A Coherence Approach to Characterizing Broadband Sound Fields in Ducts

Transcription

1 Coherence pproch to Chrcterizing Brodbnd ound Fields in Ducts Phillip JOEPH Institute of ound nd Vibrtion Reserch, University of outhmpton, Highfield O7BJ, UK. BTRCT This pper describes new mesurement technique tht llows the model mplitude distribution to be determined in ducts with men flow nd reflection bsed only mesurements of the two-point coherence mde t the duct wll. The technique is primrily pplicble to brodbnd sound field in the high frequency limit nd whose mode mplitudes re incoherent. The technique mkes the ssumption tht the reltive mode mplitude distribution is independent of frequency. Keywords: Duct coustics, Mesurement, Modl nlysis.. INTRODUCTION The brodbnd noise due to fn in modern turbofn ero engine is one of the dominnt noise sources contributing to community noise nnoynce, prticulrly t pproch. detiled understnding of its noise mechnism, n ssessment of its sound power trnsmission, rdition to the fr field, nd the design of n effective liner to ttenute this noise source, ll require detiled mesurements of its mode mplitudes. Unlike the noise t the blde pssing frequency, which typiclly comprises just few dominnt modes in ccordnce with the Tyler ofrin mode selection rule, brodbnd noise generlly comprises ll possible propgting modes. The difficulty with the mesurement of their mode mplitudes is tht, in generl s mny microphones re needed to mesure the sound field s there is numbers of modes. t frequencies close to the blde pssing frequency, for exmple, the number of modes cn redily exceed one hundred, which renders the simultneous mesurement unrelistic. pproches hve been followed to limit the number of microphones by the use of, for exmple, rotting rkes of microphone rrys where limited number of microphones re slowly rotted or trversed though the in-duct sound field. Modl mplitudes my then be deduced though the inversion of the cross spectrl pressure mtrix. Very often, however, the mtrix to be inverted cn become ill-conditioned leding to erroneous mode mplitude estimtes. prgmtic solution to determining the mode mplitude distribution in ducts hs been proposed by Lowis et ln in which single xil rry of microphones t the duct wll re used s bemformer to estimte the in-duct noise directivity. Rther thn providing informtion bout the mode mplitude for the different mode orders (m,n), the technique provides informtion bout the distribution of mode mplitudes versus in-duct propgtion ngle. For the intended pplictions listed bove, this limittion presents no difficulty s it hs long been recognized tht modes with the propgtion ngles possess ner identicl trnsmission nd rdition chrcteristics.. MODL TRNMIION Consider hrd-wlled cylindricl duct of finite-length, s sketched in Fig below, contining n xil uniform men flow moving in the positive x direction with flow speed cm (M > 0), where c is the sound speed nd M is the men flow Mch number. point on the duct cross section is represented by y = (r,) nd x denotes the xil distnce long the duct reltive to some rbitrry origin. Two microphones mounted flush to the duct wll, seprted xilly by distnce x, re used to detect the coustic pressure. The objective here is to deduce the distribution of mode mplitudes in the duct using the coustic pressure informtion t the two microphones. Inter-noise 04 Pge of 9

2 Pge of 9 Inter-noise 04 Figure. emi-infinite, hrd wlled unflnged circulr duct with ssocited co-ordinte system. Two microphones mounted flush to the duct wll, seprted xilly by distnce x, re used to detect the coustic pressure. The sound field p(x,y) in the duct stisfies the convected homogeneous wve eqution, c D Dt p 0 () where D Dt c M is the convected derivtive opertor ssocited with the men flow velocity t x cm,0,0 in the (x,y) coordinte system nd c is the sound speed in the quiescent medium. bove its cutoff frequency, t single frequency, single mode of pressure mplitude is described by p it ikx y, x e ye () where the superscript +; refers to modes propgting in the direction of flow nd - to modes propgting in the opposite direction to the flow. Eqution () in Eq. () gives k M c, k. (3,b) where M tht the corresponding mode shpe functions, defined by y 0 duct-wll boundry conditions nd the normliztion condition dy precisely t the modl cutoff frequency M c, nd tends to nd re set of eigenvlues tht re chrcteristic of the duct cross section such, lso stisfy the y. The prmeter, which we shll cll the cut-on rtio, is centrl in wht follows, nd tkes vlues between 0 s /, corresponding to modes well bove cuton. Modes propgting in the direction of the flow re represented by 0 while modes propgting in the opposite direction (ginst the flow) re represented by, 0. The in-duct sound field t ny position in the duct cross section y = (r,), xil position x, nd frequency, cn be expressed s the sum of modl components propgting in the direction of flow p, nd modes propgting opposite to the direction of flow p,, x p y, x p y, x p y, (4) m n0 where (m,n) re the usul circumferentil nd rdil mode indices. Pge of 9 Inter-noise 04

