Strip Theory Of Ship Motions. Heave & Pitch

Ship motions with forward speed

• Ship advances in the positive x-direction with constant speed [math]\displaystyle{ U\, }[/math]

• Regular waves with absolute frequency [math]\displaystyle{ \omega_0\, }[/math] and direction [math]\displaystyle{ \beta\, }[/math] are incident upon the ship

• The ship undergoes oscillatory motions in all six degrees of freedom [math]\displaystyle{ \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:

[math]\displaystyle{ \vec{U}=U\vec{L}; \quad \mbox{Forward speed} \, }[/math] [math]\displaystyle{ \vec{\dot{\xi}}_3(t)=\dot{\xi}_3(t)\vec{k}; \quad \mbox{Heave velocity (positive up)} \, }[/math] [math]\displaystyle{ \vec{\dot{\xi}}_5(t)=\dot{\xi}_5(t)\vec{j}; \quad \mbox{Pitch angular velocity (positive,bow down)} \, }[/math]

Let [math]\displaystyle{ \vec{n}(t)\, }[/math] the time-dependent unit normal vector to the instantaneous position of the ship hull. Let [math]\displaystyle{ \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:

[math]\displaystyle{ \frac{\partial\Phi}{\partial n}=\overrightarrow{V} \cdot \vec{n} \, }[/math]

where,

[math]\displaystyle{ \Phi=\Phi_3 + \Phi_5 \, }[/math] [math]\displaystyle{ \overrightarrow{V} \cong U \vec{l} + \vec{K} \left( \dot{\xi}_3 - X \dot{\xi}_5 \right) \, }[/math]

• The unsteady velocity potential [math]\displaystyle{ \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]\displaystyle{ \vec{n}(t)\, }[/math] is defined by

[math]\displaystyle{ \vec{n}(t) = \vec{n}_0 + \vec{\xi}_5 \times \vec{n}_0 \, }[/math]

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

[math]\displaystyle{ \vec{n}_0 \equiv (n_1,n_2,n_3) \, }[/math]

Keeping only the unsteady components:

[math]\displaystyle{ \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]

• Note that the steady component [math]\displaystyle{ U n_1 \, }[/math] has already been accounted for in the statement of the steady flow. Let

[math]\displaystyle{ \Phi=\mathbb{R}\mathbf{e}\left(\phi e^{i\omega t} \right), \quad \phi=\Pi_3 X_3 + \Pi_5 X_5 }[/math]

where [math]\displaystyle{ \phi_3\, }[/math] and [math]\displaystyle{ \phi_5\, }[/math] are the unit-amplitude heave & pitch complex velocity potentials in time-harmonic flow. Also:

[math]\displaystyle{ \xi_3(t) = \mathbb{R}\mathbf{e} \left\{ \Pi_3(\omega) e^{i\omega t} \right\}, \quad \xi_5(t) = \mathbb{R}\mathbf{e} \left\{ \Pi_5(\omega) e^{i\omega t} \right\} }[/math]

It follows that:

[math]\displaystyle{ \frac{\partial X_3}{\partial n} = i \omega n_3; \quad \mbox{on} \ \bar{S}_B \, }[/math] [math]\displaystyle{ \frac{\partial X_5}{\partial n} = - i \omega X n_3 + U n_3; \quad \mbox{on} \ \bar{S}_B \, }[/math]

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

Ship-hull boundary conditions

• Diffraction problem: [math]\displaystyle{ \phi=\phi_7 \, }[/math]

[math]\displaystyle{ \frac{\partial\phi_7}{\partial n} = -\frac{\partial\phi_I}{\partial n}, \quad \frac{\partial}{\partial n} \equiv \vec{n} \cdot \nabla \, }[/math]

where:

[math]\displaystyle{ \phi_I = \frac{i g A}{\omega_0} e^{KZ-iKX\cos\beta - iKY\sin\beta} \, }[/math]

• Radiation problem: [math]\displaystyle{ \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]\displaystyle{ \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:

[math]\displaystyle{ P = - \rho \left(\frac{\partial}{\partial t} - U \frac{\partial}{\partial X} \right) \Phi \, }[/math]

where [math]\displaystyle{ \Phi\, }[/math] is the respective real potential.

