A HYBRID LENGTH SCALE SIMILARITY SOLUTION FOR SWIRLING TURBULENT JETS

Transcription

1 A HYBRID LENGTH SCALE SIMILARITY SOLUTION FOR SWIRLING TURBULENT JETS Geoffey Andews Pudue Univesity Keywods: Tubulent Jet, Similaity Solution, RANS, Aeodynamics Abstact Appoximate scaling laws fo downsteam popeties of a swiling tubulent jet ae estimated using the bounday laye appoximations to the Navie-Stokes equations. The swiling jet is found to behave similaly to an equivalent nonswiling jet fo low to modeate azimuthal velocities; in this egime the chief diffeence between swiling and non-swiling jets is an inceased speading ate and an incease in entainment of ambient fluid fo the fome when compaed to the latte. As azimuthal momentum flux inceases, the flow becomes unstable and votex beakdown occus beyond a "citical" condition. Up until the citical point, the mean velocity pofiles of this flow as a function of downsteam distance can be pedicted as a function of swil using a hybid length scale model; the pedictions of this model match the available expeimental data with easonable accuacy. Intoduction. Oveview This epot will pesent the scaling analysis of tubulent jets with swil. Section 2 simplifies the Reynolds-aveaged Navie-Stokes equations in cylindical coodinate system and simplifies them using bounday laye appoximations. Section 3 is dedicated to deiving the scaling analysis based on the assumption of an undetemined similaity solution. Unlike the analyses of pevious wok, the influence of swil is consideed and used to obtain the scaling elationships fo mean velocity pofiles. Section 4 pesents compaisons between the scaling analysis and expeimental data to veify that the analysis is easonable..2 Backgound The swiling jet is a flow of geat pactical inteest, paticulaly in the field of combustos fo gas tubine engines, whee the pesence of swil enhances tubulent mixing of fuel and ai. Howeve, the poblem is not geneally well-undestood as the pesence of azimuthal velocity complicates the familia poblem of the non-swiling jet. The combination of fee jet and otating flows pesents seveal unique phenomena. Expeiments indicate that weak to modeate levels of swil incease the speading ate of the jet, epesented by a shift in the flow s vitual oigin and inceased entainment of ambient fluid into the jet. As azimuthal velocity inceases futhe, the jet foms a peculia counte-otating coe, whee fluid seveal jet diametes downsteam moves with a evesed azimuthal motion. Beyond a cetain "citical" point, the azimuthal velocity becomes ovepoweing. The bulk swiling motion of the jet distots the votex lines of the flow, esulting in a self-induced axial motion which cause the votex lines to wap aound the jet and beak up, leading to a total collapse of the voticial stuctue of the jet..3 Pio Wok Due to the inheent complexity of both geneating and measuing a swiling jet, wok on the

2 GEOFFREY ANDREWS topic has been elatively scace; cetainly, the flow is less well-undestood than its moe familia non-swiling cousin. Chigie and Chevinsky [] pesented what is aguably the most compehensive wok on the topic, investigating the effect of swil on velocity distibution. They showed that fo low swil numbes, axial velocity distibutions took the expected Gaussian shape; howeve, at highe swil numbes a counteotating votex coe fomed esulting in evesed azimuthal flow which displaced the maximum velocity fom the jet axis; fo vey stong swil, evesed flow was measued. These tends can be seen clealy in Figue below, taken fom thei 967 pape. Fig. : Axial velocity pofiles downsteam of a swiling jet fo diffeent values of swil numbe S, fom Chigie & Chevinsky [] A seies of expeimental studies by Facciolo [2] showed that swil intoduced into the jet flow inceased both the entainment of ambient fluid and the ate of speading. This study also found that the decay of centeline velocity is significantly highe than fo an iotational jet and that it inceases with highe swil numbes. Facciolo also compaed data fom a Diect Numeical Simulation (DNS) to an ealie wok by Rose [3] which measued a swiling jet using Lase Dopple Velocimety (LDV). The two sets of data geneally ageed, indicating that the intoduction of swil leads to a faste decay of axial velocity in jet flow. In the expeimental study of Shii and Geoge [4], it was shown that at low swil numbes (appoximately S <.25), thee is no significant effect on the flow as compaed to a non-swiling votex; howeve, clea diffeences appea at highe swil numbes. Othe woks suggest that beyond a cetain citical value of swil, votex beakdown occus and the flow can no longe be chaacteized as a coheent swiling jet. Howeve, in the intemediate ange of swil which lies between the non-swiling case and votex beakdown, the jet behaves similaly to an iotational jet but with slightly diffeent scaling of velocity, width, and tubulence paametes. This scaling has not been well-chaacteized beyond qualitative obsevations made in these expeiments. Seveal investigatos have studied the poblem of votex beakdown in highly swiling jets; Ruith et al. [6] pefomed a numeical simulation which aptly demonstated the phenomenon - the flow becomes inceasingly helically distubed at highe swil Reynolds numbes, ultimately becoming fully chaotic. An expeimental investigation by Billant et al [7] studied votex beakdown in swiling jets, which explained the occuence of evesed flow. They showed that a citical quantity of swil exists, and that it is independent of Reynolds numbe and jet diamete. When the jet eaches this degee of swil, beakdown stats - a stagnation point appeas in the downsteam tubulence egion and gadually moves upsteam to each an equilibium position. Thei expeiments also veified the citical swil numbe to be in good ageement with the citeion deived in Escudie and Kelle s theoy [8] based on votex beakdown in a tube. The citical swil numbe can be pedicted by: S c J (S c ) 2 J 2 (S c ) = ( t / c ) 2 Whee S c is the citical swil numbe, J and J 2 ae Bessel functions of the fist and second kind, espectively, and t and c ae the adii of the tube and the coe votex, espectively. Fo the case of the swiling jet, t can be egaded as the adius of the jet exit. D. Ewing [9] obtained similaity solutions fo ound tubulent jets with swil. His pape showed that the axial velocity and the adial velocity ae of the ode of magnitude /x, and the ode of 2

