[american institute of aeronautics and astronautics 13th aiaa/ceas aeroacoustics conference (28th...

Numerical Study on Fan Noise Generated by Rotor-Stator Interaction

Junichi Kazawa* Japan Aerospace Exploration Agency,7-44-1 Jindaiji-higashi-machi Chofu, Tokyo, Japan, 182-8522

Yasuo Horiguchi†

Foundation for promotion of Japanese Aerospace Technology, 1-16-6, Izumi-chuo Izumi-ku Sendai, Miyagi, Japan, 981-3133

Kazuhisa Saiki‡, Kazuomi Yamamoto§, Osamu Nozaki**

Japan Aerospace Exploration Agency,7-44-1 Jindaiji-higashi-machi Chofu, Tokyo, Japan, 182-8522

and

Tsutomu Oishi††

Ishikawajima-Harima Heavy Industry co,ltd., 229, Tonogaya, Mizuho-machi, Nishitama-gun, Tokyo, Japan, 190-1297

[Abstract] Numerical simulations based on the Unsteady Raynolds-Averaged Navier-Stokes equations are carried out. As a preliminary case, the case where plane waves interacts with a cascade which consists of the flat plate blades is analyzed to verify the accuracy of our numerical code. The results are compared with those calculated by theoretical analyses. Our results show good agreement with the theoretical results. Then, the fan noise generated by rotor-stator interaction is analyzed in the realistic fan. The three-dimensional behavior and the structure of 3D waves around the rotor blade row are revealed in detail by our numerical code.

Nomenclature A = amplitude of source term B = number of rotor blades D = Y-direction length of computational domain S(y,t) = noise source to generate the plane wave at position y and time t T = period of the plane wave U = flow velocity at inlet boundary V = number of stator vanes c = speed of sound at inlet boundary f = frequency of the plane wave h = the order of blade passing frequency (BPF) k = arbitrary integer number m = circumferential acoustic mode number yi,j,k = Y-coordinate of the point (i,j,k) Ω = rotational speed of the rotor blade row

* Project Researcher, Institute of Aerospace Technology, 7-44-1 JindaijiHigashi Chofu, Tokyo. Member AIAA. † 1-16-6, Izumi-chuo Izumi-ku Sendai, Miyagi. ‡ Researcher, Institute of Aerospace Technology, 7-44-1 JindaijiHigashi Chofu, Tokyo. § Section Leader, Aviation Program Group, 7-44-1 JindaijiHigashi Chofu, Tokyo. Senior Member AIAA. ** Section Leader, Institute of Aerospace Technology, 7-44-1 JindaijiHigashi Chofu, Tokyo. †† 229, Tonogaya, Mizuho-machi, Nishitama-gun, Tokyo.

American Institute of Aeronautics and Astronautics

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13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference) AIAA 2007-3681

Copyright © 2007 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

Ωm = rotational speed of the acoustic mode m α = angle of the initial wave shown in Fig.4 β = angle of the reflected wave shown in Fig.4 ε = incidence angle of the initial wave µ = radial acoustic mode number θ = flow angle at inlet boundary

I. Introduction ecently, the restriction on the noise around airports becomes severer and noise reduction is an important subject for the aircraft industries. The fan noise generated in jet engines is one of the main factors of aircraft noises. To

