at station-X:

* As <math> Y \to \pm \infty \qquad \psi_7 \sim \frac{ i g A_7^\pm}{\omega_0} e^{KZ \mp i K Y} </math>.

The hydrodynamic pressure in the diffraction problem takes the form:

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

<center><math> \mathbb{P} = - \rho \left( i\omega-U\frac{\partial}{\partial X} \right) \frac{i g A}{\omega_0} e^{-iKX\cos\beta} \left( \psi_I + \psi_7 \right) </math></center>

<center><math> \cong \rho g A e^{-iKX\cos\beta} \left( \psi_I + \psi_7 \right) </math></center>

where <math> \frac{\partial\psi_7}{\partial X} \,</math> derivatives are dropped in favor of <math> K\cos\beta\,</math>, due to slenderness.

The heave and pitch exciting forces follow simply by pressure integration

<center><math> X_i(t) = \mathrm{Re} \left\{ \mathbb{X}_i e^{i\omega t} \right\}; \quad i=3,5 </math></center>

<center><math> \mathbb{X}_3 = \rho g A \int_L \mathrm{d}X e^{-iKX\cos\beta} \int_{C(X)} \left( \psi_I + \psi_7 \right) n_3 \mathrm{d}l </math></center>

<center><math> \mathbb{X}_5 = \rho g A \int_L \mathrm{d}X (-X) e^{-iKX\cos\beta} \int_{C(X)} \left( \psi_I + \psi_7 \right) n_3 \mathrm{d}l </math></center>

where the approximation <math> n_5 = -X n_3 \,</math> was used.

* In summary, a version of strip theory was derived for slenderships which has been found to be the most rational and accurate relative to other alternatives. This is particularly true for the treatment of the diffraction problem.

* The coupled heave & pitch equation of motion take the familiar form

<center><math> \left[ - \omega^2 \left( A_{ij} + M_{ij} \right) + i\omega B_{ij} + X_{ij} \right] \pi_j = \mathbb{X}_i \left(\omega_0\right) \quad i,j = 3,5 </math></center>

where in practice:

<center><math> \omega = \left| \omega_0 - U \frac{\omega_0^2}{g} \cos\beta \right| </math></center>

And the <math>A_{ij}, \ B_{ij} \, </math> are functions of <math> \omega\,</math> defined above as integrals of their 2D counterparts <math> a_{33}\,</math> & <math>b_{33}(\omega)\,</math>.

* The heave & pitch exciting force amplitudes <math> \mathbb{X}_i(\omega_0)\,</math> are functions of <math> \omega_0\,</math>, hence not dependent on <math> U\,</math>.

* All hydrodynamic effects oscillate at <math> \omega\,</math>!

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

Strip Theory Of Ship Motions. Heave & Pitch

2010-09-10T12:07:03Z

F. Javier Arcos Zamora: /* Radiation problem */

{{Ocean Wave Interaction with Ships and Offshore Structures
| chapter title = Strip Theory Of Ship Motions. Heave & Pitch
| next chapter = [[Ship Roll-Sway-Yaw Motions]]
| previous chapter = [[Forward-Speed Ship Wave Flows]]
}}

{{incomplete pages}}

== Ship motions with forward speed ==

Ship advances in the positive x-direction with constant speed <math>U\,</math>. Regular waves with absolute frequency <math>\omega_0\,</math> and direction <math>\beta\,</math> are incident upon the ship. The ship undergoes oscillatory motions in all six degrees of freedom <math> \xi_j(t), \quad j=1,\cdots,6\,</math>. The ship seakeeping problem will be treated in the frequency domain under the assumption of linearity.

==Ship-hull boundary condition==

The relevant rigid body velocities for a ship translating with a constant forward velocity and heaving and pitching in head waves are:

<center><math> \vec{U}=U\vec{i}; \quad \mbox{Forward speed} \,</math></center>

<center><math> \vec{\dot{\xi}}_3(t)=\dot{\xi}_3(t)\vec{k}; \quad \mbox{Heave velocity (positive up)} \,</math></center>

<center><math> \vec{\dot{\xi}}_5(t)=\dot{\xi}_5(t)\vec{j}; \quad \mbox{Pitch angular velocity (positive,bow down)} \,</math></center>

Let <math> \vec{n}(t)\,</math> the time-dependent unit normal vector to the instantaneous position of the ship hull. Let <math> \vec{n_0}\,</math> be its value when the ship is at rest.

* The body boundary condition will be derived for small heave & pitch motions and will be stated on the mean position of the ship at rest.

The nonlinear boundary condition on the exact position of the ship hull is:

<center><math> \frac{\partial\Phi}{\partial n}=\overrightarrow{V} \cdot \vec{n} \,</math></center>

where,

<center><math> \Phi=\Phi_3 + \Phi_5 \,</math></center>

<center><math> \overrightarrow{V} \cong U \vec{i} + \vec{K} \left( \dot{\xi}_3 - X \dot{\xi}_5 \right) \,</math></center>

* The unsteady velocity potential <math> \Phi\,</math> has been written as the linear superposition of the heave and pitch components. The total rigid-body velocity is the sum of vertical velocity due to the ship heave and pitch.

* The normal vector<math> \vec{n}(t)\,</math> is defined by

<center><math> \vec{n}(t) = \vec{n}_0 + \vec{\xi}_5 \times \vec{n}_0 \,</math></center>

where <math> \vec{\xi}_5(t)\,</math> is the ship pitch rotation angle, and:

<center><math> \vec{n}_0 \equiv (n_1,n_2,n_3) \,</math></center>

Keeping only the unsteady components:

<center><math> \vec{V} \cdot \vec{n} = U \xi_5 n_3 + \left( \dot{\xi}_3 - X \dot{\xi_5} \right) n_3 = \frac{\partial\Phi}{\partial n} </math></center>

* Note that the steady component <math> U n_1 \,</math> has already been accounted for in the statement of the steady flow. Let

<center><math> \Phi=\mathrm{Re}\left(\phi e^{i\omega t} \right), \quad \phi=\Pi_3 X_3 + \Pi_5 X_5 </math></center>

where <math> \phi_3\,</math> and <math> \phi_5\,</math> are the unit-amplitude heave & pitch complex velocity potentials in time-harmonic flow. Also:

<center><math> \xi_3(t) = \mathrm{Re} \left\{ \Pi_3(\omega) e^{i\omega t} \right\}, \quad \xi_5(t) = \mathrm{Re} \left\{ \Pi_5(\omega) e^{i\omega t} \right\} </math></center>

It follows that:

<center><math> \frac{\partial X_3}{\partial n} = i \omega n_3; \quad \mbox{on} \ \bar{S}_B \,</math></center>

<center><math> \frac{\partial X_5}{\partial n} = - i \omega X n_3 + U n_3; \quad \mbox{on} \ \bar{S}_B \,</math></center>

* Note the forward-speed effect in the pitch boundary condition and no such effect in heave.

* Diffraction problem: <math> \phi=\phi_7 \,</math>

<center><math> \frac{\partial\phi_7}{\partial n} = -\frac{\partial\phi_I}{\partial n}, \quad \frac{\partial}{\partial n} \equiv \vec{n} \cdot \nabla \,</math></center>

where:

<center><math> \phi_I = \frac{i g A}{\omega_0} e^{KZ-iKX\cos\beta - iKY\sin\beta} \,</math></center>

* Radiation problem: <math> \phi = \phi_3 + \phi_5 \,</math>

We will consider the special but important case of heave & pitch which are coupled and important modes to study in the ship seakeeping problem.

* Heave and pitch are the only modes of interest in head waves (<math> \beta = 180^\circ \,</math>) when all other modes of motion (Roll-Sway-Yaw) are identically zero for a ship symmetric port-starboard. Surge is nonzero but generally small for slender ships and in ambient waves of small steepness.