3 A Hybid Length Scale Similaity Solution fo Swiling Tubulent Jets magnitude of azimuthal velocity is /x 2. Although his esult demonstated that the azimuthal velocity decays faste than the velocities on othe two diection, his solution did not show the influence of swil. This is the chief objective of the cuent study. 2 Govening Equations The basis fo this analysis is the Reynolds- Aveaged Navie-Stokes Equations in cylindical coodinates (x,, θ). We define the axial velocity, azimuthal velocity and adial velocity as (U,V,W), espectively. We then poceed with a Reynolds velocity decomposition, U = u + u V = v + v W = w + w whee the lowecase lettes with bas denote the mean velocity and pimed lettes ae fluctuating vaiables. Assuming incompessible flow, the density ρ should be constant; we also assume that the flow is axisymmetic and steady. Thus, ()/ t = and ()/ θ =. In addition, fo high Reynolds numbe tubulent flow, the molecula viscous tem can be neglected in compaison with the tubulent viscous tem. Based on the assumptions above, the Reynolds-aveaged Navie-Stokes equations simplify as follows: (v) + u = () v v +u v + v 2 + u v (w2 +w 2 v 2 ) = p ρ u w + v w + wv + u w = + 2 ( 2 v w ) (2) (3) u u + v u + (u v ) + u 2 = p ρ (4) Bounday laye appoximations allow us to simplify the equations futhe. We stat by nondimensionalizing the equations(*= dimensional quantity): u = u /U v = v /U w = w /U = /L x = x /L p = p /ρ U δ = δ /L <<, Choosing u = O(), = O() Assume u 2 v 2 w 2 = O(q 2 ) and q = q /U <<. Accoding to the continuity equation, Thus, (v) = u = O() L δ δ ( /δ ) ( v δ U ) = O() Fo equation (4), u =O() =O() v =O(δ) =O( δ ) >> u u v u (u v ) u 2 O() O() O(q 2 /δ) O(q 2 ) 3

4 GEOFFREY ANDREWS We can see that fo the tems to all be of the same ode of magnitude, O(q 2 ) O(δ). The tem u 2 can be neglected. Equation (4) thus becomes: ae negligible. Equation (2) be- So v v comes: u u + v u + (u v ) = p ρ Meanwhile, fo equation (2):,u v, u v v v O(δ) u v O(δ) v 2 O() u v O(δ) w 2 O() w 2 v 2 O() O() v O( δ) (5) v 2 (w2 + w 2 v 2 ) = p ρ Fo equation (3): u w O( δ) v w ( δ) v w O( δ) u w O(δ) ( 2 v w ) 2 O() Then, equation (3) becomes: ( 2 v w ) 2 = Thus, the bounday-laye appoximation povides us with the following equations: Axial Momentum: u u + v u + (u v ) = p ρ Radial Momentum: v 2 (w2 + w 2 v 2 ) = p ρ Azimuthal Momentum: 3 Scaling Analysis (6) (7) ( 2 v w ) 2 = (8) A useful metic to analyze the degee of swil pesent in a flow is the atio of azimuthal to axial momentum flux. These two quantities can be calculated by integating the axial and adial momentum bounday laye equations deived above: d dx d dx (u 2 w2 2 ) d = d dx G θ = (9) 2 (u)(w)d = d dx G θ = () The atio of these momentum fluxes nondimensionalized by the jet exit adius R gives a dimensionless swil numbe, S: S = G θ G x R The swil numbe is convenient nondimensional paamete which can descibe the degee of swil pesent in diffeent flows. Axial Velocity Fo a non-swiling jet, the centeline velocity can be expessed as a function of the axial distance downsteam though the use of a similaity function: U c (,x) = U c (x)f(η) 4

