SIMULATION OF SOUND WAVE PROPAGATION IN TURBULENT FLOWS USING A LATTICE-BOLTZMANN SCHEME PACS REFERENCE: 43.20.Mv Andreas Wlde Fraunhofer Insttut für Integrerte Schaltungen, Außenstelle EAS Zeunerstr. 38 01069 Dresden Germany Tel: (49) 351 4640 852 Fax: (49) 351 4640 703 emal: Andreas.Wlde@eas.s.fhg.de Abstract The propagaton of sound waves n nstatonary flows s often computed n two steps, whch are the determnaton of the nstatonary flow feld and calculaton of sound propagaton n the flow. Here, a alln-one approach s proposed whch s based on the Lattce-Boltzmann method. Two applcatons are presented: The soluton of a benchmark problem whch s compared to the analytcal soluton and a twodmensonal cut through an ultrasound gas flow meter. For the benchmark problem excellent agreement between theory and numercal results was obtaned. For the ultrasound gas flow meter, the results clearly ndcate the feasblty of such smulatons. The measured transt tmes of a test sgnal agree well wth rough estmatons, furthermore they exhbt a lot of detals whch are related to the nstatonary flow feld. 1 Introducton Flow acoustc problems usually can be splt nto three more or less separate parts: The determnaton of the turbulent flow tself, the propagaton of sound waves wthn a turbulent flow and n many cases the generaton of sound by the flow. The nteracton between ths parts s weak or at least only one-way for most problems at low MACH-numbers. Ths means, that the turbulent flow generates sound waves, but ths effect only plays a mnor role wth the determnaton of the nstatonary flow feld tself. Also flow structures such as eddes n a turbulent flow modfy the propagaton of sound waves, but not vce versa. Therefore the smulaton of such problems s often done n two or three steps: Frst the flow s calculated solvng the Naver-Stokes equatons, then the sound generaton s computed wth another model (acoustc analoges etc.) and fnally the propagaton effects are accounted for usng a thrd model (ray tracng or soluton of the convected wave equaton). The reasons for dong so nstead of usng a all-n-one-model are the length dspartes between the sze of flow structures and the acoustc wave lengths and the precson of the flow solvers, whch tend to be nosy. Ths means that the numercal error s too large to gather nformaton on the small scale of varablty assocated wth sound waves. On the other hand, the seres connecton of ths dfferent models poses a lot of techncal problems and the errors at each stage of the modelng process may add up nauspcously. Two of the problem parts descrbed above namely the determnaton of the turbulent flow and sound propagaton through the flow occur when the velocty of a flow s to be measured by the transt tme of sound wave through a test secton of known length. The transt tme s effected by the convecton of the sound waves and s measured as a phase dfference between the transmtted and receved sgnal, where sgnals of constant frequences are used. Ths prncple s often used n ultrasound gas flow meters. Here the sze of the strongest eddes s of the same order of magntude as wave length of the acoustc wave used for the measurement. Therefore a model whch computes the nstatonary flow and the propagaton of the test sound n one step s useful because the varablty of the measured transt tme s sought, whch s ntroduced by the turbulence. In a model where all physcal data are at hand at the same tme the correlatons between certan parameters can be tracked most easly.
