Sensing of Turbulent Flows Using Real-Time Acoustic Tomography

Transcription

1 I th Bnnal Confrn of th Nw Zaland Aoustal Soty Snsng of urbulnt Flows Usng Ral-m Aoust omograhy ravs Wns Dartmnt of Physs Unvrsty of Aukland Aukland, Nw Zaland ABSRAC hs ar s an ntroduton to th fld of aoust tomograhy, whh s th ronstruton of th tmratur and vloty flds of a flud from a ollton of son tm-of-flght masurmnts. It ams to rsnt rvous work n th ara, as wll as th urrnt rsarh bng arrd out at th Unvrsty of Aukland. A numbr of nw thnus ar rsntd, nludng full-dulx transdurs, ontnuous wav tm-of-flght masurmnt, and a mthod of ronstrutng th flds n ral-tm at hgh saml rats usng radal bass funton ntworks. INRODUCION Aoust omograhy s th ronstruton of th tmratur and/or vloty fld of a flud from a ollton of son flght tms aross a volum [][3][5]. Consdr th as of a two dmnsonal flow, suh as a Kármán vortx strt formd bhnd a ylndr, as shown n Fgur. A numbr of son transdurs ar lad n th flow, ah dnotd by an x n th fgur. Sounds ar now rodud at ah transdur. hs an b ulss, odd sgnals or ontnuous wavforms. Eah transdur also rords th rvd sgnals from ah othr transdur. h tm of flght for ah ath s dtrmnd by orrlaton or Fourr thnus (for xaml). h tm of flght for ath, btwn transdurs at ostons and s Δt ( ) + U( ) Lˆ () whr () s th son sd and U() s th twodmnsonal vloty vtor fld at oston, Lˆ s a unt vtor aralll to th sound ray, and s an ntgraton lngth along th ray s ath. h son sd s a funton of tmratur, : Rγ, () whr R s th sf gas onstant of th flud, and γ s ts sf hat rato. h roblm s to ronstrut an stmat of th tmratur and vloty fld of th flud from masurmnts of th tm of flght aross a numbr a aths. IME OF FLIGH MEASUREMEN h frst ratal roblm that travl-tm tomograhy rsnts s how to dtrmn th flght tm of th sound aross th flud. hs s th sam roblm rsntd by son anmomtrs whh tyally snd an Fgur. Smulaton of a Kármán vortx strt bng formd bhnd a ylndr wth R6. wlv transdurs ar lad n th flow, ah markd wth an x. Sound s assd along th gry lns btwn ah ar of transdurs, wth th am of stmatng th tmratur and vloty fld btwn thm basd on th son tm of flght. 7-8 Novmbr 8, Aukland, Nw Zaland

