Theory and three-dimensional numerical simulation of sound propagation along a long enclosure with side opening

Transcription

1 Theory nd three-dimensionl numericl simultion of sound propgtion long long enclosure with side opening S. H. K. CHU 1 ; S. K. TANG 2 1, 2 Deprtment of Building Services Engineering, The Hong Kong Polytechnic University, Hong Kong, Chin ABSTRACT The sound properties long long rectngulr enclosure with circulr source on the top wll nd rectngulr opening t the side wll re studied. A mthemticl model of verge pressure t n opening contributed by the sound source nd ssuming ir piston t the opening is derived in terms of the coustic mode for rigid duct. Three-dimensionl numericl modelling is conducted to compre the theoreticl prediction. The computtionl domin of interest is long rectngulr block ssocited with n opening connected to free field environment. Non-reflecting portion is pplied to ech end of the domin. Sound pressure vritions of discrete propgting mode re investigted. Interctions between the resonnce of closed duct nd effect of the opening re demonstrted t vrious frequencies. Comprisons indicte tht the theoreticl predictions give consistent results with tht of numericl models. Precision of the rdition impednce of perture is found crucil to the ccurcy of mthemticl model. This study pproch including the theory nd simultion provides constructive frmework for further exmintion of relted subjects. Keywords: Sound propgtion, Long enclosure, Side opening I-INCE Clssifiction of Subjects Number(s): INTRODUCTION Long rectngulr enclosures including ir ducts, tunnels nd the long tunnel like corridors re common fetures in building elements nd infrstructurl. Noise propgtion nd reverbertion in such long closed enclosures hve been studied in the pst few decdes (1, 2). Rectngulr opening is found populr in this kind of long enclosure. Noise propgtion long long rectngulr enclosure with rectngulr opening t the side wll is interested in this study. The present study pproch is to derive mthemticl model to compre the in-duct sound behvior of three-dimensionl numericl model. Contribution of circulr sound source on the top wll is dopted (3). Specil efforts re mde in exmining how the opening size ffecting the results. 2. THEORETICAL PREDICTION 2.1 Averge coustic pressure t the opening As illustrted in Figure 1, rectngulr ir piston of size w x h is flush-mounted on the side wll of the duct with interior of height nd width b. A circulr sound source is fixed on the top wll of the rectngulr duct. The cse for considertion is the sound field of excittion of coustic modes by the sound source with side opening in stright, hrd-wlled duct of constnt re of cross section nd infinite length. The fluid loding in term of verge pressure cting on single ir piston t the opening cn be set s follows (4, 5) r@connect.polyu.hk 2 shiu-keung.tng@polyu.edu.hk Inter-noise 2014 Pge 1 of 8

2 Pge 2 of 8 Inter-noise 2014 Figure 1 Geometry of the rectngulr side wll opening nd flush-mounted circulr sound source The verge coustic pressure t the opening contributed by the circulr source is: 1 hw p s(x, y)ds s (1) where ds=dxdy nd p s is pressure wve generted by the source The self-contribution of the ir piston t the opening: 1 VF(x, y)ds hw s (2) x 1 +w/2 +w/2 1 hw p s(x, y x 1, h/2)dxdy + 1 hw VF(x, y)dxdy = VZ r x 1 w/2 h V is the velocity to be solved nd Z r is the rdition impednce of the ir piston. Following cluses elborte the bove mthemticl model in detils. 2.2 Generl solution for the wve propgtion of wll-mounted ir piston The generl solution for the wve propgtion of flush-mounted ir piston in hrd-wlled duct of constnt cross section nd infinite length is (4): p(x, y, z, t) = ρ o 2b c ψ (y, z) ψ w/2 h (y, z )V(x, y, z, t) [H(x x iω(x x ) )e c (3) iω(x x ) + H(x x)e c ] ds (4) Pge 2 of 8 Inter-noise 2014

