Investigation of rotor–stator interaction broadband noise using a RANS-informed analytical method

Hng TONG, Lin LI, Lingfeng WANG, Luqin MAO, Weiyng QIAO,*

a School of Power and Energy, Northwestern Polytechnical University, Xi’an 710129, China

b High Speed Aerodynamics Institute, China Aerodynamics Research and Development Center, Mianyang 621000, China

KEYWORDS Acoustic analogy;Analytical methods;Rotor-stator interaction broadband noise;Strip theory;Wavy leading edge

Abstract A Reynolds-Averaged Navier Stokes (RANS)-information analytical method for predicting Rotor-Stator Interaction (RSI) broadband noise is established in this paper. First, the turbulence information is deduced from RANS simulation result. Then, the unsteady load on the stator blade is calculated using a strip theory approach based on LINearized SUBsonic unsteady flow in cascade(LINSUB)and 2-D equivalence method.In the end,the sound power of RSI broadband noise is calculated by coupling the unsteady load on the stator blade with acoustic analogy and annular duct mode. The broadband noise model part of the RANS-information analytical method is validated against the upstream sound power of an annular cascade experimental bench.Besides, the RANS-information analytical method is used in predicting RSI broadband noise of a single-stage axial fan acoustic experimental bench,the results illustrate that the RANS-information analytical method can accurately predict the RSI broadband noise in different fan working conditions. After simplification the Wave Leading Edge (WLE) stator blade, the effect of WLE stator blade on RSI broadband noise is studies.Although the simplification may bring some discrepancies,the results illustrate that the RANS-information analytical method has the capability for further studies on the broadband noise reduction with WLE stator blade.

1. Introduction

Aircraft noise not only affects the comfort of passengers, but also the life of airborne equipment. Therefore, the aircraft noise control technology has been becoming more and more important for commercial aviation.1For today’s commercial aviation industry, the dominant noise sources of aircraft are commercial aero-engine in majority flight condition. With the increase of hub ratio, the jet noise dropped significantly,while the fan noise became one of the most significant noise sources of commercial aero-engine.2

In general, tonal noise and broadband noise are the two main components of fan noise. The fan tonal noise is mainly caused by the interaction of viscous wakes caused by rotor blades and downstream stator blades.In the past,most studies on fan noise in the world were focused on tonal noise.3–6A large number of effective technologies and methods have been applied to fan tonal noise reduction, such as the selection of rotor/stator blade number to cut off the dominant acoustic modes at blade passing frequencies,7using swept stator blade and leaned blades,8–10active noise control technology by generating antiphase acoustic pressure,11,12etc.At present,the fan tonal noise has greatly reduced thank to these effective technologies and methods. As a result, it is expected that the contribution of the broadband noise will become relatively more significant than before.

Compare with fan tonal noise,the mechanism of fan broadband noise is much more complex, such as the interaction of upstream turbulence with rotor blades, the interaction of viscous wakes caused by rotor blades and downstream stator blades,the interaction of boundary layer turbulence with rotor blade,the interaction of boundary layer turbulence with stator blade, etc.13,14Among these, the most significant broadband noise source is the interaction of viscous wakes caused by rotor blades and downstream stator blades, which is called RSI broadband noise. The RSI broadband noise has always been the focus and difficulty in recent years.

Until now, many studies on fan broadband noise are still based on the method of Computational Aeroacoustics(CAA)15–17and scale-resolving simulations.18These methods show high precision, while require large calculation time and resource consumption. Therefore, these methods are not suitable to include the fan broadband noise prediction into the fan design cycles. Besides, Investigation of RSI broadband noise with experimental methods present the difficulty because of the equipment cost and complex flow conditions. After all,the analytical method to study the RSI broadband noise is still more efficient.

