Wngjin SHU, Congong CHEN, Lin DU, Xing HE, Xiofeng SUN,b

aSchool of Energy and Power Engineering, Beihang University, Beijing 100191, China

bResearch Institute of Aero-Engine, Beihang University, Beijing 100191, China

cAECC Hunan Aviation Powerplant Research Institute, Zhuzhou 412002, China

KEYWORDSContra-Rotating Open Rotors (CRORs);Interaction tonal noise;Nonlinear Harmonic(NLH);Frequency domain;Acoustic analogy

AbstractFast and accurate prediction of sound radiation of Contra-Rotating Open Rotors(CRORs) is an essential element of design methods of low-noise open rotor propulsion systems.In the present work, a previous frequency-domain model is extended to predict CROR noise.It builds explicitly the relationship between harmonic loadings and corresponding tonal noise, by which the influential parameters to noise generation can be clearly understood.The real distributions of steady and unsteady blade loadings are calculated by the Nonlinear Harmonic (NLH)method.In the present hybrid approach,both the CFD and acoustic modules are solved in the frequency domain.To assess the accuracy of the developed method, the loading noise of a CROR is calculated and compared against results by using the time-domain FW-H module of NUMECA.The predicted sound directivities by the two methods are in good agreements.The present acoustic model in the frequency domain is proven to be accurate and have high efficiency in far-field noise prediction and data processing.Furthermore, the characteristics of the CROR interaction tonal noise are analyzed and discussed.

1.Introduction

The configuration of Contra-Rotating Open Rotor (CROR)propulsion systems shows a preponderance of reductions in fuel burn and low pollutant emission relative to turbo-fan engines1,2.However, the control of noise generated by a CROR is a tough problem, which restricts the development and application of CROR airplanes.

It is well known that the major noise component of rotating blades is the loading noise if the flight Mach number is small3.The loading noise contains steady and unsteady loading noises.The steady loading noise is at the Blade Passing Frequency(BPF)and its harmonics of a Single-Rotation Propeller(SRP)4, while the frequency of the unsteady loading noise is a linear (sum and difference) combination of BPFs of the front and rear rotors.However,a difference tone has poor radiation efficiency whose mode phase speed is subsonic across the whole span of a blade, and such a mode is a cut-off mode in the case of a ducted rotor5.The noise of a CROR shows significantly higher levels and richer frequency components than those of an SRP due to rotor-rotor interactions, especially in front of the plane of rotation5.Meanwhile,the circumferential mode of an SRP is multiple of the blade number increasing with the number of harmonics; however, low circumferential mode noise can be generated by a CROR due to rotor-rotor interaction.This may greatly influence the radiation efficiency and directivity of CROR noise.

In the past decades,many experimental measurements have been set up to study the characteristics of the CROR noise.Experiments revealed that the wake introduced spikes into the pressure time history that produced higher noise levels in higher harmonics6, and the unsteady loading noise generated by rotor-rotor interactions made great contributions to the noise pattern.The noise spectrum was found to be richer in higher frequency harmonics of BPF than those of an SRP.The noise levels fore and aft of the plane of rotation were significantly higher for a CROR than those of an SRP due to aerodynamic interaction between the two rotors7.In addition,comparisons between SRP and CROR harmonic levels showed that the largest differences in noise occurred in the second and third harmonics and toward the axis of rotation8.In summary,the noise of a CROR becomes complicated due to the aerodynamic interaction between rotors, and the influential parameters to the Sound Pressure Level (SPL) and directivity pattern are of importance in the design of a CROR9.To this end, it is necessary to develop a reliable and effective acoustic model for CROR noise prediction as well as acoustic sources.

Nowadays, the most popular methods to predict CROR noise are hybrid approaches10–14, in which the unsteady flow and blade loading are computed by CFD and the far-field sound is obtained by solving the Ffowcs-Williams Hawkings equation15.One challenge is to simulate accurately and effectively the unsteady loading due to rotor-rotor interaction by CFD.To this end,the URANS(Unsteady Reynolds Averaged Navier-Stokes) equation was used by Colin et al.10,11and Delattre and Falissard14.Meanwhile,NUMECA International developed the FW-H solver coupled with the CFD module of the Nonlinear Harmonic(NLH)method which has been extensively tested against various analytical test cases16,17.The instantaneous values of static pressure can be reconstructed by cumulating the harmonics and the mean flow given by the NLH simulation(Fourier reconstruction in time).The application of NLH results in a significant reduction in the CPU time requirements and resource compared to full unsteady simulations.The NLH method and the coupled FW-H solver were applied successfully to predict CROR noise by Ferrante12and Deconinck13,18et al.