Radiation problem

[math]\displaystyle{ \Phi = \mathbb{R}\mathbf{e} \left\{ \phi e^{i\omega t} \right\} \, }[/math] [math]\displaystyle{ P = \mathbb{R}\mathbf{e} \left\{ \mathbb{P} e^{i\omega t} \right\} \, }[/math] [math]\displaystyle{ \mathbb{P} = - \rho \left( i\omega - U \frac{\partial}{\partial X} \right) \left( \Pi_3 \phi_3 + \Pi_5 \phi_5 \right) }[/math]

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:

[math]\displaystyle{ F_i(t) = \iint_{\bar{S}_B} P n_i dS, \quad i = 3,5 \, }[/math]

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

[math]\displaystyle{ n_i \equiv \vec{n} = \left(n_1,n_2,n_3\right);\quad i=1,2,3 }[/math] [math]\displaystyle{ n_i \equiv \vec{X} \times \vec{n} = \left(n_4,n_5,n_6 \right); \quad i=4,5,6 }[/math]

So for heave:

[math]\displaystyle{ n_i = n_3\, }[/math]

And pitch:

[math]\displaystyle{ n_i = n_5 = - X n_3 + X n_1 \simeq - X n_3 \left(n_1 \ll n_3 \right) \, }[/math]

Expressing [math]\displaystyle{ 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:

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

Hydrostatic restoring effects are understood to be added to [math]\displaystyle{ F_i^H\, }[/math].

Introducing complex notation:

[math]\displaystyle{ \mathbb{F}_i^H(t) = \mathbb{R}\mathbf{e} \left\{ \mathbb{F}_i e^{i\omega t} \right\} \, }[/math] [math]\displaystyle{ \mathbb{F}_i(\omega) = \left[ \omega^2 A_{ij} (\omega) - i\omega B_{ij} (\omega) \right] \Pi_j }[/math]

where the summation notation over [math]\displaystyle{ i\, }[/math] is understood hereafter.

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

[math]\displaystyle{ \mathbb{P} = \mathbb{P}_j \Pi_j \equiv \mathbb{P}_3 \Pi_3 + \mathbb{P} \Pi_5 }[/math] [math]\displaystyle{ \mathbb{F}_i = \left( \iint_{\bar{S_B} \mathbb{P}_j n_j dS \right) \Pi_j }[/math]

And:

[math]\displaystyle{ \omega^2 A_{ij} (\omega) - i\omega B_{ij}(\omega) = \iint_{\bar{S_B}} \mathbb{P}_i n_j dS }[/math]

where from Bernoulli:

[math]\displaystyle{ \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] [math]\displaystyle{ \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]

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:

[math]\displaystyle{ \frac{B}{L}, \ \frac{T}{L} = O(\varepsilon), \quad \varepsilon \ll 1 \, }[/math]

where

[math]\displaystyle{ B: \mbox{Ship maximum beam} \, }[/math] [math]\displaystyle{ T: \mbox{Ship maximum draft} \, }[/math] [math]\displaystyle{ L: \mbox{Ship water line length} \, }[/math]

• 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]\displaystyle{ \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]\displaystyle{ 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

[math]\displaystyle{ \frac{\partial}{\partial X} \Phi \ll \frac{\partial\Phi}{\partial Z}, \ \frac{\partial\Phi}{\partial Z} \, }[/math]

where [math]\displaystyle{ \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

[math]\displaystyle{ \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]

• 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:

[math]\displaystyle{ \Phi = \mathbb{R}\mathbf{e} \left\{ \phi e^{i\omega t} \right\} \, }[/math] [math]\displaystyle{ \phi = \Pi_3 X_3 + \Pi_5 X_5 \, }[/math] [math]\displaystyle{ \frac{\partial X_3}{\partial n} = i \omega n_3; \quad \mbox{on} \bar{S_B} }[/math] [math]\displaystyle{ \frac{\partial X_5}{\partial n} = - i \oemga X n_3 + U n_3; \quad \mbox{on} \ \bar{S_B} }[/math]

where now:

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

• Define the normalized potential [math]\displaystyle{ \psi_3\, }[/math]:

