CFD wake modeling using a porous disc Giorgio Crasto, Arne Reidar Gravdahl giorgio@windsim.com, arne@windsim.com WindSim AS Fjordgaten 5 N-325 Tønsberg Norway Tel. +47 33 38 8 Fax +47 33 38 8 8 http://www.windsim.com Summary of the work Single wake of a wind turbine over flat terrains is modeled with a finite-volume Computational Fluid Dynamics (CFD) Reynolds Averaged Navier-Stokes (RANS) solver. The RANS equations of an uncompressible flow are solved with a multigrid coupled algorithm; the turbulence is closed with the k-ε model. The tower and the nacelle of the wind turbine are modeled by solid cells while the rotor is modeled by a porous disc providing a resistive force which is calculated from the thrust coefficient value C T. Comparisons of CFD results are presented against wind tunnel data extracted from literature. The numerical results are further compared against three analytical models for wakes. The presented CFD model shows a very good agreement with wind tunnel data in the region between two and eight diameters downstream of the rotor while further downstream there is a tendency in slightly overestimating the wind deficit. The comparison with the analytical models is shown for four cases: two values of terrain roughness (onshore and offshore) and two values of thrust coefficient (.8 and.3). Three analytical models have been considered. The CFD simulations with the actuator disc matches very well in particular with one of the analytical models for high values of thrust coefficient (C T =.8). In the cases with low C T (C T =.3) there is not a clear tendency described by the analytical models, the actuator disc gives the same order of magnitude of the analytical models considered.
Modeling a wind turbine with a porous disc The aim of this series of simulations is to introduce in the CFD calculations over a flat terrain a complex structure like a wind turbine modeled with the concept of the actuator disc. The RANS equations are solved with a finite-volume technique; the turbulence is closed by the standard k-ε model and the algorithm of integration is a coupled multi-grid. In the present paper only simulations of a single turbine on a flat terrain are presented; the final goal of the research is to introduce the actuator disc concept in the procedure to estimate the AEP of a complete wind farm on a general orography. The wind turbine is modeled by solid cells for the tower and the nacelle, while porous cells are instead used for the swept area of the rotor, hence an actuator disc, whose concept is also sketched in Figure. By porous cells in this case are meant cells where momentum sinks are applied; in the cells constituting the actuator disc a uniform thrust per square meter t is therefore applied (pressure drop). The pressure drop t is calculated from a combination of the thrust coefficient curve and a formula to estimate the axial induction factor a. Figure. Sketch for the actuator disc and the solid cells used to model the tower and nacelle. The pressure-drop through the actuator disc is calculated from equation (), where the undisturbed velocity u is given by the Betz s theory once assumed a fixed value for the thrust coefficient, equation (4). T 2 t = = CT ρ u () A 2 u u a = u u = u and a by definition (2) a = ( C T ) from Betz s theory (3) 2 2 T t = = CT ρ u (4) A 2 a
A drawback of the present model is that the thrust coefficient is fixed while it s in the intentions of the authors to improve the model by computing the thrust coefficient starting from the velocity u extracted at the rotor position. In the presented simulations a wind turbine having the following geometrical characteristics is modeled: Rotor diameter D: 68.8 m; Hub height: 75 m. For instance, considering a roughness length z of,3 m (typical of low vegetation) all over the domain and a free stream velocity above the boundary layer (considered 5m high) of m/s it follows that the wind speed at the hub is around 8 m/s: U e = m/s at z = 5 m At the hub height for a logarithmic BL: U 75 = U 5 ln(75/z )/ln(5/z ) = 8,5 m/s u undisturbed velocity at the hub height Once chosen a wind turbine model the correct thrust coefficient C T is sampled from the relative curve like the one reported in Figure 2. In the presented simulations the thrust coefficient has been kept constant during the calculations. Ideally in the CFD runs the thrust coefficient should be updated at each iteration depending on the wind velocity at the rotor position (u ); this need will be particularly felt when the analysis will pass to the modeling of real terrains where the orography effects are present. CT.9.8.7.6.5.4.3.2. 5 5 2 25 3 undisturbed velocity [m/s] 2 Computational domains Figure 2. Example of thrust-coefficient curve for a wind turbine. Two computational domains have been employed with dimensions in the stream (x), span (y) and vertical (z) directions: Domain Lx = 4 m Easting Ly = 26 m Northing Lz = m Vertical Domain 2 Lx = 6 m Easting Ly = 26 m Northing Lz = m Vertical
In the simulations run the wind was always blowing from west. The second domain has been introduced in order to extend the region downstream of the wind turbine, enclosing the wake. The dimensions of the computational domain are also given in rotor diameters: Domain Lx = 57.4 D Domain 2 Lx = 87.2 D 3 Grid description Ly = 37.4 D Ly = 37.4 D Lz = 4.29 D Lz = 4.29 D Several grids are constructed and tested to reduce the grid dependency; the turbine is built putting together solid cells for the tower and the nacelle and porous cells to mimic the actuator disc. A perspective view of the simplified wind turbine is given in Figure 3. A second grid (Grid ) has been constructed for the Domain, more refined, in order to make an analysis of grid sensitivity. The extended domain (Domain 2) has been discretized keeping the same number of cells of Grid in the unchanged directions y and z. A top view of the extended grid (Grid 2) is given in Figure 4. Figure 3. Perspective view of the obstacle wind turbine.
