Difference between revisions of "Strip Theory Of Ship Motions. Heave & Pitch"
Line 122: | Line 122: | ||
Introducing complex notation: | Introducing complex notation: | ||
+ | |||
+ | <center><math> \mathbb{F}_i^H(t) = \mathbb{R}\mathbf{e} \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 dS \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 dS </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> | ||
+ | |||
+ | <u>Strip theory</u> | ||
+ | |||
+ | * 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. | ||
+ | |||
+ | <u>Radiation problem</u> | ||
+ | |||
+ | 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 = \mathbb{R}\mathbf{e} \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 \oemga 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{\parital \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 = \mathbb{R}\mathbf{e} \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 dl </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\mpiKy+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} dX a_{33} (X) \, </math></center> | ||
+ | |||
+ | <center><math> B_{33} = \int_{L} dX b_{33} (X) \, </math></center> | ||
+ | |||
+ | <center><math> A_{35} = - \int_{L} dX a_{33} - \frac{U}{\omega^2} B_{33} \, </math></center> | ||
+ | |||
+ | <center><math> A_{53} = - \int_{L} dX a_{33} + \frac{U}{\omega^2} B_{33} \, </math></center> | ||
+ | |||
+ | <center><math> B_{35} = - \int_{L} dX b_{33} + \frac{U}{\omega^2} A_{33} \, </math></center> | ||
+ | |||
+ | <center><math> B_{53} = - \int_{L} dX b_{33} - \frac{U}{\omega^2} A_{33} \, </math></center> | ||
+ | |||
+ | <center><math> A_{55} = \int_{L} dX X^2 a_{33} + \frac{U^2}{\omega^2} A_{33} \, </math></center> | ||
+ | |||
+ | <center><math> B_{35} = \int_{L} dX 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 dl </math></center> | ||
+ | |||
+ | <u>Diffraction problem</u> | ||
+ | |||
+ | * 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 = \mathbb{R}\mathbf{e} \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 | ||
+ | |||
+ | <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. |
Revision as of 21:24, 9 March 2007
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:
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:
where,
- 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
where [math]\displaystyle{ \vec{\xi}_5(t)\, }[/math] is the ship pitch rotation angle, and:
Keeping only the unsteady components:
- Note that the steady component [math]\displaystyle{ U n_1 \, }[/math] has already been accounted for in the statement of the steady flow. Let
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:
It follows that:
- 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]
where:
- 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:
where [math]\displaystyle{ \Phi\, }[/math] is the respective real potential.
Radiation problem
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:
The expressions derived below extend almost trivially to all other modes of motion. In general,
So for heave:
And pitch:
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:
Hydrostatic restoring effects are understood to be added to [math]\displaystyle{ F_i^H\, }[/math].
Introducing complex notation:
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:
And:
where from Bernoulli:
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:
where
- 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
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
- 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:
where now:
- Define the normalized potential [math]\displaystyle{ \psi_3\, }[/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:
- The 2D heave added-mass and damping coefficients due to a section oscillation vertically are defined by the familiar expressions
where [math]\displaystyle{ \omega = \left| \omega_0 - U \frac{\omega_0^2}{g} \cos\beta \right| \, }[/math] is the encounter frequency.
- 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
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:
.
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:
where the 2D added-mass and damping coefficients were defined above:
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:
where [math]\displaystyle{ \omega\, }[/math] is the encounter frequency, and:
Define the diffraction potential as follows:
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:
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.