Grid-generated Turbulent Flow Research Articles

Theoretical and experimental study of the spanwise effect of turbulence on the aerodynamic lift on a wing with different aspect ratios

In this paper, a combined theoretical and experimental study is carried out to investigate the spanwise effect of turbulence on the aerodynamic lift on a wing with different aspect ratios. The ratio of the mean square variance of the aerodynamic lift calculated by the commonly used strip theory and the two-wavenumber buffeting theory is analyzed comprehensively for the wings with different aspect ratios in turbulence with various integral scales. To validate the theoretical analysis and achieve a deeper understanding of the spanwise effects of turbulence, wind tunnel experiments are performed on National Advisory Committee for Aeronautics 0015 airfoils in grid-generated turbulent flows with different integral scales. The results demonstrate that it is essential to use the two-wavenumber buffeting theory to account for the spanwise effect of turbulence when calculating the aerodynamic lifts on wings with small aspect ratios, especially when in small-scale turbulence. The deviations between the equivalent two-wavenumber coherence function and the spanwise effect influence function at low reduced streamwise wavenumbers are the underlying causes for spanwise effects of turbulence. To achieve reliable wind tunnel testing results, appropriate simulations of the ratio of the turbulence integral scale to the chord are very important in the measurements of aerodynamic lifts on finite-span wing sections, especially for those with small aspect ratios.

Non-equilibrium dissipation laws in a minimal two-scale wake model

The existence of a large-scale non-equilibrium dissipation law (Cε∼Reλ−15/14) has been well documented through various experiments conducted on grid-generated turbulent wake flows. In our previous studies, we have successfully demonstrated the coexistence of a rapid non-equilibrium dissipation law (Cε∼Reλ−2) before the large-scale counterpart in homogeneous isotropic turbulence with inversed or perturbed initial conditions and accordingly developed the closure models. In this paper, we introduce a simplified minimal two-scale wake model, which serves as a simplification of grid-generated turbulence experiments. It is evident that both the large-scale and rapid non-equilibrium dissipation laws coexist in this flow, similar to our previous studies in homogeneous isotropic turbulence. Moreover, we show that the utilization or exclusion of spanwise averages does not significantly affect the qualitative characteristics of non-equilibrium behaviors. Furthermore, by investigating the turbulent/non-turbulent interface, we argue that the presence of coherent structures is not necessarily required to generate the scaling associated with non-equilibrium behavior.

Large-scale sinusoidal gust effect on aerodynamic pressures and forces on a square cylinder

Aerodynamic pressures and forces on a two-dimensional square cylinder were experimentally investigated in large-scale sinusoidal gusts. The effects of streamwise and transverse gusts with a wide range of gust amplitudes were studied at a Reynolds number of Re = 2.1 × 104. To enhance our understanding of the large-scale sinusoidal gust effect, the pressure fields and global aerodynamic forces of the square cylinder in large-scale gusts were compared with those in smooth and grid-generated turbulent flows. This study shows that both large-scale streamwise and transverse gusts can promote early pressure recovery, resulting in larger mean pressures on the side and leeward surfaces. The root mean square (RMS) of the pressures on the side surface increased and decreased with an increase in the streamwise and transverse gust amplitudes, respectively. The mean drag on the square cylinder was reduced under the effects of large-scale streamwise and transverse gusts. The increase in the streamwise and transverse gust amplitudes significantly affected the RMS of the drag and lift on the square cylinder, respectively. A proper orthogonal decomposition (POD) analysis showed that the contribution of the first POD mode switched from fluctuating lift to drag when the streamwise gust amplitude was very high. The spectral analysis demonstrated that the alternating von Kármán vortex-shedding process still dominated the fluctuating lift under the effect of large-scale gusts. In addition, the spanwise correlations of drag were significantly strong under the effects of large-scale streamwise gusts. However, the spanwise correlations of lift were reduced under the effects of large-scale gusts with high gust amplitudes.