Hanson19,20derived the well-known formulas of tonal noise prediction of an SRP based on the acoustic analogy in the frequency domain, in which a computation of the retarded time was avoided.These formulas showed explicitly the dependence of sound pressure on design parameters and loadings.It is helpful for the understanding of the mechanism and design of low noise.Further, Hanson developed a theoretical framework for analysis and prediction of noise of CRORs.It includes analytical models for the aerodynamic unsteady interaction between rotors and for the noise from the resulting unsteady blade loadings21.However, the acoustic sources of Hanson’s method were given in terms of the lift and drag harmonic coefficients at many strips, which were not discussed in detail.This was limited by the CFD capability in that period.Recently, a series of acoustic theories dedicated to the wake interaction noise of CRORs has been presented22–24, which are based on the helicoidal surface theory of Hanson.An asymptotic analytical model for wake interaction tones was established in the frequency domain, and the effect of blade sweep was investigated22.In the solution of Kingan and Parry,the unsteady aerodynamic response was obtained from a Wiener-Hopf analysis, by which the potential interaction was neglected and the practical interaction between the wake and the rear rotor blade was difficult to predict.

In this paper, a frequency-domain acoustic model25is extended for CROR noise prediction, coupled with the NLH method26for aerodynamic loading computation.Compared the hybrid approaches mentioned above, the present method can simplify data processing and reduce the required computational time and resource, because both the CFD and acoustic module are solved in the frequency domain.With the aerodynamic loading in the frequency domain of NLH methods, the derived acoustic formulas build the relationship between the harmonic loading and the corresponding unsteady loading noise.It is efficient for understanding of the mechanism of noise generation and useful for CROR noise evaluation in design.

The layout of this paper is as follows.In Section 2, the acoustic model for predicting the loading noise generated by a CROR in the frequency domain is introduced.To examine the accuracy of the present model, a real CROR geometry is introduced as an example in Section 3.Then verifications and validations of the acoustic model are demonstrated in detail in Section 4.The influential parameters to the SPL and directivity pattern are analyzed and discussed in Section 5.Primary conclusions from this investigation are given in the last section.

2.Acoustic model

This paper extends the integral formula by Yang and Wang25to a CROR model,which is based on Goldstein’s integral version of the FW-H equation27.The details are as follows, and the loading noise part can be expressed as

where U is the forward flight velocity, x1is the axial direction of coordinate system.

Eq.(4) can be written as follows in the cylindrical coordinate system:

where A is the stretched parameter of Green function Gω,with the Dirac delta function formula as

Eq.(5) can be transformed into an inhomogeneous convected wave equation as

The Green function of an inhomogeneous convected wave equation can be expressed as

in the cylindrical coordinate system, and Eq.(1) can be expressed as28

The partial derivative of the Green function of Eq.(10) in the respective direction can be expressed as

In this paper, the acoustic model of the loading noise generated by the rear blade is developed as an example.The deduction of the loading noise generated by the front blade has the same process.

For the rear blade,it is assumed that the unsteady loadings which are distributed over the blade surface are caused by the wake and potential interaction of the front blade, and the frequencies of the unsteady loadings are s1B1（Ω1+Ω2）;s1=1;2;???.The unsteady loadings on the rear blade can be expressed as27

Fig.1 Configuration of CROR.

and the summation of blade loadings on all B2blades can be expressed as

In the cylindrical coordinate system, there are forces in three directions.The acoustic pressure radiated by the axial force is taken as an example with Eq.(11) and Eq.(12) as follows:

Letting θrot=θy-Ω2τ, the limits of integration in the surface integral over the rotor blades become independent of τ.Then, we can interchange the order of the integration, so Eq.(17) becomes

does the interaction tonal noise caused by harmonic loadings exist.Eq.(18) is the same as the circumferential mode theory of an axial compressor defined by Tyler and Sofrin29.

