Planar Nearfield Acoustical Holography in High-Speed, Subsonic Flow

Transcription

1 Baltimoe, Mayland NOISE-CON Apil 19-1 Plana Neafield Acoustical Hologaphy in High-Speed, Subsonic Flow Yong-Joe Kim a) Texas A&M Univesity Depatment of Mechanical Engineeing 313 TAMU College Station, TX Hyusang Kwon b) Koea Reseach Institute of Standads and Science P.O. Box 10, Yuseong Deaon, South Koea The obective is to develop a NAH pocedue that includes the effects of the high-speed, subsonic flow of a fluid medium. Recently, the speed of a tanspotation has significantly inceased, e.g., to close to the speed of sound. As a esult, the NAH data measued with a micophone aay attached to an aicaft o tain include significant aiflow effects. Hee, the convective wave equation along with the convective Eule s equation is used to deive the poposed NAH pocedue. A mapping function between static and moving fluid medium cases is also deived fom the convective wave equation. Then, a new wave numbe filte defined by mapping the static wave numbe filte is poposed. Fo the pupose of validating the poposed NAH pocedue, a monopole simulation at Mach = -0.6 is conducted. The econstucted acoustic fields obtained by applying the poposed NAH pocedue to the simulation data match well with the exact fields. Though an expeiment with two loudspeaes pefomed in a wind tunnel at the aiflow speed of Mach = -0.1, it is also shown that the poposed NAH pocedue can be used to successfully econstuct the sound fields adiated fom the two loudspeaes. 1 INTRODUCTION Neafield Acoustical Hologaphy (NAH) is a poweful tool that can be used to visualize thee-dimensional sound fields by poecting the sound pessue data measued on a measuement suface. The NAH pocedue that includes the evanescent wave components (i.e., subsonic wave components) to impove the spatial esolution of a econstucted sound field was fist intoduced by Williams et al. in 1980s. 1-3 Since then, many eseaches have impoved the NAH pocedue and applied the impoved NAH pocedues to vaious vibo-acoustic and aeoacoustic poblems. a) addess: oeim@tamu.edu b) addess: hyusang@iss.e.

2 When the pessue data measued on a hologam suface (i.e., the measuement suface) ae poected to othe sufaces by using a NAH pocedue, the pessue data should be spatially coheent. That is, it is equied that thee is only a single coheent souce in the system of inteest o that all measuement points in a measuement apetue ae measued at the same time. The fome condition is not always satisfied since the most of eal systems have a composite souce consisting of multiple incoheent noise souces. The latte condition equies a lage numbe of micophones that completely cove a composite souce although extensive eseach has been conducted to coectly poect the sound pessue data measued only on a small patch measuement apetue. 4-8 In ode to satisfy the coheence equiement, the scan-based, multi-efeence NAH pocedue was intoduced by Hald. 9 In this pocedue, a small numbe of micophones is used to measue the sound pessue data on a patch of a complete hologam suface duing each scanning measuement while multiple efeence micophones ae fixed at thei locations thoughout the complete scanning measuements. The measued patch data ae combined to obtain complete data on the hologam suface. The combined hologam data ae then decomposed into patial sound pessue fields, each is spatially coheent. Hald fist poposed to use Singula Value Decomposition (SVD) to decompose the hologam data. 9 Then, each of all patial fields on the hologam suface is epetitively poected to othe sufaces. The total poected fields ae calculated by combining all of the poected patial fields. The scan-based, multi-efeence NAH pocedue is based on the assumption that the sound field adiated fom a composite souce is stationay duing scanning measuements. Howeve, in a eal NAH measuement, the sound field is not always stationay esulting in nonstationay effects. Fo the pupose of educing the non-stationaity effects, a souce nonstationaity compensation pocedue was intoduced by Kwon et al., povided that souce levels ae assumed to be non-stationay while thei diectivities emain unchanged duing scanning measuements. 10 In ode to obtain physically-meaningful patial fields, it is equied to place efeence micophones close to noise souces. 11 Then, each of the esulting patial fields can be associated with a specific noise souce. Howeve, it is not always possible to physically place efeence micophones close to noise souces. Kim et al. poposed to use vitual efeences of which locations ae identified whee beamfoming powes ae maximized. 11 The vitual efeence pocedue maes possible to identify physically-meaningful patial fields egadless of the physical locations of efeence micophones. When a NAH measuement is made with a micophone aay fixed on a moving tanspotation means, the measued data includes the effects of the moving fluid medium such as the Dopple Effect. Fo example, et noise data can be measued on the fuselage suface of a et aicaft duing its cuise condition (e.g., at Mach = 0.7) to visualize the et noise adiated fom its et engine to fuselage suface. Anothe example can be the tie noise data measued with a micophone aay attached to a moving vehicle. Note that the latte measuement cases can be assumed to be equivalent to the case whee thee is no motion with a noise souce and eceive while the fluid medium is in motion with a unifom velocity. Ruhala et al. poposed a plana NAH pocedue in a low-speed, moving fluid medium (e.g., below Mach = 0.1). 1 In the lowspeed case, it can be assumed that a static adiation cicle is shifted in the flow diection while its adius inceases due to the mean flow. Thus, they poposed a wave numbe filte based on the shifted and expanded adiation cicle. It was also assumed that the paticle velocities pependicula to the flow diection ae not affected by the mean flow. When the flow speed of a fluid medium is high and subsonic, the low-speed appoximations used in the NAH pocedue poposed by Ruhala et al. ae no longe valid. In this aticle, an

