# A Separated-Flow Model for 2-D Viscous Flows around Bluff Bodies Using the Panel Method
## 2 Methods
### Potential Flow Calculation  
The potential flow around the body is computed using the 2-D constant source panel method, as described in detail in [PERSON] and Plotkin [32], [PERSON] [46], and [PERSON] [50]. The incompressible flow field around an arbitrary shape is governed by mass and momentum conservation equations. With the inviscid irrotational potential flow assumption, the mass conservation equation is simplified to the Laplace equation in terms of the velocity potential \(\Phi\) and governs the potential flow independently:  
\[\
abla^{2}\Phi=0, \tag{1}\]  
The pressure field is calculated using the momentum equation (Bernoulli equation). Thus, the problem is to solve [PERSON]'s equation for one unknown scalar, which is a linear problem. For the exterior flow case, the flow should be bounded by the solid surface of the arbitrary body and no penetration is allowed through the body surface. In the present work, the Neumann boundary condition, which is imposed on the Laplace equation and specifies the values of the derivative applied at the boundary of the domain, is selected to specify a zero normal velocity component on the boundary surface directly:  
\[\
abla\Phi\cdot\mathbf{n}=0, \tag{2}\]  
where \(\mathbf{n}\) is the unit normal vector in the out-of-plane direction. The configuration surface is approximated by n straight-line panels, and each panel has a control point, normally taken as the midpoint, where the boundary conditions are applied. The perturbation velocity fields, u and v, at point P (_x_y_) included by a panel with coordinates (_x_1,\(y\)1) and (_x_2,\(y\)2) and constant source strength \(\sigma\) is calculated as follows:  
\[u=\frac{\sigma}{4\pi}\ln\frac{\left(x-x_{1}\right)^{2}+\left(y-y_{1}\right)^{ 2}}{\left(x-x_{2}\right)^{2}+\left(y-y_{2}\right)^{2}}, \tag{3}\]  
\[v=\ \frac{\sigma}{2\pi}\bigg{(}\tan^{-1}\frac{y-y_{2}}{x-x_{2}}-\tan^{-1} \frac{y-y_{1}}{x-x_{1}}\bigg{)}, \tag{4}\]  
The boundary condition is satisfied if the normal velocities of all of the control points on the panels are equal to zero. Thus, a system of linear equations is constructed and the constant source strength of each panel is obtained by solving the system:  
\[\begin{array}{c}\left[A_{ij}\right]\big{\{}\sigma_{j}\big{\}}=\{\text{RHS}_ {i}\},\\ A_{ij}=\left(u,v\right)_{ij}\cdot\mathbf{n}_{i},\\ \text{RHS}_{i}=-\left(U_{\infty},V_{\infty}\right)\cdot\mathbf{n}_{i},\end{array} \tag{5}\]  
where \(A_{ij}\) is the matrix of aerodynamic influence coefficients referring to the normal velocity at the \(i\)-th control point induced by the \(j\)-th panel, \(\sigma_{j}\) is the constant source strength of the \(j\)-th panel, \(\text{RHS}_{i}\) is the right-hand side term, which is equal to the normal velocity due to the freestream velocity (\(U_{\infty}\),\(V_{\infty}\)) at the \(i\)-th control point, and \(\mathbf{n}_{i}\) is the normal vector of the \(i\)-th panel.  
The matrix is solved here by the Gaussian elimination method and the velocity and pressure at any point in the flow field can be calculated using the calculated constant source strength. According to the Bernoulli equation, the pressure distribution is computed using the total tangential velocity at each control point by Equation (6).  
\[C_{r_{i}}=\frac{p-p_{\infty}}{\frac{1}{2}\rho Q_{\infty}{}^{2}}=\frac{-\frac{ 1}{2}\rho\big{(}Q_{\infty}{}^{2}-Q_{\infty}{}^{2}\big{)}}{\frac{1}{2}\rho Q_{ \infty}{}^{2}}=1-\frac{Q_{\infty}{}^{2}}{Q_{\infty}{}^{2}}\ \,\ Q_{i}=\left(\sum_{j=1}^{N}(u,v)_{ij}+(U_{\infty},V_{ \infty})\right)\cdot\mathbf{t}_{i}, \tag{6}\]  
where \(C_{p_{i}}\) is the pressure coefficient of the \(i\)-th control point, \(Q_{t_{i}}\) is the total tangential velocity magnitude at the \(i\)-th control point, which is 0 at the stagnation point, and \(C_{p_{i}}=1\). \(Q_{\infty}\) is the freestream velocity magnitude and \(\mathbf{t}_{i}\) is the unit tangential vector of the \(i\)-th panel.