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Research Papers: Fundamental Issues and Canonical Flows

# Quantification of Preferential Contribution of Reynolds Shear Stresses and Flux of Mean Kinetic Energy Via Conditional Sampling in a Wind Turbine ArrayPUBLIC ACCESS

[+] Author and Article Information
Hawwa Falih Kadum, Devin Knowles, Raúl Bayoán Cal

Department of Mechanical
and Materials Engineering,
Portland State University,
Portland, OR 97207

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received September 19, 2017; final manuscript received June 6, 2018; published online July 10, 2018. Assoc. Editor: Jun Chen.

J. Fluids Eng 141(2), 021201 (Jul 10, 2018) (9 pages) Paper No: FE-17-1598; doi: 10.1115/1.4040568 History: Received September 19, 2017; Revised June 06, 2018

## Abstract

Conditional statistics are employed in analyzing wake recovery and Reynolds shear stress (RSS) and flux directional out of plane component preference. Examination of vertical kinetic energy entrainment through describing and quantifying the aforementioned quantities has implications on wind farm spacing, design, and power production, and also on detecting loading variation due to turbulence. Stereographic particle image velocimetry measurements of incoming and wake flow fields are taken for a 3 × 4 model wind turbine array in a scaled wind tunnel experiment. Reynolds shear stress component is influenced by $⟨uv⟩$ component, whereas $⟨vw⟩$ is more influenced by streamwise advection of the flow; u, v, and w being streamwise, vertical, and spanwise velocity fluctuations, respectively. Relative comparison between sweep and ejection events, $ΔS⟨uiuj⟩$, shows the role of streamwise advection of momentum on RSS values and direction. It also shows their tendency to an overall balanced distribution. $⟨uw⟩$ intensities are associated with ejection elevated regions in the inflow, yet in the wake, $⟨uw⟩$ is linked with sweep dominance regions. Downward momentum flux occupies the region between hub height and top tip. Sweep events contribution to downward momentum flux is marginally greater than ejection events'. When integrated over the swept area, sweeps contribute 55% of the net downward kinetic energy flux and 45% is the ejection events contribution. Sweep dominance is related to momentum deficit as its value in near wake elevates 30% compared to inflow. Understanding these quantities can lead to improved closure models.

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## Introduction

Conditional sampling of Reynolds shear stress (RSS) has a long history in the study of boundary layers dating back to the introduction of the technique in experiments by Lu and Willmarth [1]. This method classifies the events in a uv plane into four events according to the fluctuations in the pertinent directions. This classification determines which event has the most significant contribution in Reynolds stress production. The directional preference of RSS determines the locations associated with more kinetic energy entrainment within the rotor area. Investigation of the factors influencing downward and upward kinetic energy entrainment is required for wake recovery assessments and wind farm spacing optimization. In addition, quantification of sweep and ejection events influence the stability of the turbulence loadings endured by the turbine as it was found that increased ejection events and reduced sweep events are associated with higher unsteady turbulence loads [2].

Quadrant analysis is also employed in boundary layer studies in which it was shown that the sweep events dominate near the wall while the ejection events occur more above the shear layer [3,4]. In a study meaning with urban roughness [5], Joint probability density function analysis was coupled with quadrant analysis to determine the frequency of each event occurrence. The joint probability density function spreads more toward sweep/ejection events when the aerodynamic resistance increases.

Conditional sampling technique has been applied to particle image velocimetry (PIV) data for a number of different applications. Examples include the spatial mapping of flow events in conjunction with proper orthogonal decomposition [6,7], the study of similarity of Reynolds stress profiles in turbulent boundary layers over rough walls [810], Blowing-snow transport [11], air–sea interaction aerodynamics [12], and characterization of flow over plant canopies by means of quadrant hole analysis [13]. Longo and Losada [14] used quadrant analysis on PIV data to study the structure of the boundary layer at the air–water interface of wind induced water waves. Ejection and sweep events were found to be the dominant quadrant modes, appearing at nearly an order of magnitude larger than interaction events. In Ref. [15], quadrant analysis of PIV data was performed to map the spatial distribution of dominant quadrant events in channel flow over a regular square bar roughness. A regular pattern of distinct quadrant regions was identified between roughness elements, supporting the assertion that regular flow geometries have defined regular spatial structures.

Many studies have been carried out to characterize the turbulent flow in wind turbine arrays and the effect of wind turbine array height and layout on the flow recovery [16]. Chamorro and Porté-Agel [17] found that the momentum recovery and turbulence intensity are affected by the farm layout. They also showed that the layout in a wind turbine array, along with the downstream position, affects the tip vortices impact on the flow. Lu and Porté-Agel [18] conducted a large eddy simulation investigation of the farm layout effect. Comparing between aligned and staggered farm arrangements, the later was found to allow each turbine to receive recovered flow from the wake of the preceding turbine due to greater mixing and downstream momentum advection. Similar observations were made in Ref. [19] in a cold wire experimental study in which the staggered wind farms recovered faster; however, it attained lower flux compared to an aligned wind farm. In a wind tunnel experiment meaning with surface heat flux in large-scale wind farms with various layouts [20], authors found that the staggered farms have a uniform surface heat flux distribution in the far wake. Also, compared to boundary layer flow without turbines, the staggered wind farm shows 4% net reduction in surface heat flux.

The wind farms layout effects have also been investigated using conditional sampling. Viestenz and Cal [21] collected hot wire data for a wind farm array in a wind tunnel experiment. The authors examined the conditionally sampled velocity, triple correlation, and exuberance at nine downstream locations. The ejection events were found to entrain turbulent kinetic energy into the wake as well as convect the turbulent kinetic energy out of the wake. In Ref. [22], authors investigated the transport of kinetic energy and momentum in the vertical direction for wind turbine 3 × 3 array model. This study showed that the kinetic energy flux (KEF) is correlated with the Reynolds shear stresses of an order of magnitude that is equal to the power extracted by the wind turbines. This correlation shows the importance of vertical transport in flow recovery. In Ref. [23], quadrant analysis results showed that the dominant events were sweeps and ejections, thus having the largest contribution to the vertical kinetic energy flux.