The key to the analysis methods is the selection of blade response functions. The blade response functions are mainly divided into two categories: One type of blade response function is using the Wiener-Hopf method to calculate the sound velocity potential. Mani19and Amiet20first investigated the turbulence ingestion broadband noise with Wiener-Hopf method. The effects cause by rectilinear cascade sweep blade was investigated by Glegg21with an acoustic response function.Then,the acoustic response function by Glegg21was further expanded into the simulation of cascade passage unsteady pressure by Posson et al.22–24In 2016, Masson at al.25predicted the fan broadband noise and validated by rotating cascades and strongly swirling mean flow. The other blade response function is to calculate the unsteady load on the blade surface caused by the interaction between vorticity and cascade.This method is first investigated by Smith,26Whitehead27and Kaji.28,29Based on this kind of blade response function,they simulated the interaction of vorticity and cascade and simulated the propagation of sound waves in the cascade. A bound vorticity 2-D method LINSUB proposed by Smith26has become a significant basis for the turbomachinery noise prediction.In 1995,Hanson developed a tonal noise prediction method for turbomachinery based on LINSUB code.30,31Ju et al.32,33extended the LINSUB code to the quasi-3-D prediction method and obtained a good agreement with the test results.

When using traditional analytical model, the input of turbulence information is often empirical and homogeneous.The predicted result is quite different from the actual result.In last decade, the RANS-informed analytical method to investigation the RSI noise has been the focus of many studies.To summarize,the solving steps of RANS-informed analytical methods for prediction of RSI broadband noise is as following:First,collect the turbulence information in front of stators leading edge from RANS results. Then, couple the turbulence spectrum from turbulence information with response function to calculate unsteady load on stator blade surface or velocity potential. In the end, the acoustic power is calculated by coupling the unsteady load on stator blade surface or velocity potential with acoustic analogy and annular duct mode.Compared with CAA method or scale-resolving simulations, the RANS-informed analytical method is more affordable terms of computational time and resource consumption. Compared with traditional analytical model, the RANS-informed analytical method is stricter, and show higher accuracy.

There is no doubt that the RANS-informed analytical method to study the RSI broadband noise,based on the turbulence information from RANS result and analytical method,are much more feasible. However, RANS-informed analytical method, as a new development method in the last ten years,still needs further research. The aim of the RANS-informed analytical method for investigating RSI broadband noise is to correlate the fan design parameters related to noise and develop effective noise reduction method. Therefore, it is necessary to increase the universality of this algorithm and to correlate more blade design parameters.

In recent decades, there are still a large number of scholars to achieve the purpose of noise reduction by optimizing the blade configuration.34And a more effective noise reduction methods based on bionic theory has been found in the acoustic research field. WLE blade have been widely used in the noise reduction by the interaction of turbulence and airfoil.Although there are a lot of studies on WLE blade, until now,many studies related to noise reduction with WLE blade still focused on the interaction between turbulence and airfoils.35–40Few studies by Reboul41and Tong42et al. have shown that the WLE blade is also effective for the noise reduction. However, these studies were all based on Large Eddy Simulation (LES) with large computation time and resource cost, which limited the development of studies on the noise reduction with WLE stator blade.By coupling the WLE design parameters into RANS-informed analytical method, further development of studies on the noise reduction with WLE blades become possible.

In this paper,first,a RANS-information analytical method for predicting RSI broadband noise, based on LINSUB by Smith26and 2-D equivalence method by Ju et al.,32is established.After that,the method is validated against the upstream Power Spectral Density (PSD) of sound power of an annular cascade experimental bench. Besides, the method is used in predicting RSI broadband noise of a single-stage axial fan acoustic experimental bench. In the end, the feasibility of broadband noise reduction with WLE stator blade using this quasi-3-D RANS-information analytical method is explored.

2. Mechanism of upstream turbulence-annular cascade interaction broadband noise model

An upstream turbulence-annular cascade interaction broadband noise model is established in this section. First, the upwash velocity, turbulence spectrum and radial integral length correction are briefly introduced. Then, the quasi-three dimensional method based on LINSUB proposed by Smith26and 2-D equivalence method by Ju et al.32,43is used as upstream turbulence-annular cascade acoustic response function,and the Liepmann spectrum is chosen as input turbulence to couple with acoustic response function. By coupling the unsteady load on the cascade blade with acoustic analogy and annular duct mode, the expression of sound power is derived. Besides, without loss of generality, the baseline blade and WLE blade are discretized into flat plates grid composed of radial nodes and chord nodes. In the end, after considering the effect of WLE blade,a general expression of sound power,which is suitable for various blades, is derived.

