## 2 Project Overview  
### 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-\(\epsilon\) 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  
\[\frac{\partial\rho\mathbf{u}}{\partial t}+\
abla\cdot(\rho\mathbf{uu})=\
abla\cdot\left[ \mu_{\text{eff}}(\
abla\mathbf{u}+(\
abla\mathbf{u})^{\text{T}})-\frac{2}{3}\mu_{\text {eff}}(\
abla\cdot\mathbf{u})\mathbf{I}\right]-\
abla(p+\frac{2}{3}k)-\mathbf{f} \tag{1}\]  
\[\
abla\cdot(\rho\mathbf{u})=0 \tag{2}\]  
where \(\rho\) is the fluid velocity, \(\mathbf{u}\) and \(p\) are the Reynolds-averaged velocity and pressure, respectively; \(\mathbf{f}\) is the body force resulting from the rotating wind turbine blade, and \(k\) is the kinematic energy. The notation \(\mu_{\text{eff}}\) stands for an effective viscosity coefficient defined as the sum of the molecular viscosity coefficient (denoted by \(\mu\)) and the turbulent viscosity (denoted by \(\mu_{t}\)). In the standard k-\(\epsilon\) model, \(\mu_{t}\) is computed with  
\[\mu_{t}=\rho C_{\mu}\frac{k^{2}}{\epsilon} \tag{3}\]  
where \(C_{\mu}\) is a constant (usually taken as 0.09), and \(\epsilon\) is the energy dissipation rate. The kinematic energy \(k\) and the dissipation rate \(\epsilon\) are obtained by solving the equations of  
\[\frac{\partial\rho k}{\partial t}+\
abla\cdot(\rho\mathbf{u}k)=\
abla\cdot(\mu_{ \text{eff},k}\
abla k)+P_{k}-\rho\epsilon \tag{4}\]  
\[\frac{\partial\rho\epsilon}{\partial t}+\
abla\cdot(\rho\mathbf{u}\epsilon)= \
abla\cdot(\mu_{\text{eff},\epsilon}\
abla\epsilon)+C_{\epsilon 1}\frac{ \epsilon}{k}P_{k}-C_{\epsilon 2}\frac{\epsilon^{2}}{k}\rho \tag{5}\]  
with  
\[\mu_{\text{eff},k}=\mu+\frac{\mu_{t}}{\sigma_{k}},\quad\mu_{\text{eff}, \epsilon}=\mu+\frac{\mu_{t}}{\sigma_{\epsilon}}\]  
where \(P_{k}\) is the production of \(k\) induced by the rate-of-strain tensor of \(\mathbf{u}\); \(C_{\epsilon 1}\), \(C_{\epsilon 2}\), \(\sigma_{k}\) and \(\sigma_{\epsilon}\) are the constants (respectively taken as 1.44, 1.92, 1.0 and 1.3 in this study) of the standard k-\(\epsilon\) model. The body force \(\mathbf{f}\) on the right-hand side of (1-1) is given by  
\[\mathbf{f}(\mathbf{x})=g(\mathbf{x})\mathbf{F} \tag{6}\]  
where \(\mathbf{x}\) is the three-dimensional coordinate, \(\mathbf{F}\) is the actuator line element force and \(g\) is a projection function. In this study, we employ an isotropic Gaussian projection function given by  
\[g(\mathbf{x})=\frac{1}{\Delta^{3}\pi^{3/2}}\exp\left(-\frac{\|\mathbf{x}-\mathbf{x}_{0}\|_ {2}}{\Delta^{2}}\right) \tag{7}\]  
where \(\Delta\) is the Gaussian projection width and \(\mathbf{x}_{0}\) 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.