* Bernoulli equation

The linear hydrodynamic disturbance pressure due to unsteady flow disturbances is given relative to the ship frame:

<center><math> P = - \rho \left(\frac{\partial}{\partial t} - U \frac{\partial}{\partial X} \right) \Phi \,</math></center>

where <math>\Phi\,</math> is the respective real potential.

== Radiation problem ==

<center><math> \Phi = \mathrm{Re} \left\{ \phi e^{i\omega t} \right\} \,</math></center>

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

<center><math> \mathbb{P} = - \rho \left( i\omega - U \frac{\partial}{\partial X} \right) \left( \Pi_3 \phi_3 + \Pi_5 \phi_5 \right) </math></center>

where the complex velocity potentials satisfy the 2D boundary value problems derived earlier for slender ships.

* The hydrodynamic pressure will be integrated over the ship hull to obtain the added-mass and damping coefficients next:

<center><math> F_i(t) = \iint_{\bar{S}_B} P n_i \mathrm{d}S, \quad i = 3,5 \, </math></center>

The expressions derived below extend almost trivially to all other modes of motion. In general,

<center><math> n_i \equiv \vec{n} = \left(n_1,n_2,n_3\right);\quad i=1,2,3 </math></center>

<center><math> n_i \equiv \vec{X} \times \vec{n} = \left(n_4,n_5,n_6 \right); \quad i=4,5,6 </math></center>

So for heave:

<center><math> n_i = n_3\,</math></center>

And pitch:

<center><math> n_i = n_5 = - X n_3 + X n_1 \simeq - X n_3 \left(n_1 \ll n_3 \right) \,</math></center>

Expressing <math> F_i \,</math> in terms of the heave & pitch added mass & damping coefficients when the ship is forced to oscillate in calm water, we obtain when accounting for cross-coupling effects:

<center><math> F_i^H (t) = - \sum_{j=3,5} \left[ A_{ij} \frac{\mathrm{d}^2\xi_j}{\mathrm{d}t^2} + B_{ij} \frac{\mathrm{d}\xi_j}{\mathrm{d}t} \right] </math></center>

Hydrostatic restoring effects are understood to be added to <math> F_i^H\,</math>.

Introducing complex notation:

<center><math> \mathbb{F}_i^H(t) = \mathrm{Re} \left\{ \mathbb{F}_i e^{i\omega t} \right\} \, </math></center>

<center><math> \mathbb{F}_i(\omega) = \left[ \omega^2 A_{ij} (\omega) - i\omega B_{ij} (\omega) \right] \Pi_j </math></center>

where the summation notation over <math> i\,</math> is understood hereafter.

Introducing the definition of <math> \mathbb{F}_i\,</math> in terms of the hydrodynamic pressure we obtain after some simple algebra:

<center><math> \mathbb{P} = \mathbb{P}_j \Pi_j \equiv \mathbb{P}_3 \Pi_3 + \mathbb{P} \Pi_5 </math></center>

<center><math> \mathbb{F}_i = \left( \iint_{\bar{S_B}} \mathbb{P}_j n_j \mathrm{d}S \right) \Pi_j </math></center>

And:

<center><math> \omega^2 A_{ij} (\omega) - i\omega B_{ij}(\omega) = \iint_{\bar{S_B}} \mathbb{P}_i n_j \mathrm{d}S </math></center>

where from Bernoulli:

<center><math> \mathbb{P}_3 = \left( i\omega - U \frac{\partial}{\partial X} \right) \phi_3 = \left( i\omega - U \frac{\partial}{\partial X} \right) \left( i\omega X_3 \right) </math></center>

<center><math> \mathbb{P}_5 = \left( i\omega - U \frac{\partial}{\partial X} \right) \phi_5 = \left( i\omega - U \frac{\partial}{\partial X} \right) \left( -i\omega X_3 + U X_3 \right) </math></center>

== Strip theory ==

* Strip theory is a popular approximation of the 3-D Neumann-Kelvin formulation for ships which are slender as is most often the case when vessels are expected to cruise at significant forward speeds.

The principal assumption is:

<center><math> \frac{B}{L}, \ \frac{T}{L} = O(\varepsilon), \quad \varepsilon \ll 1 \, </math></center>

where

<center><math> B: \mbox{Ship maximum beam} \, </math></center>

<center><math> T: \mbox{Ship maximum draft} \, </math></center>

<center><math> L: \mbox{Ship water line length} \, </math></center>

* The principal assumption of strip theory is that certain components of the radiation and diffraction potentials vary slowly along the ship length leading to a simplification of the n-K formulation.

* In head or bow waves where heave and pitch attain their maximum values, the encounter frequency <math>\omega\,</math> is usually high.

=== Radiation problem ===

The ship is forced to oscillate in heave & pitch in calm water while advancing at a speed <math>U\,</math>.

* Due to slenderness the variation of the flow in the x-direction is more gradual than its variation around a ship section. So

<center><math> \frac{\partial}{\partial X} \Phi \ll \frac{\partial\Phi}{\partial Y}, \ \frac{\partial\Phi}{\partial Z} \, </math></center>

where <math> \Phi = \Phi_3 + \Phi_5 \,</math>. Thus the 3D Laplace equation simplifies into a 2D form for the heave & pitch potentials. (The same argument applies to Roll-Sway-Yaw). Thus

<center><math> \left( \frac{\partial^2}{\partial Y^2} + \frac{\partial^2}{\partial X^2} \right) \Phi_j \cong 0, \ \mbox{in fluid} \ j=2, \cdots, 6 </math></center>

* The 2D equation applies for each "strip" location at station-X.

* The ship-hull condition at station-X for the heave & pitch potentials remains the same:

<center><math> \Phi = \mathrm{Re} \left\{ \phi e^{i\omega t} \right\} \,</math></center>

<center><math> \phi = \Pi_3 X_3 + \Pi_5 X_5 \,</math></center>

<center><math> \frac{\partial X_3}{\partial n} = i \omega n_3; \quad \mbox{on} \ \bar{S_B} </math></center>

<center><math> \frac{\partial X_5}{\partial n} = - i \omega X n_3 + U n_3; \quad \mbox{on} \ \bar{S_B} </math></center>

where now:

<center><math> \left( \frac{\partial^2}{\partial Y^2} + \frac{\partial^2}{\partial Z^2} \right) X_j = 0, \quad \mbox{in fluid domain} \,</math></center>

* Define the normalized potential <math> \psi_3\,</math>:

<center><math> \frac{\partial \psi_3}{\partial n} = n_3; \quad \mbox{on} \ \bar{S_B} </math></center>

<center><math> \left( \frac{\partial^2}{\partial Y^2} + \frac{\partial^2}{\partial Z^2} \right) \psi_3 = 0, \quad \ \mbox{in fluid} </math></center>

* There remains to simplify the 3D n-K free-surface condition.

* The solution of the 2D BVP for <math> \psi_3\,</math> along 30 - 40 stations used to describe the hull form of most ships can be carried out very efficiently by standard 2D panel method.

* In terms of <math> \psi_3 (Y,Z; X) \,</math> the heave & pitch potentials follow in the form:

<center><math> \phi = \Pi_3 X_3 + \Pi_5 X_5 \,</math></center>

<center><math> X_3 = i \omega \psi_3 \,</math></center>

<center><math> X_5 = ( - i \omega X + U ) \psi_3 \, </math></center>

<center><math> \Phi = \mathrm{Re} \left\{ \phi e^{i\omega t} \right\} = \Phi_3 +\Phi_5 </math></center>

* The 2D heave added-mass and damping coefficients due to a section oscillation vertically are defined by the familiar expressions

<center><math> \mbox{Station-X}: \quad a_{33} (\omega) - \frac{i}{\omega} b_{33} (\omega) = \rho \int_{C_(X)} \psi_3 n_3 \mathrm{d}l </math></center>

where <math> \omega = \left| \omega_0 - U \frac{\omega_0^2}{g} \cos\beta \right| \,</math> is the encounter frequency.

