Appl. Sci. Res. 30

M a r c h 1975

CONCENTRATION

FLUCTUATIONS

TURBULENT

IN A

JET

G. OOMS* and M. WICKS Shell Development Company, Houston, Texas 77001, U.S.A.

Abstract U s i n g S p a l d i n g ' s m o d e l of t u r b u l e n c e in a t u r b u l e n t s h e a r flow, we h a v e c a l c u l a t e d t h e r o o t - m e a n - s q u a r e v a l u e of t h e c o n c e n t r a t i o n f l u c t u a t i o n s inside a t u r b u l e n t jet. A l t h o u g h we used t h e s a m e e q u a t i o n s a n d t h e s a m e s o l u t i o n t e c h n i q u e as Spalding, we h a v e n o t b e e n able to f i n d precisely his n u m e r i c a l r e s u l t s d e r i v e d for a j e t issuing i n t o a fluid a t r e s t w i t h t h e s a m e d e n s i t y as t h e jet. T h e d i f f e r e n c e s b e t w e e n o u r n u m e r i c a l results, S p a l d i n g ' s n u m e r i c a l r e s u l t s a n d t h e e x p e r i m e n t a l d a t a of B e c k e t , H o t t e l a n d W i l l i a m s a r e f a i r l y s m a l l o n l y if t h e i n i t i a l v a l u e s of t h e t u r b u l e n c e e n e r g y a n d t h e m i x i n g l e n g t h inside t h e j e t a n d t h e t u r b u l e n c e in t h e a m b i e n t fluid are t a k e n i n t o a c c o u n t in t h e model. F o r a t u r b u l e n t j e t issuing i n t o a t u r b u l e n t l y f l o w i n g s u r r o u n d i n g s t r e a m of d i f f e r e n t d e n s i t y , we f o u n d t h a t t h e r e l a t i v e concentration fluctuations can increase considerably. This brings out the i m p o r t a n c e of t a k i n g i n t o a c c o u n t p r o p e r t y v a r i a b l e s in a n a l y s i n g t u r b , l e n t m i x i n g processes.

§ 1. Introduction

Knowledge of the concentration fluctuations in a round turbulent jet which is injected into a turbulent co-flow of a different fluid is important in many practical problems. For instance, if the two fluids are the reactants of a certain chemical reaction, the concentration fluctuations have an influence on the rate and selectivity of the reaction. Especially if the concentration fluctuations are large, the two reactants exist mainly in small chunks, completely unmixed on a molecular scale. So the rate of reaction inside a turbulent jet is lower than predicted from the time-average con* Present adress: Koninklljke Shell-Laboratorium, Amsterdam (Shell Research B.V.) The Netherlands.

-

-

381

-

-

382

O. OOMS AND M. W I C K S

centrations. This explains, for instance, the delay in the combustion of highly reactive fuels in turbulent diffusion flames [1]. An example of the adverse effect of concentration fluctuations on the selectivity of a chemical reaction is the formation of allyl chloride in the hightemperature chlorination of propylene [2]. So for practical purposes, we should like to be able to predict the magnitude of the concentration fluctuations as a function of the relevant physical parameters. Let us consider the techniques available for the calculation of these fluctuations in a turbulent flow field. If we want to describe the velocity and concentration fluctuations without using an eddy coefficient or additional transport equations, we only have the method of Orzsag and Patterson [3]. They solve the complete Navier-Stokes equations in terms of the Fourier components of the velocity field; the only assumption is that no turbulent energy is present in wave numbers larger than a certain critical wave number. The advantage of their method is that they do not have to deal with the closure problem; it has the disadvantage, however, that due to the limited amount of available computer storage and time, calculations can only be carried out for rather low Reynolds numbers. So far their method has only been used for the calculation of velocity fluctuations in a homogeneous, isotropic flow field of restricted Reynolds number. An extension to fully developed turbulent shear flows does not seem to be possible at the moment. Another possibility is the method proposed by Deardorff [4]. He makes a numerical integration in time of the three-dimensional Navier-Stokes equations for a turbulent shear flow at very large Reynolds number. A large number of grid points are used in his method, with subgrid-scale effects simulated with an eddy coefficient or additional transport equations. So far, the method has only been used for the calculation of velocity fluctuations; concentration fluctuations were not taken into account. However, such an extension seems to be possible, although the application of the method is rather complicated due to the necessity of handling a large amount of computed data. More fundamental, perhaps, is the method of Lumley [5]. He uses seventeen transport equations, but he claims that the constants occurring in these equations have universal values. So far, however, this method has not been carefully checked against experimental data; so its validity has still to be proved.