2 I th Bnnal Confrn of th Nw Zaland Aoustal Soty ultrason uls aross a short dstan and us orrlaton to dtrmn th travl tm. Howvr, thr ar a numbr of dffrns btwn a son anmomtr and th dvs usd for aoust tomograhy whh mak ths lss attratv. h frst dffrn s th ath lngth; a son anmomtr has a ath lngth of tns of mllmtrs, whl an aoust tomograhy ath lngth may b tns or hundrds of mtrs. Sony, a son anmomtr wll tyally hav sx sound aths, whl aoust tomograhy shms may hav dozns or hundrds of aths. If a ulsd systm s usd, on must b sur that ulss do no ntrfr wth ah othr. hs mans that f on wats for th rvous uls to ovr th masurmnt ara bfor sndng th nxt, th shortst tm n whh on an omlt an ntr yl of masurmnts s lmtd by th rodut of th numbr of transdurs and th tm of flght. hs s fn for son anmomtrs wth small numbrs of transdurs and short dstans, but an mak an aoust tomograhy shm unusabl for masurmnts of a turbulnt fld. On soluton to ths roblm s th us of ontnuous wavs rathr than ulss of sound. If ah transmttr smultanously rodus sound at a dffrnt fruny, th dffrnt sgnals rahng a rvr an b saratd n th Fourr doman. h has of ah rvd sgnal (alulatd from th Fourr transform) an thn usd to dtrmn th ay sn th hang n has angl, φ, s gvn by φ ωδt (3) whr ω s th sgnal fruny (n rad/s) and Δt s th ay tm. FREQUENCY DIVISION FULL DUPLE In most aoust tomograhy shms, th sakrs and mrohons wr sarat unts, thr transmttng or rvng, but not both. hs s rfrrd to as half dulx whn any on dv an only transmt or rv at any on tm. Howvr, t was dsovrd that th sakrs (Motorola KSN5a zo twtrs) ould also orat n full dulx mod, transmttng and rvng smultanously. hs allows half th numbr of sakrs to b usd and nsurs that th bams travllng n oost drtons follow th sam ath. Sn th transmttd and rvd wavs ar at dffrnt fruns ths s rfrrd to as Fruny Dvson Full Dulx, aftr th mod of rado ommunaton. h ky to saratng th hgh owr transmttd sgnal from th low owr rvd sgnal s auratly knowng th fruny of both sgnals so thy an b saratd n th Fourr doman. hs an b ahvd by synhronzng th data auston systm wth a numrally ontrolld osllator usd to gnrat th transmttd sgnals. LINEAR FORWARD PROBLEM Bfor on an attmt to solv th nvrs roblm, t s usful to smlfy th forward roblm. Frst, on assums that thr ar no larg vloty or tmratur gradnts. hs lmnats dffraton of th sound so urvd aths nd not b onsdrd. Nxt, assum that th son sd s muh largr than ts flutuatons and also muh largr than th wnd sd. Both assumtons ar gnrally saf n atmoshr flows. Basd on ths assumtons and lnarzng th flds about man valus, and U, Ostashv [] gvs Euaton as Δt l U ˆ L Δ ( ) ΔU( ) Lˆ + d l (4) whr l s th ath dstan. INVERSE PROBLEM A numbr of solutons to th nvrs roblm hav bn roosd. Wlson [7] roosd dvdng th masurmnt ara nto a grd wth a onstant valu for and U n ah grd sa. Howvr, ths mls dsontnuous flds and a larg numbr of aramtrs must b solvd for to rovd good rsoluton. Furthr dtals may b found n rfrn []. abuh [6] frst roosd usng Radal Bass Funtons for th tomograh ronstruton of a salar fld (suh as tmratur), allowng for ontnuous valus, but dd not xand th analyss to nlud vtor flds (suh as vloty). hs ston wll nlud ths analyss. A radal bass funton (RBF) ntwork s a nonlnar aramtr surfa whh s lnar n th aramtrs, allowng for asy otmzaton of ths aramtrs. A sngl radal bass funton s gvn by kr φ ( r) (5) whr r s th radus from a RBF ntr and k s a salng fator. A RBF ntwork s thn ratd from a lnar ombnaton of N r radal funtons wth dffrnt ntrs. h outut of th ntwork s thn N r k j j fˆ ( ) W j (6) or j f ˆ( ) WΦ( ) (7) whr W s a aramtr vtor of wghts. Not that, on th k and aramtrs hav bn st, fndng th W vtor s a lnar nvrson roblm. In ordr to aly a RBF ntwork to th tomograh nvrson roblm, two RBF ntworks ar usd. On s usd to stmat th tmratur and on s usd to 7-8 Novmbr 8, Aukland, Nw Zaland

3 I th Bnnal Confrn of th Nw Zaland Aoustal Soty stmat th flud vloty. h stmatd tmratur at any oston s Δ ˆ ( ) Φ ( ) W. (8) h stmatd vloty s slghtly mor omlatd. On ould st u two ntworks to mo th two vloty omonnts. Howvr, w an assum that th flow s nomrssbl (dvrgnlss) to rdu th numbr of unknown aramtrs. For a twodmnsonal flow fld, th stram funton, ψ, s dfnd as ψ Δu x y ψ Δu y x (9) () whr Δu x and Δu y ar th vloty flutuaton omonnts n th x and y drtons. In a translatd and rotatd - oordnat systm, th vloty n th - drton s ψ Δu () A radal bass funton ntwork s thn st u to stmat th stram funton: ψˆ ( ) Φ ( ) W. () ψ Now, rarrang Euaton 4 nto th form ψ l ˆ U L Δ ˆ( ) Δt d l +... Δ ˆ ˆ U( ) L + ε (3) whr ε s an rror trm. If on substtuts Euaton 8 nto th ntgral, th rsult s or Φ W (4) r N r N W j j W k k (5). (6) h ntgral of a sngl radal bass funton an b found by wrtng th ntgral wth rst to a - oordnat systm wth th sound ray along th -axs and th RBF ntr lyng on th -axs: k kj π 4k d l j j k j ( + ) d (7) [ rf ( k ) ( k )] j rf j (8) whh may b alulatd analytally from th roblm gomtry. A smlar ross s usd to trat th vloty trm n Euaton 3. In th sam - oordnat systm: ΔU( ) Lˆ j ˆ j j j ψˆ d Nr j j j j Δuˆ d (9) () Wψ Φ d () j Wψ j Φ d () j Nr j j kψ j ( j + ) Wψ j d. (3) j Nr j h drvatv and ntgral an b alulatd to b j j j πk kψ j ( j + ) k j d (4) [ rf ( k ) ( k )] j rf j, (5) whh an also b alulatd analytally from th gomtry. Euaton 3 an thn b wrttn as d ΩW + Ω ψ W ψ, (6) or, for all N aths: d Ω W + Ω ψ W ψ, (7) d ΩW, (8) whr d s a N x row vtor of data, Ω s an N x N r matrx nludng th ntgrals dvlod abov, and W s an N r x aramtr vtor to b solvd for. hs nvrs roblm an b solvd usng a numbr of mthods. If N r < > N, th roblm s undr- or ovr- 7-8 Novmbr 8, Aukland, Nw Zaland

