Wave-phase dependence of Reynolds shear stress in the wake of fixed-bottom offshore wind turbine via quadrant analysis AIP 05/01/2024 10509609 10.1063/5.0191264 Journal of Renewable and Sustainable Energy 1941-7012 16 3 C MouchrefB ViggianoO FerčákJ BossuytN AliC MeneveauD GaymeR B Cal There has been an increase in recognition of the important role that the boundary layer turbulent flow structure has on wake recovery and concomitant wind farm efficiency. Most research thus far has focused on onshore wind farms, in which the ground surface is static. With the expected growth of offshore wind farms, there is increased interest in turbulent flow structures above wavy, moving surfaces and their effects on offshore wind farms. In this study, experiments are performed to analyze the turbulent structure above the waves in the wake of a fixed-bottom model wind farm, with special emphasis on the conditional averaged Reynolds stresses, using a quadrant analysis. Phase-averaged profiles show a correlation between the Reynolds shear stresses and the curvature of the waves. Using a quadrant analysis, Reynolds stress dependence on the wave phase is observed in the phase-dependent vertical position of the turbulence events. This trend is primarily seen in quadrants 1 and 3 (correlated outward and inward interactions). Quantification of the correlation between the Reynolds shear stress events and the surface waves provides insight into the turbulent flow mechanisms that influence wake recovery throughout the wake region and should be taken into consideration in wind turbine operation and placement.

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the wave crests there is a dominance of outward and inward events. Additionally, it was noted that there was a similar frequency of occurrence for all quadrants events above crests and a very slight dominance of ejections and sweep events over troughs. Currently, little is known about the characteristics of the external flow structures that influence kinetic energy entrainment 19 . Further characterization of the Reynolds stress should be performed by analyzing the fluctuating components of the velocity.

An additional challenge for offshore wind farms is the dynamic coupling of the ocean-wind and waves. Wind speed in the lower portion of the air-water boundary layer alternates (fast wind above the swell trough and slow wind above the swell crest) resulting in oscillatory wind speed due to wave propagation. It was found that an upward flux of kinetic energy due to the accelerating wind above the swell increases the extracted wind power in offshore turbines at wind speeds of 7 m/s and 10 m/s 16 . Yang, Meneveau, and Shen 20 created a dynamic model to represent small-scale unresolved wave motions as roughness elements in the context of wall modeled large eddy simulations of offshore wind farms. They found that "waves have an appreciable effect on the wind farm performance". In order to further characterize wave phase dependence, Xiao and Yang 21 , performed a triple decomposition of the turbulent fluctuations and the phase averaged mean to define an instantaneous phase-averaged dependent fluctuation term. Additionally, in a study utilizing direct numerical simulations and triple decomposition, there was a dependence on spanwise wave-coherent velocity and vertical air velocity that was found to be in phase with the wave form 22 . Understanding the dynamic coupling between ocean waves and Reynolds shear stress can provide a better insight of offshore wind site conditions. Ferčák et al. 5 performed a similar analysis on the same dataset used in this paper, focusing on the phase-dependent dynamics of the turbine wake. More specifically, velocity and Reynolds stress profiles were used to observe wave-phase dependence. In our analyses, we further study the Reynolds shear stress by categorizing the fluctuations using quadrant analysis to reveal how the direction of the flow is affected by wave phase.

Further complications can occur for floating turbines whose motion can be correlated with the wave motion. In the present study, we focus on the problem of fixed-bottom offshore turbines and more specifically on the structure of Reynolds stress distributions in the wake of a fixed wind turbine above moving waves. The data used in this study is generated in a laboratory experiment with a single scaled wind turbine in a wind tunnel above a water tank, and measurements are taken via particle image velocimetry (PIV). Using these data, we analyze the modulations of the turbine wake associated with laboratory surrogates of deep-water ocean waves. Analytical descriptions are presented in section III, followed by details of the experimental setup and data processing techniques in section IV. Section V provides the first and second-order statistics of the flow field as well as conditional averages based on quadrant analysis. Concluding remarks are included in section VI.

where u, and v, are the instantaneous streamwise and ver-

A dependence on φ is introduced as the wave phase, while

The phase-averaged mean is not resolved from the ensem-193 ble mean, it is a composition of the ensemble mean and 3 the phase-averaged deviation. Substitution of Eq. (3) into Eq. (2), results in triple decomposition: u φ (x, y, z 0 ,t) = u(x, y, z 0 ) + ũ(x, y, z 0 , φ ) + u ′ φ (x, y, z 0 ,t), (4) providing an indication that these quantities are related.

