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Research Papers: Alternative Energy Sources

# Wind–Wave Interaction Effects on a Wind Farm Power ProductionOPEN ACCESS

[+] Author and Article Information
A. AlSam

Department of Energy Sciences,
Lund University,
P.O. Box 118,
Lund SE-221 00, Sweden
e-mail: ali.al_sam@energy.lth.se

R. Szasz

Department of Energy Sciences,
Lund University,
P.O. Box 118,
Lund SE-221 00, Sweden
e-mail: robert-zoltan.szasz@energy.lth.se

J. Revstedt

Professor
Department of Energy Sciences,
Lund University,
P.O. Box 118,
Lund SE-221 00, Sweden
e-mail: johan.revstedt@energy.lth.se

1Corresponding author.

Contributed by the Advanced Energy Systems Division of ASME for publication in the JOURNAL OF ENERGY RESOURCES TECHNOLOGY. Manuscript received November 2, 2016; final manuscript received March 10, 2017; published online May 16, 2017. Assoc. Editor: Ryo Amano.

J. Energy Resour. Technol 139(5), 051213 (May 16, 2017) (11 pages) Paper No: JERT-16-1438; doi: 10.1115/1.4036542 History: Received November 02, 2016; Revised March 10, 2017

## Abstract

In the current study, the effects of the nonlocally generated long sea surface waves (swells) on the power production of a 2 × 2 wind farm are investigated by using large-eddy simulations (LES) and actuator-line method (ALM). The short sea waves are modeled as a roughness height, while the wave-induced stress accounting for swell effects is added as an external source term to the momentum equations. The results show that the marine atmospheric boundary layers (MABLs) obtained in this study have similar characteristics as the MABLs observed during the swell conditions by many other studies. The current results indicate also that swells have significant impacts on the MABL. As a consequence of these changes in the MABL, swells moving faster than the wind and aligned with the local wind direction increase the power extraction rate.

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## Introduction

Wind power is a renewable energy source that has minor environmental impact compared to the traditional energy sources such as nuclear and fossil fuel power. The available spaces in the large bodies of water make it possible to construct very large wind farms. The offshore wind is steadier and stronger than on-land and the offshore farms have less visual impacts, which makes them an attractive choice compared to on-land wind farms. However, offshore environment characterizes by a variable surface roughness caused by the sea surface waves. To improve wind turbine (WT) design and to optimize the layout of offshore wind farms with respect to energy production and costs, knowledge of the effects of large moving ocean waves on the MABL and by this on offshore wind energy is required.

Surface waves have various effects on the overlaying winds depending on the wave to wind speed ratio (i.e., wave-age), wave steepness, and wind–wave direction misalignment [15]. Generally, waves move much slower than the overlaying winds, absorb momentum from the wind, and slow down the wind near the water surface. The wave stress acts in this case parallel to the wind direction, and the waves act as obstacles that will increase the wind drag. Such a wave field usually consists of short waves and a broad range of frequencies that make it difficult to conduct accurate measurements near the wave surface to quantify the effects of an individual wave. Instead, the bulk effects of the wave field are usually modeled as a constant aerodynamic roughness. However, the wave field in this case develops under the action of wind forcing, and therefore the aerodynamic roughness changes with wind velocity. Waves do not respond instantaneously to changes in the wind speed or direction. Therefore, the wave spectrum can include waves moving faster than the wind or waves moving in different directions than the local wind direction. Surface waves can also be nonlocally generated by far away storms as in the case of swells. In contrast to slowly moving waves, swells moving faster than the wind decrease the wind drag [3,4] due to the fact that the wave-induced stress has in this case an opposite sign to that of the turbulent stress. The magnitude of the wave-induced stress increases with the wave-age and wave steepness. At high wave-age and wave steepness [2], the wave-induced stress increases and exceeds the turbulent stress which results in a negative wind drag, i.e., a thrust, that accelerates the wind to a velocity larger than the speed of the geostrophic wind and forming a supergeostrophic wind jet. Wave statistics show that the earth’s oceans are strongly dominated by swell waves almost all the time [68]. Therefore, a better knowledge of the effect of swells on the MABL plays a decisive role in atmospheric and oceanic models and in the offshore wind energy applications.

Recently, MABL over fast moving swell simulations resolving the swell geometry and using moving mesh solver to accomplish the wave motion shows that a standalone wind turbine’s power production increases in the presence of the swell compared to a wind turbine that has the same hub height wind velocity but without swell [2].

In the current study, the impacts of swell on wind turbine wake interactions are investigated by studying the power production of a 2 × 2 wind farm. The results of two cases, with and without swells that are derived by the same driving pressure forces, are compared. One extra case is also studied in which the case without swell is forced to have the same hub height velocity as the case of swell by correcting the driving pressure force each time step.

Because of their superior features for simulating flow dominated by large-scale structures, the large-eddy simulations are widely used in the atmospheric boundary layer studies and wind farm simulations, see, for example, Refs. [2], [3], [5], and [9]. Here, the LES are used, and the wind turbine rotor is modeled by using the ALM. Resolving the 2 × 2 wind farm with reasonable accuracy using moving mesh technique is very expensive in terms of computational resources and simulation time. Instead, in this study, the wave-induced stress is added to the momentum equation as an external source for the flat cases, i.e., the typical atmospheric boundary layer over a rough surface plus an extra source term. The magnitude of the wave-induced stress at the surface and its decaying rate above the water surface are selected to resemble the MABL observed during swell. This approach reduces the computational costs significantly. The results of the MABL simulations are saved and used then as inputs for wind farm simulations.