3 impoved NAH pocedue is descibed that can be applied to the high-speed, subsonic flow conditions. In paticula, a new wave numbe filte is poposed that is defined by mapping the static wave numbe filte. It is also poposed to conside the mean flow effects on the econstucted paticle velocities in the flow diection as well as in the diections pependicula to the flow diection. Note that the poposed NAH pocedue can be applied to any subsonic and unifom flow conditions egadless of low o high Mach numbe as long as the Mach numbe is within -1 to 1 ange. Fo the pupose of validating the poposed NAH pocedue, a monopole simulation at the aiflow speed of Mach = -0.6 is conducted. An expeiment with two loudspeaes in a wind tunnel at Mach = -0.1 is also pefomed. In the following theoy sections, a spatially-coheent, patial sound pessue field on a hologam suface is assumed to be given that can be obtained by using the pocedues descibed in Refs. 10 and 11. THEORY.1 Plana NAH in Static Fluid Medium In ode to pesent the poposed NAH pocedue in a consistent and compehensive manne, conside a conventional NAH pocedue that can be applied to the static case whee the fluid medium, composite noise souce, and eceive ae not in motion. When sound pessue is measued on a measuement plane at z = z h (i.e., hologam plane), the measued pessue field can be decomposed into spatially-coheent, patial sound pessue fields Each patial field can be then expessed as a supeposition of plane wave components by applying the spatial Fouie Tansfom to the patial field, p(x,y,z h,t). The plane wave components (i.e., the sound pessue spectum) in wave numbe domain, ( x, y ), can be witten as P(,, z, ) F[ p( x, y, z, t)] (1) x y h whee F epesents the Fouie Tansfom. Note that in a eal implementation, the Fast Fouie Tansfom (FFT) is applied to spatially-sampled pessue data instead of the Fouie Tansfom. The sound pessue spectum on a econstuction suface of z = z can be calculated fom the measued spectum by multiplying the plane wave popagato: i.e., P(,, z, ) P(,, z, ) K (,, z z, ). () x y x y h p x y h In Eq. (), K p is the pessue popagato defined as whee z h K (,, z, ) exp( i z) (3) p x y z if i x y othewise x y x y. (4) In Eq. (4), is the wave numbe defined as = /c 0 whee c 0 is the speed of sound. The poected paticle velocity spectum can be obtained by applying Eule s equation into Eq. (). Then, the velocity popagato is defined as K (,, z, ) exp( i z) ( x, y, o z) (5) x y z 0c 0

