## 2 Foundation of CFD  
Computational fluid dynamics (CFD) is a tool used to model and simulate fluid and thermal behaviors numerically. CFD predicts continuum behaviors using different methods, such as the finite volume method (FVM), finite difference method (FDM), or smoothed particle hydrodynamics (SPH). Each method is based on a certain type of discretization. In our research, we use FVM, which divides the computational domain into finite volumes, which is known in practice as a mesh or grid. For these mesh elements, the CFD solver creates a discrete equation system based on the following transport equation:  
\[\frac{\partial}{\partial t}\int_{V}\rho\phi dV+\oint_{A}\rho\phi VdA-\int_{V}S _{\phi}dA-\oint_{A}\Gamma\
abla\phi dA=R \tag{1}\]  
In this equation, \(V\) denotes an arbitrary enclosed control volume, \(A\) denotes the surface of this control volume, \(\rho\phi\) is the conserved quantity (e.g., mass), \(F\) is the same quantity's flux over the \(A\) surface, \(S_{\phi}\) is the volumetric source of quantity, \(\rho\phi\) (potential energy) over volume \(V\), \(\Gamma\
abla\phi\) is the surface source of quantity, \(\rho\phi\) over surface \(A\), and \(R\) is the error of the equation (residual) [17].  
From Equation (1), the mass and the momentum equation can be derived. In a Cartesian coordinate system, these laws can be described using the following equations:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x_{i}}(\rho u_{i})=0 \tag{2}\]  
\[\frac{\partial\rho u_{i}}{\partial t}+\frac{\partial}{\partial x_{j}}\big{(} \rho u_{i}u_{j}\big{)}+\frac{\partial p}{\partial x_{i}}=\frac{\partial}{ \partial x_{j}}\Big{(}\tau_{ij}+\tau_{ij}^{R}\Big{)}+S_{i} \tag{3}\]  
In Equations (2) and (3), \(i\) and \(j\) indexes are the directions in the Cartesian coordinate system, while their values are \(i\) = 1, 2, 3 and \(j\) = 1, 2, 3, \(u\) is the fluid velocity, \(\rho\) is the fluid density, \(S_{i}\) is a mass-distributed external force per unit mass, which contains the  
Figure 1: Unconventional WTs: (**a**) dual-rotor wind turbine [14]; (**b**) wind turbine with guide baffle [15]; (**c**) Archimedes screw wind turbine [16].  
gravity (\(S_{i}^{gravity}\)), the coordinate system's rotation (\(S_{i}^{rotation}\)), and the other external forces (\(S_{i}=S_{i}^{rotation}+S_{i}^{rotation}+\ldots\)), \(\tau_{ij}\) is the viscous shear stress tensor, while \(t\) denotes time.  
Using the \(i\) and \(j\) subscripts, we can define velocities and special directions in a three-dimensional space, e.g., the velocity vector would equal as follows:  
\[\mathbf{u}=u_{x}\cdot\mathbf{e}_{x}+u_{y}\cdot\mathbf{e}_{y}+u_{z}\cdot\mathbf{ e}_{z} \tag{4}\]  
if we use the \(x\), \(y\), and \(z\) directions instead of the 1, 2, and 3.  
In a rotating reference frame, the external force from the rotation can be written as follows:  
\[S_{i}^{rotation}=-2\epsilon_{ijk}\Omega_{j}\rho u_{k}+\rho\Omega^{2}r_{i} \tag{5}\]  
where \(\epsilon_{ijk}\) is the Levy-Civita symbol, \(\Omega\) is the angular velocity, and \(r\) is the vector pointing to the axis of rotation.  
If heat transfer and/or compressibility is modeled, then conservation law is used in the modeling. However, compressibility and heat effect were not in the scope of our aims; therefore, we did not include the energy equation in our model. It must be mentioned that this study sets the focus on traditional, second-order schemes, which are available in commercial CFD software (Simcenter STAR CCM+, V2022.1), although there are higher-order schemes [18; 19] as well.