5 A Hybid Length Scale Similaity Solution fo Swiling Tubulent Jets whee η = δ Howeve, the intoduction of swil adds a second elevant length scale chaacteized by the atio of azimuthal to axial momentum flux: L = G θ G x If we assume that this swiling length scale is elevant to the scaling of the axial centeline velocity, then we can wite a geneal expession fo η as follows: η = [δ n L m ] m+n This can be substituted into the axial consevation of momentum: { d 2π } ρu 2 dθd = dx 2π 2π ρu 2 dθd = C ρu 2 c F 2 η[δ n L m ] m+n dθ[δ n L m ] m+n dη =C 2πρUc 2 [δ n L m ] m+n 2 F 2 ηdη =C Uc 2 [δ n L m ] m+n 2 =C U c [δ n L m ] m+n =C Thus, we can deduce the following about the scaling of U c : U c [δ n L m ] m+n Since fo the no-swil case, we know that the solution must collapse to U c δ, we can assume that n = and, fo the case of G θ =, m =. This gives us the following elationship: U c [δl m ] m+ () Logically, the exponent m should have some dependence on the swil numbe, as the effects of swil should be moe ponounced in a flow with stonge swil. While the swiling length scale L obviously has swil dependence too, m must be a function of the swil numbe to ensue that the swiling length scale dops out fo the nonswiling case; we assume that m(s) = fo S =. We can find this elationship by eaanging the above expession: log Radial Velocity U c =C [δl m ] m+ Uc m+ L =C δ m (U c L) m =C U c δ C U c δ =(U cl) m ( ) =mlog(u c L) C U c δ ( ) log U c δ m = C 2 log(u c L) (2) An analagous analysis pocedue can be applied to the scaling of the adial velocity. At the centeline, symmety equies that the adial velocity vanish, that is v = at =. Thus, we will use the maximum adial velocity instead of the centeline adial velocity (as was done fo the axial velocity analysis), assuming that the adial velocity at an abitay point is the poduct of the local maximum v m and a similaity function G(η). v(,x) = v m (x)g(η) Witing out the consevation of adial momentum flux using the similaity expessions fo u and v esults in the following: 2π 2πρ ρuvddθ =C uvd =C 2πρ u c Fv m G(η)η[δL m ] m+ [δl m ] m+ dη =C 2πρu c v m [δ n L m ] 2 m+n FGηdη =C u c v m [δ n L m ] 2 m+n =C 5

6 GEOFFREY ANDREWS Assuming fom above that n= and U c [δl m ] m+, we obtain the scaling of vm : Azimuthal Velocity v m [δl m ] m+ (3) The same pocedue can be followed fo azimuthal velocity. We assume that the azimuthal velocity can be expessed as the poduct of the maximum azimuthal velocity at a given point, w(x), and a similaity function H(η). w(,x) = w m (x)h(η) Intoducing u, v and η into the momentum equation (), we can simplify as follows: 2π 2πρ ρ 2 uvdθd =C 2 uvd =C 2πρ η 2 [δl m ] m+ 2 U c FW m H [δl m ] 2πρU c w m [δ n L m ] 3 m+n m+ =C FHηdη = U c w m [δ n L m ] 3 m+n =C Assuming fom above that n= and U c [δl m ] m+, we obtain the scaling of wm : w m [δl m ] 2 m+ (4) Wok by Shii [4] and othes suggests the same ode elationship between the speading ate of the asymptotic jet and the downsteam distance; as in the non-swiling case, the jet gows linealy (albeit with a diffeent ate constant). Chigie and Chevinsky suggested a linea elationship between swil numbe and jet half-angle (δ = Ax) although thei data ae not wholly consistent with this model, as seen in figue 2. Fig. 2 : Half-angle of jets with vaying degees of swil, fom Chigie & Chevinsky [] Expeimental data fom seveal authos suggests that as in the case of the non-swiling jet, jet width δ inceases linealy with downsteam distance x [2][4][9]. Hence, in the above scaling analysis, δ can be eplaced by x. Accoding to the solution, we can find that all velocity components decease as the incease of x. And when x appoaches to infinity, v decays as /x m+ 2, while u decays as /x m+. Thus, the azimuthal velocity decays significantly befoe the axial velocity - specifically, the ate of decay of azimuthal velocity is the squae of the ate of decay of axial velocity, as suggested by Ewing [9], meaning that it can be neglected at lage downsteam distances. At this point, the the flow should be almost indistinguishable fom the non-swiling jet, except fo a change in flow halfangle and vitual oigin. 4 Compaison to Expeimental Data Although detailed expeimental data on jets with vaying swil numbe ae scace in the liteatue, the wok done by Chigie and Chevinsky povides an oppotunity to evaluate the poposed scaling model. Thei data cove a ange of jets with swil numbes fom.66 to.64. Using equation 2 as deived above, it is possible to plot the swiling length scale exponent m against the swil numbe S and deduce an appoximate elationship. Doing this fo the lowe swil numbes (up to.46, beyond which swil entes the "cit- 6