CONCENTRATION FLUCTUATIONS IN A TURBULENT JET

383

Spalding [6] developed a model for the calculation of root-meansquare values of concentration fluctuations inside a turbulent boundary layer. He applied the model to a round turbulent jet injected into a stagnant fluid with a density equal to the density of the jet fluid. He found reasonably good agreement between the calculated concentration fluctuations and the experimental data of Becker, Hottel and Williams [7]. For the reasons mentioned above, Spalding's model seem to be the most practical at the moment. Therefore, in this paper, we will report on an attempt to repeat Spalding's original calculations, and then go beyond his work by applying the model to a round turbulent jet injected into a turbulent co-flow consisting of a fluid with a density different from that of the jet fluid. We will thus be able to determine quantitatively the influence of a turbulent co-flow and of a density difference between the jet fluid and the ambient fluid on the root-mean-square value of the concentration fluctuations inside a jet. § 2. Spalding's model of turbulence

A detailed discussion of Spalding's model of turbulence developed for the calculation of the root-mean-square value of concentration fluctuations in a turbulent boundary layer is given in [6] and in the book by Launder and Spalding [8]. Therefore, we shall only repeat the important equations here, and explain the meaning of their terms. According to Spalding, the concentration fluctuations in a round turbulent jet can be calculated with the aid of the following set of equations valid for axi-symmetrical turbulent boundary layers : 0

0

o~ (P~) + ~r- (ew) = o

8/

8/

Pu ~ x + pv Or

pU~-

-~ pV Or

(I)

1 8 (~ff

0/)

r

Or, = 0

Or

r

r af

Or r #elf ~

~ 0

(2)

(~)

384

G. OOMS A N D M. W I C K S

~k 3k pu -~x + p v ~r

1 r

f

~r

ak

~r

pkl.5 / ~u \2 = C.pk o .51t-~-r ) -- CD -- l -

1 6q(/~eff ~W)

~W ~W pu ~ + pv ~r

f

r

~r

aw

/ ~2u \2

pu

vx

@ OV

~r

(4)

r

~r

~r

pk °.5

r

ag

~r

~u ~

W

• = pkO.5

= c

%

g.

(6)

All variables occurring in the equations (1)-(6) are time-mean ones. C~; CD; C1; C9,; Ca; C~,; Cg: constants /: mass fraction of jet fluid g: mean-square value of mass fraction fluctuations of jet fluid k: local kinetic energy of turbulent motion l: length scale of local turbulent motion r: radial coordinate u: velocity in longitudinal direction v: velocity in radial direction W: property of local turbulence having dimension of frequency squared; it m a y be thought of as representing viscous dissipation, which is proportional to the mean square of the turbulence vorticity X: longitudinal coordinate f f e f f : effective viscosity p: fluid density, which is calculated from the jet fluid density pl and the ambient fluid density p2 with the aid of the mass fraction f af; ak; aW; ag: effective Schmidt number of respectively /, k, W and g. In the zone of flow establishment of the turbulent jet (Fig. 1), (1)-(6) are only valid in the turbulent boundary layer growing from

CONCENTRATION FLUCTUATIONS IN A TURBULENT JET

385

NTER SoFtF~CE dE]" AXIS

]

ESTABLISHMENT

ESTABLISHEDFLOW

Fig. I. Various regions inside a t u r b u l e n t jet.

the edge of the nozzle. In the zone of established flow, the inner surface of the turbulent boundary layer has reached the jet axis and the equations are valid everywhere inside the jet. In the zone of flow establishment, the inner and outer surface of the turbulent boundary layer are free boundaries. The variables in (1)-(6) reach their asymptotic free stream values there. In the zone of established flow, only the outer surface is a free boundary; the inner surface coincides with the jet axis and the space derivative of the variables with respect to the radial distance r vanishes there, so that no net turbulent transport occurs across this surface. It is, of course, the variable g of (6) in which we are interested most in this paper and which we want to calculate as a function of the relevant physical parameters, like the density difference between the jet fluid and the ambient fluid, the velocity of the ambient fluid, etc. The variable 1 in (4)-(6) represents the local average length scale of turbulence and can be related to k and W b y means of the following identity: 1~

.