## LES Framework and Case Setup

Unsteady three-dimensional flows of neutral MABL are simulated in this study using LES frame work for incompressible Navier–Stokes equations (NSE) with Boussinesq approximation for density variation. A one-equation subgrid scale (SGS) model, similar to the one used by Deardorff [10] and Moeng and Wyngaard [11], is used to close the LES filtered NSE. The turbulence length scale used in the SGS model is varied according to the MABL stability as it is suggested by Deardorff [10], where the length scale is set equal to the LES filter and reduced for regions of stable stratification.

The surface heat flux is set to zero, and the MABL is initialized with zero temperature gradient $∂T/∂z=0$ for $z, followed by a stable inversion layer $∂T/∂z=0.01 K/m$ in the free atmosphere ($z>zi$). The initial boundary layer depth is set to be $zi=400 m$. The size of the geometry is set to (2520 m, 1260 m, 800 m) in (x, y, z) directions. In the respective directions, (252, 126, 110) grid points are used to discretize the domain, where 80 points are used inside the initial depth of the MABL and 30 points for the free atmosphere. The nodes are equally distributed inside the boundary layer which results in a cell size of (10 × 10 × 5) m3. Above the boundary layer, the grids are smoothly stretched vertically toward the domain top with Δzi+1 = 1.06Δzi.

Periodic boundary conditions are applied in the horizontal directions. At the upper boundary, a free shear boundary condition is applied. At the bottom surface, the instantaneous wall stress τij is modeled by relating it to the velocity at the first cell center above the surface by using the Monin–Obukhov similarity theory (MOST). The unresolved short waves are modeled by a constant aerodynamic roughness of 0.2 mm. The von Kármán constant in the MOST is taken to be 0.4, and the Coriolis parameter is taken to be 10−4 rad/s throughout this study.

The open source computational fluid dynamic toolbox openfoam 2.1.3 is used in this study. The dynamic pressure field is obtained by solving the Poisson equation that results from taking the divergence of momentum equation and applying the continuity equation by means of the PIMPLE algorithm (merged PISO-SIMPLE) with two outer correction loops. The spatial derivatives in the NSE are approximated by a second-order central finite volume method, and the time advancement is done by second-order backward implicit scheme.

Three cases are studied: A MABL over short sea surface waves (WOS), a MABL over swells (WS), and a MABL over short sea surface waves (WOFS) that has the same hub height velocity as the swell case. The first two cases have the same driving pressure forces $−(1/ρo) (∂Pd/∂xi)$ calculated from the geostrophic balance $(−fVg,−fUg,0)$, where $(Ug,Vg,0)$ is the geostrophic wind and set to be (5, 0, 0) m/s in this study, i.e., a wind blowing along the positive x-axis with 5 m/s. The swell effect is taken into account by adding the wave-induced stress to the momentum equation. The value of the wave-induced stress depends on many parameters: wind to wave speed ratio, wave steepness, and wave direction [24,12]. Here, the wave-induced stress is taken to be $τswell=0.035 m2/s2$, which corresponding to 35% of the wall shear stress calculated from MOST. This value is in the same order of magnitude as the one used by Semedo et al. [13] and the one used by Hanley et al. [7].

The generated swell-induced stress at the surface decays fast into the overlying atmosphere. The decaying rate of the wave-induced stress is taken to be exponentially as in the irrotational wave theory $(e−2κz)$ but with a higher constant $(e−2.2κz)$ [12], where κ is the swell wave number.

To eliminate the effect of wind velocity changing on the WT thrust coefficient and following the current wind energy practice in estimating the WT power production using the wind speed at the turbine hub height as a reference velocity, the WOFS case is designed to have the same hub height velocity as in the WS case. That is done by correcting the driving pressure forces at each time step to give an average horizontal wind velocity at the hub height plane equals to that of the WS case.

In all cases, the flows are initialized from coarse meshed LES, and the simulations are run with a fixed time step of 0.25 s for a time of 18,000 s, which is more than seven large-eddy turnover times $(zi/u*i)$ sufficient to reach statistical steady-state, where the calculated friction velocity is almost constant. Here, $(u*i)$ is the initial friction velocity calculated from MOST. The statistical quantities are then sampled for approximately 9000 s.

The results of these three cases are used as inflow boundary condition for the wind farm simulations. Data in the outflow plane of these simulations are saved at a frequency of 50 Hz during 200 s. The wind farm simulations have a zero gradient boundary condition for the outlet boundary and periodic boundary conditions for the spanwise direction. The NREL 5 MW reference turbine designed by Jonkman et al. [14] is used.

