Experiments in Fluids 21 (1996) 17o 18o �9 Springer-Verlag 1996

Influence of a strong density stratification of a turbulent axisymmetric plume

L. Dehmani, D.-K. Son, L. Gbahoue, F. X. Rongere

on the entrainment

17o

Abstract The effect of entrainment and the role of the N * interface during the interaction between an axisymmetric Q turbulent mass plume and a strong stratified layer are in- r vestigated. We describe mainly the characteristics of the plume: the change in the profiles of the density, the horizontal rl component of the velocity and the corresponding intensity of turbulence, the change in the entrainment co-efficient, when r'~ the plume goes through the impingement interface, assuming Re a self-similar Gaussian property of the axial velocity compon- Ri ent and of the density difference. The influence of the u stratification on the plume angle coefficient is studied, and U compared with the results related to a homogeneous environ- v ment, obtained elsewhere. Experimental correlation on the x mean entrainment coefficient in a given plume cross-section, is xv formulated.

X :~= d

X ~ List of symbols B specific buoyancy flux at the source

( = g Q ( P a i ~ - - p . o ) I p H ~ )

d diameter of the source Fr Froude num be r= U~pHdgd (Pair--PHo) (at the

source) g magnitude of gravitational acceleration I turbulence intensity K coefficient in relation with the entrainment

( = M o B ) •m characteristic length 0.75 -0.5 m mass flow rate M momentum flux 3/o specific momentum flux at the source ( = Q Uo)

N Brunt-V~iis~/lii frequency ( N2= (Pair ~ ' -g dpo~'('~dx J/]

Received: 6April 1995/Accepted: 7March 1996

L. Dehmani 1, D.-K. Son, L. Gbahoub Laboratoire d'Etudes Thermiques de l'Ecole Nationale Supbrieure de Mbcanique et d'Abrotechnique, BP. 109, F-86960 Futuroscope Cedex, France

F. X. Rongbre Centre de recherche EDF, Dbpartement TTA, 6 Quai Wafter, F-78400 Chatou, France

Correspondence to: D.-K. Son

IPresent address: Facultb des Sciences de Tunis, Dbpartement de Physique Campus Universitaire de Tunis (Tunisia)

dimensionless N ( N * = N (Em/g) ~ volumetric flow rate (at the source = U0 S) radial coordinate

1 radial coordinate at the point where fi/tTc = -

e

1 radial coordinate at the point where A~/A~c=

e

Reynolds number (at the source Reo= Uod/VHe) Richardson number (at the source Rio = Fr-ln) vertical velocity component vertical mean velocity radial velocity component vertical coordinate distance virtual origin of the plume

dimensionless vertical coordinate distance

dimensionless parametric group, defined in (Eq. (17))

Greek symbols entrainment coefficient

p fluid density dp=p-po~ density difference

P - - P a i r = - dimensionless density PIle - - Pair

Subscripts air c He m

0 V

o0

air on the axis of the flow helium relative to minimum value or maximum value value at the source corresponding to the radial velocity ambient conditions

Superscripts dimensionless value

- mean value ' fluctuation

1 Introduction This investigation is a contribution to the understanding of the behaviour of turbulent plumes and their interaction with the surrounding media. The influence of a strong density stratification on the horizontal velocity and the entrainment of

a axi-symmetric mass plume prediction, specially prevention of fire inside closed-in structures (buildings, nuclear power stations, etc.) and of atmospheric pollution (heat rejections from industrial units, towns, forest fires, etc.).

At the early stages of fire in a closed-in area, the heat input due to the fire generally results in the creation of a hot upper layer above a cooler bottom layer. The temperature in each layer is fairly uniform due to thermal buoyancy and a thin interface often arises between the two layers. At this interface the density gradient corresponding to a strong stratification is very high; this type of situation has a strong effect on the turbulence characteristics. It is also of great interest to know how is the ambient fluid entrainment by the flow just above the fire and at the interface because this entrainment determines the mass transfer and the fuel combustion and therefore the thermal power released.

The works of Morton et al. (1956), and Morton (1958), as basic references for investigations on plumes are still valid today, even with the recent progress of the measurement techniques. These authors developed the concept of entrain- ment initially introduced by Taylor (1945). At the same period this concept was also investigated by Ricou and Spalding (1961). Later, List and Imberger (1973), and Kotsovinos and List (1977) among others, carried out further investigations.

Entrainment phenoma play a determining role in the numeric modelization of free flows and their structures, turbulent or organized. Today, we are aware of the develop- ment of the integral models (Tamanini 1981; Vachon and Champion 1986), which have the advantage of being very simple and thus, with a good knowledge of the entrainment coefficient, economizes computation time. Entrainment phe- nomena remain difficult to measure and to modelize. This was the aim of Srinivasan and Angirasa (1990), Kotsovinos (1990), Bejan (1991), Fukui et al. (1991), Kapoor and ]aluria (1993).

