Acoustic scattering mechanism and noise attenuation of circumferentially non-uniform liner with spectral-wave guide method

Wei DAI, Xuli WANG, Xioyu WANG, Gungyu ZHANG,*, Xiofeng SUN

a School of Energy and Power Engineering, Beihang University, Beijing 100191, China

b Research Institute of Aero-Engine, Beihang University, Beijing 100191, China

KEYWORDS

Abstract In this paper,a model is established with application of the spectral-wave guide method,which has higher accuracy and can serve as a rapid calculation tool for sound transmission calculations.Based on this calculation model, some numerical results of circumferentially non-uniform lined annular/circular ducts are carried out,and some physical mechanisms can be discovered.The numerical results show that periodical impedance distributions along the circumferential direction will lead to discontinuous scattered modes with regular spacing; and mirror-symmetric structure liner will converge the energy of opposite modes.Relying on this mechanism,the potential of acoustic scattering can be further developed by suppressing lower or enhancing higher order modes with expressly designed segmented liner configurations.In particular, the intrinsic mechanism of mode redistribution brought about by the non-uniform liner can be subtly utilized to attenuate broadband noise.The present work indeed shows that circumferentially non-uniform liner is conducive to the reduction of the practical broadband sound source.Furthermore, the effects of nonuniform flow are considered in the model,then distinction of noise attenuation and scattered modes energy in different flows is shown.A possible mechanism is proposed that refraction effects in complex flows lead to the distinction.These works show that the current model has profound potential and availability for the research and designs of circumferentially non-uniform liner.

1.Introduction

Due to the widespread adoption of high-bypass aero-engines on commercial aircraft, which has brought about the reduced jet noise of aero-engines,the fan noise stands out to be one of the predominant noise sources.A typical noise spectrum of a turbofan engine comprises tonal and broadband components.1,2Because the further reduction of tonal noise component is difficult,mitigating the broadband component becomes a particularly significant concern to noise elimination design for aircraft engines.Typically,acoustic liners are utilized to meet the requirements of aircraft noise control, and various technical approaches have been developed to increase the attenuation bandwidth of acoustic liners.3However,optimization of the uniform liner is limited by restriction of the liner length and weight.Therefore, the Circumferentially Non-Uniform (CNU) liner that enhances the noise elimination by the redistributions of sound energy into higher-order circumferential modes are firstly proposed and studied by Mani.4In their work,the uniformity of wall impedance is achieved by simply blocking off the peripheral sections of a uniform liner.

The understanding of the intrinsic mode scattering hitherto has not been clear.An evaluation was performed by Watson5who stated that circumferentially segmented duct liners could attenuate more sound than uniform liners with at least 50%reduction in the amount of treatment.However, how to design the circumferentially non-uniform liner to achieve the desired effect is unknown due to the lack of the knowledge of acoustic scattering.Actually, the mode scattering was expected to improve their acoustic performance by the follow-up investigations.5–15Hence the present research intends to fully understand what key role the acoustic scattering plays in noise reduction.Besides, the mode scattering also has potential in suppressing the azimuthal thermoacoustic instabilities in annular combustors, by changing the coupling of circumferential modes.For the practical application, it is very necessary to know how the segmental liner behaves differently under actual sound sources, and on what condition the sound scattering exerts great or slight influence on sound transmission.Especially, considering that real aero-engines have complex flow profiles, it is also necessary to investigate how the different segmentation designs affect the sound propagation and scattering under diverse flow conditions.To answer these questions, a fast and accurate computation model is thus required.

The early theoretical investigations on the circumferentially non-uniform liner generally adopted the assumption of uniform flow or no flow.5–13Uniquely, the separation-ofvariable technique is not applicable for the circumferentially non-uniform liner.In other words, the solution of the equations is quite complicated, the circumferential modes and radial modes usually cannot be simply isolated.So far, some analytic and numerical methods for solving this problem have been applied.Watson6and Eversman et al.9developed a Finite Element Method (FEM) scheme at early stage to obtain the sound field in duct with peripherally varying linings, namely,circumferentially non-uniform liner.Fuller10expanded the wall admittance into Fourier series while the characteristic function was expressed as the sum of specific functions where the propagation medium is at rest.Then root-finding process of algebraic equations is achieved by Muller’s method.14In Watson’s7work, the pressure in duct without flow was expanded in terms of function series of rigid duct modes and the employment of Galerkin method converted the equations satisfying the non-uniform admittance boundary condition into an eigenvalue problem to be solved.The MMPM method developed by Bi et al.12formed a constant coefficient differential matrix modelling the sound transmission in cylindrical duct without flow by expressing the pressure as a series of functions for rigid boundary condition.In comparison with threedimensional(3D)numerical calculations,semi-analytic models which reduce the problem to be two-dimensional (2D) and obtain the global solutions by boundary matching have strong superiority in the demand of computing resource.Through application of approximate hypotheses, some analytical methods, such as the Kirchhoff model by Tester et al.13were proposed as well.In recent years, the computation capacity has been powerful enough to calculate the sound field in the duct with circumferentially non-uniform impedance wall,15but most of these works focused on the effect of rigid axial splices which is normally due to the manual installation.Undoubtedly, to investigate the circumferentially non-uniform liner,the development of a new computation model is imperative.

