NUMERICAL INVESTIGATION OF AERODYNAMIC SOUND RADIATED FROM A BLUFF BODY USING LIGHTHILL TENSOR

Transcription

1 ICSV14 Carns Australa 9-1 July, 007 NUMERICAL INVESTIGATION OF AERODYNAMIC SOUND RADIATED FROM A BLUFF BODY USING LIGHTHILL TENSOR Shoj Kato 1, Hrosh Muraoka, Yuta Takahash, Tubagus Haedar, and Akyosh Ida 1 Abstract 1 Kogakun Unversty, Nakanomach. Hachoj-sh, Tokyo, , Japan skato@s.u-tokyo.ac.jp (emal address of lead author) Research Center of Computatonal Mechancs, Inc. Togosh NI-Bldg Togosh, Shnagawa-ku, Tokyo, , Japan actran@rccm.co.jp (emal address of lead author or presenter) Verfyng the effectveness of Lghthll tensor s applcaton coupled wth both Fnte Element Method (FEM) and Infnte Element Method (IEM) for aero acoustc scatterng analyss was the man purpose of ths research. The Lghthll tensor s classfed as the real source of aerodynamc sounds beng generated wthn a flud doman and understandng ts nature s crtcal n terms of nose reducton durng a product development phase. In order to evaluate both sound pressure strength and propagaton pattern, aeroacoustc felds surroundng a rectangular cylnder placed n a unform flow was tested. Lghthll tensor was taken from CFD (Front Flow/Blue, FLUENT) calculaton and the outcome was compared to ones usng Boundary Element Method (BEM) analyss based on reduced Curle s equaton along wth wnd tunnel expermental data. The man sound source was seen near the tralng edge of the cylnder at approxmately 1- tmes the length of the rectangle s sde. Although sgnfcant margns between the three methods were observed at low frequency, beyond 100Hz, the characterstc of sound pressure strength were closely matched. It became evdent that f consderaton regardng drectvty or the aerodynamc sound source scatterng needed to be taken nto account, ACTRAN (Lghthll tensor calculaton n the doman) applcaton was more promnent than utlzng reduced Curle s equaton (pressure on the surface). 1. INTRODUCTION One of the major problems n our socety s nose polluton and no matter where the sound orgnates, t s essental to understand how t s created, n order to control the magntude of ts effect. Commonly, the calculaton of the flow and sound felds are carred out separately as the relatve scalng for the two domans dffers. Also, t s stated that the aerodynamc sound s proportonal to the flow velocty order of 6 th to 8 th power. Aero acoustc nose s generated from vortces created when a flud makes contact wth a sold object n a flow feld, n ths paper a rectangular cylnder. Drect Numercal Smulaton (DNS) [1] was the obvous approach to aero acoustcs related to compressble Naver-Stokes equatons however, due to ts calculaton cost beng proportonal to Re 3 /M 4, ts use was lmted

