## Vessel theory: Wave drift and sum frequency loads |

The wave drift load and sum frequency load are second order wave loads that act on objects subject to waves. They can be applied to a vessel and are calculated based on the dimensional wave drift QTF and sum frequency QTF data. For a detailed description of the theory see Faltinsen's book. Here we give a summary and the details of the formulae used in OrcaFlex.

The wave loads on a vessel can be expressed as a sum of first order, second order and even higher order terms. The linear first order terms are usually the largest. The second order terms are nonlinear effects that are generally much smaller, but they can be significant in some cases. Available vessel input data only describe load terms up to second order.

The first order load is the sum of contributions from each *individual* wave component in the sea state; these loads excite the vessel's first order motion, which can be modelled in OrcaFlex using either displacement RAOs or load RAOs. The second order load is quadratic with wave amplitude and consists of contributions from each *pair* of wave components in the sea state. There are two types of second order contributions:

- Difference frequency terms from each wave component pair; the sum of all of these terms is called the wave drift load. The contribution from a given pair of wave components has frequency equal to the difference between the frequencies of the wave components in the pair. A wave component pair of equal frequencies (i.e. a wave component paired with itself) gives a constant (zero frequency) contribution; the sum of all these constant contributions is the mean wave drift load, which gives rise to a mean static offset of the vessel. Wave component pairs that are near each other in frequency give low frequency load contributions; these determine the slowly varying part of the wave drift load, which can excite slow drift motion of the vessel. Wave component pairs that differ more in frequency give higher frequency contributions; these are generally less important to model.
- Sum frequency terms from each component pair, the sum of which is called the sum frequency load. The contribution from a given pair of wave components has frequency equal to the sum of the frequencies of the two wave components, which can be significantly higher than the typical wave frequencies. These can be important to model for some systems, such as TLPs, where they can excite high frequency natural modes of the system, called 'ringing' modes.

These second order loads can be included in, or excluded from, the analysis by specifying that 'wave drift load (2nd order)' and 'sum frequency load (2nd order)' are in the included effects on the vessel data form calculation page. If the wave drift load is included then the mean wave drift load is included in both the static and dynamic analyses. All the other second order load contributions are time-varying loads, so they are included (if specified) only in the dynamic analysis.

QTF data are typically obtained as part of a larger set of vessel type data from a diffraction analysis, performed in the frequency domain. That data set is then input to OrcaFlex, to be used in time domain analysis. As outlined in the use of diffraction analysis data section, time domain application of the diffraction analysis data that define first order loads may generate nonlinear loads whose second order part corresponds to contributions already added into the original QTF data. These contributions to the QTF data are referred to as common second order loads, and we take care not to double-count the contribution from these loads in OrcaFlex.

Note: | Common second order loads can be avoided by application of the diffraction loads using filtered frames and heading frames as appropriate. This option corresponds to choosing the filtering vessel calculation mode. In this case, the rest of this section is not applicable - the common second order loads have been avoided in the time domain calculations, and the QTF data can be used without modification. |

The rest of this section discussing common second order loads assumes that the vessel calculation mode is chosen to be QTF modification. |

The vessel included effects determine which common second order loads may arise. OrcaFlex subtracts any applicable common second order loads from the input QTF data before dynamic calculations are performed.

Having made the required alterations to the input QTF data, common second order loads are permitted to arise naturally in the time domain simulation. This ensures that these loads are included in the analysis only once, and that their full nonlinear value is used, rather than using only the truncated second order representation that was present in the original QTF data.

Note: | The wave drift load results and sum frequency load results report the difference frequency and sum frequency QTF loads. They are calculated using the internally modified QTFs and therefore do not include the common nonlinear loads. |

In order to subtract common second order loads from the input QTF data, OrcaFlex requires additional information about the original diffraction analysis calculations. The assumed motion of the vessel hull model in the diffraction analysis is defined by the displacement RAO data, interpreted using the QTF data source conventions.

Warning: | If the vessel type has no displacement RAO data available, then no modifications will be made from the input QTFs. In order to correctly apply QTF loads, displacement RAO data should be provided to OrcaFlex – even for a calculated vessel analysis. |

The conditions under which subtraction of common second order loads becomes necessary are that

- QTF data are in use, and
- the OrcaFlex vessel primary motion includes wave frequency content.

