flow and heat transfer in round vertical buoyant

8/8/2019 Flow and Heat Transfer in Round Vertical Buoyant

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Pergamonhr. J . leut .Mrr.,., lion,,~r. Vol 37. uppI. I, pp. 51-X ,994

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0017-9310(93)E01134

Flow and heat transfer in round vertical buoyantjets0. G. MARTYNENKO and V. N. KOROVKIN

Heat and Mass Transfer Institute, Byelorussian Academy of Sciences, 25 Podlesmaya, Minsk, BelarusAbstract-The application of the algebraic stress modeling method (ASM) for calculating the flow in aturbulent vertical round jet is described. The ASM method yields algebraic equations for Reynolds stresses(u~u,) and turbulent heat flux components (u:T). Transport equations are solved for the turbulentkinetic energy k, its dissipation rate E and mean square of temperature fluctuations (T). A new versionof the model has been used allowing one to account for the effect of normal Reynolds stresses on energyredistribution in the spectrum of turbulent fluctuations. To model these effects, an additional term hasbeen included into the equation for E. A new effective approach to solving the governing system of partialdifferential equations is suggested based on the introduction of mathematical variables. Examples ofnumerical calculations are presented which are compared with the available experimental data of other

authors. It has been found that they agree satisfactorily with the results of measurements.

1. INTRODUCTIONIN RECENT years, the problem of quantitative descrip-tion of turbulent jet transfer of momentum, heat andsubstance in the field of mass forces has claimed theattention of many scientists and engineers. First ofall, the interest in this problem is stimulated by thepossibility of obtaining some insight into the natureof the phenomenon and its trends by varying initialand boundary conditions and to exert a marked influ-ence on the process. By now, a great deal of exper-imental information has been accumulated and cor-related [l-l 21 on submerged jets, plumes and buoyantjet flows. This knowledge has turned out to besufficient to develop and suggest useful mathematicalmodels of the given phenomenon. The differential-parametric models and the idea of differential-alge-braic modeling that combine the relative simplicityand universality with modern computerization haveturned out to be most popular. The results of suchmodeling have been discussed in refs. [13-l 91.

At the same time, the results of applying thesemodels to solve the problem of the development ofan axisymmetric submerged jet have proved to be farfrom reality. This justified the use of other variants ofclosing model equations, but still within the confinesof the same approach [20-221.

It is the augmentation of the equation for dis-sipation rate E with a new term, which can be inter-preted as an additional contribution to productiondue to vortex stretching under the effect of averagedflow (for the plane flow geometry this term is equal tozero, since the length of averaged vortex lines remainsunknown) [20]. According to ref. [21], energy redis-tribution between different regions of the spectrum ofturbulent fluctuations is mainly caused by trans-lational strains (or by normal Reynolds stresses).Taking this fact into consideration, the authors of

ref. [2l] have also modified the equation for E. Inref. [22] another version of the (k--E) turbulelnt modelhas been suggested which combines the propertiesof the models proposed in refs. [20, 211.

The latter points to the fact that an adequate mod-eling of turbulent jet flows is still fairly uncertainand warrants further research.

Besides, recently there has been a compelling needfor developing numerical procedures permitting theprediction of mixing processes in jet flows with highefficiency and accuracy. Such methods should com-bine stability with acceptable time expenditures, sincealong with integration of continuity, momentum andenergy equations one has to solve a large number ofequations describing the transfer of turbulent charac-teristics.

2. MATHEMATICAL MODELLet us consider a round turbulent buoyant jet,

which on the nozzle cut has the radius r9, initialvelocity uO, temperature To and fluid densitypo(po < p,) and which propagates vertically upwardin an isothermal medium under the effect of theArchimedes forces and initial momentum.

The initial system of differential equations for cal-culating the steady-state turbulent flow in the jet is ofthe following form

51


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iau dT zlcy=o: u=-=-=-ay 3~ ay (6)

u,k,c, (T*) +O, T-T,,.