As the aforementioned findings in Ref. [23] are associated with a specific downstream location, here, conditionally sampled PIV data enable observing the entire inflow/wake area as well as an integrated assessment of quantities of interest. Subsequently, conditionally sampled PIV statistics responsible for upward and downward events after the fourth turbine in a model wind turbine array are analyzed, where the velocity components are assessed including the spanwise component, for their prime importance in understanding wake recovery, unsteady loading, and in improving closure models. For more insight on wake recovery, the kinetic energy flux is examined in each quadrant, and its integrated value over the swept area is obtained. The paper is organized as follows: Sections 2 and 3 include the theory and experimental setup, respectively. Section 4 contains the results along with their discussion, and thereafter, conclusions are presented in Sec. 5.

## Theory

The transport of kinetic energy is of central importance in the context of boundary layer flow within a wind turbine array. In order to provide a framework for discussion, a simplified mean kinetic energy equation is presented as Display Formula

(1)$Uj∂12UiUi∂xj︸I=−1ρUi∂P∂xi︸II−∂Ui⟨uiuj⟩∂xj︸III+⟨uiuj⟩∂Ui∂xj︸IV−Fi︸V$

where upper- and lower-case variables differentiate mean and fluctuating quantities, respectively. The left-hand side consists of the convection of mean kinetic energy (I). The right-hand side terms are described as the rate of energy added to the flow by the mean pressure gradient (II), power associated with the energy flux (III), the production of kinetic energy (IV), and power extraction by the turbine itself (V). Terms associated with viscous diffusion and dissipation have been omitted on the grounds that these terms are negligible outside of the immediate vicinity of the rotor blades.

The mechanisms by which the flux and production terms act on the mean flow can be examined by conditional sampling of the Reynolds stress [6,2325]. The method of decomposition consists of binning the velocity fluctuations, ui, which are obtained by subtracting the mean velocity Ui from the instantaneous velocity, $uĩ$ as $ui=uĩ−Ui$, by their respective signs as shown in Fig. 1. Here, the coordinate system is considered where u, v, and w represent the streamwise, wall-normal, and spanwise fluctuating velocity components, respectively. The stresses are computed for each of these conditions resulting in four types of events, designated as follows: Q1, outward interaction; Q2, ejection; Q3, inward interaction; and Q4, sweep.

The average stresses for each event are traditionally decomposed by temporal fluctuations of the velocity components. In applying conditional sampling to spatial PIV data, the averaged shear stress for each quadrant conditionally sampled by $⟨uv⟩$ is defined as Display Formula

(2)$⟨uiuj⟩k=1N∑m=1NIk(u,v)ui,muj,m, (i≠j)$
an ensemble of N samples in quadrant k. The function, Ik(u, v) is defined as Display Formula
(3)$Ik(u,v)=1$

where k = 1, 2, 3, 4. This definition is such that the total shear stress is given by Display Formula

(4)$⟨uiuj⟩=∑k=14⟨uiuj⟩k$

Normalizing the event-averaged stresses by the total stress, the stress fractions are obtained for each quadrant as are given by Display Formula

(5)$S⟨uiuj⟩=⟨uiuj⟩k⟨ui2uj2⟩, k=1,2,3,4$

Due to the prominence of ejection and sweep events in boundary layer flows, the relative importance of these events is examined using the difference between their stress fractions as given below: Display Formula

(6)$ΔS⟨uiuj⟩=S⟨uiuj⟩,4−S⟨uiuj⟩,2$

For more complex flows, where regions of dominant events may be unclear, the exuberance includes contributions of all quadrants and is defined in Ref. [26] as Display Formula

(7)$E⟨uiuj⟩=S⟨uiuj⟩,1+S⟨uiuj⟩,3S⟨uiuj⟩,2+S⟨uiuj⟩,4$

The exuberance is a ratio of the upward rate of momentum transport to the downward rate of momentum transport (when considering the $⟨uv⟩$ stress) and provides a measure of the relative importance of interactions and sweeps/ejections.

The complexities that arise in the canopy of a large wind turbine array due to the motion of the active blade elements call for careful consideration of the out-of-plane stresses, $⟨uw⟩$ and $⟨vw⟩$. One could employ the same conditional sampling techniques discussed herein on each of these stresses, but the quadrant motions become difficult to interpret. Instead, because of the prominence of the $⟨uv⟩$ stress is expected due to the bulk motion of the flow field, the RSS, $⟨uw⟩$ and $⟨vw⟩$, are considered by conditioning on $⟨uv⟩$. With this definition, the behavior of all shear stress components in the context of upward or downward transport of momentum is characterized.

## Experimental Methods

Experiments were carried out in a closed-return low-speed wind tunnel at Portland State University (PSU). The test section is 5 m in length with a width of 1.2 m and a ceiling height of 0.8 m. The contraction ratio of the wind tunnel is 9:1 with operational velocities in the test section ranging 2–40 ms−1. Freestream turbulence was generated with a passive grid and the flow in the test section was sheared using vertical strakes to produce the desired inflow profile. Roughness elements consisting of rows of chains were used on the floor of the test section to further condition the incoming flow. Wind turbines operate in an atmospheric boundary layer characterized with an effective roughness scale yo = 4.2 mm, effective friction velocity u = 0.385 ms−1, and a boundary layer velocity deficit ΔU+ = 14.9 ms−1 [18]. The streamwise turbulence intensity has an average value of 13.5% in the interrogation area. For more details about boundary layer characterization, turbulence intensity profile, and streamwise velocity profile in linear and logarithmic scales, the reader is referred to Ref. [27].

A 3 × 4 array of model wind turbines of rotor diameter D = 120 mm and hub height H = 120 mm was installed in the test section. The data are collected behind the fourth turbine, for which the peak tip speed ratio is λ = 3.5, to ensure fully developed flow. By the fourth row, the wakes fields behind turbines do not vary significantly with the row number. The wind turbines were positioned with a streamwise spacing of 6D and a spanwise spacing of three-dimensional (3D), which contains the recovered wake of the preceding turbine which has been shown to occur 6D downstream in previous studies [28]. The study [28] also showed that 3D spanwise spacing ensures developed wake in that direction as there was no variation in the wake flow between 1.5D and 3D spacing. The experimental setup is shown in Fig. 2. The ratio of wind tunnel Reynolds number to field Reynolds number is equal to the model turbines scale. Away from turbine blade, both the wind tunnel and the field Reynolds numbers are large, suggesting that large-scale turbulence characteristics converge on Reynolds number [23]. In Ref. [29], it was found that except in the near wake area, mean velocity and turbulence statistics in wind turbine wakes were Reynolds number independent. Hence, the effects of Reynolds number on the observed statistics of large-scale turbulence are neglected. Further details of the experimental setting and wind turbine models manufacturing and scaling are provided in Refs. [27] and [30].

