Issue 
Acta Acust.
Volume 6, 2022



Article Number  24  
Number of page(s)  14  
Section  Aeroacoustics  
DOI  https://doi.org/10.1051/aacus/2022019  
Published online  22 June 2022 
Scientific Article
Acoustic scattering effect prediction of helicopter fuselage based on BEM and convective FW–H equation
Beihang University, 100191 Beijing, China
^{*} Corresponding author: yimingxu@buaa.edu.cn
Received:
9
November
2021
Accepted:
18
May
2022
The main acoustic noise source of helicopters is supposed to be the rotor blades, but the scattering effect of the helicopter fuselage sometimes cannot be ignored. For the accuracy of acoustic noise prediction and the research on the scattering affections of the helicopter fuselage, an FWH–BEM Method (FBM) based on convective FW–H equation and boundary element method (BEM) is presented for the prediction of the noise scattering effects of helicopter fuselage and the approach to the calculation of a helicopter’s acoustic noise field. In this paper, different fuselage models are adopted for the comparisons of the acoustic noise scattered by different types of fuselages. According to the discovery that helicopter fuselages with sharp edges can bring more significant acoustic scattering effects, a research on the influence of radius of curvature (RC), which reflects the sharpness of fuselage sharp edges, is also carried out. In addition, the acoustic scattering effects of the same type of fuselage but with different length, width and height ratios are also compared for discovering the influences of the fuselage size. The presented FBM is efficient to analyze the acoustic scattering effects of the helicopter fuselage and predict the acoustic noise field, taking both the rotor and the fuselage into account. Besides, the research in this paper leads to the discovery of the influence factors of the acoustic scattering effects and helps the proper selection of the fuselage in a helicopter stealth design.
Key words: Convective FW–H equation / Boundary Element Method / Acoustic scattering effects / Fuselage / Sound pressure level
© The Author(s), Published by EDP Sciences, 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
The most important characteristic signal of helicopters is acknowledged to be acoustic noise, which can be a severe problem of the acoustic stealth of helicopters. Therefore it is essential that a proper stealth design is adopted for a new helicopter, aiming at suppressing its acoustic noise and strengthening its acoustic stealth ability [1–3].
Predicting the acoustic noise [4–6] is one of the important parts of aircraft acoustic stealth technology, helping to ascertain the sound pressure level of acoustic noise and find out the main affection factors. It is commonly acknowledged that the rotor blades are regarded as the main acoustic noise source of a helicopter, whose acoustic noise prediction problem can be regarded as the calculation of the sound field of rotating sound sources. The Ffowcs Williams and Hawkings (FW–H) equation [7] is a standard approach to the solutions of this sort of problems. Taking account of the influence of medium movement (inflow), Ghorbaniasl and Lacor [8] have developed a direct calculation method of the sound field of rotating sound sources with inflow, based on convective FW–H equation, and also developed several time domain formulations for the solution of the convective FW–H equation. Ghorbaniasl et al. [9] and Huang et al. [10] have deduced convective Kirchhoff equation for the calculation of sound field.
However, not all of the sound waves travel directly in straight lines to infinity, for the existence of helicopter fuselage that brings acoustic scattering effects for its reflection of sound waves. Therefore a method is required for the calculation of the acoustic scattering effects of the helicopter fuselage. Common methods include equivalent source method (ESM) and boundary element method (BEM) [11–13]. Compared with ESM that should disperse the whole domain, BEM just requires the dispersal of the boundary, transferring three dimensional problems into twodimensional problems so that the complexity and workload can be reduced. The helicopter fuselage can be regarded as a solid boundary and thus BEM method can be a proper method adapted to the calculation of its acoustic scattering effects.
