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Academic Editor:Yuri Vladimirovich Mikhlin

School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China

Received 23 November 2015; Revised 26 April 2016; Accepted 25 September 2016; 3 January 2017

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

With wide applications of helicopter, its noise level has become one of substantial indexes of helicopter performance and acceptability [1-5]. Helicopter tail rotor is a major noise source; therefore, investigation of tail rotor noise characteristic becomes distinguishably valuable and important to control the helicopter noise level. Tail rotor is an important aerodynamic assembly of conventional configuration helicopter, and its aerodynamic characteristics could have a significant impact on performance and flying qualities of the helicopter. For this reason the analysis of the tail rotor has been one of the research priorities in the field of helicopter technology [6-8].

Due to the effect of the duct, the slipstream pattern and aerodynamic characteristic of ducted tail rotor are different from the conventional open-type rotor. It can reduce the risk of component damage and enhance the operational safety. In addition, both the duct and rotor of the ducted tail rotor can produce thrust. Thrust which is produced by the duct is due to negative pressure on the duct inlet, which contributes maximum 50% to the total. Furthermore, the noise of ducted tail rotor is lower than the conventional open-type rotor [9, 10]. Consequently, the analysis of the aerodynamic noise of the ducted tail rotor is the main topic for this paper.

In order to predict the aerodynamic noise, aerodynamic characteristics of the ducted tail rotor should be obtained firstly. Along with the development of the ducted tail rotor, many researchers have studied the aerodynamic characteristics of various types of ducted tail rotors [11-13]. In recent years, the main method which investigates the aerodynamic performance of ducted tail rotor is CFD (Computational Fluid Dynamic) technique. Among those CFD techniques, momentum source method is one of the commonly used methods. The momentum source with functional relationship to the local flow conditions represents the spanning rotor in flow field and is added to the N-S equations by UDF procedure. Cao and Yu [14] used it to solve the numerical simulation of turbulent flow around ducted tail rotor. Song et al. [15] applied it to analyze the aerodynamic characteristics of ducted tail rotor. Even though momentum source method is quite successful for the prediction of the aerodynamic performance of ducted tail rotor, detailed three-dimensional flow features cannot be accurately predicted. Because of the approximation of the blade shape parameters, the accuracy of the momentum source method is still limited. In addition, it is difficult to obtain the accurate noise source information by using the momentum source method. In this work, a CFD technique which is considered the blade shape parameters has been proposed to investigate the aerodynamic characteristics of ducted tail rotor, and the method is based on moving reference frame (MRF) model instead of momentum source method.

Calculation of the acoustic field radiated by rotating sources is a meaningful problem in the prediction of noise of aircraft rotors. The main techniques for noise prediction are based on acoustic analogy Curle equation, FW-H (Ffowcs Williams-Hawkings) equation, and the generalized treatment of Goldstein [16-18]. The inhomogeneous acoustic wave equation divides the aeroacoustic source into three types: monopole source, dipole source, and quadrupole source.

Although the aerodynamic characteristics of ducted tail rotor have been discussed by many researchers [19, 20], the investigations of the aerodynamic noise are relatively few. The prediction of rotating blade aerodynamic noise can be acquired easily by solving acoustic analogy Curle equation, but the scattering effect of the duct wall on the propagation of the sound wave is difficult to discuss and the scattering mechanism is rather complex. The acoustic boundary element method (BEM) is usually applied to predict the sound radiating and scattering field in the exterior and interior closed domain [21, 22]. Hu et al. [23] proposed a thin-body BEM, in which an imaginary interface is constructed to divide the domain into interior and exterior subdomains, and the imaginary surface is not discretized in the numerical implementation. In the present study, the thin-body boundary element method is also used to consider the scattering effect of the solid wall and to predict the far-field aerodynamic noise of the ducted tail rotor.

2. Governing Equations

Because the rotary speed is constant, in order to study the noise of the ducted tail rotor, we focus on the flow field in stable state. So the unsteady Navier-Stokes equation is applied here to calculate the flow field. In rotating coordinate system, the unsteady Navier-Stokes equation can be expressed as follows [24]: [figure omitted; refer to PDF] where V is control body, ∂V is the boundary area, and n is the normal vector. W is conservation vector, and W=^(ρ,ρ^_u0^,ρ^_v0^,ρ^_w0^,ρE)T , ρ is the density of the fluid, _u0 , _v0 , _w0 are projection of absolute velocity in relative coordinate, and E is the internal energy of the unit fluid. _HW and _HV are the flux tensors. _SV is source term.