From the above deduction, the final integral formula expressing the acoustic pressure caused by the axial force component fy1can be obtained.With the same deduction procedure, the acoustic pressures radiated by the radial force component fryand the circumferential force component fθyare also obtained as follows:

where BPF1;BPF2are the blade passing frequency of front blade and rear blade, respectively.

In summary, the integrals of the loading noise can be written as

The loading noise contains steady loading noise and unsteady loading noise.For the rear blade,when s1=0,noise at these frequencies is steady loading noise, and Fs1α（α=y1;ry;θy）in Eq.(19) are steady forces.When s1≠0,noise at these frequencies is unsteady loading noise, and Fs1α（α=y1;ry;θy）in Eq.(19) are harmonic loading (including real and imaginary parts).It is important that the frequencies of unsteady loadings exerting on the rear blade are s1B1（Ω1+Ω2）;s1=1;2;???, while the frequencies of unsteady loading noise radiated by the rear blade are s1B1Ω1+s2B2Ω2;s1=1;2;???;s2=0;1;2;???.Naturally, the unsteady loading at a single frequency can radiate more discrete frequencies of unsteady loading noise due to the rotation of the circumferential mode m=s2B2-s1B1.For example,the 1st harmonic loading of the rear blade at the frequency of B1（Ω1+Ω2）can radiate unsteady loading noise at frequencies of B1Ω1+s2B2Ω2;s2=0;1;2;???, with Eq.(20).

3.Aerodynamic module

The acoustic model of the loading noise generated by a CROR has been developed in the frequency domain.Acoustic sources are required for the acoustic model to predict the noise of a CROR.In this section, an example of CRORs is introduced,and the calculation of the distributed blade loading is briefly introduced.The aerodynamic module is based on the NLH method developed by He and Ning26,and aerodynamic results are the input of the aeroacoustic module.

The NLH method can solve a mean and finite(pre-selected)number of blade passing frequency harmonic components of the time-dependent solution of a CROR, ignoring all the higher unsteady components.The NLH method has been implemented in the Navier-Stokes solver of the FINE/Turbo which is applied in this paper and has been successfully validated and tested against basic and turbomachinery test cases30–32.This approach has also been validated in CFD simulation for aeroacoustic input of noise prediction of a CROR by Envia33.To predict the sound of a CROR,the accuracy and reliability of the CFD method to capture the aerodynamic loading are essential.It is well known that the mechanism of blade unsteady loading generation is the same for a CROR and an axial compressor.To validate the accuracy of the aerodynamic method, a reliable measurement of a rotor/stator axial compressor in annular duct is introduced for verification34.Blade parameters at design condition are shown in Table 1.

The NLH method in FINE/Turbo is applied with the first three harmonics in the simulation of rotor/stator flow.With the time reconstruction in Fourier series,the instantaneous static pressure is obtained.The unsteady force on the mid span of the stator is extracted for each time step.Results of the NLH method are compared with those of the time marching Immersed Boundary Method(IBM)by Chen et al.35and measurements by Hsu et al.34in Fig.2.Vbis the rotor blade wheel velocity;ΔF is the difference between instantaneous static pressure and average pressure.q is the density of perfect air; T′is the period of unsteady flow.

The black dot line represents the results of the NLH method in this paper.There is a good agreement between the results of the NLH method and experiment measurements indicated by the red point solid line.It is verified that theNLH method has the capability to capture unsteady forces on blade surface.

Table 1 Compressor and blade parameters at design condition.

Fig.2 Verification for NLH method.

3.1.Test case

A generic 6 × 6 CROR model is used in this paper, and the configuration is shown in Fig.3.

The diameters are 3.95 m and the numbers of blades are 6,for both the front and rear blades, as shown in Table 2.The front and rear rotors operate at a 100 r/min speed difference.The BPFs of the front and rear blades are BPF1=90 Hz and BPF2= 100 Hz, respectively.The Reynolds number is about 9×105as the reference length is the chord length of the rotor.

Fig.3 CROR geometric layout.

Table 2 CROR characteristics.

Fig.4 Sketch of computational domain.

Table 3 Computational grids case and spanwise points.

3.2.Computational domain and mesh

The CFD mesh generation tool used in this paper is the AutoGridv5TMsoftware developed by NUMECA International, which has been introduced in detail and applied to calculate the aerodynamic input for time-domain formulation for propeller noise prediction36,37.The computational domain of CFD is shown in Fig.4, where Rtipis the tip radius of blade.The far field domain in the axial direction is about 10Rtip, and the far field domain in the radial direction is about 3Rtip.