It is known that the actual flow can bring about some alterations in the acoustic property of ducted liners.16–20The convection of air in duct indeed has been confirmed to have a non-negligible effect on upstream and downstream waves.In particular, the existence of ‘‘slip” at the boundaries which is usually supposed under uniform flow is obviously never found in practice.Besides,the varying-section duct and bent duct can also cause non-uniform flow.Therefore, the assumption of uniform flow leads to biased computation results.Actually,refraction effects on transmission and attenuation of sound due to the shear flow in the boundary layer were proved to be significant by Eversman.16In practical consideration, we ought to consider the non-uniform flow profile due to the boundary layer, duct geometry and other reasons to obtain a solution closer to physical reality.Nevertheless, very little work has dealt with modelling the sound propagation inside the circumferentially non-uniform liner under non-uniform flow conditions.

In the present work, to delve into the underlying mechanism behind the circumferentially non-uniform liners and understand how significant the effects of sound scattering will be,a computational approach is presented to model the transmitted sound wave inside an annular/circular ducted circumferentially non-uniform liner with a uniform/shear flow,which is of considerable interest in the acoustic techniques related to aero-engines.When a multi-scale physical problem is investigated, high accuracy of the numerical solution is expected especially for the interpretation of small quantities,such as waves propagating through a ducted lining.The spectral method of Chebyshev Collocation (ChC) is applied together with an assumption on the solution form.21The required mesh density of spectral method to achieve acoustic solutions could be lower than that of the FEM, though the demand of the computation time may increase rapidly with increasing scales of the problem.22–23Due to the impossibility to separate the circumferential modes and radial modes, an assumption used in the study is that the solutions of the equation are separable in z direction and non-separable in the(r,θ)plane.Details of the modeling process is presented in Section 2.The validation of the method through comparisons with the result of FEM is shown in Section 3.

Based on this computational model, the mode scattering can be effectively studied with emphasis on the mechanism of its generation and acoustic energy distribution under different circumferential impedance conditions.Given a sound source close to reality, the characteristics of circumferentially non-uniform liners are also shown in the present work.All of the discussions are presented in Section 4 and Section 5,and conclusions are summarized in Section 6.

2.Model construction

2.1.Methodology

As shown in Fig.1, the model considered in this paper is an annular duct with non-uniform liner, of which the dimensionless impedance Z( θ )varies along the circumferential direction.

The fluid in the duct is assumed to be inviscid and the axial flow with axial Mach number Ma(r )is considered.Then in linear acoustic theory,the wave equation is written in dimensionless form as

where p is sound pressure, k=ω/c0is wave number,c0is acoustic velocity, vris the projection of particle velocity in radial direction.

At the outer radius (r=Rd), Ingard-Myers24boundary condition that is applicable in the cylindrical coordinate is expressed as

in which the sound pressure p and radial components of particle velocity vrsatisfy the radial momentum equation

Eliminating vrfrom Eqs.(2) and (3) gives

At the inner radius(r=Rh),no penetration boundary condition is adopted in the form of

where Eqs.(1),(4)and(5)construct a boundary value problem for the acoustic pressure to be solved.

The result of Eversman et al.9shows that the solution to the problem is a complex combination of circumferential modes and radial modes.Following Watson7and Fuller,10the pressure variation p(r,θ,z) is separable in z direction and nonseparable in transverse plane.

Substituting Eq.(6) into Eq.(1) gives

In practical application, the series ought to be truncated at m=M,where M is mesh number in circumferential direction.

In this paper,the Chebyshev Collocation(ChC)method21is employed to obtain the solution to this eigenvalue problem.The Chebyshev polynomial of the first kind is introduced as

where ～r ?[-1,1], and the practical coordinate r, as shown in Fig.2, is represented as

where N is mesh number in radial direction.

Rewriting the first term of Eq.(8)with this way at one certain angle θjleads to

The boundary conditions have not been included in the discretized Eq.(15).Substitution of derivative matrix d (1 ) into Eq.(9) yields

where d0,jand dN,jare the first row and last row of the derivative matrix respectively, and Θ is a row vector of dimension N+1, of which only the last element is one and the others are zero.The first and final rows of Eq.(15)should be replaced with Eqs.(16) and (17) respectively as below:

where subscript B denotes the inclusion of boundary effects.

