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In this paper, a comprehensive optimization approach is presented to analyze the aerodynamic, acoustic, and stealth characteristics of helicopter rotor blades in hover flight based on the genetic algorithm (GA). The aerodynamic characteristics are simulated by the blade element momentum theory. And the acoustics are computed by the Farassat theory. The stealth performances are calculated through the combination of physical optics (PO) and equivalent currents (MEC). Furthermore, an advanced geometry representation algorithm which applies the class function/shape function transformation (CST) is introduced to generate the airfoil coordinates. This method is utilized to discuss the airfoil shape in terms of server design variables. The aerodynamic, acoustic, and stealth integrated design aims to achieve the minimum radar cross section (RCS) under the constraint of aerodynamic and acoustic requirement through the adjustment of airfoil shape design variables. Two types of rotor are used to illustrate the optimization method. The results obtained in this work show that the proposed technique is effective and acceptable.

The application of the acoustic and radar stealth technology to helicopter rotors has greatly enhanced the battlefield survivability and the combat effectiveness of helicopters [

In the past few decades, the aerodynamic noise of the helicopter rotor has been calculated by a great number of researchers. A lot of numerical algorithms were proposed based on various solutions of Ffowcs Williams and Hawkings (FW-H) equation [

Currently, the integrated analyses about the aerodynamic and noise optimization, as well as the aerodynamic and stealth optimization of rotor, are discussed in some academic works. In [

The airfoil shape which significantly affects the aerodynamic, acoustic, and stealth of helicopter rotors is usually considered as a separate problem [

The CST method was proposed by Kulfan and Bussoletti [

For the upper and lower surface of the airfoil, one has

The “class function” is shown by the formula

The “shape function” can be given by the linear combination of Bernstein polynomials, that is,

Substituting the point

NACA 0012 airfoil representation.

ONERA OA213 airfoil representation.

The blade element momentum theory is a combination of the momentum theory and the blade element theory. In this method, the rotor blades are divided into a number of independent elements along the length of blade. For each section, the momentum theory is the control volume theory, and the blade element theory is the summation of the sectional thrust and torque as computed by the sectional lift and drag coefficient of the airfoil.

The thrust and the torque according to the momentum theory are as follows:

By the blade element theory, the aerodynamic lift and drag forces on the airfoil are expressed by the formula as follows:

Blade element profile.

Integrating (

To solve the total thrust or the thrust coefficient, the induced velocity

Aeroacoustic analogy can be utilized to investigate the problem of aerodynamic noise, and the FW-H equation is adopted to calculate the free-field acoustic noise. The FW-H equation is a reorganization of the Navier-Stokes equations. The derivation of the FW-H equation uses generalized function theory which is an elegant element of mathematics. The most important characteristic of the FW-H equation is that it can address acoustic propagation generated from a moving surface. Since the influence of quadrupole source is negligible for the low rotary speed blade, the simplification form of the FW-H equation is given as follows [

The total RCS of the rotor could be computed as the sum of surfaces and edges. The scattering field of the surfaces and edges is calculated by PO and MEC, respectively. The initial point of PO is the surface currents produced by an incoming electromagnetic wave. To improve the PO solution and take into account the diffraction by edges, the MEC has been proposed by Michaeli [

Stratton and Chu [

In this paper, simple diffractions by edges are treated with the usage of the formulation of equivalent currents proposed by Mitzner [

Solving (

After the RCS of

In the aerodynamic case, the rotor of [

The main parameters of the rotor.

