Reynolds Stress Anisotropy Research Articles

Improved compressor corner separation prediction using the quadratic constitutive relation

This work presents the implementation and study of the quadratic constitutive relation nonlinear eddy-viscosity model with representative compressor application, for which the corner separation has been poorly predicted with the widely used linear Boussinesq eddy-viscosity model. With the introduction of the Reynolds stress anisotropy, the secondary flow of the second kind and its effect on the corner flow can be well captured and this results in greatly improved prediction of pressure coefficient, total pressure loss coefficient and the corner separation size. Without the quadratic constitutive relation model, the separation size and loss are generally over-estimated. The mechanism of the improvement is studied using both the vortex dynamics and the momentum equation. It is proved that quadratic constitutive relation model consumes low CPU time and provides much improved compressor corner separation prediction without worsening the convergence property.

Numerical analysis of shock wave and supersonic turbulent boundary interaction between adiabatic and cold walls

ABSTRACTDirect numerical simulations of shock wave and supersonic turbulent boundary layer interaction in a 24° compression ramp with adiabatic and cold-wall temperatures are conducted. The wall temperature effects on turbulence structures and shock motions are investigated. The results are validated against previous experimental and numerical data. The effects of wall cooling on boundary layer characteristics are analysed. Statistical data show that wall cooling has a significant effect on the logarithmic region of mean velocity profile downstream the interaction region. Moreover, the influence of wall temperature on Reynolds stress anisotropy is mainly limited in the near-wall region and has little change on the outer layer. As the wall temperature decreases, the streamwise coherency of streaks increases. Based on the analysis of instantaneous Lamb vector divergence, the momentum transport between small-scale vortices on cold-wall condition is significantly enhanced. In addition, spectral analysis of wall pressure signals indicates that the location of peak of low-frequency energy shifts toward higher frequencies in cold case. Furthermore, the dynamic mode decomposition results reveal two characteristic modes, namely a low-frequency mode exhibiting the breathing motion of separation bubble and a high-frequency mode associated with the propagation of instability waves above separation bubble. The shape of dynamic modes is not sensitive to wall temperature.

A Priori Assessment of Prediction Confidence for Data-Driven Turbulence Modeling

Although Reynolds-Averaged Navier-Stokes (RANS) equations are still the dominant tool for engineering design and analysis applications involving turbulent flows, standard RANS models are known to be unreliable in many flows of engineering relevance, including flows with separation, strong pressure gradients or mean flow curvature. With increasing amounts of 3-dimensional experimental data and high fidelity simulation data from Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS), data-driven turbulence modeling has become a promising approach to increase the predictive capability of RANS simulations. Recently, a data-driven turbulence modeling approach via machine learning has been proposed to predict the Reynolds stress anisotropy of a given flow based on high fidelity data from closely related flows. In this work, the closeness of different flows is investigated to assess the prediction confidence a priori. Specifically, the Mahalanobis distance and the kernel density estimation (KDE) technique are used as metrics to quantify the distance between flow data sets in feature space. The flow over periodic hills at Re=10595 is used as the test set and seven flows with different configurations are individually used as training set. The results show that the prediction error of the Reynolds stress anisotropy is positively correlated with Mahalanobis distance and KDE distance, demonstrating that both extrapolation metrics can be used to estimate the prediction confidence a priori. A quantitative comparison using correlation coefficients shows that the Mahalanobis distance is less accurate in estimating the prediction confidence than KDE distance. The extrapolation metrics introduced in this work and the corresponding analysis provide an approach to aid in the choice of the data source and to assess the prediction confidence for data-driven turbulence modeling.

Reynolds averaged turbulence modelling using deep neural networks with embedded invariance

There exists significant demand for improved Reynolds-averaged Navier–Stokes (RANS) turbulence models that are informed by and can represent a richer set of turbulence physics. This paper presents a method of using deep neural networks to learn a model for the Reynolds stress anisotropy tensor from high-fidelity simulation data. A novel neural network architecture is proposed which uses a multiplicative layer with an invariant tensor basis to embed Galilean invariance into the predicted anisotropy tensor. It is demonstrated that this neural network architecture provides improved prediction accuracy compared with a generic neural network architecture that does not embed this invariance property. The Reynolds stress anisotropy predictions of this invariant neural network are propagated through to the velocity field for two test cases. For both test cases, significant improvement versus baseline RANS linear eddy viscosity and nonlinear eddy viscosity models is demonstrated.