3. METHOD OF SOLUTIONIntegration of equations (l)-(6) after the transition

to dimensionless variablesly= u, v= :, x2, y2,

UO Ull r0 r 0

with the given initial distributions for Al,,,To, ,, q,,(T%, integration of equations (l)-(6) was earlierperformed with the help of a number of variousnumerical schemes. Usually, in the course of solution,the flow region is covered with a rectangular gridon the X, y plane with the Ax, Ay step. Next, finite-difference equations, approximating initial differentialones are solved by the method of iterations by meansof fitting. Such analysis schemes of problem (l)-(6)have a common drawback in that they require a lotof computer time, because abandonment of iterationwhen solving each of the finite-difference equationscan lead to a situation when the general iterationprocess can turn out to be nonconvergent.

In the present work a new approach has been real-ized. It consists of the introduction of the followingmathematical variables

X=X, n=(Z[WYdY)I (8)in which the system of equations (l)-(3) acquires theform

u a u _ UdKtlU UaEdUl

2K30ae 1 23+_cf-,__+_c;YK(12F E-r/&/ F 'I= '

Flow and heat transfer in round vertical buoyant jets 53

uaKae uaEa0+2---_---_-K al? al? E all aq

* UaEaK c* dU'_----fJU_0all ll ck* all

1cg UKauao c,c:Kaeau+_pp,-_---'---Fc: E aij rj c; 0 al? all

Eaoau I1 4 Eau-c,,c,cT--- f--c c*--Cc,$,--0 r?q l? F= " U UdXYEVtldU+c,jc,- ~ -c,,cII u aq KU' (9)

where the transverse averaged velocity component v. . _.and the coordmate Y are defined by the expressions


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*(, = (2(, .T z 2 1 -c,, (2 + c;(l _(.h,)- (11)( (17 c;The proposed method has two advantages. First, inthe X, q system, the infinite interval over Y passes intothe finite interval of the variance of 9 from 0 to I (inthe case of uniform initial fields I/ and 0, I = I) and,therefore, the integration domain transforms intoband 0 d q 4 I. X >, 0. This partially eliminates thedifficulties associated with the necessity to impose arti-ficial boundary conditions in the restricted com-putational domain. Second, the employment of equa-tion (8) ensures an automatic compliance with thecondition of the initial heat momentum conservationalong the jet. This eliminates the necessity to controlin all the cross section of the flow the satisfaction ofthe equality

Mixing processes in a round jet were analyzed notby the ASM method, but by a simpler (k-a) turbulencemodel with allowance made for the contribution ofirrotational strains into turbulent energy transfer inthe spectrum of fluctuations. In this case, the shearstress was calculated using the hypothesis on turbulentviscosity

where the local values of li and 8 were determinedfrom the following transport equations :

s,, = 2n pu( T- T, )r d.r.

0Next, the flow region is split into n bands, and on and the normal stresses which appeared in equation

(2) were expressed in terms of the turbulent kineticenergy as follows

(u~)-(z.~) = c,,k.Here

each of the straight lines q = 9, (i = I. 23,. , r~+ I)the derivatives over the variable yeare replaced by theirthree-point central-difference analogs ; as a result, theinitial differential equations (9), (10) transform to asystem of 5n first-order ordinary differential equations(in the calculations it was assumed that n = 50). Byassigning initial conditions for velocity. temperature,turbulent kinetic energy, its dissipation rate and forthe mean-square temperature fluctuation at the nozzlecut (X = 0) or at a certain section X* (they can bethose modeled or taken directly from experiment), theCauchy problem is obtained which is integrated bythe standard RungeeKutta method with automaticselection of step along the dimensionless coordinateX. Let us assume that the problem has been solved,i.e. that the functions U(X.g), O(X,q), K(X,q),E(X, q). and q(X, I) have been constructed. Then, tocomplete the solution, one should resort to equality(IO) which provides the transition of the obtainedfunctions into the physical plane X, Y.

c, = 0.09, rJk = I. 0, = I .3,(8 = 1.44. ciZ = 1.92.

Table 1 presents the calculated and experimentalresults for the expansion of a round jet over the mainregion [l-3].