*Wave* frequency primary motion is a condition because diffraction analysis only considers wave frequency vessel motion as input to the calculation of QTF data. It is the application of first order diffraction analysis data in the time domain, using the instantaneous state of a vessel undergoing wave frequency motions, that is responsible for generating common second order loads.

As well as wave frequency primary motion, OrcaFlex analysis may consider only low frequency primary motion, or include no primary motion at all. It is also possible for OrcaFlex to exclude some vessel degrees of freedom from analysis, for example using calculated (3 DOF) or prescribed primary motion. The requirement for wave frequency content is applied to the degrees of freedom that are actually used by your chosen primary motion; unused degrees of freedom are considered to have no wave frequency content.

For the vessel degrees of freedom in use by primary motion, if primary motion is treated as **low frequency** only, then the displacement RAOs for those degrees of freedom will not contribute to modifications from the original QTF values.

Therefore,

- The QTF data are not modified if primary motion is
**none**. - We exclude heave, roll and pitch displacement RAOs from the calculation of common second order loads if primary motion is
**calculated (3 DOF)**or**prescribed**. The remaining surge, sway and yaw displacement RAOs will be used if the vessel primary motion is specified as having wave frequency content. - For other primary motion options, assuming that primary motion is specified as having wave frequency content, all displacement RAO degrees of freedom will contribute to the calculation of common second order loads.

Second order loads are calculated by applying the QTF data to all pairs of the wave components in the sea state, which is a superposition of a number, $n$, of the regular wave components present in all the wave trains specified. Let the properties of the $i$^{th} wave component be

$\beta_i=$ direction, relative to the heading of the vessel

$a_i=$ amplitude

$\phi_i=$ phase lag, relative to the simulation time origin, at the position of the vessel's QTF origin

$\tau_i=$ period

$\omega_i = \frac{2\pi}{\tau_i}=$ angular frequency.

The wave component elevation above mean sea level at simulation time t, at the position of the vessel's QTF origin, is therefore $a_i\cos(\omega_i t - \phi_i)$. These measurements of relative wave heading and wave elevation are provided by either the vessel primary or primary low frequency frame, depending on the selected calculation mode.

The QTFs are similar to wave load RAOs – they specify scaling and phasing that are applied to wave components to give the contributions to the wave load. But whereas RAOs are applied to each *individual* wave component to give that component's contribution to the first order wave load, QTFs are applied to each *pair* of wave components to give that pair's contribution to the second order wave load.

There are separate sets of QTF data, one for the difference frequency load (wave drift) and one for the sum frequency load, and within each of these data sets there are separate columns for each of the 6 degrees of freedom (surge, sway, heave, roll, pitch, yaw).

Consider a single degree of freedom. Then the QTF for that given degree of freedom is a complex-valued function of the directions $\beta_1$ and $\beta_2$ (relative to the vessel) and periods $\tau_1$ and $\tau_2$ of the wave components in the pair. Let us distinguish the wave drift and sum frequency QTFs by subscripts 'd' (for 'drift' or 'difference') and 's' for 'sum':

$\Qd(\beta_1,\beta_2,\tau_1,\tau_2)=$ wave drift QTF, in complex-valued form

$\Qs(\beta_1,\beta_2,\tau_1,\tau_2)=$ sum frequency QTF, in complex-valued form

Then the wave drift and sum frequency second order wave loads are sums of load contributions from all paired combinations of wave components in the sea state, given by \begin{align} \text{Wave drift load} &= \sum_{i=1}^n \sum_{j=1}^n \Re \left\{ \Qd(\beta_i,\beta_j,\tau_i,\tau_j)\ a_i\ a_j\ \exp[ \i(\omega_i-\omega_j)t - (\phi_i-\phi_j) ] \right\} \\ \text{Sum frequency load} &= \sum_{i=1}^n \sum_{j=1}^n \Re \left\{ \Qs(\beta_i,\beta_j,\tau_i,\tau_j)\ a_i\ a_j\ \exp[ \i(\omega_i+\omega_j)t - (\phi_i+\phi_j) ] \right\} \end{align} where

$\Re{.}$ denotes taking the real part of a complex number

$\Qd(\beta_i,\beta_j,\tau_i,\tau_j)$ and $\Qs(\beta_i,\beta_j,\tau_i,\tau_j)$ are the wave drift and sum frequency QTFs for interaction of wave components $i$ and $j$. These QTFs are derived from the user data; the derivation depends on how the QTF data are specified and is described below.