Figure 4. Top view of the extended mesh (Domain 2, Grid 2). 4 Results of the RANS simulations A series of simulations were run with the actuator disc; three grids over two domains, moreover the possibility to assign a value of porosity and momentum sinks also on the vertical and span direction were firstly considered. In this paper there are presented only results for what have been considered the proper settings of the model, in other words no porosity assigned to the cells modeling the rotor and a momentum sink only acting in the axial direction. The concept of porous cells and source terms in the momentum equation is used in the presented simulations. The porous cells extract momentum from the axial direction. Four cases will be presented: for two values of terrain roughness and two values of thrust coefficient, which are summarized in Table. Table. Table of cases presented. z,3 m (onshore) z, m (offshore) C T,8 a b C T,3 c d The simulations have been performed mainly with the coupled (MIGAL) solver, pointing out how the MIGAL, which is a multigrid solver, needs much less iterations to reach convergence than a segregated solver (SIMPLEST). The numerical results obtained are shown in the following paragraphs, both velocity and turbulence are investigated, comparisons with experimental and numerical data from other authors are given mainly having as a reference the state of the art analysis by Vermeer et al. [5]. 4. Comparison with wind tunnel data Vertical profiles of velocity ratios (U/U e ) downstream of the wind turbine are presented on Figure 5, comparisons are done with wind tunnel measurements from TNO and reported in Vermeer et al. [5]. The behavior in the near wake (2D downstream) is not captured; a simulation with a porous plate does not pretend to solve exactly the near wake flow. The velocity profile at 6 diameters downstream is instead fairly forecasted but observing the profiles 8 diameters downstream there are again discrepancies between the velocity profile predicted by the CFD calculations and the wind tunnel measurements by the TNO, this difference could be explained by a not sufficient turbulent
mixing predicted by the CFD calculations. Indeed the distance at which the wake is completely destroyed is highly dependent upon the ambient turbulence. TNO WT TNO WT TNO WT TNO WT -5D CFD 2D CFD 6D CFD 8D CFD 8D downstream 2D downstream 6D downstream..2.4.6.8.2 Figure 5. Vertical profiles of normalized velocity from TNO Wind Tunnel tests (black solid lines), from Vermeer et al. [5] and profiles obtained with the numerical actuator disc. 4.2 Wind deficit and turbulence along the wake s center line 4.2. Normalized wind deficit If U e is the free stream velocity (without considering Coriolis effects), firstly the velocity at hub height has to be found according to the log-law for the wind shear or from simulations on flat terrains. U e m/s; U HUB = U e ln(z HUB /z )/ln(δ ABL /z ) for a neutral stratification; U HUB = 8,5 m/s (U e = m/s; z HUB = 75 m; δ ABL = 5 m; z =,3 m). Hence, the velocity deficit is given by the difference U HUB U(x) that can be normalized against U HUB. Normalized Wind Deficit (NWD) at the hub height: U HUB U ( x) NWD = U HUB