2.1. Upwash velocity and turbulence spectrum

The expressions of upwash velocity and turbulence spectrum upstream of the stator blade are given and discussed thanks to Ju’s work.Three coordinates system(one cylindrical coordinate system(x;r;φ)as shown in Fig.1(a),two Cartesian coordinate systems (x;y;z) and (ξ;η;z) on the Face 1 as shown in Fig. 1(b) are set up. Besides, the cascade annular blades are simplified to plates without bending angle and blade thickness.

The transformation relations between the (x;y;z) and(x;r;φ) are:

The transformation relations between (x;y;z) and (ξ;η;z)are:

where θ is the flow incident angle.

The wave number vectors in (x;y;z) and (ξ;η;z) are(a;β;k3) and (k1;k2;k3), respectively. The relations of two wave number vectors are:

Fig. 1 Coordinate system for simplified blade and duct.

where ϖ(φ) is turbulence dissipation rate. The values of TKE(φ) and ϖ(φ) can be obtained from RANS results or experimental results.

The relationship between upwash velocity spectrum ^w^w*and turbulence spectrum φwwis:

where u0is mean flow velocity in ξ direction; T is time.

For 2-D model, the unsteady loads at different radial positions are gained from a series of blade strips respectively,therefore, the correlation of unsteady loads are not considered. So,the effect caused by radial wave number cannot be considered in the solution process of upwash velocity.For 3-D model,it is significant to consider the effect caused by radial wave number.

2.2. Broadband noise model

The Fourier component sound pressure at a fixed position(x0;r0;φ0) of the hard-walled fan duct could be written as:49

where Amnis modal amplitude of circumferential mode m and radial mode n; rDand rHare shroud radius and hub radius,respectively. ψmand μmnare eigenfunction and eigenvalue in hard-walled fan duct,42respectively; κmnis explained in the Ref. 42 as an intermediate variable; amnis axial wave number;^fxand ^fφrepresent the axial and circumferential unsteady load on blade surface, respectively; s represents the blade surface;The superscripts ‘‘+” and ‘‘－” denote propagating acoustic wave in downstream and upstream direction, respectively.

Using Eq. (1) and Eq. (2), Eq. (14) becomes:

Fig. 2 Comparison of LT and L′′T.

2.3. Effect of WLE blade

Fig. 3 illustrates the baseline blade and the WLE blade, the chord length of WLE blade is:

where A(r)=ε ∙c- is the WLE amplitude, and is controlled by scale factor ε;c-is the chord length of baseline blade;W is WLE wavelength.

The flow direction and wave number direction of baseline blade and WLE blade are shown in Fig. 4.

Compared to the baseline blade, following optimizations are introduced for the WLE blade. If the WLE amplitude is set as zero,the following form is the same with the 2-D equivalence method by Ju et al.

Fig. 3 Blade configuration.

Fig. 4 Flow direction and wave number vector.

(1) Optimal Mach number:when the WLE amplitude A(r)is set as zero,the length of leading edge between different radius position is: lle=abs r-r′| |.

Then, using the Eq. (20), Eq. (23) and the above optimizations, the sound power spectrum density is:

Eq. (33) is suitable for broadband noise prediction with baseline blade configuration and WLE blade configuration,which is used in our study.

3. Numerical method

A single-stage axial fan acoustic experimental bench (NPU fan) at Northwestern Polytechnical University (NPU) is used in this study.20The flow field calculation and acoustic field are carried out separately.

3.1. Flow field calculation

The NPU fan is subsonic with 19 rotor blades and 18 stator blades. The design rotation rate is 3000 r/min with massflow rate of 6.3 kg/s and total pressure ratio of 1.2. The shroud diameter is 500 mm with the hub ratio is 0.57. The chord length of the stator blade is about 90 mm. The distances between the trailing edge of the rotor blade and the leading edge of stator blade are about 50.8% chord length at the shroud,40.3%chord length at mediate span and 27.5%chord length at hub, respectively.

As Fig. 5 shown, the computational domain of NPU fan consists of rotor domain with the inlet about 2c-upstream from rotor blade leading edge and stator domain with the outlet about 2.5c- downstream from stator trailing edge. Six computational works are conducted with the mesh number ranging from 3×106to 8×106,and the wall normal mesh dimensionless size y+ is less than 1 for each kind of mesh.

