2.1. Simulation
The numerical experiments are conducted using the open-source wind farm modeling tool “Simulator for Wind Farm Applications” (SOWFA) [
28] developed by the National Renewable Energy Laboratory (NREL), together with a transient solver for incompressible turbulent flow provided in the OpenFOAM (v2206) CFD toolbox [
29]. More specifically, the standard k-ε model is employed for the turbulence modeling in unsteady Reynolds-averaged simulations (URANS), and the actuator-line model is used to model the rotating wind turbines and their wake flows. Thus, the governing equations of the fluid flow can be expressed as
where
ρ is the fluid velocity,
u and
p are the Reynolds-averaged velocity and pressure, respectively;
f is the body force resulting from the rotating wind turbine blade, and
k is the kinematic energy. The notation
μeff stands for an effective viscosity coefficient defined as the sum of the molecular viscosity coefficient (denoted by
μ) and the turbulent viscosity (denoted by
μt). In the standard k-ε model,
μt is computed with
where
Cμ is a constant (usually taken as 0.09), and
ε is the energy dissipation rate. The kinematic energy
k and the dissipation rate
ε are obtained by solving the equations of
with
where
Pk is the production of
k induced by the rate-of-strain tensor of
u;
Cε1,
Cε2,
σk and
σε are the constants (respectively taken as 1.44, 1.92, 1.0 and 1.3 in this study) of the standard k-ε model. The body force
f on the right-hand side of (1-1) is given by
where
x is the three-dimensional coordinate,
F is the actuator line element force and
g is a projection function. In this study, we employ an isotropic Gaussian projection function given by
where Δ is the Gaussian projection width and
x0 is the location at which the Gaussian projection is being applied.
In the area, two types of wind turbines are considered in the numerical experiments. The larger one is the NREL 5-MW baseline turbine, which has a rotor diameter of 126 m and a rated wind speed of 11.4 m/s. The smaller one is also a hypothetical wind turbine, which is obtained by scaling the properties of the NREL 5-MW baseline turbine in terms of a size ratio of 1:3 accordingly (yielding a rotor diameter of 42 m). The hub height of the two different turbines is assumed to be 150 m and 37 m, respectively. Two large wind turbines and three small wind turbines are arranged vertically and crosswise, as shown in
Figure 1. By changing the position of the small wind turbines, the change in the power generation of the large wind turbines downstream and the vertical velocity distribution in the flow field are analyzed. A single transverse row of three small wind turbines is arranged within the 9
D longitudinal separation between the two large wind turbines. The center small wind turbine is located on the centerline of the upstream large wind turbine, and downstream separation of the row is varied (see
Table 1). The simulated working conditions include: case 0 is the working condition when there is no small wind turbine running. The first working condition is that the small wind turbine is arranged 2
D away from the upstream large wind turbine. From the second working condition to the fourth working condition, 5
d is added one by one on the basis of the previous working condition.
The inflow of the numerical experiments is assumed to be a two-dimensional neutral equilibrium atmospheric boundary layer and employs log-law-type ground-normal inlet boundary conditions [
30,
31] with a surface roughness length (denoted by
z0) of 0.1 m. The wind velocity magnitude, kinematic energy and dissipation rate on the inflow are respectively given by
where
z is the ground-normal coordinate,
u* is the friction velocity and
κ is the von Kármán constant. All numerical experiments are carried out at the same velocity, which has a magnitude of 11.2 m/s (slightly lower than the rated wind speed of 11.4 m/s) at the hub height of the NREL 5-MW baseline turbine.
The computation domain of the numerical experiments is a rectangular of the dimension 2400 m × 960 m × 320 m in the stream-wise, cross-stream-wise and elevational directions, respectively. The employed boundary conditions corresponding to different flow variable fields are listed in
Table 2. We employ both the blockMesh and the refineMesh utilities in OpenFOAM v2206 for the mesh generation of the computational domain. More specifically, the blockMesh utility is first used to generate a hexahedral background mesh, and the refineMesh utility is then applied to selected regions to refine the mesh cells adjacent to the rotating wind turbines. The background mesh employs uniform grid spacing in the streamwise and cross-streamwise directions. In the elevational direction, the grid spacing has an exponential distribution for the z-coordinate ranging from 0 m to 20 m or 240 m to 360 m, while there is a uniform distribution for the z-coordinate ranging from 20 m to 240 m.
Figure 2 plots the three selected refinement regions inside the computational domain, in which the cell size within the 3rd level refinement region is one eighth of the background cell size in the streamwise and cross-streamwise directions.
A mesh-independence analysis is then conducted to determine to the appropriate grid spacing of the mesh. The analysis is carried out by simulating two NREL 5-MW baseline turbine (with a distance of 1134 m in the streamwise direction) on four different mesh respectively and comparing their power output afterwards. The four considered mesh employ the same mesh refinement scheme (see
Figure 2) but different grid spacing for their initial background mesh. By doing so, the grid spacing adjacent to the wind turbines in these mesh are of sizes 1.5 m, 2.0 m, 2.5 m and 3.0 m, respectively.
Table 3 summarizes the time-averaged power output (over 150 s) of the two wind turbines on these mesh. The convergence of the power output results with the decrease of grid spacing can be clearly noted from the table.
After the mesh-independence analysis, we select the mesh of case 2 (in
Table 2) for later numerical experiments. All experiments are simulated for 20,000 time-steps with a time-step size of 0.025 s, yielding a time length of 500 s in total. At the employed time-step size, the CFL number in the computational domain has a mean value around 0.05 and a maximum value around 0.15 in all experiments. Finally, the desired statistics concerning the considered wind turbines (such as the generator power of wind turbines and the mean velocity of the wake flow) are averaged over the last 6000 time-steps.