Stereo particle image velocimetry (SPIV) was used to measure the velocity fields immediately upstream and downstream the center line turbine of the fourth row in the array. The inflow conditions on the fourth row turbine can be assumed to exhibit similar behavior to the far wake of the same turbine [17].

The SPIV system supplied by LaVision consisted of a 532 nm wavelength double-pulsed Nd:Yag laser (1200 mJ pulse, 4 ns pulse duration) and two pairs of charge-coupled device cameras setup for simultaneous acquisition of both regions of interest (ROIs). Neutrally buoyant tracer particles of diethylhexyl sebacate were injected into the test section to seed the flow field. The camera setup resulted in ROIs to be 230 mm × 230 mm. Calibration of the cameras was performed using a two-plane calibration plate. Data were collected at 1 Hz frequency and the maximum particle displacement was set to be 6 pixels for each measured plane. For each ROI, 2000 SPIV image pairs were collected providing a sample size comparable to other experiments in which conditional statistics were applied to vector fields [6,25]. Vector fields were extracted from the image pairs using a multipass fast Fourier transformation-based correlation algorithm within the LaVision software with window size of 64 × 64 pixels. The resulting spatial resolution of the vector field was 1.5 mm and the uncertainty for second-order statistics was determined to be 3% using the method described by Sciacchitano and Wieneke [31].

## Results

###### Quadrant Analysis of Reynolds Shear Stresses.

The $⟨uv⟩$ Reynolds shear stress component provides a basis by which the other stresses are observed due to its role in the transport of momentum down into the turbine canopy from the freestream. Hence, it must be examined before applying conditional sampling to have an understanding of how it is influencing the momentum transfer.

The total Reynolds shear stress contours for $⟨uv⟩$ is shown in Fig. 3. The axes are normalized by the turbine diameter D = 0.12 m with the wind turbine located at a streamwise position of x/D = 0. The inflow plot represents the far wake of the third turbine, where the Reynolds shear stresses are accumulated as a consequence of prior turbines. The area above the nacelle is dominated by negative Reynolds shear stress, whereas below the nacelle has a diminutive positive stress. The inflow field does not seem to show any noticeable change with downstream direction. In the wake, $⟨uv⟩$ signs are conserved from inflow to wake as the flow passes through the turbine. This distribution is due to the vertical velocity component direction associated with the shear layer created by the rotor. Aforementioned regions are also stress maxima regions associated with mixing occurring with higher speeds due to the effect of shedding at the top and bottom tips. The stress magnitudes are increased by ≈1.5 times the inflow reaching maximum values of positive and negative Reynolds shear stress of about –0.25 m2 s−2 and 0.2 m2 s−2, respectively, with the first more present causing a downward streamwise momentum flux dominance.

Quadrant analysis as applied to the Reynolds shear stresses, $⟨uv⟩, ⟨uw⟩$, and $⟨vw⟩$ are shown in Figs. 46. The inflow stress fractions for $⟨uv⟩$ are shown in Fig. 4(a). It should be noted that the second and fourth quadrants of $⟨uv⟩$ are multiplied by a minus sign for consistency amongst the quadrant and ease of comparison. A minus sign is included in the title above the contours to indicate their downward directionality. The sweep and ejection events, Q2 and Q4, have peak magnitudes approximately twice the magnitude of the peak interaction stresses, that is 0.137 versus 0.063 m2 s−2. The regions of peak intensity for the sweep and ejection events are primarily located from the nacelle up into the freestream throughout the plane transferring momentum into the flow. This is consistent with the general trend that sweep and ejection events are prominent in high shear regions of boundary layer flows. The interaction events, Q1 and Q3, show highest stress intensities in the plane region bounded by the top and bottom rotor tip. The inward interaction is slightly more localized, with the region of stress magnitudes of approximately 0.06 m2 s−2 bounded by the nacelle and bottom rotor tip.

The wake $⟨uv⟩$ Reynolds shear stress fractions are shown in Fig. 4(b). Similar trends to the inflow conditions are present with interaction events showing peak intensities in the region from the nacelle to the bottom rotor tip, and the sweep and ejection events showing intense regions at the high shear layers around the top tip. Peak sweep and ejection events show magnitudes of stress approximately 70% higher than that of the interaction events (0.2 compared to 0.13 m2 s−2, respectively). Interestingly, interactions are bounded in the region from the nacelle to the lower rotor tip in this near wake region, while looking to the inflow, or far wake, in Fig. 4(a), the region of elevated interaction stresses has diffused to occupy the entire wake region. The interaction events occur deep in the canopy and without as much of an influence from the freestream. These regions of elevated stress are primarily caused by the motion of the rotor and the presence of the turbine mast. The persistence of the Reynolds stress and its events six diameters downstream indicates structural loads affecting the wind turbines as well as providing an indication for turbine placement [32].

The $⟨uw⟩$ RSS conditionally sampled by $⟨uv⟩$ is shown for the inflow and wake in Fig. 5. Quadrants 2 and 3 in the inflow, Fig. 5(a), show a homogenous stress signature throughout the plane at essentially negligible magnitudes of 0.016 m2 s−2 or less. Elevated stress values are observed in quadrants 1 and 4 with peak stress magnitudes of approximately 0.036 m2 s−2 distributed in the wake of the rotor. Stress peaks existence in quadrants 1 and 4 rather than quadrants 2 and 3 indicates the stress sensitivity to the sign of u component of velocity fluctuation. The relevance of $⟨uw⟩$ Reynolds shear stress in the inflow/far wake is not associated with the dominance of sweep and ejection events. However, positive advection of momentum is coincident with maximum and positive $⟨uw⟩$.

The $⟨uw⟩$ Reynolds shear stress in the near wake region, Fig. 5(b), acts similar to $⟨uv⟩$ flow events. Sweep and ejection events, Q2 and Q4, show a strongly negative stress signature in line with the top rotor tip. The mean stress levels in this region peak around –0.08 m2 s−2, with elevated stress magnitudes favoring sweep events. These stresses occupy the same region in the plane as the dominant $⟨uv⟩$ stresses for sweep and ejection events, suggesting that negative $⟨uw⟩$ stress is associated with a downward transfer of momentum. The interaction events, Q1 and Q3, of $⟨uw⟩$ both show weak positive stress, about 0.025 m2 s−2, in the region from the nacelle to the top rotor tip. Interestingly, these positive stresses, though less pronounced, also occur in the same region in the sweep and ejection quadrants. The aforementioned feature is in contrast to the behavior of $⟨uw⟩$ in the inflow since the wake experiences a streamwise velocity deficit and reduced advection of momentum in that direction. This quantity lends itself for modeling the out-of-plane RSS containing the effects due to rotation [33,34].