4 3 2 5 1 6 7 8 Fgure 1: The D2Q9-Lattce-Boltzmann model s defned on a two dmensonal orthogonal grd. The vectors x pont from one node to the -th of the 8 nearest neghbors. Together wth zero velocty ths gves 9 possble dynamc states. The Lattce-Boltzmann method s a relatvely new technque whch was ntroduced for solvng flud dynamc problems up to moderate subsonc MACH-numbers. It evolved durng the last two decades from the so-called Lattce-Gas method, whch smulate the behavor of sngle flud partcles on a lattce. Snce then the Lattce- Boltzmann method was successfully appled to problems whch nvolve hghly turbulent flows and extremely complex geometres, such as the flow around cars or through porous meda [1]. Only very few studes dealt wth the ablty of the method to smulate sound wave propagaton [2, 3, 4]. Here a frst approach s presented to smulate convecton effects on sound wave propagaton wth a Lattce-Boltzmann scheme. 2 Theory The basc dea of the Lattce-Boltzmann method s to dscretze the phase space of all the partcles that make up the flud to the effect that only a fnte number of veloctes and postons remans, whch the partcles are demanded to take at each tck of a clock countng the tme n ntervals of constant duraton. The possble postons are gven by the nodes of a grd wth suffcent symmetry, and the possble veloctes result from the restrcton, that partcles may be at rest or move from one lattce ste to one of ts nearest neghbors durng one tme nterval. In ths study an orthogonal grd s used n whch the state of a lattce ste (node) at the tme t + t s dependent on the state of the 8 stes n the drect neghborhood and the ste tself at tme t. On ths lattce the vectors x are defned whch pont from one node to ts nearest neghbor n the -th drecton (see fg. 1). The possble veloctes are thus gven by v = x t, = 0...8 whch ncludes partcles at rest wth x 0 = 0. In the followng the ndex whch ndcates the drecton of the nearest neghbor ste always ranges from 0...8. The Lattce-Boltzmann method s used to calculate partcle flows between lattce stes. The flow f ( x + x,t + t) of partcles that travel at tme t + t from a node at poston x to the -th nearest neghbor at poston x + x s gven by f ( x + x,t + t) f ( x,t) = Ω ( x,t) (1) where Ω s an operator whch descrbes the collsons of partcles n a statstcal way. If ths operator s set to zero the left hand sde of the above equaton descrbes flud partcles whch travel wthout any nteracton from one lattce ste to another. The collson operator s smplfed usng the Bhatnagar-Gross-Krook (BGK) approxmaton [6] Ω ( x,t) = 1 ( f τ ( x,t) f ( x,t) ) (2) wth f ( x,t) beng the equlbrum dstrbuton functon and τ the relaxaton tme. The macroscopc quanttes densty ρ and velocty u are defned as ρ = f ρ u = v f
To obtan conservaton of mass and momentum the equlbrum s requred to satsfy ρ = f ρ u = v f Then the equlbrum functon can be derved usng the Chapman-Enskog expanson f = ρ ( a + b v u + c( v v) 2 + d u 2) wth a,b,c,d beng constants dependent on the lattce geometry. Eq. (1) can be shown to recover the full non-lnear Naver-Stokes equatons for small local devatons from the equlbrum [3]. For the partcular lattce used here the equlbrum dstrbuton functon reads [1] ( f = ρw 1 + 3 v u + s 9 2 ( v u) 2 + 3 ) 2 u 2 wth w 0 = 4/9, w 1 = w 3 = w 5 = w 7 = 1/9 and w 2 = w 4 = w 6 = w 8 = 1/36. The speed of sound n the flud at rest s c s = 1 x 3 t and the knematc vscosty s connected to the relaxaton parameter va ν = 2τ 1 6 Here x refers to the shortest dstance between two nodes. 