<center><math> \left( i\omega - U \frac{\partial}{\partial X} \right)^2 \phi + g \phi_Z = 0, \quad Z=0 </math></center>

* The ship slenderness and the claim that <math>\omega\,</math> is usually large in head or bow waves is used to simplify the above equation as follows

<center><math> - \omega^2 \phi + g \phi_Z = 0, \quad Z=0 \,</math></center>

by assuming that <math> \omega \gg \left|U \frac{\partial}{\partial X} \right| </math>. A formal proof is lengthy and technical.

It follows that the normalized potential <math> \psi_3\,</math> also satisfies the above 2D FS condition and is thus the solution of the 2D boundary value problem stated below at station-X:

<center><math> \mbox{As} \ Y \to \infty; \quad \psi_3 \sim \frac{i g A \pm}{\omega} e^{KZ\mp iKy+i\omega t} \,</math></center>.

Upon integration along the ship length and over each cross section at station-X the 3D added-mass and damping coefficients for heave & pitch take the form:

<center><math> A_{33} = \int_{L} \mathrm{d}X a_{33} (X) \, </math></center>

<center><math> B_{33} = \int_{L} \mathrm{d}X b_{33} (X) \, </math></center>

<center><math> A_{35} = - \int_{L} \mathrm{d}X a_{33} - \frac{U}{\omega^2} B_{33} \, </math></center>

<center><math> A_{53} = - \int_{L} \mathrm{d}X a_{33} + \frac{U}{\omega^2} B_{33} \, </math></center>

<center><math> B_{35} = - \int_{L} \mathrm{d}X b_{33} + \frac{U}{\omega^2} A_{33} \, </math></center>

<center><math> B_{53} = - \int_{L} \mathrm{d}X b_{33} - \frac{U}{\omega^2} A_{33} \, </math></center>

<center><math> A_{55} = \int_{L} \mathrm{d}X X^2 a_{33} + \frac{U^2}{\omega^2} A_{33} \, </math></center>

<center><math> B_{35} = \int_{L} \mathrm{d}X X^2 b_{33} - \frac{U^2}{\omega^2} B_{33} \, </math></center>

where the 2D added-mass and damping coefficients were defined above:

<center><math> a_{33} - \frac{i}{\omega} b_{33} = \rho \int_{C(X)} \psi_3 n_3 \mathrm{d}l </math></center>

===Diffraction problem===

* We will consider heave & pitch in oblique waves. Note that in oblique waves the ship also undergoes Roll-Sway-Yaw motions which for a symmetric vessel and according to linear theory are decoupled from heave and pitch.

* Relative to the ship frame the total potential is:

<center><math> \Phi = \Phi_I + \Phi_D = \mathrm{Re} \left\{ \left( \phi_I + \phi_D \right) e^{i\omega t} \right\} </math></center>

where <math>\omega\,</math> is the encounter frequency, and:

<center><math> \phi_I = \frac{i g A}{\omega_0} e^{KZ-iKX\cos\beta-iKY\sin\beta} \quad K=\frac{\omega_0^2}{g} </math></center>

Define the diffraction potential as follows:

<center><math> \phi_0 = \frac{i g A}{\omega_0} e^{-iKX\cos\beta} \psi_7 (Y,Z;X) </math></center>

In words, factor-out the oscillatory variation <math> e^{-iKX\cos\beta} \,</math> out of the scattering potential.

* The ship slenderness approximation now justifies that:

<center><math> \frac{\partial\psi_7}{\partial X} \ll \frac{\partial\psi_7}{\partial Y}, \ \frac{\partial\psi_7}{\partial Z} \,</math></center>

Note that this is not an accurate approximation for <math> \phi_D\,</math> when <math>K=\frac{2\pi}{\lambda}\,</math> is a large quantity or when the ambient wavelength <math> \lambda\,</math> is small.

* Substituting in the 3D laplace equation and ignoring the <math> \frac{\partial\psi_7}{\partial X}, \ \frac{\partial^2\psi_7}{\partial X^2} \,</math> terms we obtain

<center><math> \left(\frac{\partial^2}{\partial Y^2} + \frac{\partial^2}{\partial Z^2} - K^2 \cos^2 \beta \right) \psi_7 \cong 0 \,</math></center>

* This is the modified 2D Helmholtz equation in most cases the <math> K^2\cos^\beta\,</math> term is not important for reasons to be discussed and the 2D Laplace equation follows.

* The body boundary condition simply states:

<center><math> \frac{\partial\phi_D}{\partial n} = - \frac{\partial\phi_I}{\partial n} \,</math></center>

with:

<center><math> \frac{\partial}{\partial n} \equiv n_1 \frac{\partial}{\partial X} + n_2 \frac{\partial}{\partial Y} + n_3 \frac{\partial}{\partial Z} </math></center>

Due to slenderness:

<center><math> n_1 \ll n_2, n_3 \,</math></center>

Invoking the slenderness approximations for the <math>\frac{\partial}{\partial X} \,</math> derivatives of <math> \psi_7\,</math>, the body boundary condition simplifies to:

<center><math> \frac{\partial\psi_7}{\partial N} = - \frac{\partial}{\partial N} \left( e^{KZ-iKY\sin\beta} \right); \ \mbox{on} \ C(X) </math></center>

with:

<center><math> \frac{\partial}{\partial N} = N_2 \frac{\partial}{\partial Y} + N_3 \frac{\partial}{\partial Z} </math></center>

The potential <math> \psi_7\,</math> satisfies the 2D Laplace equation, after dropping the <math> K^2 \cos^2\beta \,</math> term, or

<center><math> \left(\frac{\partial^2}{\partial Y^2} + \frac{\partial^2}{\partial Z^2} \right) \psi_7 = 0 \, </math></center>

And the free-surface condition derived next:

The 3D linear free surface condition for <math> \phi_D\,</math> is:

<center><math> \left( i\omega - U \frac{\partial}{\partial X} \right)^2 \phi_D + g \frac{\partial \phi_D}{\partial Z} = 0; \quad Z=0 </math></center>

The same condition is satisfied by <math> \phi_I\,</math>:

<center><math> \left( i\omega - U \frac{\partial}{\partial X} \right)^2 \phi_I + g \frac{\partial\phi_I}{\partial Z} = 0; \quad Z=0 </math></center>

Verify identically for <math> \phi_I\,</math> that:

<center><math> i \omega - U \frac{\partial}{\partial X} \equiv i \omega_0 \, </math></center>

So the forward-speed effect disappears from the free surface condition of <math> \phi_I\,</math>. The same will be shown to be approximately true for the diffraction potential <math> \phi_D\,</math>:

<center><math> \phi_D = \frac{i g A}{\omega_0} e^{- i K X \cos \beta} \psi_7(Y,Z; X) </math></center>

<center><math> \frac{\partial\phi_D}{\partial X} = \frac{i g A}{\omega_0} \left\{ (- i K \cos \beta) \psi_7 (Y,Z; X) + \frac{\partial \psi_7}{\partial X} \right\} e^{-i K X \cos \beta} </math></center>

Due to slenderness:

<center><math> \left| \frac{\partial\psi_7}{\partial X} \right| \ll |K\cos\beta| |\psi_7|,</math></center>

Therefore:

<center><math> \frac{\partial\phi_D}{\partial X} \simeq - i K \cos \beta \phi_D </math></center>

And from the free-surface condition

<center><math> \left( i\omega - U \frac{\partial}{\partial X} \right) \phi_D \simeq i \omega_0 \phi_D </math></center>

where:

<center><math> \omega = \omega_0 - U K \cos \beta \, </math></center>

So the free-surface condition for <math> \phi_D\,</math> becomes to leading order in slenderness:

<center><math> - \omega_0^2 \phi_D + g \frac{\partial\phi_D}{\partial Z} = 0, \quad Z=0 </math></center>

* So in the statement of the diffraction problem forward-speed effects are absent! The same is not true in the radiation problem even for slender ship.