(7)

The effective viscosity/~ef~ represents the proportionality constant between the turbulent shear stress in the fluid and the local value of the gradient of mean velocity 8u/gr. It is related to p, k and 1 in the following w a y /~err =

C~,pk°'5l.

(8)

386

G. OO1KS A N D M. W I C K S

Eq. (1) expresses the conservation of mass of the fluid, irrespective of the two different fluid components, while (2) expresses the conservation of the jet fluid alone. Chemical reaction is assumed to be absent. The third term on the left-hand side of (2) represents the transport due to turbulent diffusion. Eq. (3) is the equation of conservation of momentum in the longitudinal direction. In the derivation of this equation, it is assumed that the pressure in the ambient fluid is constant. The left-hand sides of (4)-(6) for k, W and g are similar to the left-hand sides of (2) and (3), but the right-hand sides are not zero; generation and dissipation terms appear there. The dissipation terms for k, W and g are similar: each is the product of a constant, the density P, the "turbulence frequency" kO.5/1, and the variable whose dissipation is in question. The source terms are also quite similar, apart from the second term on the right-hand side of (5): each is the product of a constant, the density p, the group k°.al and the square of the gradient of a relevant time-mean quantity. The values of the various constants of (1)-(6) given by Spalding E61 are slightly different from the values given by Launder and Spalding [8] (see table). We tried several combinations for the values of the constants inside the region of values attributed to the constants by Spalding, and Launder and Spalding, and we found the best agreement with the experimental data of Becker, Hottel and Williams [7] if the constants had the values given in the table. TABLE

Ca CD C1 C2 Ca Cg1

Our values

Spalding

Launder and Spalding

1.0 0.08 3.5 0.17 1.04 2.97

1.0 0.075 3.81 0.134 1.23 2.7

1.0 0.09 3.5 0.17 1.04 2.97

Cg. af a~ aw ag

Our values

Spalding

Launder and Spalding

0.19 0.7 1.0 1.0 0.7

0.134 0.7 1.0 1.0 0.7

0.19 0.7 0.9 0.9 0.7

Values of constants occurring in Spalding's model

§ 3. Solution procedure

Eqs. (1)-(6) have been solved with the aid of the finite difference technique proposed by Patankar and Spalding E9J. A mass-stream function ~ is introduced, which is defined by

CONCENTRATION FLUCTUATIONS IN A TURBULENT JET

~T c~r

-

put

387

(9)

Also, a dimensionless mass-stream function co is introduced, given by 0) = (~ry__ ~/yi)/(~irJE_ }[fI)

(10)

where g~I and g*E represent respectively the values of the massstream function ~u at the inner and at the outer surface of the turbulent boundary layer (Fig. 1). So a~ equals zero at the inner surface and unity at the outer surface. The advantage of using a~ as independent variable instead of r is that oJ(TE -- g~I) increases in proportion to the growing jet diameter. So if a certain distribution of ~o-points inside the jet is effective at the upstream end of the jet, then this distribution is also effective at the downstream end. With r as the independent variable this is not so, because if a certain distribution of r-points is effective at the upstream end, then this distribution will prove excessively fine and hence wasteful at the downstream end. A certain grid is chosen in the x -- co space. In the m-direction we choose 25 points inside the jet. In the x-direction we choose a step length proportional to the radius of the jet. So with increasing x the jet diameter increases along with the step length in the xdirection. The partial differential equations are formulated in a dimensionless way and then transformed into finite difference equations by integration of (1)-(6) over small control volumes associated with the individual grid points in the x -- ~o space. With the aid of these finite difference equations, it is possible to calculate the values of the relevant variables at the grid points on a certain "x = cons t a n t " line from the values of the variables at the grid points on the previous "x = constant" line, So, the integration of (1)-(6) proceeds by marching downstream, starting from the initial conditions at the exit of the nozzle. At each step forward in the x-direction, new values are ascribed to g*I and ~E, because fluid is entrained from the outer region into the jet. When a boundary coincides with a symmetry line, like the inner boundary in the zone of established flow (Fig. 1), the problem of specifying the entrainment rate across this boundary is easy: the mass transfer rate across it is zero. When the boundary is, however,

