Exact Eulerian Solution of the Conical Bidirectional Vortex

Transcription

1 45th AIAA/ASME/SAE/ASEE Jint Ppulsin Cnfeence & Exhibit - 5 August 9, Denve, Clad AIAA Exact Euleian Slutin f the Cnical Bidiectinal Vtex Timthy A. Babe * and Jseph Majdalani nivesity f Tennessee Space Institute, Tullahma, TN In this study, an exact Euleian slutin is deived f the bidiectinal vtex in a cnical chambe. Ou mdel is applicable t idealized epesentatins f cyclne sepaats and liquid cket engines with slwly expanding chambe css-sectins. The bulk fluid mtin is assumed t be nneactive, steady, tatinal, inviscid, and incmpessible. Ou appach is funded n the Bagg-Hawthne equatin and seeks t vecme sme f the deficiencies encunteed by Bl and Ingham (Bl, M. I. G., and Ingham, D. B., The Flw in Industial Cyclnes, Junal f Fluid Mechanics, Vl. 78, 987, pp ). Despite inevitable similaities with Bl and Ingham s mdel, us is cnstucted using a judicius famewk that cnnects the slutin t the swil numbe and the cne divegence angle. In cnsequence, a self-simila fmulatin is pduced that is independent f the cne s finite bdy length. This enables us able t chaacteize the pblem by pviding clsed-fm epesentatins f the pincipal vaiables f flw mtin. Amng the paametes f inteest, the mantle divegence angle and the maximum latitudinal velcity ae btained explicitly. The mantle cnsists f a spinning cne that sepaates the cicumfeential inflw egin (ute vtex) fm the cental utflw (inne vtex). This inteface bisects the fluid dmain at 6% f the cne s divegence half-angle. Its pecise deteminatin helps t estimate the cssflw velcity espnsible f mass tansfe, spillage as it wee, fm the ute vtex int the inne egin. Finally, esults ae illustated while vaying the cne divegence angle and spatial lcatin in bth spheical and pla cylindical cdinates. Nmenclatue A i = inlet aea a = ute injectin adius, L tan b = inne injectin adius, L tan = angula mmentum as functin f, Rsin u = stagnatin pessue head L = cne s vetical length p = pessue Q i = inlet vlumetic flw ate R = adial spheical cdinate = adial (pla) cylindical cdinate, Rsin u = velcity, u,, R u u = mean tangential (inflw) velcity W = mean axial velcity z = axial cylindical cdinate Geek = cne half-angle = mantle angle f inclinatin = tangent f half-angle, tan( ) = cnstant, csc tan( ) ln[tan( )] * Gaduate Reseach Assistant, Mechanical, Aespace and Bimedical Engineeing Depatment. Membe AIAA. H. H. Anld Chai f Excellence in Advanced Ppulsin, Mechanical, Aespace and Bimedical Engineeing Depatment. Seni Membe AIAA. Fellw ASME. Cpyight 9 by J. Majdalani. Published by the, Inc., with pemissin.

2 = density = pen aea swil numbe, a b Ai /( W) = steam functin = ati f axial t adial cdinates, z/ Subscipts and Symbls i = inlet ppety = utlet pen R,, = adial, clatitude, azimuthal cmpnent z,, = adial, axial, azimuthal cmpnent I. Intductin NTEREST in mdeling cyclnic mtins has been ecently evived, especially in ppulsive applicatins whee Iswil-diven cyclnes have becme knwn f thei elevated efficiencies and self-cling ppeties. In fact, seveal types f liquid - and hybid -9 cket engines unde develpment tday ae based n the s-called bidiectinal vtex. This bipla vtex dentes a cyclne cmpising a pai f (ute and inne) caxial, c-tating swiling steams that ae sepaated by a spinning wheel knwn as the mantle. The latte cnstitutes a tating, nntanslating shea laye alng which mass can css inwadly fm the ute, annula vtex t the inne, cental ce whee cmbustin and/ mixing can be vigusly pmted. sing cylindical cmbustin chambes, analytical mdels have been advanced by Vyas and Majdalani, Majdalani and Riensta, and Vyas, Majdalani and Chiaveini -4 f the liquid engine applicatin, and by Majdalani 5 f the hybid engine case. Cld flw expeimentatin using PIV 6-7 and numeical mdels have als been implemented unde bth cld 8 and eactive flw cnditins. 9 z Fm a histical pespective, the bidiectinal vtex cncept that is nw applied t liquid and hybid thust engines was fist implemented in industial cyclnes. In fact, ne f the ealiest analyses may be taced back t te Linden s expeimental wk n dust sepaats in the late 94s. Bth hydaulic and gaseus cyclnes wee als investigated by Kelsall and Smith, -3 espectively. Wk n cnical dust sepaats cntinues tday as dcumented in the cmpehensive studies by Peng, Hffmann and Dies, 4 Hu et al., 5 Ctes and Gil, 6 and thes. F the cnical cyclne, the ealiest theetical analysis may be attibuted t Fntein and Dijksman 7 wh nce evked semi-empiical appaches and cuve fitting t btain physically viable appximatins. 7 A me efined mdel based n the Plhausen methd was late suggested by Bl and Ingham 8 and shwn t be in fai ageement with Kelsall s measuements. A me useful appximatin f the cnical cyclne wuld late emege fm the wk f Bl and Ingham; 9 this time, they wee able t incpate ealistic bunday cnditins int thei inviscid mdel. At the utset, thei slutin was useful in epducing the veall featues f the flw simulated numeically by Hsieh and Rajamani, 3 Heksta, Deksen and Van den Akke, 3 and Deksen and Van den Akke. 3 Bl and Ingham s appach was based n the Bagg-Hawthne equatin and apppiate assumptins cncening the cnsevatin f enthalpy and angula mmentum alng inlet flw steamlines. 9 Figue. Schematic f a cnical cyclne sepaat.