Uncertainty Analysis and Data-Driven Model Advances for a Jet-in-Crossflow

For film cooling of combustor linings and turbine blades, it is critical to be able to accurately model jets-in-crossflow. Current Reynolds-averaged Navier–Stokes (RANS) models often give unsatisfactory predictions in these flows, due in large part to model form error, which cannot be resolved through calibration or tuning of model coefficients. The Boussinesq hypothesis, upon which most two-equation RANS models rely, posits the existence of a non-negative scalar eddy viscosity, which gives a linear relation between the Reynolds stresses and the mean strain rate. This model is rigorously analyzed in the context of a jet-in-crossflow using the high-fidelity large eddy simulation data of Ruiz et al. (2015, “Flow Topologies and Turbulence Scales in a Jet-in-Cross-Flow,” Phys. Fluids, 27(4), p. 045101), as well as RANS k–ϵ results for the same flow. It is shown that the RANS models fail to accurately represent the Reynolds stress anisotropy in the injection hole, along the wall, and on the lee side of the jet. Machine learning methods are developed to provide improved predictions of the Reynolds stress anisotropy in this flow.

Porous media approach for RANS simulation of compound open-channel flows with submerged vegetated floodplains

The main goal of this study is the 3D numerical simulation of river flows with submerged vegetated floodplains. Since, vegetation layers are usually dense and present a large spatial heterogeneity they are here represented as a porous media. Standard semi-empirical relations drawn for porous beds packed with non-spherical particles are used to estimate the porous media parameters based on the averaged geometry of the vegetation elements. Thus, eliminating the uncertainty arising from a bulk drag coefficient approach and allowing the use of a coarser mesh. The free flow is described by Reynolds-averaged Navier–Stokes (RANS) equations, whereas the porous media flow is described by the volumetric-average of RANS equations. The volume-of-fluid method and an anisotropic explicit algebraic Reynolds stress model are used for free-surface and turbulence closure, respectively. The simulation approach is validated against results by other authors featuring vegetated flows in horizontal and rectangular open-channel. The computed results show that the time-averaged streamwise velocity and Reynolds shear stress vertical profiles are properly simulated. The validated approach was applied to simulate compound open-channel flows with submerged vegetated floodplains and compared with data obtained in an experimental facility. The results show that the proposed porous media approach is adequate to simulate flows with submerged vegetation on the floodplains.

Influence of Eddy Momentum Fluxes on the Mean Flow of the Kuroshio Extension in a 1/10° Ocean General Circulation Model

AbstractThis study explores the role of the momentum flux divergence due to mesoscale eddies for the maintenance of the Kuroshio Extension (KE) jet. For that purpose, the zonal momentum budget in a high-resolution ocean general circulation model is examined on the basis of the temporal residual mean framework. The momentum budget analysis is performed for two control volumes: the upstream region of the KE jet flanked by the robust recirculations (33°–38°N and 142.2°–149.4°E) and the downstream region to the east (33°–38°N and 149.4°–160.0°E), both fully covering the meridional width of the KE jet. In both regions the KE jet decelerates to the east, which can be well accounted for by sum of zonal Reynolds stress and Coriolis force on mean ageostrophic flow; the former tends to decelerate the KE jet and the latter to accelerate it in the upstream region, respectively, but these effects are switched in the downstream region. The mean ageostrophic Coriolis force is partially balanced by the horizontal gradient of the eddy kinetic energy, which is the isotropic component of the Reynolds stress. The difference between these terms, that is, net ageostrophic Coriolis force, leads to the final deceleration of the KE jet in the downstream region, overwhelming the acceleration tendency of the anisotropic Reynolds stress. The authors also reinterpret the downstream decay process of an eastward jet in a previous quasigeostrophic experiment in terms of momentum and show that the same features as described above are also likely to be included in that experiment.