The non-reflecting boundary conditions based on a characteristic analysis of the linearized Euler equations38are applied at the rotor-rotor interface.

Three mesh resolutions have been performed for mesh convergence assessment as shown in Table 3.

The harmonic amplitudes over the blade surface are obtained by different mesh resolutions.Fig.5 shows the results extracted along the spanwise direction at the leading edge of two blade rows,and the aerodynamic results in this paper present a good mesh convergence behavior.

There is a fictitious shroud which separates the main channel and the far-field domain.The far-field domain is extended 3 times of the radius of the blade tip.The grid points on the surface of the blade are 67 in the spanwise direction and 55 in the chordwise direction, for each side of both the front and rear blades.The topology of the mesh around the blade in Fig.6 and the values of y+above blade surfaces are almost not exceeding 10 on the first layer of cells.

3.3.CFD settings

Fig.5 Harmonic pressure amplitudes along spanwise direction of blade surface at leading edge.

Fig.6 An O4H topology around rear blade and its leading edge at 50 % span.

Table 4 Boundary conditions.

The CFD solver used in this paper is the FINE/Turbo developed by NUMECA International.The flow model is from the NLH method, and the turbulence is modeled by the eddy-viscosity-one-equation Spalart-Allmaras model.In this paper,the first six harmonics of the blade passing frequency of the NLH method are selected.The boundary conditions of the CFD module are shown in Table 4.The blades and the hub are considered as smooth and adiabatic at the solid-wall boundaries.

With the NLH method and Fourier reconstruction in time,one can obtain the harmonic amplitude and the instantaneous values of the flow field.Fig.7 shows the instantaneous solutions of the entropy field at 50%and 90%spans,respectively.Rotor-rotor interactions can be clearly observed at both radial locations.

A mean axial thrust of 3.215 × 104N is produced by the CROR, while a mean power of 5.519 × 103N?m is required to drive them.Aerodynamic results used for the acoustic model are the time-average static pressure and the first six harmonic loadings.Higher-harmonic results and details of aerodynamic analysis are not discussed here for brevity.

4.Verifications of acoustic model

The predicted CROR sound radiation by the present frequency-domain acoustic model will be compared with the results of the time-domain model, namely the famous FW-H equation and Farassat formulation.In the rotating axial direction, 101 observers are fixed around the CROR, all of which are located 20 m (about 10 times of the blade tip radius) far away from the origin coordinates, with an interval of 1.8°,and the central observer placed off the axis above the front rotor is called Observer 1 as shown in Fig.8.

Fig.7 Entropy field at 50% and 90% spans.

Fig.8 Observer locations for aeroacoustic analysis.

The periodicity of the acoustic signal is determined by the common divisor frequency: 10 Hz (the corresponding time interval is 0.1 s18), for which the number of time steps is 512.Loading noises are firstly predicted by the application of the NLH method and the FW-H solver of NUMECA as Refs.36,37.The FW-H solver can decompose the acoustic signal into thickness noise and loading noise of the front and rear blades, respectively.For Observer 1, the time signal of the loading noise by the front blade is predicted by the FW-H solver as shown in Fig.9.The corresponding spectrum is also obtained by the FFT (Fast Fourier Transform)process.

Then the extended acoustic model in this paper, namely frequency-domain method, is applied to predict the loading noise at each interaction tonal frequency at all observer locations.The loading noises radiated by the front and rear blades are predicted respectively.Acoustic results of the two methods are compared at all tonal frequencies and 101 observer locations as below.

Fig.9 Time signal of loading noise of front blade at Observer 1.

Take the sound radiation of the rear blade as an example.Black square symbols are the loading noise predicted by the frequency-domain method while red circle symbols are the loading noise predicted by the time-domain method as shown in Fig.10, and we define f1=BPF1, f2=BPF2for brevity.Amplitudes below 40 dB are neglected.The comparison shows that there is a good agreement between the frequency- and time-domain methods at every discrete frequency at Observer 1.In fact, the agreement between the two methods for several dominant tones (i.e.,f1+f2, f1+2f2) is excellent, being less than 0.5 dB at Observer 1.

Then the directivities of the loading noise at different tonal frequencies are predicted to verify the accuracy of the present method as shown in Fig.11.There are good agreements between the frequency-and time-domain methods at all observer locations.