Wind tunnel tests on the unsteady galloping of a bridge deck with open cross section in turbulent flow

Steel–concrete composite bridge decks may risk galloping instability during their bridge-launching phase, since the normally first launched steel boxes could feature bluff and lightweight cross section. In our previous study, a bridge deck of open cross section (typically used in steel–concrete composite bridges) has been found in smooth flow highly prone to the unsteady galloping instability occurring at relatively low reduced wind speed. Considering the natural wind field is usually turbulent, motivations arise to understand what role can the incident turbulence play in the galloping instability of this open cross section deck. New wind tunnel tests were therefore carried out in grid-generated turbulent flow, characterized by an integral length scale comparable to the bridge deck height and an intensity of 9% to 15%. For the interested wind angles of attack around null, static tests first indicate the incident turbulence is able to promote an even higher galloping factor than that in smooth flow, while the Strouhal number shows a decrease. This suggests a higher proneness for the galloping and vortex induced vibration to interact. In aeroelastic tests, however, the typical unsteady galloping instability due to the interaction with vortex induced vibration has never been observed. Instead, the unsteady galloping is initiated in a more complicated manner: unrestricted oscillation arises at a reduced wind speed clearly higher than the Kármán-vortex resonance one even for a very low Scruton number (the mass-damping parameter), combined with the absence of any Kármán-vortex excitation. Nevertheless, similar to the smooth flow condition, the nosed-up 4° remains to be the most unfavorable wind angle of attack in turbulent flow, featuring the lowest galloping onset in the low Scruton number condition. Finally, dependent on the magnitude of Scruton number, it is able to conclude that the incident turbulence can either suppress or promote the galloping instability of this bridge deck.

Effects of free-stream turbulence on the near wake flow and aerodynamic forces of a square cylinder

Despite a large volume of preceding studies on the effects of free-stream turbulence (FST) on the aerodynamic forces of a square cylinder, limited knowledge is available regarding the evolutionary characteristics of flow with the change of turbulence integral length scale and turbulence intensity. This paper presents a systematic experimental study of the near wake flow and aerodynamic forces of a two-dimensional (2D) square cylinder in grid-generated turbulent flows and smooth flow. Particular attention is devoted to exploring the respective influences of turbulence integral length scale and turbulence intensity. The effect of FST with a higher-level turbulence intensity than previous investigations is also investigated. The near wake flow and aerodynamic forces are studied by particle image velocimetry (PIV) and synchronous pressure measurements. The proper orthogonal decomposition (POD) analysis is also performed to further understand the effects of FST on the coherent structures of fluctuating pressures around the square cylinder. The results show that the FST with a larger turbulence integral length scale and higher turbulence intensity can cause an increase in the vortex formation length, weaker vortex shedding process and interactions between the oppositely signed vortex rows. These changes lead to significant reductions in Reynolds normal and shear stresses, turbulence kinetic energy and fluctuating lift forces. A completely different flow pattern featured with flow reattachment and reseparation is interestingly captured on the square cylinder in a highly turbulent flow with a turbulence intensity of 20%. This flow pattern results in an obvious mean pressure recovery on the side surfaces and much larger pressure fluctuations near the leading edges and fluctuating torsional moments compared to those in smooth flow and turbulent flows with lower turbulence intensities.

Meandering of a wing-tip vortex in a grid-generated turbulent flow

The interaction of a National Advisory Committee for Aeronautics 0012 wing-tip vortex with a grid-generated turbulent flow was experimentally investigated. The experiments were conducted in the near- and mid-wake regions at three free-stream turbulence (FST) levels, viz., 0.5%, 3%, and 6%, at a Reynolds number, based on the wing chord length, cw, of 2×105. Stereoscopic particle image velocimetry measurements were carried out at four downstream positions: x/cw = 1.25, 3.25, 6.25, and 7.75. Streamwise vorticity contours showed that the wing-tip vortex decayed with increased FST and downstream distance. Turbulent surroundings were found to affect the meandering amplitude of the vortex, which increased with the dispersed scatter pattern of the vortex center motion, resulting in a meandering-induced turbulence. Meandering-corrected turbulent kinetic energy revealed the existence of a laminar core at the center of the vortex surrounded by low turbulence levels outside the core. This was attributed to the stabilizing Coriolis effects of the strong rotational motion inside the vortex core, which tend to re-laminarize the turbulent fluid crossing the periphery of the vortex. Snapshot proper orthogonal decomposition analysis on the coherent component of the velocity field revealed two dominant modes forming a helical dipole, consistent with the helical displacement of a Kelvin wave with an azimuthal wavenumber |m=1|. An analysis of the terms balancing the rate of decay of the mean enstrophy revealed that increasing FST increases the stretching of the mean enstrophy within the vortex core while it reduces both its transport and convection terms. Nonetheless, the latter contributions were larger in all cases studied acting as the main mechanism for mean enstrophy decay.