7 A Hybid Length Scale Similaity Solution fo Swiling Tubulent Jets ical" ange), eveals a powe-law elationship. ( ) m = x (5) + (.574 ) m.5..5 Exponent m vs Swil Numbe Swil Numbe (S) Fig. 3 : Powe-law fit of scaling exponent m with espect to swil numbe S.. In conjunction with equation, this function fo m allows fo compaison of the axial velocity decay pofile fo vaying levels of swil. This is shown below in figue 4. Fig. 4 : Compaison of scaling law with expeimental data [] The oveall tend of the pedicted decay pofile seems qualitatively accuate, although elatively mino discepancies seem to manifest themselves at highe swil numbes. The egion in the immediate vicinity of the jet exit - pehaps -5 jet diametes - encompasses the development of the flow; that is, the flowfield is evolving to a point of self-similaity, but has yet to each it (much like a moe canonical similaity solution such as the Blasius bounday laye o the nonswiling jet). Fathe downsteam fom this egion, the flow seems to achieve self-similaity, and the pedicted decay ates of axial velocity appea to povide an accuate model fo the physical flow. Note that a highe degee of accuacy may be possible with highe-fidelity expeimental data. Accuately measuing the azimuthal jet velocity is challenging - as this velocity is an ode of magnitude o moe less than the axial velocity, it equies highly sensitive instumentation with vey pecise positioning capabilities. Indeed, a contempoay pape by Patte and Keffe [5] cited concens with the accuacy of the Chigie and Chevinsky data and thei swil numbes. Although neithe wok calculates expeimental uncetainty, it seems easonable to suspect that the likely eos incued by measuing azimuthal velocity pofiles could have led to an inaccuate calculation of swil numbe fo one o moe of the jets. Since the poposed similaity solution has a elatively stong dependence on the swil numbe, a small discepancy in this measuement could significantly change the expected scaling behaviou; in this case, moe igoous measuement of the azimuthal velocity may incease the accuacy of the similaity solution at highe swil numbes. In any case, it appeas as though the poposed scaling of the axial velocity shows an accuate tend in the elatively fa-field (whee the flow would be expected to become self-simila). Without moe expeimental data to efeence, it is difficult to put the poposed solution on a moe solid footing, but the data fom Chigie and Chevinsky suggest cause fo optimism. Using the value of m calculated fom the axial decay data, it is also possible to apply the scaling model to the azimuthal velocity. Recalling equation (3) fom above, we expect the following elation to hold: 7

