Forward-Speed Ship Wave Flows

Neumann-Kelvin linearization of [math]\displaystyle{ U\gt 0 }[/math] ship seakeeping problem

• Let [math]\displaystyle{ \Phi(X_0,Y_0,Z_0,t) \, }[/math]be the total potential relative to the inertial coordinate system

[math]\displaystyle{ X_0 = x + U t \, }[/math]

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

[math]\displaystyle{ \frac{d\Phi}{dt} = \left( \frac{\partial}{\partial t} - U \frac{\partial}{\partial X} \right) \phi (\vec{X}, t) \, }[/math]

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

• The total potential [math]\displaystyle{ \Phi\, }[/math] consists of the sum of two components in a linearized setting

[math]\displaystyle{ \mathbf{\Phi}_{TOT} = \mathbf{\bar{\Phi}} + \mathbf{\Phi} \, }[/math]

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

• Relative to the ship frame:

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

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

[math]\displaystyle{ \phi = \phi_I + \sum_{j=1}^7 \phi_j \, }[/math]

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

Boundary-value problem for [math]\displaystyle{ \bar{\Phi} }[/math]

• Free surface condition:

[math]\displaystyle{ U^2 \bar{\Phi}_{XX} + g \bar{\Phi}_Z = 0, \quad Z=0 }[/math]

• Ship-hull condition

[math]\displaystyle{ \vec{n} \cdot \nabla\vec{\Phi} = \vec{n} \cdot \vec{U} = U n_1 \, }[/math]

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

• Far from the ship [math]\displaystyle{ \bar{\Phi}\, }[/math] represents outgoing waves which are known as the Kelvin ship wake studied earlier

• The solution for [math]\displaystyle{ \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

[math]\displaystyle{ \zeta = - \frac{1}{g} \left. \frac{d\bar{\Phi}}{dt} \right|_{\mbox{Inertial frame}} = \frac{U}{g} \left. \bar{\Phi}_X \right|_{\mbox{Ship frame}}, \quad Z=0 }[/math]

• • Hydrodynamic pressure (linear)

[math]\displaystyle{ P = - \rho \left. \frac{d\bar{\mathbf{\Phi}}}{dt} \right|_{\mbox{Inertial frame}} = \rho U \left. \bar{\Phi}_X \right|_{\mbox{Ship frame}} }[/math]

• • Hydrodynamic pressure (total)

[math]\displaystyle{ P_T = \rho \left( \frac{d\bar{\mathbf{\Phi}}}{dt} = \frac{1}{2} \nabla \bar{\mathbf{\Phi}} \cdot \nabla \bar{\mathbf{\Phi}} + g Z \right)_{\mbox{Inertial frame}} }[/math] [math]\displaystyle{ = - \rho \left( - U \bar{\Phi_X} + \frac{1}{2} \nabla \bar{\Phi} \cdot \nabla \bar{\Phi} + g Z \right)_{\mbox{Ship frame}} \, }[/math]

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

[math]\displaystyle{ \overrightarrow{F} = \iint_{\overline{S_W}} P_T \vec{n} dS \, }[/math]

The wave resistance is: [math]\displaystyle{ R_W = \vec{i} \cdot \vec{F} \, }[/math].

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

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

where [math]\displaystyle{ \varepsilon\, }[/math] is the slenderness parameter assumed to be small compared to [math]\displaystyle{ 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:

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

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

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

where [math]\displaystyle{ \omega\, }[/math] is the encounter frequency and [math]\displaystyle{ \phi\, }[/math] is any of the [math]\displaystyle{ \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]\displaystyle{ \omega\, }[/math] will lead to the popular strip theory.

• The solution for [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \omega\, }[/math].

Proof:

[math]\displaystyle{ \zeta = - \frac{1}{g} \left( \frac{\partial}{\partial t} - U \frac{\partial}{\partial X} \right) \Phi_I }[/math]

where:

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

Thus:

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

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

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

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

where the hydrodynamic coefficients [math]\displaystyle{ A_{ij}(\omega), \ B_{ij}(\omega) \, }[/math] and exciting forces are now functions of the encounter frequency [math]\displaystyle{ \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]\displaystyle{ A_{ij}(\omega), \ B_{ij}(\omega) \, }[/math] and exciting forces [math]\displaystyle{ \mathbf{X}_i (\omega) \, }[/math].

Explicitly:

[math]\displaystyle{ \omega = \omega_0 - U \frac{\omega_0^2}{g} \cos\beta \, }[/math]

• [math]\displaystyle{ \omega \gt \ \mbox{or} \ \lt 0 \, }[/math]: Both positive and negative values of [math]\displaystyle{ \omega\, }[/math] are possible. In practice will always deal with the absolute value of [math]\displaystyle{ \omega\, }[/math].

• Given the absolute wave frequency [math]\displaystyle{ \omega_0 \gt 0 \, }[/math] there exists a unique [math]\displaystyle{ \omega\, }[/math].

• Given a positive absolute encounter frequency [math]\displaystyle{ |\omega|\, }[/math], there exist possibly multiple [math]\displaystyle{ \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]\displaystyle{ \sim e^{i\omega t} \ \longrightarrow\, }[/math] output signal [math]\displaystyle{ \sim e^{i\omega t} \, }[/math].

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

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

where in deep water: [math]\displaystyle{ K = \frac{\omega_0^2}{g} \, }[/math]

Let:

[math]\displaystyle{ X_0 = x+ Ut \qquad Y_0 = y \qquad Z+0 = z \, }[/math]

It follows that:

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

Let:

[math]\displaystyle{ \omega = \omega_0 - UK\cos\beta \, }[/math]

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]\displaystyle{ O\left(A^2\right)\, }[/math].

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

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