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Theory: Damping lid |
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Theory: Damping lid |
The governing equations are based on an assumption of irrotational and inviscid (frictionless) fluid flow in the water around a vessel or offshore structure. The assumption is valid across a wide range of applications but there are situations in which it fails to give a realistic model of behaviour. In particular, inviscid fluid flow and potential theory can over-predict the response at resonances. The real-world response is often damped by viscous effects that are not captured by potential theory, which tend to over-predict the amplitude of the response.
By resonance, we mean a physical mechanism that generates large-amplitude responses at a particular wave frequency, such as
| Note: | Resonances have a physical mechanism and can be observed in measured data, albeit at a smaller amplitude than potential theory would predict. They are not to be confused with irregular frequencies, which are a mathematical and computational technicality and do not represent any physics. |
Various methods have been proposed for numerical damping of wave motions in diffraction analysis, and in computational fluid dynamics more generally (see Li 2015 for a review). OrcaWave uses a model in which a damping term proportional to the free surface elevation, $\eta$, is inserted in the free surface boundary condition (scheme 6 of Li 2015). The modified free-surface boundary condition becomes \begin{equation} \begin{aligned}%implicit alignment is {rlrlrl...} g\PD{\phi}{Z} - \omega^2 \left[1-\textrm{i}\epsilon(\vec{X})\right]\phi & = q_F(\vec{X}) & & \vec{X} \in \SF \end{aligned} \label{eqDampingLidBoundaryCondition} \end{equation} where $\epsilon$ is a dimensionless damping coefficient. Setting $\epsilon=0$ recovers the classical free surface boundary condition, while assigning a positive value, $\epsilon>0$, places a damping lid on the free surface.
There is a significant body of empirical evidence in the fields of diffraction analysis and computational fluid dynamics that the boundary condition (\ref{eqDampingLidBoundaryCondition}) has a desirable damping effect on resonant wave motions. However the theoretical motivation is rather weak, in the sense that (\ref{eqDampingLidBoundaryCondition}) does not attempt to fully capture the viscous effects that are present in the Navier-Stokes equations but absent from potential theory.
Typically the damping coefficient will be small and positive. For example, Chen 2004 recommends a value of $\epsilon=0.016$ for a damping lid in the narrow gap between two barges, a value which gives good agreement with measurement data for that particular problem. Using a small value of $\epsilon$ also ensures that the impact of the damping lid is negligible at frequencies away from the resonance, as one would wish since potential theory is a good model of the physics at these frequencies.
| Warning: | The damping parameter $\epsilon$ does not correspond to a tabulated physical property of water. The value used should be informed by experimental or measured data in a similar system to that being modelled. OrcaWave results should be inspected carefully to check that the damping lid has had the desired effect. |
Integral equations for the general boundary value problem with a damping coefficient $\epsilon(\vec X)$ on $\SF$ are given below. In principle, the damping coefficient can be any function of position $\epsilon(X, Y)$ on the free surface. OrcaWave requires that $\epsilon=0$ everywhere on $\SF$ except on the damping lid for which you provide panels, where $\epsilon$ takes a constant positive value.
| Notes: | For generality, the equations below involve surface integrals over the entire free surface, $\SF$. The damping lid can be recognised as the portion of $\SF$ where $\epsilon\neq 0$. |
| We give the equations in their most general form, including the extensions needed to remove irregular frequency effects. If these are not relevant, the integrals over $\SI$ and the equations pertaining to $\vec{X}\in \SI$ are absent. |
The potential formulation integral equations are as follows \begin{equation} \begin{aligned} 2\pi\phi(\vec{X}) + \mathcal{D}[\phi](\vec{X}) & = \mathcal{B}(\vec{X}) & & \vec{X}\in \SB \\ -4\pi \phi(\vec{X}) + \mathcal{D}[\phi](\vec{X}) & = \mathcal{B}(\vec{X}) & & \vec{X}\in \SI \\ 4\pi \phi(\vec{X}) + \mathcal{D}[\phi](\vec{X}) & = \mathcal{B}(\vec{X}) & & \vec{X}\in \SF \end{aligned} \label{eqBIE-Pot-DampingLidIRR} \end{equation} where the operators $\mathcal{B}$ & $\mathcal{D}$ are defined to condense the notation \begin{equation} \begin{aligned} \mathcal{B}(\vec{X}) = & \int_{\SB} q_B(\vec{\xi}) G \ud S_{\xi} + \int_{\SF} \frac{q_F(\vec{\xi})}{g} G \ud S_{\xi} \\ \mathcal{D}[\mu](\vec{X}) = & \CPVint{\SB}{}\mu(\vec{\xi})\PD{G}{n_{\xi}} \ud S_{\xi} -\nu\int_{\SI}\mu(\vec{\xi}) G \ud S_{\xi} +\textrm{i}\nu\int_{\SF}\epsilon(\vec{\xi})\mu(\vec{\xi}) G \ud S_{\xi} \end{aligned} \end{equation} OrcaWave obtains the unknown values of the potential by solving a matrix equation derived from (\ref{eqBIE-Pot-DampingLidIRR}), which requires an additional panel mesh to represent the damping lid.
