# A Separated-Flow Model for 2-D Viscous Flows around Bluff Bodies Using the Panel Method
## 2 Methods
### Separated-Flow Model  
The flow field around a body with separation can be identified as four regions: the potential flow region, the boundary layer, the free shear layer, and the wake. In the panel method, a simplified theoretical model of the real flow is shown in Figure 2: the body is represented by a number of constant source panels and the free shear layers emerging from the separation points of the body are represented by constant vortex panels. The wake panels are attached to the body at the body panel edges nearest to the separation point. One thing worth noting is that the time-varying wake in real flow is represented by two steady vortex sheets, so the flow characteristics calculated are thus time averaged values. There are normally two vortex sheets forming the wake behind a 2-D bluff body. In the steady model, we should have the same loss of total head across the two vortex sheets. Therefore, the vorticity values of the two vortex sheets have to be equal in magnitude and opposite in sign. This separation model is the same with the model in [PERSON]'s work [50]. In addition to the constant source panels on the body surface, a constant vortex distribution  
Figure 2: Sketch of boundary discretisation and separated-flow model. \(\gamma\) is assumed on the wake panels. The perturbation velocity induced by the constant vortex panel is different from Equations (3) and (4):  
\[u=\frac{\gamma}{2\pi}\bigg{(}\tan^{-1}\frac{y-y_{2}}{x-x_{2}}-\tan^{-1}\frac{y-y _{1}}{x-x_{1}}\bigg{)}, \tag{26}\]  
\[v=\frac{\gamma}{4\pi}\ln\frac{\left(x-x_{2}\right)^{2}+\left(y-y_{2}\right)^{2 }}{\left(x-x_{1}\right)^{2}+\left(y-y_{1}\right)^{2}}, \tag{27}\]  
Equation (25) is modified as:  
\[\big{[}A_{ij}\big{]}\big{\{}\sigma_{j}\big{\}}+\{B_{ik}\}\{\gamma_{k}\}=\{\text {RHS}_{i}-q_{i}\}, \tag{28}\]  
where \(i\) is the index number of control points including the ones on the wake panel, \(j\) is the index number of panels on the body surface, and \(k\) is the index number of wake panels, which are up to two in the case of flow around bluff bodies. \(A_{ij}\) 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 \(j\)-th panel, and \(B_{ik}\) is the matrix of the normal velocity at the \(i\)-th control point induced by the \(k\)-th wake panel with constant vorticity strength \(\gamma_{k}\). In the separated-flow region, the pressure coefficient in the base flow region is practically constant. This is taken as the base pressure coefficient. The base pressure coefficients within the separated-flow region are calculated as:  
\[C_{pb}=1-\bigg{(}\frac{Q_{t_{i}}}{Q_{\infty}}\bigg{)}^{2}-\bigg{(}\frac{\gamma _{s}}{Q_{\infty}}\bigg{)}^{2}, \tag{29}\]  
where \(C_{pb}\) is the corrected surface pressure coefficient in the recirculation region where the velocity of the \(i\)-th panel is \(Q_{t_{i}}\), and \(\gamma_{s}\) is the vorticity of the shear layer. Base pressure coefficients are sensitive to the length of the wake panels. [PERSON] (1960) and [PERSON] (1961) investigated the relationships between the wake panel lengths and the base pressure coefficients in a 2-D circular cylinder case. They observed that the shorter wake length gave good agreement with experimental pressure over the whole cylinder rather than the longer wake panels, which tend to underestimate the peak suction upstream of the separation. The essentiality of the \"correct\" wake in predicting the pressure distribution around the bluff body is self-evident due to the separation of the flow. As the \"correct\" wake length is not known a priori, researchers try to find ways to obtain it. [PERSON] (1961) used experimental base pressure coefficients to obtain the correct wake length. However, this method is difficult to implement in the absence of experimental data. [PERSON] (1960) provided several solutions to this problem. The first solution is to combine the constant pressure criterion with a multiple segment vortex panel.