* This "independence" of the diffraction problem on forward speed effects is verified very well in experiments and in fully three-dimensional solutions.

In summary, the boundary-value problem satisfied by the diffraction potential around a slender ship takes the form:

<center><math> \phi_D = \frac{ i g A}{\omega_0} e^{-iKX\cos\beta} \psi_7 (Y,Z;X) </math></center>

<center><math> \left( \frac{\partial^2}{\partial Y^2} + \frac{\partial^2}{\partial Z^2} \right) \psi_7 = 0, \quad \mbox{in fluid} \, </math></center>

<center><math> - \omega_0^2 \psi_7 +9 g \frac{\partial\psi_7}{\partial Z} = 0, \quad \mbox{on} \ Z=0 </math></center>

<center><math> \frac{\partial\psi_7}{\partial N} = - \frac{\partial}{\partial N} \left( e^{KZ-iKY\sin\beta} \right) </math></center>

Or in graphical form at station-X:

* As <math> Y \to \pm \infty \qquad \psi_7 \sim \frac{ i g A_7^\pm}{\omega_0} e^{KZ \mp i K Y} </math>.

The hydrodynamic pressure in the diffraction problem takes the form:

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

<center><math> \mathbb{P} = - \rho \left( i\omega-U\frac{\partial}{\partial X} \right) \frac{i g A}{\omega_0} e^{-iKX\cos\beta} \left( \psi_I + \psi_7 \right) </math></center>

<center><math> \cong \rho g A e^{-iKX\cos\beta} \left( \psi_I + \psi_7 \right) </math></center>

where <math> \frac{\partial\psi_7}{\partial X} \,</math> derivatives are dropped in favor of <math> K\cos\beta\,</math>, due to slenderness.

The heave and pitch exciting forces follow simply by pressure integration

<center><math> X_i(t) = \mathrm{Re} \left\{ \mathbb{X}_i e^{i\omega t} \right\}; \quad i=3,5 </math></center>

<center><math> \mathbb{X}_3 = \rho g A \int_L \mathrm{d}X e^{-iKX\cos\beta} \int_{C{X}} \left( \psi_I + \psi_7 \right) n_3 \mathrm{d}l </math></center>

<center><math> \mathbb{X}_5 = \rho g A \int_L \mathrm{d}X (-X) e^{-iKX\cos\beta} \int_{C{X}} \left( \psi_I + \psi_7 \right) n_3 \mathrm{d}l </math></center>

where the approximation <math> n_5 = -X n_3 \,</math> was used.

* In summary, a version of strip theory was derived for slenderships which has been found to be the most rational and accurate relative to other alternatives. This is particularly true for the treatment of the diffraction problem.

* The coupled heave & pitch equation of motion take the familiar form

<center><math> \left[ - \omega^2 \left( A_{ij} + M_{ij} \right) + i\omega B_{ij} + X_{ij} \right] \pi_j = \mathbb{X}_i \left(\omega_0\right) \quad i,j = 3,5 </math></center>

where in practice:

<center><math> \omega = \left| \omega_0 - U \frac{\omega_0^2}{g} \cos\beta \right| </math></center>

And the <math>A_{ij}, \ B_{ij} \, </math> are functions of <math> \omega\,</math> defined above as integrals of their 2D counterparts <math> a_{33}\,</math> & <math>b_{33}(\omega)\,</math>.

* The heave & pitch exciting force amplitudes <math> \mathbb{X}_i(\omega_0)\,</math> are functions of <math> \omega_0\,</math>, hence not dependent on <math> U\,</math>.

* All hydrodynamic effects oscillate at <math> \omega\,</math>!

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

Strip Theory Of Ship Motions. Heave & Pitch

2010-09-10T09:19:36Z

F. Javier Arcos Zamora: /* Ship-hull boundary condition */

{{Ocean Wave Interaction with Ships and Offshore Structures
| chapter title = Strip Theory Of Ship Motions. Heave & Pitch
| next chapter = [[Ship Roll-Sway-Yaw Motions]]
| previous chapter = [[Forward-Speed Ship Wave Flows]]
}}

{{incomplete pages}}

== Ship motions with forward speed ==

Ship advances in the positive x-direction with constant speed <math>U\,</math>. Regular waves with absolute frequency <math>\omega_0\,</math> and direction <math>\beta\,</math> are incident upon the ship. The ship undergoes oscillatory motions in all six degrees of freedom <math> \xi_j(t), \quad j=1,\cdots,6\,</math>. The ship seakeeping problem will be treated in the frequency domain under the assumption of linearity.

==Ship-hull boundary condition==

The relevant rigid body velocities for a ship translating with a constant forward velocity and heaving and pitching in head waves are:

<center><math> \vec{U}=U\vec{i}; \quad \mbox{Forward speed} \,</math></center>

<center><math> \vec{\dot{\xi}}_3(t)=\dot{\xi}_3(t)\vec{k}; \quad \mbox{Heave velocity (positive up)} \,</math></center>

<center><math> \vec{\dot{\xi}}_5(t)=\dot{\xi}_5(t)\vec{j}; \quad \mbox{Pitch angular velocity (positive,bow down)} \,</math></center>

Let <math> \vec{n}(t)\,</math> the time-dependent unit normal vector to the instantaneous position of the ship hull. Let <math> \vec{n_0}\,</math> be its value when the ship is at rest.

* The body boundary condition will be derived for small heave & pitch motions and will be stated on the mean position of the ship at rest.

The nonlinear boundary condition on the exact position of the ship hull is:

<center><math> \frac{\partial\Phi}{\partial n}=\overrightarrow{V} \cdot \vec{n} \,</math></center>

where,

<center><math> \Phi=\Phi_3 + \Phi_5 \,</math></center>

<center><math> \overrightarrow{V} \cong U \vec{i} + \vec{K} \left( \dot{\xi}_3 - X \dot{\xi}_5 \right) \,</math></center>

* The unsteady velocity potential <math> \Phi\,</math> has been written as the linear superposition of the heave and pitch components. The total rigid-body velocity is the sum of vertical velocity due to the ship heave and pitch.

* The normal vector<math> \vec{n}(t)\,</math> is defined by

<center><math> \vec{n}(t) = \vec{n}_0 + \vec{\xi}_5 \times \vec{n}_0 \,</math></center>

where <math> \vec{\xi}_5(t)\,</math> is the ship pitch rotation angle, and:

<center><math> \vec{n}_0 \equiv (n_1,n_2,n_3) \,</math></center>

Keeping only the unsteady components:

<center><math> \vec{V} \cdot \vec{n} = U \xi_5 n_3 + \left( \dot{\xi}_3 - X \dot{\xi_5} \right) n_3 = \frac{\partial\Phi}{\partial n} </math></center>

* Note that the steady component <math> U n_1 \,</math> has already been accounted for in the statement of the steady flow. Let

<center><math> \Phi=\mathrm{Re}\left(\phi e^{i\omega t} \right), \quad \phi=\Pi_3 X_3 + \Pi_5 X_5 </math></center>

where <math> \phi_3\,</math> and <math> \phi_5\,</math> are the unit-amplitude heave & pitch complex velocity potentials in time-harmonic flow. Also:

<center><math> \xi_3(t) = \mathrm{Re} \left\{ \Pi_3(\omega) e^{i\omega t} \right\}, \quad \xi_5(t) = \mathrm{Re} \left\{ \Pi_5(\omega) e^{i\omega t} \right\} </math></center>

It follows that:

<center><math> \frac{\partial X_3}{\partial n} = i \omega n_3; \quad \mbox{on} \ \bar{S}_B \,</math></center>

<center><math> \frac{\partial X_5}{\partial n} = - i \omega X n_3 + U n_3; \quad \mbox{on} \ \bar{S}_B \,</math></center>

* Note the forward-speed effect in the pitch boundary condition and no such effect in heave.