3 a) b) Figue. Gemetic mdel and cdinate systems used in a) the pesent analysis and b) Bl and Ingham s. 9 Othe studies, such as thse by Zha and Abahamsn, 33 sught t demnstate the effect f the upsteam bunday cnditins in the pesence f a vtex finde. Cnsideatin was als given t the diffeences between axial and sltted injectin. While sme numeical studies expled the effect f gemety n the sepaatin efficiency and/ the pefmance f cyclne sepaats, 4-6 thes sught t analyze thei instability. -3 Cnsideing that the cnical chambe is f key elevance t bth industial cyclnes and mden cncepts f liquid and hybid engines, it is the pupse f this pape t deive an exact slutin f the bidiectinal vtex in a cnical setting. sing a judicius chice f spheical cdinates, u appach will extend and cmplement the wk f Bl and Ingham 9 by evisiting the Bagg-Hawthne equatin fm which a veifiable inviscid slutin may be btained. In a cmpanin pape, 34 the same appach will be applied t the cylindical chambe f which new exact slutins will be deived unde isentpic cnditins. Nt nly will u slutins be shwn t satisfy Eule s equatin identically, but they will als exhibit the key chaacteistic paametes, such as the swil numbe, that will pemit ecnciliatin with the exact slutins in the liteatue. II. Pblem Fmulatin A. Gemety Ou cyclnic sepaat is pesented as an idealized cne with a divegence half-angle and length L. The schematic shwn in Fig. incpates bth the divegent bdy and the nn-divegent cylindical segment temed vtex finde. Ou analysis is limited t the divegent segment f this device as the vtex finde plays the le f an utlet nzzle in a cnical thust chambe. Whethe using cylindical spheical cdinates, the igin f the efeence fame is anched at the apex f the cne ( the bttm-cente f the chambe, in the case f a cylinde). Mass additin takes place tangentially at an aveage injectin speed f and vlumetic flwate Q i. The injected steam then tuns axially, thus fming a dwndaft at an aveage axial velcity f W. This inwadly diected steam geneates the ute vtex by filling the annula egin extending fm the mantle t the wall. Inside the mantle, an inne vtex is fmed thugh which fluid is caied upwadly and ut f the chambe. In this study, we ae nt cncened with the thee-dimensinal develpment f the tangential suce int an axial steam. We assume that the flw tuning pcess is immediate. As f the ute vtex in the exit plane, it is bunded by the inne and ute adii, b and a. As shwn in Fig. a, we use a ight-handed cdinate system cnsisting f a spheical adius R, a clatitude angle, and an azimuthal angle defined psitive in the diectin f swil. B. Spheical Equatins and Assumptins F the pesent mdel, the flw can be chaacteized as (i) steady, (ii) inviscid, (iii) incmpessible, (iv) tatinal, and (v) nn-eactive. When these assumptins enfced, the cnsevatin f mass and mmentum equatins becme 3

