Large eddy simulations of the flow past wind turbines: actuator line and disk modeling

Transcription

1 WIND ENERGY Wind Energ. (2014) Published online in Wiley Online Library (wileyonlinelibrary.com) RESEARCH ARTICLE Large eddy simulations of the flow past wind turbines: actuator line and disk modeling Luis A. Martínez-Tossas 1, Matthew J. Churchfield 2 and Stefano Leonardi 3 1 Johns Hopkins University, Baltimore, Maryland, USA 2 National Renewable Energy Laboratory, Golden, Colorado, USA 3 The University of Texas at Dallas, Richardson, Texas, USA ABSTRACT Large eddy simulations of the flow through wind turbines have been carried out using actuator disk and actuator line models for the turbine rotor aerodynamics. In this study, we compare the performance of these two models in producing wind turbine wakes. We also examine parameters that strongly affect the performance of these models, namely, grid resolution and the way in which the actuator force is projected onto the flow field. The proper choice of these two parameters has not been adequately addressed in previous works. We see that as the grid is coarsened, the predicted power decreases. As the width of the body force projection function is increased, the predicted power increases. The actuator disk and actuator line models produce similar wake profiles and predict power within 1% of one another when subject to the same uniform inflow. The actuator line model is able to generate flow structures near the blades such as root and tip vortices which the actuator disk model does not, but in the far wake, the predicted mean wakes are very similar. In order to perform validation against experimental data, the actuator line model output was compared with data from the wind tunnel experiment conducted at the Norwegian University of Science and Technology, Trondheim. Agreement between measured and predicted power, wake profiles, and turbulent kinetic energy has been observed for most tip speed ratios; larger discrepancies in power and thrust coefficient, though, have been found for tip speed ratios of 9 and 12. Copyright 2014 John Wiley & Sons, Ltd. KEYWORDS computational fluid dynamics; actuator line model; actuator disk model; large eddy simulations Correspondence Stefano Leonardi, Department of Mechanical Engineering,The University of Texas at Dallas, Richardson, Texas, USA. stefano.leonardi@utdallas.edu Received 8 April 2013; Revised 5 December 2013; Accepted 18 February INTRODUCTION With the shortage of fossil fuel and increasing environmental awareness, wind energy is becoming more and more important. As the market for wind energy grows, wind turbines and wind farms are becoming larger. Current utility-scale turbines extend a significant distance into the atmospheric boundary layer. Therefore, the interaction between the atmospheric boundary layer and the turbines and their wakes needs to be better understood. The turbulent wakes of upstream turbines affect the flow field of the turbines behind them, decreasing power production and increasing mechanical loading. With an improved understanding of this type of flow and the turbine response to it, wind farm developers could plan better-performing, less maintenance-intensive wind farms, and manufacturers could create better fatigue loadmitigating designs. Numerical simulations can complement experimental studies and lead to a better understanding of the flow through wind turbines and wind farms. Despite advances in parallel computing and the availability of thousands of processors in some advanced computing centers, the Reynolds numbers in this type of flow are so large (on the order of 10 8 ) that the use of direct numerical simulations is not feasible. However, large eddy simulation (LES) with an appropriate turbine aerodynamics model is feasible. Further difficulties in numerically studying wind turbines is represented by the different length scales of the problem. Wind turbines are large, on the order of hundreds of meters, with a typical spacing within a farm of around 5 10 rotor Copyright 2014 John Wiley & Sons, Ltd.