* Diffraction problem: <math> \phi=\phi_7 \,</math>

<center><math> \frac{\partial\phi_7}{\partial n} = -\frac{\partial\phi_I}{\partial n}, \quad \frac{\partial}{\partial n} \equiv \vec{n} \cdot \nabla \,</math></center>

where:

<center><math> \phi_I = \frac{i g A}{\omega_0} e^{KZ-iKX\cos\beta - iKY\sin\beta} \,</math></center>

* Radiation problem: <math> \phi = \phi_3 + \phi_5 \,</math>

We will consider the special but important case of heave & pitch which are coupled and important modes to study in the ship seakeeping problem.

* Heave and pitch are the only modes of interest in head waves (<math> \beta = 180^\circ \,</math>) when all other modes of motion (Roll-Sway-Yaw) are identically zero for a ship symmetric port-starboard. Surge is nonzero but generally small for slender ships and in ambient waves of small steepness.

* Bernoulli equation

The linear hydrodynamic disturbance pressure due to unsteady flow disturbances is given relative to the ship frame:

<center><math> P = - \rho \left(\frac{\partial}{\partial t} - U \frac{\partial}{\partial X} \right) \Phi \,</math></center>

where <math>\Phi\,</math> is the respective real potential.

== Radiation problem ==

<center><math> \Phi = \mathrm{Re} \left\{ \phi e^{i\omega t} \right\} \,</math></center>

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

<center><math> \mathbb{P} = - \rho \left( i\omega - U \frac{\partial}{\partial X} \right) \left( \Pi_3 \phi_3 + \Pi_5 \phi_5 \right) </math></center>

where the complex velocity potentials satisfy the 2D boundary value problems derived earlier for slender ships.

* The hydrodynamic pressure will be integrated over the ship hull to obtain the added-mass and damping coefficients next:

<center><math> F_i(t) = \iint_{\bar{S}_B} P n_i \mathrm{d}S, \quad i = 3,5 \, </math></center>

The expressions derived below extend almost trivially to all other modes of motion. In general,

<center><math> n_i \equiv \vec{n} = \left(n_1,n_2,n_3\right);\quad i=1,2,3 </math></center>

<center><math> n_i \equiv \vec{X} \times \vec{n} = \left(n_4,n_5,n_6 \right); \quad i=4,5,6 </math></center>

So for heave:

<center><math> n_i = n_3\,</math></center>

And pitch:

<center><math> n_i = n_5 = - X n_3 + X n_1 \simeq - X n_3 \left(n_1 \ll n_3 \right) \,</math></center>

Expressing <math> F_i \,</math> in terms of the heave & pitch added mass & damping coefficients when the ship is forced to oscillate in calm water, we obtain when accounting for cross-coupling effects:

<center><math> F_i^H (t) = - \sum_{j=3,5} \left[ A_{ij} \frac{\mathrm{d}^2\xi_j}{\mathrm{d}t^2} + B_{ij} \frac{\mathrm{d}\xi_j}{\mathrm{d}t} \right] </math></center>

Hydrostatic restoring effects are understood to be added to <math> F_i^H\,</math>.

Introducing complex notation:

<center><math> \mathbb{F}_i^H(t) = \mathrm{Re} \left\{ \mathbb{F}_i e^{i\omega t} \right\} \, </math></center>

<center><math> \mathbb{F}_i(\omega) = \left[ \omega^2 A_{ij} (\omega) - i\omega B_{ij} (\omega) \right] \Pi_j </math></center>

where the summation notation over <math> i\,</math> is understood hereafter.

Introducing the definition of <math> \mathbb{F}_i\,</math> in terms of the hydrodynamic pressure we obtain after some simple algebra:

<center><math> \mathbb{P} = \mathbb{P}_j \Pi_j \equiv \mathbb{P}_3 \Pi_3 + \mathbb{P} \Pi_5 </math></center>

<center><math> \mathbb{F}_i = \left( \iint_{\bar{S_B}} \mathbb{P}_j n_j \mathrm{d}S \right) \Pi_j </math></center>

And:

<center><math> \omega^2 A_{ij} (\omega) - i\omega B_{ij}(\omega) = \iint_{\bar{S_B}} \mathbb{P}_i n_j \mathrm{d}S </math></center>

where from Bernoulli:

<center><math> \mathbb{P}_3 = \left( i\omega - U \frac{\partial}{\partial X} \right) \phi_3 = \left( i\omega - U \frac{\partial}{\partial X} \right) \left( i\omega X_3 \right) </math></center>

<center><math> \mathbb{P}_5 = \left( i\omega - U \frac{\partial}{\partial X} \right) \phi_5 = \left( i\omega - U \frac{\partial}{\partial X} \right) \left( -i\omega X_3 + U X_3 \right) </math></center>

== Strip theory ==

* Strip theory is a popular approximation of the 3-D Neumann-Kelvin formulation for ships which are slender as is most often the case when vessels are expected to cruise at significant forward speeds.

The principal assumption is:

<center><math> \frac{B}{L}, \ \frac{T}{L} = O(\varepsilon), \quad \varepsilon \ll 1 \, </math></center>

where

<center><math> B: \mbox{Ship maximum beam} \, </math></center>

<center><math> T: \mbox{Ship maximum draft} \, </math></center>

<center><math> L: \mbox{Ship water line length} \, </math></center>

* The principal assumption of strip theory is that certain components of the radiation and diffraction potentials vary slowly along the ship length leading to a simplification of the n-K formulation.

* In head or bow waves where heave and pitch attain their maximum values, the encounter frequency <math>\omega\,</math> is usually high.

=== Radiation problem ===

The ship is forced to oscillate in heave & pitch in calm water while advancing at a speed <math>U\,</math>.

* Due to slenderness the variation of the flow in the x-direction is more gradual than its variation around a ship section. So

<center><math> \frac{\partial}{\partial X} \Phi \ll \frac{\partial\Phi}{\partial Z}, \ \frac{\partial\Phi}{\partial Z} \, </math></center>

where <math> \Phi = \Phi_3 + \Phi_5 \,</math>. Thus the 3D Laplace equation simplifies into a 2D form for the heave & pitch potentials. (The same argument applies to Roll-Sway-Yaw). Thus

<center><math> \left( \frac{\partial^2}{\partial Y^2} + \frac{\partial^2}{\partial X^2} \right) \Phi_j \cong 0, \ \mbox{in fluid} \ j=2, \cdots, 6 </math></center>

* The 2D equation applies for each "strip" location at station-X.