4 u RR u sin u R R Rsin Rsin () f cntinuity, and f Eule s spheical equatins, u u R u u u p R R Rsin R R u u u u u u u ct p R R Rsin R R R u u u u u u u u Ru p ct R R Rsin R R Rsin () In cnfmance with the they f lamina swiling flws, the absence f fictin enables us t justify the use f axisymmety abut the vetical axis. This educes Eqs. ()-() int R sin u Rsin R (cntinuity) (3) u u R u u u R p R R R R (adial) (4) u u u u u ct p R R R R R (latitudinal) (5) u u u u u u ct R R R R (azimuthal) (6) with vticity being expessible by u sin er Ru e Ru Rsin R R R R e (7) The imptance f this elatinship will sn be established. C. Bunday Cnditins Given axisymmetic cnditin with espect t the azimuth, u cnical flw field can be made t satisfy tw cnditins n the steam functin, R,. By insisting that the steam functin vanishes at bth the centeline and the cnical wall (at ), we set R, R, (8) Futheme, we assume that the tangential inlet is espnsible f mass added t the chambe. We thus let Figue 3. Spheical adius and clatitude angle cespnding t inlet cnditins. 4

5 Qi u Ri, i Ai (9) whee the spheical adius and clatitude angle cespnding t inlet cnditins ae given by (see Fig. 3) i tan al; Ri L a () We als assume that the tangential injected flw is espnsible f pducing the entie flw int the annula sectin f the ute vtex (dwndaft shwn in Fig. ). Finally, we veify that mass balance between the ute, annula vtex and inne, ce vtex is maintained. By integating the slutin ve the inlet and utlet sectins, we set t cnfim that Q Q A. i i III. Pcedue A. Steamline Pjectin We begin by using Lamb s vect identity uu uu uu t tansfm the cnvective tem in the Eule s equatin, specifically, in u u p /. The mmentum equatins becmes u ; u p/ z () whee is the fluid head. Based n the steam functin R,, the adial and clatitude velcities may be expessed as u R () R sin u (3) Rsin R Cnsideing a flw displacement ds we then pject the mmentum equatin alng a steamline. This enables us t wite ds uds ds (4) The pessue head is then btained by integating Eq. (4) alng a steamline. We ecve Benulli s fm u p/ (5) whee is cnstant alng each steamline. This basic deivatin is intended t claify the igin f which aises in the Bagg-Hawthne equatin. B. Swil Velcity Based n the mmentum equatin, we gup sin and educe Eq. (6) int u d sin sin R R dt sin (6) The mateial deivative diectly leads t the tangential velcity, sin u (7) Rsin is a fm f tangential angula mmentum that is t yet t be detemined. whee C. Radial and Tangential Vticity Relatins sing the fee vtex elatin f the tangential velcity, may be cnnected t the adial vticity, R. By substituting u int Eq. (7), we etieve d R (8) R sin R sin d whee the deivative with espect t the steam functin is delibeately used in view f ( ). Next, the tangential vticity may be extacted fm the mmentum equatin. Tansfming Eq. () int scala fm, we segegate the cmpnent and ewite it as d (9) R d 5

6 Making the necessay substitutins f the adial and tangential velcities as well as the adial vticity, we ae left with d () R R sin RsinR sin d This expessin may be cnsideably simplified and eaanged int d d () Rsin R sin d d D. Bagg-Hawthne Equatin Afte inseting the tangential vticity f Eq. (7) int Eq. (), the velcities may be eliminated thugh Eqs. () -(3). The utcme is a fm f the Bagg-Hawthne equatin (BHE) in spheical cdinates. We btain d d Rsin () RR sin R R sin Rsin d d and s sin d d R sin (Bagg-Hawthne) (3) R R sin d d will be instumental t the slutin f Eq. (3). Clealy, the ppe chice f and IV. Slutin A. Axial Inlet Cnditins The flw injected tangentially alng the peiphey must tun inwadly. We theefe take W as the aveage axial velcity at enty, whee b a, b being the inne adius f the pen gap shwn in Fig. 3. The steam functin cespnding t a tp hat pfile is well knwn t be d /d( Rsin ) WRsin. Hence, W R sin a (4) whee thugh u chice the integating cnstant makes the steam functin equal t ze at the wall. The vlumetic flw ate at the inlet then equals Qi a b W (5) Nte that the inlet steam functin is simply a unifm velcity pfile in spheical cdinates. This appach enables us t intduce an inlet bunday cnditin that will disseminate thughut the chambe. B. Relatins f and In de t link and, we cnside the inlet cnditin whee the tangential velcity entes at an aveage velcity f. Equatin (7) becmes, alng the inlet sectin, Rsin ( ) (6) whee emains cnstant alng a steamline. Next we diffeentiate Eqs. (4) and (6) with espect t Rsin t btain, at the tp f the cne, d (7) d( Rsin ) A cmbinatin f these tw expessins leads t d Rsin cnst. (8) d WRsin W This elatin gants the tangential velcity the feedm t vay with the steam functin. The ttal velcity at enty is hence u ( R, ) W (see Fig. 3). Assuming a cnstant inlet velcity, we may evisit Eq. (5) and wite i i u (, ) Ri i W p / cnst d (9) d 6

