## Vessel types: Wave drift and sum frequency QTFs |

The wave drift QTFs and sum frequency QTFs pages on the vessel type data form contain the quadratic transfer function (QTF) data that OrcaFlex uses to calculate a wave drift load and sum frequency QTF load, respectively. Wave drift load can be used to model vessel slow drift and for this reason is often referred to as slow drift load. Different draughts of the vessel type have their own separate QTF data and QTF origins.

Wave drift loads can be important for moored vessels, since they include low frequency components that might excite slow drift motion of the vessel on its moorings. Sum frequency loads include high frequency components that can be important in systems with high frequency natural modes, such as ringing of TLPs.

Note: | The wave drift load is only calculated for a vessel if the wave drift load (2 option is checked in the list of vessel included effects. The sum frequency QTF load is only calculated if the ^{nd} order)sum frequency load (2 option is checked in the list of vessel included effects. These loads will only affect the vessel motion if the vessel primary motion is set to one of the ^{nd} order)calculated options. |

QTFs, like wave load RAOs, are given in dimensional form and with respect to waves of unit amplitude. Translational QTFs have units of force per unit length squared $(F/L^2)$, and rotational QTFs of moments per unit length squared $(FL/L^2)$.

See wave drift and sum frequency loads for details of the second order load calculation in OrcaFlex.

For the wave drift load (but not for the sum frequency load) OrcaFlex supports two calculation methods: **Newman's approximation** and **full QTFs**.

The full QTF method is generally preferable, but Newman's method requires far fewer data, since it only requires the 'diagonal' QTF data that define the mean wave drift load. The Newman approximation then extrapolates the mean wave drift QTFs to approximate the off-diagonal QTF values, which allows calculation of the slowly-varying part of the wave drift load. The Newman method is also much less computationally intensive, meaning that the practicality of using full QTFs needs to be assessed on a per simulation basis: these guidance notes may be of some help in this.

The Newman method is not appropriate for the sum frequency load: in this case, the full QTF data and calculation must always be used.

The QTF origin is the point on the vessel to which the QTFs apply. The second order wave load is applied at this point. Wave drift damping (if included) is calculated using the low frequency primary motion velocity of this point

The QTF origin is defined relative to the OrcaFlex vessel axes (*not* the component directions given on the conventions page).

The second order wave load is calculated based on the wave conditions at the instantaneous primary motion position of the QTF phase origin.

Full QTF phases are defined relative to the time the wave passes this origin. For details see QTF data for full QTF method below. This phase origin does not apply to wave drift QTFs which use Newman's approximation.

The phase origin is specified by giving its coordinates with respect to the OrcaFlex vessel axes (*not* the component directions given on the conventions page). It will often be at the same point as the QTF origin, in which case the phase origin coordinates can be set to '~', meaning 'same as QTF origin'.

When Newman's method is used, you give only the diagonal terms of the full QTF matrix. These are the mean wave drift QTFs, and on theoretical grounds they must be real-valued (hence they have no phase). The effect of applying Newman's approximation is that the off-diagonal QTFs are approximated by the geometric means of the diagonal QTFs for the two directions and periods involved. For details see vessel theory: Wave drift loads and sum frequency QTFs.

The QTF data for Newman's approximation are entered in a similar way to RAO data, as follows:

- For each draught, QTF tables are specified for each of a number of wave directions. To insert a new table use the
**insert direction**button and to delete a table select that table's page and then click the**delete direction**button. To change the direction associated with a table, select that table's page and then edit the**selected direction**value. - The vessel type direction conventions apply to the QTFs.
- If the vessel type has some symmetry (see the conventions page) then OrcaFlex automatically generates QTF tables for all the reflected directions implied by that symmetry. You must provide QTF tables for enough directions for OrcaFlex to have data (either user-specified or generated based on symmetry) for directions that cover the wave directions the vessel will experience.
- Each QTF table consists of data for a range of wave periods or frequencies (depending on the convention specified). You should provide data for periods that (after allowing for Froude scaling) cover the wave periods the vessel will experience.
- The QTF data will be interpolated on direction and period as required.

Full QTF data are entered in a single table. The rows in the table may be given in any order; each row contains the following data:

- The first two columns contain the vessel-relative directions $\beta_1$ and $\beta_2$ of a pair of wave components. Directions can be positive or negative, so for example directions 270° and -90° are equivalent.
- The next two columns contain the periods $\tau_1$ and $\tau_2$ or the corresponding frequencies (in rad/s or Hz) of the pair of wave components, governed by the
**waves referred to**convention. - The remaining columns contain, for each of the six degrees of freedom, the amplitudes and phases of the QTF, $Q(\beta_1,\beta_2,\tau_1,\tau_2)$, that applies to a pair of wave components with those vessel-relative directions and periods, $(\beta_1,\tau_1)$ and $(\beta_2,\tau_2)$. These must be given in accordance with the vessel type conventions for phase and directions.