* The ship-hull condition at station-X for the heave & pitch potentials remains the same:

<center><math> \Phi = \mathrm{Re} \left\{ \phi e^{i\omega t} \right\} \,</math></center>

<center><math> \phi = \Pi_3 X_3 + \Pi_5 X_5 \,</math></center>

<center><math> \frac{\partial X_3}{\partial n} = i \omega n_3; \quad \mbox{on} \ \bar{S_B} </math></center>

<center><math> \frac{\partial X_5}{\partial n} = - i \omega X n_3 + U n_3; \quad \mbox{on} \ \bar{S_B} </math></center>

where now:

<center><math> \left( \frac{\partial^2}{\partial Y^2} + \frac{\partial^2}{\partial Z^2} \right) X_j = 0, \quad \mbox{in fluid domain} \,</math></center>

* Define the normalized potential <math> \psi_3\,</math>:

<center><math> \frac{\partial \psi_3}{\partial n} = n_3; \quad \mbox{on} \ \bar{S_B} </math></center>

<center><math> \left( \frac{\partial^2}{\partial Y^2} + \frac{\partial^2}{\partial Z^2} \right) \psi_3 = 0, \quad \ \mbox{in fluid} </math></center>

* There remains to simplify the 3D n-K free-surface condition.

* The solution of the 2D BVP for <math> \psi_3\,</math> along 30 - 40 stations used to describe the hull form of most ships can be carried out very efficiently by standard 2D panel method.

* In terms of <math> \psi_3 (Y,Z; X) \,</math> the heave & pitch potentials follow in the form:

<center><math> \phi = \Pi_3 X_3 + \Pi_5 X_5 \,</math></center>

<center><math> X_3 = i \omega \psi_3 \,</math></center>

<center><math> X_5 = ( - i \omega X + U ) \psi_3 \, </math></center>

<center><math> \Phi = \mathrm{Re} \left\{ \phi e^{i\omega t} \right\} = \Phi_3 +\Phi_5 </math></center>

* The 2D heave added-mass and damping coefficients due to a section oscillation vertically are defined by the familiar expressions

<center><math> \mbox{Station-X}: \quad a_{33} (\omega) - \frac{i}{\omega} b_{33} (\omega) = \rho \int_{C_(X)} \psi_3 n_3 \mathrm{d}l </math></center>

where <math> \omega = \left| \omega_0 - U \frac{\omega_0^2}{g} \cos\beta \right| \,</math> is the encounter frequency.

<center><math> \left( i\omega - U \frac{\partial}{\partial X} \right)^2 \phi + g \phi_Z = 0, \quad Z=0 </math></center>

* The ship slenderness and the claim that <math>\omega\,</math> is usually large in head or bow waves is used to simplify the above equation as follows

<center><math> - \omega^2 \phi + g \phi_Z = 0, \quad Z=0 \,</math></center>

by assuming that <math> \omega \gg \left|U \frac{\partial}{\partial X} \right| </math>. A formal proof is lengthy and technical.

It follows that the normalized potential <math> \psi_3\,</math> also satisfies the above 2D FS condition and is thus the solution of the 2D boundary value problem stated below at station-X:

<center><math> \mbox{As} \ Y \to \infty; \quad \psi_3 \sim \frac{i g A \pm}{\omega} e^{KZ\mp iKy+i\omega t} \,</math></center>.

Upon integration along the ship length and over each cross section at station-X the 3D added-mass and damping coefficients for heave & pitch take the form:

<center><math> A_{33} = \int_{L} \mathrm{d}X a_{33} (X) \, </math></center>

<center><math> B_{33} = \int_{L} \mathrm{d}X b_{33} (X) \, </math></center>

<center><math> A_{35} = - \int_{L} \mathrm{d}X a_{33} - \frac{U}{\omega^2} B_{33} \, </math></center>

<center><math> A_{53} = - \int_{L} \mathrm{d}X a_{33} + \frac{U}{\omega^2} B_{33} \, </math></center>

<center><math> B_{35} = - \int_{L} \mathrm{d}X b_{33} + \frac{U}{\omega^2} A_{33} \, </math></center>

<center><math> B_{53} = - \int_{L} \mathrm{d}X b_{33} - \frac{U}{\omega^2} A_{33} \, </math></center>

<center><math> A_{55} = \int_{L} \mathrm{d}X X^2 a_{33} + \frac{U^2}{\omega^2} A_{33} \, </math></center>

<center><math> B_{35} = \int_{L} \mathrm{d}X X^2 b_{33} - \frac{U^2}{\omega^2} B_{33} \, </math></center>

where the 2D added-mass and damping coefficients were defined above:

<center><math> a_{33} - \frac{i}{\omega} b_{33} = \rho \int_{C(X)} \psi_3 n_3 \mathrm{d}l </math></center>

===Diffraction problem===

* We will consider heave & pitch in oblique waves. Note that in oblique waves the ship also undergoes Roll-Sway-Yaw motions which for a symmetric vessel and according to linear theory are decoupled from heave and pitch.

* Relative to the ship frame the total potential is:

<center><math> \Phi = \Phi_I + \Phi_D = \mathrm{Re} \left\{ \left( \phi_I + \phi_D \right) e^{i\omega t} \right\} </math></center>

where <math>\omega\,</math> is the encounter frequency, and:

<center><math> \phi_I = \frac{i g A}{\omega_0} e^{KZ-iKX\cos\beta-iKY\sin\beta} \quad K=\frac{\omega_0^2}{g} </math></center>

Define the diffraction potential as follows:

<center><math> \phi_0 = \frac{i g A}{\omega_0} e^{-iKX\cos\beta} \psi_7 (Y,Z;X) </math></center>

In words, factor-out the oscillatory variation <math> e^{-iKX\cos\beta} \,</math> out of the scattering potential.

* The ship slenderness approximation now justifies that:

<center><math> \frac{\partial\psi_7}{\partial X} \ll \frac{\partial\psi_7}{\partial Y}, \ \frac{\partial\psi_7}{\partial Z} \,</math></center>

Note that this is not an accurate approximation for <math> \phi_D\,</math> when <math>K=\frac{2\pi}{\lambda}\,</math> is a large quantity or when the ambient wavelength <math> \lambda\,</math> is small.

* Substituting in the 3D laplace equation and ignoring the <math> \frac{\partial\psi_7}{\partial X}, \ \frac{\partial^2\psi_7}{\partial X^2} \,</math> terms we obtain

<center><math> \left(\frac{\partial^2}{\partial Y^2} + \frac{\partial^2}{\partial Z^2} - K^2 \cos^2 \beta \right) \psi_7 \cong 0 \,</math></center>

* This is the modified 2D Helmholtz equation in most cases the <math> K^2\cos^\beta\,</math> term is not important for reasons to be discussed and the 2D Laplace equation follows.

* The body boundary condition simply states:

<center><math> \frac{\partial\phi_D}{\partial n} = - \frac{\partial\phi_I}{\partial n} \,</math></center>

with:

<center><math> \frac{\partial}{\partial n} \equiv n_1 \frac{\partial}{\partial X} + n_2 \frac{\partial}{\partial Y} + n_3 \frac{\partial}{\partial Z} </math></center>

Due to slenderness:

<center><math> n_1 \ll n_2, n_3 \,</math></center>

Invoking the slenderness approximations for the <math>\frac{\partial}{\partial X} \,</math> derivatives of <math> \psi_7\,</math>, the body boundary condition simplifies to:

<center><math> \frac{\partial\psi_7}{\partial N} = - \frac{\partial}{\partial N} \left( e^{KZ-iKY\sin\beta} \right); \ \mbox{on} \ C(X) </math></center>

with:

<center><math> \frac{\partial}{\partial N} = N_2 \frac{\partial}{\partial Y} + N_3 \frac{\partial}{\partial Z} </math></center>

The potential <math> \psi_7\,</math> satisfies the 2D Laplace equation, after dropping the <math> K^2 \cos^2\beta \,</math> term, or

<center><math> \left(\frac{\partial^2}{\partial Y^2} + \frac{\partial^2}{\partial Z^2} \right) \psi_7 = 0 \, </math></center>