Total weight | Rotor shape | Number of blades | Radius of blades | Chord length of blades |

| ||||

9071.8kg | Rectangle | 4 | 9.144m | 0.6096m |

| ||||

Rotor solidity | Root cut ratio | Negative twist | Tip speed | Airfoil shape |

| ||||

0.085 | 0.15 | -10° | 198.12m/s | NACA 0012 |

Table

The comparison of the calculated results and the results in [

Results in Ref.[ | Calculated results | |

| ||

Thrust coefficient | 0.01438 | 0.01485 |

Torque coefficient | 0.00119 | 0.00122 |

Hover efficiency | 72.45% | 74.78% |

In the aerodynamic noise case, the Farassat 1a is utilized to compute the thickness noise and the loading noise of the Caradonna-Tung rotor (C-T rotor) (parameters of the C-T rotor are shown in Table

The main parameters of C-T rotor.

Rotor shape | Number of blades | Radius of blades | Chord length of blades |

| |||

Rectangle | 2 | 1.143m | 0.1905m |

| |||

Aspect ratio | Pitch angle | Rotation speed | Airfoil shape |

| |||

6 | 8° | 1250rpm | NACA 0012 |

The comparison of the calculated results and WOPWOP code results.

It can be seen that the thickness noise is in good coincidence with the WOPWOP code results. There are discrepancies between the calculated loading noise and the WOPWOP code results because of different methods to calculate the rotor’s aerodynamic characteristics. Since the sound pressure level of thickness noise is much bigger than the loading noise, thickness noise is only considered as the acoustic constraints.

In the RCS case, a 5-rotor is selected to illustrate the validity of the PO and MEC under the conditions of radar frequency of f=3 GHz and f=6GHz with vertical polarization, respectively. The main parameters of the rotor are displayed in Table

The main parameters of the 5-rotor.

Rotor shape | Number of blades | Radius of blades | Chord length of blades |

| |||

Rectangle | 5 | 6.595m | 0.4m |

| |||

Pitch angle | Rotation speed | Rotation period | Airfoil shape |

| |||

12° | 1250rpm | 0.048s | ONERA OA213 |

The comparison of the calculated results and the FEKO results at f=3 GHz.

The comparison of the calculated results and the FEKO results at f=6 GHz.

The comprehensive analysis about the aerodynamic, acoustic, and stealth characteristics of the rotor is an MDO issue, and the blade shapes of rotor to improve the aerodynamic performance and reduce the noise and RCS at the same time are generally inconsistent. Therefore, the key is to search the aerodynamic, acoustic, and stealth compromised results which can be described as the optimal solution. For this MDO issue, the mathematical model should be established first. Due to the influence of airfoil shape on aerodynamic, thickness noise, and scattering characteristics of rotor, the optimized rotor can be designed by the means of optimizing the airfoil shape. Subsequently, the objective function is as follows:

The flowchart of the comprehensive analysis of the aerodynamic, acoustic, and stealth of rotor based on GA.

According to Table

The range of the design variables of C-T rotor.

Variables | | | | | | | | |

| ||||||||

Lower bound | 0.1 | 0.1 | 0.1 | 0.1 | -0.1 | -0.1 | -0.1 | -0.1 |

Upper bound | 0.2 | 0.2 | 0.2 | 0.2 | -0.2 | -0.2 | -0.2 | -0.2 |

Using the blade element momentum theory, the initial C-T rotor thrust coefficient is 0.0614. The initial C-T rotor thickness noise is 96.50dB at the 1th blade pass frequency (BPF) by Farassat 1a. Figure

The comparison of the initial design variables and the optimized design variables for C-T rotor.

Variables | | | | | | | | |

| ||||||||

Initial | 0.177 | 0.1458 | 0.1462 | 0.1445 | -0.177 | -0.1458 | -0.1462 | -0.1445 |

Optimized | 0.1173 | 0.1563 | 0.1224 | 0.1956 | -0.116 | -0.1022 | -0.1886 | -0.1633 |

The comparison of airfoil between initial airfoil and optimized airfoil for C-T rotor.

Figure

The comparisons of RCS characteristics between the initial and optimized airfoil for C-T rotor.

The RCS of the C-T rotor at f=10GHz with vertical polarization for initial and optimized airfoil is shown in Figure

The comparisons of the RCS characteristics between the initial and optimized C-T rotor.