2 ICSV July 007 Carns Australa to low Reynolds number flow. Smlarly, applcaton of BEM does have advantages n terms of reduced dmensonalty, however, as ts prncple can only take nto account the surface pressure from an object, t ends to be lmted for dpole analyss problems. Curle s equaton [] or the compact Green s functons are effectve under compact problem (sze of the propagated wavelength s consderably larger than the sze of the object) when dffracton effect s non-exstent. On the contrary, the FEM doman-based methods such as FEM/IEM [3] are more sutable for solvng exteror acoustc problems due to ts sparse matrces and effcent soluton procedures such as parallel calculatons along wth enablng a natural extenson of fnte element to an unbounded doman. Geometrcal complexty does not bear any effect on the effcency as t s based on the weak varatonal formulaton and sound hard condton s automatcally accounted as a natural boundary condton. Although the FEM offers hgh accuracy results, t should be noted that the computatonal load s heavy due to the need of dscretzaton of the whole computatonal doman (large stffness matrx). Lghthll s approach [4] s one, where the aerodynamc source s frst estmated usng ts analogy from the flow results then usng t to accurately obtan the true propagaton of the acoustc wave. The correspondng aerodynamc sound sources were obtaned by CFD analyss [5] under large eddy smulaton (LES) wth dynamc Smagornsky model. Couplng ths analogy wth FEM/IEM (see fgure 1) enables to facltate the calculaton of volumetrc sound propagaton wth hgh accuracy and constranng the nfnte boundary around the nose source culmnates n reducng calculaton tme and cost respectvely. Alternatvely, the Powell sound source (product of vortcty magntude and velocty) taken from CFD calculaton could replace the Lghthll tensor for acoustc calculaton. A square cylnder (sde length of 0.0m) was placed n a unform flow at varyng angles of attack (0 to 45 degrees) to determne ts turbulent nature. Essentally, outlnng dfferences between compact (BEM) and non-compact (FEM) sound was the am of ths paper. Fgure 1. Square cylnder and ts FEM/IEM boundary separaton Ths paper s organzed as follows: After ths ntroducton, the acoustc analogy s presented n secton. Secton 3 outlnes the square cylnder problem defnton and both numercal and contour results are compared n secton 4. The penultmate secton llustrate a smlar approach used on a sde mrror of a car and the fnal sectons summarze the valdty of ths research ndcatng plausble steps for further studes.. ACOUSTIC ANALOGY Frst the contnuty (1) and Naver-stokes () equatons are consdered. ρ ρv + = 0 (1) t x

3 ICSV July 007 Carns Australa ρv ρvv j ( pδj τj) + = t x x j j () Contnuty s dfferentated by t and sound terms are added to Naver-Stokes. Followng ths, equaton () s dfferentated by x and combnng wth equaton (1) provdes the followng. ( ρ ρ ) ( ρ ρ ) 0 0 a 0 = ρvv j + δj p a0 ρ τj t x x x xj [ ( ) ] (3) The fluctuaton densty s represented by ( ρ ρ0), where ρ 0 s the mean or atmospherc densty and a denotes the speed of sound. Placng the Lghthll tensor ( T ) term on the rght 0 hand sde of equaton (3) wll defne the Lghthll s equaton. j ρ ρ ρ ρ T ( 0) ( 0) j a 0 = t x x x xj (4) The weak varatonal form of equaton (4) s appled n ACTRAN [6] as derved by Obera et al [7,8]. ( ) ( ) T Σ ρ ρ δρ+ a ρ ρ dx= dx+ nδρdγ ( x ) (5) δρ j δρ j o o o t x x xj x x Ω Ω Γ j The two terms on the rght hand sde of equaton (5) s comprsed by volumetrc and surface aerodynamc source respectvely, where δρ s a test functon, Ω denotes the computatonal doman and Γ represent the surface boundary. In a far feld, acoustc pressure p a can be represented as follows. pa ( x, t) = a ( ρ ρ )( x, t) (6) 0 0 Hence aerodynamc sound propagated n a flow feld wth sold boundary can be defned by the followng Curle s equaton (7), where P s the pressure, T s the Lghthll tensor and S represents the sold object s surface. In addton, x represents the observaton pont, y s the locaton of the sound source, n s the normal vector from the boundary and r denotes the dstance between the source and the recpent. 1 Tj ( y, t r/ a0) 3 1 np ( y, t r/ a0 ) pa ( x, t) = d y ds (7) 4π x x r 4π x r j V S j Under low Mach number flow feld, the frst term on the rght of equaton (7), whch corresponds to quadrupole sound source can be neglected n relaton to the second term (dpole sound source). If the object scale s consderably smaller compared to the wavelength of the sound source, the dpole term s space dfferental s converted to tme dfferental resultng nto what we defne as the reduced Curle s equaton (8). In ths nstant r represent the dstance