If irregular frequency effects are to be removed, then $\SI$ is included in the mesh and, after solving the matrix equation, OrcaWave discards the non-physical values of $\phi$ on $\SI$. Proof that (\ref{eqBIE-Pot-DampingLidIRR}) removes the effect of irregular frequencies can be derived by extending the arguments of Kleinman.
| Note: | As in the basic problem, OrcaWave obtains the scattered and diffraction potentials by solving an integral equation for $\phi_D$. The integral equation for $\phi_D$ is the same as (\ref{eqBIE-Pot-DampingLidIRR}), but with a simpler right-hand side: $4\pi\phi_I(\vec{X})$. The radiation potential is obtained by solving (\ref{eqBIE-Pot-DampingLidIRR}). |
The source formulation integral equations are as follows \begin{equation} \begin{aligned} \phi(\vec{X}) & = \int_{\SB+\SI+\SF} \sigma(\vec{\xi}) G \ud S_{\xi} & & \vec{X}\in \fV \end{aligned} \label{eqDampingLidSourceDefIRR} \end{equation} and the extended integral equation is \begin{equation} \begin{aligned} 2\pi \sigma(\vec{X}) + \CPVint{\SB + \SI + \SF}{} \;\;\;\;\;\;\;\;\;\;\;\sigma(\vec{\xi}) \PD{G}{n_{x}} \ud S_{\xi} & = q_B(\vec{X}) & & \vec{X} \in \SB \\ -4\pi \sigma(\vec{X}) -\nu \int_{\SB + \SI + \SF}\sigma(\vec{\xi}) G \ud S_{\xi} & = -V(\vec{X}) & & \vec{X} \in \SI \\ 4\pi \sigma(\vec{X}) +\textrm{i}\nu\epsilon(\vec X) \int_{\SB + \SI + \SF}\sigma(\vec{\xi}) G \ud S_{\xi} & = \frac{q_F(\vec{X})}{g} & & \vec{X} \in \SF \end{aligned} \label{eqBIE-Sor-DampingLidIRR} \end{equation} OrcaWave obtains the unknown values of the source by solving a matrix equation derived from (\ref{eqBIE-Sor-DampingLidIRR}).
If irregular frequency effects are to be removed, then $\SI$ is included in the mesh and $V(\vec X)$ is obtained from the solution of the potential formulation by applying the usual continuity condition on the body waterline(s), $\CWL$.
In addition to the integral equations themselves, the following equations are also modified by the presence of a damping lid and the modified boundary condition on the free surface when $\epsilon > 0$.
The scattered potential, $\phi_S$, is now a solution to the general BVP with \begin{equation} \begin{aligned} q_B & = -\frac{\partial \phi_I}{\partial n} \\ q_F & = -\textrm{i}\omega^2 \epsilon(\vec X) \phi_I(\vec X) \end{aligned} \end{equation} The non-zero forcing, $q_F$, reflects the fact that $\phi_I$, as defined, does not satisfy the boundary condition on the damping lid.
The Haskind formula for the load RAO is \begin{equation} F_i = -\textrm{i}\omega \rho \left[ \int_{\SB} \left\{(n_{\textrm{vel}})_i\phi_I - \phi_i\PD{\phi_I}{n}\right\} \ud S -\textrm{i}\nu \int_{\SF} \epsilon \phi_i \phi_I \ud S \right] \end{equation}
The formula for the body potential at a field point in the fluid domain, used to obtain sea state RAOs from the potential formulation, is \begin{equation} \begin{aligned} \phi_B(\vec X) & = \frac{1}{4\pi}\int_{\SB} \left\{\PD{\phi_B(\vec{\xi})}{n_{\xi}} G - \phi_B(\vec{\xi}) \PD{G}{n_{\xi}} \right\} \ud S_{\xi} -\frac{\textrm{i}\nu}{4\pi}\int_{\SF} \epsilon(\vec{\xi}) \Big\{\phi_B(\vec{\xi}) + \phi_I(\vec{\xi}) \Big\} G \ud S_{\xi} \\ \nabla\phi_B(\vec X) & = \frac{1}{4\pi}\int_{\SB} \left\{\PD{\phi_B(\vec{\xi})}{n_{\xi}} \nabla_x G - \phi_B(\vec{\xi}) \nabla_x \PD{G}{n_{\xi}} \right\} \ud S_{\xi} -\frac{\textrm{i}\nu}{4\pi}\int_{\SF} \epsilon(\vec{\xi}) \Big\{\phi_B(\vec{\xi}) + \phi_I(\vec{\xi}) \Big\} \nabla_x G \ud S_{\xi} \end{aligned} \end{equation} The formula for the body potential at a field point in the fluid domain, used to obtain sea state RAOs from the source formulation, is \begin{equation} \begin{aligned} \phi_B(\vec X) & = \int_{\SB+\SI+\SF} \Big\{\sigma_R(\vec{\xi}) + \sigma_S(\vec{\xi}) \Big\} G \ud S_{\xi} \\ \nabla\phi_B(\vec X) & = \int_{\SB+\SI+\SF} \Big\{\sigma_R(\vec{\xi}) + \sigma_S(\vec{\xi}) \Big\} \nabla_x G \ud S_{\xi} \end{aligned} \end{equation}
| Notes: | Field points located on the damping lid, or closer than a typical panel diameter, may give unreliable velocity results (pressure results are not affected). Alternative results for both, valid at the centroids of mesh panels, are available via panel results. |
| A field point on an edge (or vertex) of a damping lid panel will cause an error. You can check for this situation by selecting validation of panel arrangement. |
Field points located on the damping lid, or closer than a typical panel diameter, may give unreliable sea state RAOs. Therefore the panel results are extended to include results at the centroids of damping lid panels.
The pressure is given directly by the solution of the potential formulation.
If the source formulation is included in the solve type, then the velocity at panel centroids is given by \begin{equation} \begin{aligned} \nabla\phi(\vec X) = \nabla\phi_I(\vec X) + 4\pi \Big\{\sigma_R(\vec{X}) + \sigma_S(\vec{X}) \Big\}\vec{e}_Z + \int_{\SB+\SI+\SF} \Big\{\sigma_R(\vec{\xi}) + \sigma_S(\vec{\xi}) \Big\} \nabla_x G \ud S_{\xi} & & \vec{X} \in \SF \end{aligned} \end{equation}