To model the variation of impedance along the angular direction, the details of physical domain at all angles θj,j=-M,???,M should be depicted.Then the extended matrix is written as.

where diag(H )means a block diagonal matrix,each block j of the matrix is H, and Ξ is a matrix of dimension

Eq.(19) can be recast:

Fig.2 Discretized points in transverse plane.

Fig.3 Illustration of propagation waves in duct.

The linearization of non-linear eigen problem Eq.(20)with respect to:

where A,B and C denote constant matrices of the linear terms, square terms and cubic terms of, respectively.

2.2.Matching condition at interface

It is assumed that the duct is separated into several sections and the effects of impedance discontinuity are slight,25as shown in Fig.3, where the continuity of sound pressure and sound particle velocity at interface lead to.

3.Validation of model

To validate the present model,test cases will be computed and contrasted with standard examples under conditions of nonuniform flow.Firstly, a FEM result by Olivieri et al.26is employed with Mach number distribution like

where the non-dimensional boundary layer thickness δ=0?05 M0=0?5 as Fig.4 shows.The eigenvalues/k shown in Fig.5 are computed for an annular duct with hub-to-tip ratio h=0?8, and the wave number k=20.It can be seen that the difference between the two methods is negligible.Therefore,we can verify the validity of the model with shear flow.

Then a comparison of the present method and the FEM achieved by COMSOL is given for the circumferentially nonuniform liner with uniform flow.Transmission loss is defined as:

where the sound energy E includes both upstream and downstream propagating waves, which is sum of acoustic energy of all modes for upstream/downstream wave:

Fig.4 Distribution of flow in duct.

Fig.5 Eigenvalue of rigid duct with shear flow.

Fig.6 Compared result of transmission loss for a four-segment liner,Rd =0?5 m,Rh =0,L=1 m,Ma=-0.3,l=0.1,0.03,0.16,0.07 m.

where l is the depth of liner cavity and the sound source is set to be a single plane wave.In the test,the liner is a four-segment liner with external radius Rd=0?5 m and length L=1 m.The truncated number M is set as 25, and radial direction is discretized into N=12 points.A good agreement between the two methods is found in Fig.6, from which the solutions are nearly indistinguishable for uniform flow case.

Convergence analysis of the model is carried out for the dimensionless wave number k = 6.411 with uniform flow.Error is given as.

where TL0is taken as the ‘‘exact” solution.The number of radial modes in the computational model grows with radial mesh number, hence higher accuracy is achieved as well.The‘‘exact” solution in Fig.7(a) is taken for N = 100.With the default value of M being 17, the calculation errors decrease rapidly with the increase of N to a relatively stable value,and further increasing N leads to negligible changes on transmission loss.The wall impedance is described by circumferential meshes in the model.Thus,a larger M makes the modelling of segmental liners more effective; accordingly, more calculated circumferential modes arise as well.Fig.7(b) shows the convergence process of transmission loss for circumferential meshes.Similar to Fig.7(a), the transmission loss converges to the ‘‘exact” solution which is obtained for M = 101, and the slight deviation from exact solution (δTL) is less than 2.5% even for a relatively low M.In conclusion, the validity of the model is confirmed and efficient solutions can be obtained with small mesh number by this model.In addition,the demand of the radial mesh number N to model the flow will rise markedly if the existence of boundary layer is considered.

4.Results

4.1.Mechanism of acoustic scattering by circumferentially nonuniform liner

Sound scattering is commonly considered to play a significant role in the performance improvement of segmental liners.Assuming that the flow in duct is not sheared, the vanishment ofthepartialdifferentialtermin Eq.(1)willleadtotheconversion of Eq.(20)fromthreeordernonlinearityto two order nonlinearity.

where Au,Buand Cuare constant matrices of the linear terms,square terms and cubic terms offor uniform flow.Hence the dimension of the eigenvalue of Eq.(21)decreases by one third.The boundary layer thickness δ is set to be 0.

To obtain an intuitive observation of acoustic scattering,the mode energy varying along axial distances is given, and selection of impedances should be beneficial to attenuate noise and have significant difference to each other for obvious modescattering.In this paper, selection of liner parameters and sound source depends on a subsistent experiment table, and uniform flow of Ma=-0?04 is considered.Firstly, a preliminary exploration on the eight-segment liner is implemented with frequency f=700 Hz.The eight-segment liner is the combination of rigid wall and liner segments,and its configuration is outlined in Table 1 and Fig.8.The parameters of this liner are Rd=0?3 m, Rh=0, the same as sizes of the experiment table; L=1?2 m, as twice of the duct diameter, which ensure that the acoustic scattering is fully developed.It should be noted that the energy of the incident mode is set to be 100 dB,and all radial modes are contained in the corresponding case.