4 whee 0 is the static density of the fluid medium. Thus, the poected velocity spectum in the - diection can be obtained fom Eq. () by eplacing K p with K. Note that the cicle with the adius of = obtained by setting z = 0 in Eq. (4) is efeed to as the adiation cicle: i.e., x y 1. (6) Fo the plane wave components inside of the adiation cicle (efeed to as supesonic components), the z-diection wave numbe, z is a eal numbe: i.e., the fist case in Eq. (4). Thus, the popagato, K in Eq. (3) o (5) esults in only phase change between two specta at z = z and z = z h. Howeve, when a wave component is outside of the adiation cicle (that is efeed to as a subsonic component), z is an imaginay numbe: i.e., the second case in Eq. (4). As a esult, the popagato exponentially inceases o deceases the amplitude adio of the two specta. That is, the popagato exponentially inceases the amplitude of the hologam spectum in the case of z z h < 0 (i.e., bacwad NAH poection) while it exponentially deceases the amplitude when z z h > 0 (i.e., fowad NAH poection). Thus, the popagato inside of the adiation cicle epesents a popagating wave while the popagato outside of the adiation cicle epesents an evanescent wave. Note that the evanescent wave applied duing a bacwad NAH poection impoves the spatial esolution of a econstucted sound field. Howeve, the measuement noise outside of the adiation cicle is also amplified exponentially duing the bacwad poection pocedue. Fo the pupose of suppessing the measuement noise, it is ecommended to apply the wave numbe filte poposed by Kwon 13 befoe applying the bacwad NAH poection (see Appendix A). The econstucted pessue o velocity field is obtained by applying the invese Fouie Tansfom to the poected pessue o velocity spectum: e.g., p( x, y, z 1, t) F [ P(,, z, )]. (7) The afoementioned NAH pocedue is epetitively applied to all of the patial fields on the hologam suface. The poected patial fields ae then combined to obtain the total fields on the econstuction suface. Note that sound intensity field can be calculated by multiplying the poected pessue and velocity fields. x y. Plane Wave Popagation in Moving Fluid Medium Conside the (,,) coodinate system in a moving fluid medium that is coesponding to the (x,y,z) coodinate system in a static medium. When the fluid medium is moving at the unifom -diection velocity of U while a composite noise souce and eceive ae not in motion, the convective wave equation 14 can be epesented as 1 D p p (8) c Dt whee D/Dt is the total deivative (o efeed to as the mateial deivative) that is defined as 0 D U Dt t. (9) The convective Eule s equation 14 that elates acoustic pessue and paticle velocities can also be witten in tems of the total deivative: i.e.,

5 Dv 0 p (10) Dt Fo the pupose of analyzing the plane wave popagation chaacteistics in the convective fluid medium, conside the sound pessue and paticle velocity solutions epesented as and p(,,, t) Pexp[ i( t)] (11) v (,,, t) V exp[ i( t)] (1) whee the subscipt, epesents the paticle velocity diection (i.e., =,, o ). By applying the sound pessue solution of Eq. (11) into Eq. (8), the chaacteistic equation can be found as whee M is the Mach numbe defined as (1 M ) M (13) U M. (14) c 0 Similaly, by substituting Eqs. (11) and (1) into Eq. (10), the pessue and velocity can be elated as V ( ) P. (15) 0c0 M To detemine a mathematical expession fo the bode line between supesonic and subsonic components, conside the case of = 0 in Eq. (13): i.e., whee a 1 1 (16) M a, (17) 1 M and 1, (18) 1 M. (19) 1 M Eq. (16) epesents an ellipse with the semimao axis, 1 and semimino axis, centeed at the point of (-a, 0). The ellipse can be efeed to as the adiation ellipse that is coesponding to the adiation cicle in the static fluid medium: i.e., the wave components inside of the adiation ellipse epesent popagating waves while the outside wave components epesent evanescent waves. When compaed with the adiation cicle shown in Eq. (6), the cente of the adiation ellipse is shifted by -a along the flow diection. The adius of = inceases to 1 in the -