Study of the interaction of two decaying grid-generated turbulent flows

Hot-wire measurements are carried out in a decaying turbulence downstream of a grid made up of two juxtaposed perforated plates with different mesh sizes but same solidity. The two perforated plates generate two interacting (quasi–) homogeneous and isotropic decaying turbulent flows with distinct turbulence intensities and integral length scales. The interaction between these two flows leads to the development of a shearless turbulent mixing layer (STML). The main focus is on the decay of the turbulence centerline of the STML. Along the downstream distance x, the Taylor microscale Reynolds number, Reλ, remains constant, the streamwise velocity variance behaves like x−1, and the Taylor microscale (λ) varies as x1/2. This indicates that the turbulence on the centerline of the STML decays in a perfectly self-preserving manner at all scales of motion. This is further supported by the very good collapse of the velocity spectra, second-, and third-order velocity structure functions.

Aerodynamic pressures on a 5:1 rectangular cylinder in sinusoidal streamwise oscillatory flows with non-zero mean velocities

In this paper, the aerodynamic pressures on a 5:1 rectangular cylinder in sinusoidal streamwise oscillatory flows (SSOFs) with non-zero mean velocities are studied through wind tunnel experiments. First, the mean and fluctuating pressure coefficients are determined based on the measurements in SSOFs with different reduced frequencies and gust amplitudes. The results are compared with those of smooth flow and grid-generated turbulent flow. Next, the power spectral densities of surface pressures are presented to study their energy distributions in the frequency domain. Then, the spanwise correlations of surface pressures in SSOFs are compared with those of smooth flow to explore their three-dimensional characteristics. The testing results show that the reduced frequency, varying from 0.012 to 0.032 has little effect on the streamwise pressure distributions of the 5:1 rectangular cylinder, and the influences of the gust amplitude of SSOFs are significant. The maximum RMS pressure coefficient increases and the separation bubble length reduces to a large degree as the gust amplitude increases. The spectral analysis demonstrates that the energy of fluctuating pressures in SSOFs is primarily from oscillatory streamwise gusts, while it is mainly provided by vortex shedding in smooth flow. The SSOF can accelerate the vortex shedding process and the Strouhal number increases as the increased gust amplitude. The spanwise correlations of pressures are strong in the majority region of the side surface of the 5:1 rectangular cylinder in SSOFs.

Free-stream turbulence interaction with a wing-tip vortex

The interaction of a wing-tip vortex of a rectangular, square-tipped wing having a NACA 0012 airfoil section with a grid-generated turbulent flow was investigated in this paper. The experiments were conducted in the near and mid-wake regions at three free stream turbulence (FST) intensities of 0.5%, 3% and 6%, and at two Reynolds numbers, Re, based on the wing chord length, cw, of 2×105 and 3×105. Stereoscopic Particle Image Velocimetry (SPIV) and hot-wire measurements were carried out at four downstream positions, namely x/cw=1.25,3.25,6.25 and 7.75. Streamwise velocity contours showed that the wing-tip vortex decayed with increased FST and downstream distance. In the vortex core region, the streamwise velocity decelerated while the vortex adopted a wake-like profile. FST was found to decrease the vortex circulation and to increase the vortex radius and vortex meandering amplitude. When increasing Reynolds number, the grid cases showed little variation of the vortex radius and peak vorticity levels, particularly at larger downstream positions, suggesting that the effects of FST and Re number on the vortex development are nearly independent. The measured total turbulent kinetic energy (TKE) was found to be mostly due to vortex meandering. In that, total TKE levels devoid of meandering showed a virtually turbulence-free vortex core.

