4.2.8.4. Modeling Considerations¶

HydroDyn was designed as an extremely flexible tool for modeling a wide-range of hydrodynamic conditions and substructures. This section provides some general guidance to help you construct models that are compatible with HydroDyn.

4.2.8.4.1. Waves¶

Waves generated internally within HydroDyn can be regular (periodic) or irregular (stochastic) and long-crested (unidirectional) or short-crested (with wave energy spread across a range of directions). Internally, HydroDyn generates waves analytically for finite depth using first-order (linear Airy) or first- plus second-order wave theory [Sharma and Dean, 1981] with the option to include directional spreading, but wave kinematics are only computed in the domain between the flat seabed and SWL and no wave stretching or higher order wave theories are included. Modeling unidirectional sea states is often overly conservative in engineering design. Enabling the second-order terms allows one to capture some of the nonlinearities of real surface waves, permitting more accurate modeling of sea states and the associated wave loads at the expense of greater computational effort (mostly at HydroDyn initialization). The magnitude and frequency content of second-order hydrodynamic loads can excite structural natural frequencies, leading to greater ultimate and fatigue loads than can be predicted solely using first-order theory. Sum-frequency effects are important to the loading of stiff fixed-bottom structures and for the springing and ringing analysis of TLPs. Difference-frequency (mean-drift and slow-drift) effects are important to the analysis of compliant structures, including the motion analysis and mooring loads of catenary-moored floating platforms (spar buoys and semi-submersibles).

When modeling irregular sea states, we recommend that WaveTMax be set to at least 1 hour (3600 s) and that WaveDT be a value in the range between 0.1 and 1.0 s to ensure sufficient resolution of the wave spectrum and wave kinematics. When HydroDyn is coupled to FAST, WaveDT may be specified arbitrarily independently from the glue code time step of FAST. (The wave kinematics and hydrodynamic loads will be interpolated in time as necessary.)

Wave directional spreading is implemented in HydroDyn via the equal-energy method, which assumes that the directional spreading spectrum is the product of a frequency spectrum and a spreading function i.e. S(ω,β) = S(ω)D(β). Directional spreading is not permitted when using Newman’s approximation of the second-order difference-frequency potential-flow loads.

When second-order terms are optionally enabled, the second-order terms are calculated using the first-order wave-component amplitudes and extra energy is added to the wave spectrum (at the difference and sum frequencies). The second-order terms cannot be computed without also including the first-order terms.

It is important to set proper wave cut-off frequencies to minimize computational expense and to ensure that the wave kinematics and hydrodynamic loads are realistic. HydroDyn gives the user six user-defined cut-off frequencies—WvLowCOff and WvHiCOff for the low- and high-frequency cut-offs of first-order wave components, WvLowCOffD and WvHiCOffD for the low- and high-frequency cut-offs of second-order difference-frequency wave components, and WvLowCOffS and WvHiCOffS for low- and high-frequency cut-offs of second-order sum-frequency wave components—none of which have default settings. The second-order cut-offs apply directly to the physical difference and sum frequencies, not the two individual first-order frequency components of the difference and sum frequencies. Because the second-order terms are calculated using the first-order wave-component amplitudes, the second-order cut-off frequencies are used in conjunction with the first-order cut-off frequencies. However, the second-order cut-off frequencies are not used by Newman’s approximation of the second-order difference-frequency potential-flow loads, which are derived solely from first-order effects.

For the first-order wave-component cut-off frequencies, WvLowCOff may be set lower than the low-energy limit of the first-order wave spectrum to minimize computational expense. Setting a proper upper cut-off frequency (WvHiCOff) also minimizes computational expense and is important to prevent nonphysical effects when approaching of the breaking-wave limit and to avoid nonphysical wave forces at high frequencies (i.e., at short wavelengths) when using a strip-theory solution.

