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Dynamic stability is significantly important for flying quality evaluation and control system design of the advanced aircraft, and it should be considered in the initial aerodynamic design process. However, most of the conventional aerodynamic optimizations only focus on static performances and the dynamic motion has never been included. In this study, a new optimization method considering both dynamic stability and general lift-to-drag ratio performance has been developed. First, the longitudinal combined dynamic derivative based on the small amplitude oscillation method is calculated. Then, combined with the PSO (particle swarm optimization) algorithm, a dynamic stability derivative that must not be decreased is added to the constraints of optimization and the lift-drag ratio is chosen as the optimization objective. Finally, a new aerodynamic optimization method can be built. We take NACA0012 as an example to validate this method. The results show that the dynamic derivative calculation method is effective and conventional optimization design can significantly improve the lift-drag ratio. However, the dynamic stability is enormously changed at the same time. By contrast, the new optimization method can improve the lift-drag performance while maintaining the dynamic stability.

As one type of the advanced transport systems, high-performance airfoil and aircraft configurations are increasingly important for the whole design cycle of advanced aircrafts. Therefore, the development of an aerodynamic optimization method combined with CFD technology, which can determine the best aerodynamic shape and maximize the aircraft performance and flight quality under given constraints, can greatly promote aerodynamic design. In recent years, domestic and foreign research institutions have developed a variety of effective aerodynamic optimization methods and made great progress. The single-objective and single-point optimization problem of the optimization design model has become a multiobjective, multipoint optimization, and multidisciplinary optimization problem [

At present, with the improvement of aircraft performance requirements, the scope of attention of aerodynamic design extends to a wider area. One is a multidisciplinary comprehensive design combined with structural stress and reliability [

The dynamic aerodynamic characteristics of an aircraft originate from the motion of the aircraft affected by various factors. One of the most important concepts is dynamic stability [

In this paper, an aerodynamic optimization method considering dynamic stability is proposed and studied. The concept of a dynamic stability derivative is introduced into the optimization design process. Based on the CFD calculation method, the dynamic derivative is identified in real time by using the small-amplitude forced vibration process, and combined with the particle swarm optimization (PSO) algorithm, the new optimization method considering both static and dynamic aerodynamic characteristics can be much more practical during the aerodynamic design of the advanced aircraft.

The conventional aerodynamic optimization design is based on the feed-in optimization algorithm. The basic idea is to combine the flow field analysis with the optimization algorithm, take the aerodynamic characteristics as the objective function, such as the drag or lift-drag ratio, and apply aerodynamic constraints or geometric constraints to optimize the objective function directly. The commonly used optimization algorithms include genetic algorithms, particle swarm optimization, and neural network algorithms. The example used to illustrate the conventional optimization method in this paper is the aerodynamic optimization design of the particle swarm algorithm.

The particle swarm optimization (PSO) algorithm [

The particle swarm optimization algorithm simulates the flight foraging behavior of a bird swarm. The design variables in the solution space are regarded as a group of birds without volume and mass (also known as particles), and the optimal solution of the problem is regarded as the food sought by the bird swarm. All particles continuously change their flight direction and distance and guide themselves to adjust the flight state by

Therefore, the aerodynamic optimization process based on the particle swarm optimization algorithm can be described in Figure

Traditional pneumatic optimization design process.

Dynamic stability plays an important role in the evaluation of aircraft handling and stability characteristics and the design of flight control laws. In the design of aircrafts, the dynamic stability derivative (dynamic derivative) is usually used to characterize the dynamic stability. Therefore, the design value of the dynamic derivative should be reasonable. An increase in the absolute value of the dynamic derivative leads to an increase in dynamic damping, which limits the performance of fighters with high maneuverability. However, for configurations with small dynamic damping in the longitudinal and lateral-directional directions, such as flying wing vehicles, it is necessary to increase the absolute value of the dynamic derivative to reduce the cost of the control system design.

In aerodynamic optimization design, the influence of aerodynamic shape change on subsequent handling and stability characteristics and flight control systems is usually not considered, which will not only result in poor practicability of optimization results but also does not conform to the current rapid development of aircraft design ideas and concepts. Thus, we comprehensively consider the static and dynamic aerodynamic characteristics and establish a more practical integrated optimization design method.

Firstly, we take the longitudinal direction as an example to introduce the method of CFD identification of dynamic derivatives [

When the rigid vehicle oscillates at a small amplitude with low frequency

We expand equation (

When the unsteady problem is calculated long enough and you let

The reduced frequency pair

The velocity of the pitching angle

Different from the traditional optimization method, when considering the influence of the dynamic derivative, each optimization cycle not only needs to calculate the lift and drag of the static condition but also needs to further evaluate the dynamic derivative characteristics of the optimization scheme. Then, dynamic and static results are combined to determine whether the best solution is available. Its optimization process is shown in Figure

A new aerodynamic optimization design method considering the influence of dynamic derivative.