The table reveals that MM significantly increasesthe accuracy of calculations as compared to the stan-dard model (SM). At the same time, the analysisresults showed that the values of empirical constantssuggested in ref. [21]

(10)

4. DISCUSSION OF THE CALCULATEDRESULTS

Calculations were started at the outlet nozzle scc-tion (X = 0), in which uniform profiles of I!. 0 anduniform distributions of K, El. y were assumed.

U,, = I, (I,, = I. K,, = 0.05.E,, = 0.02, q. = 0.05 (12)

and continued until the profiles became self-similar.The calculations were performed in two steps : first,

hydrodynamic and thermal characteristics of forcedflow in a round jet were investigated, and then numeri-cal simulation of the effect of free convection on theturbulence structure in a vertical flow was made.

54 0. G. MAKTYNLNKO an d V. N. KOKOWII\ :

The first stage of calculations was primarily of testcharacter, since the dynamics of the round jet devcl-opmcnt was already studied within the framework ofthe modified (k-8) turbulence model (MM) [2l].

A;14c:.u -t 1

c,,=o.o9, $=I, a,,=l,c, I = 1.44, (,? = 1.90


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Flow and heat transfer in round vertical buoyant jets 55Table 1. Calculated and measured forced round jet expendingratesReference Model

PII~ 2 3 1

[II[ 3 1P I

PredictionSMMMMSRPresentinvestigationsSMMMMMExperiment

1.92 0.1151.90 0.0981.80 0.103

1.92 0.1151.90 0.1111.80 0.0890.0860.087

do not provide a satisfactory agreement between thetheory and experiment over the whole spectrum ofcharacteristics. It was established that the decrease inthe value of c,*(crr = 1.3) down to 1.80, assumed inthe models based on transport equations for Reynoldsstresses [23], eliminates this disadvantage. However,the presence ofadditional parameters c,, c,~ (assumed,as in ref. [21], equal to 0.33 and 4.44, respectively)makes it impossible to come to a definite conclusionwith respect to the value of c,, for the modified (k-s)turbulence model.

Next, different levels of K,, and E,, were prescribedand it was established that they exert a weak influenceon the dynamics of the forced jet development, withthe exception of the region adjacent to the outlet,where the initial turbulence degree strongly affects thecharacteristics. Therefore, strictly speaking, to cor-rectly reproduce the actual situation in mathematicalmodels experimental distributions of K,, and E, shouldbe assigned. Downstream of the flow (for X > 40) theprofiles of the averaged and fluctuational motion par-ameters disclose the similarity of the form :

4 6.2 (z/o),_=__ = 0.015, Y,,,, = 0.089.x,U0 x/d, u:,(13)

which conforms to the results of laboratory inves-tigations [1, 2, 121.4.2. Buoyant roundjet

At the second stage, calculations were performedfor a vertical round jet with positive buoyancyeffluxing into a stagnant medium.

Taking into account that in the process of the ASMmethod development, the values of empiricalcoefficients constantly vary, it was assumed expedientto use three groups of model constants for the analy-sis, namelyASM (I)

I-C,--cI= 0.23, c? =0.51, c, = 3.0, c,, = 0.5,

ck = 0.25, c, = 0.18, CT = 0.13,c-r, = 1.79, c,, = 1.44, cc2 = 1.80; (14a)

ASM (2)l-c,------0.23, cz=0.51, c,=2.85, c,, =0.55,c

ck = 0.25, c,, = 0.15, c,=O.ll,cr, = 1.79, c,:, = 1.44, crz2= 1.80; (14b)

ASM (3)l-c,~ = 0.23, c2 = 0.51, c, = 3.2, c,, = 0.55,cl

Ck= 0.22, c, = 0.15, c,=O.ll,CT, = 1.79, c,, = 1.44, c,,> = 1.80. (14c)