3 Inter-noise 2014 Pge 3 of 8 where ψ (y, z) = (2 δ 0m )(2 δ 0n ) cos ( mπy ) cos (nπz b ) ( c 2 ) = 1 ( ω 2 c ω ), ( ω 2 c ) = ( mπ 2 ) + ( nπ 2 b ) nd V is constnt. Since the rectngulr ir piston is mounted on the top side wll, The integrl is: (2 δ 0m )(2 δ 0n ) cos mπy where z = 0, center of x = 0 nd y = h 2 cos nπz b [H(x x )e iβ (x x ) + H(x x)e iβ (x x ) ]ds β = ds = dx dy ω c = ( ω c ) 2 ( mπ ) 2 ( nπ b ) 2 cos nπz b = 1 cos mπy h, m = 0 dy = { ( 1) m h sin ( mπh ) / (mπh ), m 0 h (5) 2.3 Self-contribution of ir piston The self-contribution of the side perture is derived s below. x +w/2 x +w/2 e iβ (x x ) dx + e iβ (x x ) dx 2 = [1 e iβw/2 cos (β iβ x)] w/2 The whole integrl is, for m = 0: for m 0, The solution, for m = 0: p(x, y, z, t) = ωh ρ ove iωt b for m 0, x (2 δ 0m )(2 δ 0n ) ( 1) m (2 δ 0m )(2 δ 0n ) (2 δ 0m )(2 δ 0n ) 1 e i( p(x, y, z, t) = ωh ρ ove iωt b sin (mπh (2 δ 0m )(2 δ 0n )( 1) m 2h iβ [1 e iβ w/2 cos (β x)] 2h [1 e iβw/2 cos (β iβ x)] sin ( mπh ) ( mπh ) w 2 ) k2 k 2 cos (x k 2 k i(k 2 k 2 ) Therefore, the verge pressure of the single ir piston: p = V hw +w/2 2 ) ψ (y, z) ) [1 e i(w 2 ) k2 k2 cos (x k 2 k 2 )] i ( mπh ) (k2 k 2 ) F(x, y)dxdy w/2 h ψ (y, z) (6) (7) (8) (9) Inter-noise 2014 Pge 3 of 8

4 Pge 4 of 8 Inter-noise w/2 w/2 cos(x k 2 k 2 ) dx = 2 k 2 k2 sin ( w 2 k2 k 2 ) cos mπy h, m = 0 dy = { ( 1) m h sin ( mπh ) / (mπh ), m 0 h One cn note tht, for m = 0: p (x 1, d/2,0) w 2 ) k2 k2 = V ωhρ (2 δ 0m )(2 δ 0n ) [w 2e i( o wb k 2 k2 i(k 2 k 2 ) for m 0, p (x 1, h/2,0) = V ωhρ (2 δ 0m )(2 δ 0n )sin 2 ( mπh 2e i( ) [w o wb k 2 k2 i ( mπh 2 ) (k 2 k 2 ) sin ( w 2 k2 k 2 )] w 2 ) k2 k2 sin ( w 2 k2 k 2 )] 2.4 Acoustic pressure t the opening contributed by the source This section describes the wve field by circulr sound source. A mthemticl model of the pressure wve generted by ssuming flush-mounted circulr piston vibrtion s the sound source for rigid duct hs been derived (3), for even n: p s (x, y, z, t) = πωr ρ ov s e iωt b (2 δ 0m )(2 δ 0n )( 1) m+n 2 (10) (11) J 1 (R k 2 k 2 m0 ) k 2 k 2 k 2 k ψ 2 (y, z)e i x k2 k2 m0 (12) The verge pressure of source contribution to the side opening is s below. x 1 +w/2 p s = 1 hw p s(x, y x 1, h/2)dxdy x 1 +w/2 x 1 w/2 e i x k2 k2 dx = 2sin (w 2 k2 k k 2 k2 x 1 w/2 cos mπy h, m = 0 dy = { ( 1) m h sin ( mπh h ) / (mπh ), m 0 One cn note tht, for m = 0: (x 1, h/2,0) p s = 2πωR ρ ov s bw ( 1)n 2 for m 0, p s (x 1, h/2,0) = 2πωR ρ ov s bw ( 1) 2m+n 2 h (2 δ 0m )(2 δ 0n )J 1 (R k 2 k 2 m0 ) (k 2 k 2 ) k 2 2 k m0 (2 δ 0m )(2 δ 0n )sin ( mπh ) J 1 (R k 2 k m0 ( mπh ) (k2 k 2 ) k 2 2 k m0 2 ) e i x 1 k2 k2 sin ( w 2 k2 k 2 ) e i x 1 k2 k2 2 ) sin ( w 2 k2 k 2 ) e i x 1 k2 k2 (13) (14) (15) Pge 4 of 8 Inter-noise 2014