3 Results In a frst seres of test calculatons the propagaton of a 2D-sound wave n a flud whch s otherwse n unform moton was smulated. Here a benchmark problem was chosen smlar to a problem posed for the second computatonal aeroacoustcs workshop on benchmark problems, Tallahassee, Florda, November, 1996. Ths problem conssts of an ntal pressure dstrbuton wth a Gaussan shape centered at x = 0 that s released n a constant unform flow wth a velocty of Mach = 0.5. Though the Mach number usually would be too hgh to apply a Lattce-Boltzmann scheme as descrbed above, n ths case the smulaton performed well up to the desred tme step. Soon after the results gven below were observed, the errors due to an nstablty grew larger than the ampltude of the sound waves. Fg. 2 shows the pressure as a functon of poston after 100 tme steps of the model. The effect of the convecton of the sound wave s clearly vsble, the whole structure s shfted towards hgher values of x where the "ground speed" of the sound wave travelng upstream s one thrd of the wave travelng downstream. The relatve devaton from the analytcal result s below 1.5%, whch clearly shows the ablty of the model to smulate the convecton effect on the sound waves. For the second set of test calculatons a two dmensonal geometry was defned whch s smlar to usual ultrasound gas flow meters. The gas flow meter s made from a U-shaped ppe where the test secton s n the lower horzontal part (see fg. 3). A sound wave that enters the test secton wll be convected by the mean flow whch results n an altered transt tme to the mcrophone at the opposte sde of the test secton. If a snusodal sound sgnal s used ths can be measured usng correlaton technques as an altered phase dfference between the sgnal from the source and the mcrophone. The boundary condtons at the nlet were normal densty and at the outlet unform flow wth constant velocty. The test sgnal was a 40 khz snusodal sound wave wth an ampltude of 0.2 Pa. Further parameters are gven n table 1. Fg. 3 also shows a snapshot of the pressure feld n absence of a mean flow. The sound wave can be seen to propagate through the test secton and also enter the nlet and outlet. Thus a complcated pattern of standng waves s generated soon after the smulaton s started. Fg. 4a shows a snapshot of the absolute value of the velocty of the flud wth a mean flow of 5 m/s and the sound source turned on. The flow enters the setup beng lamnar, but becomes turbulent when t enters the test secton. The generaton of eddes can be seen at the dscontnutes near the sound source and the
1 0.8 lne 1 lne 2 0.6 0.4 pressure 0.2 0-0.2-0.4-0.6-150 -100-50 0 50 100 150 poston x Fgure 2: Pressure as functon of the poston 100 tme steps after the release of an ntal pressure dstrbuton wth Gaussan shape, released at x = 0. The unform background flow has a Mach number of 0.5. The pressure dstrbuton s shfted to larger values values of x whch s due to the convecton effect on the sound waves. The sold lne gves the analytcal soluton, the numercal result s ndcated wth x-symbols. number of nodes 102629 dameter of the ppes 0,012 m flow velocty 1... 10 m/s Reynolds number (referred to dameter) 800... 8000 densty 1,21 kg/m 3 speed of sound (flud at rest) 340 m/s kn. vscosty 1,5x10 5 m 2 /s spatal resoluton 2x10 4 m tme step 3,4x10 7 s Table 1: Parameters of the test calculaton for the ultrasound gas flow meter mcrophone. The velocty varatons due to the sound wave are much smaller than the hydrodynamc effects and therefore cannot be seen on the mage, although the 40 khz pressure wave s well detectable n the mcrophone sgnal. In a further setup, a fxed cylnder was placed n the nlet whch generated addtonal turbulence (see fg. 4b). Ths was to study the nfluence of flow feld modfcatons on sound propagaton. To compute the phase dfference of the sgnal generated at the sound source the pressure was recorded as a tme seres at the poston of the mcrophone. The numercal experments were conducted for both setups wth mean veloctes rangng from 1 m/s to 10 m/s n steps of 1 m/s. Each smulaton ran for 300000 to 500000 tme steps, whch corresponds to a tme nterval of 0.1 to 0.17 s. In all cases the flow was turbulent, where n absence of the addtonal turbulence generator the flow became sgnfcantly nstatonary not before the test secton. Fg. 5 depcts the tme averaged phase dfference between the source and the mcrophone sgnal and ts standard devaton as a functon of the mean velocty, respectvely. Up to about 8 m/s the tme averaged phase dfference s lnearly dependent on the mean velocty. At the same tme the varablty grows wth ncreasng velocty, whch s due to stronger generaton of turbulence. At veloctes hgher than 8 m/s the varablty of the phase dfference s so large that no tme averages can be calculated. The statement also holds f a cylnder s put nto the nlet whch produces addtonal turbulence. However, the rate at whch the tme averaged phase dfference ncreases wth velocty s rased about 10% by the cylnder, whch means that the gas flow meter would delver dfferent results. A vsual nspecton of the magntude of the flow velocty shows that the flow seems to form a faster jet wth a smaller dameter when a turbulence generatng cylnder