The 5 MW NREL WT has a diameter (d = 126 m) and a 90 m hub height. The actual geometry of the turbine is not resolved. Instead, the effect of the WT on the flow is introduced as body forces in the momentum equation. The body forces are calculated by using the ALM developed by Sørensen and Shen [15]. The model accounts for the WT rotor only and does not include the hub and the nacelle. In this study, each blade is represented by 40 points, and the Gaussian projection constant is set as twice the resolution of the mesh in the vicinity of the WT as recommended by Churchfield [16]. A more detailed description of the ALM and its implementation in openfoam has been presented in Refs. [1719].

The simulated wind farm has 2 × 2 WTs, and its geometry is the same as for the previously described cases. The WTs are arranged into two columns and two rows with the first row located at 3d from the inlet and the second row is 7d distance downstream the first turbine. In the spanwise direction, the distance between the turbines is 5d. With periodic boundary condition in spanwise direction, the simulations represent an infinite number of WTs. The mesh used in the ABL simulations is refined around each turbine for the wind farm simulations in two levels. The first level is a 300 m height box of 4d wide and 20d long, and the second level is a 250 m height box of 3d wide and 20d long with a wind turbine in the center of each box. The finest mesh has a (2.5 × 2.5 × 1.25) m3 cell size resolution, and the total cell number is about 38 × 106, see Fig. 1. To ensure that the blade does not jump a whole cell during its movement at each time step, the time step is reduced to 0.02 s.

To assert that the LES of the current study have sufficient resolutions to resolve the main part of the turbulent kinetic energy spectra, the time- and space-averaged ratio of the SGS turbulent kinetic energy to the total kinetic energy of the MABL of the short sea waves (WOS) case is plotted in Fig. 2(a). As it can be seen in the figure, the LES model is resolving more than 85% of the total turbulent kinetic energy in the bulk of the boundary layer. In Fig. 2(b), the shear stress partitions (the SGS, the resolved, and the total shear stresses), normalized by the total shear stress at the sea surface, are shown. The figure implies that the contribution of SGS shear stresses to the total shear stress is a small part of the total shear stress. The turbulent energy spectrum at two probe points taken at the middle of the domain at 30 m and 180 m above the surface is calculated in Fig. 2(c). The spectrum follows to some extent the −5/3 line, which indicates that the cutoff based on the current cell size is located in the inertial subrange as it is required for a well resolved LES. A mesh sensitivity analysis is also made for the WOS case by testing two additional mesh resolutions M1 (15 × 15 × 7.5) and M3 (7.5 × 7.5 × 3.75) and compares the results to the current mesh resolution M2. The mean velocity profiles of these three resolutions are shown in Fig. 2(d). The results show that an increase/decrease of the mesh resolution by 50% results in maximum change in mean velocity profile less than 2%. The results illustrated in Fig. 2 confirm that the employed grid resolution is sufficient for the purpose of this study.

## Results

###### The Velocity Profiles.

The time- and space-averaged horizontal velocities of the three cases are presented next in Fig. 3. The MABL over short sea surface wave (the WOS and WOFS cases) resembles the flow over a flat terrain where the wind speed has a logarithmic like profile in the lower part of the atmospheric boundary layer and gradually increases to match its free atmosphere value above the boundary layer. The angle ϴ in Fig. 3(b) is the angle between the wind direction and the free stream flow direction. As it can be seen in the figure, the flow direction of the MABL over short sea surface wave cases is turned to the left of its free stream direction with maximum turning above the surface and gradually decreasing toward the boundary layer top, as it is suggested by the Ekman layer.

Contrary to short sea surface, the swell has a dramatic effect on the MABL flow. Comparing the cases of the same driving pressure forces, the one in the presence of swell (WS) and the one over short sea surface waves (WOS), the velocity in the swell case is substantially higher. The swell accelerates the overlaying wind to a velocity greater than the free atmosphere velocity at some distance above the surface. Above this velocity maximum, the wind velocity decreases slightly (i.e., negatively sheared) to match its free atmosphere velocity above the MABL. The MABL above the swell case has almost a constant velocity above . Unlike the case of MABL above short sea waves, the wind direction of the swell case does not change significantly from its free stream direction. The ABL depth dramatically reduces in the presence of swell.

Since the driving pressure force is fixed in both cases with and without swell, the geostrophic balance between the driving pressure force from one side and the result of the Coriolis and the drag forces in the other side suggests that this increase in the velocity and the reduction in flow turning in the presence of swell are a footprint of a reduction in wind drag [2,20].

These features of the MABL profiles in the presence of swell obtained in the current study are similar to the profiles measured in the Baltic Sea during swell conditions [2126] and the one obtained by LES that resolve the swell geometry [2,3]. Similar features are also obtained previously by adding the wave-induced stress as external source term as in this case but with simpler turbulence model [13,20].

The case of MABL over swell (WS) is also compared with a counterpart case over short sea waves that has the same velocity at the turbine hub height, the (WOFS) case. Altering the driving pressure force and fixing the velocity at a certain height in the MABL over short sea surface waves WOFS result in almost the same profiles as the case of MABL over short sea surface waves with fixed pressure force (WOS) case but with different magnitudes. Comparing the two cases of the same hub height velocities, WOFS and WS cases, reveals that the flow over swells has substantially higher velocities below the hub height and lower velocities above it than those obtained in the case without swell.