5 Inter-noise 2014 Pge 5 of Rdition impednce of rectngulr ir piston A flt rectngulr piston mounted flush within n infinite bffle is ssumed in the present study. The rdition impednce is s below but the width to height rtio (w/h) is neither very lrge nor very smll (1). Z = w2 θ(kw) h 2 θ(kh) iw 2 φ(kw) + ih 2 φ(kh) w 2 h 2 (16) where θ(x) = 1 4[1 J 0(x)] x 2 φ(x) = 8 πx [1 π 2x H 0(x)] J 0(x) is Bessel function of zero order; H 0(x) is Struve function of zero order. In considertion of the duct thickness of perture δ, the incident nd reflected wves re s below (6). P i = Ie ikz, P r = Re ikz At z= 0, Continuity of pressure: I+R=P 0 Continuity of volume velocity: U i+u r=u 0 I R = P 0 z g z g=ρc/s where S is the cross-section re of perture. Z = z 0 I + R = z g I R z 0 R I = Z 1 Z + 1 At z= δ, Continuity of pressure: P i+p r=p δ Continuity of volume velocity: U i+u r=u δ Ie ikδ Re ikδ z g = P δ z r e ikδ ( R R I )eikδ = e ikδ + ( I )eikδ z g z r The rdition impednce with the effect of perture thickness is: Z r = z R r = e ikδ + ( I )eikδ Z itn(kδ) z g e ikδ ( R = 1 iztn(kδ) I )eikδ 3. FINITE ELEMENT COMPUTATIONAL MODEL The numericl model is estblished in following principle (7). The Helmholtz eqution, which describes hrmonic wve propgtion in medium without dmping, is represented s below. 2 p + k 2 p = 0 (18) where p nd k denote the coustic pressure nd the wve number. The boundry condition ssocited with ll surfces of the duct follows tht of rigid wll of vnishing norml coustic prticle velocity. p n = 0 (19) wll where n represents the outwrd norml direction of boundry. The interctions between sound in the ir nd the three-dimensionl solid duct with n opening re modelled by using the finite element softwre COMSOL Multiphysics. The inhomogeneous wve eqution s below is implemented in the numericl model (8, 9). ( 1 ( p q)) ω2 p = Q (20) ρ 0 ρ 0 c2 where the dipole source q nd monopole source Q re set to be zero, ρ 0 is the fluid density, c is the speed of sound. Eq.18 is simply formed s ω/c=k. (17) Inter-noise 2014 Pge 5 of 8

6 Pge 6 of 8 Inter-noise 2014 The boundry conditions of wll, sound source nd the free field environment re introduced. The sound-hrd boundry is n ( 1 ρ 0 ( p q)) = 0 (21) This turns out to be Eq.19 for zero dipole source. The circulr sound source is set to be the norml ccelertion boundry condition. A unity inwrd ccelertion enters t the circulr boundry. The opening is connected to the free field environment. An bsorbing perfectly mtched lyer (PML) in cylindricl type is dded. It pplies pproximte the boundry t infinity in n exterior problem nd llows n outgoing wve to leve the computtionl domin with no or miniml reflections (9). PML is lso ttched to ech duct end in order to eliminte the effect of reflections t the ends of computtionl domin. PMLs bsorbing in Crtesin coordinte of x direction re set. 4. GEOMETRY DATA AND SETTINGS OF THE SIMULATED MODEL The coustic model of interest in the present study is set to be 0.8-metre long duct of internl uniform cross section re of 0.082m in width nd 0.092m in height. The spect rtio of the cross section is round 1.12 which cn reduce the opportunity of modl overlpping occurred. The duct thickness is fixed to 1mm. The sound source is in 0.046m dimeter. The centre of the opening is llocted long x-xis rbitrrily t 0.162m (x 1=9mm 18). Cses of following sizes of opening re selected for the numericl nlysis. Tble 1 Selected cses for chnge of opening size Opening Height (mm) Width (mm) Cse 1 9 (pproximte /10) 8 Cse Cse The dimeter of the free field domin is 80mm which is lrge compred to the dimension of opening (Figure 2). The PML width of the free field is 20mm. PMLs with 0.5m in length t two ends hve the sme uniform cross section of the coustic model (Figure 3). Ech cses performs numericl nlysis for series of frequency in the model. Figure 2 A section of the computtionl model Figure 3 PML t two ends of the model 5. RESULTS AND DISCUSSIONS Acoustic mode shpe cross the cross section of the duct with vrious frequencies re generted. For instnce, pttern t the resonnce ner the eigen-frequency t mode 01 of numericl model is cptured t the centre of opening s shown in Figure 4. The form of cylindricl sound rdition t the opening is noted. Pge 6 of 8 Inter-noise 2014