388

G. OOMS A N D M. W I C K S

adjacent to a free stream, like the outer surface of the jet or the inner surface in the zone of flow establishment, tile specification of the rate of entrainment is more difficult. In our solution procedure we have applied the proposal of Patankar and Spalding [9] for the entrainment rate across such a boundary. Tile finite difference equations are formulated in an implicit manner and solved b y means of tile algorithm for tri-diagonal matrices. This allows large forward steps to be made without instability. A Shell computer program* written in Fortran IV was developed, with tile aid of which tile calculations can be performed. As mentioned earlier, the difference between our problem and the problem studied by Spalding E6] is that we take into account a turbulent co-flow of the ambient fluid and a density difference between the jet fluid and tile ambient fluid. This means that in our problem the boundary conditions at tile outer surface of the jet are different from the boundary conditions applied b y Spalding. Also, the density of the fluid varies over tile jet's cross-section in our problem; it reaches the ambient fluid density at the outer surface. In Spalding's problem the density was uniform. As will be shown in the next section, an additional difference between our work and Spalding's study is that we incorporate the effect of tile initial turbulence inside the jet. § 4. and

Comparison between our numerical results, Spalding's numerical results the experimental d a t a o f Becker, Hottel and Williams

Spalding E6] says in his paper that theoretical predictions made with his model are in good agreement with the results of Becket, Hottel and Williams E7] for a turbulent jet of the same density as the density of tile ambient fluid and without a co-flow. To check our computer program we, therefore, compared its results with these same experiments. From our calculations we found that in the case of turbulent jet without a co-flow the theoretical predictions are determined b y the two following dimensionless groups: lo/Ro and k°'~/Uo, where lo, ko and Uo are the values of l, k and u at the exit from tile nozzle. Ro represents the exit radius of the nozzle. So the initial values of turbulence energy and mixing length in the jet affect the jet * A copy of this program is available from the authors on request.

CONCENTRATION FLUCTUATIONS IN A T U R B U L E N T J E T

389

--"--- EXPER[MENTS - OUR CALCULATIONS {I) [o/Ro = 0.055 koO.5/Uo= 0:0625 la/[o=O.O ka/ko= 0.0 (2)'1o/Ro=0.'105 koO-5/uo= 0, t875 to/lo=O.O ko/ko=O.O {5} [o/Ro = 0,515 koO'5/Uo= 0.5625 [a/[o=O.O kci/ko=O.O t.0

0.8

0.6

0,4

"C 0.2 3

I

]

20

I

I

40

I

I

60

I

I

80

I

I

t00 x/R o

Fig. 2. Concentration on the jet centre line as a function of the distance from the nozzle and the initial turbulence intensity.

properties. In Fig. 2 we have plotted the mass-fraction ( = concentration)/e of jet fluid on the center line of the jet as a function of the distance from the exit of the nozzle. The axial distance x is made dimensionless by dividing it by Ro. As can be seen, the calculated concentration decreases with increasing initial turbulence energy and mixing length. Only when lo/Ro ~ 0.1 and k°'5/Uo ~ 0.2, do we find a reasonably good agreement with the measured concentration, although close to the nozzle and at large distances from it there remain considerable deviations. Spalding does not mention in his paper the influence of the initial values of turbulence energy and mixing length. He only says that his calculated concentration is in acceptably good agreement with the measurements. We have tried to improve the agreement between our theoretical results and the experimental data by taking into account a small turbulent co-flow in the ambient fluid (of the order of one percent of the jet exit velocity). It is very well possible that such a co-flow was induced by the turbulent jet during the experiments, although it