6 diection and in the -diection fo the case of a subsonic flow (i.e., M < 1). A mapping function that maps the adiation cicle in the ( x, y ) domain to the adiation ellipse in the (, ) domain is then defined as 1 (, ) ( a) /, /. (0) x y The mapping of the adiation cicle to the adiation ellipse is shown in Fig. 1. (0,) y Supesonic Region x y Radiation Cicle: 1 x (,0) Subsonic Region x =( +a)/ 1 y = / Supesonic Region a Radiation Ellipse: 1 (- 1 -a,0) (-a, ) (-a,- ) (-a,0) 1 Subsonic Region ( 1 -a,0) Figue 1: Mapping of adiation cicle (M = 0) to adiation ellipse (0 < M < 1)..3 NAH Pocedue in Moving Fluid Medium When the sound pessue data ae measued in the moving fluid medium, the fowad and invese Fouie Tansfoms given in Eqs. (1) and (7) can be e-used without any modifications. The Fouie Tansfoms then elate the sound pessue fields between the (,) and (, ) domains. Fom Eqs. (13) and (15), the pessue and velocity popagatos ae defined as and whee K (,,, ) exp( ) p i (1) K (,,, ) exp( i ) (,, o ) 0c0( M ) (1 M ) M if (1 M ) M i (1 M ) M othewise (). (3)

7 Hee, a new wave numbe filte is poposed that is defined by mapping the static wave numbe filte in the ( x, y ) domain into the (, ) domain. By applying Eq. (0) to Eqs. (A1) - (A4) in Appendix A, the wave numbe filte is defined as follow: If c, Othewise, whee 1 exp[( / 1) / ]/ if W (, ) exp[(1 / c) / ]/ othewise c c. (4) 1 if W (, ) (5) 0 othewise ( a) / / (6) 1 and whee z h is the hologam height. c /(3z ) (7) h 3 MONOPOLE SIMULATION AND EXPERIMENT Fo the pupose of validating the poposed NAH pocedue in a moving fluid medium, the monopole simulation and expeiment descibed in the following sections ae conducted. 3.1 Simulation and Expeiment Setups The monopole simulation is set up to simulate the expeiment shown in Figs. and 3. An 8 5 micophone aay is used fo both the simulation and expeiment to measue and calculate sound pessue data at each scanning position fo 0 seconds at the sampling fequency of 819 Hz. The diffeences between the simulation and expeiment ae as follow; (1) Fo the simulation, two uncoelated monopole souces ae placed at the locations of the two loudspeaes used in the expeiment, i.e., (,,) = (0.3,0.5,-0.05) m and (0.5,0.1,-0.05) m; () Aiflow speed is set up to M = -0.6 fo the simulation while M = -0.1 fo the expeiment; (3) The -diection micophone spacing is d = 0.05 m in the simulation while d = 0.05 m in the expeiment; (4) To cove the almost same measuement apetue, the total numbe of scans is 6 (i.e., 8 30 measuement points) in the simulation while in the expeiment, the total numbe of scans is 3 to cove the 8 15 measuement points; (5) As shown in Figs. and 3, thee is a had suface that can be assumed as a igid bounday in the expeiment while it is assumed that thee is no gound condition (i.e., fee field) in the simulation; (6) Input signals to the loudspeaes ae diectly used as efeence signals in the expeiment while efeence signals ae calculated at the locations of two efeence micophones closely placed at the loudspeae locations in the simulation; (7) A laptop with the National Instument (NI) LabView softwae along with a NI 64 channel data acquisition system as shown in Fig. 3 ae used to acquie sound pessue data in the expeiment.