4.2.2 Analytical models Wake effects can be calculated by analytical and CFD based methods. Analytica methods are attractive as they are simpler and less computational demanding than CFD based ones; CFD models have the possibility to model in a more realistic way the wake, their 3D structure, their interaction with other wakes and with the general flow field generated by orography, roughness and obstacles. Three wake models have been considered in the present study: Model. Jensen model [3]; Model 2. Larsen model [4]; Model 3. Ishihara et al. model []. They are all single wake models calculating the normalized velocity deficit; NWD = (U-V)/U, see the definition sketch in Figure 6. Figure 6. Definition sketch wake effects. 4.2.3 Results of wind deficit The normalized wind deficit is plotted against the axial position in rotor diameters in Figure 7 for the cases a, b, c and d. The NWD predicted by the actuator disc is in the range of the one predicted by some analytical models. The discrepancies between the three analytical models increase when passing from an onshore to an offshore scenario. The NWD obtained with the actuator disc are therefore promising but a final validation has to be done against wind tunnel or full scale experiments. 4.2.4 Turbulent Intensity The turbulent intensity, which is obtained from the turbulent kinetic energy KE and the horizontal velocity, equation (5), takes into account an isotropic turbulence. 4 KE TI = 3 2 2 UCRT + VCRT x (5) The TI is plotted against the axial position in rotor diameters; qualitatively the TI follows an expected trend but comparisons with experimental results are needed for the turbulence level.
5 Conclusions An actuator disc concept is applied to model a wind turbine in RANS simulations of the atmospheric boundary layer over a flat terrain. The RANS equations are closed with the k-ε turbulence model. A grid sensitivity study has been carried out in order to set a fair level of refinement. In the final settings of the actuator disc a uniform pressure drop is applied on the disc; the value of the pressure drop is calculated from the value of thrust coefficient and axial induction factor. From comparison with WT tests the wind deficit predicted in the near wake is underestimated while the behavior of the far wake is quite well predicted; there is not a perfect match of the results downstream after 8 rotor diameters, this is probably due to a not perfect match of the ambient turbulence of the wind tunnel and the CFD, hence not enough dissipation predicted by the CFD model. The comparisons against the analytical models show a promising match, at least looking at the level of magnitude. An interesting following of the study will be the introduction of swirling in the wake and comparisons with full scale experiments. Moreover, it s interesting to introduce multiple wakes and interaction with real orography. Normalized-Wind-Deficit Normalized-Wind-Deficit.9.8.7.6.5.4.3.2. -2-2 3-2 - 2 3.9.8.7.6.5.4.3.2. case a (Ct,8; z,3m) Wake model Wake model 2 Wake model 3 case c (Ct,3; z,3m) Wake model Wake model 2 Wake model 3 Distance downstream [rotor diameters] Distance downstream [rotor diameters] Figure 7. NWD plotted against axial distance in rotor diameters. case b (Ct,8; z,m) Wake model Wake model 2 Wake model 3 case d (Ct,3; z,m) Wake model Wake model 2 Wake model 3-2 - 2 3-2 - 2 3
Turbulent Intensity [%] Turbulent Intensity [%] 9 8 7 6 5 4 3 2-2 - 2 3 9 8 7 6 5 4 3 2 a c 9 8 7 6 5 4 3 2-2 - 2 3 9 8 7 6 5 4 3 2 b d -2-2 3 Distance downstream [rotor diameters] 6 References Distance downstream [rotor diameters] Figure 8. TI plotted against axial distance in rotor diameters.. Ishihara T, Yamaguchi A, Fujino Y. Development of a New Wake Model Based on a Wind Tunnel Experiment. Global Wind Power 24. 2. Ivanell SSA. Numerical Computations of Wind Turbine Wakes. PhD thesis, KTH Mechanics, Royal Institute of Technology, Stockholm, Sweden, 25. 3. Katic I, Højstrup J, Jensen NO. A Simple Model for Cluster Efficiency. EWEC Proceedings, 7-9 October 986, Rome, Italy. 4. Larsen CG. A Simple Wake Calculation Procedure. Risø-M-276, 988. 5. Vermeer LJ, Sørensen JN, Crespo A. Wind turbine wake aerodynamics. Progress in Aerospace Sciences 23; 39 467 5. -2-2 3