2 LES of the flow past wind turbines: actuator line and disk modeling L. A. Martínez-Tossas, M. J. Churchfield and S. Leonardi diameters (D), but with a thickness of the blade on the order of 1 m. In order to resolve the full turbine geometry, ideally one would need to build a mesh with submillimeter resolution in the blade boundary layer inside a kilometer-scale computational box within which the entire farm fits. Despite the possible use of non-uniform grids or adaptive mesh refinement, the computational costs of computing a full wind farm in this way are very high. As a consequence, we use the actuator disk model (ADM) and the actuator line model (ALM) to represent the wind turbine at a reduced computational cost. These actuator turbine models mimic the wind turbines without resolving the full geometry of the blades; rather, they apply body forces, representative of the blade lift and drag forces, to the flow field. Both of these models apply a thrust and torque on the flow, as do real wind turbines. The ADM and ALM have been previously used or studied by many including Sørensen and Shen, 2 Wu and Porté-Agel, 3, 4 Porté-Agel et al., 5, 6 Lu et al., 7 Churchfield et al., 8 Troldborg, 9 and Troldborg et al. 10 Sørensen and Shen 2 found good agreement between the power predicted by the ALM and experimental power production data for a 500 kw Nordtank wind turbine. The ALM was able to capture root and tip vortices near the rotor as they are convected further downstream. In their work, Porté-Agel and Wu 5 and Wu and Porté-Agel 3 compared the performance of the ALM and ADM in computing the wake of a single turbine. Both a version of the ADM that applies only thrust and one that applies thrust and torque to the flow were tested. The results obtained with the ADM that applies thrust and torque and with the ALM were in good agreement with the experimental measurements. The wake predicted using the thrust-only ADM had the most discrepancy from the experimental findings. Wu and Porté-Agel 4 performed validation of these models finding good results. Lu et al. 7 used the ALM to simulate flow through an operation wind plant in a stable atmospheric boundary layer with good agreement with field data. Churchfield et al. 8 used the ALM coupled with a turbine structural dynamics and system dynamics code to compute wind turbine response to different types of atmospheric turbulence. Troldborg et al. 10 studied grid resolution and the force projection for the ALM. A projection function is used in the actuator models to project the forces calculated at the actuator lines and disk onto the flow field as a volumetric body force. Troldborg 9 found that the value of the projection width,, must meet the criteria 2 x to avoid numerical oscillations in the solution around the actuator lines. Later, Troldborg et al. 11 showed that the ALM, without modeling the tower or nacelle, provides results comparable with a fully resolved rotor simulation so long as there is some turbulence in the incoming flow. In the present work, the effects of grid resolution and the projection function width used to project point forces onto the flow field have been studied. The actuator forces have to be projected from the line or disk from which they originate, which have no volume, onto a volumetric mesh, requiring a projection function. Not only is this performed for numerical reasons to avoid spurious flow field oscillations caused by too concentrated a force but also for the physical reason that in reality, the blade forces are spread over the entire surface of the blade and not collapsed upon lines or a disk. The results are compared with blade element momentum (BEM) theory and to wind tunnel experimental data. Differences between the numerical results and the BEM and experimental data are discussed. 2. NUMERICAL METHOD 2.1. Solver The computational fluid dynamics (CFD) solver was implemented using the OpenFOAM (Open Field Operation and Manipulation) toolbox. 12 The OpenFOAM package is a set of C++ libraries meant for solving partial differential equations. The turbine actuator models were implemented as C++ classes called by the solver. The LES code solves the incompressible filtered Navier Stokes equations, rqu D 0, C.Qu r/qu D 1 rqp C turb r2 Qu r SGS C (2) where, Qu is the filtered velocity vector, Qp is the filtered pressure, is the kinematic viscosity, SGS D SGS.r Qu C.r Qu/ T /, SGS is the subgrid-scale viscosity and is modeled using the standard Smagorinsky subgrid-scale model 13 and Ef turb is the turbine force vector (consisting of both thrust and tangential force) that depends on the actuator line or the disk models described in the following pages. The standard Smagorinsky model 13 constant is set to C s D 0.168, which is the value theoretically derived for isotropic turbulence. 14 We use this model for its simplicity, but more advanced dynamic models are another option that we may use in the future. A finite-volume approach is used to discretize the governing equations. Second-order central-differencing is used for spatial discretization, and the second-order Crank Nicolson scheme is used for time discretization. The set of implicit equations is solved sequentially using the pressure-implicit splitting operation algorithm. 15