The $⟨vw⟩$ RSS fractions conditioned on $⟨uv⟩$ are shown in Fig. 6 for inflow and wake. The inflow/far-wake region, Fig. 6(a), shows essentially negligible mean stress levels well diffused in the plane for all quadrants. Of a note, the downward momentum quadrants, Q2 and Q4, are dominated by negative $⟨vw⟩$ and the upward momentum quadrants, Q1 and Q3, are dominated by positive $⟨vw⟩$. Hence, it can be inferred that the direction of $⟨vw⟩$ is influenced by the direction of $⟨uv⟩$ RSS component. Observing the near-wake region Fig. 6(b), the largest stress magnitudes 0.06 m2 s−2 occupy the plane region in line with the top rotor tip and are associated with sweep and ejection events. These stresses are attributed largely to the motion of the turbine blades due to the lack of the bulk component of velocity. The $⟨vw⟩$ RSS shows negligible persistence downstream, suggesting a larger dependence on the streamwise advection of the flow.

###### Relative Comparison of Sweeps and Ejections: $ΔS⟨uiuj⟩$.

Figure 7 shows the contour plots of the parameter $ΔS⟨uiuj⟩$ for $⟨uv⟩, ⟨uw⟩$, and $⟨vw⟩$, respectively. This parameter gives the prominence of sweep and ejection events according to the difference between their magnitudes. Positive values of $ΔS⟨uiuj⟩$ indicate that ejection is the dominant event, and negative values mean that sweep events are the primary contributors as both $S⟨uiuj⟩,4$ and $S⟨uiuj⟩,2$ are of negative values.

Figure 7(a) represents $ΔS⟨uv⟩$ for the incoming and wake flow. The inflow shows a sweep signature throughout the plane, and the region where sweeps are most dominant is located between hub height and top tip, $ΔS⟨uv⟩≈−0.1$. For the wake flow, $ΔS⟨uv⟩$ continues to have its peak at the same location as the incoming flow, between hub height and top tip. Comparing to Fig. 3, the sweep dominance regions are exactly the maximum negative total $⟨uv⟩$ RSS regions. Higher downward streamwise RSS result from sweep events, while lower downward streamwise RSS result from ejection events. Maximum sweep event in the near wake is 70% higher than its value in the far wake, meaning sweep dominance is associated with momentum deficit accruing in the near wake. Sweep events are still present throughout the swept area, but the difference between the two events is not as significant below the turbine hub height where $ΔS⟨uv⟩$ ranges around –0.01. The convergence between ejection and sweep events occurrence is due to the increase in ejection events, which is caused by the interaction of the flow with bottom tip. The flow interaction with the bottom tip drives the flow upward favoring positive v fluctuations which contributes to ejection in turn. Even though the wake has a sweep nature in general, ejections slightly overshadow sweep events near the shear layer after x/D = 1.2 as the flow starts recovering and consequently compensating the streamwise momentum shortfall.

$ΔS⟨uv⟩$ provides a map for which the distribution of sweep/ejection dominance is observed; the effect of this distribution on the other components of Reynolds shear stresses is at hand. The contour plots of $ΔS⟨uw⟩$ are shown in Fig. 7(b). The inflow shows that the sweep dominance region (around the top tip) is associated with reduced positive $⟨uw⟩$, whereas the region in which ejection events slightly overshadow the sweeps (below the hub height) is where $⟨uw⟩$ maxima of 0.08 occurs. This behavior supports the finding that the inflow $⟨uw⟩$ is more influenced by the streamwise advection of the flow, especially noting that the distribution of $⟨uw⟩$ is persistent downstream. On the other hand, $ΔS⟨uw⟩$ highlighting the wake displays negative $⟨uw⟩$ signature over the interrogation area signifying a sign switch between inflow and wake. The sign switch promotes the role of the direction of rotation of W component of velocity after interacting with the rotor. Furthermore, the wake in $ΔS⟨uw⟩$ reveals the influence of sweep and ejection events on $⟨uw⟩$ as its peaks of −0.08 are coincident with both ejection and sweep dominance regions around and above the hub height, and around the bottom tip, respectively. The location of dominance for sweeps highlights the ability of energy extraction and increased power production of the farm [21,23].

Figure 7(c) shows $ΔS⟨vw⟩$. Sweep events promote positive $⟨vw⟩$, and the region marginally favoring ejection events (below hub height) shows mixed negative and positive peaks of $⟨vw⟩$. The $ΔS⟨uv⟩$ distribution leads the directional preference of $⟨vw⟩$ in the inflow. In contrast, the wake $ΔS⟨uv⟩$ distribution has no effect on $⟨vw⟩$ sign or magnitude. The wake in $ΔS⟨vw⟩$ communicates maximum $⟨vw⟩$ within the momentum deficit region, yet no significant effect of sweep/ejection dominance distribution is observed. This behavior supports the balance between fluctuations, meaning as the other components of Reynolds shear stress, $⟨uv⟩$ and $⟨uw⟩$, have their intensities concentrated in the sweep and ejection dominance regions around the top tip and bottom tip for the wake flow, the $⟨vw⟩$ component is prominence at the hub height.

All in all, sweep events are the primary contributor to downward momentum transfer and $⟨uv⟩$ has the largest share in this contribution compared to $⟨uw⟩$ and $⟨vw⟩$. The sweep and ejection distribution of $ΔS⟨uv⟩$ have its significant influence on the intensity and directional preference of the remaining Reynolds shear stress components; however, it is not the only factor. The aforementioned parameters are subjected to the streamwise advection of flow momentum as well as the balance between velocity fluctuations. As the inflow sweep events are associated with elevated magnitudes of $⟨uv⟩$ and $⟨vw⟩$, the same does not apply to $⟨uw⟩$ that yields more to the downstream transport of momentum and to the ejection events despite their modest superiority over sweep events. A similar trend presents itself in wake flow of $⟨vw⟩$ Reynolds shear stress as it exhibits minimal dependence on $ΔS⟨uv⟩$.