The present hybrid method can simplify data processing and reduce the required computational time and resource,because both the CFD and acoustic modules are solved in the frequency domain.A reconstruction of the instantaneous values of static pressure over blades is not necessary.One can directly input unsteady loadings, which are obtained in the frequency domain by using the NLH method, into Eq.(19) to calculate the noise.In this paper, the same processor(AMD Ryzen 9 3950X 16-core processor) is chosen to test the time cost by the two methods.With the same CFD process and aerodynamic results, the time cost of the aeroacoustic module is evaluated.Taking the results above as an example,the time-domain method can be used to obtain the time signal and spectrum of sound pressure for 101 observers for the first six harmonics.The frequencies in the linear combination of BPF1and BPF2are extracted as results.Contrastively the frequency-domain method can directly solve the SPL of corresponding frequencies based on the acoustic model developed in this paper.The CFD module costs about 30 CPU hours, and Table 5 shows the time costs of directivities results of the rear blade by both methods.

5.Results and discussions

5.1.Amplitudes of acoustic sources

It should be noted that the rotor-rotor interaction of the front blade is dominated by the potential flow, while the harmonic loading noise of the rear blade is generated by both potential and wake interactions.The concentrated harmonic loading amplitudes by integration over the front and rear blade surfaces are expressed as

Fig.10 Prediction of loading noise by rear blade at Observer 1 by frequency- and time-domain methods.

Fig.11 Interaction tonal noise directivity of front blade.

Table 5 CPU time cost.

Fig.12 Concentrated harmonic loading amplitudes.

The noise spectrum of the rear blade radiated by steady and unsteady loadings is expressed in different symbols as shown in Fig.13.As indicated by Eq.(19), the frequency-domain method builds the relationship between the loadings and corresponding tonal noise, which requires higher cost when using the time-domain method.The steady loading noise (black square symbols) at a frequency of s2BPF2decays rapidly with the harmonic s2increasing, and the trend is similar to the results of SRP4.The overall SPL of the harmonic loading noise shows a similar trend corresponding to the amplitude of the harmonic loading in Fig.12.

Fig.13 Spectrum of loading noise by rear blade at Observer 1.

It should be emphasized that the SPL of the harmonic loading noise shows various trends even with the same harmonic loading as acoustic source, as shown with different symbols in Fig.13.The circumferential mode m plays a key role in determining the amplitudes and phases of the interaction tones.Variation of the circumferential mode m leads to great changes of the harmonic amplitude of the unsteady loading noise even radiated by the same harmonic loading.For example, the amplitude of the unsteady loading noise radiated by the 1st harmonic loading (red point symbols) reaches a peak at BPF1+2BPF2,m=6,and then falls rapidly as m increases.The trend of higher frequencies radiated by a higher harmonic loading becomes more complicated.

The acoustic source over the blade surface contains both the amplitude and phase of the harmonic loading.The 1st harmonic loading amplitude and phase of the suction side on both the front and rear blades are shown in Fig.14.The amplitudes on both the front and rear blades are in the same range; however, the phases of the suction side on the front blade are well distributed, while those of the rear blade are not well distributed, which leads to constructive interference of the front blade and destructive interference of the rear blade.Therefore,the concentrated harmonic loading amplitude of the front blade is evidently higher than that of the rear blade as shown in Fig.12.

The 1st harmonic loading of the front blade is caused by the potential flow effect of the rear blade.The amplitude is concentrated on the leading edge and top region of the pressure side, while the corresponding phase in the same region is well distributed as labeled in Fig.14(a).However,the 1st harmonic loading of the rear blade is mostly caused by the incidence wake of the front blade.The amplitude of the suction side on the rear blade is concentrated on the leading edge and blade tip, and in contrast, the corresponding phase is not well distributed as labeled in Fig.14(b).In Fig.14(c), the amplitudes and corresponding phases of the dash line on both the front and rear blades are presented in detail.Variations of the amplitudes and phases of the front blade are slight; however, the amplitudes of the rear blade are in the range from 100 Pa to 300 Pa and the corresponding phases vary from 0.75 rad to 3.5 rad.As a result,they are in phase of the harmonic loading on the front blade and out of phase on the rear blade.Even the amplitude of element of the rear blade is higher than that of the front blade in Fig.14, the concentrated harmonic loading amplitude, defined by Eq.(22), of the front blade is evidently higher than that of the rear blade as shown in Fig.12.

