A Wave Drift Force Model for Semi-Submersible Types of Floating Wind Turbines in Large Waves and Current

Abstract

The correct prediction of slowly varying wave drift loads is important for the mooring analysis of floating wind turbines (FWTs). However, present design analysis tools fail to correctly predict these loads in conditions with current and moderate and large waves. This paper presents a semi-empirical method to correct zero-current potential-flow quadratic transfer functions (QTFs) of horizontal wave drift loads in conditions with current and moderate and large waves. The method is applicable to column-stabilized types of substructures or semi-submersibles. In the first step, the potential-flow QTF is corrected for potential-flow wave–current effects by applying a heuristic method. Second, the generalized Exwave formula corrects for viscous drift effects. Viscous drift effects become important for moderate and large waves. Conditions with current in the same direction as the waves increase the viscous drift contribution further. The method is validated by comparing QTF predictions with empirical QTFs identified from model test data for the INO Windmoor semi. While potential-flow QTFs agree well with the empirical data for small seastates without current, they underestimate the wave drift loads for moderate and large seastates. Conditions with current increase the underestimation. The semi-empirical correction method significantly improves predictions.

1. Introduction

Floating wind turbine (FTW) responses to environmental loads include slowly varying resonant horizontal motions. These are in general of larger magnitude than the wave frequency motions. One of the sources of low-frequency excitation is wave drift loads; therefore, the station keeping analysis requires a good model for these loads. However, it is acknowledged that existing design analysis tools are not accurate, especially in conditions with current and large waves. For semi-submersible types of substructures, it is known that viscous drift loads become relevant as the wave amplitude increases and cannot be neglected. This has been observed for oil and gas semi-submersibles, which typically have four columns, but designs with six or eight columns are also under operation [1,2,3]. The same conclusion has been documented for semi types of substructures of floating wind turbines, which, in general, have three columns only [4,5,6,7].

Two semi-empirical approaches have been proposed that represent viscous wave drift effects. One combines potential-flow wave–structure interaction loads computed by panel methods (e.g., [8,9]) with viscous drag loads on the columns and pontoons. The latter are represented by a Morison type of load model [10], the relative velocity concept and 2D drag coefficients distributed along the columns and pontoons. The drag loads are integrated up to the instantaneous free surface elevation to properly capture viscous drift. The approach has been applied both in the context of O&G (e.g., [11]) and floating wind energy [4,7]. One advantage of the method is that the drag term simultaneously captures the viscous wave exciting loads and the viscous damping based on pre-defined, or recommended, 2D drag coefficients. There are, however, challenges. The related recommended 2D coefficients are based on data for infinitely long cylinders, while truncation effects on the semi elements and free surface effects change the coefficients significantly. Furthermore, the drag coefficients are KC-dependent, so they will change continuously along time in irregular waves. It is also not obvious that one set of coefficients is simultaneously valid for the wave frequency and the low-frequency content of the relative velocity. Good agreement between simulations and model test data can be achieved if the drag coefficients are tuned based on the model test data [4,5].

The second approach for including viscous drift effects consists of applying the Exwave formula. This formula adds a viscous drift component to potential-flow mean wave drift coefficients. It also includes one term that represents wave–current potential-flow effects in a very simplified way in case a more elaborate method is not available. An initial version of the formula was proposed by Stansberg et al. [12] and it was later improved and validated in the scope of the Exwave JIP [3]. Section 6 provides more details on the Exwave formula. One advantage of the method is that it provides reasonably good results without the need for tuning based on model test data. One limitation is that it provides mean wave drift coefficients only, while full quadratic transfer functions (QTFs) provide a better description of the wave drift loads in irregular waves.

Wave–current interactions also influence the horizontal wave drift loads. When waves and current propagate in the same direction, there is, in general, an increase in the wave drift forces compared to waves only, which cannot be neglected. This has two contributions, the first being a potential-flow contribution. An approximate method was proposed by Aranha to correct mean wave drift coefficients [13,14], which has been widely used by the industry. More elaborate methods partially solve the linear wave–current potential-flow problem, providing more consistent mean wave drift coefficients. There are a few commercial codes available with this possibility, e.g., Hydrostar [8], Muldif [15], Wadam and Wasim [16]. Achieving good numerical solutions seems to be more difficult than using Aranha’s approximation and, apparently, these more complete solutions are seldom used for design analysis. The full second-order solution including small forward speed effects (equivalent to a current) has been attempted within the research community, and provides the full second-order solution with wave–current interactions [17]. However, there are no tools available for practical engineering applications.

As mentioned above, the state-of-the-art engineering practice for including wave-induced viscous effects applies either a Morison type of model or the Exwave formula. Regarding wave–current interactions, the common practice is to include the related effects into the mean wave drift coefficients. However, it is recognized that the wave drift loads are better represented by full quadratic transfer functions (QTFs), both in terms of magnitude and phasing. The objective of the work presented herein is to propose and validate a semi-empirical method to address the limitations of existing methods described above. The proposed approach has two parts. First, it applies a heuristic method to include wave–current potential effects into zero-current full QTFs. Second, the Exwave formula is extended from correcting mean wave drift coefficients to correcting the full QTF. The updated force model is assessed by comparing predictions with QTFs identified from model test data for an FWT with a semi-submersible substructure.

