# A Separated-Flow Model for 2-D Viscous Flows around Bluff Bodies Using the Panel Method

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Potential Flow Calculation

**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 σ is calculated as follows:

_{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, σ

_{j}is the constant source strength of the j-th panel, RHS

_{i}is the right-hand side term, which is equal to the normal velocity due to the freestream velocity (U

_{∞},V

_{∞}) at the i-th control point, and

**n**

_{i}is the normal vector of the i-th panel.

**t**

_{i}is the unit tangential vector of the i-th panel.

#### 2.2. Boundary-Layer Prediction

_{e}is the streamwise potential flow velocity at the edge of the boundary layer. The remaining five parameters, θ, H, H

^{*}, 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 θ, H, and momentum thickness Reynolds number Re

_{θ}are presented as follows:

_{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}:

^{*}is approximated by the following formula:

_{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:

_{s}is the effective slip velocity and C

_{τ}is the wake-layer shear stress coefficient. This closure introduces a new variable, C

_{τ}, and according to previous research [52,58,59], this parameter is governed by Green’s lag entrainment equation [58]:

^{9}. The logarithm of the maximum amplification rate $\overline{n}$ with respect to the streamwise coordinate ξ is expressed as a function of H and θ:

#### 2.3. Separated-Flow Model

_{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, σ

_{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 γ

_{k}.

_{pb}is the corrected surface pressure coefficient in the recirculation region where the velocity of the i-th panel is ${Q}_{{t}_{i}}$, and γ

_{s}is the vorticity of the shear layer.

## 3. Results

#### 3.1. Flow Past the Cylinder

^{6}[61] are used to validate the proposed model. The separation angle predicted by the boundary-layer calculation is 105° for Re = 1 × 10

^{7}, which coincides with the experimental results. The pressure distribution on half the cylinder of this case is compared with potential flow results, Dvorak’s [50] simulations results, and Roshko’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 Dvorak’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.

#### 3.2. Flow Past the Train

#### 3.2.1. Physical Model

_{tr}= 125 mm and a total length of 8.72 D

_{tr}, which is in 1:32 scale to the real train. The cross-section profile follows the above equation in which the value of c = 62.5 mm (D

_{tr}/2) and n = 5. The nose cross-section is given by the same equation in which c follows a semi-elliptical profile with a major diameter of 1.28 D

_{tr}, while n is reduced uniformly to 2 at the nose tip. In this way, the cross-section becomes smaller and more circular toward the nose, as shown in Figure 7. The 2-D train model is discretised evenly in the following cases. The Reynolds number is 3 × 10

^{5}in the following simulations.

#### 3.2.2. Pressure Loads on the Train

^{5}. The calculation procedure follows the flowchart shown in Figure 1. As explained above, the separated flow model for simulating the viscous flow has two key parameters: the separation locations at which the vorticity panels are attached, and the “correct” wake panels with which the wake is modelled. In the former key parameter, separation locations are predicted by the boundary-layer calculation procedure as described in Section 2.2. The latter key parameter, the “correct” wake panel lengths, are calculated according to the interpolating procedure based on the criterion of zero tangential velocity increment between two points on the wake panels.

_{tr}to 4 D

_{tr}are investigated, as shown in Figure 8. For all wake lengths, the tangential velocity increases along the wake panels. The increments in tangential velocity for the wake panels decreases with the increasing of the wake panel lengths; in other words, longer wake panels lead to smaller tangential velocity differences. The “correct” wake panel lengths were determined by the interpolative procedure elaborated in Section 2.3 and validated in Section 3.1. The iteration process shown in Figure 10 starts with two assumed wake lengths, 0.75 D

_{tr}and 2 D

_{tr}, and it is converged after six iterations to 3 D

_{tr}, which is used as the “correct” wake length in evaluating the pressure distribution around the 2-D train. To validate the grid independency of the proposed model, three different panel densities: 100 panels, 200 panels, and 400 panels were tested, as shown in Figure 9. It can be seen that the distribution and value of the pressure coefficient almost coincide for all three cases. The peak values and base pressure are compared in Table 1. The discrepancies of peak values between different panel densities are mainly caused by the locations of the control points and it can be concluded that the differences between the results of 100 panels and 200 panels are small. Thus, the results of the proposed model are independent of the panel densities.

_{tr}, 1.28 D

_{tr}, and 1.6 D

_{tr}, respectively. It can be concluded from Figure 11 that the peak values that appear just downstream of the stagnation position decrease with the increment of the nose length. The peak values just upstream do not have the same trend with the increase of the nose length, but the differences between the two peak values shrink as the nose length increases. One of the main features of the train is that it is running on the rails and is close to the ground while cruising. The presence of the ground must be considered when investigating the flow around the train. The method of images is employed in this case to model a solid ground at z = 0 by placing the mirror image of the 2-D train at the negative value of the height, which is 0.15 from the ground plane, as shown in Figure 12, where the number 1 refers to the 2-D train and the number 2 refers to the image. The method of images involves the analysis of the flow over multiple bodies, even if the original problem is for flow over a single solid body. In addition, as the solid bodies are symmetric at about z = 0 at the ground plane, the no-penetration condition is satisfied. A total of 200 panels were used in the simulation, including the ground effect. Unlike the symmetrical simulation above, the introduction of the ground plane leads to the flow around the train to be asymmetrical. According to the integral boundary-layer prediction, the separation positions with the presence of ground is not symmetric on the upper and lower surface of the train. They are about 176° on the upper surface and 186° on the lower surface in this case, as shown in Figure 12.

_{tr}and for the lower wake panel is 9 D

_{tr}.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Simulation results for a circular cylinder at Re = 1 × 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.

Position | 100 Panels | 200 Panels | 300 Panels | |||
---|---|---|---|---|---|---|

C_{p} | Position | C_{p} | Position | C_{p} | Position | |

1 | −1.12 | 4.6° | −1.19 | 5.2° | −1.21 | 5.4° |

2 | −0.41 | 171.8° | −0.7 | 173.0° | −0.6 | 173.4° |

3 | −0.41 | 185° | −0.7 | 185.5° | −0.6 | 185.8° |

4 | −1.12 | 352.6° | −1.19 | 353.5° | −1.21 | 353.9° |

Base pressure | −0.34 | −0.3 | −0.36 |

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**MDPI and ACS Style**

Li, R.; Soper, D.; Xu, J.; Jia, Y.; Niu, J.; Hemida, H.
A Separated-Flow Model for 2-D Viscous Flows around Bluff Bodies Using the Panel Method. *Appl. Sci.* **2022**, *12*, 9652.
https://doi.org/10.3390/app12199652

**AMA Style**

Li R, Soper D, Xu J, Jia Y, Niu J, Hemida H.
A Separated-Flow Model for 2-D Viscous Flows around Bluff Bodies Using the Panel Method. *Applied Sciences*. 2022; 12(19):9652.
https://doi.org/10.3390/app12199652

**Chicago/Turabian Style**

Li, Rui, David Soper, Jianlin Xu, Yongxing Jia, Jiqiang Niu, and Hassan Hemida.
2022. "A Separated-Flow Model for 2-D Viscous Flows around Bluff Bodies Using the Panel Method" *Applied Sciences* 12, no. 19: 9652.
https://doi.org/10.3390/app12199652