7 Nte that the flw in questin is isentpic t the extent that the ttal enthalpy vaiatin educes t that f the stagnatin pessue head (Bagg and Hawthne 35 ). The substitutin f Eqs. (8) and (9) int BHE educe it int sin (3) R R sin W C. Steam Functin Repesentatin In seeking an exact slutin, we attempt sepaatin f vaiables and psit the ansatz, 9 RF (3) Substituting this fm int Eq. (3) gives ise t a secnd de ODE, specifically, d df F sin (3) d sin d W The slutin t Eq. (3) may be btained by setting F f sin W (33) which eadily pduces f csc ct ln csc ct (34) and s F Ksin K cs sin ln (35) W tan. whee As usual, we fix the steam functin at the axis f symmety and the sidewall. sing R R deduce F,,, we F (36) Bth K and K can be cllected, namely, K csc ln tan cscct csc tan ln tan W W K W At this pint, we define W K csccsc ct ln tan csc tan ln tan This enables us t cllapse Eq. (35) int sin F ln sin cs ln cscct csc W W and R R sin ln sin cs ln csc W W D. Velcities With the steam functin at hand, the adial and latitudinal velcities, and u, may be extacted. Thugh ppe diffeentiatin, we btain ln cs W (4) u ln sin R sin W T pduce the tangential velcity, we use Eq. (8) and slve f. Integatin endes (37) (38) (39) (4) 7

8 d d W ( ) (4) W W R sin a, which, when substituted int Eq. (4), yields At the inlet, Rsin W R sin a W a (43) Hence we have ( ) (44) a Wa Next, we may substitute int Eq. (7) t btain u a (45) Rsin Wa Nte that ne ecves u alng the inlet sectin whee a unifm flw pevails. Finally, using the actual steam functin inside the cne, we etieve, u a Rsin lncsc (46) Rsin Wa Cmpaed t the fee vtex slutin f Vyas and Majdalani the tangential velcity btained using this appach cntinues t etain the geneal fee vtex fm equiing invese vaiatin with the distance fm the axis f tatin ~( Rsin ). In additin, hweve, it exhibits a cucial dependence n the inlet velcity pfile and the spatially vaying steam functin. It can theefe be seen that the chaacteistics featues f this pcedue cnsist f (i) etaining the spatial dependence thugh the steam functin and (ii) accunting f a specific axial injectin pfile at enty. These imptant steps can be systematically applied t the gemetic settings as it will be shwn in the cmpanin pape by Majdalani. 34 V. Results and Discussin A. Cnical Swil Numbe Thugh the use f Eq. (5), the inlet axial velcity W may be eliminated in fav f the actual tangential flw ate int the chambe, Qi Ai W a b a b (47) This key substitutin gives ise t a mdified fm f the swil numbe that is applicable t u cnical mdel. 3 We thus define, a b L tan tan Ai Ai W (48) whee dentes the divegence angle f the mantle, and W /. Fthwith, the steam functin becmes R sin ln cs sin R ln csc (49) The cmpanin velcities can be similaly expessed as ln cs (5) u ln sin (5) a Rsin u ln csc Rsin a (5) This cmpletes u slutin in spheical cdinates. 8

9 B. Nmalizatin With the fcus being diected t a full cne, we can use the tignmetic eplacement, a Ltan. Inseting this expessin int the tangential velcity equatin yields L tan Rsin u ln csc Rsin Ltan (53) Equatin (53) can be used t guide u nmalizatin. We d s while attempting t adhee with the nmenclatue used in simila cntexts, such as thse by Majdalani and Riensta. We theefe take R ( ) ( ) R ; ; ( ) ; ( ) (54) a a a u u R u Qi Ai W ; u ; u ; Qi a b (55) a b whee a is the maximum adius f the cne. Given gemetic similaity at fixed divegence angle, L can be shwn t disappea in a judiciusly nmalized system, althugh esults can be pesented f a unit chambe length z L = 5. = a) z L b) = 45. = c) d) Figue 4. Cyclnic flw steamlines shwn f = and a) = 5, b) 3, c) 45, and d) 6. 9