Effects of turbulence integral scales on characteristics of fluctuating wind pressures

To investigate the effects of turbulence integral scale, four rectangular models were tested in grid-generated turbulent flows, and the characteristics of fluctuating pressures on windward surfaces were obtained. The results show that all statistical parameters except the mean pressures depend on both the longitudinal integral scale (Lux) and depth (D) of the model. The root-mean-square (rms) values of the fluctuating pressure coefficients and their correlation as well as the spanwise coherence increase with the turbulence integral scale. As the integral scale decreases, the power spectra of the fluctuating pressures in the low-frequency region, which are controlled by the quasi-steady effect, remain almost unchanged, whereas those in the high-frequency region attenuate more rapidly. To seek a potential modification method for correcting the error margin caused by an imprecisely simulated turbulence integral scale, an empirical model of aerodynamic admittance function (AAF) of fluctuating pressure was proposed with consideration of Lux /D as a key variable. Finally, using the fluctuating pressure on the model with a side ratio of 1:3 as an example, the error margins of the fluctuating pressures were discussed. It was discovered that the maximum deviation of the rms value of the fluctuating pressure coefficient reached approximately 25% when the simulated turbulence integral scale was one-sixth the target value.

Strategy for the determination of unsteady aerodynamic forces on elongated bodies in grid-generated turbulent flow

In this study, we developed a general method for estimating the unsteady aerodynamic forces on elongated bodies in grid-generated turbulent flow. The key to this method is the identification of the one-wavenumber aerodynamic transfer function (ATF) of bluff sections. Based on our previous studies, the accuracy of the strip theory in estimating the unsteady aerodynamic forces on bluff sections in isotropic homogeneous turbulence is higher than that of a thin airfoil. The thin airfoil section is therefore intended to serve as a standard configuration to calibrate the experimental parameters to satisfy the accuracy of the strip theory. This can provide a road map for the experimental determination of the unsteady aerodynamic forces on elongated bodies with different configurations in turbulent flow. To explore the deviations between the ATFs of airfoil and bluff sections, buffeting force measurement wind tunnel tests were performed on NACA 0015 airfoils and 5:1 rectangular (REC) cylinders with different length-width ratios at a 0° attack angle in grid-generated turbulence. The measured one-wavenumber ATFs of the airfoil sections correspond strongly to the theoretical predictions. In addition, it was found that the one-wavenumber ATF of the REC model is larger than Sears’ function at low reduced streamwise wavenumbers, which is explained by the difference between the lift generation mechanisms of the bluff and airfoil sections in grid-generated turbulence.

A statistical approach to the identification of the two-dimensional aerodynamic admittance of streamlined bridge decks

The two-dimensional aerodynamic admittance function (2D AAF) is of great importance in the buffeting evaluation of slender, line-like structures. This paper aims to propose a simple and practical approach to identifying the 2D AAF of streamlined bridge decks. The effects of the aspect ratio (the ratio of span to width) and turbulence characteristics on the accuracy of buffeting loads calculated from the strip assumption were investigated. It was shown that both the aspect ratio and the integral length scale of turbulence play important roles in controlling the accuracy of the strip assumption. By increasing these two parameters, the 2D AAF of a streamlined bridge deck can be obtained within an acceptable error margin. Thus, for a section model with a large enough aspect ratio in an appropriate turbulent field, the 2D AAF of a streamlined bridge deck can be easily identified by the direct measurement of the velocity fluctuations and unsteady aerodynamic forces. The identification approach was validated through wind tunnel tests of streamlined bridge decks with three different aspect ratios in grid-generated turbulent flow. The proposed statistical approach can also be used for the identification of the 2D AAF of other slender, line-like structures.

Identification and application of six-component aerodynamic admittance functions of a closed-box bridge deck

A colligated residue least square method of auto and cross spectra (CRLSMACS) is presented for identifying six-component aerodynamic admittance functions (AAFs) base on common force and pressure measurement tests in the passive grid-generated turbulent flow. The incomplete spanwise correlation of the buffeting force on the sectional model in the turbulent flow is corrected. The six-component AAFs of a flat closed-box deck of a single-tower cable-stayed bridge are identified with the presented method for the service states without/with wind barriers and the construction state under 0° wind attack angle, respectively. The buffeting responses of the bridge are then calculated for both the service and typical construction sates by using a finite element approach based on the quasi-steady buffeting theory with the tested six-component AAFs to consider the unsteady effect of buffeting forces. The calculated buffeting responses are finally compared with and found to agree well with those obtained via full bridge aeroelastic model tests, verifying the feasibility of the proposed CRLSMACS as well as the reliability of the identified six-component AAFs and the calculated buffeting responses.