For sum frequency QTFs, and for wave drift QTFs when the QTF specification method is **full QTFs**, the user data specify the QTF amplitude and phase. The complex-valued wave drift and sum frequency QTFs are then given by
\begin{align}
\Qd(\beta_1,\beta_2,\tau_1,\tau_2) &=
a_\mathrm{d}(\beta_1,\beta_2,\tau_1,\tau_2)\
\exp[ -\i\phi_\mathrm{d}(\beta_1,\beta_2,\tau_1,\tau_2) ] =
\text{ wave drift QTF, in complex-valued form} \\
\Qs(\beta_1,\beta_2,\tau_1,\tau_2) &=
a_\mathrm{s}(\beta_1,\beta_2,\tau_1,\tau_2)\
\exp[ -\i\phi_\mathrm{s}(\beta_1,\beta_2,\tau_1,\tau_2) ] =
\text{ sum frequency QTF, in complex-valued form}
\end{align}
where

$a_\mathrm{d}(\beta_1,\beta_2,\tau_1,\tau_2)=$ wave drift amplitude, as specified in the data

$\phi_\mathrm{d}(\beta_1,\beta_2,\tau_1,\tau_2)=$ wave drift phase lag, in radians, relative to the difference frequency crest

$a_\mathrm{s}(\beta_1,\beta_2,\tau_1,\tau_2)=$ sum frequency amplitude, as specified in the data

$\phi_\mathrm{s}(\beta_1,\beta_2,\tau_1,\tau_2)=$ sum frequency phase lag, in radians, relative to the sum frequency crest.

The QTF amplitudes here, $a_\mathrm{d}$ and $a_\mathrm{s}$, are as specified in the user QTF data. The phase *lag* values $\phi_\mathrm{d}$ and $\phi_\mathrm{s}$ are derived from the user's data allowing for the phase conventions specified on the vessel type conventions page.

OrcaFlex uses linear interpolation and extrapolation to derive the QTF values for wave component directions $\beta$ and periods $\tau$ which do not correspond precisely to those in the user's data.

The above theory uses the full QTF matrix, but sometimes full QTF data are not available for all pairs of wave component directions $(\beta_1,\beta_2)$ and periods $(\tau_1,\tau_2)$. Even if all of these data are available, the calculations are very computationally intensive, since they involve double summations over all wave components in the sea state. The time taken to calculate the second order wave loads using the full QTF theory is therefore proportional to $n^2$, the square of the number of wave components in the sea state.

For the wave drift load, both of these issues can be addressed by using the Newman approximation (Newman 1974). This approximation requires QTF data only for pairs of equal directions $(\beta_1{=}\beta_2)$ and of equal periods $(\tau_1{=}\tau_2)$, which are the 'diagonal' values of the full wave drift QTF function and which represent the mean wave drift load data. The Newman method then approximates the 'off-diagonal' QTF values for component pairs with unequal directions or unequal periods by using an average of the 'diagonal' QTF values for those directions and periods. The approximation method also enables the double summation to be simplified to calculating the square of a single summation, so it reduces the calculation time to being proportional to $n$ rather than $n^2$.

For the wave drift load, OrcaFlex therefore provides a choice of QTF specification method: **Newman's approximation** or **full QTFs**. For the sum frequency wave load there is no equivalent to the Newman approximation, so the full QTF data must be provided and the full double summation calculation performed.

When the Newman approximation method is used, the user data specifies only the mean wave drift QTFs, the diagonal entries in the full QTF matrix. These diagonal QTF values $\Qdiag(\beta,\tau) = \Qd(\beta,\beta,\tau,\tau)$ must be real, on theoretical grounds, so you give these values directly as a signed real value, for each of a number of wave directions and periods. A positive data value gives an in-phase QTF load contribution, a negative value corresponds to an exactly out-of-phase contribution. No separate phase values are specified, and the phase conventions specified for the vessel type do not apply.