10 ( L ). Nte that fm this pint fwad, all f u dimensinless vaiables ae tagged with vebas except f the angle vaiables. The dimensinless fms educe t R sin ln csc ; ( ) Rsin u (56) ln cs (57) u ln sin (58) u R sin ln csc (59) R sin The steamlines pescibed by Eq. (56) ae shwn in Fig. 4 at fu diffeent cne half-angles f 5, 3, 45, and 6 degees. In these plts, the ute and inne vtex egins ae sepaated by a bken line that cespnds t the mantle lcatin. The cntu cuves epesent lines f cnstant, thus illustating the dwndaft, bending, and updaft egins. C. Mantle Lcatin In de t detemine the mantle lcatin, we cnside the behavi f the adial velcity u R in and ut f the cyclne. Theetically the mantle is lcated whee, thus inducing the flw t switch plaity with espect t the cne apex between a dwnwad and upwad spial. Since u R is a functin f the clatitude angle, we slve f the t f and call this inclinatin angle. We find.6 with an abslute e that vaies between.66 and. deg f 6. The maximum e eaches.44 deg at 9. The ientatin f the mantle laye at 6% f the divegence angle is cnfimed by Badley and Pulling 36 in thei investigatin f hydaulic cyclnes. The cmplete functinal dependence f and n is illustated in Fig. 5. Nte that cespnds t the ight-hand scale and vaies between -4.4 and f 9. As f the mantle lcatin, bth exact and appximate epesentatins ae velaid and shwn t be gaphically indiscenible except nea an impbable angle f 9%. D. Equivalent Pla Values In the inteest f claity, we cnvet u fmulatins int thei pla cylindical equivalents. The elatin between spheical and cylindical cdinates fllws the standad tansfmatin matix, u sin u cs (6) uz cs u sin Substituting the values f the adial and latitudinal velcities, the cylindical adial and axial velcities emege as u and uz ln (6) The tangential velcity emains invaiant in bth cdinate systems. In de t fully cnvet the velcities, we substitute the elatins R z, sin R, cs z R, and tan z. We subsequently btain the fllwing asstment f dimensinless velcities, u (6) appx Figue 5. Mantle divegence angle and chaacteistic paamete vesus.

11 ln uz ln u (64) whee z / z/. In the same vein, given that cylindical cdinates can be simple t visualize, we cnvet the spheical steam functin, spheical adial velcity, and latitudinal velcity int pla fm; we cllect ln (65) u ln R (66) u ln (67) E. Radial Velcity Distibutin As mentined in Sectin V.C, the spheical adial velcity, u R, cntls the plaity f the flw. Simply, negative values imply dwnwad mtin wheeas psitive values cespnd t an updaft (Fig. ). In Fig. 4, the diectin and lcatin f the flw ae delineated. The ute vtex is defined by the egin in which, thus tanspting the fluid dwnwadly in a spialing fashin; cnvesely, the psitive u R egin within the inne vtex induces cnvectin f the spinning fluid upwadly and ut f the tp. Anthe featue that may be infeed fm u R cncens the physicality and behavi f the mantle. The cnical mantle esides at the lcatin whee. In Fig. 4, the line that demacates the ze axis clealy shws the angle f the mantle f each as deduced fm Eq. (57). The mantle inclinatin angle emains cnstant thughut the cne, which in tun pvides a cnstantly changing hizntal lcatin as the axial psitin is vetically inceased. This axial shifting is cnfimed thugh Eq. (66) and may be bseved in Fig. 6 whee u R is pltted at fu axial lcatins f z/ L.5,.5,.75, and, and f tw divegence angles f 3 and 45. These cuves help t delineate the inne and ute vtex egins in additin t the mantle expansin with eithe vetical mvement cne angle divegence. It shuld be emaked that, in the absence f fictin, the fced vtex egin that chaacteizes cyclnic ces cannt be fully established withut accunting f shea stesses. At the utset, the spheical adial velcity becmes unbunded as. In a viscus flw, ne pedicts a ce bunday laye t fm at the centeline, thus mitigating the bseved divegence in the velcity. One als expects a thin bunday laye t fm at the sidewall, in fulfillment f the n slip equiement. Given that this aticle is fcused n the cmplete pesentatin f exact Eule slutins, the appximate viscus analyses f the ce and sidewall layes ae defeed t a late study. F. Latitudinal Velcity Distibutin By inspectin f Eq. (58), the latitudinal velcity, u, is seen t be slely dependent n the clatitude angle and the cnical swil numbe. Being smewhat akin t the adial velcity f the bidiectinal vtex in a cylinde, the latitudinal velcity f the spheical slutin vanishes at bth the ce and the sidewall. These tw velcities, u (63) 5 = 3 z/l =. z/l =.4 z/l =.6 z/l =.8 z/l = 5 = 45 z/l =. z/l =.4 z/l =.6 z/l =.8 z/l = a) b) Figue 6. Radial velcity distibutin shwn at seveal axial psitins and divegence angles f a) 3 and b) 45.