Some basic researches based on boundary element method (BEM) have been brought out for calculations of acoustic scattering effects. Roger Michel and Kucukcoskun Korcan [14] studied the scattering effects of the solid strut member of a rotor and put forward some relevant theories. Seongkyu Lee, Kenneth Brentner and Philip Morris [15, 16] studied the scattering effects in the time domain using an equivalent source method, with a sphere solid scattering boundary and a point sound source. Curle [17] studied the influence of solid boundaries upon aerodynamic sound and the acoustic scattering effects of the solid boundaries.
Previous researches mainly focus on the theories of BEM method [18–20], and some basic verification cases of acoustic scattering effects with simple solid boundaries such as spheres and plains [21, 22]. The research presented in this paper aims at the acoustic scattering effects of the acoustic noise produced by the helicopter rotor blades with the solid boundary of helicopter fuselage, and tries to discover the influence of the variety of the fuselage types and sizes.
This research is presented in three sections. In the first section, a method based on BEM method and convective FW–H equation is presented. In the second section, helicopter models with different types of fuselages are established for the research on the influence on acoustic scattering effects of different fuselage types. In the third section, we make a research on the influence of the length, width and height of the fuselage and summarize the rules of the elements’ influence.
2. Theoretical background
2.1 Convective FW–H equation
Fundamental equations of fluid flow in an unbounded region outside the control surface without external force can be described as [7]
where ρ is the density of the fluid, u_{i} is the velocity of the fluid, and P_{ij} is the tensor of the force loading on the fluid. Equation (1) is known as the continuity equation and equation (2) is known as the momentum equation.
With the utilization of Heaviside Function, the continuity equation can be adapted as
where
H(f) is known as Heaviside Function.
Similarly, the momentum equation can be rewritten as
Let , we can obtain that
which is acknowledged as FW–H equation.
The acoustic solution of FW–H equation can be obtained as [11–13]
where p′_{T} (x, t) is thickness noise and p′_{L} (x, t) is loading noise.
Taking inflow into account, convective FW–H equation should be [18]
where
With equations (9)–(12), the analytical solutions of convective FW–H equation can be obtained as
where
and
Equation (16) indicates the vector from the sound source point on the surface of the blade to the observe point.
When M_{∞} = 0, convective FW–H equation is the same as FW–H equation. By comparing FW–H equation and convective FW–H equation, the difference caused by inflow Mach number M_{∞} can be obtained as
Therefore, equations (19) and (20) can be regarded as the influence index of inflow Mach, a quantificational expression of the affections of inflow Mach on thickness noise and loading noise.
2.2 BEM method
The common form of Helmholtz differential equation is:
where p is the acoustic pressure, ∇^{2} is the Laplace operator, k = ω/c_{0}.
The relationship of the pressure derivative and local normal velocity can be described as:
At infinity, p must satisfy Sommerfeld radiation condition:
The Green Function at free acoustic field can be introduced as:
where P is the observe point and Q is one of the field points r = Q − P.
Therefore the common form of Helmholtz boundary integral equation can be obtained as:
where
The total acoustic pressure equals to the summation of incoming acoustic pressure and scatter acoustic pressure:
For plain wave, incoming acoustic pressure appears as:
For spherical wave, incoming acoustic pressure appears as:
When p_{s} is used in Helmholtz boundary integral equation, we can obtain that
When p_{i} is used in Helmholtz boundary integral equation, we can obtain that
Subtracting equation (31) from equation (30) and combining with equation (27), we can obtain
where
Thus equation (32) can be simplified as
In equation (34), p_{i}(P) is the incidence pressure calculated by convective FW–H equation. C(P) can be obtained by equation (24). For the integral items. G is the Green function and n is the unit normal vector of a surface element.
2.3 Calculation of the sound pressure gradient
In equation (34), iρωv_{n} is the sound pressure gradient that is obtained from equation (25). The sound pressure gradient can be calculated by equation (35) [23]
where
A further simplification can obtain the analytical expression of the sound pressure gradient of convective FW–H equation