The objective of the study was to investigate the effectof free convection on the jet flow structure. It hasbeen found that the presence of a gravitational sourceresults in a non-monotonous dependence ofy, s,,/yO,sHon x, : if in the region with the predominant effectof the forced convection (x, < 0.5)(y, J_rO @)< 1,then in the zone of the predominant effect of freeconvection (x, > 5)(~,, 5,/v0,50) > 1. This means thatfor round plumes the temperature profile is narrowerthan the velocity profile. The latter agrees with theresults of experimental study [5, lo]. Note that withinthe scope of the standard ASM model (not taking intoaccount the effect of normal Reynolds stresses onthe energy redistribution in the spectrum of turbulentfluctuations) a contrary result has been obtained, i.e.in the region of a free-convective flow the velocityprofile is narrower than the temperature profile evendespite the introduction of corrections which holdonly for the case of axial symmetry (equations (5))[15, 161. Next, it has been established that ther-mogravitational convection leads to a significantincrease in the turbulent transfer level. For example,in the region of the predominance of mass forces,the Reynolds stress increases by 70%, whereas theturbulent heat flux increases by 110% as compared tosimilar characteristics in the region with x, < 0.5.

To estimate the reliability of the employed math-ematical model, the results of numerical integration(Table 2) were compared to the experimental data(Table 3) of refs. [412], in which a self-similar struc-ture of vertical round buoyant jets in uniform sur-roundings was investigated. The numerical dataobtained indicate that at F,, = 5 + 500 the studied jetflow becomes self-similar at the distance, equal,approximately, to X, z 5 for mean characteristics,and to CC,z 10 for turbulent ones. Here the behaviorof numerical solutions at the jet axis can be approxi-mated depending on X, by the known power functions

u, = A,,.x;, 8, = Aex;=, k, = A,,x:,


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0. G. MAKTYNEVW ad V. N. KOKOVW.

where the values of the found factors A,,, Ai,, A,.il,, are listed in Table 2. The table also includes thenumerical data reported by other authors [13--19.X-281 which were obtained with the use of turbulentmodels of degrees of complexity: from the simplestmodels of eddy viscosity (MEV), up to mathematicalmodels including equations for turbulent stresses andflows (MSR). On the whole, one can note a fairlysatisfactory correspondence between the experimentsand the results of the present predictions both formean characteristics, and for turbulent ones, aIthou~hthere are certain quantitative differences: the cal-culatcd values of the A, factor in the law of the meanexcess temperature attenuation were found con-sidcrably understated. For ASM(2) at the same time,the discrepancies increase (the best coincidence of thepredicted and measured results was established foithe values of model constants (14~)). The samedeficiency is also observed in calculations by the ASMmethod with the empirical corrections taken intoaccount (5) (1.5, 161. It is also worth noting that appli-cation of the model in which the equations for stresses(flows) are solved [23], only slightly improves thetheoretical prediction for the given characteristic(0, = 7.2-y, ). Next, it should bc noted that in ref.[l8] the relationship I), = I?.ls, was obtainednumerically where the values of c,) and C, wereassumed equal to 0.85 and X.2, respectively.

The relative value of the intensity of the tempera-ture fluctuations v~((7))/AT,, is less sensitive tothe variation of model constants (14). At the tlowaxis it is equal to 0.40 which corresponds to theexperiment [12]. At the same time, the form of thedistribution of ./((r))/Ai\r, differs from thatwhich is cl~ara~teristic for round plumes where,:(( T),)!AT, = J(( T),,,)/Ar,. The prohlcs ofW/(( T ))/ATC for .Y, > 5 have the dip near the jetaxis (j((T2),J:AT, = 0.43). which is however,not so large as in the region with the predominanteffect of free convection (v (( 7 ),)/AT, = 0.15,\/({ T),,)jATC = 0.25).

Recently. there has been discussions in scientificliterature regarding the value of the empirical constantc,, or R = 1 c-, , The authors of refs. [29-311 in theircalculations employed the value R = 0.56 : rck. [13.15, 161 reported that jet flows with account for massforces can be better predicted for R = 0.X. The expor-imental results of ref. [32] reveal that in jet flows withthe strong effect of free convection, the value of R ismuch lower (R z 0.25) and it is not a universalconstant. Therefore, to make the calculations of abuoyant round jet more accurate, the authors of


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Flow and heat transfer in round vertical buoyant jets 51refs. 114, 191 suggested considering the quantity Ras a function of a certain parameter characterizing aturbulent flow. In this respect, the solution of a singleequation for the dissipation rate of the quantity

E, = & ;(T)can be, apparently, a more attractive alternative (231.