###### Exuberance.

The exuberance computed by conditionally sampling $⟨uv⟩$ is shown in Fig. 8. Exuberance is the ratio between the events that contribute to upward flow to the events that contribute to downward flow. Values of exuberance with a magnitude less than −1 indicate interaction events dominance, while conversely magnitudes larger than −1 indicate stronger influence from sweeps and ejections.

The incoming flow is generally influenced by sweep and ejection events except for the region between hub height and bottom tip, where interaction events dominate. However, the values of exuberance where interaction events dominate are not very intense and are confined to the range −1 to −1.5 contributing a minimum to upward momentum transfer. The wake flow has the same trend as the inflow. Yet, the change in the exuberance values is significant. The wake flow experiences profound upward momentum movement behind the nacelle, where the interaction events are 3.5 times its inflow value. Below the bottom tip, the exuberance is about −1 or little less indicating that neither the upward nor the downward events are dominant. The null momentum transfer caused by the shear layer exists below the bottom tip, and its effect lessens after x/D = 1.2. Thereafter in the streamwise direction, the exuberance elevates above −1 showing the presence of downward momentum flux. A more pronounced downward momentum flux is associated with the other shear layer above the top tip and all the way down to the hub height, and it continues from very near wake all the way downstream.

###### Kinetic Energy Flux.

For large wind turbine arrays, the vertical flux of kinetic energy is a primary mechanism by which the turbines are supplied with energy [22]. Sweep and ejection events are the cause for downward kinetic energy flux, while the interaction events contribute to upward kinetic energy flux. The net contribution for each event across the swept area of a wind turbine is the difference between the flux at its bottom tip and the flux at its top tip.

Figures 9(a) and 9(b) represent the incoming and wake flow KEF contours, respectively. In the inflow, the interaction events have a local $⟨uv⟩U$ distributed symmetrically throughout the field of view. The even distribution results in marginal upward net kinetic energy flux across the swept area, which is the difference between the top and bottom tip value of $⟨uv⟩U$. This behavior is not present in the sweep and ejection events where $⟨uv⟩U$ increases vertically. At the hub height, $⟨uv⟩U$ magnitude increases rapidly toward the top tip, reaching a maxima of 0.5 m3s−3 resulting in downward energy flux.

In the wake, the interaction events exhibit a different behavior than in the inflow. Although there is no significant change in $⟨uv⟩U$ values between the top tip and the hub height, there is an evident change across the region from the hub height down below the bottom tip. The interaction events peak is shifting up in both magnitude and vertical position when moving downstream. The local values of $⟨uv⟩U$ are about zero at the near wake at hub height due to flow blockage caused by the rotor. As the flow moves forward, it recovers its kinetic energy gradually allowing $⟨uv⟩U$ to move gradually into this region. The sweep and ejection events show a similar trend as for having minor $⟨uv⟩U$ near wake at hub height. The kinetic energy deficit recovers downstream; hence, the downward KEF peak shifts toward the center of the turbine gradually. However, the increase of $⟨uv⟩U$ magnitude is rapid and reaches a maximum value of 0.55 m3s−3.

For a more quantitative examination of the kinetic energy flux, the integrated value of the total as well as the conditional sampled $⟨uv⟩U$ over the entire swept area is calculated as follows: Display Formula

(8)$Fnet=∫−D/2D/2∫01.6D−⟨uv⟩kU dxdy$

where D/2 is measured from the hub height in the vertical direction, and 1.6D is measured immediately behind the turbine in streamwise direction.

The integrated value of total kinetic energy flux is 19.95 m5s−3. Net KEF of the wake quadrants shows that the ejection events contribute 45% of the net downward KEF, and the sweep events contribute 55%. The obtained values of net KEF for sweep and ejection are 28.8 m5s−3 and 23.4 m5s−3, respectively. Even though the sweep events are the prime contributor to downward KEF, the contribution of the ejection events is comparable. The net KEF of outward and inward interactions are –16.3 m5s−3 and –15.8 m5s−3, respectively. The total integrated KEF is 52.2 m5s−3 downward and –32.1 m5s−3 upward, meaning 60% of the vertical transport of mean kinetic energy due to turbulence is directed downward.

## Conclusions

Conditional sampling of second- and third-order products of velocity was performed on SPIV data collected in front of and behind the centerline of a turbine located in the fourth row of a 4 × 3 model wind turbine array, aiming at assessing the effects of each event and the directional preference of momentum transfer on wake remediation and vertical entrainment for their implications on wind farm design.

The inflow varies minimally with downstream location and Reynolds shear stress fields are primarily positive and significantly increase after interacting with the rotor. $⟨uv⟩$ Reynolds shear stress contributes to the flow momentum at the top tip and withdraws from it at the bottom tip for both inflow and wake. The rotation direction of w component of the velocity is reversed due to interacting with the turbine. The $⟨uv⟩$ Reynolds shear stress is the highest at the top tip where shedding occurs, and slightly below the hub height where the hub blocks the way of the momentum forcing it to shift downward. $⟨uv⟩$ component is the major contributor to transport of the flow momentum having a maximum value of –0.25 m2 s−2 since it consists of the two primary velocity components u and v, whereas the lesser velocity component w intervention in the formation of the other Reynolds shear stress components is behind the relatively lower input they add to the flow.

The conditionally sampled Reynolds shear stresses allow closer look at the effect of each stress component on the wake flow recovery and momentum transport. The $⟨uv⟩$ Reynolds shear stress is contributing more to the sweep events than to the ejection events aiding in the wake recovery process as well as energy extraction. The sweep and ejection events are the dominant events both in the incoming and wake flow agreeing with the findings by Viestenz and Cal [21] and Hamilton et al. [23]. Furthermore, they have higher intensities but similar distribution of the incoming flow where both sweep and ejection events are at their maximum near the shear layer. Even though the incoming $⟨uw⟩$ component is influence by the u velocity sign, the wake $⟨uw⟩$ is affected by $⟨uv⟩$ events. $⟨vw⟩$ component is affected by the downstream advection of the flow rather than by $⟨uv⟩$ events.

The $ΔS⟨uv⟩$ provides the map of sweep/ejection dominance regions. For the inflow, sweep dominance is associated with the upper shear layer and its intensity is elevated 30% after interacting with the rotor. The area below hub height is almost equally packed with sweep and ejection events, though ejection events are slightly higher. However, in the wake flow, the ejection events are notably higher at this region. The sweep and ejection distribution of $ΔS⟨uv⟩$ plays prime role in leading the intensity and directional preference of the other components of Reynolds shear stress. Yet, the impact of the streamwise advection of flow momentum and the balance between velocity fluctuations is significant as well.