For the unsteady loading noises generated by the front and rear blades with the same harmonic number and circumferential mode,the sound radiation is mostly dominated by acoustic source.The sound directivities at 190 Hz (s1=1;s2=1) of both the front and rear blades are shown in Fig.15.The SPL of the front blade is higher than that of the rear blade within the range of 0°-55° and 125°-180° in polar angle.The sound radiation of the front blade is also significant.

Fig.15 Sound directivities at 190 Hz （BPF1+BPF2）of both frontandrearblades.

From Figs.12-15, it is emphasized that the SPL of the CROR is not only simply dominated by the strength of the harmonic loadings as indicated by Eq.(19), but also significantly influenced by the circumferential mode and loading distribution.

5.2.Distribution of harmonic loadings

To investigate the effect of the distribution of harmonic loadings, the blade surface is divided into strips in the spanwise direction in this paper.Distributed loadings in each strip are concentrated on a point in the center of element, and each point forms the line source model.For the rear blade, when s1= 1 and s2= 1, an unsteady loading noise is generated by the 1st harmonic pressure, and the frequency is f1= 190 Hz.The wavelength λ1is longer than the spanwise length, and the distribution of the 1st harmonic pressure amplitude is shown in Fig.16.

Results of the line source model and the blade source are compared in Fig.17.There is a good agreement between the line and blade sources.Actually, the wavelength is much longer than the chord length, and the blade source is acoustically compact.The line source model is reasonable in this case.

Nevertheless,when s1=4 and s2=4,an unsteady loading noise is generated by the 4th harmonic pressure, and the frequency is f2= 760 Hz.The distributions of the 4th harmonic pressure amplitude and wavelength λ2are shown in Fig.18.

Fig.14 1st harmonic pressure amplitude and corresponding phase on blade surface.

Fig.16 1st harmonic amplitudes of rear blade and line source model.

Fig.17 Comparison of noise directivities at 190 Hz.

The directivity of the unsteady loading noise at 760 Hz is shown in Fig.19.There is an evident difference between the line and blade sources at most observer locations.In this case,the wavelength is not long enough compared to the chord length, so the blade is not acoustically compact in the chordwise direction.The line source model is not reasonable.

Fig.18 4th harmonic amplitudes of rear blade and line source model.

Fig.19 Comparison of noise directivities at 760 Hz（4BPF1+4BPF2）.

More interaction tonal noises of the rear blade at three observer locations are shown in Fig.20.The amplitudes below 50 dB are neglected for brevity.A comparison between the blade and line source models shows a big difference, even larger than 5 dB at several interaction tonal frequencies.The drawback of the line source model reveals that it must consider the distribution of harmonic loadings for interaction tonal noise at all observer locations.A blade cannot be treated simply by strips assumption.

5.3.Sound interference between harmonic loadings in three directions

The sound pressure can be easily decomposed into contributions of radial, circumferential, and axial directions by using the developed model in Eq.(19).The loading noise at a frequency can be separately obtained for the three components of forces in three directions without additional effort and cost, which is plotted in Fig.21 for the front blade at 100 Hz.It is found that the SPL radiated by the radial loading is the smallest while the circumferential force-related noise dominates the SPL at this frequency.It should be noted that the total SPL is lower than the noises generated by the circumferential loading in the range of 0°-95° and the axial loading in the range of 0°-80°.This is because the noise signal radiated by the loading from each direction has a different phase and amplitude, which may lead to destructive interference in the superposition of sound.Naturally, this mechanism can be applied to reduce the SPL in the design of a CROR.

The decomposition of the unsteady loading noise of the front blade at 380 Hz is shown in Fig.22.At this frequency,s1=1,s2=1, and the circumferential mode m=s2B2-s1B1=0 in Eq.(18), which leads to an acoustic pressure radiated by the circumferential force pθy=0 in Eq.(19).This means that the circumferential mode propagating in free space is zero, and observations at 0° and 180° polar angles have strong amplitudes.This phenomenon was observed in Block et al.’s experiment9.