8 [m] Micophones with windshields [m] 0.3 Scan 1 Scan Scan 3 Loudspeaes m/s Ai Flow Rigid Gound 0.05 m [m] 0.06 m [m] Figue : Setch of expeimental setup. 3. Results and Discussion Figue 3: Photos of expeimental setup. The measued sound pessue data on the hologam suface at = 0.06 m ae shown in Fig. 4. Figue 5 shows the imaginay pat of wave numbe, and the weighting function, W of the poposed wave numbe filte that is used fo bacwad NAH poections when M = -0.6 and f = 1.5 Hz (see Eqs. (3) and (4)). The adiation cicle in the static fluid medium case (i.e., no flow case) is ovelaid as the blac, solid-lined cicle in Fig. 5. In Fig. 5(a), the adiation ellipse can be identified at the edge of the supesonic egion whee the imaginay pat of is equal to zeo. Note that the cente of the adiation ellipse is on the positive -axis due to the negative fluid flow, i.e., M = -0.6 (see Eqs. (16) and (17)). As you can see in Fig. 5(b), the wave numbe filte that is used fo a bacwad poection passes supesonic components while it educes the most of subsonic components with the smooth tansition aound the edge of the adiation ellipse. Figue 6 shows the exact and econstucted total sound pessue fields at f = 1.5 Hz. The exact sound pessue fields ae shown in Figs. 6(a) and 6(b) while the econstucted sound pessue fields pesented in Figs. 6(c) and 6(d). Note the left two plots, i.e., Figs. 6(a) and 6(c) show the esults at the souce suface (i.e., = 0 m) while the ight two plots, i.e., Figs. 6(b) and 6(d) show the esults at the sliced suface of = 0.5 m. As you can see, the econstucted fields

9 ae almost identical with the exact fields on these two sufaces except the edges of the measuement apetue (see the edges of Figs. 6(a) and 6(c)). Note that the poposed NAH poection equies that the measued sound pessue along the edges of the measuement apetue should be vey small to educe tuncation eos. Howeve, as shown in Fig. 4, the sound pessue in Fig. 4 at the top and bottom edges, in paticula, at (,) = (0.3,0.35) m and (0.5,0.0) m cannot be negligible esulting in the diffeences along the edges of the measuement apetue between the exact and econstucted sound pessue fields. Figue 7 shows the sound pessue fields on the souce suface at = 0 that ae econstucted fom expeimental data when M = As you can see, each sound pessue field epesents the sound pessue field adiated fom one of the loudspeaes. The locations of the highest sound pessue peas in Fig. 7 ae also in line with the locations of the loudspeaes. 4 CONCLUSIONS In this pape, the impoved NAH pocedue is descibed that can be used to poect the sound pessue data measued in a moving fluid medium with a unifom and subsonic velocity. In the latte case, a composite noise souce and eceive ae assumed to be not in motion. In paticula, the new wave numbe filte is poposed that is defined by mapping the static wave numbe filte. The poposed velocity popagatos ae also ecommended in ode to include the mean fluid effects on econstucted paticle velocities even in the diections pependicula to the flow diection. By applying the poposed NAH pocedue to the monopole simulation data at M = -0.6, it is shown that the poposed pocedue can successfully epoduce the exact sound fields. Though the expeiment with the two loudspeaes pefomed in the wind tunnel at M = -0.1, it is also shown that the poposed NAH pocedue can be used to identify the loudspeae locations and thei adiation pattens. In the nea futue, it is planned to apply the poposed NAH pocedue to the et noise data measued on the fuselage suface of a et aicaft duing its cuise condition (e.g., at M = 0.7). It is also expected to develop moe applications of the poposed NAH pocedue in the futue. REFERENCES 1. J. D. Maynad, E. G. Williams, and Y. Lee, Neafield acoustic hologaphy: I. Theoy of genealized hologaphy and the development of NAH, Jounal of Acoustical Society of Ameica, Vol. 78, (1985).. W. A. Veonesi and J. D. Maynad, Neafield acoustic hologaphy (NAH): II. Hologaphic econstuction algoithms and compute implementation, Jounal of Acoustical Society of Ameica, Vol. 81, (1985). 3. E. G. Williams, Fouie Acoustics: Sound Radiation and Neafield Acoustical Hologaphy, Academic Pess, (1999). 4. J. Hald, Patch nea-field acoustical hologaphy using a new statistically optimal method, Poceedings of Inte-Noise 003, pp , J. Hald, Patch nea-field acoustical hologaphy using a new statistically optimal method, Büel & Kæ Technical Review, No. 1, pp , Y. T. Cho, J. S. Bolton, and J. Hald, Souce visualization by using statistically optimized nea-field acoustical hologaphy in cylindical coodinates, Jounal of Acoustical Society of Ameica, Vol. 118(4), pp , E. G. Williams, Continuation of acoustic nea-fields, Jounal of Acoustical Society of Ameica, Vol. 113, pp , 003.