12 u - - u - - z/l = a) 3 4 b) -.5 = 3 u z/l =. z/l =.4 -. z/l =.6 z/l =.8 z/l = c) -.5 = 45 u z/l =. z/l =.4 -. z/l =.6 z/l =.8 z/l = d) Figue 7. Latitudinal velcity ve a) a ange f as a functin f ; b) a ange f at the tp f the cne ( z=l); a ange f axial psitins and divegence angles f c) 3 and d) 45. and u, shae the ability t link the inne and ute vtex egins thugh mass tanspt acss the mantle. Since at, the cnnectin acss the mantle depends nly n the latitudinal velcity. The magnitude f u vaies ve a ange f cne divegence angles, as shwn in Fig. 7. Hweve, these cuves exhibit simila pfiles. This is especially visible in Figs. 7c and 7d whee the vaiatin f u is shwn at fu axial css-sectins and tw divegence angles. Eveywhee between the axis and the wall, u etains a negative value that is indicative f inwad flw twad the cne axis. In the vicinity f the mantle, a maximum u can be detemined. Inteestingly, the maximum adial velcity f the bidiectinal vtex in a cylinde als ccus within clse pximity f the mantle. Hee, the maximum u appeas at a cnstant angle given that Eq. (58) is a functin f nly. This angle may be calculated fm the deivative with espect t, namely, du d sinln tan tan (68) max One gets cs max ln tan max sec max (69) Hence f evey thee exists a cespnding max which can be btained numeically. A pactically equivalent analytical t can be expessed in piecewise fashin, 3 pln e / ; ;.3873 (.9deg) max (7) 3 pln e / ; 5 whee 4 4ln. 3 Altenatively, a simple asympttic appximatin f max may be extacted in degees and witten as: max [deg] [deg]; deg (7) As f the cssflw velcity, it cincides with the mantle lcatin and may theefe be btained thugh the simple substitutin f ; ne btains

13 max a) max max, appx u / max max u u b) css u / css appx css = Figue 8. Vaiatin with f a) the maximum latitudinal speed and b) the cssflw velcity alng with thei cespnding lci. u a) 5 = 3 z/l =. z/l =.4 z/l =.6 z/l =.8 z/l = u b) 5 = 45 z/l =. z/l =.4 z/l =.6 z/l =.8 z/l = Figue 9. Tangential velcity distibutin shwn at seveal axial psitins and divegence angles f a) 3 and b) sin ln tan tan u css (7) whee is in adian. The cssflw velcity alng the mantle pemits a cnstant supply f mass tanspt, spillage as it wee, fm the ute, annula steam t the ce egin. Bth u max / and max, including the appximate expessin given by Eq. (7), ae illustated in Fig. 8a. The cssflw velcity given in bth exact and appximate fms by Eq. (7), ae displayed side-by-side in Fig. 8b alng with the mantle lci thugh which the cssing ccus. It may be seen that the maximum latitudinal velcity mis the cssflw velcity s clsely that velaying them can peclude visual discenment. This behavi is inteesting because each f these velcities stands at a diffeent angle, as shwn in Fig. 8. G. Tangential Velcity Figue 9 illustates the behavi f the swil velcity in a pla plane shwn at fu equally spaced altitudes and tw divegence angles f a) 3, and b) 45. As it may be infeed fm the plts and cnfimed thugh Eq. (64), u diminishes with the distance fm the cne axis while beaing a weake dependence n the inlet pfile and spatial vaiatin f the steam functin. This gants the mtin added sensitivity t the inlet cnditins, especially when cmpaed t the inviscid fee vtex mdel f Vyas and Majdalani (whee u slely depends n the aveage inlet velcity). In bth mdels, hweve, the puely inviscid fm gws t unbunded levels at the ce and fails t accmmdate the velcity adheence cnditin that must be secued at the walls. This esult is unsupising, being a chaacteistic featue f mst swil dminated fictinless flws (see Bl and Ingham, 9 Havey, 37 Leibvich ). The axial velcity shwn in Fig. exhibits simila featues. Depending n the vetical distance fm the apex, u z csses the blique mantle while switching plaity. H. Pessue and Vticity Evaluatin The pessue may be diectly evaluated fm Eule s mmentum equatin. sing cylindical cdinates f ease f efeencing, we btain: 3

14 uz 5 = 3 z/l =. z/l =.4 z/l =.6 z/l =.8 z/l = uz 5 = 45 z/l =. z/l =.4 z/l =.6 z/l =.8 z/l = a) b) Figue. Axial velcity distibutin shwn at seveal axial psitins and divegence angles f a) 3 and b) p -4 z/l =. z/l =.4-6 z/l =.6 z/l =.8-8 z/l = 4 z/l =. z/l =.4 z/l =.6 z/l =.8 z/l = a) b) Figue. Radial distibutin f a) pessue efeenced t the apex and b) ttal vticity at seveal axial psitins and divegence angle f 45. z high intensity ce vticity p z acsinh( z / ) z / z / acsinh 3 3 p z z z acsinh( z / ) acsinh (74) z z Taking the nmalized p as u baseline at the apex f the cne, Eqs. (73)-(74) may be patially integated t yield, p p p, whee Figue. Isvticity lines f a divegence angle f 45. (73) 4