Anisotropic Reynolds stress tensor representation in shear flows using DNS and experimental data

ABSTRACTTensorial decompositions and projections are used to study the performance of algebraic non-linear models and predict the anisotropy of the Reynolds stresses. Direct numerical simulation (DNS) data for plane channel flows at friction Reynolds number (Reτ = 180, 395, 590, 1000), and for the boundary layer using both DNS (Reτ = 359, 830, 1271) and experimental data (Reτ = 2680, 3891, 4941, 7164) are used to build and evaluate the models. These data are projected into tensorial basis formed from the symmetric part of mean velocity gradient and non-persistence-of-straining tensor. Six models are proposed and their performances are investigated. The scalar coefficients for these six different levels of approximations of the Reynolds stress tensor are derived, and made dimensionless using the classical turbulent scales, the kinetic turbulent energy (κ) and its dissipation rate (ε). The dimensionless coefficients are then coupled with classical wall functions. One model is selected by comparing the predicted Reynolds stress components with experimental and DNS data, presenting a good prediction for the shear and normal Reynolds stresses.

On the turbulent flow in piston engines: Coupling of statistical theory quantities and instantaneous turbulence

Planar particle image velocimetry (PIV) and tomographic PIV (TPIV) measurements are utilized to analyze turbulent statistical theory quantities and the instantaneous turbulence within a single-cylinder optical engine. Measurements are performed during the intake and mid-compression stroke at 800 and 1500 RPM. TPIV facilitates the evaluation of spatially resolved Reynolds stress tensor (RST) distributions, anisotropic Reynolds stress invariants, and instantaneous turbulent vortical structures. The RST analysis describes distributions of individual velocity fluctuation components that arise from unsteady turbulent flow behavior as well as cycle-to-cycle variability (CCV). A conditional analysis, for which instantaneous PIV images are sampled by their tumble center location, reveals that CCV and turbulence have similar contributions to RST distributions at the mean tumble center, but turbulence is dominant in regions peripheral to the tumble center. Analysis of the anisotropic Reynolds stress invariants reveals the spatial distribution of axisymmetric expansion, axisymmetric contraction, and 3D isotropy within the cylinder. Findings indicate that the mid-compression flow exhibits a higher tendency toward 3D isotropy than the intake flow. A novel post-processing algorithm is utilized to classify the geometry of instantaneous turbulent vortical structures and evaluate their frequency of occurrence within the cylinder. Findings are coupled with statistical theory quantities to provide a comprehensive understanding of the distribution of turbulent velocity components, the distribution of anisotropic states of turbulence, and compare the turbulent vortical flow distribution that is theoretically expected to what is experimentally observed. The analyses reveal requisites of important turbulent flow quantities and discern their sensitivity to the local flow topography and engine operation.

Seismic Sounding of Convection in the Sun

Thermal convection is the dominant mechanism of energy transport in the outer envelope of the Sun (one-third by radius). It drives global fluid circulations and magnetic fields observed on the solar surface. Vigorous surface convection excites a broadband spectrum of acoustic waves that propagate within the interior and set up modal resonances. These acoustic waves, also called seismic waves in this context, are observed at the surface of the Sun by space- and ground-based telescopes. Seismic sounding, the study of these seismic waves to infer the internal properties of the Sun, constitutes helioseismology. Here we review our knowledge of solar convection, especially that obtained through seismic inference. Several characteristics of solar convection, such as differential rotation, anisotropic Reynolds stresses, the influence of rotation on convection, and supergranulation, are considered. On larger scales, several inferences suggest that convective velocities are substantially smaller than those predicted by theory and simulations. This discrepancy challenges the models of internal differential rotation that rely on convective stresses as a driving mechanism and provide an important benchmark for numerical simulations.

On the compressibility effects in mixing layers

Previous studies of compressible flows carried out in the past few years have shown that the pressure-strain is the main indicator of the structural compressibility effects. Undoubtedly, this terms plays a key role toward strongly changing magnitude of the turbulent Reynolds stress anisotropy. On the other hand, the incompressible models of the pressure-strain correlation have not correctly predicted compressible turbulence at high speed shear flow. Consequently, a correction of these models is needed for precise prediction of compressibility effects. In the present work, a compressibility correction of the widely used incompressible Launder Reece and Rodi model making their standard coefficients dependent on the turbulent and convective Mach numbers is proposed. The ability of the model to predict the developed mixing layers in different cases from experiments of Goebel and Dutton is examined. The predicted results with the proposed model are compared with DNS and experimental data and those obtained by the compressible model of Adumitroiae et al. and the original LRR model. The results show that the essential compressibility effects on mixing layers are well captured by the proposed model.