An experimental investigation of surface pressures in separated and reattaching flows: effects of freestream turbulence and leading edge geometry

Flow separation and reattachment over bluff bodies is frequently encountered in a wide range of engineering practice. This study presents an experimental investigation on the characteristics of surface pressures over bluff bodies in separated and reattaching flows, with emphasis placed on the effects of freestream turbulence and leading edge geometry. Measurements of mean, r.m.s and peak suction pressures on blunt flat plates with different leading edge shapes are conducted in both smooth and grid-generated turbulent flows in wind tunnel test, and comparative analyses are carried out to unveil the dependence of the distributions of mean, r.m.s and peak suction pressure coefficients on turbulence intensity, turbulence integral length scale and leading edge geometry. The experimental results show that the distributions of mean pressure coefficient respond strongly to changes of turbulence intensity, but with little effect on turbulence integral length scale except in the leading edge region for the cases there are large differences of the parameter. The distributions of r.m.s and peak suction coefficients demonstrate a close correlation on both the turbulence intensity and integral length scale. In addition, the distributions of surface pressures are also related evidently on the leading edge geometry. The mean reattachment length exhibits a monotonic increase as the leading edge angle increases, while the maximum magnitudes of r.m.s and peak suction pressure coefficients are found to increase with increasing leading edge angle.

A stochastic Lagrangian micromixing model for the dispersion of reactive scalars in turbulent flows: role of concentration fluctuations and improvements to the conserved scalar theory under non-homogeneous conditions

Several reaction schemes, based on the conserved scalar theory, are implemented within a stochastic Lagrangian micromixing model to simulate the dispersion of reactive scalars in turbulent flows. In particular, the formulation of the reaction-dominated limit (RDL) reaction scheme is here extended to improve the model performance under non-homogeneous conditions (NHRDL scheme). The validation of the stochastic model is obtained by comparison with the available measurements of reactive pollutant concentrations in a grid-generated turbulent flow. This test case describes the dispersion of two atmospheric reactant species (NO and O3) and their reaction product (NO2) in an unbounded turbulent flow. Model inter-comparisons are also assessed, by considering the results of state-of-the-art models for pollutant dispersion. The present validation shows that RDL reaction scheme provides a systematic overestimation (relative error of ca. 85% around the centreline) in computing the local reactant consumption/production rate, whereas the NHRDL scheme drastically reduces this gap (relative error lower than 5% around the centreline). In terms of NO2 production (or reactant consumption), neglecting concentration fluctuations determines overestimations of the product mean of around 100% and a NO2 local production of one order of magnitude higher than the reference simulation. In terms of standard deviations, the concentration fluctuations of both the passive and reactive scalars are generally of the same order of magnitude or up to 1 or 2 orders of magnitudes higher than the corresponding ensemble mean values, except for the background reactant close to the plume edges. The study highlights the importance of modelling pollutant reactions depending on the instantaneous instead of the mean concentrations of the reactants, thus quantifying the role of the turbulent fluctuations of concentration, in terms of scalar statistics (mean, standard deviation, intensity of fluctuations, skewness and kurtosis of concentration, segregation coefficient, simulated reaction rate). This stochastic particle method represents an efficient numerical technique to solve the convection–diffusion equation for reactive scalars and involves several application fields: micro-scale air quality (urban and street-canyon scales), accidental releases, impact of odours, water quality and fluid flow industrial processes (e.g. combustion).

Parametric analysis of fluidic energy harvesters in grid turbulence

Even though it is omnipresent in nature, there has not been much research in the literature involving turbulence as an energy source for piezoelectric fluidic harvesters. In the present work, a grid-generated turbulence forcing function model which we derived previously is employed in the single degree-of-freedom electromechanical equations to find the power output and tip displacement of piezoelectric cantilever beams. Additionally, we utilize simplified, deterministic models of the turbulence forcing function to obtain closed-form expressions for the power output. These theoretical models are studied using experiments that involve separately placing a hot-wire anemometer probe and a short PVDF beam in flows where turbulence is generated by means of passive and semi-passive grids. From a parametric study of the deterministic models, we show that a white noise forcing function best mimics the experimental data. Furthermore, our parametric study of the response spectrum of a generic fluidic harvester in grid-generated turbulent flow shows that optimum power output is attained for beams placed closer to the grid with a low natural frequency and damping ratio and a large electromechanical coupling coefficient.