4 I th Bnnal Confrn of th Nw Zaland Aoustal Soty dtrmnd and a last-suars soluton may b usd, suh as th sudonvrs ( Ω Ω) Ω d W (9) W Ad (3) whr sursrt dnots a matrx transos. Not that th omutatonally ntnsv matrx nvrson nd only b rformd whn th roblm gomtry hangs, not for vry flght-tm masurmnt. hs allows th tmratur and vloty flds to b ukly dtrmnd n ral tm at usful saml rats for turbulnt flows. Fgur. argt vloty and tmratur fld from smulaton. SIMULAION SUDY A smulaton xaml s rsntd hr to dmonstrat th rforman of th systm. h Matlab od s avalabl on rust. A flow fld was alulatd usng th Grrs Flow Solvr s Kármán vortx strt xaml [htt://gfs.sourforg.nt], on fram of whh s shown n Fgur. 5 frams of data wr gnratd, usng th smulaton s trar varabl to rrsnt a tmratur wth a K flutuaton rang. A tomograh array was onstrutd downstram of th ylndr wth transdurs ostond on a rl. h nvrs roblm usd 5 RBF ntrs randomly dstrbutd ovr th masurmnt ara wth k. for ah. h A matrx n Euaton 3 was ralulatd. For ah masurmnt fram th wght matrx was thn dtrmnd and th vloty and tmratur flds wr alulatd on a both a oars 5 x 5 grd and a fnr x grd. Fgur 3. Vloty and tmratur fld ronstrutd by aoust tomograhy. On fram of th Grrs targt data s shown n Fgur and th orrsondng ronstrutd data s shown n Fgur 3. h rror n tmratur s shown n Fgur 4 and th rror n vloty s shown n Fgur 5. Not that th drton of th vloty rror s gnrally rndular to th ath lns, whh s to b xtd sn th only vloty nformaton avalabl s aralll to th aths. h root man suard (RMS) rror aross th masurmnt ara for ths fram s.34 K for tmratur and.8 m/s for vloty. For all 5 frams, th man RMS rror s.5 K for tmratur and. m/s for vloty. Fgur 4. Error n ronstrutd tmratur fld. 7-8 Novmbr 8, Aukland, Nw Zaland

5 I th Bnnal Confrn of th Nw Zaland Aoustal Soty us for ral-tm alulaton and dslay of turbulnt data. REFERENCES [] S. Bray, Atmoshr Aoust Rmot Snsng, CRC Prss, Boa Raton, 8. Fgur 5. Error n ronstrutd vloty fld. Not that th rror s tyally rndular to th sound aths, sally at th dgs of th masurmnt ara. An mortant fatur of ths algorthm s ts otntal for raltm oraton at hgh saml rats. h omutaton tm of ah stag of th alulaton was masurd. It should b notd that Matlab s not a artularly fast languag, so on an xt that thr sgnfant s room for mrovmnt. h ntal stu tm for th omutaton was 5.6 s on a modrn omutr wth an Intl Cor Duo 64 rossor runnng at.3 GHz. hs nluds gnraton of th A matrx as wll as smlar matrs for alulaton of th tmratur and vloty flds from th W matrx. h alulaton of th W matrx took 5.9 μs r fram of data, whl th alulaton of th tmratur and flow flds took 59 μs. hs orrsonds to a maxmum fram rat of 97 frams/s to rform both sts. hs fram rat s xssv for most atmoshr flows, allowng for muh slowr mbddd hardwar to b usd. CONCLUSION hs ar rsnts a nw mthod of ronstrutng flow and tmratur flds of a flud from aoust tm of flght data. It was shown to b fast nough to [] H. Braun and A. Hauk, omograh Ronstruton of Vtor Flds, IEEE ransatons on Sgnal Prossng, vol. 39, no., , 99. [3] I. Jovanov, L. Sbaz, and M. Vttrl, Aoust omograhy Mthod for Masurng mratur and Wnd Vloty, IEEE Intrnatonal Confrn on Aousts, Sh, and Sgnal Prossng. IEEE, 6. [4] V. Ostashv, S. Vhrn, D. Wlson, A. Zmann, and G. Godk, Rnt Progrss n Aoust omograhy of th Atmoshr, IOP Confrn Srs: Earth and Envronmntal Sn, vol., no., 8. [5] J. Prn, omograh Ronstruton of 3-D Vtor Flds, IEEE Intrnatonal Confrn on Aousts, Sh, and Sgnal Prossng, vol. 5, , 993. [6] H. abuh,. Myosh, H. Ihbash, and K. Ohno, Comutrzd tomograhy wth radal bass funtons ntwork: a nuro-fuzzy aroah, IEEE Intrnatonal Confrn on Nural Ntworks, vol.5, , 995. [7] D. Wlson and D. homson, Aoust omograh Montorng of th Atmoshr Surfa Layr, Journal of Atmoshr and Oan hnology, vol., no. 3, , Novmbr 8, Aukland, Nw Zaland

6 Errata Euaton 5: j πk k j [ ( ) ( )] rf k k j rf j, (5) should rad j πk k j [ ( ) ( )] rf k k j rf j. (5)