390

G . OO?vIS A N D

M. WICKS

------ EXPERIMENTS - OUR CALCULATIONS (1) I o / R o = 0.t05 koO'5/Uo=0.1875 t a / [ o = O . O ko/ko=O.O (2) to/Ro = 0.t05 koO'5/Uo=0.1875 [ o / I o = 1.0 k o / k o = 0 . 5 (3) to/Ro = 0.105 koO'5/Uo = 0,1875 I.o/Lo = 1.0 k o / k o = 1.0 fc '1.0

-I 0.8

0.6

0,4

0.2

2 3 0

I

I 20

I

I 40

I

I 60

I

i 80

I

I "100 x/Ro

Fig. 3. Concentration on the jet centre line as a function of the distance from the nozzle and the ambient turbulence intensity. is not m e n t i o n e d in Becker, H o t t e l and Williams' paper. Indeed, we f o u n d t h a t a t u r b u l e n t co-flow reduced the concentration, b u t at the same time gave too small widths of the jet unless the t u r b u lence level was assumed to be unrealistically high. So, a t u r b u l e n t co-flow, obviously, does not improve the agreement. However, when we assumed t h a t there was no co-flow b u t still some t u r b u lence in the a m b i e n t fluid, it was found t h a t the concentration decreased and t h a t also the width of the jet changed in the right direction (as will be shown later on). An example is shown in Fig. 3, where the c o n c e n t r a t i o n is p l o t t e d as a function of the distance from the nozzle for different turbulence energy levels ka in the a m b i e n t fluid. Yet, as can be seen, it is still not possible to m a t c h the e x p e r i m e n t a l c o n c e n t r a t i o n curve precisely. In Fig. 4 we h a v e p l o t t e d the dimensionless width of the jet as a function of the distance from the nozzle. As a characteristic length for the width we have chosen the radius ro.5,f, where the concent r a t i o n is at half its centre line value. As can be seen here again,

CONCENTRATION FLUCTUATIONS IN A TURBULENT JET - - - - - - EXPERIMENTS - OUR CALCULATIONS (t)1o/Ro=0.035 kO'5/uo=O.0625 (2) 1o/Ro=0.105 kO-5/uo=O.t875 (5)lo/Ro=O.315 kO.5/uo=O.5625

391

la/[o=O.O ka/ko=O.O la/[o=O.O ka/ko=O.O [a/Io-=O.O ko/ko=O.O

r0.5,f/Ro tO

//

o

//

a

6

o

I

0

L 2O

I

t 4O

I

I 60

1

I 80

I

i tO0

x/Ro

Fig. 4. W i d t h of t h e jet as ~ function of tile distance from the nozzle and the initial t u r b u l e n c e intensity. ---EXPERIMENT - - OUR CALCULATIONS (t) Io/Ro=0.105 kO5/uo=O.1875 [e/Io:O.O ka/ko=O.O (2) Io/Ro=O. I05 koO.5/Uo=O.1875 [a/Io=LO ka/ko=0.5 (3) Lo/Ro=0.105 koO5/uo=O.t875 [a/[o=t.0 ka/ko='LO ro.5,f/Ro tC

i C

0

I

I

20

I

I

40

I

I

60

I

1

80

I

I ~00 X/RO

Fig. 5. W i d t h of t h e jet as a function of the distance f r o m t h e nozzle and t h e a m b i e n t t u rbulence intensity.

392

G. OOMS A N D M. W I C K S

the agreement between theory and experiment is not very good and there is a considerable influence of the initial turbulence energy and mixing length. The width increases with increasing k o" 0 5/Uo and lo/Ro. Spalding says in his paper that the predicted angle of spread of the jet is in good agreement with experiments. Again, we have tried to improve the agreement between our calculated and the measured results by taking into account a certain turbulence level in the ambient fluid. The results is shown in Fig. 5, where we have plotted the jet width as a function of distance for different turbulence energy levels ka in the ambient fluid. Although ambient turbulence increases the width of the jet, agreement between theory and experiment is still not perfect. In Fig. 6 we have plotted the relative concentration fluctuation intensity goo.5//e on the jet centre line as a function of the distance from the nozzle. Here again the influence of k°o'5/Uo and lo/Ro is considerable. Again the agreement between theory and experiment