15 z( z z) acsinh( z / ) acsinh( z / ) p acsinh acsinh (75) Figue a illustates the behavi f p at seveal axial statins and. It is clea that the pessue vaiatin is dminated by its leading de tem, a esult that is als chaacteistic f the bidiectinal vtex in a cylinde. Nte that the pessue diffeence is negligible at the wall and lagest nea the centeline. Vticity in this pblem may als be evaluated using u. We btain (76) ( ) acsinh / acsinh ; z (77) acsinh / The adial vaiatin f the ttal vticity is displayed in Fig. b at seveal fixed lcatins. The vticity lines cnfim the duality f adial psitins that yield the same value f at given z. This may be attibuted t the tanspt f vticity alng lping steamlines. As f the magnitude f vticity, it inceases as the axis f tatin is appached, especially inside an appximately % adius. Hee t, the ve-amplificatin at the igin is caused by the absence f viscus damping. T cmpensate f the deficiencies assciated with u Eule slutin, the additin f apppiate viscus cectins have t be judiciusly cnsideed thugh the use f bunday laye teatment that tightly intetwines with asympttic analysis. With the steam functin and velcities at hand, it is hped that viscus effects will be addessed in fthcming study, in additin t the pssible fms f slutin. Numeical and expeimental investigatins ae als hped t be achieved f the pupse f veificatin and validatin. VI. Cnclusins This wk evisits the pblem aising in the cntext f a bidiectinal vtex in a cnical chambe. Immediate applicatins include industial cyclne sepaats mdified vesins f the vtex liquid and hybid cket engines. Stating with the spheical Bagg-Hawthne equatin, an exact and veifiable Eule slutin is deived that vecmes sme f the deficiencies and limitatins f pevius mathematical mdels f cnical cyclnes. Ou esults ae nt nly pesented in spheical cdinates, but als in pla cylindical fm t facilitate cssefeencing. Thugh a judicius chice f nmalizatin paametes a univesal, self-simila fmulatin is pduced that is independent f the cne s vetical dimensin. The ensuing analysis enables us t identify key chaacteistic paametes such as: (.) the mantle inclinatin at 6% f the cne s divegence half-angle; (.) the cssflw velcity alng the mantle inteface u css ; and (3.) the maximum latitudinal velcity u and its lcus. max The latte is eminiscent f the adial velcity in a ight-cylindical chambe. It is inteesting that u theetical pedictin f is fully suppted by expeimental measuements. 36 On this nte, seveal asympttic appximatins ae pvided and sme appea in a piecewise fm that depends n a cutff half-angle f. deg. Afte expessing the steam functin in pla cylindical fm, ( ln ), we ae able t deduce the fundamental expessin linking the tangential angula mmentum, u, and the steam functin. This functin plays a cental le in the Bagg-Hawthne equatin as it leads t a slutin that is capable f satisfying the pblem s physical cnstaints. It thus cmplements pevius studies such as thse by Bl and Ingham 9 and Majdalani and Riensta. In hindsight, this fm culd have been psited at the beginning f the analysis t pecipitate the slutin me apidly. Simila fms f may be substituted in seeking slutins m m t this geneal class f pblems including B, B B, B B, B, B,... Alng simila lines, a genealizatin beynd cnst may be attempted in the seach f me elabate flw mtins. Mdels m m exhibiting vaiable H, H H, H H, H, H,... and cmbinatins theef may be wthwhile t cnside. While Majdalani and Riensta have initiated the quest f altenate flwfield m epesentatins, including nnlinea elatins between vticity and, much explaty wk emains ahead. The esulting appximatins may find suitable applicatins beynd the ealm f injectin and swil diven mtins. 5

