Vessel theory: Frequency domain

$\newcommand{\enc}{{_\mathrm{e}}}$ $\newcommand{\dL}{d_\mathrm{L}}$ $\newcommand{\dT}{d_\mathrm{T}}$

Wave drift load process

When conducting frequency domain analysis at low frequency solution frequencies, OrcaFlex will solve for the model's response at the user specified wave drift frequencies and wind component frequencies. The only external loading the model attracts at wave drift frequencies is the wave drift load (2nd order) for vessel objects which include it. Unfortunately, due to the second order nature of the wave drift load, there is no linear mapping between the wave elevation process and the wave drift load process. Instead, the wave drift load cross-spectral density matrix, at wave drift frequency $\mu$, can be calculated, in multi-directional seas, based on a generalisation of the methods given by Pinkster and Langley, as \begin{equation} L_{ij}(\mu) = 8 \sum_m \sum_n \int_{-\infty}^\infty c(i,j)\ Q_i(\beta_m,\beta_n, f+\mu,f)\ Q_j(\beta_m,\beta_n, f+\mu,f)\ S_m(f+\mu)\ S_n(f)\,\ud\! f \end{equation} where

$i$ and $j$ represent the $i$th and $j$th degrees of freedom of the vessel

$Q_i(\beta_m,\beta_n, f{+}\mu,f)$ is the QTF, in the vessel's $i$th degree of freedom, calculated from the diagonal QTF data, $Q_\mathrm{d}(\beta_m, f{+}\mu)$ and $Q_\mathrm{d}(\beta_n, f)$, by Newman's approximation method using the geometric mean

$S_m(f)$ is the wave elevation spectrum in the $m$th wave direction, $\beta_m$

$c(i,j)$ is a scaling factor which, for a single vessel, simply takes the value 1 independently of the values of $i$ and $j$.

For models containing multiple vessels, this method for calculating the wave drift load cross-spectral density matrix can be further generalised. The indices $i$ and $j$ now range over all low frequency calculated vessel degrees of freedom, for vessels which include wave drift load. The scaling factor $c(i,j)$ is now given by \begin{equation} c(i,j) = \begin{cases} 1, & \text{if $i$ and $j$ are for the same vessel} \\ \exp\left\{-\mathrm{i} \left(\vec{k}_m(f+\mu)-\vec{k}_n(f)\right) \left(\vec{x}_\mathrm{A}-\vec{x}_\mathrm{B}\right)\right\}, & \text{if $i$ and $j$ are for different vessels A and B.} \end{cases} \end{equation} Here

$\vec{k}_m$ and $\vec{k}_n$ are the vector-valued wave numbers associated with the $m$th and $n$th wave directions respectively

$\vec{x}_\mathrm{A}$ and $\vec{x}_\mathrm{B}$ are the vector-valued positions of vessel A and vessel B respectively

$c(i,j)$ is now complex-valued when $i$ and $j$ represent different vessels.

The integral over frequency, $f$, is numerically evaluated using a composite Simpson's rule with a user-defined sub-interval count. The limits of the integral are replaced with the minimum and maximum wave frequency component band ranges, given by the wave spectrum discretisation algorithm, evaluated across all wave trains.

Finally, eigen decomposition is used to decompose the wave drift load cross-spectral density matrix, $\mat{L}(\mu)$, into $N$ independent complex vector-valued wave drift load processes, where $N$ is the number of non-zero eigenvalues. The response due to each independent wave drift load process is then solved separately in the usual way, by applying a series of linear mappings from each process through to the response process. This means that, for each wave drift frequency, OrcaFlex will repeat the solve $N$ times, resulting in a total solution count of ($N \times$ number of wave drift frequencies) + (number of wind component frequencies).

When wave drift damping is included, and a non-zero current speed is present, the QTF values used in the calculation of the wave drift load are inclusive of encounter effects evaluated at the static current speed.

Note: Only QTF data for Newman's approximation method are supported in the frequency domain. The full QTF method is not supported.

Wave drift damping

If a vessel object includes both wave drift load (2nd order) and wave drift damping, a frequency-independent wave drift damping matrix $\mat{C}$ is calculated, in multi-directional seas, using a generalisation of the method given by Molin\begin{equation} C_{ij} = \sum_m \int_{-\infty}^\infty 2\ D_{ij}(\beta_m,f)\ S_m(f)\,\ud\!f \end{equation} where

$D_{ij}(\beta_m,f)$ are the coefficients of the frequency-dependent wave drift damping matrix $\mat{D}$ for the $m^\text{th}$ wave direction, which are internally calculated from the QTF data for Newman's approximation method

$S_m(f)$ is the wave elevation spectrum in the $m^\text{th}$ wave direction, $\beta_m$

The integral over frequency, $f$, is numerically evaluated using a composite Simpson's rule with a user-defined sub-interval count. The limits of the integral are replaced with the minimum and maximum wave frequency component band ranges, given by the wave spectrum discretisation algorithm, for each individual wave direction in the summation.