The flow field calculation works are finished by using the CFX 17.1 code. The Reynolds Averages Naviers Stokes equations with the k-ω Shear Stress Transport (SST) turbulence model are solved in the rotor domain and stator domain.The second order backward Euler scheme with the high resolution scheme are used in the calculations. 101325 Pa is set as inlet total pressure boundary conditions. In the flow field calculation works, the static pressure outlet conditions are changed to correspond with the experimental results. Adiabatic no-slip condition is set for the hub, shroud and blade.

Fig. 5 Sketch of NPU fan computational domain.

For RANS-informed analytical method, the turbulence information is obtained using CFD methods. The interested turbulence information is used for mesh independence study.

In order to obtain the turbulence information in front of the stator blade accurately,a plane(Plane 1 in Fig.5)is placed about 1 mm in front of the stator blade leading edge.As Fig.6 shown, when mesh number is greater than 5×106, the circumferential average TKE and circumferential average TLS change little. Finally, the mesh number of 8×106is selected to obtain the turbulence information.

As Fig. 7 shown, the mesh independence study was also conducted for A9W15 stator blade configuration. Four simulations with mesh numbers ranging from 6.7×106to 9.7×106are conducted. As for A9W15 stator blade configuration, the turbulence information is obtained from rotor-stator interface,and the circumferential average TKE and circumferential average TLS change little.

Finally, the mesh number of 9.7×106is selected to obtain the turbulence information and to compare the turbulence information differences between A9W15 stator configuration and Baseline stator configuration. And the configuration differences of A9W15 stator configuration and Baseline stator configuration are shown in Fig. 8.

Fig. 6 Mesh independence study of baseline stator blade configuration.

3.2. Acoustic field calculation

The acoustic field calculation works are based on Eq. (33).Before using the RANS-informed analytical method,the stator blade is need to be simplify using the strip theory,which means the simplified blade have no thickness or bending angle.Fig.9 is the sketch of the stator blade section shape at 50%span.In the prediction of RSI broadband noise, only the noise source term related to the stator blade unsteady load is considered.The other noise term due to the stator blade thickness, stator blade bending angle and potential field around stator blade are not model.So there are two key for predicting RSI broadband noise, one is the chord length, the other one is the flow incidence Angle,which is related to normal vector.In strip theory, the stagger angle is chose as the flow incident angle.

Firstly, two different WLE stator blade are established to investigate the effect of WLE amplitude on RSI broadband noise: A9W15 stator blade (the length of WLE amplitude is 9 mm, and WLE wavelength is 15 mm), and A18W15 stator blade (the length of WLE amplitude is 18 mm, and WLE wavelength is 15 mm), as shown in Figs. 10 (b) and (c). The parameters of these two WLE stator blades are selected based on previous studies.38–42Moreover,in Tong’s study,these two kinds of WLE stator blades have investigated using a hybrid LES/acoustic analogy method (Goldstein equations) method.Then, in order to investigate the effect of WLE wavelength on RSI broadband noise, the A9W30 stator blade is established by doubling the WLE wavelength of the A9W15 stator blade, and the A9W7.5 stator blade is established by reduced the WLE wavelength of the A9W15 stator blade. Finally, a total of 5 kinds of stator blade configurations are used in the study.

Fig. 7 Mesh independence study of A9W15 stator blade configuration.

Fig. 8 Comparison of stator domain.

Fig. 9 Sketch of stator blade section shape.

According to the previous introduction, the three kinds of stator blades are simplify to flat plate without thickness and bending angle. Then the stator blades are discretized into flat plates composed of radial nodes and chord nodes, which contains coordinate information and the flow incident angle information, as shown in Fig. 10. The grid nodes is related to Eq.(30).

Fig. 10 Stator blade configuration and nodes using strip theory.

4. Numerical results

4.1. Validation for broadband noise model

In order to study the turbulence-stator interaction broadband noise without the effect of rotor-self noise,Posson and Roger51set up an annular cascade experimental bench. The shroud diameter is 460 mm, and the hub ratio is 0.65. The chord length of the annular cascade blade is about 25 mm. Two different annular cascades(stator blades number of Cascade 1 is:49,stator blades number of Cascade 2 is:98)and two different turbulence-generating grids (Grid 1: turbulence intensity is of 3%, Grid 2: turbulence intensity is of 5.5%). The mean axial velocity upstream of the cascade is 80 m/s. To validate the broadband noise model, the turbulence from experiment is used as the input.51

Fig. 11 illustrates the comparison of the annular cascade experimental result and the predicted results with Hanson’s model,52Posson’s model51and present theory.The broadband noise predicted results using present method follow the same trend as the other two model. Three models are all sensitive to sound mode from cut-off state to cut-on state in the frequency range below 1 kHz, which is due to the small downstream max cut-on mode order of (m, n).