3 Inter-noise 04 Pge 3 of 9 3. RELTIONHIP BETWEEN MODE MPLITUDE DITRIBUTION ND COHERENCE The coustic pressure cross spectrum between two points seprted xilly long the duct wll y r,, t xil distnces x nd x + x, my written s (5) T *, y, x, x E py, x p y x x, where E{} denotes the expecttion nd the coustic pressures refer to Fourier Trnsforms of the pressure time series tken over time durtion T. For incoherent excittion of the sound field we tret the mode * mplitudes s uncorrelted rndom vribles so tht E 0. We further ssume tht the sme mode propgting in opposite directions re lso uncorrelted such tht, E * 0. ubstituting Eqs () nd (4) into Eqs (5) nd invoking the uncorrelted mode ssumptions bove leds to, (6), e m, n ikx ikx y E e E y Work by Rice 3, nd more recent work by Joseph et l 4, hve shown tht there re physiclly importnt clss of source distributions for which the reltive mode mplitude distribution is independent of frequency nd only function of the cut off rtio (equivlently, mode propgtion ngle, see eqution (5) below). Well known exmples include uniform distribution of monopole sources, xil dipole sources nd equl energy per mode 4. In these, nd mny other source distributions, we my write, where frequency nd E is the frequency-dependent source strength with dimensions of pressure squred per unit specifies the reltive distribution of non-dimensionl men squre mode mplitudes, which depends only on. list of some physiclly importnt exmples re listed in the ppendix. The ssumption of the seprbility of E into purely frequency dependent term nd mode distribution term (which controls the sptil vrition of the sound field) is centrl to the vlidity of the technique. The split between the two terms in Eq. (7) is essentilly rbitrry. For resons tht will become cler below, we define with the normliztion property, m, n (7) (8) We now denote the mplitude of wves propgting ginst the direction of flow (i.e., reflected modes in this cse) by negtive rgument so tht, 0, 0 (9) In the high frequency limit (k = /c > 0 hs been found to be sufficient, where is the duct rdius), we my tret s continuous vrible so tht the discrete summtion over in Eq. (6) my be replced by n integrtion over. The normliztion condition of Eq. (8) my therefore be written s n d (0) where n() the modl density function introduced to tke ccount the distribution of modes cross their rnge of - vlues, defined by, Inter-noise 04 Pge 3 of 9