where
The elements mentioned above are all known qualities and the only unknown quality is the total sound pressure p, which is the solution target.
2.4 Matrix form of the sound pressure calculation
The discretization of the integral can transfer equations (34)–(52):
Based on the same surface element dS_{j}, every integral/summation element marked j on the left of the equal sign in equation (52) equals to the one marked j on the right side. Therefore equation(52) can be written as:
Equation (36) can be written as the matrix form:
where
Let , , .
Equation (37) can be written as the form of:
Then the total sound pressure vector p can be obtained by equation (56), which is known as the matrix form of the sound pressure calculation.
3. Models and simulation environments
3.1 Geometry models of the helicopters
The geometry models of the fuselages are established by adopting the parametric modeling method (PMM), combined with the real fuselage shape. Main geometry parameters of the fuselages are confirmed as the decisive factors of the helicopter fuselage and the rotor.
Figure 1 shows the threeview drawing of the simplified helicopter geometry model and the main decisive parameters are outlined. L_{h} is the length of the whole helicopter. L_{f} is the length of the main body of fuselage. W_{f} is the width of the fuselage. H_{f} is the height of the fuselage. H_{r} is the height of the rotor that refers to the distance from the coordinate origin to the center of rotor. R_{r} is the radius of the rotor. L_{b} is the length of the blades.
Figure 1 Establishment of the simplified helicopter geometry model (including rotor and fuselage). 
The parameters listed in Figure 1 and Table 1 can be the main geometry parameters of the approximate design of a helicopter. The size and shape of the helicopter can be preliminary determined with the main geometry parameters and the primary geometry model of the helicopter can be given. Then the primary geometry model can be corrected to the final geometry model according to other detailed information.
Nomenclature of the physical quantities.
Three typical helicopter fuselage, OH6, HUEY and RAH66, are selected for the research on the scattering effect of different fuselage types.
The OH6 fuselage model established in this paper is shown in Figure 2, according to the design of the OH6 helicopter [24]. The model mainly includes the main part of the OH6 fuselage while the tail of the fuselage is simplified.
Figure 2 Helicopter model with OH6 fuselage. 
The HUEY fuselage model established in this paper is shown in Figure 3, according to the design of the HUEY helicopter [25]. The fuselage model mainly includes the main part of the OH6 fuselage while the tail of the fuselage is simplified.
Figure 3 Helicopter model with HUEY fuselage. 
The RAH66 fuselage model established in this paper is shown in Figure 4, according to the design of the RAH66 stealth helicopter [26]. The fuselage model mainly includes the main part of the OH6 fuselage while the tail of the fuselage is simplified.
Figure 4 Helicopter model with RAH66 fuselage. 
In the simulation case shown in Figure 5, the acoustic noise detector receives the acoustic noise waves produced by helicopter at the pitching angle θ = 45°, and the azimuth angle from 0° to 360°. The distance from the helicopter to the detector is 10 m. The inflow Mach number is M_{∞} = 0.2.
Figure 5 Schematic of simulation case. 
4. Results and discussions
4.1 Approach to the results
In this paper, the theories and the methodologies enumerated in Section 2 are used for the calculation of the acoustic noise generated by the helicopter rotor and the scattering effects of the helicopter fuselage. Figure 6 is created to intuitively illustrate the whole progress from the model establishment to the result calculation, and also the combination of the methodologies and the results.
Figure 6 Schematic diagram and flow chat of the methodology approach. 
According to Figure 6, the methodology approach to the final results (sound pressure and SPL) can be described as 4 steps.
Step 1: Establishment of simulation environment.
Environment parameters, as is shown in Figure 4, is fixed in this step. The environment parameters mainly includes the relative location and the distance from the helicopter to the observer, the height and attitude angles of the helicopter, and the flight speed (inflow Mach number, M_{∞}) of the helicopter, reflecting the whole simulation environment.