The off-diagonal wave drift QTF values, $\Qd(\beta_1,\beta_2,\tau_1,\tau_2)$ where $\beta_1{\neq}\beta_2$ or $\tau_1{\neq}\tau_2$, are approximated from the given diagonal QTFs that are on the same 'row' and 'column'. Newman's original approximation method used the arithmetic mean of the two diagonal QTFs, but OrcaFlex implements the following variant formulation developed later by Standing, Brendling and Wilson, which uses instead the geometric mean \begin{equation} \Qd(\beta_1,\beta_2,\tau_1,\tau_2) = \begin{cases} \sgn(\Qdiag(\beta_1,\tau_1))\ \lvert\Qdiag(\beta_1,\tau_1)\ \Qdiag(\beta_2,\tau_2)\rvert^{1/2} &\text{ if }\sgn(\Qdiag(\beta_1,\tau_1)) = \sgn(\Qdiag(\beta_2,\tau_2)) \\ %%surd gets misaligned, so use ^(1/2} or \sqrt 0 &\text{ if } \sgn(\Qdiag(\beta_1,\tau_1)) \neq \sgn(\Qdiag(\beta_2,\tau_2)) \end{cases} \end{equation} where $\lvert x\rvert$ denotes absolute value and $\sgn(x)$ is the signum, or sign, function.

The justification for the various forms of Newman approximation is that

- On theoretical grounds, the near-diagonal QTFs should be continuous in both $\tau_1$ and $\tau_2$. They should therefore be well-approximated by the mean of the diagonal values on the same row and column.
- Providing there is not a significant spread of wave directions, then the further-from-diagonal QTF values are less important since they give higher frequency contributions. For most vessels, their large inertia means that high frequency loads do not have much effect.

For a sea state with $n$ components, we have seen that the computational load is proportional to $n$ with the Newman approximation method but to $n^2$ with the full QTF method. Simulation might therefore be considerably slower if the full QTF method is used in combination with a large number of wave components.

On the other hand, this performance penalty only applies to the vessel calculation, and in most simulations the run time is usually determined to a greater extent by the other objects in the model (for example calculations for lines). In practice, this means that the only reliable way to determine how practical it is to use full QTFs is to run both with Newman's approximation and with full QTFs to see how the run times are affected for your particular model.

This test is easy to do, since the performance when using full QTFs is not particularly sensitive to the amount of QTF data, or to the data values (unless they are all zero) – the performance depends much more on the total number of wave components in the model. So you can easily run trial simulations with full QTFs by simply entering a small amount of dummy full QTF data, with arbitrary but small non-zero values, to see what effect this has on simulation run time before deciding whether to obtain and use correct full QTF data.

OrcaFlex uses complex-valued linear interpolation to derive the QTF values, $\Qd(\beta_1,\beta_2,\tau_1,\tau_2)$ and $\Qs(\beta_1,\beta_2,\tau_1,\tau_2)$, to use for directions $\beta_1,\beta_2$ or periods $\tau_1,\tau_2$ that are not present in the user's data. This interpolation is done as follows:

- Firstly, if, for any specified direction or direction pair, the user data do not specify the limiting QTF value for period $\tau{=}0$ or for period $\tau{=}\infty$, then a default limiting value is assumed at that period.
- OrcaFlex then obtains the QTF value, $Q$, for the interaction of each pair of wave components, by determining the specified directions and periods either side of the wave component directions and periods and using linear interpolation between the specified values. This interpolation is done by treating the complex-valued QTF $Q$ as a linear function of the directions and periods, with the exception that, if one of the periods is infinite, then the interpolation towards that limit is done according to frequency instead of according to period. In other words, for $\tau$ greater than the largest finite period given in the data, $Q$ is treated as a linear function of direction and
*frequency*, rather than linear with direction and*period*. - For wave drift QTFs specified using the Newman approximation, interpolation is only needed for the diagonal QTF, $\Qdiag(\beta,\tau)$, which is a function of two variables only. In this case the interpolation is done using two separate 1-dimensional interpolations, the first being on period $\tau$ (or on frequency if necessary), and the second on direction $\beta$.
- For bi-directional full QTFs (both wave drift and sum frequency), we have a QTF $Q(\beta_1,\beta_2,\tau_1,\tau_2)$ which is a function of four variables. The interpolation here is done using a 4-dimensional linear interpolation in the space whose coordinates are $\beta_1$, $\beta_2$, $\tau_1$ and $\tau_2$.
- Uni-directional full QTFs (both wave drift and sum frequency) have the form $Q(\beta_1,\beta_1,\tau_1,\tau_2)$, a function of only 3 variables, since the QTF is only needed for wave component pairs with equal directions $\beta_1$. So in this case the interpolation is done using a 3-dimensional linear interpolation space whose coordinates are $\beta_1,\tau_1,\tau_2$.