Large eddy simulation of turbulent axially rotating pipe and swirling jet flows

Fully-developed, turbulent rotating pipe flow and swirling jet flow, emitted from the pipe, into open quiescent ambient are investigated numerically using large eddy simulation. Simulations are performed at various rotation rates and Reynolds numbers. Time-averaged large eddy simulation results are compared to experimental and simulation data from previous studies. Pipe flow results show deformation of the turbulent mean axial velocity profile towards the laminar-type Poiseuille profile, with increased rotation. The Reynolds stress anisotropy tensor experiences a component-level redistribution due to pipe rotation. Turbulent energy is transferred from the axial component to the tangential component as rotation is increased. The Reynolds stress anisotropy invariant map also shows a movement away from the one-component limit in the buffer layer, with increased rotation. Exit conditions for the pipe flow simulation are utilized as inlet conditions for the jet flow simulation. Jet flow without swirl and at a swirl rate of S = 0.5 are investigated. Swirl is observed to change the characteristics of the jet flow field, leading to increased jet spread and velocity decay, and a corresponding decrease in the length of the jet potential core. The Reynolds stress anisotropy invariant map shows that the turbulent stress field, with or without rotation straddles the axi-symmetric limit.

Numerical simulation of a spatially developing accelerating boundary layer over roughness

The direct numerical simulation of an accelerating boundary layer over a rough wall has been carried out to investigate the coupling between the effects of roughness and strong free-stream acceleration. While the favourable pressure gradient is sufficient to achieve quasi-laminarization on a smooth wall, the flow reversion is prevented on a rough wall, and a higher friction coefficient, a faster increase of turbulence intensity compared to the free-stream velocity and more isotropic turbulence near the wall are observed. The logarithmic region of the mean-velocity profile presents an initial decrease in slope as in the smooth case, but soon recovers, as the fully rough regime is reached and a new overlap region is established. A strong coupling between the roughness and acceleration effects develops as roughness leads to more responsive turbulence and prevents the strong acceleration from stabilizing the turbulence, and the acceleration intensifies the velocity scale of the wake field (i.e. the near-wall spatial heterogeneity of the time-averaged velocity distribution). The combined effect is a ‘rougher’ surface as the flow accelerates. In addition, the link between the local values of the free stream and the near-wall velocity depends on the flow history; this explains the different flow responses observed in previous studies, in terms of friction coefficient, turbulent kinetic energy and Reynolds-stress anisotropy. This study elucidates the near-wall flow dynamics, which may be used to explain other non-canonical flows over rough walls.

Comparative computational study of turbulent flow in a 90° pipe elbow

Turbulent flow through a 90° pipe elbow in a range of moderately high Reynolds numbers between 14,000 and 34,000 is studied computationally using wall-resolved large-eddy simulation (LES) as well as various RANS (Reynolds-averaged Navier–Stokes) models aiming at a comparative assessment to illustrate benefits and drawbacks of different computational approaches for the considered case. The RANS models applied in the framework of this study include both the basic low Reynolds number k–ε-model, see Launder and Sharma (1974), as well as a near-wall second moment closure model as proposed by Jakirlić and Maduta (2015). Accordingly, the respective results for the mean flow fields being subjected to strong pressure variations – thus causing locally alternating flow acceleration and deceleration which are correspondingly reflected in a distinct Reynolds stress anisotropy variability – are analyzed in detail along with experimental data by Kalpakli and Örlü (2013). Furthermore, characteristic secondary motions resembling the axially oriented counter-rotating Dean vortices, see Bradshaw (1987), are further investigated and their predictability using LES is assessed.

Numerical correlation for the pressure drop in Stirling engine heat exchangers

New correlation equations, to be valid for the pressure drop and heat exchange calculation under the developing transitional reciprocating flow encountered in Stirling heat exchangers are numerically derived. Reynolds-Averaged Navier–Stokes (RANS) equations based turbulence models are used to analyse laminar to turbulent reciprocating flow, focussing on the onset of turbulence and transitional reciprocating flow regime. The relative performance of four turbulence models in more accurately capturing the characteristics of the flow of interest is assessed in relation to overcoming the problems identified in previous numerical studies. The simulation results are compared with published and well-known experimental data for reciprocating pipe flows, indicating that the effects of the turbulence anisotropy need to be taken into account in order to accurately predict the laminar to turbulent transition. The anisotropic Reynolds stress turbulence model is selected as a best choice among the tested turbulence models for analysis of this transitory phenomenon based on the comparative qualitative and quantitative results. This model is used to evaluate the heat transfer and pressure drop and propose new correlations considering the working and dimensional characteristics of Stirling heat exchangers: 100 ≤ Reω ≤ 600, A0 ≤ 600, βcri > 761 and 40 ≤ L/D ≤ 120. These correlation equations reduce the unsteady 2D behaviour in reciprocating pipe flow into a manageable form that can be incorporated into Stirling engine performance codes. It is believed that the validated numerical model can be used with confidence for studying the transitional reciprocating flow and the obtained correlations, can be applied as a cost effective solution for the development of Stirling engine heat exchangers.