3 L. A. Martínez-Tossas, M. J. Churchfield and S. Leonardi LES of the flow past wind turbines: actuator line and disk modeling (a) Commercial scale turbine (b) Actuator line representation (c) Flow visualization (d) Commercial scale turbine (e) Actuator disk representation (f) Flow visualization Figure 1. A schematic showing the relation between a real turbine rotor (left) and an ALM/ADM (middle), along with a visualization of the flow created by the ALM/ADM (right), where the blue isosurface is of the second invariant of the velocity-gradient tensor and the contours are of streamwise velocity. Photo credit: Senu Sirnivas/NREL. Figure 2. A two-dimensional section of the blade with the respective airfoil shape at that location Actuator line model With the ALM, turbine blades are represented by a force distribution on a line that extends from the hub to the tip of the blade. For example, a three-bladed turbine is represented by three lines of virtual forces rotating with the angular speed of the rotor (Figure 1). However, these smooth lines of force have to be applied to the volumetric grid surrounding the actuator line. The continuous actuator lines are discretized into lines made up of discrete elements or segments. The properties of the blade at the location of the center of each actuator element are found. These properties include chord length, twist angle and airfoil type. Each airfoil type has a corresponding look-up table with lift and drag coefficients as a function of angle of attack. The wind vectors experienced by each element are also found and used to calculate angle of attack, velocity magnitude and, ultimately, lift and drag at each element. Figures 2 and 3 show the section of the blade that the actuator point represents and the local coordinates system with the force projection. The body force equal and opposite to the lift and drag is imposed in the momentum equation. The lift and drag at each element are calculated on the basis of the local wind speed and angle of attack L D 1 2 C L. /Urel 2 cw, (3) D D 1 2 C D. /Urel 2 cw (4)

4 LES of the flow past wind turbines: actuator line and disk modeling L. A. Martínez-Tossas, M. J. Churchfield and S. Leonardi Figure 3. The two-dimensional airfoil section is shown where U x 0 and U y 0 are the velocity vector components projected onto the local coordinate system, U rel is the magnitude of the velocity projected onto the plane and the added rotational component, r is the radius,! is the rotational speed of the rotor, is the angle of attack and tw is the twist angle. where is the local angle of attack, C L. / is the lift coefficient, C D. / is the drag coefficient, is the density, U rel is the local wind speed, c is the chord and w is the width of the actuator section. These forces are then projected onto the flow field by means of the Gaussian function: F Ef turb D E 3 3=2 expœ.r=/2 (5) where the vector EF D EL C ED and r is the distance from the location of the actuator element center to the point where the force is applied. Thus, the force at each actuator element is distributed on a spherical region of the computational mesh surrounding the element proportional to. The summation of the spherical projections along the actuator line forms a cylindrical-like region of body force following the line. This is in part to avoid numerical instabilities that would result from the abrupt application of force along a line. It also spreads the force over a distance similar to the blade chord, which is more physically realistic. It is important that the actuator elements be short enough in length that the spherical regions of body force overlap each other sufficiently to produce a continuous distribution of force along the blade. On the basis of our previous experience, this requires at least 40 actuator elements per blade. A time step constraint is applied in which the actuator line tip may not pass through more than one grid cell per time step to maintain a smooth application of body force. In the next sections, we discuss in detail the dependence of power production on the value of Actuator disk model The ADM simulates the turbine rotor as a smooth application of force over the area that the blades sweep during rotation. The implementation takes into account both the axial and tangential forces of the disk, similar to the ADM with rotation from Porté-Agel et al. 5 and Wu and Porté-Agel. 3 This is carried out by dividing the rotor swept area into many elements similar to the discretization of the ALM. In fact, we implemented the ADM as an ALM with many lines to form a well-resolved disk. Figure 1 shows the schematic of the ADM in which each actuator point is the center of an element in the disk and the visualization is an example of the wake and vortex system created by the ADM (notice the lack of tip/root vortices). The ALM has more asymmetric wake breakup than the ADM, but in the far wake, they look basically the same. The force at each element in the ADM is scaled by a solidity factor D NA B A r D N N sector (6)