Step 2: Establishment of geometry model and mesh generation.
Main geometry parameters mentioned in Section 3 is assigned in this step for the design of the helicopter geometry models. The size and the primary shape can be figured out by the main geometry parameters, and then the shape is finally decided by the detailed information such as the airfoil of the blades, the specific shape of the fuselage, etc. Mesh generation is operated after the establishment of the geometry models, for the discretization of the surfaces of the models.

Step 3: Application of method based on FW–H and BEM.
The combined methodology mainly consisting of FW–H equation and BEM helps transfer the geometry models under the specific environment into the matrixes for calculations. Firstly, the acoustic noise generated by the rotor and the sound pressure gradient is obtained by using convective FW–H equation. Then BEM is applied for the acquisition of the Green Function, the surface normal vector, etc. Finally, the acquired physical quantities are recombined into the matrixes for calculations.
Step 4: Calculation of sound pressure and SPL.
After the acquisition of the matrixes for calculations, the calculations of the final results can be easily carried out by program languages. The sound pressure of the rotor noise can be firstly obtained and the sound pressure levels (SPL) can be obtained by Fourier transformation.
4.2 Influence of fuselage types
The scattering effects of different fuselage types can be observed from Figures 7 and 8.
Figure 7 Scattering effects of different types of fuselage (polar graph). 
Figure 8 Scattering effects of different types of fuselage (XY graph). 
It can be illustrated from Figures 7, 8 and Table 2 that fuselage RAH66 has more obvious scattering effects than the other two traditional fuselages, OH6 and HUEY, with the superiority of 2.79 dB in total sound pressure level and 7.47 dB in maximum sound pressure level, indicating that the acoustic noise scattering effects vary from different fuselage types.
Total and maximum SPL of helicopters with different fuselages.
As a type of radarstealth fuselage, RAH66 fuselage consists of pieces of plains, causing sharp edges on the fuselage surface. OH6 fuselage and HUEY fuselage, however, are made up of smooth surfaces with few sharp edges. Thus an inference can be presented that the sharp edges of a helicopter fuselage can cause obvious acoustic scattering effects.
For the verification of the inference above, another two helicopter models are established, one equipped with cuboid fuselage with sharp edges and the other equipped with capsule fuselage without sharp edges, as is shown in Figures 9 and 10.
Figure 9 Helicopter model with cuboid fuselage. 
Figure 10 Helicopter model with capsule fuselage. 
The acoustic scattering effects of cuboid and capsule fuselage are shown in Figures 11 and 12.
Figure 11 Acoustic noise scattering effects of cuboid fuselage and capsule fuselage. 
Figure 12 Acoustic noise scattering effects of cuboid fuselage and capsule fuselage. 
It can be obtained from Figures 11, 12 and Table 3 that the sound pressure level of acoustic noise scattered by cuboid fuselage apparently differs from that of the rotor blades while the acoustic noise scattered by capsule fuselage is almost the same as that of the rotor blades, verifying that helicopter fuselages with substantial sharp edges causes more obvious scattering effects.
Total and maximum SPL of helicopters with cuboid and capsule fuselages.
4.3 Influence of radius of curvature of fuselage sharp edges
For quantitative studies on the influences of fuselage sharp edges, the radius of curvature (RC) of fuselage sharp edges is introduced. The variation progress from the cuboid fuselage to the capsule fuselage can be regarded to be the increase of RC of fuselage sharp edges, as is shown in Figure 13.
Figure 13 Effects of the increase of RC of fuselage sharp edges. 
For the cuboid fuselage mentioned above, the RC of the sharp edges is fixed at 100 mm, 200 mm, 300 mm, 400 mm and 500 mm, and the scattering effects of these fuselages with different RC is calculated by the method based on FW–H equation and BEM.
The acoustic scattering effects of cuboid fuselages with different RC are shown in Figure 14.
Figure 14 Acoustic scattering effects of cuboid fuselages with different RC. 
From Table 4, it can be seen that with the increase of RC, the maximum SPL shows monotone decreasing while the total SPL changes irregularly. It can be indicated from the variation of the maximum SPL that the increase of RC of the fuselage sharp edges makes contributions to the suppression of the fuselage scattering effects. As for the total SPL, it can reflects the approximate level of the SPL. However, its variation tendencies cannot reflect the influence on the scattering effects. The reason can be described that the existence of the outlier points such as some obviously low minimum points can pull down the average value, causing that the data group with higher maximum SPL and more significant scattering effects might have lower total SPL.
Maximum and total SPLs of helicopters with cuboid fuselages of different RC.
In order to preliminarily find the function relationship between the RC of the fuselage sharp edge and the maximum SPL, several assumptions are made according to the shapes of SPL–RC graph, (1/SPL) –RC graph, etc. It is finally discovered that Y = Y(SPL) is linear with X = X(RC), which can be described as:
In equation (57), SPL represents the maximum sound pressure level and RC represents the radius of curvature. Coefficients waiting for determination are a, b and k.
Figure 15 shows the scatter diagram of Y and X, indicating the linear relationship between them. According to least square method, the linear fitting equation can be obtained as
with
Figure 15 The scatter diagram and the linear fitting of Y and X. 
Besides, we can obtain the linearity relevance modulus equals to 0.9936, verifying the linear relationship between Y and X.
Therefore the estimated function relationship can be described as
where the unit of SPL is dB and the unit of RC is mm.