2. Case Study and Model Tests

The substructure of the INO Windmoor floating offshore wind turbine is a semi-submersible consisting of three vertical columns connected with rectangular cross-section pontoons. The tower is centered with one of the columns and it supports a 12 MW wind turbine (for more details, see [18]). Figure 1 shows the FOWT model under testing in the ocean basin and

Table 1. Main characteristics of the INO Windmoor floating wind turbine.

Parameter	Unit	Value
Column diameter	[m]	15.0
Column height	[m]	31.0
Pontoon width	[m]	10.0
Pontoon height	[m]	4.0
Center–center distance	[m]	61.0
Draft	[m]	15.5
Displacement	[t]	14,124
Long. center of gravity (LCG) *	[m]	0.0
Trans. center of gravity (TCG) *	[m]	0.0
Vert. center of gravity (VCG) **	[m]	19.4
Long. metacentric height (pitch)	[m]	9.52
Transv. metacentric height (roll)	[m]	9.53
Roll radius of gyr. (Rxx)	[m]	43.6
Pitch radius of gyr. (Ryy)	[m]	44.0
Yaw radius of gyr. (Rzz)	[m]	29.9

* with regard to the floater geometric center. ** with regard to the baseline.

presents its main characteristics.

Model tests were carried out during 2020 in the ocean basin at SINTEF with a 1:40 scaled model of the FOWT. The full-scale water depth was 150 m. The model was moored using a simplified horizontal mooring system with three lines, one end attached to the model at the deck level and the other end to the sides of the ocean basin. The very thin lines were always above the water to avoid hydrodynamic loads. They had springs at the ocean basin end. The mooring system was designed to achieve the full-depth mooring system natural frequencies of the horizontal motions.

Table 2. Motions’ natural periods and frequencies.

	Tn [s]	ωn [rad/s]	fn [Hz]
Surge	94.9	0.066	0.011
Sway	94.6	0.066	0.011
Heave	16.3	0.385	0.061
Roll	28.1	0.224	0.036
Pitch	30.1	0.209	0.033
Yaw	49.6	0.127	0.020

presents the motions’ natural periods identified from the model test data. Furthermore, the horizontal restoring properties are linear. The aim was to remove any complexity due to mooring non-linearity and damping. The model was instrumented to measure the platform motions in six degrees of freedom, the relative wave elevation at six locations and the mooring line tensions at fairlead. Reference [19] provides more detailed information about the tests.

The objective of the model tests was three-fold, namely to investigate (a) the low-frequency wave drift loads and motion responses, (b) the coupling between the aerodynamic and hydrodynamic loads and (c) the experimental uncertainty. This paper focuses on the wave drift loads only. A wide range of tests were performed to investigate wave drift loads. These include conditions with waves only and with waves and current propagating in the same direction. The irregular seastates range from small to medium and severe conditions. A JONSWAP wave spectrum was used to generate the irregular waves, except for one small condition, where the waves follow a broad-banded spectrum with nearly constant energy over a broad range of frequencies. Two wave directions were tested, namely 0 and 90 degrees (see the wave propagation convention). Some seastates were repeated for several different realizations (different seeds) to reduce the sample variability of the QTFs of wave drift loads to be identified.

Table 3. List test selected for identification of wave drift loads.

Test No.	Heading (deg.)	Uc (m/s)	Hs (m)	Tp (s)	Gamma
4010	0	0	4.0	4.5-20	-
4210, 4222, 4223, 4224, 4225	0	0	3.7	7.0	4.9
4230, 4240, 4241, 4242, 4243, 4244	0	0	6.2	9.0	4.9
4270	0	0	3.7	12.0	1.0
4290, 4292	0	0	6.2	12.0	1.2
4662, 4670, 4673, 4674, 4675, 4676, 4677	0	0	11.0	12.0	4.9
4480	90	0	6.2	9.0	1.9
4490	90	0	6.2	12.0	1.2
4510, 4520, 4521, 4523, 4523, 4525	0	0	2.0	7.0	1.1
4560	0	0	15.0	14.0	4.9
4250, 4260, 4261, 4262, 4263	0	1.2	6.2	9.0	4.9
4340, 4350, 4351, 4352, 4353, 4354	0	1.2	11.0	12.0	4.9
4530	0	1.2	2.0	7.0	1.1

lists the tests used in the present study. $H_S$ and $T_P$ stand for the seastate significant wave height and wave peak period, $U_c$ represents the current velocity and γ is the peak enhancement factor.

3. Identification of Empirical QTFs

The force model for wave drift loads proposed herein will be assessed by comparing numerical and empirical QTFs. The empirical QTFs are identified by applying cross bi-spectral analysis to the model test data, namely to the measured horizontal motion responses. The resulting wave drift coefficients represent mostly the quadratic contents in the empirical signals of wave drift loads; however, they may also include higher-order contributions. Details on the method can be consulted in [20,21]. As a summary, the procedure follows four steps:

(a)

First, the measured motions are low-pass-filtered to extract the low-frequency (LF) component.