And the free-surface condition derived next:

The 3D linear free surface condition for <math> \phi_D\,</math> is:

<center><math> \left( i\omega - U \frac{\partial}{\partial X} \right)^2 \phi_D + g \frac{\partial \phi_D}{\partial Z} = 0; \quad Z=0 </math></center>

The same condition is satisfied by <math> \phi_I\,</math>:

<center><math> \left( i\omega - U \frac{\partial}{\partial X} \right)^2 \phi_I + g \frac{\partial\phi_I}{\partial Z} = 0; \quad Z=0 </math></center>

Verify identically for <math> \phi_I\,</math> that:

<center><math> i \omega - U \frac{\partial}{\partial X} \equiv i \omega_0 \, </math></center>

So the forward-speed effect disappears from the free surface condition of <math> \phi_I\,</math>. The same will be shown to be approximately true for the diffraction potential <math> \phi_D\,</math>:

<center><math> \phi_D = \frac{i g A}{\omega_0} e^{- i K X \cos \beta} \psi_7(Y,Z; X) </math></center>

<center><math> \frac{\partial\phi_D}{\partial X} = \frac{i g A}{\omega_0} \left\{ (- i K \cos \beta) \psi_7 (Y,Z; X) + \frac{\partial \psi_7}{\partial X} \right\} e^{-i K X \cos \beta} </math></center>

Due to slenderness:

<center><math> \left| \frac{\partial\psi_7}{\partial X} \right| \ll |K\cos\beta| |\psi_7|,</math></center>

Therefore:

<center><math> \frac{\partial\phi_D}{\partial X} \simeq - i K \cos \beta \phi_D </math></center>

And from the free-surface condition

<center><math> \left( i\omega - U \frac{\partial}{\partial X} \right) \phi_D \simeq i \omega_0 \phi_D </math></center>

where:

<center><math> \omega = \omega_0 - U K \cos \beta \, </math></center>

So the free-surface condition for <math> \phi_D\,</math> becomes to leading order in slenderness:

<center><math> - \omega_0^2 \phi_D + g \frac{\partial\phi_D}{\partial Z} = 0, \quad Z=0 </math></center>

* So in the statement of the diffraction problem forward-speed effects are absent! The same is not true in the radiation problem even for slender ship.

* This "independence" of the diffraction problem on forward speed effects is verified very well in experiments and in fully three-dimensional solutions.

In summary, the boundary-value problem satisfied by the diffraction potential around a slender ship takes the form:

<center><math> \phi_D = \frac{ i g A}{\omega_0} e^{-iKX\cos\beta} \psi_7 (Y,Z;X) </math></center>

<center><math> \left( \frac{\partial^2}{\partial Y^2} + \frac{\partial^2}{\partial Z^2} \right) \psi_7 = 0, \quad \mbox{in fluid} \, </math></center>

<center><math> - \omega_0^2 \psi_7 +9 g \frac{\partial\psi_7}{\partial Z} = 0, \quad \mbox{on} \ Z=0 </math></center>

<center><math> \frac{\partial\psi_7}{\partial N} = - \frac{\partial}{\partial N} \left( e^{KZ-iKY\sin\beta} \right) </math></center>

Or in graphical form at station-X:

* As <math> Y \to \pm \infty \qquad \psi_7 \sim \frac{ i g A_7^\pm}{\omega_0} e^{KZ \mp i K Y} </math>.

The hydrodynamic pressure in the diffraction problem takes the form:

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

<center><math> \mathbb{P} = - \rho \left( i\omega-U\frac{\partial}{\partial X} \right) \frac{i g A}{\omega_0} e^{-iKX\cos\beta} \left( \psi_I + \psi_7 \right) </math></center>

<center><math> \cong \rho g A e^{-iKX\cos\beta} \left( \psi_I + \psi_7 \right) </math></center>

where <math> \frac{\partial\psi_7}{\partial X} \,</math> derivatives are dropped in favor of <math> K\cos\beta\,</math>, due to slenderness.

The heave and pitch exciting forces follow simply by pressure integration

<center><math> X_i(t) = \mathrm{Re} \left\{ \mathbb{X}_i e^{i\omega t} \right\}; \quad i=3,5 </math></center>

<center><math> \mathbb{X}_3 = \rho g A \int_L \mathrm{d}X e^{-iKX\cos\beta} \int_{C{X}} \left( \psi_I + \psi_7 \right) n_3 \mathrm{d}l </math></center>

<center><math> \mathbb{X}_5 = \rho g A \int_L \mathrm{d}X (-X) e^{-iKX\cos\beta} \int_{C{X}} \left( \psi_I + \psi_7 \right) n_3 \mathrm{d}l </math></center>

where the approximation <math> n_5 = -X n_3 \,</math> was used.

* In summary, a version of strip theory was derived for slenderships which has been found to be the most rational and accurate relative to other alternatives. This is particularly true for the treatment of the diffraction problem.

* The coupled heave & pitch equation of motion take the familiar form

<center><math> \left[ - \omega^2 \left( A_{ij} + M_{ij} \right) + i\omega B_{ij} + X_{ij} \right] \pi_j = \mathbb{X}_i \left(\omega_0\right) \quad i,j = 3,5 </math></center>

where in practice:

<center><math> \omega = \left| \omega_0 - U \frac{\omega_0^2}{g} \cos\beta \right| </math></center>

And the <math>A_{ij}, \ B_{ij} \, </math> are functions of <math> \omega\,</math> defined above as integrals of their 2D counterparts <math> a_{33}\,</math> & <math>b_{33}(\omega)\,</math>.

* The heave & pitch exciting force amplitudes <math> \mathbb{X}_i(\omega_0)\,</math> are functions of <math> \omega_0\,</math>, hence not dependent on <math> U\,</math>.

* All hydrodynamic effects oscillate at <math> \omega\,</math>!

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

Forward-Speed Ship Wave Flows

2010-09-09T16:33:35Z

F. Javier Arcos Zamora: /* Boundary-value problem for time-Harmonic velocity potential */

{{Ocean Wave Interaction with Ships and Offshore Structures
| chapter title = Forward-Speed Ship Wave Flows
| next chapter = [[Strip Theory Of Ship Motions. Heave & Pitch]]
| previous chapter = [[Wave Scattering By A Vertical Circular Cylinder]]
}}

{{incomplete pages}}

== Linear free surface conditions ==

The linear free surface conditions are as follows,

<center><math> \frac{\mathrm{d}^2\Phi}{\mathrm{d}t^2} + g \frac{\mathrm{d}\Phi}{\mathrm{d}z} = 0, \quad z=0 </math></center>

We can calculate the second derivative above as
<center><math>
\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}
</math></center>

When no ambient waves are present <math> \frac{\partial\bar{\Phi}}{\partial t} = 0 \, </math> and we obtain the free surface condition for the steady Kelvin ship wave problem
<center><math> U^2 \frac{\partial^2 \bar{\Phi}}{\partial x^2}
+ g \frac{\partial\bar{\Phi}}{\partial z} = 0, \qquad z=0 </math></center>
This is the famous Neumann-Kelvin free-surface condition governing the linear steady wave pattern generated by a translating ship.