Equation (1) is discretized by using Jameson cell-centered finite volume method. An explicit time integration algorithm based on the multistep Runge-Kutta is used to advance the solution in time. The linear system obtained by using the above method is solved by applying Gauss-Seidel method. The standard k-[straight epsilon] model is adopted here because of its reasonable accuracy and fewer resources. The moving reference frame is also used to solve the problem of rotating. The far-field boundary is treated as the nonreflecting boundary condition. In order to accelerate the convergence, a local time step and implicit residue smoothing have been used. The detailed procedure for solving the governing equations can be found in [25].

3. Free Field of Rotor Noise

Lighthill aeroacoustic analogy method is used to investigate the generation of the aerodynamic noise. The general equation is given by [16] [figure omitted; refer to PDF] where _c0 is the speed of sound and ^{p[variant prime]} is the acoustic pressure. The terms on the right-hand side of (2) are interpreted as source terms. q denotes the volume source per unit volume per unit time, and ∂q/∂t is the monopole source. _fi denotes the force per unit volume, -∂_fi /∂_xi means the dipole source. _Tij is the Lighthill stress tensor and ^∂2_Tij /∂_xi ∂_xj denotes the quadrupole source.

After the CFD calculation, the aerodynamic noise at observation point is calculated using (2) by referring to the time history of the blade loading of the CFD results. Since a low rotational speed is considered for the tail rotor, the monopole source and dipole source are dominant, and quadrupole source could be negligible. A brief description will be given later to explain that the loading noise generated by dipole source is much bigger than the thickness noise generated by the monopole source for the tail rotor. Then (2) can be reduced to [figure omitted; refer to PDF] The free space solution of (3) is [26] [figure omitted; refer to PDF] where G(x,y,t-τ) is Green function given by [figure omitted; refer to PDF] where r=(x-y) denotes the distance between the observer location x and source location y. The relation between observe time t and source time τ is τ=t-r/_c0 . The square bracket means evaluation at the retarded time τ.

Using the equality [figure omitted; refer to PDF] (4) can be written as [figure omitted; refer to PDF] By using the Fourier transform and its inverse transform, force _fi , sound pressure ^{p[variant prime]} (x,t), and G(x,y,t-τ) can be given in the frequency domain as [figure omitted; refer to PDF] where k=ω/_c0 is called the wave number and j=-1 the complex unit. Substituting (10) into (7), we obtain [figure omitted; refer to PDF] Comparing (11) with (9), we have [figure omitted; refer to PDF] Differentiating (12) with regard to the direction of the normal unit vector n(x), we get [figure omitted; refer to PDF] where P(y,ω) is the static pressure on the source surface in frequency domain. The function G(x,y,ω) and its derivations ∂G(x,y,ω)/∂n(y) and ^∂2 G(x,y,ω)/∂n(x)∂n(y) in above equations are expressed as follows: [figure omitted; refer to PDF]

4. Thin-Body Acoustic Boundary Element Method (BEM)

In this section, a thin-body boundary integral formulation is applied to calculate the far-field sound pressure. Due to the scattering effect of the solid wall in the duct, the total sound pressure is acquired as the sum of the incident and scattered pressure [figure omitted; refer to PDF] where ^{PI[variant prime]} and ^{PS[variant prime]} are incident and scattered sound pressure in the frequency domain, respectively. The incident sound pressure can be obtained by using (12), and the scattered sound pressure will be got by using BEM. The calculated domain is shown in Figure 1; the surface of the duct is denoted by S and an imaginary surface s is added to close the duct and divide the computational domain into an interior subdomain ^D+ and an exterior subdomain ^D- . The sound pressure on the outside of the surface S+s is denoted by ^{P[variant prime]-} and that on the inside is denoted by ^{P[variant prime]+} . The integral equation can be used to each subdomain [23] [figure omitted; refer to PDF] where _n1 and _n2 are normal unit vectors at the two sides of the wall and ^C+ (x) and ^C- (x) are the two constants that depend on the position of x [figure omitted; refer to PDF] Adding (16) gives the thin-body boundary integral equation [figure omitted; refer to PDF] where ∂/∂_n1 =-∂/∂_n2 =∂/∂n, and the continuous boundary conditions of the pressure and its partial derivation on the imaginary surface s are used [figure omitted; refer to PDF] The assumption of acoustic rigid boundary conditions are applied over the entire surface S [figure omitted; refer to PDF] Then (18) reduce to [figure omitted; refer to PDF] Equation (22) is not sufficient to obtain the two unknowns ^{P[variant prime]+} (x,ω) and ^{P[variant prime]-} (x,ω). Differentiating (21) with regard to the direction of normal vector n(x), it can be transformed into [figure omitted; refer to PDF] The problem of scattering by the duct can be dealt with by initially solving (23) to calculate the sound pressure jump ^{P[variant prime]+} (y,ω)-^{P[variant prime]-} (y,ω) on the surface S, and afterwards evaluating the acoustic pressure at any field point. Then the acoustic pressure on both sides of the duct could be easily got.