8 GEOFFREY ANDREWS w m [δl m ] 2 m+ = x Given the limited data available fom Chigie and Chevinsky, we find a least-squaes linea fit given by w m = 2.4x This elationship is shown in the figue below. Azimuthal velocity (wm) Scaling Paamete [δl m ] /m+ Fig. 5 : Azimuthal velocity measuements plotted against poposed scaling paamete []. Indeed, the scaling appeas to be appoximately linea (although with so few data points it is had to daw fim conclusions). Unfotunately, the limited data fom Chigie and Chevinsky ae all that could be found in the liteatue. Regadless, it would appea that the azimuthal velocity decays an ode of magnitude moe apidly than the axial o adial velocities. 5 Conclusions Based on the available expeimental data, the bounday laye scaling analysis pesented in this wok appeas to offe a easonable desciption of the fa-field decay of swiling jets. As mentioned peviously, the scacity of detailed expeimental data is a significant limitation, but the poposed solution based on a hybid length scale compised of the jet width δ and the swiling length scale L appeas to match the available data with easonable accuacy. The decay of axial velocity is well-pedicted by the model fo low and modeate values of the swil numbe. As Chigie and Chevinsky only povide thee usable measuements fo the decay of azimuthal velocity at sub-citical swil numbes, it is difficult to daw a definite conclusion egading the validity of this scaling theoy. Howeve, given the limitations of the available data (thei scacity as well as the equipment and instumentation used to collect them), the fit between expeimental data and the scaling solution is encouaging. Oveall, the scaling analysis pesented in this epot appeas to pesent a cedible model fo the scaling of velocity pofiles in a tubulent swiling jet. As othe authos have noted, the scaling of mean axial and adial velocity pofiles is linea in x as in the case of the non-swiling jet; the key diffeences fo the flow of the swiling jet ae a highe decay constant and the obvious addition of azimuthal velocity; this decays at a highe ate than the axial and adial velocities which is detemined by the nondimensional swil numbe. The chief limitation of this theoy is the lack of expeimental data available fo igoous compaison. As noted above, seveal authos have studied the case of the swiling jet expeimentally; howeve, the diffeent methods used by each have esulted in a wide vaiety of elatively naow data sets athe than the moe compehensive measuements of flow popeties which ae necessay to validate the poposed similaity solution. Few woks in the liteatue study jets with a vaiety of swil numbes; few pesent compehensive data on the decay of each velocity component and pofiles thoughout the fa field; only one wok pesents both in enough detail fo pactical use in this setting. In addition, diffeent expeiments in the liteatue use vaying appoaches to geneate and measue swil. All of these factos geatly complicate the evaluation of this theoy, and futhe expeimental wok o a seies of DNS computations could geatly elucidate the emaining uncetainty suounding the question of the swiling jet. Nonetheless, based on the available data, the hybid length scale similaity solution appeas to be a elatively accuate model fo descibing the scaling of swiling tubulent jets in the mid- to fa-field domain. 8

9 A Hybid Length Scale Similaity Solution fo Swiling Tubulent Jets 6 Acknowledgements The autho would like to thank Pofesso Geg Blaisdell fo his guidance duing the couse of this wok, Pofesso Jonathan Poggie fo being a suppotive adviso and mento, and fellow student Ang Li fo his unpublished contibutions to the poject. The autho would also like to expess his gatitude to the Ameican Institute of Aeonautics and Astonautics (AIAA) fo choosing this pape fo submission to ICAS Refeences Refeences [] Chigie N and Chevinsky A. Expeimental Investigation of Swiling Votex Motion in Jets, Jounal of Applied Mechanics, Vol. 34, No. 2, pp , 967. [2] Facciolo L, Tillmak N, Talamelli, A. and Henik, A.P. A Study of Swiling Tubulent Pipe and Jet Flows, Physics of Fluids, Vol.9, No.3, pp.35-5, 27 [3] Rose, W. A Swiling Round Tubulent Jet: - Mean-Flow Measuements, Jounal of Applied Mechanics 29.4 pp , 962. [4] Shii A, Geoge W, and Naughton W. An Expeimental Study of the Fa-Field of Incompessible Swiling Jets, AIAA Jounal, Vol. 46, No, 8. pp , 26. [5] Patte B and Keffe J. The Swiling Tubulent Jet, Jounal of Basic Engineeing, 94.4, pp , 972. ASME Pape 72-FE-8. [6] Ruith M, Chen P, Meibug E, and Maxwothy T. Thee-dimensional votex beakdown in swiling jets and wakes: diect numeical simulation, Jounal of Fluid Mechanics Vol. 486, pp , 23. [7] Billant P, Chomaz J, and Heue P. Expeimental Study of Votex Beakdown in Swiling Jets, Jounal of Fluid Mechanics, Vol. 376 pp , 998. [8] Escudie M and Kelle J. Votex Beakdown: a Two-Stage Tansition, Bown Bovei Reseach Cente, Baden, Switzeland, 983. [9] Ewing D. Decay of Round Tubulent Jets with Swil, Fouth Intenational Symposium on Engineeing Tubulence Modelling and Expeiments, pp , Ajaccio, Cosica, 999. [] Batchelo G. Steady Axisymmetic Flow with Swil, An Intoduction to Fluid Dynamics, Cambidge Univesity Pess, Cambidge, Contact Autho Addess geoffeymgandews@gmail.com Copyight Statement The authos confim that they, and/o thei company o oganization, hold copyight on all of the oiginal mateial included in this pape. The authos also confim that they have obtained pemission, fom the copyight holde of any thid paty mateial included in this pape, to publish it as pat of thei pape. The authos confim that they give pemission, o have obtained pemission fom the copyight holde of this pape, fo the publication and distibution of this pape as pat of the ICAS poceedings o as individual off-pints fom the poceedings. 9