Citial Condition o Flow Tansition in a Full- Developed Annulus Flow Hua-Shu Dou,, Boo Cheong hoo, and He Mann Tsai. Temase Laboatoy, National Univesity o Singapoe, Singapoe 96. Depatment o Mehanial Engineeing, National Univesity o Singapoe, Singapoe 96 tsldh@nus.edu.sg; huashudou@yahoo.om The low tansition to tubulene still lagely emains an unsolved luid dynamis poblem. The stability o paallel lows is still a eseah subjet o the tubulent tansition o studies despite muh wos in the liteatue. In pevious wos, we have shown that the low instability in wall shea lows depends on the elative magnitude o enegy gadient and the enegy loss o a given distubane. Fo pipe and plane Poiseuille lows, we have demonstated that the subitial tansition to tubulene o wall bounded paallel lows ous at a onsistent value o the enegy gadient paamete. In this study, the itial ondition o subitial tansition in annulus low is alulated o vaious adius atios. The itial low ate and itial eynolds numbe ae given o vaious adius atios. These esults have wide impliations o many luid delivey devies. eywods: Subitial tansition; Full-developed low; Annulus; Citial low ate; adius atio. INTODUCTION Annulus low passage is widely used in aeo-engines, tubomahiney and vaious hemial industial devies. The low dynamis in the annulus has signiiant inluene on the peomane and low eiieny o the devies. The low in the annulus may be lamina o tubulent depending on the behaviou o the main low and the distubane level. Geneally, the distubane level in tubomahiney is vey high so that the low is liely to be tubulene in most iumstanes. Howeve, the low may be ept lamina i the low passage is aeully designed. The detemination o the itial ondition o the low tansition in annulus is theeoe o geat inteest. Flow instability and tubulene tansition have been hallenging topis o luid dynamists o moe than a entuy. This poblem has not been solved ully owing to the many diiulties enounted [-]. Thee exist a ew theoies o the analysis o low instabilities. These ae linea stability analysis, enegy method, weely nonlinea method, seonday instability theoy, and othes [-]. Linea stability analysis was suessul o some lows suh as Taylo-Coutte low and ayleigh-benad low. But, it is ailed o some lows suh as plane Poiseuille low, pipe Poiseuille low, and plane Couette low [-]. The pipe Poiseuille low (Hagen-Poiseuille) is stable by linea stability analysis o all the eynolds numbe e. Howeve, expeiments showed that the low would beome tubulene i e (ρud/µ) exeeds a value o about. Linea stability analysis o plane paallel low gives a itial eynolds numbe e (ρ u h/µ) o 577, while expeiments show that tansition to tubulene ous at the eynolds numbe o ode [3-4]. Fo plane Couette low, the low is stable o all e om linea theoy onsideation, but the expeiments show a itial value o e o 37. The enegy method uses the eynolds-o equation and involves the integation o enegy o the entie low domain. Using the enegy method, the itial e is 88, 6 and 4 espetively o plane Poiseuille low, pipe Poiseuille low and plane Couette low []. Theeoe, the non-uniomity in using this method to pedit the low tansition is still imminent. On the othe hand, the ouene o stability is stitly a loal behaviou and the low duing the tansition is intemittent. The ist ouene o the low instability geneally taes plae in the most dangeous positions as seen in the omation o tubulene spot, the ylinde wae, and the dynami stall on the aioil with lage atta angle. Hene, a method onsideing the loal low behaviou may be the oet appoah. The weely nonlinea method [5] and the seonday instability theoy [6] seem to give bette esults than the abovementioned othe methods and an explain some phenomena; howeve thee is still disepany with expeiments. eently, Dou [7] poposed a new appoah to study low instability and tubulent tansition. It is demonstated that the plane and the pipe Poiseuille lows have a onsistent itial ondition at the subitial tansition ondition detemined by expeiments. In this pape, ollowing the poposed piniple o enegy gadient theoy, the itial ondition o subitial tansition in annulus low is alulated o vaious adius atios. The itial low ate and itial eynolds numbe ae also obtained o vaious adius atios.
