# The Aerodynamic Analysis of a Rotating Wind Turbine by Viscous-Coupled 3D Panel Method
## 2 Numerical Formulation
### The Separated Wake
#### 2.3.2 The Thick Wake Method (Current Method)  
Much like the above \"double wake\" method, the proposed \"thick wake\" method treats the separated wake as a region of constant total pressure, \"such that the pressure along the section of the airfoil's surface between the separation point and the trailing edge, henceforth referred to as the separation region, equals that along the streamlines bounding the wake, henceforth referred to as separation streamlines\" [6].  
We now recall the \"displacement body\" model, which is a well-established method of computing the viscous flow field around a body by offsetting the inviscid surface streamline. The current method extends this \"displacement body\" approach to include the separated wake, by \"treating the separation streamlines bounding the separated wake as an extension of the displacement body streamline\" [6]. The configuration of these separation streamlines is iteratively adjusted until the desired constant pressure distribution is achieved. By treating the separated wake as \"an extension of the displacement body\", the current method is able to model the \"thickness\" of the separated wake as per that of the body, namely, by employing source distributions, thereby forgoing the extraneous calculations required by the double wake method for the \"off-body induced velocity field\" and the \"pressure jump in the separated wake\".  
The separation points are determined via the 2D boundary layer analysis of Section 2.2, applied in the axial and tangential directions. The thick wake method then adopts the initial wake geometry illustrated in Figure 2, transformed to cylindrical coordinates, such that the undisturbed relative flow velocity \(U_{\text{rel}}\) is equal to the vector sum of the free stream velocity \(U_{\infty}\) and the rotor's local rotational velocity \(-\omega r_{i}\) (Figure 3), and the wake length \(WL\) is a helical arc aligned with the relative flow. The wake shape, which is constrained in the \(\xi\)- (parallel with \(U_{\text{rel}}\)) and \(\eta\)- (spanwise) directions, is then adjusted according to the local pressure values, Equation (6), by moving the \(i^{\text{th}}\) wake panel vertex in the \(\zeta\)-direction (perpendicular to \(U_{\text{rel}}\)) from position \(\zeta_{i}^{n}\) to \(\zeta_{i}^{n+1}\), as calculated by the \(n^{\text{th}}\) iteration of the following Quasi-Newton method:  
\[\zeta_{i}^{n+1}=\zeta_{i}^{n}-\frac{\zeta_{i}^{n}-\zeta_{i}^{n-1}}{\Delta C_{ p,i}^{n}-\Delta C_{p,i}^{n-1}}\Delta C_{p,i}^{n} \tag{21}\]  
where \(\Delta C_{p}\) is the difference between the pressure coefficients at the separation point and the \(i^{\text{th}}\) wake panel. Convergence is achieved when all predicted pressure values in the separated wake are within 5% of the pressure at the separation point, which is typically very close to zero, such that a 5% tolerance along the separated wake corresponds to tolerances from as low as 0.1% to no more than 1% in terms of peak-to-trough pressure differentials.  
Finally, by treating the separated wake as an extension of the blade surface, the trailing edge of the \"body\" is now located at the intersection of the thick and thin wakes. (This separated wake \"trailing  
Figure 3: Illustration of airfoil and separated wake in cylindrical coordinate system.  
edge\" is visible in Figure 4, which also shows the separated \"thick wakes\", as predicted by the current method, and helical thin wakes, aligned with the local flow, trailing behind several airfoil sections of one of the wind turbine's rotor blades.) Manipulating the pressure around the \"trailing edge\" allows us to impose the Morino Kutta condition without having to implement an iterative pressure correction, as is common practice where strong 3D effects exist.