The cases including short sea surface waves exhibit higher vertical shear and veer comparing to the case over swell. Unlike the case without swell, the velocity profile is negatively sheared in the rotor area in the presence of the swell. Table 1 summarizes the wind shear and veer of the three cases, both between the hub height and the surface, and between the turbine blades highest and lowest positions. The overall increase in wind velocity across the turbine swept area due to swell is about 20% in the cases of the same driving pressure forces and about 1.5% for the cases of the same hub height velocity.

###### The Turbulence Profiles.

The total turbulence quantities (resolved plus subgrid scale) are illustrated in Fig. 4. The figure reveals a substantial decrease in the atmospheric turbulence in the presence of swell over the whole boundary layer depth. This reduction in the turbulence must be due to the extensive decrease in the wind shear and veer, where for a neutral boundary layer the atmospheric turbulence is mainly generated mechanically by wind shear.

The turbulent stress components of the WOS and the WS cases are further examined in Figs. 5(a) and 5(b), respectively. The stresses are presented normalized by their corresponding turbulent kinetic energy. As it can be seen in the figure, the components of the stress tensor in the WS case are closer to each other than the WOS case, which implies that the presence of swell increases the turbulence isotropy slightly. To examine the turbulence isotropy of the two cases, the anisotropy invariant map of the kinematic shear stresses is plotted in the turbulence triangle [27] in Fig. 5(c). The stress components of the WOS case are mainly located to the right limit of the triangle which indicates that the turbulence in the WOS case has an axisymmetric (prolate) shape with one large eigenvalue. In the WS case, the turbulence near the surface has a spherical isotropic shape, where the eigenvalues of the normalized stress tensor are almost zero. Above the surface, the turbulence has an axisymmetric (prolate) shape, similar to the WOS case, with one large eigenvalue. The first invariant (η) of the normalized stress tensor decreases, while the second invariant (ς) increases with the height up to about the height of the velocity maximum (25 m above the surface), which means that the turbulence goes more toward the two-component axisymmetric (the base of the triangle) near the velocity maximum. Further above the velocity maximum, the turbulence returns gradually to the spherical isotropic shape following the axisymmetric (prolate) line. There are regions of alternating increasing/decreasing stress magnitudes in the WS case in Fig. 5(b), while they generally decrease with the height in the WOS case in Fig. 5(a) (except for the vv component which increases due to the spiral profile of the Ekman layer). More interesting is the presence of a region of positive streamwise shear stress (uw) in the WS case. This positive shear stress region is believed to be caused by the swell-induced stress τswell, which acts against the turbulent shear stress [2,3,12]. To test this hypothesis, the shear stress (uw) is further decomposed into viscous, turbulence (filtered and SGS), and swell components. The viscous term is neglected here due to the high Reynolds number. The other components are displayed for the lower part of the MABL of the WS case in Fig. 6. The partition of the total stress is noticeably different from the one of the MABL over short sea waves (see Fig. 2(b)). The total shear stress and its components τgs and τsgs of the WS case are much lower than the counterpart stresses of the WOS case. Both the filtered turbulent shear stress τgs and the swell-induced stress τswell of the WS case have their maximum at the surface, and they are almost equal in magnitude. Therefore, the total shear stress at the surface is very small, and it is mainly due to the subgrid scale stress. The low shear stress at the surface results in a low velocity gradient above the surface. The reduced velocity gradient produces less turbulence, and therefore the τgs also reduces. The τswell decays exponentially above the surface, and it acts against the result of τgs and τsgs. At some height above the surface, the τswell balances the turbulent stress, and the total stress becomes zero. Above this height, the turbulence stress becomes even lower than the swell-induced stress, and the total stress changes its sign (i.e., becomes positive, upward momentum transfer). This upward momentum accelerates the wind to a velocity higher than its free stream velocity and results in so-called supergeostrophic wind jet that can be seen in the velocity profile.

To explore whether the turbulent structures are affected by the presence of the swell or not, the fluctuation velocities of two planes parallel to the surface at 25 m and 90 m of the WS case are further analyzed and compared with planes of the WOS case. In Fig. 7, the streamwise and vertical fluctuations in a plane taken at 25 m height show that in the presence of the swell the turbulent activity is much lower. In both cases, some longitudinal strikes are identified in the flow direction. However, the WS case exhibits sparse events of larger structures. This might be due to the absence of a strong flow shear that can break down these structures in the WS case. These structures break down further up in the boundary layer as it can be seen in Fig. 8 for a plane 90 m above the surface.

The turbulence quadrant analysis of these planes is presented next in Fig. 9, and the probability of occurrence of each quadrant is summarized in Table 2. The quadrant analysis is an attempt to further extricate the structure of turbulent events, where the shear stress is decomposed into four quadrant events (Q1–Q4). These quadrants are numbered conventionally:

• Quadrant 1 (Q1): u > 0, w > 0 (outward interaction)

• Quadrant 2 (Q2): u < 0, w > 0 (ejection or burst)

• Quadrant 3 (Q3): u < 0, w < 0 (inward interaction)

• Quadrant 4 (Q4): u > 0, w < 0 (sweep)