ENEGY GADIENT THEOY Dou [7] poposed an enegy gadient theoy with the aim to laiy the mehanism o tansition om lamina to tubulene o wall bounded shea lows. Hee, we shall give a shot disussion /desiption o a bette undestanding o the pesent wo. In the theoy, the whole low ield is teated as an enegy ield. It is suggested that the gadient o total enegy in the tansvese dietion o the main low and the visous ition in the steamwise dietion dominate the instability phenomena and hene the low tansition o a given distubane. The enegy gadient in the tansvese dietion has the potential to ampliy a veloity distubane, while the visous ition loss in the steamwise dietion an esist and absob this distubane. The low instability o the tansition to tubulene depends on the elative magnitude o these two oles o enegy gadient ampliiation and visous ition damping to the initial distubane. It is noted that the enegy loss ( H / s ) in unit length in steamwise dietion due to visous ition is equal to the gadient o total enegy ( E / s ) o unit volume luid in steamwise dietion o pessue diven lows. As suh, a new dimensionless paamete, (the atio o the enegy gadient in the tansvese dietion to that in the steamwise dietion), is deined to haateie the stability o the base low o the ase whee thee is no wo input, E / n. (a) E / s Hee, E p + ρv + ρgξ is the total enegy o inompessible lows with ξ as the oodinate in the dietion o gavitational ield, n denotes the dietion nomal to the steamwise dietion and s denotes the steamwise dietion. Futhemoe, ρ is the luid density, g is the gavity aeleation, V is the veloity, and p is the hydodynami pessue. Fo shea diven lows (thee is wo input), the alulation o an be obtained by the atio o the enegy gadient in tansvese dietion and the enegy loss in unit length along the steamline [8], E / n. (b) H / s As suh, the paamete in Eq.() is a ield vaiable. Thus, the distibution o in the low ield and the haateistis o distubane may seve as the means to desibe i thee is a distubane ampliiation o deay in the low. It is suggested that the low instability an ist ou at the position o whih is onstued to be the most dangeous position. Thus, o a given distubane, the ouene o instability depends on the magnitude o this dimensionless paamete and the itial ondition is detemined by the imum value o in the low. Fo a given low geomety and luid popeties, when the imum o in the low ield is lage than a itial value, it is expeted that instability an ou o etain initial distubane [7]. Fo a given low, is popotional to the global eynolds numbe. A lage value o has the ability to ampliy the distubane, and vie vesa. The analysis has suggested that the tansition to tubulene is due to the enegy gadient and the distubane ampliiation[7], athe than just the linea eigenvalue instability type as stated in [3,4]. Both Teethen et al. [3] and Gossmann [4] ommented that the natue o the onset-o-tubulene mehanism in paallel shea lows must be dieent om an eigenvalue instability o linea equations o small distubane. In at, inite distubane is needed o the tubulene initiation in the ange o inite e as ound in expeiments [9]. Dou [7] demonstated that the iteion obtained has a onsistent value at the subitial ondition o tansition detemined by the expeimental data o plane Poiseuille low, pipe Poiseuille low as well as plane Couette low (see Table ). The data in Table may suggest that the subitial ondition o tansition o Poiseuille lows ous at 385 and the subitial tansition in paallel lows taes plae at a value o 37-385. This inding uthe suggests that the low instability is liely esulted om the ation o enegy gadients, and not due stitly to the eigenvalue instability o linea equations. Using enegy gadient theoy, it is also demonstated that the visous low with an inletional veloity poile is unstable o both twodimensional and axisymmetial lows []. Fo plane Poiseuille low, this said position whee > should then be the most dangeous loation o low beadown, whih has been onimed by Nishioa et al s expeiment []. Nishioa et al's [] expeiments o plane Poiseuille low showed details o the low beadown. The measued instantaneous veloity distibutions indiate that the ist osillation o the veloity ous at y/h.5~.6. Fo pipe low, in a eent study, Wedin and eswell [] showed that thee is the pesene o the "shoulde" in the veloity poile at about /.6 om thei solution o the taveling waves. They suggested that this oesponds to whee the ast steas o taveling waves eah om the wall. It an be onstued that this ind o veloity poile as obtained by simulation is simila to that o Nishioa et al's expeiments o hannel lows []. The loation o the "shoulde" is about the same as that o. Aoding to