achieve a quiet jet engine, fan noise reduction is very important and it is necessary to develop the numerical technique of high-accurate fan noise prediction. The fan noise is classified into broadband noise, buzz-saw noise and rotor-stator interaction noise. In the present study, the rotor-stator interaction noise is analyzed with numerical method based on Unsteady Reynolds-Averaged Navier-Stokes (URANS) equations. The rotor-stator interaction noise is generated by the interaction between the rotor wakes and the stator vanes. When the rotor wakes interact with the stator vanes, the pressure fluctuation is generated around the stator vanes. This pressure fluctuation forms the acoustic waves and propagate upstream and downstream of the stator vanes. Regarding the prediction of the rotor-stator interaction noise, Tsuchiya, et al1 have developed a hybrid method. This method is the hybrid of the non-linear calculation and linear analysis. In this method, the unsteady load acted on the stator vanes is obtained by URANS and this unsteady load is treated as the noise source. The propagation of the rotor-stator interaction noise is calculated by the three-dimensional linear theory. Although this method can predict the generation and propagation of the rotor-stator interaction noise, the reflection and transmission of the acoustic waves at the rotor blade row can not be simulated. URANS calculation is necessary for the prediction of the rotor-stator interaction noise including the reflection and transmission of the acoustic waves at the rotor blade row. Some researchers have carried out URANS calculations to analyze rotor-stator interaction noises so far2-3. Their results show the possibility to predict rotor-stator interaction noises by URANS calculation but details of acoustic fields around a fan is not described. In this paper, analyses of two cases are conducted with regard to the prediction of the rotor-stator interaction noise with UPACS4 which is a CFD code being developed in Japan Aerospace Exploration Agency. One is a case where an acoustic wave propagates through a cascade with flat plate blades. The results are compared with those calculated theoretically by Kaji and Okazaki5. The other is a case where fan noise is generated by the rotor-stator interaction in a realistic fan. The circumferential acoustic modes are extracted by Fast Fourier Transform (FFT) applied to the pressure fluctuation, and the radial modes of each circumferential modes are obtained by Bessel Function6. From these results, the acoustic fields around the rotor blade row are analyzed in detail.

R

II. Interaction between a Cascade with Flat Plate Blades and Plane Wave A case where an acoustic wave propagates through a cascade with flat plate blades is analyzed. The pressure

fluctuation generated by rotor-stator interaction forms acoustic wave which propagates upstream and downstream of the stator. This wave is transmitted and reflected by the rotor blades. To verify the accuracy of our numerical code for analyzing the interaction between the acoustic wave and rotor blade row, our numerical results are compared with the theoretical analysis results calculated by Kaji and Okazaki5, who investigated transmission and reflection of sound waves by a single blade row in a two-dimensional subsonic flow fields by the thin airfoil theory based on the singular solution method.

Figure 1. Computational domain and Boundary Conditions

A. Generation of Plane Wave and Flow Conditions Euler equations are adopted as the

governing equations, 3rd-order MUSCL scheme and 3rd-order Runge-Kutta explicit time marching scheme are used. Figure 1 shows computational domain. Buffer regions are adopted at the inlet and outlet boundaries to avoid the reflection of the wave at these boundaries. Stagger angle and solidity of the cascade is 60 degrees and unity, respectively. The inlet Mach number is 0.5 and incidence


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angle is 0 degree. All length scales are reduced by chord length of the flat plate blade. Noise sources are located in the downstream of the cascade adding the source term to the right hand side of the

equation of continuity. To generate the plane wave with frequency of f, noise sources, S(y, t), are located at each grid points on the line of constant X as follows;

( ) ( )( )DyftAtyS kji ,,2sin, += π (1)

where, A is amplitude of source term, f is frequency of the wave, yi,j,k is Y-coordinate of the point (i,j,k) and D is Y-direction length of the computational domain. The angle of the plane wave can be controlled by changing D.

The angle of plane wave measured from X axis, ε , is obtained by following equation;

( )( ) εεθ sincos =−− DTUc (2)

where, ε is positive in the counter-clockwise direction, c, U, and θ are the speed of sound, flow velocity and flow angle at the inlet boundary, respectively. T is period of the plane wave.

In the present study, three plane waves are analyzed. One is a plane parallel to the cascade, ε = 0 degree, and the other three cases are those with ε is 33.5, 53.6 degrees, where ε is calculated by Eq.(2). The wave length of these waves is 8. Figure 2 shows the instantaneous static pressure contours when plane wave with ε is 0 degree and 33.5 degrees of ε is given in the flow field.

B. Comparison between the Results by UPACS and the Theoretical Analysis Results In these cases, the amplitude of the reflected wave could not be calculated because the initial wave interacts with

the reflected wave. To extract the reflected wave, the case without the flat plate blade is calculated. The amplitude of the reflected wave can be calculated by the difference between the results in the case with flat plate blade and those without flat plate blade. Figure 3 shows the instantaneous pressure fluctuation contours of the reflected wave in the case where ε is 0 degree and 33.5 degrees. The reflected wave can be clearly seen in each cases. In the literature 5 the angle of reflected wave, β, as shown in Fig.4, can be calculated by the following equation;

( ) ( ) ( ) ( )θααθββ −−=++ sincos1sincos1 MM (3)

ε = 0 degree

ε = 33.5 degrees

Figure 2. Instantaneous Static Pressure Contours of Initial Wave

ε = 0 degree

ε = 33.5 degrees

Figure 3 Instantaneous Static Pressure Fluctuation Contours of Reflected Wave


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where, α is the incidence angle of the initial wave shown in Fig.4.