----- EXPERIMENTS -OUR CALCULATIONS (t) I0/Ro=0.0:35 1',o0'5/uo=0.0625 toito;O.O ko/ko=O0 (21 Io/Ro=OJ05 koO'5/uo=O.J875 la/Io=O.O ka/ko=O0 (3) [o/Ro=0.345 koO'5/Uo=0.5625 l(l/to=O0 ko/ko = 0 0

~co.% 0.2:4

O.20

3 2

jjJJ

0.16

0.12

0.08

0.04

I 20

I 40

I

I 60

I

[

I

80 x/~ o

Fig. 6. Relative concentration fluctuation on the jet centre line as a function of the distance from the nozzle and the initial turbulence intensity.

CONCENTRATION FLUCTUATIONS IN A T U R B U L E N T J E T

- - - - EXPERIMENTS -OUR CALCULATIONS (1) [o/Ro=O.I05 kO'5/Uo=~O.1875 [o/[o=O.O (2) Io/Ro=OA05 koO'5/Uo=Od875ta/Io=l.0 (3) Io/Ro=OA05 koO'5/uo=OJ875 [0/[o=4.0 gcO'5/fc 0.24

0"0

393

ka/ke=O.O ko/ko=O.5 kg/ko= t.0

'

°°sIi, 0.04

I

20

I

I

40

I

I

60

I

1

80

r

I

100 xtR o

Fig. 7. Relative concentration fluctuation oll the jet centre line as a function of the distance from the nozzle and the ambient turbulence intensity. is not too good, although the theory seems to give a good first estimate of the concentration fluctuations. We have no explanation to offer for the essential difference between our predictions and the experimental results for x/Ro > 15, except that m a y be the experimental results of Becker, Hottel and Williams must be doubted. Spalding says in his paper that he found the value of 0.221 for g°'5//e and that this value compares well with the value of 0.222 reported by Becker, Hottel and Williams. However, he does not report the complete curve given in Fig. 6. Again we have investigated the influence of a certain turbulence level in the ambient fluid. In Fig. 7 we have plotted the relative concentration fluctuation intensity as a function of the distance for different turbulence levels in the ambient fluid. As can be seen, taking into account the ambient turbulence does not materially improve the agreement between theory and experiment. The absolute value of the root-mean-square value of the concentration fluctuations g0.5 can, of course, be found from a combination of, for instance, Fig. 2 and Fig. 6.

394

a . OOMS AND M. WICKS - - - EXPERIMENTS -OUR CALCULATIONS (1) I o/Ro=O.105 koO'5/uo=O.1875 [a/Lo=O.O (2) [o{'Ro=O.!05 " koOS/uo=O.IB75 Ia/[o=t.O

ka/ko=O.O ka/ko=0.5

u/u c 10

0.8

O.E

0.4

0.2

~,.\

0

I

0

0.5

I

I

1.0

t.5

J

2.0

2.5 r/r 0.5,u

Fig. 8. V e l o c i t y profile in a cross-section of t h e t u r b u l e n t jet.

In Fig. 8 we have plotted the similarity profile of the velocity in a cross-section of the jet. The velocity u is normalized by dividing it by the velocity ue on the centreline of the jet. The radial distance r is divided by the radius ro.5,u, where the velocity is at half its centre line value. Close to the center line there is a good agreement between theory and experiment. However, at large distances from the center line, the theoretically predicted velocity is too low. Spalding reported essentially perfect agreement between his calculated and the measured velocity profile and did not find the deviation shown in Fig. 8. Again we have tried to improve the agreement between our theoretical results and the measurements by taking into account an ambient turbulence. The result is shown in Fig. 8, where the similarity profile is plotted for the case of a turbulent ambient fluid. Indeed, ambient turbulence makes a longer tail on the computed profile at large distances from the centre line. Yet, it was not possible to match the experimental curve precisely. When we calculated the velocity profile for a turbulent flowing ambient fluid, we found that a similarity profile no longer exists,