The exuberance indicates that the inflow is mainly sweep ejection impacted flow except for the region between hub height and bottom tip that is influenced by interaction events though not highly pronounced. Although the exuberance conserves its distribution in the wake flow, its values elevate significantly. At the hub height, the downward momentum is 3.5 times the sweep and ejection events. The upper shear layer is the cause of downward momentum transfer, whereas the flow movement below the bottom tip area is generally neutral. KE is being transferred downward more than upward in both incoming and wake flows as shown in Refs. [22] and [23]. The interaction events contribution to upward kinetic energy transfer is marginal in the inflow, but more pronounced in the wake. Also, the mean kinetic energy flux increases downstream as it is not evident at the very near wake. The sweep events contribution to the wake recovery is slightly higher than the ejection events. The integrated kinetic energy flux over the swept area shows that sweep events contribution to the downward KEF is marginally higher than the ejection contribution, where they contribute 55% and 45% of the total integrated downward KEF, respectively. The percentage of mean kinetic energy that is being entrained into the flow via vertical transport is 60%. The development of these quantities have significance in obtaining power output and loading on the wind turbines, and consequently their impact on unsteady loading analysis as sweep and ejection events are sensitive to turbulence loading fluctuations.

## Funding Data

• The National Science Foundation (Grant No. NSF-CBET-1034581).

## References

Lu, S. S. , and Willmarth, W. W. , 1973, “ Measurements of the Structure of the Reynolds Stress in a Turbulent Boundary Layer,” J. Fluid Mech., 60(3), pp. 481–511.
Shen, S. , and Leclerc, M. , 1973, “ Modelling the Turbulence Structure in the Canopy Layer,” Agric. For. Meteorol., 87(1), pp. 3–25.
Guan, D. , Agarwal, P. , and Chiew, Y. M. , 2018, “ Quadrant Analysis of Turbulence in a Rectangular Cavity With Large Aspect Ratios,” J. Hydraulic Eng., 144(7), p. 04018035.
Qi, M. , Li, J. , Chen, Q. , and Zhang, Q. , 2018, “ Roughness Effects on Near-Wall Turbulence Modeling for Open-Channel Flows,” J. Hydraulic Res., 37(2), pp. 1–14.
Mo, Z. , and Liu, C. H. , 2018, “ A Wind Tunnel Study of Ventilation Mechanism Over Hypothetical Urban Roughness: The Role of Intermittent Motion Scales,” Building Environ., 135, pp. 94–103.
Roussinova, V. , Shinneeb, A.-M. , and Balachandar, R. , 2009, “ Investigation of Fluid Structures in a Smooth Open-Channel Flow Using Proper Orthogonal Decomposition,” J. Hydraulic Eng., 136(10), pp. 143–154.
Viggiano, B. , Dib, T. , Ali, N. , Mastin, L. G. , Cal, R. B. , and Solovitz, S. A. , 2018, “ Turbulence, Entrainment and Low-Order Description of a Transitional Variable-Density Jet,” J. Fluid Mech., 836, pp. 1009–1049.
Wu, Y. , and Christensen, K. T. , 2007, “ Outer-Layer Similarity in the Presence of a Practical Rough-Wall Topography,” Phys. Fluids, 19(8), p. 085108.
Volino, R. J. , Schultz, M. P. , and Pratt, C. M. , 2001, “ Conditional Sampling in a Transitional Boundary Layer Under High Free-Stream Turbulence Conditions,” ASME Paper No. 2001-GT-0192.
Nolan, K. P. , and Zaki, T. A. , 2013, “ Conditional Sampling of Transitional Boundary Layers in Pressure Gradients,” J. Fluid Mech., 728(7), pp. 306–339.
Aksamit, N. O. , and Pomeroy, J. W. , 2018, “ The Effect of Coherent Structures in the Atmospheric Surface Layer on Blowing-Snow Transport,” Boundary-Layer Meteorol., 167(2), pp. 211–233.
Buckley, M. P. , and Veron, F. , 2018, “ The Turbulent Airflow Over Wind Generated Surface Waves,” Eur. J. Mech.-B/Fluids, (in Press).
Zhu, W. , Van Hout, R. , and Katz, J. , 2007, “ PIV Measurements in the Atmospheric Boundary Layer Within and Above a Mature Corn Canopy—Part II: Quadrant-Hole Analysis,” J. Atmospheric Sci., 64(8), pp. 2825–2838.
Longo, S. , and Losada, M. A. , 2012, “ Turbulent Structure of Air Flow Over Wind-Induced Gravity Waves,” Exp. Fluids, 53(2), pp. 369–390.
Pokrajac, D. , Campbell, L. J. , Nikora, V. , Manes, C. , and McEwan, I. , 2007, “ Quadrant Analysis of Persistent Spatial Velocity Perturbations Over Square-Bar Roughness,” Exp. Fluids, 42(3), pp. 413–423.
Craig, A. E. , Dabiri, J. O. , and Koseff, J. R. , 2016, “ Flow Kinematics in Variable-Height Rotating Cylinder Arrays,” ASME J. Fluids Eng., 138(11), p. 111203.
Chamorro, L. P. , and Porté-Agel, F. , 2011, “ Turbulent Flow Inside and above a Wind Farm: A Wind-Tunnel Study,” Energies, 4(11), pp. 1916–1936.
Lu, H. , and Porté-Agel, F. , 2015, “ On the Impact of Wind Farms on a Convective Atmospheric Boundary Layer,” Boundary-Layer Meteorol., 157(1), pp. 81–96.
Markfort, C. D. , Zhang, W. , and Porté-Agel, F. , 2012, “ Turbulent Flow and Scalar Transport Through and Over Aligned and Staggered Wind Farms,” J. Turbul., 13(1), p. N33.
Zhang, W. , Markfort, C. D. , and Porté-Agel, F. , 2013, “ Experimental Study of the Impact of Large-Scale Wind Farms on Land-Atmosphere Exchanges,” Environ. Res. Lett., 8(1), p. 015002.
Viestenz, K. , and Cal, R. B. , 2016, “ Streamwise Evolution of Statistical Events in a Model Wind-Turbine Array,” Boundary-Layer Meteorol., 158(2), pp. 209–227.
Cal, R. B. , Lebron, J. , Castillo, L. , Kang, H. S. , and Meneveau, C. , 2010, “ Experimental Study of the Horizontally Averaged Flow Structure in a Model Wind-Turbine Array Boundary Layer,” J. Renewable Sustainable Energy, 2(1), p. 013106.
Hamilton, N. , Kang, H. S. , Meneveau, C. , and Cal, R. B. , 2012, “ Statistical Analysis of Kinetic Energy Entrainment in a Model Wind Turbine Array Boundary Layer,” J. Renewable Sustainable Energy, 4(6), p. 063105.
Raupach, M. R. , 1981, “ Conditional Statistics of Reynolds Stress in Rough-Wall and Smooth-Wall Turbulent Boundary Layers,” J. Fluid Mech., 108(1), pp. 363–382.
Nolan, K. P. , Walsh, E. J. , and McEligot, D. M. , 2010, “ Quadrant Analysis of a Transitional Boundary Layer Subject to Free-Stream Turbulence,” J. Fluid Mech., 658, pp. 310–335.
Shaw, R. H. , Tavangar, J. , and Ward, D. P. , 1983, “ Structure of the Reynolds Stress in a Canopy Layer,” J. Clim. Appl. Meteorol., 22(11), pp. 1922–1931.
Hamilton, N. , Melius, M. , and Cal, R. B. , 2015, “ Wind Turbine Boundary Layer Arrays for Cartesian and Staggered Configurations-Part I, Flow Field and Power Measurements,” Wind Energy, 18(2), pp. 277–295.
Ali, N. , Hamilton, N. , DeLucia, D. , and Cal, R. B. , 2018, “ Assessing Spacing Impact on Coherent Features in a Wind Turbine Array Boundary Layer,” Wind Energy Sci., 3(1), p. 43.
Chamorro, L. P. , Arndt, R. E. A. , and Sotiropoulos, F. , 2012, “ Reynolds Number Dependence of Turbulence Statistics in the Wake of Wind Turbines,” Wind Energy, 15(5), pp. 733–742.
Hamilton, N. , and Cal, R. B. , 2015, “ Anisotropy of the Reynolds Stress Tensor in the Wakes of Wind Turbine Arrays in Cartesian Arrangements With Counter-Rotating Rotors,” Wind Energy, 27(1), p. 015102.
Sciacchitano, A. , and Wieneke, B. , 2016, “ PIV Uncertainty Propagation,” Meas. Sci. Technol., 27(8), p. 084006.
Thomsen, K. , and Sørensen, P. , 1999, “ Fatigue Loads for Wind Turbines Operating in Wakes,” J. Wind Eng. Ind. Aerodyn., 80(1–2), pp. 121–136.
Camp, E. H. , and Cal, R. B. , 2016, “ Mean Kinetic Energy Transport and Event Classification in a Model Wind Turbine Array Versus an Array of Porous Disks: Energy Budget and Octant Analysis,” Phys. Rev. Fluids, 1(4), p. 044404.
Katul, G. , Hsieh, C. I. , Kuhn, G. , Ellsworth, D. , and Nie, D. , 1997, “ Turbulent Eddy Motion at the Forest-Atmosphere Interface,” J. Geophys. Res.: Atmospheres, 102(D12), pp. 13409–13421.
View article in PDF format.