The estimated function relationship shows the approximate variation law of the maximum SPL related to the radius of curvature of fuselage sharp edges, and can be used for the estimation for the scattering effects of fuselages with similar shapes but different radius of curvature of fuselage sharp edges.
4.4 Influence of length, width and height of the fuselage
As is shown in Figures 16–18, 12 other models are established based on RAH66 model for the research on the variation effects of helicopter fuselage’s length, width and height. In Figure 16, the length of the fuselage is multiplied by 0.8, 0.9, 1.1, 1.2. In Figure 17, the width of the fuselage is multiplied by 0.8, 0.9, 1.1, 1.2. In Figure 18, the height of the fuselage is multiplied by 0.8, 0.9, 1.1, 1.2.
Figure 16 Fuselages with different length ratios. 
Figure 17 Fuselages with different width ratios. 
Figure 18 Fuselages with different height ratios. 
The scattering effect of fuselages of different length ratios is shown in Figure 19 and Table 4.
Figure 19 Scattering effect of fuselages of different length ratios. 
Figure 19 indicates that the scattering effects of fuselages with different length obviously differ from each other. Besides, the scattering effects appear to be more significant with a longer fuselage. It can also be seen from Table 5 that the total sound pressure level and the maximum sound pressure level raises as the length of the fuselage increases. The main reason for the result can be illustrated that a longer fuselage has a larger scattering surface, letting more sound waves reflect at the fuselage surface rather than propagate directly, and therefore make more chances for sound waves to get to the position of the detector.
Total and maximum SPL of helicopters with fuselages of different length ratios.
The scattering effect of fuselages of different width ratios is shown in Figure 20 and Table 5.
Figure 20 Scattering effect of fuselages with different width ratios. 
The considerable difference of acoustic noise scattering effects can be seen from Figure 20 that the scattering effects of fuselages with different width. But the scattering effects seem not to be more significant with a wider fuselage. It can also be seen from Table 6 that the total sound pressure level and the maximum sound pressure level sometimes rise as the fuselage expands in width, but sometimes fall.
Total and maximum SPL of helicopters with fuselages of different width ratios.
The scattering effect of fuselages of different height ratios is shown in Figure 21 and Table 7.
Figure 21 Scattering effect of fuselages with different height ratios. 
Total and maximum SPL of helicopters with fuselages of different height ratios.
Similarly, the considerable difference of acoustic noise scattering effects can be seen from Figure 21 that the scattering effects of fuselages with different height. But the scattering effects seem not to be more significant with a higher fuselage. It can also be seen from Table 6 that the total sound pressure level and the maximum sound pressure level sometimes rise as the fuselage expands in height, but sometimes fall.
The main reason for this can be explained that the affection on the scattering effects of the normal directions of surface elements, which affect the travel directions of scattered sound waves, must be taken into consideration. Multiplied by the same ratio numbers, the change of width and height bring much more variations to surface element normal directions than the change of length. Thus the affection of the surface element normal directions cannot be ignored.
To sum up, the acoustic scattering effects of helicopter fuselage can be an essential element of acoustic stealth that cannot be ignored. Considering the acoustic stealth ability of the designed helicopter, fuselages with massive sharp edges should be avoided. In addition, the length of helicopter fuselage should be limited and proper width and height of helicopter fuselage should be selected in order to take account of both acoustic stealth ability and capacity of helicopter fuselage.
Conclusions
Based on the presented FWHBEM Method (FBM), the SPL of the noise generated by the rotor and scattered by the fuselage is predicted and the factors affecting the scattering effects are studied and discussed in detail. According to the results listed above, the following conclusions can be obtained:
The acoustic scattering effect of helicopter fuselage sometimes cannot be ignored for the reason that it varies with different types of fuselages, especially some radarstealth fuselages. The total SPL of the acoustic noise of RAH66 fuselage reaches 83.14 dB, with the superiority of 2.79 dB to that of OH6 fuselage and HUEY fuselage.
Helicopter fuselages with sharp edges bring obvious scattering effects to the acoustic noise of helicopters while fuselages with few sharp edges have little scattering influence on the acoustic noise. The maximum SPL decreases when the RC of fuselage sharp edges increases. It can be obtained from the studies in this paper that for cuboid helicopter fuselages, the reciprocal of the maximum SPL is linear to the natural log of RC + a (a is a constant number). The estimated function relationship can be used for the estimation for the scattering effects of fuselages with similar shapes but different radius of curvature of fuselage sharp edges.
The length, width and height of helicopter fuselage can also be factors that affect the acoustic noise scattering effects. The scattering effect of a helicopter’s acoustic noise appears to be more significant with a longer, wider or higher fuselage. Therefore the size of the helicopter should be limited, taking both the acoustic scattering and the volume of the fuselage into account.
Conflict of interest
The authors declare that they have no conflict of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant No. 12004027).
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Cite this article as: Wang Z. Huang J. & Yi M. 2022. Acoustic scattering effect prediction of helicopter fuselage based on BEM and convective FW–H equation. Acta Acustica, 6, 24.
All Tables
All Figures
Figure 1 Establishment of the simplified helicopter geometry model (including rotor and fuselage). 