Figure 1: A diagram of acoustic scattering by a thin-body.

[figure omitted; refer to PDF]

The integral Equations (21)-(23) cannot be directly calculated; a discretized scheme based on BEM should be used to calculate the unknown value ^{P[variant prime]+} (y,ω)-^{P[variant prime]-} (y,ω). The simplest constant boundary element is applied in this paper and the thin-body surface Sis discretized into N elements. Each element has one node which is located in the center of the element. Then, (23) can be transformed into a system of algebraic equations. [figure omitted; refer to PDF] where _[varphi]i =∂^{PI[variant prime]} (_xi ,ω)/∂n(_xi ), _Aj =^∂2 G(_xi ,_yj ,ω)/∂n(_yj )∂n(_xi ), and _{[straight phi]j} =^{P[variant prime]+} (_yj ,ω)-^{P[variant prime]-} (_yj ,ω); _xi and _yj are the center of element i, j, respectively. d_Sj is the area of the element j. Therefore, the form of matrix of (24) is written as [figure omitted; refer to PDF] Equation (25) may be solved easily by using the software Mathematica. When the unknown [straight phi] is calculated, the acoustic pressure at any field point can be obtained through (21) and (22).

5. Numerical Results and Discussions

5.1. Geometric Configuration and Mesh Generation

To demonstrate the effectiveness of the present method, we consider a model of ducted tail rotor at TsAGI, which has been installed on the Ka-60 helicopter. Figure 2 shows CATIA geometric model of TsAGI ducted tail rotor. Its geometric dimensions are tabulated in Table 1 as reported in [27].

Table 1: Geometric dimensions of the ducted tail rotor.

Rotor	Radius	297 mm
Number of blades	11
Solidity	0.4951
Twist angle	-12 deg
Tip speed	74.6 m/s
Airfoil	NACA23012 (r /R = 0.35~1.0)

Duct	Tip clearance gap	0.01R
Inlet radius	0.2R
Diffuser length	0.7R
Diffuser angle	4 degrees

Figure 2: Geometric model of TsAGI ducted tail rotor.

[figure omitted; refer to PDF]

A structured mesh is generated around the duct of the TsAGI model, and the surface quadrilateral for the all configuration is presented in Figure 3. The tail rotor of the TsAGI model is meshed with an unstructured mesh which is shown in Figure 4, and an adaptive encryption technique is adopted to mesh the leading edge and the trailing edge of the tail rotor.

Figure 3: Surface quadrilateral for the duct.

[figure omitted; refer to PDF]

Figure 4: Surface quadrilateral for the tail rotor.

[figure omitted; refer to PDF]

5.2. Aerodynamic Performance of the Ducted Tail Rotor

An initial condition is made for the ducted rail rotor at a tip Mach number of 0.22 and a collective pitch angle of 40 degree at the blade root. At this moment, the predicted thrust from the present calculation is 9.547 kg. The rotor thrust in [28] and [29] are 9.54 kg and 9.69 kg, respectively. The coincidence indicates that our method can investigate aerodynamic performance of ducted tail rotor effectively. The predicted sectional thrust distribution along the rotor in Figure 5 is also compared with the momentum source theory and the vortex theory [27], which shows a good agreement.

Figure 5: Comparison of the spanwise distribution of the blade loading.

[figure omitted; refer to PDF]

In Figures 6 and 7, spanwise distribution of the induced downwash and the circumferential induced velocity are compared with the experiment and the results obtained by vortex theory [27]. It is also shown that the induced downwash and the circumferential induced velocity are generally in good coincidence with the experiment results. The results obtained by our method also compare well with those of the vortex theory. In addition, the present results indicate better comparison than those of the vortex theory when approaching toward the tip.

Figure 6: Comparison of the spanwise distribution of the induced downwash.

[figure omitted; refer to PDF]

Figure 7: Comparison of the spanwise distribution of the circumferential induced velocity.

