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Mixing length and two-equation turbulence model: K. Morgan et al .

kinetic energ. I

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Mixing length and two-equation turbulence m odel: K Morgan et al .

in which the ith element of fi is

Bi [

u.l (13)In the formation of the non-linear equation (12) all

terms except those containing surface integrals areretained on the left hand side. Jt js immedja1ely

apparent that a suitable iterative technique is necessaryto obtain a solution to the matrix equation. Themethod adopted by the authors is to replace termssuch as U(aU/ax), where U is the value at the nthiteration, by Q du/Zxj, where an overbar denotes aquantity evaluated at the (n - 1)th iteration. Therequired nth iteration value of pT is given byC2s2((8D/&)l. The process is usually started byassuming zero velocity within the solution domain,although this is not a rigid requirement.

Results

The geometry of the solution domain considered

and the boundary conditions employed are illustratedin Figure 2 A ~~ypic~~%nik &x-w_~< dkc~c..is&iion OCthe d@mdn u%ng 52 e>emen>s js also shown jn Z%>Sfigure.

ymmetry

Figure i Solution domam and boundary conditrons for coaxialjet flow

In the initial region of the jet a potential core ofconstran? V&S&~ V> ex&s. T&e cak&a%& a&9velocity profiles at various downstream distances inthis initial region are displayed in Figure 3 for the caseUJU, = 0.25. This figure also shows the experimentalresults obtained by Abramovicht7 for an axisymmetricsubmerged jet ano jt can he seen rkiat goob agreemenfhas been obtained.

I IuI --= 0Iv

E ,& -0.1 (q-

1 -24 5 Xl Iil

AOX0

00x 0

@do*

a =x

-1.0 -CD8 -0b -09 -D;? D 02 D4 D b D B I D W

k r MAr,)

Figure 3 Axial velocity in initial region of jet

For the main region of the jet, Abramovich showsthat experimental results for the axial velocity closelyfollow the curve:

Z&+1 _ isi. (14)

rlr,12

Figure 4 Axial vetocity profile in main region for U,,/U, = 0.25

element method correctly predicts the form of the jetflow for the case when U,/UJ = 0.25. For the sameratio of velocities, the development of the axial velocityprofile with distance downstream is shown in Figure 5.

i.i

2x/d

. 5A IO. 20 50c 1000 150L 250

0.20 I 2 3 4 5 6 7 8

2 r l d

Figure 5 Calculated development of axial velocity profile forU,lUJ = 0.25

Again good agreement with the values abstracted fromthe curves of Figures 3 and 4 was obtained for U,/U,= 0.5, 0.75.

Figure 6 illustrates a comparison for UO/UJ = 0.25between the present model, the finite difference modelQ[ Launde,~ et al.? ad cxqxLmxext3 fix t v&c 4aecay ofvelbcily on tfie axis ofsymmetry with &stancedownstream. It can be seen that the finite elementmethod produces similar results to those obtained

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Mixing length and two-equation turbulence model: K. Morgan et al .

I I0 2 4 6 8

Xl

Figure 6 Variation of velocity along axis for lJ,/lJ, = 0.25. (0).experImenta?; (---), finite difference,( x), present model

from the finite difference technique and both comparefavourably with the experimental results.

The results of this section indicate that the finiteelement method can, for the particular viscosity modelchosen, be used to analyse free turbulent shear flows. Abetter comparison with experiment should result fromthe use of more sophisticated models for the turbulentviscosity8, the development of such models in thefinite element context is described below.

Improved turbulent viscosity models

The defects in the mixing length model for turbulentviscosity have been fully described by Launder andSpalding. Prandt16 and Kolmogorov were the first tosuggest how the local nature of the mixing lengthformula for evaluating pT could be improved. They

proposed the relationship:

where k is the time-averaged turbulence kinetic energyand postulated that k should be obtained by thesolution of a separate differential transport equationwhich can be derived from the Navier-Stokesequations and may be written in the form:

where ok and CD are taken to be constant for fullyturbulent flows.

If the length scale variation 1 can be specified thenthis method may be used for flows in which separationand recirculation occurs e.g. Runcha12. However, theinherent requirement for the algebraic specification ofthe length scale means that, in practice, this model isonly slightly superior to the mixing length modeldescribed previously.

In the so-called two-equation models of turbulence(references 7-l l), the length scale itself is determinedby the solution of a differential equation. Again bysuitable manipulation of the Navier-Stokes equation it

can be shown that:

a k1212 ak+G )2 ax j

+ C,k~l~ 2 + 2) - C,k3/2 17j

where g1,cr2 and C, are assumed to be constants forfully turbulent flows and C:, is allowed to vary in thevicinity of a wall.

Fully developed pipe nnd channel ,flowsThe simultaneous solution, by the finite element

method, of the conservation equations (1) and (2) andthe transport equations (16) and (17) is a difficultproblem. As a first stage in the development of such aprocess, fully developed turbulent flow in pipes andchannels was analysed. In such instances the governing

equations are simplified, since both the mean flow andthe turbulence remain the same at successive sections.

Within the above context the relevant equationsare:

(18)

C,k312 o-pl= 19)

12k12 01+ pra-c a r

+ C 12k112au 2

PH

_ car m

k3/2 = ) 20)

where a = 0 for channel flow, c( = 1 for pipe flow andin equation (20):

C:, = c, + c, l/y)q 21)

where C,, C, and q are constants for fully turbulentflows and y denotes distance from the wall.