Similar to the traditional method, the new aerodynamic optimization method is still set as a single objective, which takes the dynamic derivative effect as a constraint to join the optimization model. In the design of the aircraft, the dynamic derivative is different from the resistance characteristics and its value has no obvious limit range. The dynamic derivative balances the dynamic maneuverability and stability, so the ideal constraint condition is that the dynamic derivative characteristics of the optimized configuration are basically consistent with the original configuration. For this reason, when the dynamic derivative is added to the constraint in this paper, it is expressed as the absolute value that does not decrease.

The model used to verify the dynamic derivative calculation method in this paper is the international standard model basic Finner missile [

Finner missile configuration.

Computational grid.

Surface grid

Overall grid

We calculated the longitudinal combined dynamic derivatives of the model at supersonic speed. The Mach number is

To realize the movement of the object surface, this paper adopts rigid dynamic grid technology, which makes the whole grid perform a rigid movement. Compared with previous mesh deformation and unstructured mesh reconstruction methods, this method is simpler in form and more practical for single-degree-of-freedom small-amplitude vibration. To accurately capture the dynamic aerodynamic performance, we use the

The hysteresis loop of the unsteady pitching moment coefficient calculated by the small amplitude forced sinusoidal vibration method in this paper is shown in Figure

Hysteresis loop of pitching moment coefficient.

According to the calculation method in this paper, the longitudinal combined dynamic derivatives can be calculated as shown in Table

Longitudinal combined dynamic derivative.

Initial angle of attack | |||
---|---|---|---|

0 | 0.004852 | −0.136386 | −511.48 |

The dynamic derivative test results in reference [

The optimization design model includes three elements: objective function, constraint conditions, and design variables. For the conventional aerodynamic optimization design of an airfoil, after the design point is generally determined, the aerodynamic coefficient that represents the lift-to-drag is selected as the objective function, the thickness is taken as constraints, and the geometric shape of the airfoil is taken as the design variable.

In this paper, the PSO algorithm is used to optimize the aerodynamic performance of the NACA0012 airfoil. The parameterization of the airfoil adopts the CST method [

The Mach number is

The design objectives and constraints are as follows:

The results of the optimization design are presented in Figures

Geometric shape after conventional optimization.

Pressure coefficient distribution of airfoil after conventional optimization.

Aerodynamic parameters after conventional optimization.

Parameters | Original airfoil | Conventional optimization |
---|---|---|

0.406184 | 0.6082 | |

0.01240452 | 0.012 | |

Lift-to-drag ratio | 32.74 | 50.68 |

12% | 12.6% | |

−2.11 | 16.32 |

Table

Conventional airfoil optimization design does not consider the dynamic characteristics of airfoils. In this section, the dynamic derivative of the airfoil in the optimization is calculated in real time using the above dynamic derivative calculation method. The value is introduced into the optimization design of the airfoil as a constraint, and the absolute value of the dynamic derivative (

The design objectives and constraints are as follows:

The geometric and pressure distributions of the final optimization results are shown in Figures

Geometric shape after conventional optimization.

Pressure coefficient distribution of airfoil after conventional optimization.

Comparison of optimization results.

Parameter | Original airfoil | Conventional optimization | Considering dynamic derivative optimization |
---|---|---|---|

0.406184 | 0.6082 | 0.45586 | |

0.012404 | 0.012 | 0.011 | |

Lift-to-drag ratio | 32.74 | 50.68 | 41.44 |

−2.11 | 16.32 | −2.38 | |

12% | 12.6% | 14.5% |

Table

The new optimization method considers both the static and dynamic aerodynamic performances of the airfoil; thus, it should calculate the unsteady aerodynamics of the dynamic motion in every iteration. The time-cost can be much more higher than the original optimization. In this study, a convergent conventional optimization costs only 10 hours, while the new optimization using the same algorithm needs more than 80 hours. Therefore, how to reduce the time cost of the unsteady optimization should be further investigated.

In this study, the PSO algorithm is used for the conventional optimization of a two-dimensional airfoil and the optimization calculation considering the dynamic stability characteristics. The conclusions are as follows:

Dynamic stability plays an important role in the dynamic performance of aircrafts. The calculation method based on small amplitude forced vibration in this paper has high accuracy in calculating the longitudinal dynamic stability derivative. In addition, this method can be extended to the calculation of the dynamic derivative of the lateral direction

Conventional optimization methods can greatly improve the lift and drag characteristics of the NACA0012 airfoil. However, few constraints lead to dramatic changes in the dynamic characteristics of optimized airfoils. The dynamic stability is enormously changed at the same time

The optimization design considering the influence of dynamic stability fully considers the constraint of the dynamic derivative, which can improve the lift-drag performance while maintaining dynamic stability

The new optimization method developed by combining the dynamic stability analysis process can improve the static and dynamic performances of an aircraft, which has strong practicability in advanced aircraft design.

The data will be provided if anyone needed.

The authors declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work; there is no professional or other personal interest of any nature or kind in any products, service, and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled.

The authors would like to acknowledge the support of the Fundamental Research Funds for the Central Universities (Grant no. G2020KY05115) and Natural Science Basic Research Program of Shaanxi (Program no. 2021JQ-084).