The QTF phase values specify the phase timing of the peak positive QTF load contribution in the positive direction given by the direction conventions, relative to the time that the wave crest, trough or zero down- or up-crossing occurs at the primary motion position of the QTF phase origin. Whether the phases are leads or lags, in degrees or radians, relative to trough, crest, zero down-crossing or zero up-crossing, are all determined by the phase conventions. The resulting total QTF load is then applied at the QTF origin (which may, if necessary, differ from the QTF phase origin).

You should provide data for directions and periods that (after allowing for Froude scaling) cover the wave periods the vessel will experience. The QTFs for wave component pairs with other directions and periods are obtained using linear interpolation on period and direction; for details see vessel theory: Wave drift loads and sum frequency QTFs.

The number of rows in the table can be altered by changing the **number of QTFs**. Specific rows can be removed by highlighting cells in the appropriate rows and pressing delete. The full QTF table will normally have a very large number of rows. If we denote by $\beta_i, i{=}1,\ldots,n_\beta$ the directions, and by $\tau_j, j{=}1,\ldots,n_\tau$ the periods given, then the complete full QTF data table for these directions and periods would have $n_\beta^2\,n_\tau^2$ rows. So for example, if $n_\beta{=}12$ (30° intervals covering the full 360°) and $n_\tau{=}26$ (1s intervals covering periods from 5s to 30s, say), then the complete full QTF table would have $12^2\,26^2{=}97344$ rows. This amount of data clearly cannot be typed in. It can however be imported or pasted into the table from a separate source, such as Excel.

However, the amount of data required will be reduced if it falls into one or more of the following three categories.

Some diffraction programs only output QTFs for pairs of wave components with the same direction, whereas others also output QTFs for pairs of wave components whose directions differ. We call these two types of QTF data **unidirectional** and **bidirectional**, respectively.

OrcaFlex allows you to enter QTF data of either type, and automatically detects whether you specify unidirectional or bidirectional data, as follows:

- OrcaFlex will treat the QTF data as
**bidirectional**if there are*any*rows in the full QTFs table with $\beta_1\neq\beta_2$. In this case you must provide data for*all*$n_\beta^2$ combinations of directions. - OrcaFlex will treat the QTF data as
**unidirectional**if*all*rows have $\beta_1=\beta_2$. In this case the QTFs are being specified only for the $n_\beta$ direction pairs $(\beta_i,\beta_i),\ i=1,\ldots,n_\beta$, so the number of rows in the full QTF table is reduced by a factor of $1/n_\beta$. But for any pair of wave components that have differing directions, $\beta_1\neq\beta_2$, OrcaFlex will now have to approximate, using $Q(\beta_1,\beta_2,\tau_1,\tau_2) \approx Q(\beta_\textrm{mean},\beta_\textrm{mean},\tau_1,\tau_2)$ for that pair, where $\beta_\textrm{mean}$ is the direction that is half way (modulo 360°) between $\beta_1$ and $\beta_2$. It is therefore preferable to provide bidirectional, rather than unidirectional, full QTF data when the sea state includes multiple wave directions.

In both cases, OrcaFlex will use linear interpolation to obtain the QTFs for wave component directions not present in the table. This interpolation includes wrapping through 360°, so for example if the directions given are (-90°, -60°, -30°, 0°, 30°, 60°, 90°) then for direction 180° OrcaFlex will linearly interpolate between the QTFs given for directions 90° and -90°, since in the full circle of directions 180° lies halfway between those directions. The range of directions should therefore cover the range of wave component directions present in the sea state, and with a sufficiently small direction interval for the accuracy wanted.

OrcaFlex will use any **symmetry** specified by the vessel type conventions.

For unidirectional QTF data *only*, this reduces the number of directions $n_\beta$ required, according to the symmetry convention as follows:

**xz plane**: the directions must all be on the same side of the xz plane. That is, either $0\degree\leq(\beta \bmod 360\degree)\leq180\degree$ for all directions $\beta$, or $180\degree\leq(\beta \bmod 360\degree)\leq360\degree$ for all $\beta$. The QTFs for the other half-plane will be derived using the symmetry condition.**yz plane**: the directions must all be on the same side of the yz plane. That is, either $-90\degree\leq(\beta \bmod 360\degree)\leq90\degree$ for all directions $\beta$, or $90\degree\leq(\beta \bmod 360\degree)\leq270\degree$ for all $\beta$. The QTFs for the other half-plane will be derived using the symmetry condition.**xz and yz planes**: the directions must all be in the same quadrant. That is, one of the following must hold*for all*$\beta$: $0\degree\leq(\beta \bmod 360\degree)\leq90\degree$, or $90\degree\leq(\beta \bmod 360\degree)\leq180\degree$, or $180\degree\leq(\beta \bmod 360\degree)\leq270\degree$, or $270\degree\leq(\beta \bmod 360\degree)\leq360\degree$. The QTFs for the other quadrants will be derived using the symmetry condition.**Circular**: only one wave direction must be specified. The QTFs for all other directions will be derived using the symmetry condition.