Here, z 0 is introduced to denote that the velocity is taken at a given plane directly behind the turbine. Therefore additional conditioning of the velocity signals is required to quantify the influence of the wave-wind interface on the wake dynamics.

Analysis of dominant contributions to the Reynolds shear stress can be characterized through a conditional sampling technique called quadrant analysis 16,24 . Velocity fluctuations are characterized into four types of events based on the respective signs of streamwise and wall-normal fluctuating velocity components designated as followed: Q1, outward in-

. These four events can be represented visually by creating a plane spanned by orthogonal axes with u ′ φ on the abscissa and v ′ φ as the ordinate see figure 1 18 .

where m denotes the quadrant, i.e., m= 1, 2, 3, and 4, k is a given snapshot signal and N is the total number of snapshots in the data set. The function

Quadrant events can be physically interpreted as revealing the directionality of the flow in comparison to the mean flow.

For example, ejections represent turbulent bursts upwards at velocities slower than the mean, while sweeps represent turbulent burst downward in the positive streamwise direction 16 .

Events in quadrants 2 and 4 cause a downward convection of streamwise momentum while quadrants 1 and 3 provide an upward flux 18 .

IV. EXPERIMENTAL SETUP

Experiments

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The inflow conditions, measured without the wind turbine 243 in the wind tunnel, fit well into an idealized neutral turbulent 244 boundary layer common for wind tunnel experiments 17,26 .

245

The roughness length, found by fitting log-law to the mean 4 velocity profile, is 0.5×10 -4 and 0.6×10 On-shore turbine wake flow presents large structures at the top tip in Q2 and Q4 16 , as is observed for the fixed bottom turbine cases presented in figure 6 as well. In addition, structures are observed at the bottom of the interrogation area and cause inward and outward interactions which are less commonly observed in on-shore turbine wakes. This addition is likely due to wave-wind interactions causing correlation of the streamwise and spanwise fluctuations.

To interpret the effects of the phase on the conditioned flow, quadrant analysis is applied and profiles are compared.

Figures 7 and8 depict quadrant analysis of the Reynolds shear stress profiles (u ′ φ i ,v ′ φ i ) at a low wind tunnel speed (v L ) of 2.5 m/s at long (ω 1.25 , figure 7) and short (ω 2.00 , figure 8)

wave conditions for all four phases. The profiles are obtained by time averaging then spatial averaging over the respective streamwise locations to investigate near to far-wake dynamics. The profiles are normalized by the averaged magnitude of ⟨|u ′ v ′ |⟩ for each given case. Reynolds shear stress in all four quadrants and all conditions dissipates moving downstream as undulations disappear and magnitude increases.

Negative peak magnitudes occur in quadrants 2 and 4 between the rotor and the top tip of the turbine blade. Positive larger amplitude that is associated with ω = 2.0 in compar-449 ison to ω = 1.25. This higher amplitude extends its effect on the wake behavior and therefore produces clear variation of the above hub height as a function of the wave phase. In all four quadrants, larger peak shear stress values are seen in the 2.00 Hz wave condition( figure 8) compared to the 1.25

wave Hz condition( figure 7). For onshore wind, peak sweep and ejection events show magnitudes of stress approximately 70% higher than that of the interaction events 19 . Peak sweep and ejection events (Q2 and Q4) show magnitudes of stress approximately 45-60% percent higher than that of the interaction events (Q1 and Q3). Higher magnitude of interaction events is likely due to interactions at the air-water interface.

The dependence on the wave for increased stresses near the wave-wind interface could cause increased vibrations and fatigue to the turbines and the mast when operating in offshore, fixed bottom conditions. This is most noticeable by the large stresses observed in the far-field, the PIV plane 4D downstream of the turbine and farthest away from the inflow, for Q1 and Q3 in figures 7 and 8, which occur just above the bottom tip of the rotor. Q1 and Q3 turbulent events can influence power production of consecutive turbines. Understanding how phase influences Reynolds shear stress can provide insights on wake recovery which can be used to maximize power extraction in downstream turbines through wake deviation, control strategies, and turbine spacing.

Differences in sweeps and ejections are correlated to gradients of second-order correlation terms, and can therefore utilized in second-order closure models 34 . Figure 9 Quantifying the difference magnitude between Q2 and Q4 events as well as Q1 and Q3 events provides information on the location of the flow mechanisms, which influence wake recovery.