4 Pge 4 of 9 Inter-noise 04 N N n, d lim 0 n (,b) where N() is the number of modes with vlues of between - nd nd N is the totl number of propgting modes t frequency k, i.e., N d. Rice hs shown tht in cylindricl duct with uniform men flow, the totl number of propgting modes N tkes the high-frequency limiting vlue 3, /, k / k () Following Rice, nd re-expressed in terms of cuton rtio by Joseph et l 4,5, the high-k symptotic density function n, is given by, n (3) Note tht Eq. (3) differs by fctor of ½ from the expression originlly presented by Joseph et l 4, which ssumes distribution modes propgting from one direction only. Eqution (3) indictes scrcity of modes tht re just cut-on ( 0 ) compred with higher popultion of modes tht re well cut on, ( ). implifictions to Eq. (6) for the pressure cross spectrum t the duct wll re obtined by replcing by its verge vlue t the duct wll 4, verged over ll vlues of mode indices m nd n, y y y. (4) Tking the verge incurs gretest error for modes with the lrgest m vlues whose vlues of y re concentrted t the duct wll. These modes re comprtively scrce, however (with m = 0 hving the lrgest number of rdil modes nd hence being most common), nd hence the pproximtion of Eq. (4) introduces negligible error compred with the exct clcultion of Eq. (6). ubstituting Eq. (7) for, Eq. (3) for k, nd tking the high frequency limit in the sense of Eq. (0), leds to n integrl expression for the pressure cross spectrum, y, x between two microphones seprted xilly by distnce x t the duct wll y, involving only the cutoff rtio nd the frequency-dependent source strength, ˆ, ˆ y im ˆ ˆ e e i n which is function only of the non-dimensionl frequency, ˆ d, (5) ˆ x / c, (6) Note tht the source term ˆ hs lso been written s function of ˆ which is permissible since is source term nd therefore unrelted to x nd so there is no difficulty in non-dimensionlising the source frequency with respect to this rbitrry distnce. consequence of mking the seprbility ssumption of Eq. (7) is tht the cross spectrum is only function of the non-dimensionl frequency, ˆ. Thus, cross spectr mesured t the duct wll for different seprtion distnces x, plotted ginstˆ, should collpse provided tht this seprbility ssumption is met. This property therefore provides simple test of the vlidity of Eq. (7). In prctice, however, the coherence mesurement will be ffected by non-coustic pressure contributions from flow noise t the microphones. In prctice, therefore, steps should be tken to minimize contmintion by flow noise by, for exmple, recessing the miscrophones into the duct wll. Pge 4 of 9 Inter-noise 04

5 Inter-noise 04 Pge 5 of 9 Interprettion of. Putting x 0 in Eq (6) yields the pressure Power pectrl Density,y t ny point over the duct cross section y. verging the result over the duct cross section re nd tking the high frequency limit yields, ˆ, y d y ˆ n d y d y (7) f Noting the normliztion property of the mode shpe functions, dy mplitude normliztion property of Eq. (0), Eq. (7) reduces to, f y nd the mode ˆ ˆ,y d y (8) The source strength ˆ therefore hs the interprettion s the high frequency noise pressure spectrum verged over the duct cross sectionl re, per mode. Joseph et l 4 hs shown tht, in the high frequency limit, the pressure PD verged over the duct cross section, ˆ, is hlf the pressure Power pectrl Density (PD) mesured t the duct wll ˆ, y, i.e., ubstituting Eq. (9) into (5) leds to, ˆ, ˆ y (9) im ˆ i ˆ ˆ ˆ e e In Eq. (0), nd ll future results, the dependence on y is dropped since it is now understood tht ll ˆ nd mesurements re mde t the duct wll. Finlly we mke the pproximtion tht ˆ hence ˆ ˆ ˆ, since x is usully very smll (typiclly few centimeters), nd set the upper limit of integrtion to infinity (since 0 finl result is, d (0) for corresponding to cutoff modes). The im ˆ i ˆ e e ˆ d () Eqution () represents Fourier Trnsform reltionship between the weighted normlized mode mplitude distribution function nd the complex coherence function ˆ ˆ ˆ ( 0 ˆ ˆ ˆ, ) () The men squre mode mplitude distribution, with the normliztion property of Eq (8), my therefore be ˆ redily deduced from the inverse Fourier Trnsform of the complex coherence function weighted by e im. ˆ im ˆ i ˆ e e d ˆ (3) Eqution (3) is the min result of this pper. It suggest tht the normlized mode mplitude distribution my in principle be deduced using just two microphones for ny incoherent multi-mode sound field whose ˆ cross spectr collpses on the non-dimensionl frequency ˆ x / c. The phse fctor e im serves s Lorentz trnsformtion into the reference frme moving with the flow such tht the mplitude distribution Inter-noise 04 Pge 5 of 9