In the text 
Figure 2 Helicopter model with OH6 fuselage. 

In the text 
Figure 3 Helicopter model with HUEY fuselage. 

In the text 
Figure 4 Helicopter model with RAH66 fuselage. 

In the text 
Figure 5 Schematic of simulation case. 

In the text 
Figure 6 Schematic diagram and flow chat of the methodology approach. 

In the text 
Figure 7 Scattering effects of different types of fuselage (polar graph). 

In the text 
Figure 8 Scattering effects of different types of fuselage (XY graph). 

In the text 
Figure 9 Helicopter model with cuboid fuselage. 

In the text 
Figure 10 Helicopter model with capsule fuselage. 

In the text 
Figure 11 Acoustic noise scattering effects of cuboid fuselage and capsule fuselage. 

In the text 
Figure 12 Acoustic noise scattering effects of cuboid fuselage and capsule fuselage. 

In the text 
Figure 13 Effects of the increase of RC of fuselage sharp edges. 

In the text 
Figure 14 Acoustic scattering effects of cuboid fuselages with different RC. 

In the text 
Figure 15 The scatter diagram and the linear fitting of Y and X. 

In the text 
Figure 16 Fuselages with different length ratios. 

In the text 
Figure 17 Fuselages with different width ratios. 

In the text 
Figure 18 Fuselages with different height ratios. 

In the text 
Figure 19 Scattering effect of fuselages of different length ratios. 

In the text 
Figure 20 Scattering effect of fuselages with different width ratios. 

In the text 
Figure 21 Scattering effect of fuselages with different height ratios. 

In the text 
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