# Aerodynamic Characteristics of a Square Cylinder with Vertical-Axis Wind Turbines at Corners
## 2 Numerical Simulation Configurations  
### Numerical Method  
RANS [16; 17; 18] treats the vortices of different scales equally in the flow field and smoothens the details of the temporal and spatial variations in pulsation motion through average operation. The turbulence model we used is the SST \(k\)-\(\omega\). The continuity equation and momentum equation for compressible flow are expressed as:  
\[\
abla\cdot\mathbf{U}=0 \tag{1}\]  
\[\frac{\partial\mathbf{U}}{\partial t}+\
abla\cdot(\mathbf{U}\mathbf{U})=-\frac{1}{\rho} \
abla P+\
abla\cdot(\mathbf{v}\
abla\mathbf{U})-\
abla\cdot\tau \tag{2}\]  
where \(\mathbf{U}\) and \(P\) are the averaged velocity and pressure, respectively; \(\rho\) shows air density and \(v\) refers to eddy viscosity; \(\tau\) is the Reynolds stress which can be expressed as \(\tau_{ij}=v_{t}(\partial u_{i}/\partial x_{j}+\partial u_{j}/\partial x_{i})- 2/3 pk\delta_{ij}\) and \(\delta_{ij}\) is the Kronecker delta. According to [18; 19], the control equations of SST \(k\)-\(\omega\) are expressed as follows:  
\[\frac{\partial\rho k}{\partial t}+u_{i}\frac{\partial\rho k}{\partial x_{i}} =\frac{\partial}{\partial x_{j}}\Bigg{[}\Gamma_{k}\frac{\partial k}{ \partial x_{j}}\Bigg{]}+G_{k}-Y_{k} \tag{3}\]  
\[\frac{\partial\rho\omega}{\partial t}+u_{i}\frac{\partial\rho k\omega}{ \partial x_{i}}=\frac{\partial}{\partial x_{j}}\Bigg{[}\Gamma_{\omega}\frac{ \partial\omega}{\partial x_{j}}\Bigg{]}+G_{\omega}-Y_{\omega}+D_{\omega} \tag{4}\]where \(\rho=1.225\) kg/m\({}^{3}\); \(k\) is turbulent kinetic energy; \(\omega\) is the dissipation rate of turbulent kinetic energy; \(u_{i}\) is the velocity component and subscripts \(i\), \(j=x\), \(y\), \(z\); \(G_{w}\) is the term of turbulent kinetic energy generation caused by the velocity gradient; \(\Gamma_{k}\) and \(\Gamma_{\omega}\) are the convection terms for \(k\) and \(\omega\), respectively; \(Y_{k}\) and \(Y_{w}\) are the effective diffusion terms of \(k\) and \(w\) caused by turbulence, respectively; and \(D_{\omega}\) is the cross-convection term (\(D_{\omega}=2(1-F1)\rho\sigma_{\omega 2}\frac{1}{\omega}\frac{ \partial k}{\partial x_{j}}\frac{\partial\omega}{\partial x_{j}}\)). The coefficient of eddy viscosity \(\
u_{t}\) can be obtained by:  
\[\
u_{t}=a_{1}\frac{k}{\max(a_{1}\omega,\Omega F_{2})} \tag{5}\]  
\[F_{2}=\tanh(\arg_{2}^{2}),\arg_{2}=\max(2\frac{\sqrt{k}}{0.09\omega y},\frac{ 500\
u}{y^{2}\omega}) \tag{6}\]  
where \(\Omega\) is the absolute value of vorticity, \(F_{2}\) is a function that is one for boundary-layer flows and zero for free shear layers, and \(a_{1}=0.31\) is the empirical coefficient. The transition from the \(k\)-\(\omega\) model near the wall to the \(k\)-\(\epsilon\) model far from the wall is controlled by mixed function \(F_{1}\):  
\[F_{1}=\tanh(\arg_{1}^{4}) \tag{7}\]  
\[\arg_{1}=\min(\max(2\frac{\sqrt{k}}{0.09\omega y},\frac{500\
u}{y^{2}\omega}); \frac{4\rho\sigma_{\omega 2}k}{CD_{kw}y^{2}}) \tag{8}\]  
\[CD_{kw}=\max(2\rho\sigma_{\omega 2}\frac{1}{\omega}\frac{\partial k}{\partial x _{j}}\frac{\partial\omega}{\partial x_{j}};10^{-20}) \tag{9}\]  
where \(y\) is the distance to the next surface, \(\
u_{t}\) is the eddy viscosity, and \(CD_{kw}\) is the positive portion of the cross-diffusion term. Model coefficient \(\sigma_{\omega 2}\) is 0.856. The open-source computational fluid dynamics package OpenFOAM is used for numerical simulations in this study. The finite volume method is adopted to solve the unsteady incompressible Navier-Stokes equation. The PIMPLE scheme is used to solve the pressure-velocity coupling. The second order implicit scheme is selected for time discretization. Gauss linear upwind scheme is used for spatial discretization. The time step is set as 1.0 \(\times\) 10\({}^{-6}\) s. In terms of time step selection, to assess how the size of the time steps may affect the results, three time steps of 2 \(\times\) 10\({}^{-5}\), 1.0 \(\times\) 10\({}^{-6}\), and 1.0 \(\times\) 10\({}^{-7}\) s are tested.