(b)

Second, a one-degree-of-freedom dynamic system is assumed to represent correctly the LF system dynamics, for example, the LF surge motion.

$$\ddot{x}\left(t\right)+2\xi {\omega }_n\dot{x}\left(t\right)+{\omega }_n^{2}x\left(t\right)={\displaystyle \frac{1}{m}}{g}^{\left(2\right)}\left(t\right)$$

where $\ddot{x}, \dot{x} \mathrm{a}\mathrm{n}\mathrm{d} x$ are the motion low-frequency acceleration, velocity and displacement, $\xi =c/2m{\omega }_n$, c and m are the system damping coefficient and mass, ${\omega }_n$ is the natural frequency and $g^{\left(2\right)}$ is the second-order wave load. ${\omega }_n$ and m are known from the model tests, while $\xi$ is estimated iteratively. The second-order wave load, $g^{\left(2\right)}\left(t\right)$, is the only unknown and is estimated by solving the equation.

(c)

Third, calculate the cross bi-spectrum of $g^{\left(2\right)}\left(t\right)$ with respect to $\zeta \left(t\right)$, namely $S_{\zeta \zeta g}\left(f_m,f_n\right)$, where $\zeta \left(t\right)$ is the measured wave elevation.

(d)

Finally, the difference frequency wave exciting the QTF matrix is given by

$$H^{\left(2\right)}\left({\omega }_m,{\omega }_n\right)={\displaystyle \frac{S_{\zeta \zeta g}\left({\omega }_m,{\omega }_n\right)}{S_{\zeta \zeta }\left({\omega }_m\right)S_{\zeta \zeta }\left({\omega }_n\right)}}$$

where $H^{\left(2\right)}\left({\omega }_m,{\omega }_n\right)$ is the wave drift force coefficient corresponding to the wave frequency pair $\left({\omega }_m,{\omega }_n\right)$, and $S_{\zeta \zeta }\left({\omega }_m\right)$ represents the wave spectrum.

This method has been applied to identify the QTFs of surge wave forces for all the tests of Table 3. Validation of the identified empirical QTF is performed for every test case and is part of the identification workflow. The procedure compares the experimental responses with the ones reconstructed from the empirical QTF, namely the LF second-order loads and LF motions in terms of spectra and time signals. Good agreement between “measured” and reconstructed quantities means a good identification of empirical QTFs.

Validation of the identification procedure can also be achieved by comparing empirical QTFs with potential-flow QTFs for wave conditions within the range of applicability of the second-order potential-flow theory. This is the case of seastates with small-amplitude waves and no current. In this case, one expects a good agreement between empirical and numerical QTFs.

Figure 2 presents the surge QTF of the wave drift loads for a small seastate without current (Hs = 2.0 m, Tp = 7 s), namely the real and the imaginary parts (upper row of plots and lower row of plots, respectively). Different columns of plots correspond to different difference frequencies, $df=f_2-f_1$ with $f_2>f_1$. $df$ ranges from 0 to 0.015 Hz, while the surge natural frequency for this vessel is 0.011 Hz. The gray lines stand for second-order potential-flow results and the black lines for empirical results. The empirical QTFs correspond to averaged results for six seed wave realizations, corresponding to tests 4510 to 4525. We observe a very good agreement between the empirical and the potential-flow QTFs for both the real part and the imaginary part. This observation gives confidence on the procedure to identify the empirical QTFs from model test data.

4. Potential-Flow Numerical Model

The potential-flow radiation/diffraction calculations were performed with Hydrostar v8.2.0 [8]. The first-order solution gives the motion response amplitude operators (RAOs), among other first-order results. Mean wave drift force coefficients are calculated from the first-order results.

The second-order solution at the difference frequency for pairs of harmonic waves gives the QTFs of wave drift forces. Wave–current effects are not included. The control surface method is applied, or middle field method according to [8,22], and requires a control surface surrounding the hull and closed at the free surface. Expressions for the quadratic forces are evaluated on the control surface instead of over the hull surface, with advantages in terms of accuracy for the estimation of the required fluid velocity terms. The free surface integral in the second-order solution is included. The related numerical solution requires meshing of the free surface on a circle centered with the platform hull. Figure 3 shows the hull mesh, the control mesh and the free surface mesh, while

Table 4. Number of low-order panels (half hull).

Surface	No. of Panels
Hull	5606
Control surface	2982
Free surf.mesh	2532
Total	11,120

presents the number of corresponding first order panels. The total number of panels is 11,120, where symmetry about the y = 0 plane is applied, meaning that this number corresponds to half meshes only.

A numerical convergent test of the free surface mesh has not been performed for this specific semi-submersible. However, it has been performed for another semi-submersible, and the experience is used to prepare the present free surface mesh. Furthermore, one should note that the effects of the free surface integral, which uses the free surface mesh, become relevant only for moderate and high difference frequencies and for high wave frequencies. For difference frequencies corresponding to horizontal low-frequency motions of the platform and for wave frequencies corresponding to the relevant irregular seastates, the free surface integral effects are very small.

Additional stiffness coefficients in surge, sway and yaw represent the mooring system effects. Finally, additional damping coefficients are applied in all modes of motion to represent linearized viscous damping effects and, in this way, limit the RAOs’ resonant peaks to realistic values. This is important since the QTFs depend on the wave frequency motions; therefore, the level of damping will affect the QTF prediction, especially around the motions’ resonance frequencies.