5 L. A. Martínez-Tossas, M. J. Churchfield and S. Leonardi LES of the flow past wind turbines: actuator line and disk modeling Figure 4. A cross section of the domain and mesh that has a turbine/wake-local resolution of 4.2 m. The turbine is placed in the center of the domain. where N is the number of blades, A B is the area of an individual actuator disk section (i.e. r r), A r is the swept area of that actuator disk section and N sector is the number of azimuthal sectors into which the disk is divided. The lift and drag forces on each actuator disk element are given by L D 1 2 C L. /Urel 2 cw, (7) D D 1 2 C D. /Urel 2 cw. (8) The time step constraint used for the ALM does not apply to the ADM since there is no discrete tip moving through the flow. 3. RESULTS 3.1. NREL 5 MW turbine subject to uniform inflow Large eddy simulation of the flow through the 5 MW reference turbine designed by Jonkman et al. 16 at the National Renewable Energy Laboratory was performed. This horizontal-axis upwind turbine has three blades, a rotor diameter of 126 m, a hub height of 90 m, a rated wind speed of 11.4 m s 1 and a rated power production of 5 MW. Simulations are performed with a non-turbulent uniform inflow condition of 8 m s 1. The rotational speed of the rotor is fixed at RPM giving a tip speed ratio of 7.55, this turbine s normal region 2 tip speed ratio. These conditions were chosen because they provide the optimum power coefficient of the turbine, extracting maximum energy from the flow. This is when the wakes are strongest and wake effects are most important. As is shown in Figure 4, we used a cubic mesh started with a background grid of 16.8 m resolution. Regions of successive refinement are introduced into the background grid to provide higher resolution around the turbine and its wake. The domain is 10 D wide in all directions with the turbine rotor placed in the geometric center of the domain. The finest region extends 1 D upstream of the turbine and 5 D downstream. It extends 1 D beyond the edge of the rotor in the other two directions. The inflow conditions are a uniform, non-turbulent velocity of 8 m s 1 and a zero normal gradient of pressure. The lateral boundary conditions are periodic. The outflow boundary conditions are a zero normal gradient of velocity and a fixed pressure. The standard Smagorinksy model does not require boundary conditions on subgrid-scale viscosity. Cases were run for four different values of the parameter ranging from 4.2 to 10.5 m. Three different grid refinement levels were used with the finest x D y D z D 4.2, 2.1 and Mean velocities reported in this section are based on 10 min of averaging time past an initial 200 s time to allow start-up transients to pass. Aerodynamic power output predicted by the ALM and ADM is shown as a function of grid resolution for different values of in Figure 5. Power is calculated by multiplying the rotor speed by the total torque; electrical efficiency is not taken into consideration. The power output increases but seems to converge as the grid is refined. We used Richardson extrapolation to estimate the power production with the leading order discretization error removed. As becomes larger, the predicted power increases. The increase is not linear, though. For a fixed rate of increase of, the rate of increase of predicted power seems to approach zero. The larger the value, the smaller the rate of change of predicted power as a function of grid resolution. When D 4.2 m, the difference in power predicted on the finest grid and the coarsest grid is 300 kw. When D 10.5 m, this difference is only 40 kw. Power production is clearly more grid dependent for smaller