When enabling second-order potential-flow theory, a setting of WvLowCOffD = 0 is advised to avoid eliminating the mean-drift term (second-order wave kinematics do not have a nonzero mean). WvHiCOffD need not be set higher than the peak-spectral frequency of the first-order wave spectrum (ωp = 2π/WaveTp) to minimize computational expense. WvLowCOffS need not be set lower than the peak-spectral frequency of the first-order wave spectrum (ωp = 2π/WaveTp) to minimize computational expense. Setting a proper upper cut-off frequency (WvHiCOffS) also minimizes computational expense and is important to (1) ensure convergence of the second-order summations, (2) avoid unphysical “bumps” in the wave troughs, (3) prevent nonphysical effects when approaching of the breaking-wave limit, and (4) avoid nonphysical wave forces at high frequencies (i.e., at short wavelengths) when using a strip-theory solution.

For all models with internally generated wave data, if you want to run different time-domain incident wave realizations for given boundary conditions (of significant wave height, and peak-spectral period, etc.), you should change one or both wave seeds (WaveSeed(1) and WavedSeed(2)) between simulations.

Wave elevations or full wave kinematics can also be generated externally and used within HydroDyn.

WaveMod = 5 allows the use of externally generated wave-elevation time series, which is useful if you want HydroDyn to simulate specific wave transient events where the wave-elevation time series is known a priori e.g. to match wave-elevation measurements taken from a wave tank or open-ocean test. Internally, HydroDyn will compute an FFT of the provided wave-elevation time series to store the amplitudes and phases of each frequency component, and use those in place of a wave energy spectrum and random seeds to internally derive the hydrodynamic loads in the potential-flow solution or the wave kinematics used in the strip-theory solution. The wave-elevation time series specified is assumed to be of first order and long-crested, but is not checked for physical correctness. The time series must be at least WaveTMax in length and not less than the total simulation time and the time step must match WaveDT. When second-order terms are optionally enabled, the second-order terms are calculated using the wave-component amplitudes derived from the provided wave-elevation time series and extra energy is added to the wave energy spectrum (at the difference and sum frequencies). Using higher order wave data may produce erroneous results; alternatively, WvLowCOff and WvHiCOff can be used to filter out energy outside of the first-order wave energy range. The wave-elevation time series output by HydroDyn will only match the specified time series identically if the second-order terms are disabled and the cut-off frequencies are outside the range of wave energy.

WaveMod =6 allows the use of full externally generated wave kinematics for use with the strip-theory solution (but not the potential-flow solution), completely bypassing HydroDyn’s internal wave models. This feature is useful if you want HydroDyn to make use of wave kinematics data derived outside of HydroDyn a priori e.g. from a separate numerical tool, perhaps bypassing some of HydroDyn’s internal wave modeling limitations. To use this feature, it is the burden of the user to generate wave kinematics data at each of HydroDyn’s time steps and analysis nodes. HydroDyn will not interpolate the data; as such, when HydroDyn is coupled to FAST, WaveDT must equal the glue code time step of FAST. Before generating the wave kinematics data externally, users should identify all of the internal analysis nodes by running HydroDyn and generating the summary file—see Section 4.2.8.3.3. The fluid domain at each time step are specified by the use of numeric values and nonnumeric strings in the wave data input files. The wave kinematics data specified are not limited to the domain between a flat seabed and SWL and may consider wave stretching, higher-order wave theories, or an uneven seabed. The specified wave kinematics data are not processed (filtered, etc.) or checked for physical correctness. The wave kinematics output by HydroDyn should match the specified data identically.

You can generate up to 9 wave elevation outputs (at different points on the SWL plane) when HydroDyn is coupled to FAST or a large grid of wave elevations when running HydroDyn standalone. While the second-order effects are included when enabled, the wave elevations output from HydroDyn will only include the second-order terms when the second-order wave kinematics are enabled.

4.2.8.4.2. Strip-Theory Model Discretization¶

A user will define the geometry of a structure modeled with strip theory in HydroDyn using joints and members. Members in HydroDyn are assumed to be straight circular (and possibly tapered) cylinders. Members can be further subdivided using MDivSize, which HydroDyn will internally use to subdivide members into multiple elements (and nodes). HydroDyn may further refine the geometry at the free surface, flat seabed, marine-growth region, and filled-fluid free surface.