The events in quadrants 2 and 4 contribute positively to the downward momentum flux, while the events in quadrants 1 and 3 contribute negatively. The planes of the MABL over short sea waves’ case have the majority of events in the second and fourth quadrants (i.e., negative momentum transport). The ratio between the ellipse two axis is smaller in the 90 m plane than the 25 m plane, which means that the turbulence is more isotropic. The planes of the MABL for the swell case show more events in the first and third quadrants which confirm the existence of a positive momentum transport in both planes. Similarly to the WOS case, the lower the plane, the closer the major axis to the x-axis which implies the presence of the longitudinal structures in the streamwise direction near the surface. While the higher the plane, the more isotropic turbulence is noticed. The 90 m and 25 m planes of the WS case show almost the same probability of the outward events (the first quadrant). However, there is an increase in the probability of the inward interaction events (the third quadrant) for the 90 m plane, where the inward interaction is the dominant event in 90 m while the outward is the dominant event in the 25 m plane.

###### The Wind Farm Results.

As it is mentioned before, the results of these cases are utilized to initialize the flow and as inlet boundary conditions of the wind farm simulations. The three-dimensional vortical structures in the turbine wakes are illustrated in Fig. 10, where isosurfaces of the λ2 criterion are shown for the three cases.

The wind turbine wake for the case of the MABL over swell (WS) is substantially different for the cases of the MABL over short sea waves (WOS). The wake is longer in the presence of swell with larger vortical structures comparing to that of the MABL over short sea waves. Due to the larger veer in the mean velocity profile of the case without swell, the wake of the upstream wind turbines is turned away from the downstream turbines’ direction. While in the case of swell, the downstream turbines are operating in the wake region of the upstream turbines. On the other hand, the differences between the cases of the same hub height velocities, the case over short sea waves WOFS and the case over swell WS, are not significant. However, one can still identify some differences in the presence of swell, where larger structures with larger distance between the structures behind the upstream turbines are noticed in the WS case comparing to that of WOFS case.

The flow velocities of the three cases at a plane parallel to the surface and at the hub height are shown in Figs. 11 and 12.

As it can be seen in the figures, for the same driving pressure force, there is a large differences in the flow between the case of the same driving pressure force, without swell and with swell, while small differences are noticed between the cases of the same hub height velocities. The mean velocities exhibit longer near wake region behind the first row of WTs in the presence of swell comparing to the case without swell of the same hub height velocity. This is also noticed previously for standalone WT simulations [2]. The velocities of the second row WT’s wake are almost the same for the cases of the same hub height velocity. One can also see in the perpendicular plane, see Fig. 12, that there is a region of high velocities above the surface in the WS case and that the WT’s wake shifted upward compared to the WOFS case.

The turbine power productions of the three cases are depicted in Fig. 13 and summarized in Table 3. For the same driving pressure force, there is an extensive increase in the overall power production (about 48%) in the presence of swell compared to the short surface sea waves’ case. The power production of the upwind turbines in the WS case is almost double the power production of the counterpart turbines in the WOS case (see Fig. 13(b)), due to the substantial increase in the wind velocity. However, because of the flow direction, the downstream wind turbines are not affected by the upstream wind turbines in the WOS case, while they are operating in the wake of the upstream WTs in the WS case. That can be seen in Fig. 13(c), where the ratio of the downstream to upstream power production is almost 1 for the WOS case while it is about 0.45 for the WS case. This reduces the differences in wind farm overall power production between the two cases.

For the same hub height velocities, there is only 1.5% increase in the integral velocities over the turbine swept area of the WS case compared to the WOFS case. This difference in flow velocity results in about 4.3% increase in the power production of the upstream turbines. Unlike the cases over short sea waves, the case over swell does not show a larger scale variation in the upstream turbines’ power productions magnitude over time. More remarkable is the increase in downstream wind turbines’ power production of the WS case compared to the WOFS case by almost 9.9%. We attribute this increase mainly to the supergeostrophic wind jet that is formed in the WS case due to the presence of swell. The high velocity above the surface in the WS case feeds the upstream turbine wake continuously with upward momentum and reduces the velocity deficit. The other reason to this increase is the decrease in wind shear and veer, where the wind turbine power production is found to increase with decreasing wind shear, e.g., Ref. [28].

## Conclusions

The impacts of nonlocally generated long sea surface waves (swell) on the MABL and by this on a 2 × 2 wind farm are investigated by using the LES and the ALM. The bulk effects of the short sea surface waves are taken into account as a roughness height, while the swell-induced stress is added as an external term into the momentum equations. The simulation results exhibit similar features of the MABL as the one measured during the swell conditions and the MABLs over swell obtained previously by other LES simulations.

The results show that the swell has extensive effects on the MABL. Swells are found to increase the velocity in the whole boundary layer and decrease both the wind shear and veer. As a result, the overall wind farm power extraction rates increase in the presence of swell. It is worth noting that these results are obtained by adding a swell-induced stress that corresponds to a very fast swell that propagates faster than the local wind and in the local wind direction. Different results might be obtained for other swell-induced stresses, since it depends on the swell to wind velocity ratio, swell steepness, wind–swell direction misalignment, and many other parameters.

## Acknowledgements

This work is a part of the Offwind project which is a collaboration between IRIS, SINTEF, Aalborg University, Lund University, Vattenfall, Norsk Vind Energi AS, and MEGAJOULE. Financial support for Offwind was provided by Nordic Energy Research. The simulations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at the Center for scientific and technical computing at Lund University (LUNARC).