[figure omitted; refer to PDF]

Figure 8 shows the distribution of the pressure on the surface of the ducted tail rotor in hover. It is obvious that the pressure changes violently on the inlet of the duct while few on the other zones remain the same.

Figure 8: Pressure distribution for the surface of the ducted tail rotor in hover.

[figure omitted; refer to PDF]

A grid sensitivity analysis is given for a steady-state CFD simulation. The induced downwash, the circumferential induced velocity, and the induced attack angle are analyzed for the hybrid grids which are obtained by steadily increasing the meshing parameters. Their mean error (_emean ) and maximum error (_emax ) with respect to the reference grid are established in Table 2. D denotes the max grid cell size.

Table 2: Mean and maximum values of the induced downwash, velocity, and attack angle for different grid densities.

Meshing parameters	[...]Induced downwash	Circumferential induced velocity	Induced attack angle
Grid cells	D (mm)	_{e m e a n}	_{e m a x}	_{e m e a n}	_{e m a x}	_{e m e a n}	_{e m a x}
48 k	1.335	1.028	3.374	0.799	1.376	0.542	0.835
96 k	0.861	0.914	3.021	0.723	1.224	0.407	0.801
258 k	0.742	0.857	2.853	0.583	1.011	0.380	0.788
480 k	0.604	0.777	2.634	0.371	0.969	0.236	0.728
712 k	0.359	0.513	2.028	0.126	0.808	0.105	0.519
837 k	0.147	0.434	1.634	0.101	0.761	0.095	0.353
968 k	0.082	0.125	1.008	0.098	0.486	0.088	0.104

From Table 2, we can see that the errors for all three flow variables are smaller and smaller when the grid cells increase. The small errors also display a converging trend.

To verify the effectiveness and the accuracy of the CFD method, the analytical solution of a dipole source has been investigated. A dipole has an axis, which is the line connecting the centers of the two monopoles (pulsating sphere with radius of a) in Figure 9. The two monopoles are positioned in the z direction at a distance of d=2.5a from the center (d is the distance between the two monopole points). The pressure fluctuation induced by the dipole source for an observer positioned at (x,y,z) is [figure omitted; refer to PDF] where _[varphi]0 =ta^n-1 [...](1/ka), _r1 =^x2 +^y2 +^(z+0.5d)2 , and _r2 =^x2 +^y2 +^(z-0.5d)2 .

Figure 9: Dipole source and spherical are shown.

[figure omitted; refer to PDF]

To perform the flow field characteristic of the dipole source, we put the dipole source into the center of a sphere (the radius _rs of the sphere to be 3.25a). The speed of sound _c0 is 340 m/s. The density for medium is 1.2 kg/m³ . The angular velocity of the source is 1020 rad/s. The other parameters are A=4π^a2 U, a=0.01 m, and U=8 m/s. The value of the spherical pressure at any point can be acquired by using the CFD method and (26), respectively.

The calculation is performed for this case, and the spherical pressure at any point is calculated by using CFD method and analytic method. The CFD results are compared with those of the analytical solution in Figure 10. From Figure 10, we can see that the CFD results are in perfect coincidence with the analytical solution.

Figure 10: The spherical pressure at any point for different method.

[figure omitted; refer to PDF]

5.3. Aerodynamic Noise of the Ducted Tail Rotor

The sound pressure is expressed as dB (decibels) and the predicted SPLs (sound pressure levels) are given by the following: [figure omitted; refer to PDF] where _Pe denotes predicted pressure and _Pr denotes the reference pressure and equals 2 × 10^-5 Pa.

Before investigating the aerodynamic noise of the TsAGI ducted tail rotor, we firstly consider the radiation sound field generated by its isolate tail rotor. Figures 11 and 12 are the directivity of SPLs of the thickness noise and the loading noise, respectively. They can be obtained by using Farassat 1A formulation [26]. From the two figures, we can find that the thickness noise generated by the monopole source is much smaller than the loading noise generated by the dipole source for the isolate tail rotor. For the ducted tail rotor, the loading noise is dominant in rotation noise. The main reason is that the number of tail rotors is too high, and the size of tail rotors is too small. Therefore, the loading noise generated by the dipole source is considered only for the following discussions.

Figure 11: Directivity of SPLs of the thickness noise at different distance.

[figure omitted; refer to PDF]

Figure 12: Directivity of SPLs of the loading noise at different distance.

