EFFECT OF FRINGE DIVERGENCE IN FLUID ACCELERATION MEASUREMENT USING LDA

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1 Fachtagung asemethen n e Stömungsmesstechnk Septembe, Unvestät Kalsuhe EFFECT OF FRINGE DIVERGENCE IN FUID ACCEERATION MEASUREMENT USING DA Sejng Chun, Hlge Nbach, Nls Damaschke, Nklay Semetnv, Cam Tpea Depatment Mechancal Engneeng, Kea Avance Insttute Scence an Technlgy, 35-7, Daejen, Kea Fachgebet Stömungslehe un Aeynamk, Technsche Unvestät Damstat, Petesenstaße 3, 687 Damstat, Gemany E-mal : schun@sla.tu-amstat.e. Intuctn The lase Dpple technque s well-establshe as a velcty measuement technque hgh pecsn lw velcty. Recently, the lase Dpple technque has als been use t measue acceleatn lu patcles (ehmann et al. ). Acceleatn s nteestng m a lu mechancs pnt vew, snce the Nave Stkes equatns, speccally the let-hanse, ae mulate n tems lu acceleatn. Futhe, thee ae seveal avenues t estmatng the sspatn ate usng the acceleatn (Mann et al. 999). Hweve such measuements place atnal emans n the esgn the ptcal system; n patcula nge nn-unmty must be hel belw abut.% t av systematc es. Relatns epessng nge vegence as a unctn the ptcal paametes the system have been gven n the lteatue (Mles 996, Mles an Wte 996); hweve, ect use these mulae t mnme nge vegence lea ethe t vey lage measuement vlumes t etemely hgh ntesectn angles (ehmann et al ). Ths lemma can be eslve by usng an -as ecevng aangement, n whch the measuement vlume s tuncate by a pnhle n nt the etectn plane. In the pesent stuy an ptcal esgn stuy s peme ptmng lase Dpple systems lu acceleatn measuements. Ths s llwe by labaty valatn usng a un ee jet an a stagnatn lw, tw lws n whch ethe lu acceleatn has been pevusly measue n whch the acceleatn s knwn analytcally. A 9 egee -as ecevng angle s use wth a pnhle.. Desgn Cnseatns Fnge Dvegence Fnge vegence s ene as a elatve e the nge spacng at abtay pstns, δ wth espect t the nge spacng at the gn, δ. ( δ δ )/ δ () Thee ae tw man epenences δ t be cnsee, lngtunal an tansvese. The lngtunal epenency s gven by (Mles 996).

2 + R R l l cs cs cs sn δ () Hee, s the wavelength tw lase beams, s the ull ntesectn angle the beams, s the beam amete bee the tansmttng lens, s the beam sepaatn bee the tansmttng lens, s the pstn beam wast n nt the tansmttng lens, s the cal length tansmttng lens, R l s the Raylegh length, an ẑ s the nmale beam as,.e.,. F an eal beam algnment, +, cs sn eal δ. (3) The tansvese nge spacng epenency s gven by (Mles 996), R l 5 cs sn δ () whee s the nmale tansvese as,.e.,. F an eal beam algnment, the tansvese nge spacng becmes sn, δ eal. (5) Fllwng the entn nge vegence n Eq. (), the lngtunal nge vegence wth an eal beam algnment s then cs (6) The tansvese nge vegence wth the eal beam algnment becmes (7) whch means n nge vegence n ths stuatn. s symmetc abut an the vegence n the ectn mnates ve the tansvese vegence,. F lu acceleatn measuements, shul be less than abut.%. A esgn mula a spece nge vegence s theee ( ).5.5 / cs tan /. (8)

3 nmale cal length [/] nmale beam wast aus [_/].%.5%.% Fg. : Desgn cuve nge vegence (ts ncate esgn pnts) / / (mm) (mm) w (mm) (eg) (%) (%) Case Case Case Table : Desgn pnt tansmttng ptcs Fg. splays esgn cuves eent lmtng nge vegence values. The aeas belw the sl lne n Fg. epesent cntns une whch the nge vegence s less than.%. The black ts ente esgn pnts t be use n the pesent stuy. Thee esgn pnts have been selecte an pescbe n me etal n Table. All the paametes ae btane usng the abve equatn. Bth an have been calculate by the nge spacng mula wthut appmatn cmpasn pupses (Mles 996). + lr + lr δ +, (9) sn tan + lr + lr Hee, an (, ) ae lcal beam cnates t escbe tw nput lase beams. l R (, ) s the Raylegh length cespnng t each lase beam. In ths case, ect evatn nge vegence mula wul be much me cmplcate than Eqs. (6) an (7). The cmplete nge vegence by usng Eq. (9) shul gve g ageement wth Eqs. (6) an (7). In Table, Case splays the stuatn when the se the measuement vlume s athe lage an n lnge a pnt measuement. Because the beam aus bee tansmttng lens s vey small, the beam aus ate the lens becmes lage accng t (Albecht et al. 3).