## Nomenclature

• ALM =

actuator-line method

• d =

turbine diameter (m)

• f =

Coriolis parameter (rad/s)

• g =

geostrophic

• GS =

filtered or gridscale

• h =

turbine hub height

• hm =

rotor lowest position

• hp =

rotor highest position

• i =

initial value

• I =

turbulent intensity

• K =

subgrid scale kinetic energy (m2/s2)

• LES =

large-eddy simulations

• MABL =

marine atmospheric boundary layer

• MOST =

Monin–Obukhov similarity theory

• NSE =

Navier–Stokes equations

• o =

reference value

• P =

pressure (Pa)

• SGS =

subgrid scale value

• tot =

total (grid plus subgrid) value

• U =

subgrid scale velocity (m/s)

• WT =

wind turbine

• Z =

latitude (m)

• ϴ =

flow direction angle with x-axis (deg)

• κ =

wave number

• ρ =

density (kg/m3)

• τ =

kinematic shear stress (m2/s2)

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Moeng, C.-H. , and Wyngaard, J. C. , 1988, “ Spectral Analysis of Large-Eddy Simulations of Convective Boundary Layer,” J. Atmos. Sci., 45(23), pp. 3573–3587.
Sullivan, P. P. , and McWilliams, J. C. , 2010, “ Dynamics of Winds and Currents Coupled to Surface Waves,” Annu. Rev. Fluid Mech., 42(1), pp. 19–42.
Semedo, A. , Saetra, Ø. , Rutgersson, A. , Kahma, K. K. , and Pettersson, H. , 2009, “ Wave-Induced Wind in the Marine Boundary Layer,” J. Atmos. Sci., 66(8), pp. 2256–2271.
Jonkman, J. , Butterfield, S. , Musial, W. , and Scott, G. , 2009, “ Definition of a 5-MW Reference Wind Turbine for Offshore System Development,” Technical Report No. NREL/TP-500-38060.
Sørensen, J. N. , and Shen, W. Z. , 2002, “ Numerical Modeling of Wind Turbine Wakes,” ASME J. Fluids Eng., 124(2), pp. 393–399.
Churchfield, M. , 2011, “ Wind Energy/Atmospheric Boundary Layer Tools and Tutorials,” Sixth OpenFOAM Workshop, State College, PA, June 13–16, pp. 1–70.
Churchfield, M. J. , Moriarty, P. J. , Vijayakumar, G. , and Brasseur, J. G. , 2010, “ Wind Energy-Related Atmospheric Boundary-Layer Large-Eddy Simulation Using OpenFOAM,” Technical Report No. NREL CP-500-48905.
Lee, S. , Churchfield, M. , Moriarty, P. , Jonkman, J. , and Michalakes, J. , 2011, “ Atmospheric and Wake Turbulence Impacts on Wind Turbine Fatigue Loading,” Technical Report No. NREL CP-5000-53567.
Churchfield, M. J. , Lee, S. , Moriarty, P. , Martinez, L. A. , Leonardi, S. , Vijayakumar, G. , and Brasseur, J. G. , 2012, “ A Large-Eddy Simulation of Wind-Plant Aerodynamics,” Technical Report No. NREL CP-5000-53554.
Hanley, K. , and Belcher, S. , 2008, “ Wave-Driven Wind Jets in the Marine Atmospheric Boundary Layer,” J. Atmos. Sci., 65(8), pp. 2646–2660.
Högström, U. , Smedman, A. , Sahlée, E. , Drennan, W. , Kahma, K. , Pettersson, H. , and Zhang, F. , 2009, “ The Atmospheric Boundary Layer During Swell: A Field Study and Interpretation of the Turbulent Kinetic Energy Budget for High Wave Ages,” J. Atmos. Sci., 66(9), pp. 2764–2779.
Smedman, A. , Tjernström, M. , and Högström, U. , 1994, “ The Near-Neutral Marine Atmospheric Boundary Layer With No Surface Shearing Stress: A Case Study,” J. Atmos. Sci., 51(23), pp. 3399–3411.
Smedman, A.-S. , Högström, U. , and Bergström, H. , 1997, “ The Turbulence Regime of a Very Stable Marine Airflow With Quasi-Frictional Decoupling,” J. Geophys. Res., 102(C9), pp. 21049–21059.
Smedman, A. , Tjernström, S. , Högström, U. , Bergström, H. , Rutgersson, A. , Kahma, K. K. , and Pettersson, H. , 1999, “ A Case Study of Air-Sea Interaction During Swell Conditions,” J. Geophys. Res., 104(C11), pp. 25833–25851.
Smedman, A. , Tjernström, M. , and Sjöblom, A. , 2003, “ A Note on Velocity Spectra in the Marine Boundary Layer,” Boundary-Layer Meteorol., 109(1), pp. 27–48.
Smedman, A. , Högström, U. , and Sahlée, E. , 2009, “ Observational Study of Marine Atmospheric Boundary Layer Characteristics During Swell,” J. Atmos. Sci., 66(9), pp. 2747–2763.
Pope, S. , 2000, Turbulent Flows, Cambridge University Press, Cambridge, UK.
Wagner, R. , Courtney, M. , Gottschall, J. , and Lindelöw-Marsden, P. , 2011, “ Accounting for the Speed Shear in Wind Turbine Power Performance Measurement,” Wind Energy, 14(8), pp. 993–1004.
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## References

