# A Separated-Flow Model for 2-D Viscous Flows around Bluff Bodies Using the Panel Method
## 2 Methods
### Boundary-Layer Prediction
\begin{cases}\quad 1.505+\frac{4}{R\epsilon_{\theta}}+\Big{(}0.165-\frac{1 _{6}}{R\epsilon_{\theta}1^{/2}}\Big{)}\frac{(H_{0}-H)^{1.6}}{H},\;H<H_{0}\\ 1.505+\frac{4}{R\epsilon_{\theta}}+\Big{(}\frac{0.04}{H}+\frac{0.007}{H-H_{0}+ 4/\ln R\epsilon_{\theta}}\Big{)}(H-H_{0})^{2},\;H>H_{0}\end{cases} , \tag{13}\]  
where \(H_{0}=\min\Big{(}3+\frac{400}{R\epsilon_{\theta}},4\Big{)}\). However, the Coles out-layer profile is not sufficient to obtain the remaining closure for \(C_{D}\) because this parameter also depends on the Reynolds stresses. Based on the characteristics of equilibrium turbulent flows, the dissipation coefficient \(C_{D}\) can be approximated by a sum of the wall layer and the wake layer contribution:  
\[C_{D}=\frac{C_{f}}{2}U_{s}+C_{\tau}(1-U_{s}), \tag{14}\]  
\[U_{s}=\frac{H^{*}}{2}+\left(1-\frac{H-1}{0.75H}\right), \tag{15}\]  
where \(U_{s}\) is the effective slip velocity and \(C_{\tau}\) is the wake-layer shear stress coefficient. This closure introduces a new variable, \(C_{\tau}\), and according to previous research [52; 58; 59], this parameter is governed by [PERSON]'s lag entrainment equation [58]:  
\[\frac{\delta}{C_{\tau}}\frac{dC_{\tau}}{d\xi}=5.6\Big{(}C_{\text{req}}^{1/2}- C_{\tau}^{1/2}\Big{)}+2\Big{(}\frac{\delta}{u_{e}}\frac{du_{e}}{d\xi} \Big{)}_{\text{eq}}-2\frac{\delta}{u_{e}}\frac{du_{e}}{d\xi}, \tag{16}\]  
\[C_{\text{req}}=0.015\frac{H^{*}}{1-U_{s}}\Big{(}\frac{H-1}{H}\Big{)}^{3}, \tag{17}\]  
\[\left(\frac{\delta}{u_{e}}\frac{du_{e}}{d\xi}\right)_{\text{eq}}=\frac{\delta /\theta}{0.75H}\Bigg{[}\frac{C_{f}}{2}-\frac{1}{44.89}\bigg{(}\frac{H-1}{H} \bigg{)}^{2}\Bigg{]}, \tag{18}\]  
\[\delta=\bigg{(}3.15+\frac{1.72}{H-1}\bigg{)}\theta+H\theta, \tag{19}\]  
where \(\delta\) is the boundary-layer thickness, and the subscript 'eq' refers to the equilibrium state. The envelope method is used in the present work [52; 56; 57] for the boundary-layer transition, which assumes that the transition occurs when the most unstable Tollmien-Schlichting wave in the boundary layer has grown by some factor, usually taken to be \(\epsilon^{9}\). The logarithm of the maximum amplification rate \(\overline{n}\) with respect to the streamwise coordinate \(\xi\) is expressed as a function of \(H\) and \(\theta\):  
\[\theta\frac{d\overline{n}}{d\xi}= \begin{cases}\quad 0,\;R\epsilon_{\theta}<R \epsilon_{\theta 0}\\ =\frac{1}{\overline{\theta}}\frac{d\overline{n}}{dKe_{\theta}}\Big{(}\frac{dKe_ {\theta}}{d\xi}\Big{)},\;R\epsilon_{\theta}>R\epsilon_{\theta 0}\end{cases} , \tag{20}\]  
\[\frac{d\overline{n}}{dRe_{\theta}}=0.028(H-1)-0.0345\exp\Bigg{[}-\bigg{(} \frac{3.87}{H-1}-2.52\bigg{)}^{2}\Bigg{]}, \tag{21}\]  
\[\theta\frac{dRe_{\theta}}{d\xi}=-0.05+\frac{2.7}{H-1}-\frac{5.5}{(H-1)^{2}}+ \frac{3}{(H-1)^{3}}+0.1\exp\bigg{(}\frac{-20}{H-1}\bigg{)}, \tag{22}\]  
\[\log_{10}R\epsilon_{\theta 0}=\frac{2.492}{(H-1)^{0.43}}+0.7\bigg{[}tanh \bigg{(}\frac{14}{H-1}-9.24\bigg{)}+1\bigg{]}, \tag{23}\]  
The entire boundary-layer region is divided into a laminar interval and a turbulent interval, separated by a transition location \(\xi_{tr}\). The momentum equation and the shape parameter equation are solved for the entire boundary layer. The amplification equation is solved for the laminar interval, while the lag equation is solved for the turbulent interval. The transition onset location \(\xi_{tr}\) does not have to be on one of the edge points--it can be solved implicitly by the interpolation of its neighbour. Weighted averages of the laminar and turbulent subintervals are the actual governing equations of the transition interval. To define the development of the turbulence based on the onset location, [PERSON] and [PERSON] [52] suggested using 0.7 times its equilibrium value as the initial value of \(\sqrt{C_{\tau}}\) at \(\xi_{tr}\). In this work, the boundary layer is calculated in MATLAB and the the _fsolve_ function is used to solve the system of nonlinear equations.