4 ICSV July 007 Carns Australa between the recever and the centre of the sold object. 1 x pa( x, t) = np(, / y t r a0) ds (8) 4π a r t 0 S By defnng the Lghthll s tensor wthn the Helmholtz equaton (solved n the frequency doman), the surroundng sound pressure dstrbuton around the square cylnder and drectvty can be determned. 3. NUMERICAL CONDITIONS Two separate meshes sutable for flow and acoustc analyss were created to facltate the evaluaton process n terms of capturng the vortcty nature of the flow feld and the resultng sound wave propagaton nature respectvely. The flow feld mesh comprsed of.9 mllon elements and the Reynolds number was set to as llustrated below. Also, the LES calculaton results were used as a base for the reduced Curle s equaton applcaton to obtan the acoustc feature derved from surface pressure fluctuaton. Fgure. Aerodynamc boundary condton and grd structure Fve angles of attacks for the cylnder were tested (0, 10, 15, 30 and 45 degrees) and the spanwse length of the cylnder was set 15 tmes the sde length of the square. Boundary condtons were placed as to recreate the wnd tunnel settng and LES was appled to obtan the Lghthll tensor necessary for acoustc studes. Dmensons of the analyss doman and cylnder remaned unchanged, however the densty of the mesh was reduced to elements as shown n fgure 3. Fgure 3. Acoustc boundary condton and grd structure Unlke the aerodynamc mesh, the acoustc mesh does not need smlar densty as long as the elements sze are small enough to capture the wavelength of the relevant frequency. Nonreflectve boundary (IE) was placed for the surroundng except for the floor and cylnder whch

5 ICSV July 007 Carns Australa had the wall property (reflectve boundary). Acoustc smulatons were carred out across varyng Strouhal numbers (St = fl/u) havng relatve frequences between 49.5Hz to 734.5Hz n the goal of evaluatng sound propagaton nature, drectvty, peak frequency component and general effcency of the three dfferent methods. 4. RESULTS AND COMPARISON LES analyss llustrated that as the angle of attack attan 15 degrees, the separated flow re-attaches at the bottom of the cylnder whch decreases the sound source strength behnd and on the surface of the sold object as shown n fgure 4. Ths ndcates that the propagatng vortcty phase s not symmetrc between the top and bottom of the cylnder at ths partcular angle contrary to the others. Fgure 4. Velocty vector and Lghthll tensor dstrbuton ( log T j ) around the cylnder In essence the drectvty wll always be n a dpole shape when the reduced Curle s equaton s appled. Fgure 5 reterates that under smlar analyss usng a commercal BEM code, the drectvty s hardly affected by the angle of attack of the cylnder snce the analyss can be assumed as beng a compact body havng no dffracton effects. Fgure 5. Dstrbuton of sound pressure radated from a square cylnder (St = 0.13)

6 ICSV July 007 Carns Australa In contrast, as the Lghthll tensor represents the vortcty movement of the flow, the drectvty llustrate a quadrupole and dpole characterstcs under FEM condtons wth a compact body. Analyss method s effcences (reduced Curle s equaton, FEM, wnd tunnel experment) were compared usng the sound pressure level obtaned from a pont 1m lateral (Y drecton) to the square cylnder as demonstrated n Fgure 6. Fgure 6. Sound pressure level comparson between angle of attack at 0 degree and 15 degrees Both graphs have smlar curve patterns n terms of sound pressure fluctuaton ncludng the peak frequency at around 00Hz (St = 0.13) and except for the lower frequency regon; all three methods were wthn +10dB of each other. The mss algnment between the expermental and numercal analyss methods below 100Hz was probably due to the background nose created wthn the wnd tunnel. Followng these results, a slghtly more complex geometry was tested usng the Lghthll tensor applcaton to nvestgate the drectvty fluctuaton. 5. FURTHER STUDY A sde mrror of heght 0.3m and 0.1m radus was placed n a unform flow feld of 55m/s for LES calculaton usng FLUENT. Grd structures for both flow and sound analyss are shown below. Fgure 6. Aerodynamc and Acoustc mesh dmensons Smlar to the square cylnder case, the acoustc mesh (approxmately elements) was less dense compared to the aerodynamc mesh comprsng of.8 mllon elements. Although the frequency was set at 1000Hz (wavelength of 0.34m), the element sze was small enough to

