# Adjoint-Based Aerodynamic Design Optimization and Drag Reduction Analysis of a Military Transport Aircraft Afterbody

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## Abstract

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## 1. Introduction

#### 1.1. Related Works

#### 1.2. State of the Art

## 2. Geometry and Grid

#### 2.1. Afterbody Design Parameters

#### 2.2. Model and Grid Generation

#### 2.3. Grid Sensitivity

## 3. CFD Simulation Method and Validation

## 4. Optimization Framework

## 5. Analysis Method of Afterbody Vortex System

#### 5.1. Boundary Layer Extraction from RANS Solutions

#### 5.2. Vortex-Tracking Method

#### 5.3. The Vorticity Loop Model and Vortex-Induced Force

## 6. Optimization Problem

## 7. Optimization Results

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$\mathcal{A}$ | Residual |

${A}_{n}$ | The projection area of a vorticity loop in n direction. |

${b}_{r,0}$ | The center distance of two counter-rotating wake vortex tubes |

${C}_{D,vi}$ | Vortex-induded drag coefficient |

${C}_{d}$ | Drag coefficient |

${C}_{dp}$ | Pressure drag coefficient |

${C}_{dv}$ | Viscous drag coefficient |

${C}_{f}$ | Friction drag coefficient |

${C}_{l}$ | Lift coefficient |

${C}_{p}$ | Pressure coefficient |

c | Constraints |

$\mathcal{D}$ | Artificial viscosity |

${\overline{D}}_{v}$ | Time-average induced drag |

${d}_{A},{d}_{B}$ | Diameter |

F | Aerodynamic force |

${F}_{s}$ | Safety ratio in GCI theory |

${\mathit{F}}_{\mathit{I}},{\mathit{F}}_{\mathit{V}}$ | The inviscid flux term and the viscous flux term |

${h}_{B}$ | The distance between the zero longitudinal down point of a certain normal section of the afterbody and the maximum width point |

I | Object function |

${\overline{L}}_{v}$ | Time-average induced lift |

${l}_{A}$ | Length of afterbody |

${l}_{F}$ | Length of fuselage |

M | Mach number |

p | Pressure |

$\mathit{Q}$ | Conserved variables |

q | The order of convergence |

$Re$ | Reynolds number |

$R,{R}_{max}$ | Defined distance for near-roundness |

${R}_{v}$ | Enclosed control volume |

r | The position vector pointing to the fluid element from the origin |

${r}_{g}$ | The grid refinement ratio |

t | Time |

${u}_{vw}$ | The vortex wash speed in the vertical direction |

U | Streamwise velocity |

V | Volume |

W | Flatness |

w | The maximum width of the section |

$x,y,z$ | Cartesian coordinates |

${\mathit{x}}_{\mathit{s}},{\mathit{x}}_{\mathit{v}}$ | Surface mesh and volume mesh |

$\alpha $ | Angle of attack |

$\beta $ | Upswept angle |

$\zeta $ | Contraction ratio |

$\eta $ | Nondimensional span location |

$\lambda $ | Fineness |

$\varphi $ | Near-roundness |

${\delta}_{e}$ | Boundary layer thickness |

${\delta}_{1},{\delta}_{2}$ | Displacement and momentum boundary layer thickness |

$\omega $ | Vorticity |

$\rho $ | Density |

${\Gamma}_{v}$ | Induced circulation |

$\mathsf{\Omega}$ | Vorticity tensor |

${\tau}_{w}$ | Shear stress on the wall |

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**Figure 7.**The comparison of pressure distribution between ADflow solution and experiment ($\eta =2y/b$ is the span).

**Figure 15.**Surface comparison of military transport aircraft optimization (blue: initial; red: optimization).

**Figure 19.**Comparison of surface, pressure coefficient, displacement thickness, and momentum thickness in eight cross-sectional positions of military aircraft.

**Figure 20.**Comparison of crossflow velocity profile in eight cross-sectional positions of military aircraft.

Parameters | Value |
---|---|

Reference area (${\mathrm{m}}^{2}$) | 162.10 |

Fuselage length ${l}_{F}\left(\mathrm{m}\right)$ | 28.97 |

Maximum diameter ${d}_{A}\left(\mathrm{m}\right)$ | 4.035 |

Afterbody length ${l}_{A}\left(\mathrm{m}\right)$ | 12.568 |

Upswept angle $\beta {(}^{\circ})$ | 24 |

Fineness $\lambda $ | 3.11 |

Contraction ratio $\zeta $ | 0.095 |

Mesh Level | Mesh Size | ${\mathit{C}}_{\mathit{d}}$ (Counts) | GCI/% | q |
---|---|---|---|---|

L2 | 1,396,665 | 72.36 | / | |

L1 | 2,782,585 | 70.62 | 10.497 | |

L0.5 | 4,936,345 | 68.37 | 6.384 | 2.6 |

(a) Section 1–6 | ||||||
---|---|---|---|---|---|---|

ID | 1 | 2 | 3 | 4 | 5 | 6 |

Position (m) | 16.414 | 17.241 | 18.047 | 18.954 | 19.966 | 20.975 |

Percentage (%) | 0 | 6.58 | 12.99 | 20.20 | 28.25 | 36.29 |

Flatness | 0.9089 | 0.8858 | 0.8195 | 0.7140 | 0.5954 | 0.5021 |

Near-roundness | 0.9089 | 0.8945 | 0.8491 | 0.7701 | 0.6697 | 0.5751 |

(b) Section 7–12 | ||||||

ID | 7 | 8 | 9 | 10 | 11 | 12 |

Position (m) | 21.977 | 23.084 | 24.078 | 25.096 | 26.191 | 27.474 |

Percentage (%) | 44.26 | 53.07 | 60.98 | 69.08 | 77.79 | 88.00 |

Flatness | 0.4345 | 0.3507 | 0.2803 | 0.1722 | 0.1602 | 0.2132 |

Near-roundness | 0.4822 | 0.3671 | 0.2803 | 0.2081 | 0.2159 | 0.2791 |

Configuration | Drag Coefficient (${\mathit{C}}_{\mathit{D}}$) | Lift Coefficient (${\mathit{C}}_{\mathit{L}}$) |
---|---|---|

Initial | 0.00707 | −0.0144 |

Optimized | 0.00543 | −0.0049 |

$\Delta $/% | 16.4 counts/−23.2% | 65.97% |

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**MDPI and ACS Style**

Rao, H.; Chen, Y.; Shi, Y.; Yang, T.; Liu, H. Adjoint-Based Aerodynamic Design Optimization and Drag Reduction Analysis of a Military Transport Aircraft Afterbody. *Aerospace* **2023**, *10*, 331.
https://doi.org/10.3390/aerospace10040331

**AMA Style**

Rao H, Chen Y, Shi Y, Yang T, Liu H. Adjoint-Based Aerodynamic Design Optimization and Drag Reduction Analysis of a Military Transport Aircraft Afterbody. *Aerospace*. 2023; 10(4):331.
https://doi.org/10.3390/aerospace10040331

**Chicago/Turabian Style**

Rao, Hanyue, Yifu Chen, Yayun Shi, Tihao Yang, and Hongyang Liu. 2023. "Adjoint-Based Aerodynamic Design Optimization and Drag Reduction Analysis of a Military Transport Aircraft Afterbody" *Aerospace* 10, no. 4: 331.
https://doi.org/10.3390/aerospace10040331