Due to the exponential decay of hydrodynamic loads with depth, a higher resolution near the water free surface is required to capture hydrodynamic loading as waves oscillate about SWL. It is recommended, for instance, that the HydroDyn discretization not exceed element lengths of 0.5 m in the region of the free surface (5 to 10 m above and below SWL), 1.0 m between 25 and 50 m depth, and 2.0 m in deeper waters. When HydroDyn is coupled to SubDyn through FAST for the analysis of fixed-bottom systems, it is recommended that the length ratio between elements of HydroDyn and SubDyn not exceed 10 to 1.

4.2.8.4.3. Domain for Strip-Theory Hydrodynamic Load Calculations¶

Part of the automated geometry refinement mentioned in the above section deals with splitting of input members into sub-elements such that both of the resulting nodes at the element ends lie within the discrete domains described in the following sections.

4.2.8.4.3.1.1. Inertia, Added Mass, Buoyancy, Marine-Growth Weight, Marine-Growth Mass Inertia¶

These loads are generated at a node as long as PropPot = FALSE, the Z-coordinate is in the range [–WtrDpth,MSL2SWL], and the element the node is connected to is in the water. When WaveMod = 6, the domain is determined by the use of numeric values and nonnumeric strings in the wave data input files.

4.2.8.4.3.1.2. Viscous Drag¶

These loads are generated at a node as long as the Z-coordinate is in the range [–WtrDpth, MSL2SWL] and the element the node is connected to is in the water. When WaveMod = 6, the domain is determined by the use of numeric values and nonnumeric strings in the wave data input files.

4.2.8.4.3.1.3. Filled Buoyancy, Filled Mass Inertia¶

These loads are generated at a node as long as the Z-coordinate is in the range [–WtrDpth, FillFSLoc] and the element the node is connected to is in the filled fluid.

Lumped loads at member ends (axial effects) are only calculated at user-specified joints, and not at joints HydroDyn may automatically create as part its solution process. For example, if you want axial effects at a marine-growth boundary, you must explicitly set a joint at that location.

These loads are generated at a node as long as PropPot = FALSE and the Z-coordinate is in the range [–WtrDpth,MSL2SWL]. When WaveMod = 6, the domain is determined by the use of numeric values and nonnumeric strings in the wave data input files.

4.2.8.4.3.2.2. Axial Drag¶

These loads are generated at a node as long as the Z-coordinate is in the range [–WtrDpth,MSL2SWL]. When WaveMod = 6, the domain is determined by the use of numeric values and nonnumeric strings in the wave data input files.

4.2.8.4.3.2.3. Filled Buoyancy¶

These loads are generated at a node as long as the Z-coordinate is in the range [–WtrDpth,FillFSLoc]

4.2.8.4.4. Strip-Theory Hydrodynamic Coefficients¶

The strip-theory solution of HydroDyn is dependent, among other factors, on user-specified hydrodynamic coefficients, including viscous-drag coefficients, Cd, added-mass coefficients, Ca, and dynamic-pressure coefficients, Cp, for transverse and axial (Ax) loads distributed along members and for axial lumped loads at member ends (joints). There are no default settings for these coefficients in HydroDyn. In general, these coefficients are dependent on many factors, including Reynold’s number (Re), Keulegan-Carpenter number (KC), surface roughness, substructure geometry, and location relative to the free surface, among others. In practice, the coefficients are (1) selected from tables derived from measurements of flow past cylinders, (2) calculated through high-fidelity computational fluid dynamics (CFD) solutions, or (3) tuned to match experimental results. A value of 1.0 is a plausible guess for all coefficients in the absence of any other information.

While the strip-theory solution assumes circular cross sections, the hydrodynamic coefficients can include shape corrections; however, there is no distinction made in HydroDyn between different transverse directions.

Please note that added-mass coefficients in HydroDyn influence both the added-mass loads and the scattering component of the fluid-inertia loads. For the coefficients associated with transverse loads distributed along members, note that $$C_{P} + C_{A} = C_{M}$$, the inertia coefficient. For the distributed loads along members, there are separate set of hydrodynamic coefficients both with and without marine growth (MG).