[figure omitted; refer to PDF]

Through the analysis of aerodynamic performance of TsAGI ducted tail rotor, we can acquire the unsteady flow properties and the enough pressure data of the surface of the rotor. Then they are transformed into data in frequency domain by applying fast Fourier transform method. The far-field sound pressure is calculated using Curle's analogy in the frequency domain. Before using the BEM method to study the radiation and propagation of the sound source, we should use a simplified configuration which is shown in Figure 13 to consider the scattering effect of the duct. From Figure 13, we know that the surface is constructed by quadrilateral mesh. The maximum size of the mesh is 0.023 m and the number of elements is 2584. In this section, we use a point force method which regards the rotors as a series of points to investigate the scattering effect of the duct. The method can be applied in the prediction of the noise when the force fluctuations of the rotors are obtained.

Figure 13: Mesh of duct used in BEM.

[figure omitted; refer to PDF]

In the present article, we just focus on rotation noise of the ducted tail rotor. The dominant noise in rotation noise is blade passing frequency (BPF) noise and its harmonic noise [30]. And the BPF of the ducted tail rotor is 2400 × 11/60 = 440 Hz.

Acoustic pressure in frequency domain is calculated by combining Curle equation and the thin-body BEM. For different observation distance, the directivity of SPLs at the BPF and 2BPF are shown in Figures 14 and 15, respectively. Free field is also rotor alone. The more interesting result is the comparison of the total field and the free field. This provides information about the scattering effect of the duct. For the angle (180 to 360 degrees), upstream of the rotor (180 to 270 degrees) is slightly louder due to scattering. And at the angle (270 to 360 degrees), it is quieter. From Figure 14(a), we can see clearly that the sound pressure at inlet is bigger than that of the outlet. From Figures 14(b), 14(c), and 15, we can also find the directivity of the incident sound pressure is symmetrical as the properties of the dipole sound source. These characteristics are consistent with the fact.

Figure 14: Noise directivity pattern at the BPF: (a) R=1 m; (b) R=5 m; (c) R=3 m.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Figure 15: Noise directivity pattern at the 2BPF: R=3 m.

[figure omitted; refer to PDF]

For 30 m/s forward flight velocity, the directivity of SPLs at the BPF is shown in Figure 16. The polar plot is made at a distance of ten times the rotor radius (10R ) away from the center of the duct. Compared with the SPLs in Figure 14, the SPLs for forward flight are bigger than the SPLs for hover, and the noise directivity is also changed.

Figure 16: Noise directivity pattern at the BPF: 30 m/s forward flight.

[figure omitted; refer to PDF]

The directivity of SPLs at the BPF for rotational speed 2100 rpm and 2400 rpm are displayed in Figure 17. The SPLs of rotational speed 2400 rpm are bigger than that of 2100 rpm; this can be noticed by taking a closer look at Figure 17. The directivity of SPLs at the BPF for various observation distances is presented in Figure 18. From Figure 18, we can find that the SPLs become more and more small when the observation distance increases.

Figure 17: Directivity of SPLs of the ducted tail rotor at different rotational speed.

[figure omitted; refer to PDF]

Figure 18: Directivity of SPLs of the ducted tail rotor at different distance.

[figure omitted; refer to PDF]

In order to illustrate the validation of the scheme, the results obtained in this paper are compared with the results obtained by commercial software Virtual.Lab Acoustics (VLA) [30]. The comparisons are shown in Figure 19.

Figure 19: The comparison of the SPLs at different distance: (a) R=1 m; (b) R=3 m.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

In Figure 19, at the majority of azimuth of the ducted tail rotor, the SPLs obtained by our method are in good agreement with those obtained by using VLA.

6. Conclusions

The aim of this paper is to develop the Curle/thin-body BEM method for predicting the aerodynamic noise of the ducted tail rotor numerically. A CFD method based on MRF model is used to study the aerodynamic performance of ducted tail rotor. Calculations are made for TsAGI ducted tail rotor, and the results are compared with the known results and the analytical solution of a dipole source for validation. The unsteady blade force is also obtained and transferred to the Curle equation to calculate the sound source without the duct. In order to consider the scattering effect of the duct, a thin-body BEM is proposed to predict the far-field aerodynamic noise. The calculated SPLs at BPF accord well with the results obtained by VLA and show that our method can effectively predict the far-field SPLs of the ducted tail rotor. The numerical results also show that the scattering effect of the duct should be included in the prediction of the noise. The control of the inlet noise is advisable for the ducted tail rotor noise reduction.

Nomenclature