As shown in Fig.11(a),as for the configuration of Cascade 1 & Gird 1, the predicted results of all three models are agree well with the experimental result in the frequency range of 1.2 kHz to 6 kHz. On the contrary, the predicted results are not agree well with the experimental result in the frequency range below 1 kHz, which is may due to the dominated noise source is the interaction of boundary layer turbulence with annular cascade blade rather than the interaction of turbulence generated by grids with annular cascade blade. Similarly, the other figures also illustrate this phenomenon when mode(1,0) is cut-off.

As shown in Fig.11(b),as for the configuration of Cascade 1 & Gird 2, the predicted results by present theory is more accurate than the other model in the frequency range of 2 kHz to 6 kHz, while the Hanson’s method and Posson’s method both predict lower sound power.As shown in Figs.11(c) and (d), as for the configuration of Cascade 2, the present method predict a higher sound power in the frequency range of 1 kHz–4 kHz,the discrepancy is also exists when Ju et al. predict the fan-wake/Outlet-Guide-Vane (OGV) interaction broadband noise of NASA SDT fan.32

By comparing the annular cascade experimental result and the predicted results,the present method shows more accurate than Hanson’s method and Posson’s method for Cascade 1 configuration,but higher sound power in high frequency range for Cascade 2 configuration.Generally,the accurate prediction results could be obtained with the input of turbulence. Therefore,the present theory could be used in investigation of rotor–stator interaction broadband noise.

4.2. Results for NPU fan with baseline stator blade

In the experimental works,three work conditions with rotation rate of 3000, 2700 and 2400 r/min are conducted. As Fig. 12 shown, the red circle represent the experimental work conditions with rotation rate. The inlet duct in front of the rotor–stator stage is long, the massflow rate is smaller than design point due to the development of shroud and hub boundary layers. To correspond with the experiment, the flow field calculation by CFD is set to the same boundary condition.

Base on the analysis of acoustic propagation in the circular and annular duct, it could be obtained that the upstream max cut-on mode order of(m,n)for NPU fan is shown in Fig.13.It could be seen from Fig. 13, only circumferential mode m=0 could propagate if the frequency is less than 397 Hz. When the frequency is greater than 397 Hz,the circumferential mode|m |=1 begins to propagate. In the frequency range of 1–3000 Hz, the max cut-on mode order of circumferential mode is |m |=12, and the max cut-on mode order of radial mode is n=3.

In order to obtain the turbulence information in front of the stator blade accurately,a plane(Plane 1 in Fig.5)is placed 1 mm in front of the stator leading edge. The circumferential average TKE and circumferential average TLS are obtained from the plane. Fig. 14 shows the circumferential average TKE and circumferential average TLS for the rotation rate of 3000, 2700 and 2400 r/min, these turbulence information varies significantly along the radial direction. In the present calculation, those non-homogeneous turbulence information are used as input to present broadband noise model.

Fig. 11 Comparison of experimental result and predicted results for annular cascade experimental bench.

Fig. 12 Total performance of NPU fan and work conditions.

Fig.15 illustrates the comparison of upstream sound power result of single-stage axial fan acoustic experimental bench using CC-MDM53,54and the predicted results using RANSinformed analytical method for the rotation rate of 3000,2700 and 2400 r/min, respectively. The experimental measurement results are the total sound energy of all sound sources propagating to the rotating microphone array segment,such as rotor self-noise, RSI tonal noise, RSI broadband noise, etc.The stators are unable to remove,so the‘‘rotor-alone”configuration can’t be implemented, which lead to the fact the rotor self-noise cannot be measure from experiment. According to the experimental results of SDT fan,55the rotor self-noise dominates the broadband noise sound power in low frequency range, the RSI broadband noise dominates the broadband noise sound power in other frequency range, which is about 1–3 dB lower than total sound energy.