Numerical Study of Bubbly Flow Using the Second-Order Moment Turbulence Model and the Population Balance Method

A comprehensive computational fluid dynamics (CFD) turbulence model, coupled with a population balance method (PBM), is presented to simulate the two-phase gas–liquid bubbly flow. The simulation results are in good agreement with the experimental data. Furthermore, considering the bubble size distribution including the drag force and lift force, the results showed that the calculated results are in good agreement with the experimental measurements. The studies reveal the liquid recirculation and bubble uprising flow patterns, and anisotropic liquid and bubble normal Reynolds stresses. Moreover, both the liquid velocity gradient and the bubble–liquid interaction are important for the generation of liquid turbulence.

Modeling of turbulent dissipation and its validation in periodically stratified region in the Liverpool Bay and in the North Sea

The present work explores the applicability of an alternative eddy viscosity formulation in numerical models dealing with the dynamics of the coastal ocean. The formulation is based on the Reynolds stress anisotropy–anisotropy being an important tool for capturing turbulent mixing. Initially idealized entrainment scenarios are evaluated that are typical for shelf seas viz. entrainment in linearly stratified and two-layer fluids caused by surface wind stress or barotropic pressure gradient-driven bottom stress. An attempt is made to simulate the realistic semi-diurnal cycle of turbulent dissipation in Liverpool Bay Region of Freshwater Inflow (ROFI) in the Irish Sea characterized by strong horizontal gradients and interactions with tidal flow. Turbulent dissipation cycles with a 25-h period using free-falling light yo-yo (FLY) dissipation profiler exhibits a strong asymmetry between ebb and flood. The above dynamics involving tidal straining during the ebb and mixing during the flood has been simulated using k– $$ \varepsilon $$ and the alternative formulated turbulence scheme in a one-dimensional (1-D) dynamic model. The model is forced with observed tidal flow and horizontal gradients of temperature and salinity. Simulated dissipation cycles show good agreement with observation. The present work also involves a comparison of dissipation rate measurements in northern North Sea using the abovementioned turbulence schemes—the measurements being taken using free-falling shear probes and CTD (conductivity, temperature, and depth) sensors. The main forcing provided for the upper and bottom boundary layers are atmospheric forcing and tides, respectively. To compare the observations and model results, quantitative error measurements have also been studied which reveal the applicability of the alternative turbulence scheme.

Anisotropy and shear-layer edge dynamics of statistically unsteady, stratified turbulence

Direct numerical simulation data of an evolving Kelvin-Helmholtz instability have been analyzed in order to characterize the dynamic and kinematic response of shear-generated turbulent flow to imposed stable stratification. Particular emphasis was put on anisotropy and shear-layer edge dynamics in the net kinetic energy decay phase of the Kelvin-Helmholtz evolution. Results indicate a faster increase of small-scale anisotropy compared to large-scale anisotropy. Also, the anisotropy of thermal dissipation differs significantly from that of viscous dissipation. It is found that the Reynolds stress anisotropy increases up to a stratification level roughly corresponding to Rig ≈ 0.4, but subsequently decreases for higher levels of stratification, most likely due to relaminarization. Coherent large-scale turbulence structures are cylindrical in the center of the shear layer, whereas they become ellipsoidal in the strongly stratified edge-layer region. The structures of the Reynolds stresses are highly one-componental in the center and turn two-componental as stratification increases. Stratification affects all scales, but it seems to affect larger scales to a higher degree than smaller scales and thermal scales more strongly than momentum scales. The effect of strong stable stratification at the edge of the shear layer is highly reminiscent of the non-local pressure effects of solid walls. However, the kinematic blocking inherently associated with impermeable walls is not observed in the edge layer. Vertical momentum flux reversal is found in part of the shear layer. The roles of shear and buoyant production of turbulence kinetic energy are exchanged, and shear production is transferring energy into the mean flow field, which contributes to relaminarization. The change in dynamics near the edge of the shear layer has important implications for predictive turbulence model formulations.

