Theory: Hydrodynamic drag

Hydrodynamic drag forces depend on $\vec{v}$, the relative velocity between the fluid and an object\begin{equation} \vec{v} = \vec{v}_{\textrm{fluid}} - \vec{v}_{\textrm{object}} \end{equation} The dependence on $\vec{v}$ is quadratic, so the forces are second-order in the perturbation expansion that underlies an OrcaWave diffraction analysis. Therefore, strictly speaking, drag loads do not contribute to the first-order equation of motion or displacement RAOs. In practice, however, drag loads can be very significant for some bodies. This is because real waves have finite amplitude, whereas the perturbation expansion (and the clean separation of terms into first order, second order etc.) is valid in the limit of infinitesimal wave steepness.

If drag loads are significant, it may be desirable to include drag effects in the calculation of displacement RAOs. To do this, the drag must be approximated by a linear function of $\vec{v}$. Drag linearisation is the process of obtaining a suitable linear approximation.

Hydrodynamic drag can be included in an OrcaWave model by attaching Morison elements to one or more body.

Drag linearisation

Essentially the aim is to replace the nonlinear model for the drag force $\vec{f}$ \begin{equation} \vec{f} \propto \vec{v} \vert\vec{v}\vert \end{equation} with a linear model of the form \begin{equation} \label{eqLinearDragModel} \vec{f} \propto \mat{L}_1 \vec{v} \end{equation} Such a linear model (\ref{eqLinearDragModel}) gives drag loads which can contribute to the first-order equation of motion because $\vec{v}$ is itself a first-order quantity.

OrcaWave uses the equivalent linearisation method, also known as stochastic linearisation, to determine $\mat{L}_1$. This must be performed in the context of a stochastic, or random wave, analysis, i.e. where the sea state is defined by a wave spectrum.

Equivalent linearisation calculates the linear drag model such that some measure of the difference between the linear and nonlinear drag is minimised. The measure chosen here is the mean square error, and the method employed is the so-called minimum mean square error (MMSE) method as detailed by Langley and Leira. The matrix $\mat{L}_1$ is calculated to minimise the mean square of the error $\vec{\epsilon}$ between the linear and nonlinear models \begin{equation} \PD{\E\left[\vec{\epsilon}^T\vec{\epsilon}\right]}{\mat{L}_1} = 0 \end{equation} where \begin{equation} \vec{\epsilon} = \mat{L}_1 \vec{v} - \vec{v} \vert\vec{v}\vert \end{equation} and $\E$ is the expectation operator. Solving this equation for $\mat{L}_1$ yields \begin{equation} \label{L1} \mat{L}_1 = \mat{\Sigma}_\mathrm{v}^{-1} \E\left[\vert\vec{v}\vert \vec{v} \vec{v}^T \right] \end{equation} where $\mat{\Sigma}_\mathrm{v}$ is the relative velocity covariance matrix. The expectations in (\ref{L1}) are of functions of the relative velocity: they can be calculated, assuming the components of the relative velocity are distributed with the multivariate normal distribution, by using Gauss-Hermite quadrature to evaluate the integral \begin{equation} \E\left[f(\vec{v})\right] = \int_{-\infty}^\infty f(\vec{v})\, \P(\mat{\Sigma}_\mathrm{v})\, \ud\vec{v} \end{equation} where $\P$ is the multivariate normal distribution.

The matrix $\mat{L}_1$ obtained from (\ref{L1}) is a function of the covariance matrix of the relative velocity vector. Unfortunately, this in turn is a function of the body motion, and so an iterative approach becomes necessary. The procedure is as follows:

1. Estimate the matrix $\mat{L}_1$, assuming an initial guess of zero response.
2. Solve for the body motion using the current estimate of $\mat{L}_1$.
3. Update the estimate for $\mat{L}_1$, using the latest body motion.
4. If the convergence criteria are not satisfied then return to step 2.

This process continues until the convergence criteria are met or the maximum number of iterations is reached.

Convergence criteria

The drag linearisation is deemed to have converged if the relative error in the linear drag matrix, measured between the current iteration and the previous iteration, is less than the user-specified tolerance $\textit{tol}$ \begin{equation} \label{Convergence} \lVert\mat{L}_1 - \mat{L}_1^\textrm{prev}\rVert \lt \max\big\{ \lVert\mat{L}_1\rVert,\ \lVert\mat{L}_1^\textrm{prev}\rVert \big\} . \textit{tol} \end{equation} Inequality (\ref{Convergence}) must be satisfied for all linear drag matrices in the model for convergence to be achieved.

Validity of $\mat{L}_1$

After MMSE linearisation has converged, the constant matrix $\mat{L}_1$ is used to obtain a linear drag model. It is important to note that the matrix $\mat{L}_1$ is only valid for a single sea state – the one used to evaluate the expectations above. Using a different energy spectrum, or a different wave direction (relative to the owner's heading), will give a different linear drag model.

The dependence on wave direction means that OrcaWave must perform a separate drag linearisation for each wave heading specified in the model. For this reason, the wave spectrum in the model has no direction of its own.