## References

Lu, S. S. , and Willmarth, W. W. , 1973, “ Measurements of the Structure of the Reynolds Stress in a Turbulent Boundary Layer,” J. Fluid Mech., 60(3), pp. 481–511.
Shen, S. , and Leclerc, M. , 1973, “ Modelling the Turbulence Structure in the Canopy Layer,” Agric. For. Meteorol., 87(1), pp. 3–25.
Guan, D. , Agarwal, P. , and Chiew, Y. M. , 2018, “ Quadrant Analysis of Turbulence in a Rectangular Cavity With Large Aspect Ratios,” J. Hydraulic Eng., 144(7), p. 04018035.
Qi, M. , Li, J. , Chen, Q. , and Zhang, Q. , 2018, “ Roughness Effects on Near-Wall Turbulence Modeling for Open-Channel Flows,” J. Hydraulic Res., 37(2), pp. 1–14.
Mo, Z. , and Liu, C. H. , 2018, “ A Wind Tunnel Study of Ventilation Mechanism Over Hypothetical Urban Roughness: The Role of Intermittent Motion Scales,” Building Environ., 135, pp. 94–103.
Roussinova, V. , Shinneeb, A.-M. , and Balachandar, R. , 2009, “ Investigation of Fluid Structures in a Smooth Open-Channel Flow Using Proper Orthogonal Decomposition,” J. Hydraulic Eng., 136(10), pp. 143–154.
Viggiano, B. , Dib, T. , Ali, N. , Mastin, L. G. , Cal, R. B. , and Solovitz, S. A. , 2018, “ Turbulence, Entrainment and Low-Order Description of a Transitional Variable-Density Jet,” J. Fluid Mech., 836, pp. 1009–1049.
Wu, Y. , and Christensen, K. T. , 2007, “ Outer-Layer Similarity in the Presence of a Practical Rough-Wall Topography,” Phys. Fluids, 19(8), p. 085108.
Volino, R. J. , Schultz, M. P. , and Pratt, C. M. , 2001, “ Conditional Sampling in a Transitional Boundary Layer Under High Free-Stream Turbulence Conditions,” ASME Paper No. 2001-GT-0192.
Nolan, K. P. , and Zaki, T. A. , 2013, “ Conditional Sampling of Transitional Boundary Layers in Pressure Gradients,” J. Fluid Mech., 728(7), pp. 306–339.
Aksamit, N. O. , and Pomeroy, J. W. , 2018, “ The Effect of Coherent Structures in the Atmospheric Surface Layer on Blowing-Snow Transport,” Boundary-Layer Meteorol., 167(2), pp. 211–233.
Buckley, M. P. , and Veron, F. , 2018, “ The Turbulent Airflow Over Wind Generated Surface Waves,” Eur. J. Mech.-B/Fluids, (in Press).
Zhu, W. , Van Hout, R. , and Katz, J. , 2007, “ PIV Measurements in the Atmospheric Boundary Layer Within and Above a Mature Corn Canopy—Part II: Quadrant-Hole Analysis,” J. Atmospheric Sci., 64(8), pp. 2825–2838.
Longo, S. , and Losada, M. A. , 2012, “ Turbulent Structure of Air Flow Over Wind-Induced Gravity Waves,” Exp. Fluids, 53(2), pp. 369–390.
Pokrajac, D. , Campbell, L. J. , Nikora, V. , Manes, C. , and McEwan, I. , 2007, “ Quadrant Analysis of Persistent Spatial Velocity Perturbations Over Square-Bar Roughness,” Exp. Fluids, 42(3), pp. 413–423.
Craig, A. E. , Dabiri, J. O. , and Koseff, J. R. , 2016, “ Flow Kinematics in Variable-Height Rotating Cylinder Arrays,” ASME J. Fluids Eng., 138(11), p. 111203.
Chamorro, L. P. , and Porté-Agel, F. , 2011, “ Turbulent Flow Inside and above a Wind Farm: A Wind-Tunnel Study,” Energies, 4(11), pp. 1916–1936.
Lu, H. , and Porté-Agel, F. , 2015, “ On the Impact of Wind Farms on a Convective Atmospheric Boundary Layer,” Boundary-Layer Meteorol., 157(1), pp. 81–96.
Markfort, C. D. , Zhang, W. , and Porté-Agel, F. , 2012, “ Turbulent Flow and Scalar Transport Through and Over Aligned and Staggered Wind Farms,” J. Turbul., 13(1), p. N33.
Zhang, W. , Markfort, C. D. , and Porté-Agel, F. , 2013, “ Experimental Study of the Impact of Large-Scale Wind Farms on Land-Atmosphere Exchanges,” Environ. Res. Lett., 8(1), p. 015002.
Viestenz, K. , and Cal, R. B. , 2016, “ Streamwise Evolution of Statistical Events in a Model Wind-Turbine Array,” Boundary-Layer Meteorol., 158(2), pp. 209–227.