Fig. 13 Number of cut-on modes (m, n) as function of frequency.

Fig.14 Turbulence information obtained from RANS results in different fan working conditions.

In the predicted results,the upstream sound power is about 1–3 dB lower than the experiment in the frequency range of 0.4 kHz–2 kHz.The discrepancies can be attributed to the fact there are still other sound source,although the RSI broadband noise is dominant.In the frequency range higher than 2300 Hz,the broadband noise upstream sound power are inaccurate for the result that the microphone numbers is not enough to ensure the conditions of the error transfer matrix is in the normal range.56Generally, the RSI broadband noise predicted results using RANS-informed analytical method follow the same trend as the experimental result. Besides, after the circumferential mode|m |=1 begins to propagate, the sound power results predicted using RANS-informed analytical method are about 1–4 dB lower than experimental results,which is consistent with the experimental results of SDT fan.55Furthermore, the upstream sound power difference caused by different rotation rate can also be predicted accurately.This indicates that the using the turbulence information from RANS result is feasible, and radial unhomogeneous turbulence plays a significant role in turbulence-stator interaction broadband noise.

4.3. Results for NPU fan with WLE stator blade

In the study of WLE stator blade to achieve low RSI broadband noise, only the work conditions with rotation rate of 3000 r/min is considered. Fig. 16 illustrates the comparison of turbulence information of baseline stator blade configuration and A9W15 stator blade configuration. Plane 1 in Fig. 5 is not suitable for obtaining turbulence information of A9W15 stator blade configuration because the Plane 1 intersects leading edge of A9W15 stator blade.To compare the differences, the turbulence information is obtained from rotorstator interface. As shown in Fig. 16, the change of stator blades does not have an obvious influence on the upstream turbulence. The upstream turbulent flow is mainly generated by rotor blades. In this study, the geometric configuration of the rotor blades was not changed. Without loss of generality,the input turbulence information included TKE and TLS are all obtained from the RANS result of baseline blade configuration,as the red line shown in Fig. 14.The acoustic field is calculated with the five kinds of grids, as shown in Fig. 10.

Fig. 17 illustrates the comparison of stream-wise vorticity on leading edge of baseline blade configuration and A9W15 blade configuration. It can be seen that there is no obvious stream-wise vorticity structure on leading edge as for the baseline stator blade. While, there is obvious stream-wise vorticity structure on leading edge as for the A9W15 stator blade. The counter vorticity structure generated by wave peak develops span-wave due to streamline separation caused by wave peak.The counter vorticity structure generated by wave trough develops circumferential direction.

Fig.18 illustrates the comparison of upstream sound power predicted results with baseline stator blade and WLE stator blade.It is observed that all four WLE stator blade configurations can effectively reduce the RSI broadband noise in the frequency range below 1 kHz. Compared with A9W15 stator blade configuration,A18W15 stator blade configuration shows more effective RSI broadband noise reduction in the frequency range below 2 kHz. Compared with A9W15 stator blade configuration, A9W7.5 stator blade configuration also shows more effective RSI broadband noise reduction in the frequency range below 2 kHz.

Fig. 15 Comparison of experimental results and predicted results for NPU fan.

Fig.16 Comparison of turbulence information of baseline blade configuration and A9W15 blade configuration.

Fig. 18 Comparison of predicted results of baseline blade and WLE blade.

In general, all four WLE stator blade configurations could significantly reduce the RSI broadband noise in the low to medium frequency range, but resulted in an increase in noise in the higher frequency range, which is consistent with some experiment studies.57In fact, large scale vortexes is associated with the noise in low frequency range,the small scale vortexes is associated with the noise in high frequency range. The increase of RSI broadband noise in high frequency range is associated with a phenomenon that the large scale vortexes can be broken into small scale vortex with WLE blade configuration. On the contrary, in Tong’s study,42the numerical result using LES/Goldstein method shows both two kinds of WLE blade could reduce broadband noise in all frequency range.

The discrepancy may be attributed to following reasons:

(1) The treatment of WLE blade is simplistic,which lead to an inaccuracy of the RANS-informed analytical method in broadband noise prediction with WLE stator blade configuration.

(2) The hybrid LES/Goldstein equations method proposed by Tong et al.42may not be possible to accurately predict RSI broadband noise.