6 LES of the flow past wind turbines: actuator line and disk modeling L. A. Martínez-Tossas, M. J. Churchfield and S. Leonardi Figure 5. Power output as a function of grid resolution for different values. (a) Angle of attack (b) Axial velocity Figure 6. Angle of attack and axial velocity as a function of non-dimensional blade length. values of. This hints that grid dependence is highly influenced by the grid s ability to well resolve the Gaussian-projected body force. Simulations for D 2.1 m with the ALM were run with grid resolutions of x D 1.05 and m. This value of is limited to the coarsest grid resolution of 1.05 m because numerical instabilities appear when <2 x. The trend is followed showing that a smaller will produce less power. The BEM solution for power output is MW. This value lies between the finest cases with D 4.2 and 2.1 m. The ALM is more sensitive than the ADM to the grid resolution. This is perhaps not surprising because the forces of the ADM are distributed in a wide area and not on a thin line as with the ALM. For fine resolutions, the ALM and ADM present very similar power production. Figure 6 shows time-averaged angle of attack (left) and axial velocity (right) for the ALM along the blade for different values of. The axial velocity at the location of the actuator point is higher for larger values of.when is small, a larger force over a smaller volume opposes the flow when it reaches the actuator point. On the other hand, larger values of apply a smaller force over a larger volume, not decelerating it as much at the velocity sampling point as a larger force. For larger values of, the axial velocity at the actuator point is higher, creating a higher angle of attack. Higher angle of attack and velocity lead to higher lift, thus increasing the power produced by the turbine. Mean wind speed contours for the ALM are shown in Figure 7. This shows, qualitatively, the difference in the wake that the value of produces. A smaller value of produces thinner shear layers. Mean streamwise velocity profiles, 1 and 4 D downstream, are shown in Figures 8 and 9. The velocity in the center of the wake is greater than at larger radii. This is expected because the hub and nacelle are not included in the model, and thus, it is the effect of the blades only. The blades are also not as loaded at the root as farther outboard, so they do not decelerate the flow there so much. Mean speed profiles are similar for different values of. The main difference observed is in the middle of the wake where there is a jet. A larger value of projects the forces further beyond the root, leaving a small gap between the blades for the jet to form. This leaves less space for the jet as the value is increased. As the value decreases, the shear layer becomes thinner and more unstable at the rotor tip and root, causing a faster transition to turbulence. This can be observed in Figure 10, in which instantaneous vorticity magnitude contours are shown for a grid resolution of x D 1.05 m; a smaller value of causes earlier transition to turbulence. This is perhaps not surprising because a large smooths the force distribution and, as a consequence, the velocity gradients that generate turbulence. It is also clear that the body force projection width strongly influences the distance over which tip and root vortices remain coherent. The larger the value of the projection

7 L. A. Martínez-Tossas, M. J. Churchfield and S. Leonardi LES of the flow past wind turbines: actuator line and disk modeling Figure 7. Mean velocity (m s 1 ) contours in a plane through the turbine hub predicted by the ALM with a mesh resolution of x D 1.05 m. The position of the turbine rotor is indicated by the black line. width, the more diffused these structures appear and the less distance downstream they remain coherent. The structure of the instantaneous ALM wake is not symmetric with respect to the center of the wake because the vortical structures produced by individual blades are helical and asymmetric across the wake. This asymmetry causes asymmetric roll-up of Kelvin Helmholtz-like instabilities in the outer shear layer of the wake. As the grid is refined, vorticity contours show smaller structures that are filtered in coarser grids. The simulations with x D D 4.2 m are subject to numerical oscillations. These instabilities are due to the fact that = x D 1, and this causes numerical instabilities because the grid is too coarse relative to the body force width. A lower bound of 2 x should be used for an oscillation-free solution Comparison with wind tunnel experiment at NUST A validation of the ALM has been performed comparing the numerical results with the experiments performed at the low-speed closed-return wind tunnel of the Department of Energy and Process Engineering at The Norwegian University of Science and Technology, Trondheim. 1, 17 The tunnel has a test section of 1.85 m (height) 2.71 m (width) m (length). The test section height is adjustable to account for wall boundary layer growth and maintain a zero pressure gradient. For most of the experimental runs, the wind tunnel speed was set to 10 m s 1, and the turbulence intensity based on the streamwise velocity fluctuations was 0.3%. The study aims to numerically simulate the experimental setup. The turbine modeled has a rotor with a diameter of m and a tower with a height of m. The blades use the NREL S826 airfoil along the entire span. Two-dimensional lift and drag coefficients were obtained from the NREL S826 report 18 and corrected for stall delay using the NREL-developed tool AirfoilPrep. 19 The ALM does not include the hub, the tower or the nacelle and takes into account only the blades of the turbine. The turbine is placed 3.66 m from the tunnel inlet and is centered laterally. Simulations were performed for tip speed ratios () of3,4,6,9and12. Because the rotor swept area is roughly 12% of the wind tunnel test section cross section area, blockage effects should be considered. We used no slip boundary conditions on the walls of the tunnel to mimic the blockage effect due to the presence of the turbine and the tunnel walls. The inflow conditions are of uniform, non-turbulent velocity (we did not add any synthetic turbulence to try to match the experimental wind tunnel s 0.3% turbulence intensity) and zero normal gradient of pressure, and the outflow conditions are of zero normal gradient of velocity and fixed pressure. The values for and grid resolution x were adjusted on the basis of the results from the extensive numerical study described in the previous section. A fine resolution near the rotor of x=d D 0.01 was used. The value of was adjusted to =D = 0.02, which is 2 x. A smaller would have required a smaller x, which is beyond the reach of our current computational facilities. Mean quantities reported in this section are based on 10 min of averaging time past an initial 200 s time to allow start-up transients to pass. Power and thrust coefficients are shown as a function of tip speed ratio in Figure 11. The power and thrust coefficients are calculated on the basis of aerodynamic power and thrust C P D Power, C 1 T D Thrust 2 U AU2 1 (9)

