# Performance Characteristics of an Orthopter-Type Vertical Axis Wind Turbine in Shear Flows
## 3 Numerical Approach
### Governing Equations and Discretization Method  
The flow field around the wind turbine was assumed to be incompressible and isothermal. The DDES turbulence mode treats near-wall region in a manner like a Reynolds-averaged Navier-Stokes (RANS) turbulence model and treats the rest of the flow field in a manner like a large-eddy simulation (LES) turbulence model [34]. This model has the potential to achieve higher accuracy than RANS models and save a large number of computing resources compared with pure LES models. The governing equations for the CFD simulation with the DDES turbulence model based on the Spalart-Allmaras (SA) model are the continuity equation:\[\frac{\partial u_{i}}{\partial x_{i}}=0, \tag{9}\]  
the Navier-Stokes equation:  
\[\frac{\partial u_{i}}{\partial t}+\frac{\partial u_{i}u_{j}}{\partial x_{j}}=- \frac{1}{\rho}\frac{\partial p}{\partial x_{i}}+\
u\frac{\partial}{\partial x_{ j}}\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{ \partial x_{i}}-\frac{2}{3}\delta_{ij}\frac{\partial u_{i}}{\partial x_{j}} \right)-\frac{\partial}{\partial x_{j}}\left[\vec{i}\left(\frac{\partial u_{i} }{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right)-\frac{2}{3}k \delta_{ij}\right] \tag{10}\]  
and the transport equation for the kinematic eddy viscosity \(\widetilde{\
u}\):  
\[\frac{\partial\widetilde{\
u}}{\partial t}+\frac{\partial\widetilde{\
u}u_{i }}{\partial x_{i}}=C_{b1}\vec{i}\left(S+\frac{\widetilde{\
u}}{\kappa^{2}d_{ DDES}2}\left(1-\frac{\chi}{1+xf_{i1}}\right)\right)+\frac{1}{\widetilde{\
u}} \left[\frac{\partial}{\partial x_{j}}\left(\
u+\widetilde{\
u}\right)\frac{ \partial\widetilde{\
u}}{\partial x_{j}}\right]+C_{b2}\left(\frac{\partial \widetilde{\
u}}{\partial x_{j}}\right)^{2}\right]-C_{w1}f_{w}\left(\frac{ \widetilde{\
u}}{d_{DDES}}\right)^{2} \tag{11}\]  
where \(u_{i}\) is the wind-velocity component in the \(x_{i}\) direction; \(p\) is the pressure; \(\
u\) is the kinematic viscosity; \(t\) is the time; \(\rho\) is the air density; \(k\) is the turbulence kinetic energy; \(\delta_{ij}\) is the Kronecker delta; \(d_{DDES}\) is the DDES length scale; \(\chi\) is \(\widetilde{(\
u}/\
u)\); \(S\) is a scalar measure of the deformation tensor; \(f_{\
u 1}\) and \(f_{w}\) are damping functions; and \(C_{b1}\), \(C_{b2}\), \(C_{w1}\), \(\sigma_{\widetilde{\
u}}\) and \(\kappa\) are constants.  
The DDES length scale is computed by:  
\[d_{DDES}=d-f_{d}max\ (0,d-c_{des}\Delta_{max}) \tag{12}\]  
where \(d\) is the distance to the closest wall; \(c_{des}\) is the empirical constant; and \(\Delta_{max}\) is the maximum edge length of the local computational cell, i.e., \(\Delta_{max}=max\ (\Delta_{x},\ \Delta_{y},\ \Delta_{z})\). The switching between the RANS and the LES mode depends on the following shielding function;  
\[f_{d}=1-tanh\left((8\pi_{d})^{3}\right) \tag{13}\]  
and  
\[r_{d}=\frac{\widetilde{\
u}}{\sqrt{u_{i,j}u_{i,j}\kappa^{2}d^{2}}} \tag{14}\]  
where \(u_{i,j}\) is the velocity gradient. The damping functions and closure coefficients are as follows:  
\[f_{\
u 1}=\frac{\chi^{3}}{c_{\
u 1}^{3}+\chi^{3}},\ f_{w}=g\left[\frac{1+c_{ w3}^{6}}{g^{6}+c_{w3}^{6}}\right]^{1/6},\ g=r+c_{w2}(r^{6}-r),\ c_{w1}=\frac{c_{b1}}{\kappa^{2}}+\frac{(1+c_{ b2})}{\sigma_{\widetilde{\
u}}}, \tag{15}\]  
\[c_{b1}=0.1355,\ c_{b2}=0.622,\ c_{\
u 1}=7.1,\ c_{w2}=0.3,\ c_{w3}=2.0,\ \sigma_{ \widetilde{\
u}}=\frac{2}{3},\ c_{des}=0.65,\ \kappa=0.4187. \tag{16}\]  
The governing equations are discretized by the finite-volume method. The advection terms of the Navier-Stokes equations are discretized by the bounded central-difference scheme. The advection term of the transportation equations for \(\widetilde{\
u}\) is discretized by a second-order upwind scheme. Other spatial derivatives are discretized by the central-difference scheme. The time integration is performed using the second-order implicit method.