It could be seen from Fig. 18, the RSI broadband noise reduction effect of A9W7.5 stator blade configuration and A18W15 stator blade configuration are similar. It means that the ratio of WLE amplitude and WLE wavelength is closely related to noise reduction effect.

The discrepancy may be attributed to following reasons:

Fig. 17 Comparison of stream-wise vorticity on leading edge of baseline blade configuration and A9W15 blade configuration.

(1) The treatment of WLE blade is simplistic,which lead to an inaccuracy of the RANS-informed analytical method in broadband noise prediction with WLE stator blade configuration.

(2) The hybrid LES/Goldstein equations method proposed by Tong et al. may not be possible to accurately predict RSI broadband noise.

It could be seen from Fig. 18, the RSI broadband noise reduction is sensitive to both WLE amplitude and WLE wavelength. Larger WLE amplitude and WLE wavelength provide better RSI noise reduction in low frequency range.WLE stator blade could effectively transferred the sound power in lower frequencies to sound power in higher frequencies. Among the existing noise control methods, there are few methods that can reduce the low frequency noise.So,the WLE blade configuration is very useful and potential. Meanwhile, it could be seen that the RSI broadband noise reduction effect of A9W7.5 stator blade configuration and A18W15 stator blade configuration are similar. It means that the ratio of WLE amplitude and WLE wavelength is closely related to noise reduction effect. Generally, the RSI broadband noise reduction investigated by RANS-informed analytical method is consistent with the studies by experimental method and numerical method.

5. Conclusions

In this paper, a RANS-information analytical method for turbulence-stator interaction broadband noise is established,based on LINSUB and 2-D Equivalence method. The stator can be discretized into flat plate composed of radial nodes and chord nodes, which contains coordinate information and the flow incident angle information. The results show that the RANS-informed analytical method could be used in predicting rotor–stator interaction broadband noise.

(1) The broadband noise method has been compared with Hanson’s model and Posson’s model in a single annular cascade. The same with Hanson’s model and Posson’s model, the downstream sound power spectra are underestimated in low-frequency range,whereas a good agreement is obtained at medium-frequency range.As for the configuration of Cascade 1, this method is more accurate than the other model. As for the configuration of Cascade 2, this method are a little overestimate than Hanson’s model and Posson’s model in medium-tohigh frequency range.

(2) As for RSI broadband noise prediction, the turbulence information obtained from RANS results is used as the input for analytical method.The RANS results show that these turbulence information varies significantly along the radial direction. Therefore, it is significant to consider the non-homogeneous turbulence when using RANS-informed analytical method. The RSI broadband noise predicted results show the same trend as experimental result. Upstream sound power by RANSinformed analytical method is about 1–3 dB lower than experimental results because the other noise sources are not considered.Furthermore,the upstream sound power difference caused by different rotation rate can also be predicted accurately.

(3) Based on the comparison of experimental results and predicted results, we can find that the dominant noise sources in low frequency range are rotor-self noise and stator-self noise, especially when the circumferential mode |m |=1 is cut-off.

(4) Based on RANS-informed analytical method, the RSI broadband noise reduction effect of WLE stator blade is studied. The input turbulence information included TKE and TLS is the same for the baseline blade. It’s the same as other studies on WLE stator blade,the predicted results illustrate that the A18W15 stator blade configuration has more RSI broadband noise reduction potential, which declared that the RSI broadband noise reduction is sensitive to WLE amplitude. The RANSinformed analytical method can be further improved in RSI broadband noise prediction with WLE blade, the predicted results illustrate that the A18W15 stator blade configuration and A9W7.5 has more RSI broadband noise reduction potential in low frequency range.

(5) By comparing the flow field results with the predicted results, it can be seen that the increase of RSI broadband noise in high frequency range is associated with a phenomenon that the large scale vortexes can be broken into small scale vortex with WLE blade configuration.WLE stator blade could effectively transferred the sound power in lower frequencies to sound power in higher frequencies. Among the existing noise control methods, there are few methods that can reduce the low frequency noise. So, the WLE blade configuration is very useful and potential.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This study was co-supported by the National Science and Technology Major Project, China (No. 2017-Ⅱ-0008-0022)and the National Natural Science Foundation of China(Nos. 51936010 and 51776174).