Fig.7 Convergence of the method for four-segment liners with parameter k=6?411.

Table 1 Liner configurations of eight-segment liner.

Fig.8 Cross section of eight-segment liner.Blue parts are rigid walls and red parts are liner segments given in Table 1.

Analysis results for this phenomenon are presented in Fig.9, with different incident modes of m=0&2.It is seen from Figs.9(a) and (c) that mode scattering is observed at the entrance of lining duct, and both the original wave and generated waves are absorbed simultaneously in the propagation process.At inlet of the duct, there is a sharp minimum in acoustic energy of the scattered modes, which is ascribed to the change of the direction of net sound energy.13In Figs.9(b)and(d),the relation of generated modes and incident mode can be formulated as m′=m0+p ?Nc, where m′is the order of generated mode, m0is of original mode.Ncis the period number of the axial symmetry of the segment liner, or in other words,the cycle number of Z(θ )in the whole circumference; and Nc=4 in this situation.p can be any integer.In addition, energy of the existing modes with opposite order numbers, for example, m=±4, keeps the same in Figs.9(a)and (b); or m=±2, tends to be the same in Figs.9(c) and(d); we describe it as ‘‘symmetry”.If we check the configuration of the liner shown in Fig.8, ‘‘periodicity” (axial symmetry) and ‘‘symmetry” (mirror symmetry) will also be revealed, which reminds us some relationships exist between the liner configurations and acoustic scattering.

Then we design a non-axisymmetric nine-segment liner with mirror symmetry, and its configuration is outlined in Table 2 and Fig.10 with incident modes of m=2, which is actually an aperiodic eight-segment liner similar to the liner in Fig.8.Mode energy of each mode is indicated in Fig.11,from whichwe notice that the all modes are generated, which is different from the phenomena of the axisymmetric eight-segment liner as shown in Fig.9.Besides, energy of m=±2, as the modes with opposite order numbers, still has the tendency to be the same,which is similar to the axisymmetric eight-segment liner.Considering the situation of fully developed acoustic scattering, energy of the opposite-order modes should be the same,because the liner is mirror-symmetric and the two modes are also mirror-symmetric, which means they will attenuate equally.Otherwise, if energy of the two modes is unequal, it will scatter more energy of larger mode to the small one, then the larger one will decrease faster until energy of two modes reaches to a same value.Therefore, it can be affirmed that a mirror-symmetric configuration of liner will affect the mode energy by converging energy of the opposite modes; and the orders of the generated modes depend on the distribution of liner/rigid segments.

Table 2 Liner configurations of nine-segment liner.

Fig.9 Mode energy distribution(a)(c)along axial distance;and(b)(d)at the outlet;for eight-segment liner.The mode order of incident wave is m = 0 for (a)(b) and m=+2 for (c)(d), f = 700 Hz, Ma=-0?04.

Fig.10 Cross section of nine-segment liner.Blue parts are rigid walls and red parts are liner segments given in Table 2.

Another design with axial symmetry but without mirror symmetry is a twelve-segment liner shown by Fig.12 and Table 3.The results shown in Figs.13(a) and (b) of forward liner should be primarily considered.Scattered modes appear with a spacing of Nc=4; energy of the opposite modes does not converge to a same value.Hence, we can confirm that the orders of scattered modes are directly related to axialsymmetric characteristics of segmented liner,rather than numbers of splices or segments; a mirror-dissymmetric liner will cause energy of the opposite modes becoming different.Then we change the direction of liner, and the result of reverse liner is in Figs.13(c) and (d).Compared with the forward liner,mode with maximum energy changes to m=-2, instead of m=2 when the liner is forward,and transmission loss of noise also has a little decrease.Although the change is small, it suggests a method to influence the energy of generated modes and improve noise reduction by varying configurations of segmented liner.

Fig.12 Cross section of twelve-segment liner.Blue parts are rigid walls, green parts are liner segments with Z =1.10 +0.60i,red parts with Z = 0.69–0.72i.

Table 3 Liner configurations of twelve-segment liner.

In the discussion above, truncation numbers M and N are much greater than the needs for cut-on modes, so that the acoustic energy redistribution and dissipation related to cutoff modes can be manifested.It can be predicted that effect of truncated cut-off modes is much smaller than that of the considered cut-off modes,which means the conclusion of convergence analysis in Section 3 is still tenable.However, the minimum values of M and N for convergence is not pursued in the present work.Detailed discussion of effects on acoustic energy redistribution and dissipation with different M and N is in Appendix A.

Fig.11 Mode energy distribution(a)along axial distance;and(b)at the outlet for nine-segment liner.The mode order of incident wave is m=+2 for f = 700 Hz, Ma=-0?04.