CONCENTRATION FLUCTUATIONS IN A TURBULENT JET

395

unless the velocity of the co-flow decreases with increasing distance from the nozzle. However, then the question arises how to choose such a velocity distribution. Moreover, the additional assumption of a non-constant velocity of the co-flow would imply a nonconstant ambient pressure, so that the pressure term may no longer be ignored in the momentum-balance equation. Our conclusion is that although we used the same equations and the same solution technique as Spalding, we have not been able to match precisely his results. However, if the initial values of turbulence energy and mixing length in the jet and the turbulence in the ambient fluid are adapted, the differences between our results, Spalding's results, and the experimental data of Becker, Hottel and Williams are fairly small. So, our computer program developed after Spalding's model at least gives a reliable first estimate of the values of the relevant jet variables. § 5. I n f l u e n c e of a t u r b u l e n t l y flowing s u r r o u n d i n g s t r e a m and o f a density difference o n t h e c o n c e n t r a t i o n f l u c t u a t i o n s

We will now go beyond Spalding's theoretical results by taking into account a turbulent co-flow in the ambient fluid or a density difference between the jet fluid and the ambient fluid. As far as we know, there are no experimentM data for g°'5//e for a jet with a turbulent co-flow, so that it is not possible to check our predicted concentration fluctuations for that case. First, we investigated the influence of a turbulently co-flowing ambient fluid. We assumed that the mixing length in the ambient fluid la was the same as or ten times as large as the mixing length inside the jet at the exit from the nozzle, so la/lo = 1 or la/lo = 10. Moreover, it was assumed that the square root of the turbulence energy divided by the time-mean axial fluid velocity was the same in the ambient fluid as inside the jet at the exit from the nozzle; ka.o5~us = ko.O5/Uo = 0.1875. Of course, m a n y other combinations are possible and can be calculated equally well. But the purpose of this paper is to show the possibilities of our computer program and to give some general results, rather than to supply the results of a detailed investigation of the influence of a co-flow or of a density difference. In Fig. 9 we have plotted the relative concentration fluctuation intensity as a function of the distance from the nozzle for three different values of the co-flow ua in the ambient

396

G. OOMS AND M. WICKS gco.5/f c

//t~luo=

0A0

036

/ /

I

/ /

/

/

//"

] /

032

0.4

f

/ ua/uo = 0.2

/

ua/uo~,O.4

0.28

0.24

Ua/Uo=

0.0

O,46

0.t2

Q08

0.04

0

--Io/Ro=OA05 koO'5/uo=O.1875[aAo= t.0 kO'5/oo:OAB75 ---1o/Ro=0.t05 koO.5/Uo=O.18"P5[a/Io=tO.O kOS/ha=O.'i875 I

I

20

I

[

40

I

I

60

I

1

80

I

T

t O0 x/Ro

Fig. 9. R e l a t i v e c o n c e n t r a t i o n f l u c t u a t i o n on t h e jet centre line as a f u n c t i o n of t h e distance from tile nozzle and the a m b i e n t flow.

fluid. As can be seen, the relative concentration fluctuations increase considerably with increasing co-flow and with increasing mixing length in the ambient fluid. In Fig. 10 we have plotted the relative concentration fluctuation intensity as a function of the distance from the nozzle for different values of the ratio of the density of the ambient fluid O~ and the density of the jet fluid p1 for the case of an ambient fluid at rest. As can be seen, with increasing density of the ambient fluid the relative concentration fluctuations increase considerably, especially close to the nozzle. As a point of general interest we have also calculated the influence of a density difference on the axial distribution of time-mean concentration and of time-mean velocity. In Figs. 11 and 12 we have plotted, respectively, the mass fraction of jet fluid on the jet center line/e and the relative axial velocity on the jet center fine ue/uo as a function of the distance from the nozzle for three values of re~P1. As can be seen, the influence of a