16 Acknwledgments This pject was funded by the Natinal Science Fundatin thugh gant N. CMMI-35358, D Eduad A. Misawa, Pgam Diect. Refeences Majdalani, J., and Vyas, A. B., Rtatinal Axisymmetic Mean Flw f the Vtex Injectin Hybid Rcket Engine, AIAA Pape , July 4. Knuth, W. H., Chiaveini, M. J., Saue, J. A., and Game, D. J., Slid-Fuel Regessin Rate Behavi f Vtex Hybid Rcket Engines, Junal f Ppulsin and Pwe, Vl. 8, N. 3,, pp Knuth, W. H., Chiaveini, M. J., Game, D. J., and Saue, J. A., Final Rept n Gas-Fed, Vtex Injectin Hybid Rcket Engine- a Phase II SBIR Pject, Obital Technlgical Cpatin, NASA Cntact N. NAS8-975 Rept. OTC-GS55-FR-99-, Madisn, Wiscnsin, Januay Knuth, W. H., Chiaveini, M. J., Game, D. J., and Saue, J. A., Final Rept n Vtex Cmbustin Ramjet- a Phase I SBIR Pject, Obital Technlgical Cpatin, NASA Cntact N. NAS Rept. OTC-GS75- FR-99-, June Knuth, W. H., Chiaveini, M. J., Game, D. J., and Saue, J. A., Slid-Fuel Regessin Rate and Cmbustin Behavi f Vtex Hybid Rcket Engines, AIAA Pape 99-38, July Knuth, W. H., Chiaveini, M. J., Game, D. J., Saue, J. A., St. Clai, C. P., Whitesides, R. H., and Dill, R. A., Peliminay Cmputatinal Fluid Dynamics Analysis f the Vtex Hybid Rcket Chambe and Nzzle Flwfield, AIAA Pape , July Knuth, W. H., Game, D. J., Chiaveini, M. J., and Saue, J. A., Develpment and Testing f Vtex Diven, High Regessin Rate Hybid Rcket Engines, AIAA Pape , July Knuth, W. H., Chiaveini, M. J., Game, D. J., and Saue, J. A., Expeimental Investigatin f a Vtex- Diven High-Regessin Rate Hybid Rcket Engine, AIAA Pape , July Knuth, W. H., Bemwski, P. A., Game, D. J., Majdalani, J., and Rthbaue, W. J., Gas-Fed, Vtex Injectin Hybid Rcket Engine, NASA Mashall Space Flight Cente, SBIR Phase I Final Technical Rept. NASA/MSFC Cntact NAS8-4679, Huntsville, AL, August 996. Vyas, A. B., and Majdalani, J., Exact Slutin f the Bidiectinal Vtex, AIAA Junal, Vl. 44, N., 6, pp Majdalani, J., and Riensta, S. W., On the Bidiectinal Vtex and Othe Similaity Slutins in Spheical Cdinates, Junal f Applied Mathematics and Physics (ZAMP), Vl. 58, N., 7, pp Vyas, A. B., Majdalani, J., and Chiaveini, M. J., The Bidiectinal Vtex. Pat : An Exact Inviscid Slutin, AIAA Pape 3-55, July 3. 3 Vyas, A. B., Majdalani, J., and Chiaveini, M. J., The Bidiectinal Vtex. Pat : Viscus Ce Cectins, AIAA Pape 3-553, July 3. 4 Vyas, A. B., Majdalani, J., and Chiaveini, M. J., The Bidiectinal Vtex. Pat 3: Multiple Slutins, AIAA Pape 3-554, July 3. 5 Majdalani, J., Vtex Injectin Hybid Rckets, Fundamentals f Hybid Rcket Cmbustin and Ppulsin, edited by K. Ku and M. J. Chiaveini, AIAA Pgess in Astnautics and Aenautics, Washingtn, DC, 7, pp Andesn, M. H., Valenzuela, R., Rm, C. J., Bnazza, R., and Chiaveini, M. J., Vtex Chambe Flw Field Chaacteizatin f Gelled Ppellant Cmbust Applicatins, AIAA Pape , July 3. 7 Rm, C. J., Andesn, M. H., and Chiaveini, M. J., Cld Flw Analysis f a Vtex Chambe Engine f Gelled Ppellant Cmbust Applicatins, AIAA Pape , July 4. 8 Fang, D., Majdalani, J., and Chiaveini, M. J., Simulatin f the Cld-Wall Swil Diven Cmbustin Chambe, AIAA Pape 3-555, July 3. 9 Fang, D., Majdalani, J., and Chiaveini, M. J., Ht Flw Mdel f the Vtex Cld Wall Liquid Rcket, AIAA Pape , July 4. te Linden, A. J., Investigatins int Cyclne Dust Cllects, Pceedings f the Institutin f Mechanical Enginees, Vl. 6, 949, pp Kelsall, D. F., A Study f Mtin f Slid Paticles in a Hydaulic Cyclne, Tansactins f the Institutin f Chemical Enginees, Vl. 3, 95, pp Smith, J. L., An Expeimental Study f the Vtex in the Cyclne Sepaat, Junal f Basic Engineeing- Tansactins f the ASME, 96, pp

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