The default infinite-period limiting value is $Q{=}0$. The effect of this is that if the user data do not contain a QTF value for $\tau=\infty$ then, for periods greater than the longest period given in the data, the data will be linearly extrapolated (on frequency) towards zero QTF at infinite period (zero frequency). This default long-period QTF limit is theoretically correct for a freely floating body: since the body behaves like a cork and does not disturb the propagating wave, the second order wave load tends to zero for long periods (low frequencies).

At the short-period end, the data are *truncated*: if QTF values are not given for $\tau{=}0$, then the default zero-period limiting value is taken to be equal to the $Q$-value for the lowest period given in the data.

Warning: | QTF extrapolation could introduce significant errors if a significant amount of wave energy is outside the range of wave periods given by the QTF data. In addition, the default zero QTF value for long-period waves is only theoretically valid for a free-floating vessel: it could be inappropriate for a fixed or moored vessel, especially one which is firmly moored. If the default zero and infinite period limits are not suitable for your model, you should therefore take care to specify appropriate limiting QTFs in the data. |

OrcaFlex will calculate the wave drift damping effect on the wave drift load if both 'wave drift load' and 'wave drift damping' are among the included effects.

The wave drift damping is calculated using an encounter effects approach developed by Molin from Aranha's original analysis in deep water, but extended according to the results of Malenica *et al* to be applicable to all water depths. The velocity used in this calculation is the vessel low frequency velocity relative to the current, so the wave drift damping includes both the current effect on wave drift load and the damping effect on vessel low frequency motion.

Note: | These encounter effects are only applied to the wave drift load, not to the sum frequency load. OrcaFlex also only applies wave drift damping in the surge and sway directions, since there is not normally significant slow drift motion in heave, roll or pitch, and there is not yet a widely accepted method of modelling yaw wave drift damping. |

Molin uses the same form of Newman's approximation as OrcaFlex. The effect of wave drift damping is implemented by using modified wave drift QTF values, $\Qde(\beta,\beta,\tau,\tau)$, where the extra subscript 'e' has been added to denote that they allow for the encounter effects. The modified diagonal QTF values are given by \begin{equation} \label{Qde_Newman} \Qde(\beta,\beta,\tau,\tau) = A\enc\ \Qd(\beta\enc,\beta\enc,\tau\enc,\tau\enc) \end{equation} where

$A\enc = 1 + \bigl( \omega \PD\alpha\omega-2 \bigr) \frac{U_\mathrm{L}}{C_\mathrm{g}}=$ Aranha scaling factor

$\beta\enc = \beta + \frac{U_\mathrm{T}}{C_\mathrm{g}}=$ encounter heading

$\tau\enc = \frac{2\pi}{\omega\enc}=$ encounter period

$\omega\enc = \omega\left( 1-\frac{U_\mathrm{L}}{C_\mathrm{p}} \right)=$ encounter frequency

$C_\mathrm{p} = \frac{\omega}{k}=$ wave phase velocity

$C_\mathrm{g} = \PD{\omega}{k}=$ wave group velocity

$k=$ wave number

$\alpha = \frac{C_\mathrm{g}}{C_\mathrm{p}}$

$U=$ vessel low frequency velocity $-$ current velocity, at the wave drift QTF origin

$U_\mathrm{L}, U_\mathrm{T}=$ components of U in the wave component direction and transverse to that direction, respectively.