[math]\displaystyle{ \frac{\parital \psi_3}{\partial n} = n_3; \quad \mbox{on} \ \bar{S_B} }[/math] [math]\displaystyle{ \left( \frac{\partial^2}{\partial Y^2} + \frac{\partial^2}{\partial Z^2} \right) \psi_3 = 0, \quad \ \mbox{in fluid} }[/math]

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

• The solution of the 2D BVP for [math]\displaystyle{ \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]\displaystyle{ \psi_3 (Y,Z; X) \, }[/math] the heave & pitch potentials follow in the form:

[math]\displaystyle{ \phi = \Pi_3 X_3 + \Pi_5 X_5 \, }[/math] [math]\displaystyle{ X_3 = i \omega \psi_3 \, }[/math] [math]\displaystyle{ X_5 = ( - i \omega X + U ) \psi_3 \, }[/math] [math]\displaystyle{ \Phi = \mathbb{R}\mathbf{e} \left\{ \phi e^{i\omega t} \right\} = \Phi_3 +\Phi_5 }[/math]

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

[math]\displaystyle{ \mbox{Station-X}: \quad a_{33} (\omega) - \frac{i}{\omega} b_{33} (\omega} = \rho \int_{C_(X)} \psi_3 n_3 dl }[/math]

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

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

• The ship slenderness and the claim that [math]\displaystyle{ \omega\, }[/math] is usually large in head or bow waves is used to simplify the above equation as follows

[math]\displaystyle{ - \omega^2 \phi + g \phi_Z = 0, \quad Z=0 \, }[/math]

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

It follows that the normalized potential [math]\displaystyle{ \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:

[math]\displaystyle{ \mbox{As} \ Y \to \infty; \quad \psi_3 \sim \frac{i g A \pm}{\omega} e^{KZ\mpiKy+i\omega t} \, }[/math]

.

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:

[math]\displaystyle{ A_{33} = \int_{L} dX a_{33} (X) \, }[/math] [math]\displaystyle{ B_{33} = \int_{L} dX b_{33} (X) \, }[/math] [math]\displaystyle{ A_{35} = - \int_{L} dX a_{33} - \frac{U}{\omega^2} B_{33} \, }[/math] [math]\displaystyle{ A_{53} = - \int_{L} dX a_{33} + \frac{U}{\omega^2} B_{33} \, }[/math] [math]\displaystyle{ B_{35} = - \int_{L} dX b_{33} + \frac{U}{\omega^2} A_{33} \, }[/math] [math]\displaystyle{ B_{53} = - \int_{L} dX b_{33} - \frac{U}{\omega^2} A_{33} \, }[/math] [math]\displaystyle{ A_{55} = \int_{L} dX X^2 a_{33} + \frac{U^2}{\omega^2} A_{33} \, }[/math] [math]\displaystyle{ B_{35} = \int_{L} dX X^2 b_{33} - \frac{U^2}{\omega^2} B_{33} \, }[/math]

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

[math]\displaystyle{ a_{33} - \frac{i}{\omega} b_{33} = \rho \int_{C(X)} \psi_3 n_3 dl }[/math]

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:

[math]\displaystyle{ \Phi = \Phi_I + \Phi_D = \mathbb{R}\mathbf{e} \left\{ \left( \phi_I + \phi_D \right) e^{i\omega t} \right\} }[/math]

where [math]\displaystyle{ \omega\, }[/math] is the encounter frequency, and:

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

Define the diffraction potential as follows:

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

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

• The ship slenderness approximation now justifies that:

[math]\displaystyle{ \frac{\partial\psi_7}{\partial X} \ll \frac{\partial\psi_7}{\partial Y}, \ \frac{\partial\psi_7}{\partial Z} \, }[/math]

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

• Substituting in the 3D laplace equation and ignoring the [math]\displaystyle{ \frac{\partial\psi_7}{\partial X}, \ \frac{\partial^2\psi_7}{\partial X^2} \, }[/math] terms we obtain

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

• This is the modified 2D Helmholtz equation in most cases the [math]\displaystyle{ K^2\cos^\beta\, }[/math] term is not important for reasons to be discussed and the 2D Laplace equation follows.