Cal, R. B. , Lebron, J. , Castillo, L. , Kang, H. S. , and Meneveau, C. , 2010, “ Experimental Study of the Horizontally Averaged Flow Structure in a Model Wind-Turbine Array Boundary Layer,” J. Renewable Sustainable Energy, 2(1), p. 013106.
Hamilton, N. , Kang, H. S. , Meneveau, C. , and Cal, R. B. , 2012, “ Statistical Analysis of Kinetic Energy Entrainment in a Model Wind Turbine Array Boundary Layer,” J. Renewable Sustainable Energy, 4(6), p. 063105.
Raupach, M. R. , 1981, “ Conditional Statistics of Reynolds Stress in Rough-Wall and Smooth-Wall Turbulent Boundary Layers,” J. Fluid Mech., 108(1), pp. 363–382.
Nolan, K. P. , Walsh, E. J. , and McEligot, D. M. , 2010, “ Quadrant Analysis of a Transitional Boundary Layer Subject to Free-Stream Turbulence,” J. Fluid Mech., 658, pp. 310–335.
Shaw, R. H. , Tavangar, J. , and Ward, D. P. , 1983, “ Structure of the Reynolds Stress in a Canopy Layer,” J. Clim. Appl. Meteorol., 22(11), pp. 1922–1931.
Hamilton, N. , Melius, M. , and Cal, R. B. , 2015, “ Wind Turbine Boundary Layer Arrays for Cartesian and Staggered Configurations-Part I, Flow Field and Power Measurements,” Wind Energy, 18(2), pp. 277–295.
Ali, N. , Hamilton, N. , DeLucia, D. , and Cal, R. B. , 2018, “ Assessing Spacing Impact on Coherent Features in a Wind Turbine Array Boundary Layer,” Wind Energy Sci., 3(1), p. 43.
Chamorro, L. P. , Arndt, R. E. A. , and Sotiropoulos, F. , 2012, “ Reynolds Number Dependence of Turbulence Statistics in the Wake of Wind Turbines,” Wind Energy, 15(5), pp. 733–742.
Hamilton, N. , and Cal, R. B. , 2015, “ Anisotropy of the Reynolds Stress Tensor in the Wakes of Wind Turbine Arrays in Cartesian Arrangements With Counter-Rotating Rotors,” Wind Energy, 27(1), p. 015102.
Sciacchitano, A. , and Wieneke, B. , 2016, “ PIV Uncertainty Propagation,” Meas. Sci. Technol., 27(8), p. 084006.
Thomsen, K. , and Sørensen, P. , 1999, “ Fatigue Loads for Wind Turbines Operating in Wakes,” J. Wind Eng. Ind. Aerodyn., 80(1–2), pp. 121–136.
Camp, E. H. , and Cal, R. B. , 2016, “ Mean Kinetic Energy Transport and Event Classification in a Model Wind Turbine Array Versus an Array of Porous Disks: Energy Budget and Octant Analysis,” Phys. Rev. Fluids, 1(4), p. 044404.
Katul, G. , Hsieh, C. I. , Kuhn, G. , Ellsworth, D. , and Nie, D. , 1997, “ Turbulent Eddy Motion at the Forest-Atmosphere Interface,” J. Geophys. Res.: Atmospheres, 102(D12), pp. 13409–13421.

## Figures

Fig. 1

Conditional sampling of the fluctuating components of velocity

Fig. 2

Diagram of the experimental setup as viewed from the side. The diagram is not to scale.

Fig. 3

Total ⟨uv⟩ Reynolds shear stresses. Units are m2s−2. The horizontal lines at y/D = 0.5, 1, and 1.5 refer to bottom tip, hub height, and top tip location, respectively.

Fig. 7

ΔS⟨uiuj⟩=S⟨uiuj⟩,4−S⟨uiuj⟩,2. The inflow to the left and the wake to the right: (a) ΔS⟨uv⟩, (b) ΔS⟨uw⟩, and (c) ΔS⟨uw⟩.

Fig. 6

Stress fractions for ⟨uw⟩. conditionally sampled by ⟨uw⟩. Units are m2s−2: (a) inflow and (b) wake. Quadrants as in Fig.4.

Fig. 5

Stress fractions for ⟨uw⟩. conditionally sampled by ⟨uv⟩. Units are m2s−2: (a) inflow and (b) wake. Quadrants as in Fig.4.

Fig. 4

Stress fractions for ⟨uv⟩. Units are m2s−2: (a) inflow and (b) wake. Q1 is the top right contour, Q2 is the top left contour, Q3 is the bottom left contour, and Q4 is the bottom right contour.

Fig. 9

Conditionally averaged Fk=−⟨uv⟩kU for (a) incoming flow and (b) wake flow

Fig. 8

The exuberance, E⟨uv⟩=(S1+S3)=(S2+S4). Momentum transfer is primarily upward for values of E < −1 and conversely momentum transfer is directed downward for E > −1.

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