For water of depth $h$ we have

$\omega = \sqrt{g\ k \tanh(kh)}$

$g=$ acceleration due to gravity

$C_\mathrm{p} = \sqrt{\frac{g\tanh(kh)}{k}}$

$\alpha = \frac12\left( 1+\frac{2kh}{\sinh(2kh)} \right)$

$\PD{\alpha}{\omega} = \frac{h}{C_\mathrm{g}\sinh(2kh)} \left(1-\frac{2kh}{\tanh(2kh)}\right)$

In the case of deep water, these reduce to

$\omega = \sqrt{gk}$

$C_\mathrm{p} = \sqrt{\frac{g}{k}}$

$\alpha = \frac12$

$\PD{\alpha}{\omega} = 0$

which, finally, gives the terms required by equation (\ref{Qde_Newman})
\begin{align}
A\enc &= 1 - \frac{4\omega}{g}U_\mathrm{L} \text{ = Aranha scaling factor} \label{Aranha}\\
\beta\enc &= \beta + \frac{2\omega}{g}U_\mathrm{T} \text{ = encounter heading} \\
\omega\enc &= \omega - \frac{\omega^2}{g}U_\mathrm{L} \text{ = encounter frequency.} \label{EncounterFreq}
\end{align}
When wave drift damping is included, OrcaFlex uses these modified QTF values, $\Qde(\beta,\beta,\tau,\tau)$, for surge and sway. The *unmodified* QTFs are always used for heave, roll, pitch and yaw, and of course for all degrees of freedom if wave drift damping is not included.

By modifying the diagonal values of the QTFs in this way, before applying Newman's approximation, OrcaFlex is able to incorporate the time-varying effect of wave drift damping, not just the constant mean value.

OrcaFlex uses the same approach for full QTFS as for the Newman approximation method, taking the QTF values for the encounter directions and encounter periods of the two wave components in each pair and scaling by an Aranha factor.

However, in the literature the Aranha factor is only defined for the mean drift QTFs, where the two components in the pair have the same frequencies. For full QTFs, we must extend this to wave component pairs with different frequencies. In OrcaFlex we do so by using the geometric mean of the mean drift Aranha factors for the two components \begin{equation} \Qde(\beta_1,\beta_2,\tau_1,\tau_2) = A\enc\ \Qd(\beta_{1\mathrm{e}},\beta_{2\mathrm{e}},\tau_{1\mathrm{e}},\tau_{2\mathrm{e}}) \end{equation} where

$A\enc = \sqrt{A_{1\mathrm{e}} A_{2\mathrm{e}}}=$ geometric mean of the mean drift Aranha scaling factors for the two wave components

$A_{i\mathrm{e}} = 1 + \bigl( \omega_i \PD{\alpha_i}{\omega_i}-2 \bigr) \frac{U_\mathrm{L}}{C_{\mathrm{g},i}}=$ mean drift Aranha factor for component $i{=}1,2$

$\beta_{i\mathrm{e}} = \beta + \frac{U_\mathrm{T}}{C_{\mathrm{g},i}}=$ encounter direction for component $i{=}1,2$

$\tau_{i\mathrm{e}} = \frac{2\pi}{\omega_{i\mathrm{e}}}=$ encounter period for component $i{=}1,2$

$\omega_{i\mathrm{e}} = \omega_i \left( 1-\frac{U_\mathrm{L}}{C_{\mathrm{p},i}} \right)=$ encounter frequency for component $i{=}1,2$.

The theory of wave drift damping assumes that the vessel velocity is small compared to the wave speed; if it is not, then these formulae can result in unreasonable encounter effects. In particular, we see from equation (\ref{Aranha}) that the Aranha scaling factor in deep water will be negative if $U_\mathrm{L}$ exceeds a quarter of the wave crest speed, $g/\omega$. For even higher vessel speeds, the encounter frequency, $\omega\enc$ in equation (\ref{EncounterFreq}), could become negative, which is clearly not physically meaningful.

At this point the assumptions have broken down and the theory is no longer valid. To avoid this problem, for both Newman QTFs and full QTFs, OrcaFlex limits the Aranha factor to be non-negative. This effectively says that the wave drift damping effect for a given wave component can be no greater than (and opposite to) the wave drift load itself for that component. In other words, for a sufficiently fast-moving vessel and slow-moving wave component, the wave drift load contribution can be completely cancelled out, *but not reversed*, by the wave drift damping. This is only likely to arise for short-wave components, since they travel more slowly. OrcaFlex will issue a warning if this happens for a significant proportion of the wave energy in the sea state.