4 w s the beam wast aus ate the tansmttng lens. In Case, w () π w. 5mm s elatvely lage cmpae wth the the tw cases shwn n Table. Theee, the se measuement vlume mght be teate as vlumetc athe than pnt-wse amng the thee esgn pnts. Case cespns t a hghly cusse system wth a lage ntesectn angle. A lage ntesectn angle actually vlates the assumptns mae n evng the nge spacng mulae (Mles an Wte 996; Mles 996). Case 3 s a me typcal ptcal aangement, usng an -as ecevng ptcs an a pnhle. Whle the pnhle s nmally nclue t emve backgun nse cmpnents, n ths case t als tuncates the measuement vlume n the ectn an elmnates ptns wth hgh nge vegence. Case an 3 gve vey lw value at, whch ncates the assumptns use t eve Eq. (8) ae cect. Fgs. llustate the nge vegence each esgn case. Fg. splays the nge vegence n Case. The lngtunal nge vegence eaches.5% at the ege the measuement vlume, whle the tansvese nge vegence s essentally e. Fg. 3 splays the nge vegence n Case. Sgncant evatn m the appmate nge vegence mulae s un, because a vey lage ntesectn angle s use. In ths case, the tansvese nge vegence seems t mnate the nge vegence. Hweve, ths esults smply shws the vlatn assumptns mentne abve. Fg. splays the nge vegence n Case 3. The lngtunal nge vegence ncates a nge vegence.% at the ege the measuement vlume. By settng the cten at.%, the useul measuement ange alng the beam as s un t be.5mm. 5mm. Ths value s calculate m the llwng equatn. ma () cs actan Hee, ẑ ma s the mamum lcatn whee the equement a gven nge vegence s satse. Theee, a slt a pnhle, whch has an mage wth at the measuement vlume less than.3 mm s necessay t lmt the lngtunal nge vegence belw.%.

5 . nge vegence [%] nmale cnates [ an ] lngtunal vegence tansvese vegence Fg. : Fnge vegence case. nge vegence [%] nmale cnates [ an ] lngtunal vegence tansvese vegence Fg. 3: Fnge vegence case. nge vegence [%] nmale cnates [ an ] lngtunal vegence tansvese vegence Fg. : Fnge vegence case 3

6 3. Futhe Cntent Pape Measuements have been peme n a un ee jet an n a stagnatn lw. Analytcal lu velcty an acceleatn ples ae gven by an ey vscsty appmatn. The tubulent stuctue a un ee jet s als well-cumente n the lteatue, n patcula the epenence sspatn ate, hence the lu acceleatn s knwn n tems nnmensnal aal pstn. Als n the stagnatn lw the epecte acceleatn s knwn m the mean velcty changes alng the stagnatn steamlne. Sme uthe emaks wll be mae abut the use lu acceleatn nmatn n the els. F nstance, n tems lw nse eseach, the gaent lu acceleatn s als mptant because t gves the ple cmpnent secn-e evatves the ghthll stess tens T j. Accng t the ghthll s acustc analgy, the wave equatn s eve as llws. ρ' c t T j ρ' j ( ρuu ) j j a + ( c ρ' ) Hee, ρ s ensty, ρ ' s ensty vaatn, u s lu velcty, a s lu acceleatn, c s the spee sun, ρ ' s the ensty vaatn n an acustc suce egn, an s the aplacan peat. Even thugh the last tw tems n the ght-han-se ae teate as ple an mnple sun suces n the sl suaces, they can be als cnsee as suces by lu mtn. The gaent agangan lu acceleatn pves ne acustc suces n ths case. Theee, the gaent lu acceleatn wll be als measue as a st step t evaluate the magntue ghthll stess tens n the ee-a jet an the stagnatn lw. () Acknwlegement Ths wk was suppte by the Pst-ctal Fellwshp Pgam Kea Scence & Engneeng Funatn (KOSEF). teatue Albecht, H. -E., Bys, M., Damaschke, N., Tpea, C., 3: ase Dpple an Phase Dpple Measuement Technques, Spnge Velag ehmann, B., Nbach, H., Tpea, C., : Measuement Acceleatn Usng the ase Dpple Technque, Measuement Scence Technllgy, 3, pp Mann, J., Søen, O., Anesen, J. S., 999. Epemental Stuy Relatve, Tubulent Dusn, Rsø-R-6(EN), Rept Rsø Natnal abaty, Rskle, Denmak. Mles, P. C., Wte, P. O., 996: Evaluatn the Gaussan Beam Mel Pectn DV Fnge Fels, Pceengs 9 th Intenatnal Sympsum n Applcatn ase Technlgy t Flu Mechancs, sbn, Ptugal, pp. 8 Mles, P. C., 996: Gemety the Fnge Fel Fme n the Intesectn Tw Gaussan Beams, Apple Optcs, 35, pp