At the same time, this surely, will cause new prac-tical difficulties.Accordingly, it is interesting to compare the aboveresults for (T2) with the similar value, numericallyobtained within the range of the standard ASMmodel. Therefore, a number of additional calculationswere conducted, in which the system of equations (9)was integrated, but the effect of normal stresses wasnot taken into account. The values of constants (4)were employed taking into account the empiricalrelations of the form ofequation (5). It has been foundthat the results of numerical integration at R = 0.8significantly overpredict the relative rate of flue-tuations, but for X, > 10, the maximum value ofJ((T))/AT, lies on the central line and the profilesof temperature fluctuations have the form typical forround plumes [4].

5. CONCLUSIONA modified version of an algebraic turbulence

model is presented which takes into considerationthe effect of normal Reynolds stresses on the energyredistribution in the spectrum of turbulent fluc-tuations. The modification has been carried out byaccounting additional terms in the dissipation rateequation. This version of the ASM model is appliedfor predicting the development of turbulent roundforced and buoyant vertical jets. It is shown that themodel enables one to predict these jet flows with theaccuracy sufficient for practical application. In par-ticular, the model adequately accounts for the effectof free convection on the turbulent jet transfer mech-anism and correctly describes the evolution dynamicsof turbulent momentum and heat fluxes at the distancex. and also their variation along the coordinate I.

REFERENCES

1.2.

3.

4.

5.

I. Wygnanski and H. Fiedler, Some measurements in theself-preserving jet. J. Fluid Mech. 38, 577-612 (1969).E. J. List, Mechanics of turbulent buoyant jets andplumes. In Turbulent Buoyant Jets and Pkmes (Editedby W. Rodi), pp. l-68. Pergamon Press, New York(1982).P. N. Papanicoiaou and E. J. List, Investigations ofround vertical turbulent buoyant jets, J. Fluid Mech.195,341L391 (1988).P. N. Papanicolaou and E. J. List, Statistical and spectralproperties of tracer concentration in round buoyant jet.Int. J. Hear Mass Trumf~r 30, 1059-1071 ( 1987).W. K. George. R. L. Alpert and F. Tamanini, Tur-bulence measurements in an axisymmetric buoyantplume, Int. J. Ficul Mass Transfer 20, 1155-l 154 (1977).


8/8

0. G. MAKIY~~ENKO and V. N. KOKOVKINx

h

7.

x.9.

IO.

I I.

12.

13.

14.

15.

16.

17.

IX.

V. D. Zimin and P. G. Frik. Averaged temperature tieldsin axisymmetrical turbulent streams over localized heatsources, III.. AX[I~I. .Vl~lrliSSSR, MckII. Zhidk. Gax 2,199 703 (1977).F. Ogino, H. Takeuchi, I. Kudo and T. Mizushina.Heated jet discharged vertically into amhicnts of uniformand linear temperature profiles. hrr. J. H~~LI/Mtr.s.r Tr~nz.c-1~1.23, 15x1 158X (1980).N. E. Kotsovinos, Temperature measurements in a tur-bulent round jet. Int. J. Hcur Mass Transftv 28, 771777 (1985).P. D. Bcuther, S. P. Cappand W. K. George, Momentumand temperature balance measurements in an axisym-metric turbulent plume, ASME Publication. 79-HT-42(1979).H. Nakagome and M. Hirata, The structure of ambientdiffusion in an axisymmetric thermal plume. In HearTram/k and Turhulenr Buoyum Conwction, Vol. I, pp.361 372. Washington (1977).H. Roust. C. S. Yih and H. W. Humpreys. Gravitationalconvection from a boundary source. TLIIKF4, 201 210(1952).J. C. Chen and W. Rodi. Ver/iwI Turhuknt Bu~~vrm/Jc/.r : ,4 Reck of E.~pwirncw/tr/ Du/cr. Pergamon Press.Oxford (1980).M. S. Hossain and W. Rodi. A turbulent model forbuoyant flows and its application to vertical buoyantjets. In Turbulent Jels and Plume.s (Edited by W. Rodi).pp. I21 178. Pergamon Press, New York (1982).F. Tamanini, The effect of buoyancy on the turbulentstructure of vertical round jets. Trunx. ASME, .I. HCNITrtru.sfi,r 100, 659-664 (197X).C. J. Chen and C. H. Chen, On prediction and unifiedcorrection for decay of vertical buoyant jets. Tram.ASME. J. Hcu, TranJf>r 101, 532 537 (1979).C. J. Chen and C. P. Nikitopoulos, On the near fieldcharacteristics of axisymmetric turbulent buoyant jets.ltz/. J. Heat Maw Trunsfcr 22, 245-255 (1979).A. P. Netyukhailo and Z. I. Kanevskiy, Prediction ofvertical axisymmetric buoyant jets, Irr. VUZOI~, Encr-yc,riku 5, I IO-1 I3 (1983).V. P. Prokhodko. Numerical calculation of turbulentjets and axisymmetric vertical buoyant jets, J. En.9n.yP/y.v. 47(3), 493 (1984).

