# The Aerodynamic Analysis of a Rotating Wind Turbine by Viscous-Coupled 3D Panel Method
## 2 Numerical Formulation
### The Boundary Layer  
Within the boundary layer, an order of magnitude analysis may be applied to the 2D Navier-Stokes equations to obtain Equations (7) and (8). These two equations, together with the 2D continuity equation, Equation (9), comprise the Prandtl boundary layer equations.  
\[\rho\left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right)= -\frac{\partial p}{\partial x}+\mu\frac{\partial^{2}u}{\partial y^{2}} \tag{7}\]  
\[\frac{\partial p}{\partial y}=0 \tag{8}\]  
\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0 \tag{9}\]  
where \(x\) is measured parallel to the body surface streamwise, and \(y\) normal to it.  
A few extra parameters which are integral to the description of the boundary layer are the boundary layer displacement thickness \(\delta^{*}\), Equation (10), and momentum thickness \(\theta\), Equation (11), defined as the distances by which the inviscid surface streamline must be displaced to compensate for, respectively, the reductions in flow rate and total momentum in the boundary layer. The shear stress and skin friction coefficient are defined by Equations (12) and (13),  
\[\delta^{*}\equiv\int_{0}^{\infty}\biggl{(}1-\frac{u}{U_{e}}\biggr{)}dy \tag{10}\]  
\[\theta\equiv\int_{0}^{\infty}\frac{u}{U_{e}}\biggl{(}1-\frac{u}{U_{e}}\biggr{)}dy \tag{11}\]  
\[\tau=\mu\frac{\partial u}{\partial y} \tag{12}\]  
\[c_{f}\equiv\frac{\tau_{w}}{\frac{1}{2}\rho U_{e}^{2}} \tag{13}\]  
where \(U_{e}\) is the flow speed outside the boundary layer, predicted by the inviscid analysis, and \(\tau_{w}\) is the shear stress at the body surface.  
Substituting Equations (10)-(13) into Equations (7)-(9) gives the von Karman momentum integral equation:  
\[\frac{d\theta}{dx}+\frac{\theta}{U_{e}}(2+H)\frac{dU_{e}}{dx}=\frac{1}{2}c_{f} \tag{14}\]  
where \(H=\frac{\delta^{*}}{\theta}\) is a shape factor. The separation point may be determined by supplementing Equation (14) with additional relations. For the computation of the laminar and turbulent separation points, the present study adopts [PERSON]'s [24] and [PERSON]'s methods [25], respectively.  
By rearranging the von Karman momentum integral equation, Equation (14), [PERSON] showed that \(\theta^{2}\) is very well approximated by:  
\[\theta^{2}=0.45\frac{\
u}{U_{e}^{6}}\int_{0}^{x}U_{e}^{5}dx \tag{15}\]  
He then introduced a new parameter \(\lambda\equiv\frac{\theta^{2}}{v}\frac{dU_{e}}{dx}\), which may be calculated via (15), and went on to show that laminar-to-turbulent flow transition occurs at \(\lambda<-0.09\).  
Concerning the turbulent boundary layer, [PERSON] considered the entrainment of fluid from the free stream into the turbulent boundary layer, where the entrainment velocity \(E\) may be written in terms of the displacement thickness, Equation (16), via a turbulent shape factor, \(H_{1}=\frac{\delta^{*}-\delta}{\theta}\):\[E=\frac{d}{dx}(U_{e}\theta H_{1}) \tag{16}\]  
The dimensionless entrainment velocity \(E/U_{e}\) may then be calculated via the following empirical formulae, as fit by [PERSON] and [PERSON] [26]:  
\[\frac{1}{U_{e}}\frac{d}{dx}(U_{e}\theta H_{1})=0.0306(H_{1}-3)^{-0.617} \tag{17}\]  
\[\text{where }H_{1}=3.3+0.8234(H-1.1)^{-1.287}\text{ for }H\leq 1.6 \tag{18}\]  
\[\text{and }H_{1}=3.3+1.5501(H-0.6778)^{-3.064}\text{ for }H>1.6\]  
To solve for \(c_{f}\), [PERSON] adopted the Ludwig-Tillman skin-friction law:  
\[c_{f}=0.246\times 10^{-0.678H}Re_{\phi}{}^{-0.268} \tag{19}\]  
\[Re_{\phi}=\frac{U_{e}\theta}{\
u} \tag{20}\]  
and ordinary differential Equations (15) and (17) are solved for \(\theta\) and \(H_{1}\) via a 2 nd order Runge-Kutta method. Turbulent separation is assumed to occur when \(H_{1}\) falls below 3.3 [27].