8 LES of the flow past wind turbines: actuator line and disk modeling L. A. Martínez-Tossas, M. J. Churchfield and S. Leonardi Figure 8. Mean velocities in the cross-stream direction that is 1 D behind the turbine rotor are shown, as predicted by the ALM (left column) and ADM (right column). Each row shows plots that are the result of using different values of the body force projection width (). where is the fluid density, A is the rotor area and U 1 is the inflow velocity. The maximum computed power coefficient of 0.47 occurs at a tip speed ratio of 6. This corresponds to the highest induction factor, as would be expected. For the case with the lowest tip speed ratio, the predicted power coefficient is nearly the same as that of the experiment, but it is then underpredicted for the tip speed ratio of 4 case. For higher tip speed ratios, the power is overpredicted by up to 25%. The thrust coefficient is underpredicted by 20 25% across the tip speed ratio range.

9 L. A. Martínez-Tossas, M. J. Churchfield and S. Leonardi LES of the flow past wind turbines: actuator line and disk modeling Figure 9. Mean velocities in the cross-stream direction that is 4 D behind the turbine rotor are shown, as predicted by the ALM (left column) and ADM (right column). Each row shows plots that are the result of using different values of the body force projection width ().

10 LES of the flow past wind turbines: actuator line and disk modeling L. A. Martínez-Tossas, M. J. Churchfield and S. Leonardi Figure 10. Vorticity (1 s 1 ) contours in a plane through the turbine hub predicted by the ALM (left column) and ADM (right column) with a mesh resolution of x D 1.05 m. The position of the turbine rotor is indicated by the black line. Each row shows contours resulting from using different values of the body force projection width ( ). Figure 11. Power and thrust coefficients as a function of tip speed ratio.

11 L. A. Martínez-Tossas, M. J. Churchfield and S. Leonardi LES of the flow past wind turbines: actuator line and disk modeling There are probably several contributors to the discrepancies between computed and measured power and, especially, thrust. One possible explanation is that actuator methods are not blade-resolving (by blade-resolving, we mean the blade geometry is not explicitly captured with the computational mesh), so they rely upon airfoil tables of lift and drag versus angle of attack. Such tables are usually derived from experiments using two-dimensional airfoil sections in wind tunnels. A wind turbine blade, especially a smaller one (such as the one used in this wind tunnel experiment) that is not of as high aspect ratio as large utility turbines, produces a highly three-dimensional flow field in which rotational effects and tip effects are important and substantially modify lift and drag. Corrections to two-dimensional airfoil data for three-dimensional and rotational effects exist, and we applied one, 19 but those corrections are empirical and cannot be expected to work well under all conditions. It is probable that the airfoil data we used do not agree well with the behavior of the actual blade used in this experiment. In fact, Krogstad and Lund 1 show that a blade element momentum calculation of this wind tunnel experiment s rotor predicted power quite well but underpredicted thrust. Similar with the ALM, blade element momentum theory relies upon airfoil lift and drag tables. Krogstad and Lund 1 also show that a blade-resolving CFD computation of the wind tunnel rotor well predict both power and thrust. Since the CFD computation was blade-resolving, it does not rely upon lift and drag tables but rather generates the blade-induced flow completely on its own, including all the three-dimensional and rotational effects. Another possible explanation for the discrepancy between the power and thrust coefficients predicted by the ALM and that of the measurements is that our ALM and ADM implementation does not yet include the effects of the nacelle. Flow is allowed to pass through the center of the rotor disk where the nacelle is in reality. In reality, the oncoming wind flows around the nacelle and thus causes a different flow effect through the rotor, especially in the inboard section, than we are able to capture. We would expect that the inclusion of a nacelle in the model would cause an increase in axial velocity through the inboard blade sections. This will increase angle of attack and also drag. Lift may decrease also unless the increase in angle of attack causes stalled conditions. Regardless, the increased axial flow should always increase drag. This increased drag directly increases thrust. The change in lift, though, must be multiplied by radius from the rotor centerline to obtain the change in torque, which then changes power (power equals torque multiplied by rotor speed, and we are considering a fixed rotor speed, so power scales with torque). At the inboard sections, the radius is small, so changes to lift make small changes to power. Furthermore, the most inboard section is cylindrical, which produces only drag. Figure 12. Mean velocity profiles 1 and 3 D behind the rotor.