4.2.8.4.5. Impact of Substructure Motions on Loads¶

In general, HydroDyn assumes that structural motions of the substructure are small, such that (1) small-angle assumptions apply to structural rotations, (2) the frequency-to-time-domain-based potential-flow solution can be split into uncoupled hydrostatic, radiation, and diffraction solutions, and (3) the hydrodynamic loads dependent on wave kinematics (both from diffraction loads in the potential-flow solution and from the fluid-inertia and viscous-drag loads in the strip-theory solution) can be computed using wave kinematics solved at the undisplaced position of the substructure (the wave kinematics are not recomputed at the displaced position). Nevertheless, HydroDyn uses the substructure motions in the following calculations:

• The structural displacements of the WRP are used in the calculation of the hydrostatic loads (i.e., the change in buoyancy with substructure displacement) in the potential-flow solution.

• The structural velocities and accelerations of the WRP are used in the calculation of the wave-radiation loads (i.e., the radiation memory effect and added mass) in the potential-flow solution.

• The structural displacements and velocities of the WRP are used in the calculation of the additional platform loads (via the Platform Additional Stiffness and Damping).

• The structural velocities of the substructure nodes are used in the calculation of the viscous-drag loads in the strip-theory solution (e.g., the relative form of Morison’s equation is applied).

• The structural accelerations of the substructure nodes are used in the calculation of the added-mass, marine-growth mass inertia, and filled-fluid mass inertia loads in the strip-theory solution.

• When coupled to FAST, the hydrodynamic loads computed by HydroDyn are applied to the displaced position of the substructure (i.e., the displaced platform in ElastoDyn and/or the displaced substructure in SubDyn), but are based on wave kinematics at the undisplaced position.

4.2.8.4.7. Fixed-Bottom Substructures¶

When modeling a fixed-bottom system, the use of a strip-theory (Morison) only model is recommended. When HydroDyn is coupled to FAST, SubDyn is used for the substructure structural dynamics.

All members that are embedded into the seabed (e.g., through piles or suction buckets) must have a joint that is located below the water depth. For example, if the water depth is set to 20 m, and you are modeling a fixed-bottom monopile, then the bottom-most joint needs to have a Z-coordinate such that m. This configuration avoids having HydroDyn apply static pressure loads on the bottom of the structure.

Gravity-based foundations should be modeled such that the lowest joint(s) are located exactly at the prescribed water depth. In other words, the lowest Z-coordinate should be set to m if the water depth is set to 20 m. This configuration allows for static pressure loads to be applied at the bottom of the gravity-base structure.

4.2.8.4.8. Floating Platforms¶

When modeling a floating system, you may use potential-flow theory only, strip-theory (Morison) only, or a hybrid model containing both.

Potential-flow theory based on frequency-to-time-domain transforms is enabled when PotMod is set to 1. In this case, you must run WAMIT (or equivalent) in a pre-processing step and HydroDyn will use the WAMIT output files—see Section 4.2.8.4.8.4 for guidance. For a potential-flow-only model, do not create any strip-theory joints or members in the input file. The WAMIT model should account for all of the members in the floating substructure, and Morison’s equation is neglected in this case.

For a strip-theory-only model, set PotMod to FALSE and create one or more strip-theory members in the input file. Marine growth and nonzero MSL2SWL (the offset between still-water and mean-sea level) may only be included in strip-theory-only models.

A hybrid model is formed when both PotMod is TRUE and you have defined one or more strip-theory members. The potential-flow model created can consider all of the Morison members in the floating substructure, or just some. Specify whether certain members of the structure are considered in the potential-flow model by setting the PropPot flag for each member. The state of the PropPot flag for a given member determines which components of the strip-theory equations are applied.

When using either the strip-theory-only or hybrid approaches, filled fluid (flooding or ballasting) may be added to the strip-theory members. Also, the hydrostatic restoring matrix must be entered manually for the strip-theory members—see Section 4.2.8.4.8.3 for guidance.

Please note that current-induced water velocity only induces hydrodynamic loads in HydroDyn through the viscous-drag terms (both distributed and lumped) of strip-theory members. Current is not used in the potential-flow solution. Thus, modeling the effects of current requires the use of a strip-theory-only or hybrid approach.

