How far the wake of a wind farm can persist for?

Guodan Dong a , b , Zhaobin Li a , b , Jianhua Qin a , b , Xiaolei Yang a , b , ∗

a The State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China b School of Engineering Sciences, University of Chinese Academy of Sciences, Beijing 10 0 049, China

Keywords:Wind farm wake Large-eddy simulation Actuator disk Wind turbine wake

ABSTRACT With the increased penetration of wind energy in our nation’s energy portfolio, wind farms are placed in a way close to each other. Thus, their wakes have to be fully considered in the design and operation of a wind farm. In this study, we investigate the wake of a wind farm using large-eddy simulation with wind turbine rotor modelled by the actuator disk model. The simulated results show that the wake of a wind farm can persist for a long distance in its downstream. For the considered wind farm layout, the velocity in the wake recovers 95% of that of the undisturbed inflow at 55 rotor diameters downstream from its last row, suggesting that the wake of a wind farm should be fully considered in the optimal design and operation for its downstream wind farms.

Wind turbines are often clustered as a wind farm to extract energy from the wind. A wind farm leaves a wake in its downstream as converting wind kinetic energy into electricity, which has to be taken into account properly when designing wind farms in regions with multiple planned farms, in order to correctly estimate the annual energy production and structure loads [1] . The wake of a single wind turbine has been extensively studied in the literature [2–4] . However, investigations on the wake of a wind farm are relatively limited.

The wind farm wake, consisting of flow structures ranging from wind turbine scale ( ∼100 m) up to meteorological scale ( ∼100 km) [5] , is affected by wind turbine wakes, their interaction with the atmospheric turbulence and several other factors (e.g.,terrain topology, wind farm operations, and meteorologic condition) [6] . It has been shown that a wind farm wake can propagate downstream for tens of kilometers [7] . Field measurements of the Horns-Rev wind farm by satellite synthetic aperture radar [8] reveal that a speed loss of more than 2% can still be observed within a range of 5 to 20 km downstream. For the German Bight offshore wind farm, aerial measurements found that the wake speed deficit can reach up to 40% immediately downstream the wind-farm, and the wake can propagates for tens of kilometers. A nine-year longterm measurement using met tower and supervisory control and data acquisition (SCADA) data in the North sea wind farm cluster confirmed evidently that the wake of the upstream wind farm reduces the wind speed, enhances the turbulence, and increases the structural load for their downstream wind farms [9] .

Besides measurements, numerical simulations with different fidelity have also been carried out in the literature. To study the regional or global effects of wind farms, the whole wind farm is often modelled as equivalent surface roughness [10] or as an elevated sink of momentum and a source of turbulent kinetic energy[11,12] . Modeling an individual turbine wake in mesoscale simulations as an elevated sink of momentum and a source of turbulent kinetic energy were also reported in the literature [13,14] .These modeling techniques without resolving wind turbine wakes allow the investigation of regional to global effects of wind farms with affordable comput ational resources. In the meanwhile, resolving turbine wake can provide more details of the wind-farm wake[15] , which is essential for the design of wind farm layout and operation, but is more resource demanding. Most of the farm-scale simulations have been done with Reynolds-Averaged Navier-Stokes(RANS) method [16] or with large-eddy simulation (LES) utilizing a wake region less than 10 km [17,18] , which is relatively short compared to the distance that the wake effect can reach as reported from measurement [8] .

In this work, we present a preliminary investigation of wind farm wake with LES, to probe into the flow details of the wind farm wake. An idealized wind farm layout consisting of 40 turbines is studied in a large computational domain, allowing analyzing the wind-farm wake over 15 km long.

Fig. 1. Layout of the simulated wind farm and the actuator disk model for the rotor of the wind turbine.

The turbulent flow over a wind farm is simulated using LES via the VFS-Wind code [19] , in which the filtered Navier-Stokes equations for incompressible flows in the curvilinear coordinate are solved with the unresolved subgrid-scale stress modelled using the dynamic Smagorinsky model [20] . A second-order accurate central differencing scheme is employed for the spatial discretization, and the fractional step method [21] is employed for the time integration. The nonlinear momentum equation is solved with a Jacobian-Free Newton-Krylov method [22] . The Poisson equation is solved with a Generalized Minimal Residual (GMRES) method with an algebraic multigrid acceleration.

The wind turbine is parameterized by the actuator disk (AD)model with uniformly distributed axial forces [23] . The AD model,which neglects the geometry detail of individual wind turbine blades, represents the rotating blades as a fixed two-dimensional porous disk exerting a uniform thrust on the flow. In the present AD model, the axial thrust force per unit areafT(t)is computed as follows:

whereρis the density of air,Udis the streamwise velocity averaged over the actuator disk, andC′T= 4a/(1 −a)(whereais the axial induction factor). The above expression is derived based on the conventional definition for thrust and the one-dimensional momentum theory. The same approach has been employed in the literature for the simulation of wind farms [24,25] .

As shown in Fig. 1 , the simulated wind farm consists of 40 turbines arranged into 4 rows with 10 turbines in each row, and placed with the streamwise and spanwise spacings 15Dand 5D,respectively, in an area of an area of 45D×45D, whereDis the rotor diameterD= 100 m of the employed wind turbine with hub heightzhub =D. This layout is designed using the method presented in Ref. [26] for maximizing the power output efficiency. In the simulation, the axial induction factor is fixed ata= 0.25 for all turbines. The employed computational domain is rectangular (as shown Fig. 2 ) and has a size ofLx×Ly×Lz= 225D×148.8D×10D,withx,yandzdenoting the streamwise, spanwise and vertical directions, respectively. The domain is discretized by a Cartesian grid with grid nodes ofNx×Ny×Nz= 1126 ×1488 ×152 . The grid is uniform in thexandydirections with the grid spacingΔx=D/ 5 andΔy=D/ 10 . In thezdirection, the grid is uniform near the ground (z∈(0,1.5D)) withΔz=D/ 50 and is gradually stretched to the top boundary. This mesh resolution is accurate enough to capture wind turbine wake and wind farm wake according to Yang et al. [17,25] .