the pesent theoy, this "shoulde" may then be intiately elated to the distibution o enegy gadient. The solution o taveling waves has been onimed by expeiments eently [3]. Flow type e expession Eigenvalue analysis, Enegy Expeiments, at e (om e method e expeiments), e Pipe Poiseuille e ρ UD / µ Stable o all e 8.5 385 Plane Poiseuille e ρ UL / µ 7696 68.7 35 389 e ρ uh / µ 577 49.6 389 Plane Couette e ρ Uh / µ Stable o all e.7 37 37 Table Compaison o the itial eynolds numbe and the enegy gadient paamete o plane Poiseuille low and pipe Poiseuille low as well as o plane Couette low [7,8]. U is the aveaged veloity, u the veloity at the midplane o the hannel, D the diamete o the pipe, h the hal-width o the hannel o plane Poiseuille low (Lh) and plane Couette low. Fo Plane Poiseuille low and pipe Poiseuille low, the ous at y/h.5774, and /.5774, espetively. Fo Plane Couette low, the ous at y/h.. VELOCITY DISTIBUTION AND FLOW ATE IN THE ANNULUS The dieential equation o the ull-developed low in an annulus (Fig.) o an inompessible Newtonian luid (negleting gavity oe) in the ylindial oodinates (,) is p u ρ + µ () whee ρ is the luid density, u is the axial veloity, p is the hydodynami pessue, and µ is the dynami visosity o the luid. Integating the above equation and using the bounday ondition at the walls o inne and oute ylindes, the veloity distibution along the adius an be obtained as [4] u p 4µ + ln / ln u + b ln ( ), (3) whee / is the adius atio (, ), and u ylinde o the pipe Poiseuille low and b ( )/ ln( / ) The low ate an be integated as [4] Q u The aveage veloity in the annulus is. p 4µ 4 ( ) ( ). Hee, u is the veloity at the axis o the u π d π. (4) ln(/ ) U π Q π ( ) u ( ) ( + ln(/ ). (5) The eynolds numbe based on aveage veloity is theeoe 3
Uρ( ) ρu e µ µ ρu ( ) ( ) ( ) ( + µ ln(/ ). (6) CALCULATION OF IN THE ANNULUS The paamete o the low in an annulus is using Eq.() du du d u ρ u / µ + d. (7) d d Intoduing Eq.(3) into Eq.(7) and simpliying, the ollowing equations o paamete an be deived, Let ρu µ e + b ln b. (8) ( / ) ρu ρ( u / ) (9) µ µ be the eynolds numbe o the vitual pipe low, Eq.(8) an be uthe expessed as b e + bln. () ( / ) When b, the above equation simpliies to the pipe low expession [4]. Equation () an be also e-witten as whee e,, () b, + b ln. () ( / ) The distibutions o (, ) minimum o (, ) / vesus / o vaious ae shown in Fig.. It is ound that thee is a imum o / along / o given. Theeoe, we now om Eq.() that thee is a imum o minimum o o given e, whih oesponds to those o, espetively. Now, we ee these values to as o whih is lage in the magnitude between the imum and minimum o them. We have om Eq.() ρu e,, µ CITICAL EYNOLDS NUMBE OF THE ANNULUS. (3) As we now om enegy gadient theoy, the dominating paamete detemining the itial ondition o instability is. Dou [7] demonstated that the itial value ( ) o at subitial ondition o wall bounded paallel lows (pessue diven lows) is a onsistent quantity o 385. Given this itial value o the subitial tansition ondition, the itial values o u, U, and e an be alulated o any adius atio o the annulus. As long as the values o these paametes ae below the itial values, the low in the annulus an be maintained as a lamina low egadless o the level o distubane. Obviously, this obsevation has signiiant impliations in engineeing. 4