4.2.8.4.8.1. Undisplaced Position for Floating Systems¶

The HydroDyn model (geometry, etc.) is defined about the undisplaced position of the substructure. For floating systems, it is important for solution accuracy for the undisplaced position to coincide with the static-equilibrium position in the platform-heave (vertical) direction in the absence of loading from wind, waves, and current. As such, the undisplaced position of the substructure should be defined such that the external buoyancy from displaced water balances with the weight of the system (including the weight of the rotor-nacelle assembly, tower and substructure) and mooring system pretension following the equation below. In this equation, is the water density, is gravity, is the undisplaced volume of the floating platform (found in the HydroDyn summary file), is the total mass of the system (found in the ElastoDyn summary), and is the mooring system pretension (found in e.g. the MAP summary file). The effects of marine growth, filled fluid (flooding and/or ballasting), and the additional static force (AddFX0) should also be taken into consideration in this force balance, where appropriate.

(4.178)$\rho g V_{0} - m_{Total} g - T_{Mooring} = 0$

4.2.8.4.8.2. Initial Conditions for Floating Systems¶

Because the initial conditions used for dynamic simulations typically have an effect on the response statistics during the beginning of the simulation period, an appropriate amount of initial data should be eliminated from consideration in any post-processing analysis. This initial condition solution is more important for floating offshore wind turbines because floating systems typically have long natural periods of the floating substructure and low damping. The appropriate time to eliminate should be chosen such that initial numeric transient effects have sufficiently decayed and the floating substructure has reached a quasi-stationary position. To decrease this initial time in each simulation, it is suggested that the initial conditions of the model (especially blade-pitch angle, rotor speed, substructure surge, and substructure pitch in ElastoDyn) be initialized according to the specific prevalent wind, wave, current, and operational conditions.

4.2.8.4.8.3. Hydrostatic Restoring for Strip-Theory Members of Floating Systems¶

One notable absence from the list calculations in HydroDyn that make use of substructure motions—see Section 4.2.8.4.3—is that the substructure buoyancy in the strip-theory solution is not recomputed based on the displaced position of the substructure. While the change in buoyancy is likely negligible for fixed-bottom systems, for floating systems modeled using a strip-theory solution, the change in buoyancy with displacement is likely important and should not be neglected. In this latter case, the user should manually calculate the 6x6 linear hydrostatic restoring matrix associated with the strip-theory members and enter this as the additional linear restoring (stiffness) matrix, AddCLin. (The static buoyancy of the strip-theory members is automatically calculated and applied within HydroDyn.)

In its most general form, the 6x6 linear hydrostatic restoring matrix of a floating platform is given by the equation below.

(4.179)$\begin{split}\text{AddCLin} = \left[ \begin{array}{cccccc} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \rho g A_{0} & \rho g \iint_{A_{0}} ydA & -\rho g \iint_{A_{0}} xdA & 0 \\ 0 & 0 & \rho g \iint_{A_{0}} ydA & \rho g \iint_{A_{0}} y^2dA + \rho g V_{0} z_{b} - m_{mg}gz_{mg} - m_{f}gz_{f} & -\rho g \iint_{A_{0}} xydA & -\rho g V_{0} x_{b} + m_{mg}gx_{mg} + m_{f}gx_{f} \\ 0 & 0 & -\rho g \iint_{A_{0}} xdA & -\rho g \iint_{A_{0}} xydA & \rho g \iint_{A_{0}} x^2dA + \rho g V_{0} z_{b} - m_{mg}gz_{mg} - m_{f}gz_{f} & -\rho g V_{0} y_{b} + m_{mg}gy_{mg} + m_{f}gy_{f} \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array} \right]\end{split}$

where:

• $$\rho$$ is water density (kg/m3)

• $$g$$ is gravity (m/s2)

• $$A_{0}$$ is undisplaced waterplane area of platform (m2)

• $$V_{0}$$ is undisplaced volume of platform (m3)

• $$(x_{b}, y_{b}, z_{b})$$ is coordinates of the center of buoyancy of the undisplaced platform (m)