Bernoulli equation:

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

Free surface elevation:

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

In the Kelvin ship wave problem
<center><math> \frac{\partial\phi}{\partial t} = 0 </math></center>

Hence
<center><math> \zeta = \frac{U}{g} \frac{\partial\bar{\Phi}}{\partial x}, \qquad z=0 </math></center>

== Neumann-Kelvin linearization of <math>U>0</math> ship seakeeping problem ==

Let <math> \Phi(X_0,Y_0,Z_0,t) \,</math>be the total potential relative to the inertial coordinate system

<center><math> X_0 = x + U t \, </math></center>

Let <math> \Phi( X,Y,Z,t) \, </math> be the same potential expressed relative to the translating frame. It was shown before that

<center><math> \frac{\mathrm{d}\Phi}{\mathrm{d}t} = \left( \frac{\partial}{\partial t} - U \frac{\partial}{\partial X} \right) \phi (\vec{X}, t) \, </math></center>

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

The total potential <math> \Phi\,</math> consists of the sum of two components in a linearized setting

<center><math> \mathbf{\Phi}_{TOT} = \mathbf{\bar{\Phi}} + \mathbf{\Phi} \,</math></center>

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

Relative to the ship frame:

<center><math> \mathbf{\bar{\Phi}} = \bar{\Phi} (X,Y,Z) \, </math></center>

<center><math> \mathbf{\Phi} = \Phi(X,Y,Z,t) = \mathbf{Re} \left\{ \phi(X,Y,Z)e^{i\omega t} \right\} \,</math></center>

where <math>\omega\,</math> is the encounter frequency and

<center><math> \phi = \phi_I + \sum_{j=1}^7 \phi_j \,</math></center>

with <math> \phi_j, \ j=1,\cdots,6\,</math> being the radiation and <math> \phi_7\,</math> being the diffraction potentials.

== Boundary-value problem for <math> \bar{\Phi}</math> ==

Free surface condition:

<center><math> U^2 \bar{\Phi}_{XX} + g \bar{\Phi}_Z = 0, \quad Z=0 </math></center>

Ship-hull condition

<center><math> \vec{n} \cdot \nabla\vec{\Phi} = \vec{n} \cdot \vec{U} = U n_1 \, </math></center>

where <math> \vec{n} = ( n_1, n_2, n_3 ) \, </math> is the unit vector pointing inside the ship hull.

Far from the ship <math> \bar{\Phi}\,</math> represents outgoing waves which are known as the Kelvin ship wake studied earlier.

The solution for <math> \bar{\Phi}\,</math> 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

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

*Hydrodynamic pressure (linear)

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

*Hydrodynamic pressure (total)

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

<center><math> = - \rho \left( - U \bar{\Phi_X} + \frac{1}{2} \nabla \bar{\Phi} \cdot \nabla \bar{\Phi} + g Z \right)_{\mbox{Ship frame}} \,</math></center>

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

<center><math> \overrightarrow{F} = \iint_{\overline{S_W}} P_T \vec{n} \mathrm{d}S \, </math></center>

The wave resistance is: <math> R_W = \vec{i} \cdot \vec{F} \, </math>.

We will derive boundary value problems for the potentials <math> \bar{\Phi}\,</math> and <math> \Phi\,</math> 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 <math> B\,</math> is the ship beam, <math> T\,</math> its draft and <math> L\,</math> its length we will assume that:

<center><math> \frac{B}{L}, \quad \frac{T}{L} = O (\varepsilon), \quad \varepsilon \ll 1 \,</math></center>

where <math> \varepsilon\,</math> is the slenderness parameter assumed to be small compared to <math> 1 \,</math>.

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:

<center><math> \left( \frac{\partial}{\partial t} - U \frac{\partial}{\partial X} \right)^2 \Phi + g \Phi_Z = 0, \quad Z=0 \, </math></center>

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

<center><math> \Phi = \mathbf{Re} \left\{ \phi e^{i\omega t} \right\} \, </math></center>

<center><math> \left( i\omega - U \frac{\partial}{\partial X} \right)^2 \phi + g \phi_Z = 0 , \quad Z=0 \, </math></center>

where <math> \omega\, </math> is the encounter frequency and <math> \phi\,</math> is any of the <math> \phi_j\,</math> 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 <math>\omega\,</math> will lead to the popular strip theory.

* The solution for <math> \bar{\Phi}\,</math> 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 <math>\overline{S_W}\,</math> 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 <math>\omega\,</math>.

<u>Proof</u>:

<center><math> \zeta = - \frac{1}{g} \left( \frac{\partial}{\partial t} - U \frac{\partial}{\partial X} \right) \Phi_I </math></center>

where:

<center><math> \Phi_I = \mathbf{Re} \left\{ \frac{i g A}{\omega_0} e^{KZ-iKX\cos\beta-iKY\sin\beta+i\omega t} \right\} </math></center>

<center><math> \frac{\partial}{\partial t} = i \omega, \qquad \frac{\partial}{\partial X} = - i K \cos \beta </math></center>

<center><math> \frac{\partial}{\partial t} - U \frac{\partial}{\partial X} = i \left( \omega + K U \cos \beta \right) = i \omega_0 \, </math></center>

Thus:

<center><math> \zeta = \mathbf{Re} \left\{ A e^{-iKX\cos\beta-iKY\sin\beta+i\omega t} \right\} \,</math></center>

<center><math> = \mathbf{Re} \left\{ A e^{i\omega t} \right\}_{X=Y=0} </math></center>

where <math>\Pi_j(\omega) \,</math> is the complex amplitude of the vessel motion in mode <math>-j\,</math>, a function of the frequency of encounter <math> \omega\,</math>, known as the response amplitude operator (RAO).

The ship equations of motion follow as in the <math> U=0\,</math> case using linear system theory:

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

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

* Inertia & hydrostatic matrices the same as in the zero-speed case

* Will derive BVP's governing the coefficients <math>A_{ij}(\omega), \ B_{ij}(\omega) \, </math> and exciting forces <math>\mathbf{X}_i (\omega) \, </math>.

Explicitly:

<center><math> \omega = \omega_0 - U \frac{\omega_0^2}{g} \cos\beta \, </math></center>

* <math> \omega > \ \mbox{or} \ < 0 \, </math>: Both positive and negative values of <math> \omega\,</math> are possible. In practice will always deal with the absolute value of <math> \omega\,</math>.

* Given the absolute wave frequency <math> \omega_0 > 0 \, </math> there exists a unique <math> \omega\,</math>.

* Given a positive absolute encounter frequency <math> |\omega|\,</math>, there exist possibly multiple <math> \omega_0\,</math>'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 <math>\sim e^{i\omega t} \ \longrightarrow\,</math> output signal <math> \sim e^{i\omega t} \,</math>.

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

<center><math> \Phi_I = \mathbf{Re} \left\{ \phi_I \right\} \,</math></center>

<center><math> \phi_I = \frac{i g A}{\omega_0} e^{KZ_0 - iKX_0 \cos\beta - iKY_0 \sin\beta +i\omega_0 t} \, </math></center>

where in deep water: <math> K = \frac{\omega_0^2}{g} \, </math>

Let:

<center><math> X_0 = x+ Ut \qquad Y_0 = y \qquad Z_0 = z \,</math></center>

It follows that:

<center><math> \phi_I = \frac{i g A}{\omega_0} e^{ Kz - iKx\cos\beta - iKy\sin\beta + i (\omega_0 -UK \cos\beta) t} </math></center>

Let:

<center><math> \omega = \omega_0 - UK\cos\beta \,</math></center>

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 <math>O\left(A^2\right)\,</math>.

* When Taylor expanding the free-surface and body-boundary condition about <math>Z=0\,</math> and <math>\overrightarrow{S_B}\,</math> respectively, the steady flow potential <math> \bar{\Phi}\simeq 0 \,</math>.

* For ships with appreciable thickness a better approximation for <math>\bar\Phi\,</math> is that of the double-body flow disturbance such that <math> \overrightarrow{\Phi_Z} = 0\,</math> on <math> Z=0\,</math> and <math> \overrightarrow{\Phi_n} = U n\,</math>, on <math>\overrightarrow{S_B}\,</math>, 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.

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