Fig.13 Mode energy distribution (a)(c) along axial distance and (b)(d) at the outlet for (a)(b) forward twelve-segment liner and (c)(d)reverse twelve-segment liner.The mode order of incident wave is m=+2 for f = 700 Hz, Ma=-0?04.

Fig.14 Comparison of three configurations in transmission loss with incident mode m = 1, Rd =0?3 m, Rh =0, L=1?2 m,Ma=-0?04.

4.2.Comparison of noise attenuation effects of circumferentially non-uniform liner and uniform liner

To compare lining performance of circumferentially nonuniform liner with uniform liner, some liner configurations are proposed to investigate lining performance in the computation model in which the impedance is obtained by impedance model developed by Xin.28Because of the linear assumption of acoustic dynamical system, linear superposition of acoustic modes can be performed, thus mode analysis is an effective means for obtaining a good understanding of sound transmission in aero-engines.Obviously, performance of CNU(Circumferentially Non-Uniform) liner is not better than the two uniform liners at the design points, but the property of liner in a broad band frequency deserves discussion.Cases of single incident mode at different frequencies are presented below,where t,d and σ respectively are thickness,orifice diameter and perforation ratio of perforated plate.

For low order incident mode m = 1, the Uniform liner 1(U1) is designed for f=700 Hz, and the Uniform liner 2(U2) is for f=1500 Hz, where liner is optimized for the best performance,as shown in Fig.14 and Table 4.The Circumferentially Non-Uniform liner 1 (CNU1) consists of these two uniform liners, which is expected to have better performance in the frequency range of 700–1500 Hz.A configuration similarto Fig.8 is adopted to scatter specific modes.The acoustic features of the uniform liners are changed by segmentation as shown in Fig.14.The number of resonance frequency is doubled for CNU liner and related to that of uniform liners.The attenuation value for CNU liner is between the two uniform liners, exception exists nearby 1250 Hz where CNU liner performs better than both uniform liners.And in the frequency range of 1500–2000 Hz, attenuation of CNU liner is close to that of U2 liner.

Table 4 Liner configurations for mode m = 1.

Fig.15 Comparison of three configurations in transmission loss with incident mode m = 6, Rd =0?3 m, Rh =0, L=1?2 m,Ma=-0?04.

Table 5 Liner configurations for mode m = 6.

Although liner designs are almost the same, for higher order incident mode m = 6, there is not the same exception.The Uniform liner 3 (U3) is designed for f=1400 Hz, and the Uniform liner 4 (U4) is for f=2100 Hz, where the liner performance is also optimized,as shown in Fig.15 and Table 5.The Circumferentially Non-Uniform liner 2 (CNU2) consists of these two uniform liners.The acoustic features of the uniform liners are changed by segmentation as shown in Fig.15.Different from the previous situation, attenuation value for the CNU liner is lower than both the uniform liners around 1600 Hz.

Why do circumferentially non-uniform liners have performance as previously mentioned? One of the reasons is that some segments behave to reduce the noise at their resonance state, while the others may work at the condition deviating from the best working states,which means the highly effective area for a mode is reduced.Considering mode m = 1 at frequency f=700 Hz,only half of the liner segments are beneficial for reducing noise.Another reason is because acoustic scattering changes the distribution of mode energy.The work of Rice29–31argues that the sound attenuation in a lined duct is dependent on the modes’cut-off ratios.The wave of higher circumferential order generally takes on lower cut-off ratio that represents the easiness to absorb or reflect for the same liner.Thus, the energy transfer to higher order modes is attenuated more smoothly.Considering about mode m = 1 at frequency f = 1250 Hz, an analysis of mode energy distribution is presented in Fig.16.It is seen that extra modes are generated and a little potion of mode m=1 energy is allocated to mode m= -3,+5 and other higher order modes.Those modes are either cut-off modes which should damp rapidly along axial direction, or higher cut-on modes which should be absorbed faster by liners.The phenomenon in the frequency range of 1500–2000 Hz can also be similarly explained.Thus, acoustic scattering will give rise to positive effect to noise reduction with low order modes, and may offset the loss caused by segmenting.However, the acoustic energy may be redistributed into some waves of lower orders as well, which may lead to adverse effects.For another case of mode m= 6 at frequency f = 1600 Hz, the situation is different.The analysis of mode energy distribution shown in Fig.17 points out that part of the higher order mode m = 6 energy is allocated to the lower order modes m = ±2.Lower order modes will be more difficult to attenuate than the higher incident mode,and therefore,acoustic scattering will cause negative effect to noise reduction for high order modes.From this case, it is concluded that reducing the energy of lower order modes, and scattering energy to higher order modes, will be a significant concern to further reinforce/mitigate the positive/negative effects of acoustic scattering on sound attenuation.