Finite element solutionThe flow in the immediate vicinity of the wall

exhibits a transition from laminar flow at the wall tofully turbulent flow some distance away. In equations(19), (20) and (21), the quantities which have beenstated to be constant for fully turbulent flows arefound to vary with the local Reynolds number in thenear-wall region. For this reason the solution domainconsidered does not extend to the wall but terminatesinside the fully turbulent region. Following Ng andSpalding 4 the near-wall boundary conditions appliedare:

u = [~)s[2.51n~($)o5 + 5.51

k = z,/C;~I

(22)

I= o.4cg25 , J

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Mixing length and two-equat ion turbulence m odel : K. Morgan et a l .

where ~~ is the wall shear stress and is directly relatedto the applied pressure gradient.

The finite element discretisation of the solutiondomain adopted for both pipe and channel flow isshown in Figures 7a and Sa. With this discretisation,equations (18)-(20) may be written in the matrix formof equation (11) where the variable now becomes:

(23)

The solution of this matrix equation again has to beaccomplished by an iterative method, with the non-linear terms receiving similar treatment to thatdescribed in the section on the finite element solutiontechnique.

ResultsThe calculations were performed using the values

given for the constants in equations (19)-(20) by Ngand Spalding14. Figure Tb shows a comparison

b

o25L__---r500.2 0.4 O-6 0.8 I.00

2fld

C X Too

Xx

I 0.075

0.121 /*

x

0.09- x x 0.025X X X

.w.xX

/

X ..- ._._--X

O-06. -? 1 0/

XIX

Y I0 0.2 04 0.6 o-8 I.0

*r/d

Figure 7 Fully developed pipe flow, RN = 5 x 1 05. d = 1, a fmiteelement mesh and boundary conditions. b. plot of U/U,, and

/?a; ( x), present model; ( , experiment.13 c. plot of2v,/d,fi. x), present model; ( , experiment13

I-OO-

075 -

i:s 0.50.-3

o25i5o02 0.4 0.6 0.8 I.00

lh

0 100C X

x

x

X---L

i 0.075

I X 0.025X X

X

1 I I I I I0 0.2 0.4 O-6 0.8 I-O

f lh

Figure 8 Fully developed channel flow; R.V = 30,800, h = 0.5. a,finite element mesh and boundary conditions. b, plot of U/lJcL

and k/&; ( , experiment; ( x), present model. c, plot of

v,/hJp/Fx, and l/h; x), present model

between the predictions of the finite element model andLaufers 3 experimental results. This figure shows thevelocity and turbulent kinetic energy distribution forfully developed pipe flow at a Reynolds number, R,N, of5 x 10. The turbulent viscosity distribution iscompared with experiment in Figure 7c and thepredicted length scale variation is also shown in thisfigure.

Figure 8 b and c) displays the results of acorresponding calculation for fully developed flow in achannel at a Reynolds number, R,, of 30800. It can beseen that the finite element predictions are in goodagreement with experimental observations. Ng andSpalding14 obtained similar agreement by solving thetwo-dimensional form of equations (18)-(20) using the

finite difference method. This was achieved byemploying forward integration in the stream-wisedirection until the fully-developed flow stage wasreached.

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Conclusions

The present paper and previous work havedemonstrated that the finite element method, usingsimple turbulent viscosity models, can be successfullyemployed in the analysis of both free turbulent shearflows and flows of the wall-turbulence type. Asuccessful analysis of fully developed turbulent flowusing a more sophisticated viscosity model has also

been achieved.The incorporation of a two-equation turbulence

model into a finite element solution procedure for two-dimensional flows is currently being investigated.

Acknowledgements

The authors wish to thank the Science ResearchCouncil for the financial support, in the form of aResearch Studentship, which made the presentinvestigation possible.

References

1 Taylor, C. rt al. A numerical analysis of turbulent flow inpipes (In press)

234

5

6

7

89

10

II

1213

14I5

1617

18

I920

Van Driest, E. R. J. Aero. Sci. 1956, 23, 1007Forstal, W. and Shapiro, A. H. .I. Appl. Mrch. 1950, 17, 399Prandtl, L. Bericht tiber Untersuchungen zur ausgebildetenTurbulenz Ztmwr. 1925, 5, 136Launder, B. E. and Spalding, D. B. Lectures in mathematicalmodels of turbulence Academic Press, London, 1972Prandtl, L. Uber ein neues Formelsystem fur die ausgebildctcTurbulenz Nachr. Akd. Wis.s. Giittingen, 1945Kolmogorov, A. N. I-r. Aktrd. auk SSSR, Ser Phrs. 1942, 6.No. l-2, 56Chou, P. Y. Qwt. Appl. Muth. 1945, 3. 38Rotta. J. Z. PhJ,.s.. 1951, 129, 547 and 131, 51Rotta. S. Proc. AGARD Cot~f. 7urhulent Shem Flmvs, London,1971Spalding, D. B. Imperial College, Heat Transfer Section, Rep.EF/TN/AjIh. 1969Laufer, J. 1Y4C,4. Rep. 1053, 1951Lamer, J . The structure of turbulence in fully developed pipeflow NACA Rep. 1174. 1954Ng, K. H. and Spalding, D. B. Phrs. Flrrids. 1972, 13. 20Goldstein, S. (Ed). Modern developments in fluid dynamicsVol. I, Dover Publications, London, 1965Taylor, C. and Hood. P. Cwnp. trrtd F/&s, 1973, 1, 73Abramovich, G. N. The theory of turbulent jets. M.I.T. Press,1963Launder, B. E. et d. Proc. Cm/. Free Turhule~~t Shear Flows,NASA-SPJZI, 1. 1X1, 1973Wolfshtein, M. 7wn.s. A.S.M.E., J . B&C Engng, 1970. 92, 915Runchal, A. K. Imperial College, Heat Transfer Section, Rep.EF:R:OiI. 1968

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