Al Sam, A. , Szasz, R. , and Revstedt, J. , 2014, “ The Effect of Moving Waves on Neutral Marine Atmospheric Boundary Layer,” ITM Web Conf., 2, p. 01003.
AlSam, A. , Szasz, R. , and Revstedt, J. , 2015, “ The Influence of Sea Waves on Offshore Wind Turbine Aerodynamics,” ASME J. Energy Resour. Technol., 137(5), p. 051209.
Sullivan, P. , Edson, J. , Hristov, T. , and McWilliams, J. , 2008, “ Large-Eddy Simulations and Observations of Atmospheric Marine Boundary Layers Above Nonequilibrium Surface Waves,” J. Atmos. Sci., 65(4), pp. 1225–1245.
Sullivan, P. P. , McWilliams, J. C. , and Moeng, C.-H. , 2000, “ Simulation of Turbulent Flow Over Idealized Water Waves,” J. Fluid Mech., 404, pp. 47–85.
Nilsson, E. O. , Rutgersson, A. , Smedman, A.-S. , and Sullivan, P. P. , 2012, “ Convective Boundary-Layer Structure in the Presence of Wind-Following Swell,” Q. J. R. Meteorol. Soc., 138(667), pp. 1476–1489.
Semedo, A. , Sušeli, K. , Rutgersson, A. , and Sterl, A. , 2011, “ A Global View on the Wind Sea and Swell Climate and Variability From ERA-40,” J. Clim., 24(5), pp. 1461–1479.
Hanley, K. E. , Belcher, S. E. , and Sullivan, P. P. , 2010, “ A Global Climatology of Wind–Wave Interaction,” J. Phys. Oceanogr., 40(6), pp. 1263–1282.
Chen, G. , Chapron, B. , Ezraty, R. , and Vandemark, D. , 2002, “ A Global View of Swell and Wind Sea Climate in the Ocean by Satellite Altimeter and Scatterometer,” J. Atmos. Oceanic Technol., 19(11), pp. 1849–1859.
Ivanell, S. , Mikkelsen, R. , Sørensen, J. N. , and Henningson, D. , 2008, “ Three-Dimensional Actuator Disc Modelling of Wind Farm Wake Interaction,” European Wind Energy Conferences and Exhibition (EWEC), Brussels, Belgium, Mar. 31–Apr. 3, pp. 3038–3047.
Deardorff, J. , 1980, “ Stratocumulus-Capped Mixed Layers Derived From a Three-Dimensional Model,” Boundary-Layer Meteorol., 18(4), pp. 495–527.
Moeng, C.-H. , and Wyngaard, J. C. , 1988, “ Spectral Analysis of Large-Eddy Simulations of Convective Boundary Layer,” J. Atmos. Sci., 45(23), pp. 3573–3587.
Sullivan, P. P. , and McWilliams, J. C. , 2010, “ Dynamics of Winds and Currents Coupled to Surface Waves,” Annu. Rev. Fluid Mech., 42(1), pp. 19–42.
Semedo, A. , Saetra, Ø. , Rutgersson, A. , Kahma, K. K. , and Pettersson, H. , 2009, “ Wave-Induced Wind in the Marine Boundary Layer,” J. Atmos. Sci., 66(8), pp. 2256–2271.
Jonkman, J. , Butterfield, S. , Musial, W. , and Scott, G. , 2009, “ Definition of a 5-MW Reference Wind Turbine for Offshore System Development,” Technical Report No. NREL/TP-500-38060.
Sørensen, J. N. , and Shen, W. Z. , 2002, “ Numerical Modeling of Wind Turbine Wakes,” ASME J. Fluids Eng., 124(2), pp. 393–399.
Churchfield, M. , 2011, “ Wind Energy/Atmospheric Boundary Layer Tools and Tutorials,” Sixth OpenFOAM Workshop, State College, PA, June 13–16, pp. 1–70.
Churchfield, M. J. , Moriarty, P. J. , Vijayakumar, G. , and Brasseur, J. G. , 2010, “ Wind Energy-Related Atmospheric Boundary-Layer Large-Eddy Simulation Using OpenFOAM,” Technical Report No. NREL CP-500-48905.
Lee, S. , Churchfield, M. , Moriarty, P. , Jonkman, J. , and Michalakes, J. , 2011, “ Atmospheric and Wake Turbulence Impacts on Wind Turbine Fatigue Loading,” Technical Report No. NREL CP-5000-53567.
Churchfield, M. J. , Lee, S. , Moriarty, P. , Martinez, L. A. , Leonardi, S. , Vijayakumar, G. , and Brasseur, J. G. , 2012, “ A Large-Eddy Simulation of Wind-Plant Aerodynamics,” Technical Report No. NREL CP-5000-53554.
Hanley, K. , and Belcher, S. , 2008, “ Wave-Driven Wind Jets in the Marine Atmospheric Boundary Layer,” J. Atmos. Sci., 65(8), pp. 2646–2660.
Högström, U. , Smedman, A. , Sahlée, E. , Drennan, W. , Kahma, K. , Pettersson, H. , and Zhang, F. , 2009, “ The Atmospheric Boundary Layer During Swell: A Field Study and Interpretation of the Turbulent Kinetic Energy Budget for High Wave Ages,” J. Atmos. Sci., 66(9), pp. 2764–2779.
Smedman, A. , Tjernström, M. , and Högström, U. , 1994, “ The Near-Neutral Marine Atmospheric Boundary Layer With No Surface Shearing Stress: A Case Study,” J. Atmos. Sci., 51(23), pp. 3399–3411.
Smedman, A.-S. , Högström, U. , and Bergström, H. , 1997, “ The Turbulence Regime of a Very Stable Marine Airflow With Quasi-Frictional Decoupling,” J. Geophys. Res., 102(C9), pp. 21049–21059.
Smedman, A. , Tjernström, S. , Högström, U. , Bergström, H. , Rutgersson, A. , Kahma, K. K. , and Pettersson, H. , 1999, “ A Case Study of Air-Sea Interaction During Swell Conditions,” J. Geophys. Res., 104(C11), pp. 25833–25851.
Smedman, A. , Tjernström, M. , and Sjöblom, A. , 2003, “ A Note on Velocity Spectra in the Marine Boundary Layer,” Boundary-Layer Meteorol., 109(1), pp. 27–48.
Smedman, A. , Högström, U. , and Sahlée, E. , 2009, “ Observational Study of Marine Atmospheric Boundary Layer Characteristics During Swell,” J. Atmos. Sci., 66(9), pp. 2747–2763.
Pope, S. , 2000, Turbulent Flows, Cambridge University Press, Cambridge, UK.
Wagner, R. , Courtney, M. , Gottschall, J. , and Lindelöw-Marsden, P. , 2011, “ Accounting for the Speed Shear in Wind Turbine Power Performance Measurement,” Wind Energy, 14(8), pp. 993–1004.