The frequency-dependent wave drift damping matrix, $\mat{D}(\beta,f)$, for wave direction $\beta$, is calculated as \begin{equation} \begin{aligned} %%for only one equation number D_{11}(\beta,f) &= \left[ \cos\beta \cdot \dL + \sin\beta \cdot \dT \right] Q_{\mathrm{d}1}(\beta\enc,f\enc) \\ D_{12}(\beta,f) &= \left[ \sin\beta \cdot \dL - \cos\beta \cdot \dT \right] Q_{\mathrm{d}1}(\beta\enc,f\enc) \\ D_{21}(\beta,f) &= \left[ \cos\beta \cdot \dL + \sin\beta \cdot \dT \right] Q_{\mathrm{d}2}(\beta\enc,f\enc) \\ D_{22}(\beta,f) &= \left[ \sin\beta \cdot \dL - \cos\beta \cdot \dT \right] Q_{\mathrm{d}2}(\beta\enc,f\enc) \end{aligned} \end{equation} where \begin{equation} \begin{aligned} %%for only one equation number \dL &= -\left[\frac{1}{C_\mathrm{g}}\left(f\PD{\alpha}{f}-2 \right) - A\enc\frac{f}{C_\mathrm{p}}\PD{}{f\enc}\right] \\ \dT &= -\frac{A\enc}{C_\mathrm{g}} \PD{}{\beta\enc} \end{aligned} \end{equation} $A\enc$, $f\enc$, and $\beta\enc$ are the encounter factor, encounter frequency, and encounter direction evaluated at the static current speed. $Q_{\mathrm{d}{i}}(\beta\enc,f\enc)$ is the user-specified diagonal QTF data for Newman's approximation method, in the vessel's $i^\text{th}$ degree of freedom. $C_\mathrm{p}$, $C_\mathrm{g}$, and $\alpha$ are the wave phase velocity, wave group velocity, and their ratio. As is the case in time domain application, the wave drift damping is only applied at low frequency in the vessel's surge $(i{=}1)$ and sway $(i{=}2)$ degrees of freedom.

Frequency-Dependent added mass and damping

To implement frequency-dependent added mass and damping in the frequency domain, there is no requirement to calculate the impulse response function. Instead, the added mass and damping load can be simply included by pointwise product, so at each solution frequency $f$ we calculate \begin{equation} \left[-(2\pi f)^2 \mat{A}(f) + \mathrm{i}2\pi f \mat{B}(f)\right] \vec{x}(f) \end{equation} where $\mat{A}(f)$ is the frequency dependent added mass matrix, $\mat{B}(f)$ the frequency dependent damping matrix, and $\vec{x}(f)$ the vessel's response.

The data for $\mat{A}(f)$ and $\mat{B}(f)$ are given at specific user-defined frequencies $f_i, i=1,2,3,…,m$ which, in general, will not coincide with the desired solution frequencies. We therefore assume the following forms for added mass and damping, to allow us to interpolate or extrapolate them to obtain values at each of the solution frequencies.

For damping,

For added mass,

If the infinite frequency added mass has been specified, it is removed from the data before the interpolate or extrapolate takes place. Frequency-dependent added mass and damping are applied at both wave and low frequencies.

Note: The frequency-dependent added mass and damping data should ideally be specified for the full range of frequencies to be analysed, or if this is not possible then for the widest range available. If the data do not cover the full range, OrcaFlex is forced to extrapolate as described above and, since the extrapolated added mass values are derived (via the Kramers-Kronig relations) from the damping data over the whole frequency range, the quality of the added mass extrapolation will be dependent on the quality and range of the specified damping data.

Current and wind load

Unlike the other quadratic drag models in OrcaFlex, which are linearised using equivalent linearisation, the OCIMF current and wind loading on the vessel is linearised, if included, simply by taking the Jacobian matrix of the load relationship, i.e. the tangent damping defined at the static state of the system. This simplified method is chosen because the directional dependence of the data does not make it amenable to the equivalent linearisation scheme.

Current and wind load are only applied at low frequencies.

Note: For high mean flow speeds, which is generally the case for wind, the tangent damping method yields a good linear approximation of the quadratic model. However, for low mean flow speeds, which can be the case for current, then the damping can be underestimated by this linearisation scheme; in the limit of zero mean flow speed there will be no damping contribution from OCIMF current and wind loading.

Although included with the current load, the drag due to yaw rate model has no directional dependence and is linearised using the equivalent linearisation scheme. Because this effect is associated with low frequencies only, when solving with combined linearisation the wave frequency motion does not influence the linearisation of drag due to yaw rate.