β

α

θ

ε

Reflected Wave

Initial Wave

Transmitted

Flat Plate

X

Y

Wave

Flow

Figure 4. Co-ordinates in the computation

Table 1 is the comparison of the reflected wave angle between the results calculated by Eq.(3) and the present results obtained by UPACS4. Figure 5 shows the transmission and reflection ratio. These ratio are defined as the pressure amplitude of the transmitted wave or the reflected wave divided by that of the initial wave. The pressure amplitude of each wave is calculated by Fast Fourier Transform (FFT). The results by Kaji and Okazaki are shown in the figure for comparison. The horizontal axis is the incidence angle of the plane wave, and the vertical axis is transmission and reflection ratio. The symbols indicate the present results and the lines are those calculated by Kaji and Okazaki. The reflection ratio is decreased as incidence angle of plane wave approaches stagger angle. In Table 1 and Fig.5, the present results and theoretical analysis results show good

mod45, Newcomper aboFigroto

Fig

Table 1. Comparison of the Angle of Reflected Wave ε (deg) 0 33.5 53.6

Theory -60 -113.3 -140.2 β (deg) Present -60 -113 -141

agreement.

III. Analysis of Rotor-Stator Interaction Noise in the Realistic Fan URANS calculation is carried out to analyze the rotor-stator interaction noise in the realistic fan. The cascade el is the baseline model which Tsuchiya et al.1 analyzed. The number of rotor blades and stator vanes is 18 and respectively in this model. 2nd-order upwind scheme and 2nd-order Euler implicit time marching scheme with ton iteration are used, and Spalart-Allmaras turbulence model is used in our analysis. Figure 6 shows the putational grid. To capture the Blade Passing Frequency (BPF) wave, grid density is from 20 to 30 grid points wave length in the rotor section and the stator section. Total number of grid points in computational domain is ut twelve millions. Rotational speed of the rotor was 80% of the design point which is the approach condition. ure 7 shows the total pressure contours on the tip, mid-span and hub plane in this case. It can be clearly seen that r wakes interacts with stator vanes. On the tip plane, shock waves exist on the suction side of the rotor blades.

Figure 6. Computational Grid

−50 500

0.5

1

Transmitted wave

Reflected wave

Incidence of initial wave

Tran

smis

sion

/Ref

lect

ion

ratio

0

Kaji and Okazakipresent

ure 5. Reflection and Transmission Ratio


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Tip Plane Mid-Span Plane Hub Plane

Figure 7. Total Pressure Contours


Figure 8. Instantaneous Pressure Fluctuation Contours

A. Circumferential Mode Analyses The unsteady interaction between the rotor wakes and the stator vanes generates pressure fluctuations around the

stator vanes. These pressure fluctuations form acoustic waves. Figure 8 shows the instantaneous pressure fluctuation contours on the tip, mid-span and hub plane. This pressure fluctuation is defined as the point-wise difference between the instantaneous static pressure and the time-averaged one in the rotor section and the stator section, respectively. Lines indicate the wave front which is dominant in the results. In the stator section, waves are generated around the stator vanes and propagate to upstream and downstream of the stator vanes on the tip, mid-span and hub plane. In the rotor section, the waves propagate from the downstream of the rotor blade row and these upcoming waves interact with the rotor blades. As a clue to understand three-dimensional wave propagation and reflection, the perspective of the linearized theory written in the previous chapter is applied to the results. As shown in Fig.5, when the difference between the stagger angle and the incidence angle of the wave to the rotor blades is small, the reflection ratio is small and the transmission ratio is large. On the hub plane, the difference of these angle is small and most of the upcoming waves propagate through the rotor blades. On the mid-span and tip plane, the reflection of the waves are clearly seen in the rotor section since the difference between the stagger angle and the incidence of the wave to the rotor blades is large. The upcoming waves interact with the reflected wave, and the acoustic field between the rotor blades and the stator vanes is complex. On the mid-span plane, waves propagate to the upstream direction of the rotor blade row. These are transmitted wave through the rotor blades. On the tip plane, the wave propagating through the rotor blades is very weak since the shock wave blocks the wave propagation to the upstream of the rotor blades.

