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Vessel theory: Time domain panel pressure |
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Vessel theory: Time domain panel pressure |
The hydrostatic stiffness load, wave load and added mass and damping load on a vessel in OrcaFlex are all usually derived from the results of a first-order diffraction analysis. Panel pressure refers to a disaggregation of these loads onto a set of panels that describe the vessel hull – typically the same panels that defined the mesh in the diffraction analysis. Panel pressure gives the distribution, over the hull, of the water pressure which generates the loads. This, in turn, can be used as an input to a structural analysis of the hull.
Panel pressure is available as a standard result from most diffraction analysis software, including OrcaWave. However, those results are relevant only for the frequency domain motion considered by the diffraction software, specifically the vessel motion that corresponds to displacement RAOs. If the above loads are included in a time domain dynamic analysis, vessel motion is generally different from the displacement RAOs and it is necessary to perform a different calculation to determine the corresponding time domain panel pressure.
Time domain panel pressure results can be calculated by combining
See vessel results for details of the process.
The theory for calculating time domain panel pressure begins with an expansion of the total water pressure as a series in wave amplitude. See, for example, equation (4.11) of Lee (1995). The pressure $p(t)$ combines three distinct contributions, each caused by a different physical mechanism and associated with a different vessel load in OrcaFlex \begin{equation} p(t) = p_H(t) + p_D(t) + p_R(t) \end{equation} where
$p_H(t)$ is hydrostatic panel pressure, associated with hydrostatic stiffness load.
$p_D(t)$ is diffraction panel pressure, associated with wave load.
$p_R(t)$ is radiation panel pressure, associated with added mass and damping load.
Given that these three OrcaFlex loads are based on first-order diffraction analysis, in principle the time-domain vessel loads computed by OrcaFlex are only valid for small changes in the position and orientation of a vessel relative to its datum position (i.e. the heave, roll and pitch for which the diffraction analysis was performed). Therefore we restrict the theory presented here to pressure terms that are zeroth and first order in wave amplitude. We do not consider any terms at second or higher order.
The hydrostatic contribution to panel pressure is associated with the hydrostatic stiffness load on the vessel. The load on the whole vessel is encapsulated in the hydrostatic stiffness matrix, which specifies how the net weight + buoyancy load on the vessel changes due to small changes in the vessel's position and orientation.
The water pressure that generates the buoyancy contribution to the stiffness load is given by the hydrostatic terms in equation (4.11) of Lee (1995). Restricting to terms that are zeroth and first order in wave amplitude, we have \begin{equation} \label{eq:p_H} p_H(t) = -\rho g \left(z + p_1 + p_2 y - p_3 x\right) \end{equation} where
$\rho$ is the water density.
$g$ is the acceleration due to gravity.
$(x, y, z)$ is the position of the panel centroid when the vessel is in its datum position, expressed relative to the stiffness origin and with respect to the vessel's diffraction frame axes. In other words, it is the position expressed in OrcaWave's body coordinates.
$\vec{p} = (p_1, p_2, p_3)$ is a 3-vector containing the heave position and roll and pitch angles (in radians) at time $t$, as defined in the stiffness theory for vessels.
| Note: | If the panel is a dipole panel in the diffraction analysis, the hydrostatic contribution to the panel pressure jump is $\Delta p_H = 0$. |
The diffraction contribution to panel pressure is associated with the wave load on the vessel. The load on the whole vessel is encapsulated in the load RAOs, which specify the first-order wave force and moment due to incident waves of given period and direction.
The water pressure that generates the wave load is given by the first-order terms in equation (4.11) of Lee (1995) due to the diffraction potential $\Phi_D\fo$, i.e. \begin{equation} \label{eq:p_D} p_D(t) = -\rho \PD{}{t} \Phi_D\fo \end{equation} where
The complex potential $\phi_D\fo(\omega)$ in the frequency domain can be evaluated at the panel centroid during the diffraction analysis.
The transformation to $\Phi_D\fo(t)$ in the time domain is performed using the same approach as for load RAO data, i.e. by interpolating the diffraction results on wave period and direction, and summing linear combinations for all the wave components in the sea state.
