# A Separated-Flow Model for 2-D Viscous Flows around Bluff Bodies Using the Panel Method
## 3 Results  
### Flow Past the Cylinder  
To validate the present separated-flow model and the prediction of the separation locations, the simulations for the viscous flows over a 2-D circular cylinder at high Reynolds numbers (Re) have been carried out using the developed method. The results of the present model are verified and validated using the simulation and experiment results of previous studies from the literature [50, 61].  
In the separated-flow model, the body is represented by a certain number of constant-source panels and the free shear layers by uniform vorticity panels aligned in the free-stream direction, attached to the body at the source panel edges nearest to the separation points. The flow field simulated using the presented model is shown in Figure 4a. For bodies with symmetrical axes parallel to the freestream, the two wake panels will have the same length, and the separation positions on the upper and lower surface are the same. The separation points are predicted by boundary-layer calculations, and the wake lengths, which are highly relevant to the base pressure, are determined by the solution proposed by [PERSON] [50]. [PERSON]'s experiment of flows around the circular cylinder at Re = 8.4 \(\times\) 10\({}^{6}\)[61] are used to validate the proposed model. The separation angle predicted by the boundary-layer calculation is 105\({}^{\circ}\) for Re = 1 \(\times\) 10\({}^{7}\), which coincides with the experimental results. The pressure distribution on half the cylinder of this case is compared with potential flow results, [PERSON]'s [50] simulations results, and [PERSON]'s experimental results [61], as shown in Figure 4b. One thing worth mentioning is that the present method corrects the pressure coefficient at the separation positions by assuming the velocity there is zero. This resolves the \"overshoot\" in pressure just upstream of the separation point in [PERSON]'s results [50]. The independence of the pressure distribution on panel densities is validated by comparing the results of the 40-Panel case and 120-Panel case. Both simulations agree well with the experimental results.  
In order to determine the \"correct\" wake length, the distributions of the tangential velocity increment calculated according to Figure 3 on a circular cylinder with various wake lengths are studied, as shown in Figure 5. The tangential velocities along the wake panels with lengths ranging from 0.75D to 1.5D are investigated, where D refers to the diameter of the 2-D circular cylinder. It is shown that the shorter the wake panel, the larger the increment. For longer wake panels, the tangential velocity first decreases slightly and then increases along the wake panels. Evaluating the tangential velocity increment for  
Figure 3: Calculation of tangential velocity increment along the wake panel.  
two wake length assumptions enables a wake length to be interpolated, which gives a zero-velocity increment. The iteration process shown in Figure 6 starts with two assumed wake lengths, 0.75D and 2D, and it is converged after three iterations to 1.5D. This validates that the proposed interpolative procedure can be used to determine the \"correct\" wake length for the 2-D bluff bodies in evaluating their pressure distributions using the presented separated-flow model.  
Figure 4: Simulation results for a circular cylinder at \(Re=1\times 10^{7}\): (**a**) flow field around the circular cylinder; (**b**) pressure distribution on half of the cylinder.  
Figure 5: Distributions of tangential velocity increment on a circular cylinder for various wake lengths.  
Figure 6: Results from the iterative wake length procedure applied to a circular cylinder.