## Figures

Fig. 1

The geometry and the mesh used to simulate the wind farm

Fig. 2

LES resolution and subgrid scale stresses’ contribution of the WOS case: (a) the ratio of the subgrid (SGS) to total (tot) turbulent kinetic energy, (b) the filtered, SGS, and total turbulent kinematic shear stresses normalized by the total kinematic shear stress at the surface, (c) the power spectra of the streamwise fluctuation velocity at the two probe points, and (d) the time- and space-averaged horizontal wind speed of three different mesh resolutions

Fig. 3

Time- and space-average of (a) the horizontal wind speeds and (b) the horizontal wind speed directions. The black lines are a schematic depiction of the studied wind turbine.

Fig. 4

Time- and space-average of the atmospheric turbulence: (a) the turbulent kinetic energy, (b) the turbulent intensity, and (c) the magnitude of the turbulent kinematic shear stress

Fig. 5

The kinematic shear stresses normalized by their kinetic energy of (a) the WOS case and (b) the WS case, and (c) the anisotropy invariant maps of the kinematic shear stresses of the two cases

Fig. 6

The total streamwise kinematic turbulent shear stress (uw) and its components of the WS case

Fig. 7

Velocity fluctuations (u in the upper part and w in the lower part) of a plane at 25 m above the surface. The left side is the WOS case, and the right side is the WS case. Note that the color scale of the WOS case is four times larger than the WS case.

Fig. 8

Velocity fluctuations (u in the upper part and w in the lower part) of a plane at 90 m above the surface. The left side is the WOS case, and the right side is the WS case. Note that the color scale of the WOS case is four times larger than the WS case.

Fig. 9

The turbulence quadrant analysis of 25 m (upper part) and 90 m (lower part) planes above the surface. The left side is the WOS case, and the right side is the WS case.

Fig. 10

Isosurfaces of the λ2 criterion colored by the mean velocity values. The cases are (a) WOS, (b) WOFS, and (c) WS.

Fig. 11

The 2 × 2 wind farm velocities, (a)–(c) is the instantaneous velocity and (d)–(f) is the mean velocity, of a horizontal plane parallel to the surface taken at the hub height plane. The cases are WOS ((a) and (d)), WOFS ((b) and (e)), and WS ((c) and (f)) from the left to the right, respectively.

Fig. 12

The 2 × 2 wind farm velocities, (a)–(c) is the instantaneous velocities and (d)–(f) is the mean velocities, of a vertical plane perpendicular to the WTs in the right column of wind farm. The cases are WOS ((a) and (d)), WOFS ((b) and (e)), and WS ((c) and (f)).

Fig. 13

The turbine power productions: (a) the power production of the whole wind farm, (b) the power productions of the upstream turbines, and (c) the ratio between the upstream and downstream turbines’ power production

## Tables

Table 1 Summary of the velocity magnitude and direction at the studied turbine rotor plane
Table 2 Summary of the turbulence quadrant analysis of two planes 25 m and 90 m above the surface
Table 3 Summary of the wind farm power production. Pw is the power production; Pwupstream and Pwdownstream are the averaged power production of the first and second rows of WTs, respectively.

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