6 Pge 6 of 9 Inter-noise 04 for right nd left-trveling propgting modes is now symmetric in, lying in the rnge. In this section we vlidte the principles set out bove by number of numericl exmples to illustrte the effectiveness of the technique in deducing, bsed only on the complex coherence mesurement t the duct wll, the mode mplitude distribution nd trnsmitted sound power for incident nd reflected modes, nd the fr field pressure directivity. We consider the idelized cse of duct in which ll the modes propgting towrds the end of the duct contin equl sound power. The normlized mode mplitude distribution is obtined by setting W 0 = in Eq. (8) nd normlizing ccording to Eq. (b), M (36) M /3 In hrd wlled cylindricl duct the mode shpe function re of the form r Jmk r r/, where J m re Bessel functions of the st kind or order m, kr is the n th sttionry vlue of J m nd re constnts chosen to stisfy the normlistion condition presented bove. Modl pressure reflection coefficients of the form R exp / re ssumed in the simultions, so tht R, where specifies the rte t which the reflection coefficient diminished s the modes is excited well bove cut off. This reflection coefficient model is consistent with other more ccurte models nd is designed to ensure tht modes t cutoff, = 0, re perfectly reflected, with the reflection coefficient reducing s the modes become incresing cuton s frequency is incresed. pecil cses; zero Mch number, rbitrry reflection We first consider the cse of M = 0 since it llows nlytic expression to be derived nd compred ginst exct numericl predictions. For the cse of Equl energy per Mode, the mode mplitude distribution my be obtined by setting the sound power in ech mode equl to unity, W W in Eq. (8), / N 0 (37) exp / N 0 where N is the fctor designed to ensure tht is correctly normlized ccording to Eq. (0) nd equls, N e /. The complex coherence function is obtined from substituting Eqs (37) into () to give ˆ e e / e ˆ i ˆ ˆ i ˆ sin cosh i cos sinh e ˆ i 0 sin ˆ ˆ Figures nd b show comprison of the coherence mgnitude nd phse respectively, evluted t the duct wll for n equl energy per mode sound field computed from the exct modl summtion of Eqs. (6 nd ) (blue curve) with the nlytic expression of Eq. (38) (blck dshed curve). Comprison re shown = for the four reflection fctors, = 0,, nd 5. Note tht the curves hve been seprted for ese of redbility. (38).5 5 Mgnitude Coherence.5 = = 5 = = Coherence phse (rds) = = = 5 = kx/² kx/² Figure. Comprison of exct nd theoreticl coherence function mgnitude nd phse for M = 0 t different levels of reflectivity, = 0,, nd 5. Note tht the curves hve been seprted for ese of viewing. Pge 6 of 9 Inter-noise 04

7 Inter-noise 04 Pge 7 of 9 Oscilltions in the exct clcultion rise from the behvior of the spectr t the modl cuton frequencies. Here the pressure mplitude tends to infinity s the cutoff frequency is pproched. s the modl reflectivity is incresed (by reducing ) the coherence mgnitude nd phse both exhibit greter vribility. Limiting cses cse of the coherence function my be obtined for the cse of perfect modl reflectivity =0, nd when the reflectivity is zero, i.e., the duct my be ssumed to infinite. In the ltter cse, putting into Eq. (38) yields ˆ exp ˆ sin ˆ (39) ˆ i which is in close greeement with the exct clcultion shown in figures nd b. The ducted sound field my now be regrded s one-sided (or hemi-diffuse sound field). The phse dely between the two microphones which vries with frequency s x / c, i.e., precisely hlf the rte of purely plne wve. Precisely this behvior is observed in figure nd b for the cse of lest reflectivity, = 5. When ll modes re perfectly reflected t the end of the duct, 0, nd Eq. (38) tends to sin ˆ ˆ (40) ˆ In this cse, where ech incident hs equl sound power nd is perfectly reflected incoherently, the coherence function is identicl to tht of diffuse sound field in which energy is rriving from ll ngles eqully. Clerly, therefore, there is no phse vrition between the two microphones, s shown in figure b, where phse jumps cn be observed due to unwrpping issues. Mode mplitude distribution Figure 3 shows comprison of the exct men squre mode mplitude distribution versus of Eq. (37) with tht deduced by inversion of the complex coherence functions plotted in figures by the use of Eq. (3) = 0 X = X = X = 5 Reflected modes Incident modes Figure 3. Comprison of the exct (blue curves) nd inverted mode mplitude distribution (red curves) for four reflectivity fctors t M = 0. greement between the exct nd inverted mode mplitude distribution is generlly excellent except ner the extreme vlue of = 0, where the modes re well cuton, nd, corresponding to modes tht re close to cutoff. Errors re prticulrly gret for the very well cuton modes. This is likely to be due to the choice of equl energy per mode model chosen for the simultion since the mode distribution becomes singulr t = 0, which clerly cnnot be recovered from Eq. (3) using numericl integrtion. Errors re lso pronounced for the ner-cutoff modes prticulrly for the cse of lest reflectivity rising from numericl errors in the evlution of Eq. (3). Inter-noise 04 Pge 7 of 9