To analyze the rotor-stator interaction noise, FFT is applied to the pressure fluctuation data to extract BPF and its higher harmonic components. The circumferential acoustic mode, m, defined as Eq.(4) is introduced.

kVhBm −= (4)


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where, h is the order of BPF, B and V are the number of rotor blades and stator vanes, respectively and k is an arbitrary integer number. m is the theoretical number of the wave in the circumferential direction. Rotational speed of the circumferential acoustic mode m, is calculated theoretically as follows;

mhBm /Ω=Ω (5)

where, Ω is rotational speed of the rotor blade row. When m is positive, the wave of mode m rotates in the same direction of the rotor rotation.

The fundamental frequency of the rotor-stator interaction noise is BPF. Resulting BPF waves are not described in this paper since this frequency waves are cut-off mode. Figure 9 shows the pressure fluctuation contours of 2BPF on the tip, mid-span and hub plane in the stator section. Solid lines indicate the 2BPF wave front. The circumferential acoustic mode of this wave is clearly -9 (single peak in each 5 stator vanes out of 45 stator vanes) and the found rotational speed of this mode is about four times faster than the speed of the rotor rotation in the opposite direction of the rotor rotation. These numerical results from Fig.9 are in accordance with the theoretical acoustic mode of 2BPF and the rotational speed of this mode when k is unity in Eq.(4).

The wave generated around the stator vanes propagates through and gets reflected at the rotor blade row. Figure 10 shows the pressure fluctuation contours of 2BPF and 3BPF components on the tip, mid-span and hub plane in the upstream and downstream region of the rotor blade row. The circumferential mode of 2BPF is -9 and that of 3BPF is 9. The results in one-ninth sector (40 degrees) of the annulus are presented in the figure. The number of the wave in the circumferential direction is unity in the figure when m is -9. In the upstream region, 2BPF and 3BPF waves propagate to the upstream direction both on the mid-span and hub plane. On the tip plane, the 2BPF and 3BPF waves propagating to the upstream are weak because of the shock wave blocking as written in the above. Figure 11 shows the pressure fluctuation contours of 4BPF components whose m is 27 in the upstream and downstream region. In the upstream region of the rotor blades, the waves propagating to the upstream thorough the rotor blades are


Figure 9. Pressure Fluctuation Contours of 2BPF around Stator

Figure 11. Amplitude of 4BPF Components around Rotor Blade Row

2BPF 3BPF

Figure 10. Amplitude of 2BPF and 3BPF Components around Rotor Blade Row


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observed both on the mid-span and hub plane. While in the downstream region of rotor blades, the waves are not clear on the mid-span, and the waves on the hub plane seems propagating to the downstream not, upstream. So 4BPF waves corresponding to the waves in the upstream region both on the mid-span and hub plane seems absent in the downstream region. On account of the presence of strong 2BPF waves in the downstream as shown in Fig.10, one possible explanation is follows. When the wave whose the circumferential acoustic mode is m interacts with the rotor blade row, another circumferential acoustic mode defined as m’, is generated as following equation;

nBmm ±=' (n=1,2,3...) (6)

where, n is arbitrary integer number. In this case, m of 2BPF is -9. m’ is 27 when n is unity. The acoustic mode of 4BPF shown in Fig.11 is 27. Thus 4BPF wave is generated by interaction between 2BPF wave and rotor blade row.

B. Radial Mode Analyses To analyze the acoustic field around the rotor blades in more detail, radial modes are extracted by Bessel

Function6. In this paper, (m, µ) mode is defined as the wave whose circumferential mode is m and the radial mode is µ. Figure 12 shows the 2BPF components whose m is -9 on the tip, mid-span, hub plane in the upstream and downstream region of the rotor blades. Figure 13 shows these components on the radial cross section. The radial modes of µ from 0 to 6 are shown in these figure. In the downstream region, (-9, 3) and (-9, 4) modes are dominant, while in the upstream region, (-9, 2), (-9, 3), (-9, 4), (-9, 5) modes are almost equally dominant. (-9, 3) and (-9, 4) modes are considerably decayed in the upstream while the waves propagate through the rotor blades.