12 LES of the flow past wind turbines: actuator line and disk modeling L. A. Martínez-Tossas, M. J. Churchfield and S. Leonardi Figure 13. Mean turbulent kinetic energy profiles 1 and 3 D behind the rotor. Figure 12 shows wake profiles for cases with tip speed ratios 3 and 6, 1 and 3 D behind the rotor. Agreement between the computed mean wake profile and that of the experiment is good on the outer half of each side of the blade jy=rj > 0.5. The profiles differ significantly for jy=rj < 0.5. This difference is because the nacelle is not a part of the ALM, so its wake is not captured. Figure 13 shows turbulent kinetic energy horizontal profiles behind the turbine. Profiles 1 D behind turbine rotor show peaks of turbulent kinetic energy at the edges and center of the wake. These peaks are probably an apparent turbulent kinetic energy caused by the periodic passage of tip and root vortices, which are seen by a measurement probe as fluctuations in the flow. The peaks from the simulations are not nearly as great as those of the experiment. Although we have not quantified it, the tip and root vortices resolved in the simulations may not be as strong as the real ones created in the experiment, which would cause weaker fluctuations as observed at a fixed location and, hence, lower turbulent kinetic energy. As the wake progresses downstream, the shear layer breaks down at the center and edges of the wake, and the turbulent kinetic energy is smaller and because of the actual turbulence, not apparent turbulence from tip/root vortex passage. Agreement is found between the simulations and experimental data on the outer half of each side of the blade jy=rj > 0.5. The region where jy=rj < 0.5 differs more. This is caused by the shear layer instabilities of the jet passing through the rotor in the simulations. This may, again, be a result of the fact that the ALM does not include tower and nacelle effects, both of which would create wakes. Rather, the ALM predicts a turbulence-generating jet along the centerline. 4. CONCLUSIONS In this work, we used LES to compute the flow through actuator representations of wind turbine rotors. We compare both the actuator line and ADMs. Such models are an attractive alternative to resolving the full rotor geometry because they require fewer grid cells and do not require as small grid dimensions, allowing for larger time steps. This efficiency comes at the expense of resolving the fine details of the blade boundary layer, but if the objective is to model the far wake, where the blade boundary layer details are forgotten, this trade-off is a reasonable one. For example, actuator turbine models are