Fig.16 Mode energy distribution along axial distance and at outlet for CNU1 liner.The mode order of incident wave is m = 1 for f = 1250 Hz,Ma=-0?04.

Fig.17 Mode energy distribution along axial distance and at outlet for CNU2 liner.The mode order of incident wave is m = 6 for f = 1600 Hz,Ma=-0?04.

Table 6 Liner configurations for actual sound source.

4.3.Attenuation of broadband noise by circumferentially nonuniform liner

As mentioned in the introduction, the fan noise becomes a prominent part of aero-engine noise as turbofan engines with high bypass ratio are widely used in civil aircrafts.Mitigating broadband multimode noise becomes a particularly important concern to the noise attenuation of aero-engines.Since the broadband noise is the superposition of many modes,the scattering is a synthesis effect of each mode in the sound source.Besides, the response of acoustic scattering to different order of waves alters as well.All of these make the scattering become extremely complex.

The performances of circumferential segmentation process under an actual sound source are meaningful for practical application.Now, it is assumed that the energy of each mode is not equal anymore, and the mode distribution in sound source is associated with the predicting result by Zhang et al.32The frequency of the source is up to 8000 Hz, and the highest order of contained mode is m=41,which is time consuming to computing for many numerical methods.Performances of the three configurations as Table 6 are illustrated in Fig.18.It is seen again that CNU liner exhibits a compromised property of two uniform liners and good performance of broadband liners.The maximum attenuation at some discrete frequencies is not exactly the pursuit of broadband liners but the whole noise reduction for a wide frequency range.The peak of the segmental liners is less than 25 dB, but the minimum value is near 5 dB,and in a large part as 1000–5000 Hz is above 15 dB.

Fig.18 Performances of three configurations for a given real sound source, Rd =0?28 m, Rh =0?14 m, L=1 m, Ma=0?355.

4.4.Effects of the complex flow

4.4.1.Convection effects of mean flow

Acoustic liners are generally applied in the inlet, bypass duct and core nozzle.The actual working conditions for a liner placed in different positions are different, thus performances are not the same as well.The air flows against the direction of acoustic wave propagation at inlet of engine but the case is exactly the opposite at core nozzle.Firstly,the effect of convection on sound attenuation for three liners in Table 7 is shown in Fig.19, where positive Mach number indicates the sound propagating downstream while the negative suggests the one propagating upstream.For mode m=1, it is evident that altering the velocity of the mean flow changes the acoustic performance of the liners.The noise for negative Mach number tends to be attenuated more easily, especially in low frequency.In high frequency, the noise for positive Mach number is instead absorbed more easily, because of properties of uniform liners.

One possible explanation for the phenomenon is that when the media flows against the propagating wave,the sound wave is compressed due to Doppler effects,thus wave period in axial direction decreases.It can be reflected by the change of axial wave numberin rigid duct.For a certain sound wave,when Mach number is smaller, the larger axial wave numberwill bring about a shorter wave period.The interaction times between sound waves and liners become more in certain axial length.On the contrary,the flow moving downstream will lead to smaller wave numberand the reduction of noise attenuation.

Acoustic scattering, in fact, is also influenced by flow conditions.For circumferentially non-uniform liner 4, frequency point f=1000 Hz for mode m=1 are selected.At this point,transmission loss of two conditions is close to each other,hence the influences on scattering by flow condition become visible.Mode energy distribution at the inlet and along axial distance at frequency f=1000 Hz for different Mach numbers is plotted in Fig.20.The energy of other modes is excited by sound scattering, and developed rapidly for negative Machnumber, as shown in Fig.20.For positive Mach number, in particular,there is a gentle energy growth of these waves along axial distance.For example,the mode m=-3 for Ma=-0?3 is around 73 dB at inlet of the duct, and grows rapidly in the front part; but for Ma=0?3, the mode begins as 71 dB and grows slower in the front part, but reaches 86 dB, the same as Ma=-0?3 at the end.The energy growth of upstream propagation indicates that the generated modes continually obtain energy from the mode m=1,which is attenuated in lining region, accompanied by losing its energy through scattering the energy to the waves of other modes.There is no doubt that the transferred energy to higher order modes should be absorbed more easily.Therefore,in large part of frequency spectrum,where the attenuation is small,acoustic scattering makes an additional attribution to the sound attenuation for negative Mach number.

Table 7 Liner configurations for |Ma|=0.3.

Fig.19 Comparison of attenuation on noise propagating in different directions for uniform and circumferentially non-uniform liners, Rd =0?3 m, Rh =0, L=0?6 m.Mode m = 1.