On a PLIF quantification methodology in a nonlinear dye response regime

A new technique of planar laser-induced fluorescence calibration is presented in this work. It accounts for a nonlinear dye response at high concentrations, an illumination light attenuation and a secondary fluorescence’s influence in particular. An analytical approximation of a generic solution of the Beer–Lambert law is provided and utilized for effective concentration evaluation. These features make the technique particularly well suited for high concentration measurements, or those with a large range of concentration values, c, present (i.e. a high dynamic range of c). The method is applied to data gathered in a water flume experiment where a stream of a fluorescent dye (rhodamine 6G) was released into a grid-generated turbulent flow. Based on these results, it is shown that the illumination attenuation and the secondary fluorescence introduce a significant error into the data quantification (up to 15 and 80 %, respectively, for the case considered in this work) unless properly accounted for.

Experimental analysis of a stochastic model for estimating wind-borne compact debris trajectory in turbulent winds

The impact of flying debris against building envelopes during high winds is a major source of structural damage. For example, damage produced by Hurricanes Katrina and Ike in the United States on the facades of tall buildings, located in urban areas, has been documented. It is therefore of relevance to analyze the vulnerability of tall buildings to debris-induced non-structural damage in the general context of performance-based wind engineering. In order to analyze the random trajectory of debris in highly turbulent winds, a numerical model combined with a probability-based algorithm was recently proposed by the authors (Moghim and Caracoglia, 2013). This model investigates the trajectory of “compact debris”, defined as point-mass objects of negligible mass moments of inertia and for which the aerodynamics is predominantly controlled by the drag force. The model replicates both the inherent randomness in debris properties and the effect of wind shear and atmospheric turbulence to estimate debris trajectory and the likelihood of impact against vertical building facades in a probabilistic setting.This paper describes the comparison between numerical model results and wind tunnel experiments. Tests were carried out in the Northeastern University׳s small scale wind tunnel in both smooth flow and grid-generated turbulent flow. The motion of spheres and cubes, simulating compact debris objects, was investigated in two dimensions (2D) on a vertical plane.The 2D motion of compact objects of various sizes was captured by a high-speed digital camera at different flow speeds. Experimental results showed to be consistent with numerical simulations. They also confirmed that not only mean flow speed but also turbulence features can have a non-negligible effect on the trajectory of compact objects.

Effects of Free-Stream Turbulence and Reynolds Number on the Aerodynamic Characteristics of a Semicylindrical Roof

The influences of free-stream turbulence and Reynolds number on the aerodynamic characteristics of a semicylindrical roof have been investigated experimentally under uniform flow conditions. Grids were used to generate homogeneous turbulent flows with turbulence intensities varying from 5.7 to 12.2%. The diameters, D, of the semicylindrical roof models were 0.2 and 0.6 m. The Reynolds numbers based on D ranged from 6.90×104 to 8.28×105 in smooth flow and from approximately 6.06×104 to 4.79×105 in grid-generated turbulent flow. Measurements of the surface pressure indicated that, in smooth flow, the mean and RMS pressure distributions became Reynolds number independent when R>4.14×105. The introduction of turbulence has caused a premature transition of the separated shear layer from laminar to turbulent. Increasing free-stream turbulence helps to remarkably reduce the mean and fluctuating drag force at lower Reynolds number, R<1.46×105. There is also a relatively significant reduction in the mean lift force as the turbulence intensity increases when R>1.46×105. The data also suggested that the effects of turbulence essentially depend on the turbulence intensity and the ratio of turbulence length scale to cylinder diameter. The small-scale turbulence was better able to interact with the surface boundary layer before and after separation and thus dramatically increased the pressure fluctuations in both the top and wake regions.

Turbulence effects on the wake flow and power production of a horizontal-axis wind turbine

This study experimentally investigated the effects of ambient turbulence on the wake flows and power production of a horizontal-axis wind turbine. The approaching flows included low-turbulence smooth flow and grid-generated turbulent flow. The profiles of time-averaged velocity, turbulence intensity and Reynolds stress from the intermediate to the far-wake regions were measured and compared for smooth and turbulent flows. Based on the measured data, prediction models for the centerline velocity deficit, turbulence intensity, wake radius and velocity profile were proposed. In addition, the experimental results showed that the power productions in the grid-generated turbulent flows were slightly higher than that in the smooth flow. But the power loss due to the velocity deficit in the wake flow was larger than 50% when the downwind distance was less than 12D (D is the rotor diameter). An empirical relation between the power production and the downwind distance x and lateral distance y was proposed.