CONCENTRATION FLUCTUATIONS IN A TURBULENT J E T

397

gcO-5/fc 0.32

,F

0,28

0.24 "p2/Pl =3

-p2/pt

0.20

o.~e I" I

/

/1

=

-P2/e~ =°,33

0.t2

0.08 Lo/Ro = 0.105 koO.5/Uo=0,t875: I.o/[o:O.O I~/ko= 0.0 0,04

0 ~ 0

20

40

60

80

I00 ×/%

Fig. I0. R e l a t i v e c o n c e n t r a t i o n f l u c t u a t i o n o n t h e j e t c e n t r e line as a f u n c t i o n of t h e d i s t a n c e f r o m t h e nozzle a n d t h e d e n s i t y difference. [O/RO = 0.105 koO'5/Uo=0.1875 la/Io ~ 0.0 ka/ko=O.O fc t,0

0.8

0.6

0.4 P2@t =0.333

O.Z

~

~

e2/pl = I

p2/p1 =3 0

0

, 20

T 40

[ 60

l 80

x/R o

Fig. 1 1. C o n c e n t r a t i o n o n t h e j e t c e n t r e line as a f u n c t i o n of t h e d i s t a n c e f r o m t h e nozzle a n d t h e d e n s i t y difference.

398

G. OOMS AND M. WICKS

[o/F~o = 0.105 koO'5/Uo= 0.1875 [ o / t o = 0.0

ko/ko = 0 . 0

Uc/Uo ~.0 r - -

0.8

0.6

~p~21pt

OA.

=

0.333

0.2

pz/pt=3 0 ~ 0

20

40

60

80 x/R o

Fig. 12. Relative axial velocity on the jet centre line as a function of the distance from the nozzle and the density difference. density difference is rather strong; the concentration and velocity decrease considerably with increasing density of the ambient fluid. § 6. Concluding remarks

From our calculations we can conclude that for the calculation of the root-mean-square value of concentration fluctuations in a turbulent jet, Spalding's method is useful and can be applied fairly easily. The agreement between theoretical predictions made by this method and experimental data is encouraging, although we found an essential difference between our predictions and the experimental results for X/Ro > 15. In our calculations we have disregarded the effect of intermittency in the boundary region of the jet. It is very well possible that for instance the calculated similarity profile of the velocity in a crosssection of the jet (see Fig. 8) would be in better agreement with the experiments if intermittency had been taken into account.

C O N C E N T R A T I O N F L U C T U A T I O N S IN A T U R B U L E N T J E T

399

A basic disadvantage of Spalding's method is that the constants occurring in (1)-(6) have been determined by adapting numerical predictions to experimental results. This means that extrapolation outside the field of experiments for which the values of the constants have been determined is somewhat uncertain. Another imperfection of Spalding's method is that turbulent transport of a certain quantity is described by means of a "gradienttype" term of this quantity. This is only justified if the region of influence of the quantity is small compared with the characteristic length scale for a change in the mean value of this quantity. Received 18 June 1974 In final form 31 October 1974

REFERENCES [I] I2] [3] E4] [5] [6] [7] [8] [9]

HAWTHORNE, W. R., D. S. WEDDEL and H. C. HOTTEL, Third Symposium on Combustion, Williams and Wilkins, Baltimore, 1959, p. 266. DYKYJ, J., P. KLUCOVSKY, J. HASPRA and I. ONDRUS, Int. Chem. Eng. 2 (1962) 57. ORSZAG, S. A. and G. S. PATTERSON, Statistical Models and Turbulence, Lecture Notes in Physics, Springer Verlag, Berlin, 1972. DEARDORFF, J. W., J. of Fluid Mech. 41 (1970) 453. LUMLEY, J. L., I.A.H.R. Int. Syrup. on Stratified Flows, Novosibirsk, I972. SPALDI~'O, D. B., Chem. Eng. Sci. 26 (1971) 95. BECKER, H. A., H. C. HOTTEL and G. C. WILLIAIvIS,J. of Fluid Mech. 30 (1967) 285. LAUNDER, B. E. and D. B. SPALDING, Mathematical Models of Turbulence, Academic Press, London and New York, 1972. PATANKAR,S. V. and D. B. SPALDING,Heat and Mass Transfer in Boundary Layers, Intertext Books, London, 1970.