8 Pge 8 of 9 Inter-noise EXPERIMENTL PPLICTION We now pply the mode mplitude mesurement technique to some coherence dt obtined in the bypss section of the necom fn rig t Germny, shown below in figure X. The men flow Mch number ws The microphones were mounted flush to the duct wll nd no ttempt ws mde to shield the microphones from the turbulent boundry lyer t the duct wll. Figure 4. chemtic of the fn rig nd mesurement section in the bypss section x=0.07m - Mgnitude Coherence x=0.054m x=0.6m Phse Coherence (rds) x=0.07m x=0.054m x=0.6m x=0.43m x=0.08m 0. 0 x=0.43m x=0.08m kx/² kx/² Figure 5 nd b show the mgnitude nd phse of the mesured coherence for the five seprtion distnces, x = 0.07m, 0.054m, 0.08m, 0.8m, nd 0.43m, plotted ginst normlized seprtion distnce. The mgnitude nd phse of the coherence function re in resonble greement except for the two lrgest seprtion distnces where the mgnitude of the coherence is generlly very smll due to boundry lyer noise. The corresponding inverted mode mplitude distribution function is shown below in figure 6. Pge 8 of 9 Inter-noise 04

9 Inter-noise 04 Pge 9 of x=0.054m 0 0 x=0.6m 0 - x=0.43m 0 - x=0.08m x=0.07m Figure 6. Modl mplitude distribution versus cutoff rtio deduced from the complex coherence function mesured in the bypss section of fn rig for 5 different seprtion distnces. s for the coherence estimtes the mode mplitude estimtes re resonbly consistent except for those obtined from the lrgest seprtion distnces. 5. CONCLUION This pper hs described new method for determining the mode mplitude distribution in multi-mode brodbnd sound field in ducts in the presence of uniform men flow nd reflections. The novelty of the techniue is tht it requires only mesurements of the complex coherence function mde t the duct wll. The technique is vlid in the high frequency limit nd is restricted to cses where the reltive mode mplitude distribution is independent of frequency. REFERENCE. E. J. Rice 978 I Journl. 6, Multimodl fr-field coustic rdition pttern using mode cutoff rtio.. C.L. Morfey 97 Journl of ound nd Vibrtion 4, ound trnsmission nd genertion in ducts with flow. 3. E. J. Rice 978 I Journl. 6, Multimodl fr-field coustic rdition pttern using mode cutoff rtio. 4. P. Joseph,, C.L. Morfey, C.R. Lowis. Multi-mode sound trnsmission in ducts with flow. Journl of ound nd Vibrtion 64 (003) P. Joseph nd C. L. Morfey 999 Journl of the cousticl society of meric 05(5), Multi-mode rdition from n unflnged, semi-infinite circulr duct. 6.. nyoko, P. Joseph nd. Mclpine. Multi-mode rdition from ducts with flow. J. coust. oc. m. 7 4, pril 00 Inter-noise 04 Pge 9 of 9