Figure 12. Amplitude of Radial Mode Components (2BPF, m = -9) on the Tip, Mid-Span, Hub Plane

Figure 13. Amplitude of Radial Mode Components (2BPF, m = -9) on the Radial Cross Section


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Figures 14 and 15 show the 4BPF components whose m is 27 on circumferential cross section and radial cross section, respectively. In the upstream region, (27, 4) and (27, 5) modes are dominant. These waves propagate to the upstream of the rotor blade row. In the downstream region, the acoustic mode of µ from 1 to 4 are dominant. The waves of (27, 1) and (27, 2) modes are reflected waves because these waves propagate to the downstream. Propagation direction of the waves (27, 3) and (27, 4) is not clear since the wave angle of (27, 3) and (27, 4) is almost same as the rotational direction. (27, 6) is the only mode which propagates to the upstream in the downstream region. Although the transmission and reflection of (27, 6) mode by the rotor blades should generate the reflected waves of µ from 1 to 4 in the downstream region and the transmitted (27, 3) and (27, 4) modes in the upstream region, the amplitude of reflected waves is larger than that of (27, 6) mode. This is unusual. So the explanation at the end of section A seems appropriate; interaction between (-9, 3) or (-9, 4) mode of 2BPF waves (see Figs.12, 13) and rotor blade row generates the waves of µ from 1 to 4 in the downstream region and (27, 3) and (27, 4) modes of 4BPF from Fig.14 in the upstream region.

IV. Concluding Remarks Analyses of two cases were conducted with regard to the prediction of the rotor-stator interaction noise with

UPACS. To verify the accuracy of our numerical code applied to the interaction between the acoustic wave and the rotor blade row, the case where the cascade which consists of flat plate blades interacts with plane wave was analyzed. The reflection ratio, transmission ratio and the angle of the reflection wave of our results were compared

Figure 14. Amplitude of Radial Mode Components (4BPF, m = 27) on the Tip, Mid-Span, Hub Plane

Figure 15. Amplitude of Radial Mode Components (4BPF, m = 27) on the Radial Cross Section


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with those calculated by a theoretical method. Our results and the theoretical analysis results showed good agreement.

URANS calculation was carried out to analyze the rotor-stator interaction noise in the realistic fan. FFT was applied to the results to extract the circumferential modes. The number of circumferential acoustic mode and the rotational speed of the modes conformed with those calculated by the linear theory. To elucidate the three-dimensional nature of the acoustic fields, radial modes were also extracted by Bessel Function. From the results, the behavior of dominant BPF waves around the rotor blade row was revealed.

References 1Tsuchiya, N., et al., “Low Noise FEGV Designed by Numerical Method Based on CFD,” Proceedings of ASME Turbo Expo 2004 [CD-ROM], GT2004-53239, Vienna, Austria, 2004. 2Rumsey, C. L., et al.: “Ducted-Fan Engine Acoustic Predictions Using a Navier-Stokes Code,” Journal of Sound and Vibration, Vol.213, No.4 , 1998, pp.643-664. 3Biedron, R. T., et al., “Predicting the Rotor-Stator Interaction Acoustics of a Ducted Fan Engine,” AIAA Paper 2001-0664, 2001. 4Yamane, T., Yamamoto, K., Enomoto, S., Yamazaki, H., Takaki, R., and Iwamiya, T., “Development of a Common CFD Platform - UPACS -,” in Parallel Computational Fluid Dynamics - Proceedings of the Parallel CFD 2000 Conference, Trondheim, Norway, Elsevier Science B. V., 2001, 257-264. 5Kaji, S., Okazaki, T., “Propagation of Sound Waves through a Blade Row I & II,” Journal of Sound and Vibration, Vol.11, No.3, 1970, pp.339-375. 6Tyler, J. M., Sofrin, T.G., “Axial Flow Compressor Noise Studies,” SAE Transactions, Vol.70, pp.209-332, 1962


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