4.4.2.Effects of non-uniform flow

In the above research, the ducted mean flow is assumed to be uniform.But the simplification inevitably neglects the effects due to the non-uniform flow,for example,caused by boundary layers,variable section inlet and inlet distortion.The following content will discuss the impact of non-uniform flow on the acoustic performance of circumferentially non-uniform liner.In comparison with the computation for uniform flow, the model including the non-uniform flow makes the dimension of the matrix solved larger,thus it demands slightly more time to finish the calculation under the circumstance of non-uniform flow.

Fig.20 Mode energy distribution along axial distance for CNU4 liner.The mode order of incident wave is m=1 for f=1000 Hz.

Preliminarily, three liners in Table 7 are considered in the test, the profile of the flow is given by Eq.(26).The thickness of the boundary layer is δ=0?05(5%of the radius).In another condition, a profile of flow (not boundary layer) is set as δ=0?5 to correspond exaggerated non-uniform flow, as shown in Fig.21.The sound source of mode m=1 is given.Effects of non-uniform flow on transmission loss of uniform liners U7 and U8 are presented in Fig.22.A slight influence in noise attenuation can be found when non-uniform flow is considered, while its positivity or negativity is correlated with frequency and flow direction.For Ma=-0?3 situation, noise attenuation increases at high frequency, but decreases at low frequency in non-uniform flow condition; vice versa, reversed effects happen for Ma=0?3.Resonance peaks of the curves are not altered significantly.

One possible mechanism attempts to explain current phenomenon: when sound waves propagating in non-uniform flow, refraction effects happen, which deflect the incident waves to the duct wall for downstream propagation, or to the duct center for upstream propagation.This may change the spatial distribution of acoustic energy and the incident angles of sound waves to liner wall,then noise attenuation will be affected.

Fig.21 Distribution of non-uniform flow in duct (δ=0?5).

For circumferentially non-uniform liners, effects of nonuniform flow on the attenuation capacity are still nonnegligible.Changes on transmission loss is illustrated in Fig.22 where the sound source of mode m=1 is given.Similar to uniform liners, refraction effects caused by non-uniform flow will affect sound scattering and noise attenuation of circumferentially non-uniform liners.It is seen that nonuniform flow affects the noise attenuation positively or negatively with different frequency and flow direction.Besides,non-uniform flow will intervene effects of sound scattering.From Figs.23(a)-(b), it can be seen that the energy of modes excited by sound scattering is higher and developing faster in non-uniform flow for negative Mach number,but uncorrelated with frequency.Conversely,for positive Mach number,effects of sound scattering are weakened in non-uniform flow, as Fig.23(c) shown.

Though the relationship between sound scattering/noise attenuation and non-uniform flow is found,actual mechanisms deserve further research and discussion through experiment or CFD.From the research above, the computational model is practicable for the research of non-uniform flow.In practical situation, distribution of non-uniform flow may be different from the assumption of the above example, and actual effect will also change.Then, more exact results and conclusions can be found by the current model.

5.Further analysis and discussion

In summary, the acoustic system of circumferentially segmented structure is investigated by the development of a fast-computational model including multiple control variables.Due to applications of the spectral approach, the present model possesses higher calculation accuracy and thus provides a reliable and rapid tool for the acoustic design of circumferentially non-uniform liners which should have required massive computing power.

Fig.22 Effects of non-uniform flow on uniform and circumferentially non-uniform liners.The mode order of incident wave is m = 1.

Fig.23 Mode energy distribution along axial distance for CNU4 liner.The mode order of incident wave is m=1.Flows are mean flow and non-uniform flow (δ = 0.5).

Especially, this work has made a thorough exploration to figure out the significance of the mode scattering to sound propagation.For a single incident mode, the tests of several configurations presented in the paper show that the acoustic ability of segmental liners is related to the acoustic scattering,the attenuation is more powerfully affected by a stronger acoustic scattering.A segmented liner can scatter acoustic waves to specific order of modes by axial-symmetric configurations, or adjust the energy weight of different generated modes by mirror-symmetric configurations.The specific influence of acoustic scattering on sound propagation depends on the generated modes.In particular,if the high-power acoustic waves are transformed into the scattering modes that are easily damped, the improvement of sound attenuation can be accordingly obtained; on the contrary, the generation of the lower order propagating modes may also lead to adverse effects on the sound attenuation.Suppressing the generation of lower order modes through structure design thus becomes a significant concern to tap the potential of acoustic scattering for further improving the attenuation.

In spite of the complex attenuation features even for a single incident mode, the mechanism of acoustic mode scattering by circumferentially non-uniform liner can be still applied to suppress broadband noise sources,which are more common in practical problems.In fact,the control of broadband noise requires a wider frequency range instead of pursuing the reduction of the maximum peak corresponding to some specific frequency.This just needs different mode energy distributions, whereas the design of circumferentially segmented structure can be used to adjust the mode energy distributions by rich scattering effects.Indeed, the obtained solutions in the present study show the superior performance of circumferentially non-uniform liners on the reduction of actual broadband noise which was associated with the predicting results by Zhang et al.32,the attenuation is larger than 5 dB in the whole spectrum,and over 15 dB in a large frequency range between 1000 to 5000 Hz.