• $$m_{mg}$$ is total mass of marine growth (kg)

• $$(x_{mg}, y_{mg}, z_{mg})$$ is coordinates of the center of mass of the undisplaced marine growth mass (m)

• $$m_{f}$$ is total mass of ballasting/flooding (kg)

• $$(x_{f}, y_{f}, z_{f})$$ is coordinates of the center of mass of the undisplaced filled fluid (flooding or ballasting) mass (m)

The equation above can be simplified when the floating platform has one or more planes of symmetry. That is, $$\iint_{A_{0}} ydA = 0$$, $$\iint_{A_{0}} xydA = 0$$, $$y_{b} = 0$$, $$y_{mg} = 0$$, $$y_{f} = 0$$, and if the $$x-z$$ plane of the platform is a symmetry plane. Likewise, $$\iint_{A_{0}} xdA = 0$$, $$\iint_{A_{0}} xydA = 0$$, $$x_{b} = 0$$, $$x_{mg} = 0$$, $$x_{f} = 0$$, and if the $$y-z$$ plane of the platform is a symmetry plane.

The undisplaced coordinates of the center of buoyancy, $$(x_{b}, y_{b}, z_{b})$$, center of marine-growth mass, $$(x_{mg}, y_{mg}, z_{mg})$$, and center of filled-fluid mass, $$(x_{f}, y_{f}, z_{f})$$, are in the global inertial-frame coordinate system. Most of these parameters can be derived from data found in the HydroDyn summary file. While the equation above makes use of several area integrals, the integrals can often be easily estimated by hand for platforms composed of one or more circular members piercing the waterplane (still-water free surface).

The waterplane area of the undisplaced platform, $$A_{0}$$, affects the hydrostatic load because the displaced volume of the fluid changes with changes in the platform displacement. Similarly, the location of the center of buoyancy of the platform affects the hydrostatic load because its vector position changes with platform displacement and because the cross product of the buoyancy force with the vector position produces hydrostatic moments about the WRP. $$A_{0}$$, $$V_{0}$$, and $$(x_{b}, y_{b}, z_{b})$$ should be based on the external volume of the platform, including marine-growth thickness. The marine-growth mass and filled-fluid mass also have a direct effect of the hydrostatic restoring because of the moments produced about the WRP.

In classical marine hydrostatics, the effects of body weight are often lumped with the effects of hydrostatics when defining the hydrostatic-restoring matrix; for example, when it is defined in terms of metacentric heights. However, when HydroDyn is coupled to FAST, the body-weight terms (other than the marine-growth and filled-fluid mass within HydroDyn) are automatically accounted for by ElastoDyn, and so, are not included here.

4.2.8.4.8.4. Floating Systems Modeled with Potential Flow¶

Frequency-dependent hydrodynamic coefficients are needed before running the potential-flow solution in HydroDyn using PotMod = 1. An external pre-processing tool should be used to generate the appropriate frequency-dependent hydrodynamic coefficients. The naming in this manual has focused on WAMIT [LN06], but other frequency-domain wave-body interaction panel codes can be used that produce similar data. However, in the end, the WAMIT format is what is expected by HydroDyn.

For the first-order potential-flow solution, HydroDyn requires data from the WAMIT files with .1, .3, and .hst extensions. When creating these files, one should keep in mind:

• The .1 file must contain the 6×6 added-mass matrix at infinite frequency (period = zero). Additionally, the .1 file must contain the 6×6 damping matrix over a large range from low frequency to high frequency (the damping should approach zero at both ends of the range). A range of 0.0 to 5.0 rad/s with a discretization of 0.05 rad/s is suggested.

• The .3 file must contain the first-order wave-excitation (diffraction) loads (3 forces and 3 moments) per unit wave amplitude across frequencies and directions where there is wave energy. A range of 0.0 to 5.0 rad/s with a discretization of 0.05 rad/s is suggested and the direction should be specified across the desired range—the full direction range of (-180 to 180] degrees with a discretization of 10 degrees is suggested. While the .3 file contains both the magnitude/phase and real/imaginary components of the first-order wave-excitation loads, only the latter are used by HydroDyn.