We first show the instantaneous streamwise velocity field in Fig. 3 , with zoomed-in Fig. 3 b and 3 c for better visualization of wind turbine wakes inside the wind farm and the near wake of the wind farm, respectively. As seen, the wake of the wind turbine is well captured in the present simulation. The flow within the farm is composed of low speed wind turbine wakes and relatively high wind speed regions located between neighboring turbine columns.In the downstream of the farm, complex flow structures featured by the interplay of turbine wakes appear.

Fig. 2. Schematic of the employed computational domain. The origin ”A” is indicted by a blue dot. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 3. ( a ) Contours of the instantaneous streamwise velocity on the x −y plane at z hub , y −z plane at x = 55 D (the last row of the farm), and z −x plane at y = 96 . 8 D (one of the columns along the edge of the farm). ( b ) and ( c ) Zoomed-in contours showing the turbine wake inside the wind farm and the near wake of the wind farm, respectively.

Fig. 4. ( a ) Contours of the time-averaged streamwise velocity on the x −y plane at z hub . ( b ) and ( c ) Zoomed-in contours showing the turbine wakes inside the farm and in near wake of the farm, respectively.

Fig. 5. Streamwise variations of spanwise-averaged streamwise velocity. Here the black solid line labelled as ”farm wake averaged” corresponds to the streamwise velocity averaged from y/D = 52 to y/D = 97 D on the horizontal plane at z = z hub . For other profiles, the streamwise velocity is averaged in the spanwise direction over a distance of D with each row of turbines in the center.

Fig. 6. Spanwise profiles of the time-averaged steamwise velocity deficit at different streamwise locations. The vertical dashed line indicates ΔU/U hub = 0 . 05 for 95 percents of wake recovery.

The time-averaged streamwise velocity field on the horizontal plane located at the hub height are then examined in Fig. 4 . In the spanwise direction, the time-averaged flow field is heterogeneous because of the turbine wakes within the wind farm and the fairly complex wake of the wind farm. In general, the velocity in the wake recovers quickly to that close toUhub, e.g., the one located aty= 86.85Dand 91.85D. However, taking the two neighboring columns of the farm located aty= 51.85Dand 56.85Dfor example, coalescence of the two wakes behind columns appear and the generated new wake persist for a very long distance in the downstream. In addition, the recovery of the wake will be mitigated due to the inhomogeneous inflow field in the spanwise direction, in which the low speed streaks from the inflow reinforce turbine wakes in the wind farm. Streamwise variations of the spanwise-averaged streamwise velocity are then presented in Fig. 5 . Downstream variations because of turbine wakes occur for all the profiles. Figure 5 The figure indicate that the farm-wise averaged profile and wake-wise averaged profiles are different within the wind farm in its near wake. From 55Ddownstream of the last row of the wind farm, the wind farm averaged streamwise velocity is higher than 98% ofUhub, while those from the wakewise averaged streamwise velocity are much lower than 98% ofUhub.

At last, the spanwise profiles of the streamwise velocity deficit,which is computed by subtracting the time-averaged streamwise velocity at the inlet, are shown in Fig. 6 . The 10D,25D,40Dand 55Dare the positions of the 1–4 rows of the wind farm, respectively. The black solid line (ΔU= 0.05 ) in the figure indicates that the velocity in the wake recovers 95% of that at the inlet. As seen,at approximately 110Dfrom the inlet, i.e. 55Ddownwind of the last row of the wind farm, the velocity deficit in the wake decreases to 5% ofUhub. From 55Dto 75Ddownstream of the last row, the velocity deficit, on the other hand, does not decrease significantly. As moving to further downstream locations, the velocity deficit decreases to approximately 2.5% ofUhubat 165D(which corresponds to 16.5 km for this simulated case) downstream of the last row of the wind farm. In addition, periodic distributions of the streamwise velocity deficit are within the wind farm and the near wake of the farm propagates until 75Ddownstream of the last row of the wind farm. At further downstream locations, although the periodic distribution are not pronounced any more, the influences of wind turbine wakes exist even at 165Ddownstream of the last row of the wind farm.

In this work, we investigated the wake from a wind farm using large-eddy simulation with the rotor modelled using an actuator disk model. In the simulated wind farm, the streamwise and spanwise turbine spacings are 15Dand 5D, respectively, which are designed using the method presented in Ref. [26] . The simulated results show that the streamwise velocity recovers 95% of the undisturbed incoming wind speed at 55Ddownstream of the last row of the wind farm. The influences of wind turbine wakes persist for a very long distance, which are still observed even at 165Ddownstream of the last row of the wind farm. An important suggestion from this work is that the wake from the upstream wind farm should be considered in the optimal design and control of downstream wind farms, even their distance is as far as 55D, which corresponds to 5.5 km for the simulated case.

It is noted that we considered an idealized setup for only one turbine layout. Future work will be carried out for different layouts with different turbine spacings. Factors including the Coriolis force and other atmospheric factors (e.g., thermal stratification) will also be taken into account.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Nos. 11988102, 12172360) and Institute of Mechanics and Chinese Academy of Sciences.