# A Separated-Flow Model for 2-D Viscous Flows around Bluff Bodies Using the Panel Method
## 2 Methods
### Boundary-Layer Prediction  
The separated flow can be calculated with the separated-flow model [46; 47; 48; 50]. The separated shear layers of the bluff body are represented by two vortex panels for which strength is calculated using the panel method. However, the separation positions should be prescribed according to experimental results or high-fidelity numerical simulations. Both measures are costly and time consuming. Thus, the separation positions are normally considered as unknowns for design problems that need to be economic both in terms of expense and time. To tackle this, the boundary-layer integral methods are used as an alternative approach in this work, which are relatively simple and fast in predicting the boundary-layer characteristics [51; 52; 53; 54; 55]. To represent flows with limited separation regions accurately, viscous equations should be adopted in the present work. Compatibility between laminar and turbulent formulations is necessary to represent the transition in a well-ordered and analytically continuous manner. The fewest additional variables related to viscosity should be added to the global Newton solution procedure for solving the boundary-layer parameters to ensure the computational efficiency. The two-equation integral formulation [51; 52; 55] based on dissipation closure is suitable for developing the numerical model. The transition is predicted by spatial amplification theory [56; 57] together with an extra lag equation [58; 59]. Compared to one-equation methods, such as [PERSON] method [54], two-equation methods possess the capability of describing thin separated regions and are suitable for the development of the proposed model. The formulations to represent the incompressible boundary layer in this work employ standard integral momentum and kinetic energy shape parameter equations, based on the formulations presented in [PERSON] and Giles [52] with slight modifications for incompressible flow applications [60]:  
\[\frac{d\theta}{d\xi}+(2+H)\frac{\theta}{u_{e}}\frac{du_{e}}{d\xi}=\frac{C_{f} }{2}, \tag{7}\]  
\[\theta\frac{dH^{*}}{d\xi}+H^{*}(1-H)\frac{\theta}{u_{e}}\frac{du_{e}}{d\xi}=2 C_{D}-H^{*}\frac{C_{f}}{2}, \tag{8}\]  
where \(\xi\) is the streamwise coordinate, and \(u_{e}\) is the streamwise potential flow velocity at the edge of the boundary layer. The remaining five parameters, \(\theta\), \(H\), \(H^{\prime}\), \(C_{f}\), and \(C_{D}\), in the two equations are unknowns, which are the boundary-layer momentum thickness, the shape parameter, the kinetic energy shape parameter, the skin friction coefficient, and the dissipation coefficient, respectively. Thus, additional functional dependencies must be assumed to close the integral boundary-layer equations. For the incompressible laminar flow, the relationships are approximated based on the Falkner-Skan one-parameter profile family and the expressions of \(H^{*}\), \(C_{f}\), and \(C_{D}\) in terms of \(\theta\), \(H\), and momentum thickness Reynolds number \(Re_{\theta}\) are presented as follows:  
\[H^{*}= \begin{cases}1.515+0.076\frac{(H-4)^{2}}{H},\ H<4,\\ 1.515+0.040\frac{(H-4)^{2}}{H},\ H>4\end{cases} , \tag{9}\]  
\[Re_{\theta}\frac{C_{f}}{2}= \begin{cases}-0.066+0.066\frac{(6.2-H)^{15}}{H-1}, \ H<6.2,\\ -0.066+0.066\frac{(H-6)^{2}}{(H-4)^{2}},\ H>6.2\end{cases} , \tag{10}\]  
\[Re_{\theta}\frac{2C_{D}}{H^{*}}= \begin{cases}0.207+0.00205(4-H)^{5.5},\ H<4\\ 0.207-0.100\frac{(H-4)^{2}}{H^{2}},\ H>4\end{cases} , \tag{11}\]  
For the incompressible turbulent flow, closure functions for \(C_{f}\) and \(H^{*}\) are obtained from the Coles out-layer profile. The Coles skin friction formula is used as the closure correlation for \(C_{f}\):  
\[C_{f}=\frac{0.3e^{-1.33H}}{\left(\log_{10}Re_{\theta}\right)^{1.74+0.31H}}, \tag{12}\]and \(H^{*}\) is approximated by the following formula:  
\[H^{*}=