• The .hst file should account for the restoring provided by buoyancy, but not the restoring provided by body mass or moorings. (The hydrostatic file is not frequency dependent.) An important thing to keep in mind is that the pitch and roll restoring of a floating body depends on the vertical distance between the center of buoyancy and center of mass of the body. In WAMIT, the vertical center of gravity (VCG) is used to determine the pitch and roll restoring associated with platform weight, and WAMIT will include these effects in the restoring matrix that it outputs (the .hst file). However, the ElastoDyn module of FAST intrinsically accounts for the platform weight’s influence on the pitch and roll restoring if the platform weight and center-of-mass location are defined appropriately. To avoid double booking these terms, it is important to neglect these terms in WAMIT. This can be achieved by setting VCG to zero when solving the first-order problem in WAMIT.

The second-order WAMIT files only need to pre-calculated if a second-order potential-flow option is enabled in HydroDyn. For the second-order mean-drift solution, or for Standing et al.’s extension to Newman’s approximation to the mean- and slow-drift solution, HydroDyn requires WAMIT files with .7, .8. .9, .10d, .11d, or .12d extensions. For the second-order full difference-frequency solution of the mean- and slow-drift terms, HydroDyn requires WAMIT files with .10d, .11d, or .12d extension. For the second-order full sum-frequency solution, HydroDyn requires WAMIT files with .10s, .11s, or .12s extensions. When creating any of these files, one should keep in mind:

• The second-order frequency-domain solution is dependent on first-order body motions, whose accuracy is impacted by properly setting the 6×6 rigid-body mass matrix and center of gravity of the complete floating wind system and the 6×6 mooring system restoring matrix. So, while the body center of gravity and mooring stiffness should be zeroed when creating the first-order WAMIT files, they should not be zeroed when creating the second-order WAMIT files. (Thus, obtaining the first-order and second-order WAMIT files requires distinct WAMIT runs.)

• The .7, .8, and .9 files contain the diagonal of the difference-frequency QTF, based on the first-order potential-flow solution. The files contain the second-order mean-drift loads (3 forces and 3 moments) per unit wave amplitude squared at each first-order wave frequency and pair of wave directions, across a range of frequencies and a range of direction pairs. While the .7, .8, and .9 files contains both the magnitude/phase and real/imaginary components of the second-order wave-excitation loads, only the latter are used by HydroDyn.

• The 10d, .11d, and .12d, or .10s, .11s, and .12s files contain the full difference- and sum-frequency QTFs, respectively, based on the first-order or first- plus second-order potential-flow solutions. The files contain the second-order wave-excitation (diffraction) loads (3 forces and 3 moments) per unit wave amplitude squared at each pair of first-order wave frequencies and directions, across a range of frequency and direction pairs. While the 10d, .11d,.12d, .10s, .11s, and .12s files contains both the magnitude/phase and real/imaginary components of the second-order wave-excitation loads, only the latter are used by HydroDyn.

• The frequencies and directions in the WAMIT files do not need to be evenly spaced.

• The discretization of the first set of directions does not need to be the same as the discretization of the second set of directions; however, the matrix of direction pairs must be fully populated (not sparse). Both sets of directions should span across the desired range—the full direction range of (-180 to 180] degrees with a discretization of 10 degrees is suggested.

• The frequencies should span the range where there is first-order wave energy and the frequency discretization should be such that the differences and sums between pairs of frequencies span the range where there is second-order wave energy. A range of 0.25 to 2.75 rad/s with a discretization of 0.05 rad/s is suggested.

• Second-order hydrodynamic theory dictates that difference-frequency QTFs are conjugate symmetric between frequency pairs and sum-frequency QTFs are symmetric between frequency pairs. Due to this symmetry, the QTFs (the 10d, .11d, or .12d, .10s, .11s, and .12s files) may be upper triangular, lower triangular, a mix of upper and lower triangular terms, or full; however, after applying the symmetry, the matrix of frequency pairs must be fully populated (not sparse). When an element of the QTF is supplied together with its symmetric pairing, HydroDyn will warn the user if the QTF is not properly symmetric.