7 ICSV July 007 Carns Australa accommodate up to 500Hz n order to maxmze the calculaton effcency. IE were placed on the ellpsodal surface of the acoustc grd. Fgure 7. Pressure and Powell sound source contours from FLUENT Approxmately 0000 teratons were carred out ( t = 6x10-5 ) to determne the wake characterstcs and the correspondng Lghthll tensor around the sde mrror shown n fgure 7. Applyng ths nto ACTRAN provded contours seen n fgure 8, where the sound seemed to be propagatng from a pont behnd the mrror. Fgure 8. Sound wave propagaton by the sde mrror (front and rear vew) Maxmum sound pressure level was calculated to be around 135dB and the hgh nose regon s manly concentrated at the back of the mrror. It s also evdent that sound pressure strength was dmnshed towards the front sde as the mrror surface dffracted the oncomng waves as llustrated n fgure 9. Fgure 9. Sound pressure level and Ampltude contour (sde vew)

8 ICSV July 007 Carns Australa 6. CONCLUSIONS Sound was generated from the rear of the cylnder at approxmately one to twce the sde length of the rectangular cylnder followng LES calculaton and the applcaton of Lghthll tensor allows the proper vsualzaton of sound drectvty and sound wave dstrbuton. When the cylnder was at angle of attack of 15 degrees, the re-attachment of the separated flow culmnated n a reducton effect upstream of the cylnder n both drectvty and dffracton. FEM provded both quadrupole and dpole drectvty characterstc at varyng angles unlke the reduced Curle s equaton cases where t contnuously generated dpole nature. However, all three methods showed smlarty n terms of numercal output concernng sound pressure levels across varable frequency range especally for the peak frequency value, whch stayed constant regardless of the angle of the cylnder at around 00Hz. It s therefore vald to state that compact (BEM) and non-compact (FEM) sound source do not dffer n terms of numercal result, hence the applcaton would be dependent on whether t s necessary to account for the sound source s dynamc movement and the accurate capturng of drectvty pattern. As a suggeston to further cement ths fndng, the case could be analysed at a small angle of attack and frequency ncrement along wth other calculaton methods for result refnement. Applyng a more complex geometry wll also be nvestgated. REFERENCES [1] C. Kato, M. Kaho, and A. Manabe, "Industral Applcaton of LES n Mechancal Engneerng" (nvted paper), DNS/LES Progress and Challenges, pp , Greyden Press (001-11). [] N. Curle, The nfluence of Sold Boundares on Aerodynamc Sound, Proc. Roy. Soc (London), Volume A 31, No. 1187, pp (1955). [3] R.J. Astley, G.J. Macaulay and J.P. Coyette, Mapped wave envelope elements for acoustc radaton and scatterng, JSV, 170 (1), (1994) [4] M.J. Lghthll, Waves n fluds, CAMBRIDGE UNIVERSITY PRESS, [5] K. Ohnsh, H. Zhang, T. Tomohro, H.Kayama, Kamsu-cho, M.Nawa, N.Tanguch, The evdence of the nose analyss technque by LES usng general-purpose code FrontFlow, JSFM, Vol. 18, (004) [6] Free-Feld-Technologes-S.A., Actran 006 Aeroacoustc Solutons: Actran/TM and Actran/LA - User s Manual, Belgum, 006. [7] A. Obera, F. Ronaldkn and T. Hughes, Computatonal Procedures for Determnng structural-acoustc Response due to Hydrodynamc Sources, Comput. Methods Appl. Mech. Engrg, Volume 190, pp (000). [8] A. Obera, F. Ronaldkn and T. Hughes, Computaton of Tralng-Edge Nose due to Turbulent Floe over an aerofol, Volume 40, pp (00). [9] M.S. Howe, Theory of Vortex Sound, CAMBRIDGE UNIVERSITY PRESS, 199.