7 mgnitude (plne mode) mgnitude (plne mode) Inter-noise 2014 Pge 7 of 8 Figure 4 Pttern t the resonnce frequency, mode 01, 9mm height opening In-duct modl decomposition (10, 11) is dopted to nlyse the computtionl dt of sound pressure over the concerned internl cross sections. The intervl of cross section long x direction is 9mm nd the frequency resolution is 10Hz. Dt from 37 sections nd 11 frequencies (totl 407 points) were collected for ech cses. Results obtined re considered s the trget vlues for comprison. R=0.023, ρ 0=1.25kg/m 3, c=343m/s, =0.092m nd b=0.082m. The source norml ccelertion is unity nd V s = iωv s, V s is i/(kc). The coustic velocity norml direction to the opening V of Eq.3 cn be solved by section 2 mthemticlly. Thus, the predicted sound field long the duct cn be clculted by the contribution of the opening Eq.7, 8 nd the force from source Eq.12. The root men squre (RMS) percentge error is pplied s the mesure of ccurcy of vlidtion which is defined by the formul. Results of vrious cses re shown in Tble 2. RMS percentge error = 100% 1 n n (T t P t t=1 ) 2 T t n is number of fitted points T t re the trget vlues obtined from 3-D numericl model P t re the prediction vlues clculted by mthemticl model Tble 2 RMS percentge error of dt from COMSOL ginst MATLAB Cse Opening size (mm) Frequency rnge Hz 1 9 x 8 (without duct thickness correction) 0.52% 1.22% 1b 9 x % 0.74% x % 1.52% 3 18 x % 2.50% Frequency rnge Hz Figure 5, 6 illustrte the findings of Cse 1 nd 1b t frequency rnge of Hz (plne mode). Comprisons show tht the theoreticl predictions provide good greement with the results from numericl models. The presence of effect of duct thickness t the rdition impednce of perture gives better ccurcy on the mthemticl model MAT MAT MAT260 MAT270 MAT MAT260 MAT270 MAT MAT290 MAT300 MAT MAT290 MAT300 MAT MAT320 MAT330 MAT340 MAT MAT320 MAT330 MAT340 MAT350 element no. (x direction) element no. (x direction) Figure 5 Cse 1 (without duct thickness) Figure 6 Cse 1b Inter-noise 2014 Pge 7 of 8

8 mgnitude (mode 01) Pge 8 of 8 Inter-noise 2014 The error increses s opening size enlrges. Figure 7 illustrtes the worst cse in the present study which is 2.5% error. Such devition my be due to the existence of uneven coustic velocity nd sound diffrction t the opening in numericl modelling. It is expected tht coustic velocity of x-component could be hppened especilly in propgting mode element no. (x direction) Figure 7 Cse 3 t Hz MAT1850 MAT1860 MAT1870 MAT1880 MAT1890 MAT1900 MAT1910 MAT1920 MAT1930 MAT1940 MAT CONCLUSIONS The cousticl properties in long rectngulr duct in the presence of rectngulr side wll opening nd circulr sound source re reviewed. A mthemticl model of interior sound field is derived. Three-dimensionl numericl model is lso implemented to compre with the predictions by the mthemticl model. Vlues clculted from theoreticl prediction give consistent cousticl behviour with tht of numericl model. Results re convinced tht precision of the rdition impednce of perture is importnt to the ccurcy of theoreticl prediction. The present study pproch formulte construction frmework for further investigtion of relted subjects such s understnding of sound propgtion long prtil/ discontinuous enclosure. ACKNOWLEDGEMENTS S. H. K. Chu is supported by grnts from The Hong Kong Polytechnic University nd Fr Est Consulting Engineers Limited. REFERENCES 1. Morse PM, Ingrd KU. Theoreticl Acoustics. New York: McGrw Hill; Dok PE. Excittion, trnsmission nd rdition of sound from source distributions in hrd-wlled ducts of finite length (I): the effects of duct cross-section geometry nd source distribution spce-time pttern. J. Sound Vib. 1973; 31: Chu SHK, Tng SK. Theoreticl prediction nd experimentl modl nlysis of sound propgtion long long enclosure. Proc. ICSV19, Vilnius, Lithuni, July 8-12, Hung L. A theory of rective control of low-frequency duct noise. J. Sound Vib. 2000; 238(4): Tng SK. Nrrow sidebrnch rrys for low frequency duct noise control. J. Acoust. Soc. Am. 2012; 132 (5): Kinsler LE, Frey AR, Coppens AB, Snders JV. Fundmentls of Acoustics. 4th ed. New York: Wiley; Chu SHK, Tng SK. Three-dimensionl numericl modelling of sound propgtion in long prtil enclosure. Proc. Inter-noise 2013; September 2013; Innsbruck, Austri Floody SE, Venegs R, Leighton FC. Optiml design of slit resontors for coustic norml mode control in rectngulr room. Proc. COMSOL Conference; Pris COMSOL AB. COMSOL Multiphysics Modelling Guide. Ver Morse PM, Feshbch H. Methods of Theoreticl Physics. Vol. 1. New York: McGrw Hill; Abom M. Modl decomposition in ducts bsed on trnsfer function mesurements between microphone pirs. J. Sound Vib. 1989; 135(1): Pge 8 of 8 Inter-noise 2014