Table 5. Additional damping coefficients.

Mode	Rel. Damp [-]
Surge	0.048
Sway	0.048
Heave	0.054
Roll	0.047
Pitch	0.047
Yaw	0.03

presents the additional relative damping coefficients defined as the actual damping coefficient normalized by the critical damping.

5. Potential-Flow Wave–Current Effects

Potential-flow wave–current interaction effects change the wave drift loads, as compared to conditions with waves only. While a full second-order solution is still not available for design analysis, an approximate method has been proposed by Kim et al. [23]. Herein, we use an approach to include wave–current interaction effects into the zero-current QTFs of surge and sway based on a proposal by Kim et al. [23]. The formulation for deep-water and bi-chromatic waves propagating in the same direction is presented in the following paragraphs.

Figure 4 defines two inertial and Cartesian coordinate systems, namely (x_G, y_G, z_G), which is an Earth-fixed global coordinate system, and (x, y, z), which is the platform coordinate system following the low-frequency yaw motion. The z-axis points upwards. The two coordinate systems are coincident when the heading, ${\alpha }_b$, is zero. ${\beta }_w$ and ${\beta }_c$ are the wave heading and the current heading in the vessel coordinate system, respectively. ${\overrightarrow{U}}_c$ is the current velocity vector. The wave heading and the current heading in the global coordinate system are

$${\theta }_w={\beta }_w+{\alpha }_b$$

(1)

$${\theta }_c={\beta }_c+{\alpha }_b$$

(2)

The wave drift force coefficients for surge and sway (k = 1, 2) related to two harmonic waves traveling in the same direction in the presence of current, $Q_k^c$, are represented in terms of the corresponding zero-current wave drift coefficients, $Q_k^0$:

$$Q_k^c\left({\omega }_i,{\omega }_j,{\beta }_w\right)=A_e{Q}_k^0\left({\omega }_{ie},{\omega }_{je},{\beta }_{we}\right)$$

(3)

where ${\omega }_i,{\omega }_j$ are the frequencies of the two harmonic wave components, ${\omega }_{ie},{\omega }_{je}$ are the corresponding encounter frequencies, ${\beta }_w$ is the wave heading (see Figure 4), ${\beta }_{we}$ is the apparent wave heading (or wave encounter heading) and $A_e$ is an amplification factor due to wave–current interactions. The amplification factor, the apparent wave heading and the encounter frequencies are calculated by

$${\omega }_{ie}={\omega }_i+{\displaystyle \frac{{\omega }_i^2}{g}}{U}_c^L$$

(4)

$${\omega }_{je}={\omega }_j+{\displaystyle \frac{{\omega }_j^2}{g}}{U}_c^L$$

(5)

$$A_e=\sqrt{A_{ie}{A}_{je}}$$

(6)

$$A_{ie}=1-4{\displaystyle \frac{{\omega }_i}{g}}{U}_c^L$$

(7)

$$A_{je}=1-4{\displaystyle \frac{{\omega }_j}{g}}{U}_c^L$$

(8)

$${\beta }_{we}={\beta }_w+2{\displaystyle \frac{\omega }{g}}{U}_c^T$$

(9)

$$U_c^L=U_{c}cos\left({\beta }_c-{\beta }_w\right)$$

(10)

$$U_c^T=U_{c}sin\left({\beta }_c-{\beta }_w\right)$$

(11)

$U_c^L$ and $U_c^T$ are the components of $U_c$ in the wave direction and perpendicular to the wave direction and ${\beta }_c$ is the current heading according to Figure 4.

This method was applied to the INO Windmoor semi and the result presented in Figure 5 for a wave heading of 0 degrees. Solid lines represent the zero-current potential-flow surge QTF, while the dashed lines represent the potential-flow QTF corrected by the wave–current correction method introduced above for a current velocity of 1.2 m/s and collinear with the waves. The results are presented for several constant difference frequency diagonals of the QTF and as function of the lower frequency of the pair f₁.

The wave–current effects increase the wave drift loads significantly for frequencies above around 0.10 Hz. Furthermore, the effects are felt mostly by the real part of the QTF.

6. Viscous Drift Effects

A semi-empirical correction formula to estimate the mean wave drift force coefficients in regular waves on column-based semi-submersibles has been previously proposed in the scope of the Exwave pilot project [12]. The correction accounts for viscous effects and wave–current interaction effects and it was pointed out as an alternative while more advanced and commonly accepted procedures are not in place. Viscous drift forces are particularly important in high seastates and conditions with combined waves and current. The formula has been updated, generalized and validated in the scope of Exwave JIP [3]. The surge and sway wave drift force coefficients for frequency ω are given by

$$f_{Dx}\left(\omega ,U_c,H_s\right)=f_{Dx}^{pot}\left({\omega }_e\right)\left(1+C_p{U}_{c}cos{\mathsf{\beta }}_{wc}\right)+f_{Dx}^v\left(\omega ,U_c,H_s\right)$$

(12)

$$f_{Dy}\left(\omega ,U_c,H_s\right)=f_{Dx}^{pot}\left({\omega }_e\right)\left(1+C_p{U}_{c}cos{\mathsf{\beta }}_{wc}\right)+f_{Dy}^v\left(\omega ,U_c,H_s\right)$$

(13)

where the first term and second terms represent potential-flow wave drift loads and viscous drift loads respectively. The first term includes a simple correction for wave–current effects, where the value $C_p=0.25 \mathrm{s}/\mathrm{m}$ is derived from Aranha’s formula [13] for a wave period of 10 s and a collinear current of 1 m/s (note that, in the present work, we apply the wave–current formulation described in Section 5 and not the simple correction above).

The surge and sway viscous drift contributions are given by

$$f_{Dx}^v\left(\omega ,U_c,H_s\right)=B\left(G{U}_{c}cos{\mathsf{\beta }}_{wc}+H_s\right)cos{\mathsf{\beta }}_w$$

(14)

$$f_{Dy}^v\left(\omega ,U_c,H_s\right)=B\left(G{U}_{c}cos{\mathsf{\beta }}_{wc}+H_s\right)sin{\mathsf{\beta }}_w$$

(15)

Additionally,

○

$f_{Dx}\left(\omega ,U_c,H_s\right)=F_{Dx,y}\left(\omega ,U_c,H_s\right)/{\left({\zeta }^{\left(1\right)}\right)}^2$ is the mean wave drift force, corresponding to a seastate with $H_s$, normalized by the wave amplitude squared.

○

$f_{Dx,y}^v\left(\omega ,U_c,H_s\right)=F_{Dx,y}^v\left(\omega ,U_c,H_s\right)/{\left({\zeta }^{\left(1\right)}\right)}^2$ is the mean viscous wave drift force normalized by the wave amplitude squared.

○

${\omega }_e=\omega -{\omega }^2/g{U}_{c}cos{\mathsf{\beta }}_w$.

○

$C_p$ is a potential-flow wave–current interaction coefficient, previously assumed as 0.25 s/m.

○

$U_c$ is the current velocity.

○

${\beta }_w$ is the wave heading angle.

○

${\beta }_{wc}$ is the angle between wave and current directions.

○

$B=k^{\prime }{d}_{sum}p$, with $k^{\prime }={\displaystyle \frac{k}{\left[1+{\left(k.L\right)}^{-2}\right]}}$ and $d_{sum}=N{D}_0$.

○

$D_0=\sqrt{{\displaystyle \frac{4A_{wp}}{N\pi }}}$, $A_{wp}$= water plane area.

○

L is the platform length.

○

$p=e^{-0.95{\left(k{D}_0\right)}^3} \left[\mathrm{k}\mathrm{N}/{\mathrm{m}}^3\right]$ and $D_0$ is the columns’ equivalent diameter.

○

$k={\displaystyle \frac{2\pi }{\lambda }}$ is the wave number and $\lambda$ is the wavelength.

○

$G=10 \left[\mathrm{s}\right]$ represents a viscous wave–current factor determined empirically.

The viscous drift formula includes the significant wave height Hs; therefore, it is a third-order load, i.e., the resulting drift coefficients include a contribution that increases linearly with Hs.

The viscous drift load in regular waves represents the oscillatory horizontal drag force averaged over one wave cycle. The drag force is due to separation of the flow around the columns. The formula includes semi-empirical terms that attenuate the viscous drift component for high wave frequencies, since drag forces reduce for short wavelengths (small wave particle trajectories), and for low wave frequencies, since the platform tends to follow the waves for very long wave lengths and the wave drift forces reduce to zero. The empirical parameters were determined empirically based on the model test data.

The Exwave formula is extended in the present work from a mean viscous wave drift component in harmonic waves to a full pseudo viscous drift QTFs of surge and sway in bi-chromatic waves. The following assumption is used: the viscous wave drift effects influence the real part of the QTF only; therefore, the imaginary part of the potential-flow QTFs is kept unchanged. This assumption is further discussed below.

Based on the former assumption, the “generalized Exwave formula” represents the viscous drift contribution to the surge and sway QTFs given by

$${\left\{f_{Dx}^v\left({\omega }_i,{\omega }_j,U_c,H_s\right)\right\}}^{Re}=B\left(G{U}_{c}cos{\mathsf{\beta }}_{wc}+H_s\right)cos\mathsf{\beta }$$

(16)

$${\left\{f_{Dx}^v\left({\omega }_i,{\omega }_j,U_c,H_s\right)\right\}}^{Im}=0$$

(17)

$${\left\{f_{Dy}^v\left({\omega }_i,{\omega }_j,U_c,H_s\right)\right\}}^{Re}=B\left(G{U}_{c}cos{\mathsf{\beta }}_{wc}+H_s\right)sin\mathsf{\beta }$$

(18)

$${\left\{f_{Dy}^v\left({\omega }_i,{\omega }_j,U_c,H_s\right)\right\}}^{Im}=0$$

(19)

where

$${\left\{f_{Dx,y}^v\left({\omega }_i,{\omega }_j,U_c,H_s\right)\right\}}^{Re}={\left\{F_{Dx,y}^v\left({\omega }_i,{\omega }_j,U_c,H_s\right)\right\}}^{Re}/{\zeta }_i^{\left(1\right)}{\zeta }_j^{\left(1\right)}$$

(20)

and $F_{Dx,y}^v$ is the viscous drift force induced by two harmonic waves propagating in the same direction (bi-chromatic wave) with frequencies ${\omega }_i,{\omega }_j$ and amplitudes ${\zeta }_i^{\left(1\right)}, {\zeta }_j^{\left(1\right)}$ in the presence of a collinear current $U_c$. The viscous drift correction is seastate-dependent, namely, it depends on $H_s$. The coefficients B and G are given above, where B depends on the wave number k. In the case of bi-chromatic waves, we have two wave numbers. For small difference frequencies, as are typical of the horizontal motions’ natural frequencies, it makes a small difference to use either ${\omega }_i \mathrm{o}\mathrm{r} {\omega }_j$ to calculate the coefficient B. A recommended approach consists of using the average frequency, $\left({\omega }_i+{\omega }_j\right)/2$, to determine the wave number.

Regarding the assumption that the viscous drift contributes to the real part of the QTF only, observation of empirical surge QTFs and comparison with zero-current potential-flow predictions seem to indicate that the viscous drift effects influence mainly the real part of the QTFs. This is illustrated in Figure 6 and Figure 7, with four QTFs off-diagonals corresponding to constant difference frequencies from 0 to 0.015 Hz, while the natural frequency of the surge is 0.01 Hz. Figure 6 includes the real part and Figure 7 the imaginary part. Upper plots represent conditions without current and the lower plots with a current of 1.2 m/s collinear with the waves, while the seastate $H_s$ and $T_p$ is the same for the two cases. Finally, the gray lines represent zero-current potential-flow predictions and the black line the empirical QTFs.

As mentioned, the plots of Figure 6 and Figure 7, together with the observation that potential-flow wave–current effects influence mostly the real part of the QTF (Figure 5), suggest that the viscous drift effects affect mainly the real part of the QTF. These viscous drift effects are mainly due to drag loads on the wave piercing columns, but also on the pontoons in the presence of current. The current greatly amplifies the viscous wave drift as compared with similar wave conditions without current.

One aspect that is not discussed nor included in the formulation above is the effect of possible vortex-induced motions (VIMs) due to current incidence. Periodic shedding of vortices under current may induce transverse and inline motions with a magnitude in the same order as the columns’ dimensions, although inline motions are of smaller amplitude (see [24] for a good review). Furthermore, transverse motions increase the inline mean hydrodynamic loads. Regarding the effect of waves, transverse VIM may be greatly attenuated, or it may show amplitudes like those of conditions with current only, depending on the wave characteristics [25]. Identifying inline VIM in the presence of waves from model tests does not seem possible since the measured motions result from a combination of second-order wave drift effects together with any possible vortex-shedding effects. The authors in [25] discuss model tests with a semi-submersible under current and irregular waves and conclude that the inline motions are induced by wave effects (and not vortex shedding).

7. Summary of Method and Comparisons between Empirical and Numerical Results

This section assesses the new semi-empirical wave drift force model by comparing numerical predictions with QTFs identified from model test data. However, we start by summarizing the method to correct potential-flow QTFs of horizontal wave drift loads for wave–current effects and viscous drift effects. The procedure is illustrated in the flowchart of Figure 8. It starts with the calculation of potential-flow QTFs. Second, the wave–current interaction effects are introduced into the QTFs by the method described in Section 5. Finally, the additional viscous drift component is calculated according to the method of Section 6 and added to the former QTF. The result is a pseudo QTF representing wave drift coefficients normalized by the products of bi-chromatic wave amplitudes; however, these QTFs are seastate-dependent due to the viscous drift contribution.

The first set of plots, Figure 9, shows results for four moderate and severe seastates without current, where each row of plots corresponds to one seastate. As before, each column of plots corresponds to one difference frequency of the QTFs or one diagonal of the QTFs. The wave drift coefficients are shown as a function of f₁, the first frequency of the bi-chromatic waves. Gray lines represent the potential-flow (uncorrected) QTF, blue lines the QTFs corrected for viscous drift effects and the black lines the empirical QTFs.

One should note that the empirical QTFs can be identified from the model test data only within the wave frequency where the seastate has energy. This is the reason for why their wave frequency range is limited and dependent on the wave spectrum frequency range. Furthermore, the empirical QTFs are subjected to some uncertainty or random noise, which is related to the finite duration of the time series. This is observed especially for very low wave frequencies and high wave frequencies, where the estimations are not reliable. Overall, the empirical QTFs are to be compared with numerical predictions in a qualitative manner.

Figure 9 shows a consistent underprediction of wave drift loads, especially for wave frequencies below around 0.1 Hz. The discrepancies increase for larger seastates, where the actual wave loads can exceed the predictions by more than double. The predictions improve significantly when viscous drift effects are included.

Figure 10 and Figure 11 present the QTFs for two seastates with current, one moderate and one severe seastate. In this case, we present the real part, the imaginary part and the absolute value of the QTFs in three rows of plots. The plots include three sets of results, namely potential flow (gray lines), potential flow corrected for potential-flow wave–current effects (pink lines) and potential flow corrected for wave–current and viscous drift effects (blue lines). One feature that stands out from the plots is the very large underestimation of the zero-current potential-flow results. The underestimation is on the real part only, which is related to the fact that viscous drift effects have a real part only. The numerical results indicate that including potential-flow wave–current effects increases the wave drift forces for wave frequencies above around 0.1 Hz, mostly the real part, but the result still underpredicts the actual loads. Correcting the zero-current potential-flow surge QTFs with the heuristic approximation for moderate and severe seastates brings the predictions closer to the empirical data; however, there is still a significant underestimation of the wave drift forces. The reason is that the viscous wave drift effects are still missing.

Adding the viscous drift component given by the generalized Exwave formula to the former numerical results brings the predictions close to the empirical QTFs. Overall, the method proposed herein to correct zero-current potential-flow QTFs of the horizontal wave drift loads for conditions with current and large waves significantly improves the predictions, especially in conditions with current. One advantage of the method is that it does not require model test data to achieve realistic results. Note that no adjustments in formulae presented before [3] have been introduced here. However, if model test data are available, the numerical model may be fine-tuned, if needed, by adjusting the factor G and L of Equations (16) and (18).

Table 6. Normalized root mean squared error of predicted values compared to empirical data for four diagonals of the QTF amplitude and for four series of model tests. Comparison between potential-flow predictions (uncorr.) and potential-flow predictions corrected for viscous drift effects (visc. drift).

	f2 − f1 = 0 [Hz]		f2 − f1 = 0.005 [Hz]		f2 − f1 = 0.01 [Hz]		f2 − f1 = 0.015 [Hz]
Test	Uncorr.	Visc. Drift	Uncorr.	Visc. Drift	Uncorr.	Visc. Drift	Uncorr.	Visc. Drift
4230–4244	0.26	0.19	0.24	0.18	0.25	0.22	0.24	0.27
4290–4292	0.36	0.29	0.35	0.24	0.38	0.25	0.43	0.28
4662–4677	0.43	0.24	0.35	0.24	0.29	0.3	0.29	0.36
4560	0.52	0.24	0.45	0.26	0.41	0.23	0.45	0.13

and

Table 7. Normalized root mean squared error of predicted values compared to empirical data for four diagonals of the QTF amplitude and for four series of model tests. Comparison between potential-flow predictions (uncorr.) and potential-flow predictions corrected for wave–current effects and viscous drift (w.c. and visc. drift).

	f2 − f1 = 0 [Hz]		f2 − f1 = 0.005 [Hz]		f2 − f1 = 0.01 [Hz]		f2 − f1 = 0.015 [Hz]
Test	Uncorr.	w.c. and Visc.	Uncorr.	w.c. and Visc. Drift	Uncorr.	w.c. and Visc. Drift	Uncorr.	w.c. and Visc. Drift
4250–4263	0.55	0.12	0.54	0.1	0.55	0.11	0.57	0.15
4340–4354	0.7	0.2	0.65	0.17	0.59	0.17	0.54	0.2

present the root mean square error of the numerical predictions as compared to the empirical wave drift coefficients. The first table corresponds to model test cases without current and the second table to the cases with current. The errors are presented for four diagonals of the surge wave drift load QTFs and for predictions based on potential-flow QTFs (“uncorr.”), predictions based on potential-flow QTFs corrected for viscous drift effects (“visc. drift”) and predictions based on potential-flow QTFs corrected for wave–current effects and viscous drift (“w.c. & visc. drift”).

The root mean square error (RMSE) is the standard deviation of the residuals. The residuals represent how far from the empirical values the predictions are:

$$RMSE=\sqrt{{\sum }_{i=1}^n{\displaystyle \frac{{\left({\widehat{x}}_i-x_i\right)}^2}{n}}}$$

(21)

where ${\widehat{x}}_i$ is the predicted values, $x_i$ is the empirical values and n is the number of data points. The analysis is performed within a frequency range where the seastates have energy, namely within the frequency range where the wave energy is larger than 10% of the wave spectra peak energy. The reason for this is that the empirical QTF estimates have large uncertainty for frequencies where the wave energy is very small. The results in Table 6 and Table 7 represent the RMSE normalized by the mean value of the wave drift coefficients corresponding to the diagonal under analysis.

The results show a consistent reduction in the prediction error when the corrections are introduced. The reduction in error increases with the seastate severity and it especially reduces for conditions with current, where viscous drift effects are more important. For the cases with current, the estimated error reduces to around 20% to 30% compared to potential-flow predictions.

One should note that the standard deviation of the residuals would not be zero even if the numerical predictions were excellent and representing the physical phenomena fully correctly. The reason for this is that the empirical QTFs inherently have some random noise or uncertainty due to the finite duration of the time series used for their identification.

8. Limitations of the Method

Compared to state-of-the-art second-order potential-flow codes, the method proposed herein significantly improves the predictions of horizontal wave drift loads in moderate and severe seastates with current. However, it is a semi-empirical method and therefore with limitations. The first part consists of correcting the zero-current potential-flow QTF for wave–current interactions. The method may be considered as a generalization of Aranha’s proposal to correct mean wave drift coefficients only [ref]. Aranha’s formula is derived for fixed bodies and assumes that the current velocity is small and that the Brard number is also small ($U_c\omega /g$ smaller than around, e.g., 0.15). The drift force in harmonic waves and in deep water is given by correcting the zero-current force by a frequency shift and dynamic effect as

$$F_k^c\left(\omega ,U_c\right)=\left(1-{\displaystyle \frac{4\omega }{g}}{U}_c\right)F_k^0\left({\omega }_e,0\right)$$

(22)

It has been shown that Aranha’s formula provides very accurate results for fixed structures compared to analytical results [26] and compared to results from complete wave–current formulations [27]. The main limitation of Aranha’s formula is that it does not represent wave frequency body motions. The authors in [27] compared surge mean wave drift coefficients from the simple formula and from a wave–current formulation for an FPSO in head waves. The results were quite similar up to a $\lambda /L_{pp}$ of around 0.4, which covers most of the range of interest for mooring analysis ($\lambda \mathrm{a}\mathrm{n}\mathrm{d} L_{pp}$ stand for the wave length and ship’s length between perpendiculars). There are deviations between the two sets of results for high frequencies.

The formulation presented herein to represent QTFs with wave–current effects in terms of QTFs without current is a generalization of Aranha’s method for all diagonals of the QTF and for the imaginary part as well. The authors expect the limitations of the approximation to be at least the same as those identified for Aranha’s formula: applicable to small current velocities, low Brard numbers and low to moderate wave frequencies.

The second part of the method consists of including viscous drift effects into the potential-flow QTFs. This is achieved by generalizing a viscous drift formula derived and calibrated for regular waves. The basic viscous drift load is an analytical result representing the drag load on a vertical cylinder averaged over one wave cycle. The cross-sectional loads are integrated up to the instantaneous wave elevation assuming a drag coefficient of 1. The actual drag loads are, at least, KC-dependent (KC—Keulegan–Carpenter), Reynolds-number-dependent and dependent on the columns’ finite length. Fixing the drag coefficient introduces a large uncertainty. The effects of the KC number are partly included by a semi-empirical term that reduces viscous drift effects for short wave lengths. Another empirical term reduces the viscous drift loads for long waves, where the body tends to follow the motion of the waves. The described formula represents the shape of the viscous drift load as a function of the wave frequency. Finally, an empirical constant is used to tune the magnitude of the viscous loads. This has been tuned by comparing predictions of empirical coefficients identified from model tests for the Exwave semi-submersible [3].

In the present work, we generalize the method from the correction of mean wave drift coefficients to full QTFs under the assumption that the semi-empirical formula can be applied to bi-chromatic waves. Summarizing the limitations of this simplistic method, the KC dependence of the viscous slowly varying loads is very simplistic and has been calibrated for one specific semi-submersible. The same can be said with respect to the viscous amplification factor. It is encouraging to observe that the method, which was calibrated for a semi with four columns, works well for a different semi in this paper that has three columns. Both have columns with a circular cross-section and with similar diameters. Application of the method to semi-submersibles with significantly different designs should be exercised with caution, such as, for example, those with a rectangular type of column cross-section.

Although there are strong limitations and the method is not based on a consistent theoretical background, it definitely provides a significant improvement compared to using potential-flow results alone. The industry does not seem to have a practical alternative presently. Application of computational fluid dynamics solvers is not realistic when the design needs to be based on numerous three-hour simulations, each with seed variation for sound statistics of the mooring line tensions. This is especially relevant for floating wind turbines, where the solution needs to be fully coupled, including control, aeroelastic responses and full turbulent wind, within a volume around the turbine.

9. Conclusions

The comparison between zero-current potential-flow surge QTFs and corresponding ones identified from model test data shows quite good agreement for conditions with small waves and no current. The case study consists of a semi-submersible substructure of a floating wind turbine. Discrepancies increase for moderate and high seastates, especially for frequencies lower than around 0.1 Hz with underestimation by the potential-flow tool. This is inline with the observation by other authors, as described in the introduction. The differences are significant for severe seastates and the reason for this is viscous wave drift loads. Conditions with current increase the difference between predictions and empirical data. In this case, wave–current potential-flow effects add a contribution to the wave drift loads and the viscous wave drift component is amplified in the presence of current.

This paper presents a semi-empirical procedure to correct zero-current potential-flow quadratic transfer functions (QTFs) of horizontal wave drift loads for conditions with current and moderate and large waves. The method is applicable to semi-submersible types of structures and it includes two components: a heuristic correction for wave–current potential-flow effects and another one for viscous drift effects. The correction method improves the predictions as compared to the test data for the cases without current, especially in the most severe seastates, and at the low-frequency range, where the seastates have most of the energy. For conditions with current, the wave–current potential-flow correction bring the predictions closer to the empirical data for frequencies above 0.1 Hz, but there is still significant underprediction. Correcting further for viscous drift effects results in predictions in good agreement with the empirical QTFs.

One advantage of the method is that it does not require tuning based on model test data to achieve reasonably good results. This was the case with the present FWT substructure, where quite good results were achieved without tuning. However, if model test data are available, it is possible to fine-tune the numerical model to improve the predictions.