13 L. A. Martínez-Tossas, M. J. Churchfield and S. Leonardi LES of the flow past wind turbines: actuator line and disk modeling well suited to full wind farm computations where there may be tens or hundreds of turbines. Although these methods have been applied to wind turbine flows for the past decade, the details of their use have not been well outlined. The aim of this work is to shed more light on such details. We examined the effect of grid refinement and body force projection width on the predicted power and wakes. When subject to non-turbulent uniform inflow, we observed that the power predicted by both the ALM and ADM can depend on grid resolution unless a sufficient resolution relative to the projection function width is used. Grid independence appears to occur when the grid resolution is such that x <=5. We make this statement because when changing the grid resolution from x D =5 to=10, only a roughly 0.5% change in predicted power was observed. We also observed, similar to Troldborg, 9 that if x >=2, spurious numerical oscillations form at the rotor disk. Power predicted by both the ALM and ADM is highly dependent on the projection function width,. In this study, we used a three-dimensional Gaussian projection function, the purpose of which is to convert the actuator element forces, which are concentrated at points at the center of each element, into volumetric body forces that can enter the momentum equations in a straightforward way with the finite-volume discretization used here. The only convergence that seems to occur is that as approaches infinity, the body force is so diluted that it would have no effect on the flow field, meaning the velocity sampled at each actuator element would be that of the freestream. The results would be unrealistic, though, as a finite power exceeding the Betz limit would be predicted with no wake generation. On the other end of the spectrum, though, as is decreased, the predicted power decreases with no indication of convergence. As is decreased, the body force becomes more concentrated such that locally, the flow is more decelerated. The velocity sampled at the actuator point is hence lower, meaning that the predicted angle of attack and lift are lower, resulting in lower power production. Both the ALM and ADM give power predictions within 1% of one another for the same grid and value of. Although they predict very similar power, the ALM and ADM do create noticeably different near-field instantaneous wakes. The fact that the ALM contains individual lines of force means that it can create tip and root vortices that spiral downstream into the wake. The ADM does not create the tip and root vortices. This difference also causes a subtle difference in the way the wake shear layer breaks down into turbulence. The ADM wake breakdown starts with axisymmetric Kelvin Helmholtz-like vortex roll-up. The tip and root vortices created by the ALM break the axisymmetry. Even with this difference, the mean wake profiles are nearly indistinguishable. The ALM was then used to compute a wind tunnel wind turbine study in which power, thrust and mean wake profiles were measured. Up to tip speed ratios of 6, power is predicted within 10% of the experiment. Above a tip speed ratio of 6, the predicted power production is less in agreement with the experiment. The predicted thrust coefficient is always underpredicted. We attribute the thrust discrepancy to the lack of nacelle effects in the actuator models and to the use of lift and drag look-up tables. In reality, a nacelle modifies the axial velocity through the inboard blade sections as compared with the actuator methods used here that model only the blades, allowing flow to pass through the rotor center. It is probable that the effect of the nacelle on the inboard flow is part of the cause of the thrust discrepancy. A probably larger contributor to the discrepancy is the fact that the wind turbine used in the wind tunnel experiment has low aspect ratio blades (compared with large utility-scale turbines), and three-dimensional and rotational effects are likely strong. We use airfoil lift and drag look-up tables derived from two-dimensional section measurements. Although we apply a three-dimensional and rotational effect correction, such corrections are empirical and do not always work well. Wake velocity profiles are in good agreement with the experimental data only from the rotor midspan to the rotor edge. There are major differences inboard, probably due to the fact that we do not model the nacelle nor capture its work. The same is observed for turbulent kinetic energy in the wake. One exception to this occurs 1 D behind the turbine operating at a tip speed ratio of 6. At that operation point and location, we underpredict the turbulent kinetic energy at the wake edge by an order of magnitude, and the reason is not clear. Overall, we have shown the behavior of the ALM and ADM with respect to grid resolution and projection function width. We have shown that a grid resolution of x <=5 is necessary for the predicted power to be grid independent. We also show that should be around the size of the characteristic blade chord length, although we do not give a definitive value. Most likely, should vary with span and should be some ratio of the local chord length. Our comparison with the wind tunnel experiment shows that more work needs to be carried out on implementing a model for the wind turbine tower and nacelle to assess their impact on the turbine s predicted performance and wake profiles. ACKNOWLEDGEMENTS Professor Per-Åge Krogstad is acknowledged for useful discussion. Luis A. Martínez-Tossas was supported by contract AFC with the National Renewable Energy Laboratory. Computer time was provided by the National Renewable Energy Laboratory with the Red Mesa high-performance computing system. LM-T and SL acknowledge support from NSF PIRE grant #

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