10 8. M. Lee and J. S. Bolton, Reconstuction of souce distibutions fom sound pessues measued ove discontinuous egion: Multipatch hologaphy and intepolation, Jounal of Acoustical Society of Ameica, Vol. 11, pp , J. Hald, STSF a unique technique fo scan-based Neafield acoustical hologaphy without estiction on coheence, Büel & Kæ Technical Review, No. 1, pp. 1 50, (1989). 10. Y.-J. Kim, J. S. Bolton, and H.-S. Kwon, Compensation fo souce nonstationaity in multiefeence, scan-based nea-field acoustical hologaphy, Jounal of Acoustical Society of Ameica, Vol. 113(1), pp , Y.-J. Kim, J. S. Bolton, and H.-S. Kwon, Patial sound field decomposition in multiefeence nea-field acoustical hologaphy by using optimally located vitual efeences, Jounal of Acoustical Society of Ameica, Vol. 115(4), pp , R. J. Ruhala and D. C. Swanson, Plana nea-field acoustical hologaphy in a moving medium, Jounal of Acoustical Society of Ameica, Vol. 1(), pp , H.-S. Kwon, Sound visualization by using enhanced plana acoustic hologaphic econstuction, Ph.D. Thesis, Koea Advanced Institute of Science and Technologies, L.E. Kinsle, A.R. Fey, A.B. Coppens, and J.V. Sandes, Fundamentals of Acoustics, 3d Ed., Wiley, New Yo, P. M. Mose and K. U. Ingad, Theoetical Acoustics, McGaw-Hill, Inc., Appendix A: Wave Numbe Filte in Static Fluid Medium The wave numbe filte poposed by Kwon 13 is pesented below: If c, Othewise, whee 1 exp[( / 1) / ]/ if W ( x, y) exp[(1 / c) / ]/ othewise c c. (A1) 1 if W ( x, y ) (A) 0 othewise (A3) x y and c /(3z ) (A4) h whee z h is the hologam height (i.e., the distance fom the souce suface to measuement suface). Note that the wave numbe filte is the function of x and y fo a given fequency (z h is detemined when the expeiment is pefomed): i.e., is the vaiable, and and c ae the constants fo the given conditions.

11 Figue 4: Measued sound pessue fields at f = 1.5 Hz on the hologam suface (monopole simulation at M = -0.6): (a) Fist sound pessue field and (b) Second sound pessue field. Figue 5: Two quantities in wave numbe domain when M = -0.6 and f = 1.5 Hz (the blac, solid-lined cicle epesents the static adiation cicle): (a) Imaginay pat of wave numbe, (see Eq. (3)) and (b) Weighting function, W of the poposed wave numbe filte (see Eq. (4)).

12 Figue 6: Total sound pessue fields at f = 1.5 Hz (monopole simulation at M = -0.6): (a) Exact sound pessue field on the suface at = 0 m, (b) Exact sound pessue field on the suface at = 0.5 m, (c) Reconstucted sound pessue field on the suface at = 0 m, and (d) Reconstucted sound pessue field on the suface at = 0.5 m. Figue 7: Reconstucted sound pessue fields on the souce suface of = 0 m at f = 1.5 Hz (expeiment at M = -0.1): (a) Fist sound pessue field and (b) Second sound pessue field.