The aerodynamic characteristic and the noise of the C-T rotor are displayed in Table

The comparisons of the aerodynamic characteristic and the noise between initial and optimized for C-T rotor.

Thrust /N | Thrust coefficient | SPL/dB | |

| |||

Initial | 1099.7 | 0.0614 | 96.50 |

Optimized | 1256.3 | 0.0701 | 95.08 |

The comparisons of the thickness noise between the initial and optimized C-T rotor for time domain.

The comparisons of the thickness noise between the initial and optimized C-T rotor for frequency domain.

Since the airfoil of the 5-rotor is ONERA OA213 (see Table

The range of the design variables of 5-rotor.

Variables | Lower bound | Upper bound |

| ||

| 0.2 | 0.3 |

| 0.2 | 0.3 |

| 0.1 | 0.2 |

| 0.4 | 0.5 |

| -0.1 | -0.2 |

| -0.01 | -0.15 |

| -0.01 | -0.1 |

| -0.01 | -0.1 |

| -0.1 | -0.2 |

| -0.01 | -0.1 |

| 0.002 | 0.003 |

| -0.002 | -0.003 |

Utilizing the same method to optimize this airfoil, one can compare the initial airfoil and the optimized airfoil, which is given by Figure

The comparison of airfoil between initial airfoil and optimized airfoil for 5-rotor.

The comparisons of the design variables between the initial airfoil and the optimized airfoil are displayed in Table

The comparison of the initial design variables and optimized design variables for 5-rotor.

Variables | Initial | Optimized |

| ||

| 0.2708 | 0.2079 |

| 0.2876 | 0.2598 |

| 0.1293 | 0.1718 |

| 0.4124 | 0.4688 |

| -0.1048 | -0.1220 |

| -0.0881 | -0.0202 |

| -0.0668 | -0.0565 |

| -0.0524 | -0.0586 |

| -0.1609 | -0.1604 |

| -0.0475 | -0.0816 |

| 0.0021 | 0.0023 |

| -0.0021 | -0.0022 |

Figure

The comparisons of the RCS characteristics between the initial and optimized airfoil for C-T rotor.

The RCS of the 5-rotor at f=10GHz with vertical polarization for the initial and optimized airfoil is displayed in Figure

The comparisons of the RCS characteristics between the initial and optimized 5-rotor.

The aerodynamic characteristic and the noise of 5-rotor are presented in Table

The comparisons of the aerodynamic characteristic and the noise between the initial and optimized 5-rotor.

Thrust /N | Thrust coefficient | SPL/dB | |

| |||

Initial | 40424 | 0.0364 | 92.35 |

Optimized | 42448 | 0.0382 | 91.99 |

The comparisons of the thickness noise between the initial and optimized 5-rotor for time domain.

The comparisons of the thickness noise between the initial and optimized 5-rotor for frequency domain.

Generally, results indicate that the airfoil shape design of rotor with high aerodynamic performances, low noise, and low scattering characteristics has been given, which shows that the optimization strategy in this article is feasible and credible.

An automated process for the comprehensive optimization design of the aerodynamic, acoustic, and stealth characteristics of the rotor is developed in the article. The airfoil curve is represented by utilizing the CST method with few design variables, which is reasonable for the performance of the optimization problem. The aerodynamic, acoustic, and stealth characteristics of the rotor in hover are simulated effectively by utilizing the blade element momentum theory, Farassat 1a, PO, and MEC, respectively. Optimizing the design variables of airfoil by objective function with constraints, based on GA, a new airfoil could be acquired. Adopting the new airfoil to the rotor, another new rotor with high aerodynamic characteristics, low noise, and low RCS would be achieved. The proposed comprehensive optimization design method is suitable for the preliminary design phase where there is a need for quick estimation in consideration of the aerodynamic, acoustic, and stealth factors.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.