nlet outlet sound source test secton mcrophone Fgure 3: Geometry of the ultrasound gas flow meter. The flow enters at the upper left end and exts at the upper rght end. The sound source s stuated at the lower left end of the test secton whle the mcrophone s on the lower rght. The colors ndcates the pressure at an early stage of the smulaton whle the mean flow s set to zero. The red and blue areas correspond to pressure maxma and mnma, respectvely. s present n the nlet. Ths ncreased velocty compared to the lamnar case mght be responsble for the observed effect on the phase dfference measurements. 4 Conclusons A 2D smulaton of convecton effects on sound waves was performed usng a Lattce-Boltzmann scheme. Appled to a benchmark problem the results acheved wth the Lattce-Boltzmann code showed excellent agreement wth the analytc soluton avalable for that case. As a further test an ultrasound gas flow meter geometry was used to perform a flow smulaton along wth an evaluaton of the phase dfference between a source and a mcrophone sgnal, whch serves as a measure for the mean flud velocty. The tme averaged phase dfference shows a lnear dependency on the mean velocty. By ntroducton of an addtonal turbulence generator n the nlet of the flow meter, an alteraton of proportonalty factor between the tme averaged phase dfference and the mean velocty was acheved. Although ths result has to be confrmed by experments n future, t ndcates that the method s potentally senstve enough to predct the effects of dfferent nflow condton on the measurements of ultrasound gas flow meters. Acknowledgments Ths study was supported by the Deutsche Forschungsgemenschaft through the SFB 358 Automatserter Systementwurf and by the BMBF wthn the project EKOSAS under grant 16 SV 1161/7. References [1] S. Chen and G. Doolen, Lattce Boltzmann method for flud flows, Annual Revew of Flud Mechancs, vol. 30, pp. 300 364, 1998. [2] J. M. Buck, C. A. Greated, and D. M. Campbell, Lattce BGK smulaton of sound waves, Europhyscs Letters, vol. 43, no. 3, pp. 235 240, 1998. [3] J. M. Buck, C. L. Buckley, C. A. Greated, and J. Glbert, Lattce Boltzmann BGK smulaton of nonlnear sound waves: the development of a shock front, Journal of Physcs A: Mathematcal and General, vol. 33, pp. 3917 3928, 2000. [4] D. Haydock and J. Yeomans, Lattce Boltzmann smulatons of acoustc streamng, Journal of Physcs A: Mathematcs and General, vol. 34, pp. 5201 5213, 2001. [5] D. Wolf-Gladrow, Lattce-Gas Cellular Automata and Lattce-Boltzmann Models. Sprnger-Verlag Berln Hedelberg, 2000. [6] P. L. Bhatnagar, E. P. Gross, and M. Krook, A model for collson processes n gases, Physcal Revew, vol. 94, pp. 511 525, 1954.
a: b: Fgure 4: a: Absolute value of the flud velocty after 300000 tme steps. Blue color ndcates flud at rest whle red means 10 m/s. The mean flow velocty was 5 m/s. b: Same as a, but wth cylnder generatng turbulence n the nlet. 0.4 1.6 0.2 1.4 0 1.2 phase dfference n rad 0.2 0.4 0.6 0.8 1 1.2 lamnar turbulent sgma n rad 1 0.8 0.6 0.4 0.2 lamnar turbulent 1.4 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 mean velocty n m/s mean velocty n m/s Fgure 5: Left: Phase dfference of the source and the mcrophone sgnal as a functon of mean velocty. Up to 8 m/s the phase dfference s approxmately lnear dependent on the mean velocty. Rght: At hgher mean veloctes the standard devaton of the phase dfference σ becomes too large to evaluate the phase dfference correctly. The label turbulent and lamnar refer to the nlet condtons,.e. wth or wthout turbulence generator n the nlet.