For bidirectional full QTFs, vessel symmetry results in more complicated relationships which do not lead to simple reductions in the number of QTF directions required. Bidirectional QTFs therefore require the QTFs for *all* direction combinations, and the directions *must* cover the full 360° range of directions.

Warning: | If the vessel type symmetry defined in the conventions includes a plane of symmetry, then bidirectional full QTFs must cover the full 360° range of directions. It is not sufficient to only specify directions on one side of the symmetry plane in this case. And, of course, the QTF data should obey the defined symmetry. |

QTF theory shows that the complete complex-value sum frequency QTF matrix must be symmetric and the complete complex value wave drift (difference frequency) QTF matrix must be conjugate-symmetric. That is, for any $i,j$ we have the following symmetry equations for the complex-valued QTFs \begin{equation} \begin{aligned} Q_\mathrm{d}(\beta_i,\beta_j,\tau_i,\tau_j) &= Q_\mathrm{d}(\beta_j,\beta_i,\tau_j,\tau_i)^* \\ Q_\mathrm{s}(\beta_i,\beta_j,\tau_i,\tau_j) &= Q_\mathrm{s}(\beta_j,\beta_i,\tau_j,\tau_i) \end{aligned} \label{symmetry} \end{equation} where $Q_\mathrm{d}$ is the wave drift QTF, $Q_\mathrm{s}$ is the sum frequency QTF, and superscript $^*$ denotes the complex conjugate. For details of the definition of these complex value QTFs see theory for full QTFs.

OrcaFlex takes advantage of these symmetry relations, and allows you to choose which data you enter from the following:

**upper triangle only**: all period combinations with $\tau_2\geq\tau_1$**lower triangle only**: all period combinations with $\tau_2\leq\tau_1$**both upper and lower triangles**: the whole matrix; that is, all combinations of periods.

If only the upper or lower triangle is given, not both, then OrcaFlex will use the symmetry equations (\ref{symmetry}) to obtain the QTFs from the other triangle. This reduces the amount of data required by almost a factor of 2. OrcaFlex will also enforce the condition that difference frequency QTFs are real-valued when $i{=}j$. If the whole matrix is given, then OrcaFlex ensures that the symmetry conditions hold by using recalculated QTFs $\bar{Q}_\mathrm{d}$ and $\bar{Q}_\mathrm{s}$ defined as \begin{equation} \begin{aligned} \bar{Q}_\mathrm{d}(\beta_i,\beta_j,\tau_i,\tau_j) &= \tfrac12 [ Q_\mathrm{d}(\beta_i,\beta_j,\tau_i,\tau_j) + Q_\mathrm{d}(\beta_j,\beta_i,\tau_j,\tau_i)^* ] \\ \bar{Q}_\mathrm{s}(\beta_i,\beta_j,\tau_i,\tau_j) &= \tfrac12 [ Q_\mathrm{s}(\beta_i,\beta_j,\tau_i,\tau_j) + Q_\mathrm{s}(\beta_j,\beta_i,\tau_j,\tau_i) ] \end{aligned} \end{equation}

The full QTF calculation involves a double summation over all pairs of wave components in the sea state. This can be very time consuming, and many of the resulting QTF load contributions might have frequencies that are not significant in the situation being modelled. In this case, you can reduce the amount of computation by specifying a **cutoff period**. OrcaFlex will then omit QTF load contributions that are beyond that cutoff period, as follows:

- For the wave drift QTF load, it is generally only the low frequency, long-period contributions that are significant, so OrcaFlex will suppress contributions whose periods are
**less than**the wave drift cutoff period. To include the entire wave drift QTF load, set this cutoff period to zero. - In the case of sum frequency QTFs, usually only the high frequency, short-period contributions are of interest, so OrcaFlex will suppress those whose periods are
**greater than**the sum frequency cutoff period. To include the whole sum QTF load, set this cutoff period to infinity.

Note that OrcaFlex tapers the cut off, by gradually scaling down (instead of suppressing completely) contributions whose periods are up to 10% beyond the specified cutoff period. This is necessary to avoid the QTF load being a discontinuous function of the wave component frequencies. The behaviour of the cut off is illustrated by the following figure:

Figure: | The application of the QTF cut off period |

All of the above data can generally be obtained from the results of a diffraction program. OrcaFlex can import these data directly from the output files of AQWA and WAMIT, and for Newman QTFs many other diffraction program output file formats can be imported if OrcaFlex-specific markers are first added to the file.

Warning: | The settings on the conventions page apply to all draughts and they apply to both the vessel type's RAOs and to its wave drift and sum frequency QTFs. If your RAO and QTF data are in different files and use different conventions you will therefore need to take account of this, for example by defining the different convention systems for each in the files from which they are imported. |

OrcaFlex may modify the input QTFs before use, in order to make best use of diffraction analysis data. There are specific settings on the conventions page which affect only this modification, the QTF data source conventions. OrcaFlex will set some of the QTF data source conventions for import from a WAMIT input file, but not for import from AQWA.