Numerical study of the flow interference between tandem cylinders employing non-linear hybrid URANS–LES methods

We present a systematic assessment of hybrid URANS/LES models based on linear and non-linear eddy viscosity formulations. Flow past tandem circular cylinders are considered at a subcritical Reynolds number of 1.66×105 for three different spacing L to diameter D ratios: L/D=1.4, 3.0, and 3.7. The cylinder aspect ratio and the blockage factor for the computational domain are 3 and 5%, respectively. This test problem is considered due to the rich flow physics of wake-turbulence and wake-body interaction depending on L/D ratio and the availability of high fidelity experimental data for comparisons. Initial simulations for L/D=3.7 will be performed using the non-linear hybrid models and their linear counterparts to determine the most optimum model. It is shown that the unified time-scale based non-linear model, which provides a better representation of the modeled Reynolds stress anisotropy, shows the best agreement with the experimental measurements. For the spacing ratio L/D=3.0, the non-linear hybrid model accurately predicts the drag inversion and the bi-stable behavior of the wake in the gap between the cylinders similar to experiments. The bi-stable state significantly enhances the turbulent kinetic energy production in the gap between the cylinders. Finally, for the small gap ratio L/D=1.4, it will be shown that the vortex shedding on the front cylinder ceases and the vortex shedding from the back cylinder is weak. No dominant vortex shedding frequency exists at this L/D ratio. The accurate representation of the anisotropy of the modeled stress tensor is relevant for wall and wake turbulence around multiple bluff bodies in proximity. The use of the non-linear hybrid model provides improved predictions of the force coefficients and the turbulent wake physics for such flow scenarios.

Anisotropy of the Reynolds stress tensor in the wakes of wind turbine arrays in Cartesian arrangements with counter-rotating rotors

A 4 × 3 wind turbine array in a Cartesian arrangement was constructed in a wind tunnel setting with four configurations based on the rotational sense of the rotor blades. The fourth row of devices is considered to be in the fully developed turbine canopy for a Cartesian arrangement. Measurements of the flow field were made with stereo particle-image velocimetry immediately upstream and downstream of the selected model turbines. Rotational sense of the turbine blades is evident in the mean spanwise velocity W and the Reynolds shear stress −vw¯. The flux of kinetic energy is shown to be of greater magnitude following turbines in arrays where direction of rotation of the blades varies. Invariants of the normalized Reynolds stress anisotropy tensor (η and ξ) are plotted in the Lumley triangle and indicate that distinct characters of turbulence exist in regions of the wake following the nacelle and the rotor blade tips. Eigendecomposition of the tensor yields principle components and corresponding coordinate system transformations. Characteristic spheroids representing the balance of components in the normalized anisotropy tensor are composed with the eigenvalues yielding shapes predicted by the Lumley triangle. Rotation of the coordinate system defined by the eigenvectors demonstrates trends in the streamwise coordinate following the rotors, especially trailing the top-tip of the rotor and below the hub. Direction of rotation of rotor blades is shown by the orientation of characteristic spheroids according to principle axes. In the inflows of exit row turbines, the normalized Reynolds stress anisotropy tensor shows cumulative effects of the upstream turbines, tending toward prolate shapes for uniform rotational sense, oblate spheroids for streamwise organization of rotational senses, and a mixture of characteristic shapes when the rotation varies by row. Comparison between the invariants of the Reynolds stress anisotropy tensor and terms from the mean mechanical energy equation indicate correlation between the degree of anisotropy and the regions of the wind turbine wakes where turbulence kinetic energy is produced. The flux of kinetic energy into the momentum-deficit area of the wake from above the canopy is associated with prolate characteristic spheroids. Flux upward into the wake from below the rotor area is associated with oblate characteristic spheroids. Turbulence in the region of the flow directly following the nacelle of the wind turbines demonstrates greater isotropy than regions following the rotor blades. The power and power coefficients for wind turbines indicate that flow structures on the order of magnitude of the spanwise turbine spacing that increase turbine efficiency depending on particular array configuration.