To make better use of the single or multi-modes scattering effects, it is still necessary to further understand how various complex parameters affect the sound scattering and acoustic mode attenuation.For these aspects, it is noted that the response of scattering to different flow conditions are calculated for a deeper understanding of this mechanism.Results show that the phenomenon of scattering is different for the wave propagating in two directions.Mode scattering for downstream propagation is gentle, but for upstream propagation scattered mode energy grows to maximum quickly.In addition, non-uniform flow caused by various factors influences the mode scattering as well.Although resonance peaks of the curves are not altered significantly, visible influences on attenuation and sound scattering are observed.The effects of the flow condition are proved to be non-negligible;however,it is too difficult to design a liner directly with complex conditions.At the present stage,modifications of complex flow conditions to optimization results are still meaningful, though the logical modifications need to be calculated according to the investigations of practical and engineering examples.

The next step is to validate the results and conclusions by experimental results,and some unclear mechanisms in complex flow are expected to be clarified through experiments or CFDs.

6.Conclusions

(1) A computational model for circumferentially nonuniform liners under uniform flow is proposed in the present paper.The model is validated by a comparison with FEM results and a convergence analysis.

(2) The transmission loss spectra and mode energy distribution are calculated to explore the attenuation characteristics of the liners.The discussions are mainly focused on the intrinsic scattering mechanism to obtain a further knowledge of the circumferentially segmented liners.It is found that scattering effects are related to the configuration of the liner, and also change the sound attenuation ability of the liner.With specific configuration,CNU liners will scatter specific modes, or change mode energy distributions, so that the potential of sound attenuation can be further developed by suppressing lower or enhancing higher order modes with expressly designed segmented liner configurations.Moreover, circumferentially non-uniform liner is conducive to reduce the practical broadband sound source.

(3) Results of the present numerical simulation also show that the performance of acoustic scattering is affected by flow conditions.Actual mechanisms of effect of flow seem to be complex,but one possible mechanism is that refraction effects happening in non-uniform flow may change the spatial distribution of acoustic energy and the incident angles of sound waves to liner wall, which will affect the interactions of sound waves and liner, in other words, sound scattering and noise attenuation.

In summary,the current rapid calculation method shows its availability for research,evaluation and design of circumferentially non-uniform liners,even in complex flow conditions.The understanding of acoustic scattering studied in this paper, for example,specific design and arrangement of liner segment configurations, will also conduce to the design and application of circumferentially non-uniform liners.

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 work was supported by the National Natural Science Foundation of China (No.52106038) and the Science Center for Gas Turbine Project of China (No.P2022-B-Π-013-001).

Appendix A.Effects of truncation numbers on acoustic energy redistribution and dissipation When we calculate numerical results of CNU liners, the redistribution and dissipation of mode energy, especially those of cut-off modes, are considerable.If some cut-off modes are neglected, errors or mistakes of results will appear.The number of modes considered in calculation is truncated by truncation numbers, M and N.So, some numerical results with different M and N are in computation to show the effects of truncation numbers on acoustic energy redistribution and dissipation.

The result shown in Fig.A1(a) is the same with that in Fig.9(b), with M = 48 and N = 17.So that the acoustic energy redistribution and dissipation related to modes｜m ｜≤23 are considered.The modes ｜m ｜≥3 are cut-off modes at the current condition, so scattered modes m=±4 are cutoff.Those modes with order ｜m ｜≥8 are 15–20 dB smaller than incident mode m=0 and main scattered modes m=±4.Energy is hard to be redistributed from m=0,±4 to modes far away, and other higher order modes are weak, so it can be surmised that effects of truncated cut-off modes(｜m ｜>23) is also smaller than main scattered modes and can be ignored reasonably.

And the results with different M and N are shown in Figs.A1(b)-(c).It is observed that effects of truncation numbers M and N are small to m=0 and m=±4,but slightly larger to high order modes.When truncation numbers increase,the results of mode energy seem to be convergence, as Table A1 shows.This suggests us to choose proper M and N, considering the highest cut-on mode or most concerned modes.

Fig.A1 Mode energy distribution at outlet for eight-segment liner.The mode order of incident wave is m = 0, f = 700 Hz,Ma=-0?04.Truncation numbers are (a) M = 48, N = 17; (b)M = 24, N = 9; (c) M = 72, N = 25.

Table A1 Mode energy at outlet with different truncation numbers.