At the itial ondition, om Eq(3), we have the itial value o u given as µ u. (4) ρ Thus, at the itial ondition, the itial low ate is obtained om Eq.(4) as µ Q π. (5) ρ ln(/ ) The aveage veloity at itial low ate is om Eq.(5), 4 ( ) ( ) U π The itial eynolds numbe is om Eq.(6), Q π µ ρ ( ) ( + ln(/ ). (6) e Uρ( ) ρu µ µ ( ) ( + ln(/ ). (7) At the itial ondition, given as 385, and an be ound om Eq.() (see Fig.). Thus the itial values o Q, U and e vesus the adius atio an be obtained. The itial low ate and itial eynolds numbe ae shown in Fig.3 and Fig.4 espetively. It is ound that the itial value o the e ineases with the adius atio () o the annulus when the atio is lage than a value o.8. Below.8, the itial value o the e is less than that o the Poiseuille low o iula pipe. Theeoe, we an say that the inne ylinde initiates instability o <.8 and enhane stability o >.8. Similaly, the itial value o the Q ineases with the adius atio () o the annulus when the atio is lage than a value o.. Below., the itial value o the Q is less than that o the Poiseuille low o iula pipe. The impliation is that we an hange the itial value o e by inseting an inne ylinde at the ente o a pipe and thus ontol the ouene o tubulene. In this way, tubulene low in a pipe an be aused to elaminaise in an annulus. This hange may inease the aveaged veloity in the passage, but the dag oe is not ineased signiiantly i is small. This is beause the ition oeiient o lamina low is lowe than that o a tubulent low. As above disussed, the adius atio o the inne to the oute ylindes has impotant inluene on the subitial ondition o tansition to tubulene. This inding an be applied to many industial poesses and luid tanspotations. CONCLUSIONS The itial ondition o the instability o ull-developed low in an annulus is given. The iteion is based on the enegy gadient theoy o paallel low instability. The itial low ate and itial eynolds numbe inease with the adius atio o lage adius atio annulus, but thee is a tuning point in the ange o vey small adius atio o about <.. It is lea that inne ylinde eates instability o <.; and enhane stability o >.. Following this piniple, the low tansition an be ontolled by hanging the adius atio o a given oute adius. The idea an be employed in the design o luid low devies to ontol the low status lie dag edution (by eeping to lamina low) o inease the mixing o the luid media (by hanging to tubulent low). (Blan line : pt) EFEENCES (Blan line : pt.) [] Shmid, P.J., Henningson, D. S.: Stability and tansition in shea lows, Spinge-Velag, () [] Dain, P. G., eid, W. H.: Hydodynami Stability, Cambidge Univesity Pess, nd Ed., (4) [3] Teethen, L. N., Teethen, A. E., eddy, S. C., Disoll, T. A.: Hydodynami stability without eigenvalues, Siene, Vol.6, pp.578-584, (993) [4] Gossmann, S.: The onset o shea low tubulene, eviews o moden physis, Vol.7, pp.63-68, () [5] Stuat, J.T.: Nonlinea Stability Theoy, Annu. ev. o Fluid Meh., Vol.3, pp.347-37, (97) [6] Bayly, B. J., Osag, S. A.: Hebet Th. Instability mehanism in shea-low tansition, Annu. ev. Fluid Meh., 988, : 359-39 5
[7] Dou, H.-S.: Enegy gadient theoy o hydodynami instability, Tehnial epot o National Univesity o Singapoe,. Also, pesented at The Thid Intenational Coneene on Nonlinea Siene, Singapoe, 3 June -- July, 4. http://axiv.og/abs/nlin.cd/549 [8] Dou, H.-S., hoo, B.C., Phan-Thien, N., and Yeo,.S.: Instability o plane Couette low, http://axiv.og/abs/nlin.cd/548 [9] Dabyshie, A.G., Mullin, T.: Tansition to tubulene in onstant-mass-lux pipe low, J. Fluid Meh., Vol.89, pp.83-4, (995) [] Dou, H-S.: Visous instability o inletional veloity poile, eent Advanes in Fluid Mehanis, Po. o the 4th Inte. Con. on Fluid Meh., July ~3, 4, Dalian, China; Tsinghua Univesity Pess & Spinge-Velag, pp. 76-79, (4) [] Nishioa, M., Iida, S., Ihiawa, Y.: An expeimental investigation o the stability o plane Poiseuille low, J. Fluid Meh, Vol.7, pp.73-75, (975) [] Wedin, H., eswell,..: Exat oheent stutues in pipe low: tavelling wave solutions, J. Fluid Meh., Vol.58, pp.333-37, (4) [3] Ho, B., van Doone, C.W.H., Westeweel, J., Nieuwstadt, F.T.M., Faisst, H., Ehadt, B., Wedin, H., eswell,.., Walee, F.: Expeimental obsevation o nonlinea taveling waves in tubulent pipe low, Siene, Vol.35, pp.594-598, (4) [4] Papanastasiou, T.C., Geogiou, G.C., Alexandou, A.N.: Visous luid low, CC Pess, (999).8.6.4 (/). Fig. Seth o the low in an annulus. ; -. -.4....4.6.8.5.5.75 / Fig. Funtion (/) along the adius atio / o vaious adius atios. 35 3 unstable.5 5 unstable e stable Q.5 stable 5 5.5..4.6.8 Fig.3 eynolds numbe at itial ondition o annulus low at vaious adius atio. The out adius is ept onstant...4.6.8 Fig.4 Flow ate at itial ondition o annulus low at vaious adius atio. The oute adius is ept onstant. 6