I).

2 0 .

I.

22.

23.

24.

25.

26.

17.

28.

29.30.

31.

32.

J. C. 1:. Pereira and J. M. P. Rocha. Prediclion (11.no,,-isothermal turbulent free flow with an algebraic sccond-moment closure model? N~m~rric,. Me/h. Lrrmirror wu/Turhu lwrt Flon ~: Proc ~.h/l/ /ur. Couf.. Vol. 6. pp. 353 364(19X9).S. B. Pope, An explanation of the turbulent round-Jetplane jet anomaly. AfAA J. 16(3). 279 281 (lY7X).K. Hanjalic and B. E. Launder, Sensitizing the dis-sipation equation to irro~ational strains, J. Fluid, Engnq102,34 40 (1980).S. H. Sohn, D. H. Choi and M. K. Chung. Calculationof plane-of-symmetry houndary layers with a modified(li E)-model, AIAA J. 29(3). 395~400 (1991).M. R. Malin and B. A. Younis, Calculation of turbulentbuoyant plumes with a Reynolds stress and heat fluxtransport closure, Int. J. Hwt ,Mu.F,sTronsfkr 33, 22472264 (1990).0. G. Martynenko, V. N. Korovkin and Yu. A. Soko-vishin. Asymptotic investigation of turbulent buoyantjet, Wiirm$- uml Sto~lhert~acpng 21, 199-207 (19X;).Yu. A. Gostintsev, 0. 0. Yedgorov and V. V. Lazarev.Turhulmr Jcr Flo~x m lhc S/ratified .4imo.vphcrc~.Chernogolovka (1990).0. G. Martynenko, V. N. Korovkm and Yu. A. Soko-vishin. The Theory of Buownt Jets and Wc1kc.v.Izd.Nauka Tekh., Minsk (1991).0. G. Martyncnko, V. N. Korovkin and Yu. A. Soko-vishin, A swirling vertical turbulent buoyant jet. In Cur-wn/ R~~scarch irl Hea, trnd MUS.F Transf&, pp. 139--l 46.Washington (1986).G. Wilks and R. Hunt, The axisymmetric turbulentbuoyant jet& numerical solution for a scmiempiricaltheory. Nunrtjr. Hpcrr Transfer 9, 495 509 (19X6).D. B. Spalding. Concentration fluctuation in a roundturbulent free jet, Chem. Engng Sci. 26,95 107 (1971).M. R. Malin. Turbulence modeling for flow and heattransfer in jets. Ph.D. Thesis. University of London(1986).0. G. Martynenko and V. N. Korovkin, Calculation ofvertical mixing in plane algebraic model of turbulence../. Tlrwm. Sci. 1, I OX- I I 3 (1992).A. Shahbir and D. B. Taulbee. Evaluation ofturhulenccmodels for predicting buoyant flows, Truns. ISME. J.Hen/ Trtm.sfi,r 4, 945-952 (I 990).

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