For periods outside the range included in the diffraction analysis, interpolation is performed between the shortest/longest period and the theoretical limits for zero period ($p_D = 0$) and infinite period ($p_D = \rho g$).
| Note: | For a dipole panel, the pressure jump is $\Delta p_D = 0$ in both theoretical limits (zero and infinite period). |
The radiation contribution to panel pressure is associated with the added mass and damping load on the vessel. The load on the whole vessel is encapsulated in the added mass and damping matrices, which represent the fluid loads on the vessel due to wave radiation effects.
The water pressure that generates the wave radiation load is given by the first-order terms in equation (4.11) of Lee (1995) due to the radiation potential $\Phi_R\fo$, i.e. \begin{equation} \label{eq:p_R:simple} p_R(t) = -\rho \PD{}{t} \Phi_R\fo \end{equation} where
The complex potential $\phi_R\fo(\omega)$ in the frequency domain can be evaluated at the panel centroid during the diffraction analysis.
The transformation to $\Phi_R\fo(t)$ in the time domain is performed using the same approach as for frequency-dependent added mass and damping data, i.e. via impulse response and convolution.
The impulse response and convolution for panel pressure closely follows the analogous theory for vessel load, giving \begin{align} p_R(t) = & \label{eq:p_R} -\vec{a}(\infty) \cdot \vec{x}''(t) -\int_{\tau=0}^\infty \vec{\IRF}(\tau) \cdot \vec{x}'(t-\tau) \,\ud\tau \\ \vec{\IRF}(\tau) = & \; \label{eq:IRF} c(\tau) \int_{f=0}^\infty 4 \, \vec{b}(f) \cos(2\pi f \tau) \,\ud\! f \\ a_i(\infty) = & \; \rho \, \phi_i(\infty) \\[6pt] b_i(f) = & -\Re[ p_i(f) ] \end{align} Here
The index $i$ represents vessel degrees of freedom. So $i = 1 \ldots 6$ for a standard vessel, whereas $i = 1 \ldots 6N$ for a vessel in a multibody group.
The vector $\vec{a}(\infty)$, analogous to infinite-frequency added mass, is obtained from the infinite-frequency panel potential $\phi_i(\infty)$, evaluated at the panel centroid during the diffraction analysis.
The vector $\vec{b}(f)$, analogous to frequency-dependent damping at frequency $f$, is obtained from the radiation components of decomposed panel pressure, evaluated at the panel centroid during the diffraction analysis.
All other quantities in equations (\ref{eq:p_R}) and (\ref{eq:IRF}) are the same as in the analogous theory for vessel load.
For frequencies outside the range included in the diffraction analysis, $\vec{b}(f)$ is assumed to decay to zero in the same manner as the damping matrix (i.e. $\vec{b}(f) \propto f$ as $f \rightarrow 0$ and $\vec{b}(f) \propto f^{-3}$ as $f \rightarrow \infty$).
| Note: | In principle, the convolution integral in equation (\ref{eq:p_R}) spans the entire history of vessel motion. In practice, OrcaFlex integrates vessel motion over at most $T_c$ seconds, where $T_c$ is the cutoff time. |
Given a completed OrcaFlex time domain simulation, calculating panel pressure is effectively a disaggregation of vessel loads into contributions per panel. Equivalently, the vessel loads could be obtained by aggregating panel pressures. Thus consistency between the loads and the panel pressure is a natural way to validate the calculations.
The force exerted by the water on each panel is simply the pressure multiplied by the panel's normal vector and area. The load on a whole vessel is obtained by integrating (summing) panel loads over the wet portion of the vessel hull. This approach is, for example, the basis for the diffraction load RAOs formula in OrcaWave. In the following sections we outline some of the more detailed technicalities that must be considered to perform a consistency check.
The buoyancy contribution to the vessel hydrostatic stiffness load can be reconstructed by summing the forces arising from the hydrostatic panel pressure $p_H(t)$. The following points are relevant to this summation:
The vessel wave load can be reconstructed by summing the forces arising from the diffraction panel pressure $p_D(t)$. The following points are relevant to this summation:
The vessel added mass and damping load can be reconstructed by summing the forces arising from the radiation panel pressure $p_R(t)$. The following points are relevant to this summation: