Investigation of the influence of sinusoidal internal waves on a SPAR buoy structure

Offshore wind has a great potential as a competitive source of renewable energy, especially in deep waters where wind speeds are more consistent and fewer restrictions apply for running large wind turbines. Previous analyses of Floating Offshore Wind Turbines (FOWTs) mostly considered obvious sources of loading: surface waves, currents, wind and mooring. However, in some deep-water locations, internal waves can occur and these have been shown to significantly affect floating structures. Since the hydrodynamic response of an FOWT governs the structure’s general stability, the aim of this research is to investigate the impact of sinusoidal internal waves on the platform motion of a free-floating SPAR-type cylinder. A potential flow model and Morison’s equation are applied numerically to calculate the forces acting on a free-floating cylinder in an oscillating flow. The commercial Finite Element Analysis software OrcaFlex is verified by the potential flow model of the oscillating flow and is then used to mimic sinusoidal internal waves acting on a free-floating cylinder in a stratified flow. Three different internal wave amplitudes and peak velocities are analysed, and the nine resulting cases are investigated for the oscillating and stratified flow each. It has been found that the pitch rotations of the SPAR cylinder were small (< 0.1°) in all cases and can likely be disregarded. The surge displacements of the free-floating cylinder were substantial in both oscillating and stratified flows, with maximum surge magnitudes of 423m and 120m, respectively. Therefore, significant additional mooring loads due to internal waves could be sustained by SPAR-type FOWTs.


Introduction
With the ever-increasing threat of climate change and need to move away from fossil fuels, considerable effort is put into the development of renewable energy technologies.Following the rapid expansion of offshore wind farms over the past decade which has exploited shallow nearshore regions, attention is now turning to floating offshore wind turbines which can be deployed in greater water depths [1].

FOWTs
Offshore wind has great potential as a competitive source of renewable energy, especially in deep waters with more consistent wind speeds and fewer restrictions for running large wind turbines [2].Research and investment into Floating Offshore Wind Turbines (FOWTs) has risen notably in recent years, with an ambition by several European countries to deliver over 10 GW of floating wind power by 2030 [3].
The hydrodynamic response of an FOWT is one of the most important aspects in its design because it governs the structure's general stability [4].Wave and wind loads acting on the platform produce forces and moments which destabilise the FOWT.Existing platform designs for the FOWT industry can be divided into three preeminent categories determined by their method of attaining static stability; buoyancy (e.g., semi-submersible platforms), mooring (e.g., tension leg platforms) or ballast (e.g., single point anchor reservoir -SPARplatforms) [4].Their stability depends on the displaced volume of water, on the mooring lines or on the additional weight to shift the centre of gravity below the centre of buoyancy, respectively [5].In practice, combinations of these methods are used to stabilise an FOWT.
The research presented here considers a SPAR floating structure due to its high stability when compared to other buoy platforms in deep waters, as it consists of a long draft in the shape of a slender cylinder (reaching 120m) with catenary mooring attached directly to the platform [6].

Internal waves
Ordinarily in FOWT analysis, the most obvious sources of loading are considered: surface waves, currents, wind and mooring [7].However, in some deep-water locations, internal waves can occur and these have been shown to significantly affect floating structures, e.g., a floating drilling platform in the Andaman Sea was moved by 30.48 metres and rotated by 90 degrees [8].Contrary to tidal currents, internal waves move at time-varying and higher frequencies.They occur beneath the ocean surface, e.g., at 50m depths, at a pycnocline which is the interface between two layers of different water density [9].
The water layer above the pycnocline is of lower density than the layer below it.The different water densities are either caused by a variation in salinity or water temperature [10].The layering is called a stratification.When the water is strongly stratified, i.e., a sharp density change along the interface, any excitation of the pycnocline tends to generate an internal wave propagating away from that region [11].The vertical displacement of the pycnocline creates a restoring force that initiates waves on the density interface [9].As internal waves propagate along the pycnocline, they create circular water particle motions, with their largest radius at the pycnocline and smaller radii further away [9].Due to this circular motion, the flow directions above and below the pycnocline are opposite in direction.
The most common type of internal waves is a solitary wave or soliton which is a non-sinusoidal and nonlinear wave [12].Solitary wave packets consist of several oscillations of internal solitons confined to a limited area in the ocean [12].The typical wave periods of internal waves range between tens of minutes to several hours, and their wavelengths reach hundreds of metres to tens of kilometres [10].The internal wave height, which is measured from its peak to trough, frequently surpasses 50 metres, e.g., in the northern region of the South China Sea internal solitary wave heights over 100 metres have been recorded and internal waves can arise almost all year round [13].Since internal waves with large heights can produce large local loads and bending moments on offshore structures, they are suspected of causing substantial movement and severe damage to offshore structures [14], which is a serious safety concern [8].In the South China Sea, a cable in the Liuhua 11-1 oil field was broken as a result of sudden powerful internal waves [8].The world's most advanced semi-submersible oil drilling platform in deep waters, the "Offshore Oil 981", was installed by the China National Offshore Oil Corporation in 2012 and is notably the first oil platform design which considered the impact of internal waves [9].
When considering the relatively novel FOWT concepts, internal wave loads have been neither fully analysed nor extensively researched.Song et al. [15] created a numerical model in the time-domain to determine the impact of internal solitary waves on a fixed platform and on a floating SPAR platform, albeit not a FOWT but an oil-drilling platform with superstructures attached to the top of a SPAR buoy with mooring lines.It was discovered that the horizontal displacement of the SPAR buoy caused by internal waves was substantial and significantly higher than that due to surface waves even though the additional horizontal force is lower.Therefore, it was highlighted that considerable safety risks to the marine structures are posed due to the low-frequency effect of internal solitary waves.
As well as causing horizontal displacements, in some cases the long draft of a floating SPAR wind turbine may penetrate the pycnocline, and the opposing, oscillatory flow above and below the pycnocline will lead to an overturning moment.Hence, the aim of this research is to investigate the impact of sinusoidal internal waves on the platform motion of a free-floating SPAR-type cylinder.

Methodology
Since internal ocean waves are caused by two layers of different density, the flow velocity  depends on the depth of the pycnocline [9].The location of the pycnocline is defined by the thickness of the upper and bottom layer, ℎ and , respectively, between the free surface and seabed.The fluid density in the top layer is  ℎ and the density in the bottom layer is   .The numerical domain for this study uses a 3-dimensional coordinate system, with the -axis in the horizontal plane, the -axis in the vertical plane and the y-axis perpendicular to these axes, see Figure 1.The internal wave direction is used to distinguish between the different motions, i.e., surge, sway and heave movements are in the ,  and -directions, respectively, whereas roll, pitch and yaw rotations are around the ,  and -axis, respectively [16].
In this study, sinusoidal internal ocean waves using the long wave approximation in shallow water [11] are used to model internal waves.This approximation is still applicable for the deep sea with the assumption that the internal wavelength  = 2  is much larger than the fluid depth [18].When considering the internal wave number , it is notable that , ℎ, and  are all dimensionless and small, i.e., , ℎ, and  ≪ 1 [11].The resulting fluid velocities in the -direction can be expressed as: in the top and bottom layer, respectively. amp is the amplitude for linear internal waves measured from the undisturbed pycnocline to the wave's peak, which is equal to half the internal wave height.The dispersion relation between the internal angular wave frequency,  = 2  , where  is the internal wave period in seconds, and the wave number  is defined as: where  is the acceleration due to gravity.Thus, the internal wave celerity (or phase speed),  =   =   , represents the propagation speed of the internal wave crest in the -direction [14].For simplicity in section 3 below, the peak velocities of the sinusoidal internal wave motion are defined as: in the top and bottom layer, respectively.Differentiating equations (1) and (2) with respect to time  yields the corresponding fluid accelerations in the -direction in the top and bottom layer, respectively:

Internal wave forces
Morison's equation was first derived to model the effect of oceanic surface waves on floating structures [19].Here, it is used to compute the force induced by internal waves acting on a free-floating cylinder: with  and ̇ the internal wave velocity and acceleration, respectively; ̇ and ̈ the cylinder velocity and acceleration, respectively;  the cylinder radius;  the neutrally-buoyant cylinder draft beneath the free undisturbed surface;  w the average of the fluid density, i.e.,  w =  ℎ +  2 ;  A the added mass coefficient;  D the drag coefficient;  M the hydrodynamic inertia coefficient and  M = 1 +  A [20].

Numerical model
The commercial Finite Element Analysis software OrcaFlex [21] is used to model a floating cylinder in sinusoidal internal waves.As internal waves are a novel source of load when considering FOWTs, there are limitations in the existing software.The aim is to use OrcaFlex to determine the cylinder's movement in the -direction (surge) and its rotation around the -axis (pitch).The approach to simulate internal waves in OrcaFlex comprises two steps.Firstly, the drag force component of the internal wave is modelled using OrcaFlex' built-in current model.OrcaFlex' current speed in the -direction is defined by an external function in Python, based on the velocity equations ( 1) and ( 2), in combination with a depth profile to simulate a current varying in time, direction and depth.Secondly, an applied force in the -direction acting on the cylinder is implemented as a global load in OrcaFlex to simulate the fluid inertia component of the internal wave.This inertial force is defined by two external Python functions based on the acceleration equations (6) and ( 7) substituted into the fluid inertia component of equation (8).The fluid inertia force is then applied as a global load in the built-in 6D buoy datasheet in OrcaFlex, acting in different directions and depths while varying in time.This second step is necessary as OrcaFlex neglects current acceleration, and therefore, disregards any inertial forces caused by a current.Hereafter, this two-step method is named the "adapted OrcaFlex approach".
Within OrcaFlex, an approximation of equations (1), ( 2), ( 6) and ( 7) is needed since the -position of the SPAR buoy is not known at all timesteps.The approximation is to evaluate these equations by setting the -value to 0. Equation ( 9) can be derived mathematically for the position of the SPAR buoy in an oscillating flow.Based on this equation, the assumption of  = 0 is appropriate where   ≪ : This assumption captures the amplitude of the SPAR buoy oscillation in the surge direction as , consists of a mean drift and a second harmonic oscillation.Once the buoy is moored, its surge will naturally be smaller, which is an additional justification for the assumption  = 0.

Results
The results in this section present the surge and pitch motions of a free-floating and neutrally buoyant cylinder, see Figure 1.The surge movement is defined as the horizontal cylinder displacement along the -axis from  = 0 measured in metres [m].The pitch rotation is defined as the rotational cylinder displacement around the -axis measured in degrees [°].The origin of the cylinder axis is at the origin of the undisturbed free fluid surface at (, , ) = (0,0,0).Two scenarios are modelled: • Sinusoidal internal waves, where the cylinder's draft penetrates the pycnocline ( > ℎ), hereafter referred to as the "stratified flow".• An oscillatory flow with frequencies and velocities equivalent to those in the upper layer of the sinusoidal internal waves in the stratified flow.This is effectively the case where the cylinder draft is smaller than the pycnocline depth ( < ℎ), hereafter referred to as the "oscillating flow".
The numerical simulations reported in the sections below investigate two variable parameters: the internal wave period, , and the internal wave amplitude,  amp .The input data is shown in Table 1.The set variables for all cases are:  = 9.81 m s 2 ,  D = 0.6,  A = 1 and hence,  M = 1 +  A = 2; with the cylinder parameters:  = 4.40m,  = 120m,   = 7.47 × 10 6 kg and the centre of mass at  = −89.92m.These values are based on the dimensions of the OC3-Hywind SPAR-buoy prototype [22] but are adapted for a neutrally buoyant free-floating cylinder with a constant diameter.= 1025 kg m 2 is used in Morison's equation (8) for both instances: the stratified and oscillating flow.All internal wavelengths  are much larger than the total fluid depth of 200m for the cases considered here, satisfying the conditions described in section 2.1 for the long wave approximation in shallow water [11].The internal wave celerity  is 1.9 m s for all cases, which satisfies the condition   ≪  required for the approximation of  = 0 described in section 2.3.

Validation of the adapted OrcaFlex approach for modelling the oscillating flow
This section validates the adapted OrcaFlex approach described in section 2.3 by simulating a freefloating cylinder in the oscillating flow.The adapted OrcaFlex approach is verified numerically for all cases in Table 1 against solving Morison's equation for a free-floating cylinder in an oscillating potential flow which is given by the equations in section 2. .The validation results for case A1 in the oscillating flow are presented here.
The fluid and cylinder velocities are a perfect fit when comparing OrcaFlex results to the potential flow model based on equation (1), as seen in Figure 2. Noticeably, the free-floating cylinder velocities in all the oscillating flow cases of Table 1 are equal to their corresponding fluid velocities.Since OrcaFlex does not model the current acceleration, the applied load is used as an input to OrcaFlex to simulate the effect of the oscillating flow on a free-floating cylinder.OrcaFlex calculates the cylinder acceleration using that applied load.The results from OrcaFlex and the potential flow model agree for both cylinder accelerations based on equation (6), see Figure 3. Thus, the adapted OrcaFlex approach for modelling cylinder motions in the oscillating flow is validated, as is the assumption  = 0 for the presented cases.

The oscillating flow
The results presented in this section are for a free-floating cylinder in the oscillating flow for all cases in Table 1, using the adapted OrcaFlex approach described in section 2.3.The resulting surge and pitch motions are compared for all A-cases in Figures 4 and 5, respectively.The larger the internal wave amplitude   , the larger the cylinder motion is in both surge and pitch.For all cases A, B and C, the peaks and troughs of the surge occur at the same time  depending on the case's internal wave period , and similarly for the pitch.
The surge and pitch motions are compared for all 1-cases in Figures 6 and 7, respectively.By normalising time using the internal wave period, it is evident the larger the period, the further the buoy moves in surge but the smaller the pitch rotation.These observations apply for all three   values.

The stratified flow
The results presented in this section are obtained for sinusoidal internal waves acting on a free-floating cylinder in the stratified flow for all cases of Table 1, using the adapted OrcaFlex approach described in section 2.3.Using the adapted OrcaFlex approach to simulate the stratified flow cases, it is notable that the total applied force in the -direction and corresponding cylinder velocities are smaller than for the equivalent oscillating flow cases.This is expected because the opposing flow directions either side of the pycnocline will lead to a smaller net horizontal force.While it was shown that the cylinder and fluid velocity match in an oscillating flow, this is not true for the stratified case.The undisturbed cylinder draft is 50m in the top layer and 70m in the bottom layer of the stratified flow and so it is influenced more by the lower layer fluid velocity.The cylinder velocity is consistently lower than the fluid velocity.The resulting surge and pitch motions are compared for all A-cases for the stratified flow in Figures 8 and 9, respectively.The larger the internal wave amplitude   , the larger the buoy motion is in both surge and pitch.Interestingly, the larger   , the earlier the surge peaks appear but the later the pitch peaks emerge.These observations also apply to all B and C-cases.The surge and pitch motions are compared for all 1-cases for the stratified flow in Figures 10 and 11, respectively.It can be seen the larger the internal wave period, the further the cylinder moves in surge but the smaller the pitch rotation.These observations apply for all three values of   .

Discussion
The maximum magnitude of the surge and maximum pitch motions are compared for all cases of Table 1 in the oscillating flow in Figures 12 and 13, respectively.The maximum magnitude of the cylinder surge movement is defined as the total oscillation measured from peak-to-trough of the surge oscillation to account for the slight asymmetry in surge motion.By contrast, the maximum cylinder pitch rotation is measured from zero to the peak of the pitch oscillation.As described in section 2.3, OrcaFlex captures the amplitude of the SPAR buoy surge oscillation, which can be described mathematically as in the oscillating flow and which is half the magnitude of the cylinder surge movement.Thus, the maximum magnitude of the surge results by OrcaFlex for the oscillating flow can be compared with the mathematical results of the surge amplitude.All cases for the oscillating flow have been tested and agree within ±1m, e.g., for case A1, the surge amplitude is    = 0.664 0.00331×1.9= 106m which is half of the maximum magnitude of the cylinder surge movement of 212m.Therefore, it can be concluded that OrcaFlex' surge movement results are correct for the oscillating flow cases.Inferring from Figure 12, the larger the internal wave amplitude and period,   and , the larger the maximum surge magnitude is for the floating cylinder.Interestingly, a positive linear correlation can be observed between  and the resulting maximum surge magnitudes.This can be explained by the mathematical relation described above between  and the surge amplitude: as  increases,  and  decrease according to the dispersion relation in equation (3), hence causing a rise in the surge amplitude.In Figure 13, the pitch of the floating cylinder is at its maximum for the combination of the largest   with the smallest value of .A weak negative linear correlation can be seen between  and the resulting maximum pitch rotations.This could be due to higher wave periods, and hence lower internal angular wave frequencies , resulting in smaller flow acceleration ̇, see equations ( 6) and (7).This leads to larger peak inertial force on the cylinder for the smaller wave period, causing smaller pitch rotations.
The maximum magnitude of the surge and the maximum pitch motions are compared for all cases in Table 1 for the stratified flow in Figures 14 and 15, respectively.Notably, the larger   and , the larger the maximum surge magnitude is for the floating cylinder, see Figure 14.This indicates a positive linear correlation between  and the resulting maximum surge magnitudes, which is due to flows of comparable velocities acting on the free-floating cylinder for longer and hence causing larger surge movements.In contrast, the pitch rotation of the floating cylinder is at its maximum for the combination of the largest value of   with the smallest .The analysis of Figures 12 and 14 shows that the maximum magnitudes of the surge movements are larger for the oscillating flow than the stratified flow, e.g., the maximum surge magnitude of case A1 reduces from 212m to 60m.The largest maximum magnitude of the surge movements is in case C1 in both oscillating (423m) and stratified flows (120m).However, the largest proportional decrease in surge movement is for case A3, where the surge decreases by a factor of 3.8 from 91m in the oscillating flow to 24m in the stratified flow.In contrast, the maximum pitch rotations are larger for the stratified flow when compared to the oscillating flow, e.g., the maximum pitch rotation of case A1 rises from 0.024° to 0.058° which is the largest maximum pitch rotation in both oscillating and stratified flows.However, Case C1 achieves the largest proportional increase in its pitch rotation by a factor of 4.25, as it rises from 0.012° in the oscillating flow to 0.051° in the stratified flow.
The pitch rotations around the -axis are symmetrical around 0° in both oscillating and stratified flows, whereas the surge movements are only symmetrical around  = 0m in the oscillating flow.The asymmetrical surge movements in the stratified flow, as exemplified in Figures 8 and 10, are due to the different depths of the top and bottom fluid layers (50m and 150m, respectively) and the resulting different flow forces along the -axis in these two layers.Since the peak velocities are higher in the top layer than in the bottom, i.e., | ℎ peak | > |  peak |, and in the positive -direction, the cylinder experiences a higher positive net force which equals positive forces minus negative forces acting on the cylinder along the -axis.Thus, in the stratified flow, the cylinder surge movement from  = 0m is larger in the positive direction.
The observed surge movements are large and require further research because there is the potential for significant, sustained mooring forces from internal waves on all types of moored platforms.The significance of mooring forces has already been observed for a moored OC3-Hywind SPAR buoy FOWT in the low frequency region of irregular surface waves where surge resonance peaks appeared due to the surge restoring force of its mooring system [23].Internal waves have low frequencies, and the resulting extra mooring loads are expected to be even larger when the platform is contained entirely within the upper layer of the stratified fluid, i.e., similar to the oscillating flow investigated here.Even though the unstretched length of catenary mooring lines can be as long as 902m [23], an accepted surge movement of a single FOWT is between 20-50m around a central point [24].As floating offshore wind farms are in development, the surge movement will have to be restricted, depending on the optimal distances between individual FOWTs within offshore floating wind farms.
Although the absolute pitch rotations presented here become more prominent in the stratified flow when compared to the oscillating flow, they are small in both.More importantly, they are smaller by an order of magnitude than the pitch motions observed for a moored SPAR FOWT due to surface waves in irregular sea states [23].Thus, it is expected that pitch motions due to internal waves are unlikely to be a major design consideration for SPAR FOWTs.
Possible limitations of this research are the adapted OrcaFlex approach applying the drag and inertia fluid forces separately and investigating only sinusoidal internal waves.Yet, this approach is considered appropriate for initially assessing the implications of internal waves acting on a free-floating cylinder.

Conclusion
This research investigated the effects of internal waves acting on a free-floating cylinder.In conclusion, the rotations were small (less than 0.1°) in the cases considered in this study for free-floating SPAR cylinders and can likely be disregarded.Semi-submersible FOWTs could be at a higher risk of surge motions and yaw rotations than SPAR buoys, as their geometry is similar to that of oil platforms which have experienced significant surge and yaw motions due to internal waves in the past.Additionally, the surge displacements of a free-floating cylinder were substantial in both oscillating and stratified flows, with maximum surge magnitudes of 423m and 120m, respectively.Therefore, significant additional mooring loads due to internal waves could be sustained by SPAR-type FOWTs.Further research is needed for the effects of internal waves when coupled with surface waves, wind and mooring lines.

Figure 1 .
Figure 1.Sketch of a sinusoidal internal wave acting on a free-floating cylinder.
magnitude as 2 ×    .The correction to this motion scales with (    ) 2

Figure 2 .
Figure 2. Case A1: Time series of the fluid and cylinder flow velocities simulated by the potential flow model (=ℎ) and by OrcaFlex (=) in the oscillating flow.

Figure 3 .
Figure 3. Case A1: Time series of the fluid and cylinder flow accelerations simulated by the potential flow model (=ℎ) and by OrcaFlex (=) in the oscillating flow.

Figure 4 .
Figure 4. Time series of the cylinder surge movement for  = 1000s and varying  amp in the oscillating flow.

Figure 5 .
Figure 5.Time series of the cylinder pitch rotation for  = 1000s and varying  amp in the oscillating flow.

Figure 6 .
Figure 6.Normalised time series of the cylinder surge movement for  amp = 17.5m and varying  in the oscillating flow.

Figure 7 .
Figure 7. Normalised time series of the cylinder pitch rotation for  amp = 17.5m and varying  in the oscillating flow.

Figure 8 .
Figure 8.Time series of the cylinder surge movement for  = 1000s and varying  amp in the stratified flow.

Figure 9 .
Figure 9.Time series of the cylinder pitch rotation for  = 1000s and varying  amp in the stratified flow.

Figure 10 .
Figure 10.Normalised time series of the cylinder surge movement for  amp = 17.5m and varying  in the stratified flow.

Figure 11 .
Figure 11.Normalised time series of the cylinder pitch rotation for  amp = 17.5m and varying  in the stratified flow.

Figure 12 .
Figure 12.Bar plot of the maximum magnitude of cylinder surge movement: the oscillating flow.

Figure 13 .
Figure 13.Bar plot of the maximum cylinder pitch rotation: the oscillating flow.

Figure 14 .
Figure 14.Bar plot of the maximum magnitude of cylinder surge movement: the stratified flow.

Figure 15 .
Figure 15.Bar plot of the maximum cylinder pitch rotation: the stratified flow.

Table 1 .
Input data for the numerical simulations for the oscillating and stratified flow, where    is only applicable in the latter.