The SPL is dominated by the axial loading when m=0,which shows a typical sound directivity of a dipole in the axial direction.The noise radiated by the radial loading can be negligible except in the vicinity of the planes of rotation.This is partly due to the sharp fall of noise generated by the axial loading in this range.The directivity at the circumferential mode m=0 reveals that there is a clear difference on the SPL between the axis direction and plane of rotation.It is important to choose suitable observer locations for an estimation of the Overall Sound Pressure Level (OASPL), especially in experimental measurements.

Fig.20 Comparison between models at different observer locations.

Fig.21 Directivity of unsteady loading noise and contributions of three directional loadings at 100 Hz of front blade.

Fig.22 Decomposition of sound pressure at three directions at 380 Hz of front blade.

5.4.Interference of two blade rows

The interaction tonal noise by the CROR is the sum of the harmonic sound pressures of the front and rear blades at the same frequency.However, the interaction tonal noise at frequency s1B1Ω1+s2B2Ω2of the front and rear blades is radiated from different harmonic loadings.The noise at frequency s1B1Ω1+s2B2Ω2of the front blade is radiated by the s2harmonic loading, while that of the rear blade is radiated by the s1harmonic loading.With an increase of the harmonic number,the amplitude of the harmonic loading decreases as shown in Fig.12.At most tonal frequencies, the total SPL is dominated by the rear blade if s1

For a tone with the same harmonic number, for example,s1=s2=1, the amplitudes of the harmonic loadings exerted on the front and rear blades are in the same order as shown in Fig.12.This results in a close SPL in the sound directivities of the front and rear blades as shown in Fig.23(a).There exists a constructive interference at a polar angle of 45°-135°.Meanwhile, the total SPL is smaller than that of the front blade at polar angles of 0°-45° and 150°-180°, for which the design of the CROR is beneficial for interference of two blade rows.

Fig.23 Interferences at several tonal noises.

Results in Fig.23 show that the interaction tonal noise of the CROR might be interfered constructively or destructively at different polar angles, which is essentially determined by Eq.(19).This is much more complicated than noise of an SRP.The integral of harmonic loading amplitudes, as shown in Fig.12,is helpful in estimating the contribution of the front and rear blades to a tone noise with a combination of s1and s2,although the sound signal is also influenced by the interference among strips of blades.

6.Conclusions

In this paper, a hybrid CFD-acoustic method is extended for prediction of loading noise generated by a CROR in the frequency domain.The real distributions of steady and unsteady loadings are calculated by the NLH method.In the present hybrid approach, both the CFD and acoustic modules are solved in the frequency domain.Data processing is simplified,and the required computational time and resource are reduced.The formulas build an explicit relationship between harmonic loading and corresponding tonal noise in the frequency domain of a CROR.With the same acoustic sources by the NLH method, the acoustic model shows good agreements on the sound directivities of all tonal loading noises compared with the FW-H solver in the time domain.

Then, the characteristics of CROR interaction tonal noise are discussed.The amplitudes of acoustic sources are analyzed.The integrals of harmonic loadings show decreasing trends with an increase of the harmonic number, but the trends of the corresponding noise spectrum are more complex.The potential flow effect on the front blade proves to be important.Results show that the simplification of acoustic sources with strips is unreasonable for high frequencies of interaction tonal noise.The blades of a CROR cannot be treated as acoustically compact.The contribution of harmonic loadings in each direction to the sound directivity is analyzed in detail with different combinations of s1and s2.The characteristics of sound radiation,especially when the circumferential mode m=0,are analyzed by the integrals of the acoustic model.The noise generated by the radial force cannot be completely ignored for m=0.The complexity of the CROR noise is not only in spectrum but also in sound directivity.Compared with an SRP, more attentions should be paid to the sound radiation along the axial direction for a CROR.Finally, the contribution and interference of the front and rear blades are analyzed.The plots of sound directivities show that, at most tonal frequencies, the total SPL is dominated by the rear blade if s1

The preponderance of this approach is that the model explains the relationship between the harmonic loading and the corresponding tonal noise directly, and data processing is simple.This approach shows a great efficiency and accuracy,and it can be applied to prediction of loading noise with an accurate input.For further investigation, experimental results are required for further verification of this approach.

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 Natural Science Foundation of China (Nos.52022009 and 51790514), the National Science and Technology Major Project, China (No.2017-II-003-0015), and the Key Laboratory Foundation,China (No.2021-JCJQ-LB-062-0102).