It can be seen from this reference to the relevant literature that the variation of the entrainment of a turbulent plume from the source to a strong stratification has not been studied in detail. In this paper, an experimental investigation on this type of flow, particularly on entrainment, is reported. Experiments have been carried out for three configurations, corresponding to different conditions where the buoyancy and the stratifica- tion effects have varied. The experimental values of the horizontal velocity and the entrainment coefficient were determined.

2 Experimental arrangement The flow could be simulated by a helium-air axisymmetric jet with a weak initial density and a low initial velocity, (Dehmani (1990)). The ambient stratification was carried out by placing a 0.8 x 0.8 m x 0.3 m box at a height of about two nozzle diameters above the source. Dehmani et al. (1989) observed three zones, when using a box open at the bottom and fed by the flow (Fig. 1).

In the area near the source, theplume zone, the weak density flow is controlled by the initial conditions in a homogeneous air density ambient medium. Above, at about two nozzle diameters from the source is the neutral zone, fed by a mixture of helium and entrained air whose concentration varies with the flow strength. The density within this zone is fairly

uniform. Between the plume and the neutral zones which have different uniform densities, we find an interaction zone, i.e. the interface, where the turbulence characteristics are very much modified by a strong gradient of density.

A more detailed investigation on the density variations can be found in Dehmani et al. (1989). Here the density was measured by continuous sampling of the helium-air mixture and the velocity by laser Doppler anemometry. Three ex- perimental configurations were investigated (Table 1). A con- figuration (A) with very strong pure plume-like buoyancy, corresponds to the minimum helium flow rate available with our experimental equipment. For a better knowledge of the force of inertia, a configuration is set up (C), with the initial momentum corresponding to the maximum flow rate value available. Lastly, an intermediate configuration (B), with a strong contribution from the forces of buoyancy and of inertia, is added.

Air-Helium mixture vent

l a . . . . ~ / ; /

/ Stratification

box Mixture vent

/ ozzle

x/d

utral I

Plume zone

Interface

Poo

Fig. 1. Experimental set up and physical model. 1 glass-sight cabin; la stratification box; 2 nozzle-tube; 2a Helium reservoir

Table 1. Experimental configurations

Configuration u0 (m/s) 1000Fr0 Reo

(A) 0.06 1.16 27.31 (B) 0.19 10.54 82.43 (C) 0.45 63.12 201.71

171

172

To each configuration corresponds a particular ambient stratification. The vertical dimensionless ambient density distribution, Fig. 2a, shows a zone with rapid variation of ~ , between two zones where this parameter is fairly constant.

To characterize this stratification, the Brunt-V~iisfil~i fre- quency, N 2, is introduced and calculated from the density gradient. The stability of the ambient is characterized by the sign of the coefficient N 2, which is negative if the stratification is unstable, zero if it is neutral and positive it is stable. The stratification is neutral inside the two areas with a constant density and stable in the interaction zone where the ambient density variations are linear. The coefficient N 2, varies with the configurations: equal to 10.89 s -2 for configuration (A), 57.70 s-2 for configuration (C), while configuration (B) reaches 31.18 s 2. These values of the stratification coefficient are in a range two series greater than certain of the values proposed in the literature; for instance, Chen and Rodi (1980) found that N z ranged between 0.2 s -2 and 0.5s z for salt water and be- tween 0.2 s -2 and 0.3 s 2 for hot air.

The density varies very significantly in the narrow zone of the interaction whereas in the neutral zone it remains constant.

The change of the dimensionless density difference between the flow axis and the ambient is shown in Fig. 2b. First, ( ~ --'~) decreases rapidly, implying a significant decrease in the buoyancy forces, becomes equal to zero in the neutral zone

where the flow progress is due solely to the forces of inertia. A study of the three above mentioned configurations shows that by increasing the injected flow we have stronger buoyancy forces characterized by a greater difference ( f ~ - f). The variations of the dimensionless governing parameters, the local Froude and Reynolds numbers are given in Fig. 3. These parameters were calculated from the centreline velocity and the reference length rl, except at ~--0.05 where we chose instead the radius of the nozzle because at this point the Gaussian profile was not well established.

Inside the area near the source till ~ = 1.5, Fig. 3a shows that the Reynolds number can be estimated by a linear increase, while greater values are obtained for configuration (B). This increase of the influence of the forces of inertia is probably due to a smaller air entrainment, as we shall see below. Above this area, the ambient stratification is stable; the local Reynolds number ceases to increase. The variations of the Froude number Fr show (Fig. 3b), that the balance between the forces of inertia and the forces of buoyancy (Fr= 1) is reached faster for about ~ = 1 by configuration (C) than by configurations (A) and (B), because of the high value of the centreline velocity due to a big disparity of density between flow and ambient. Inside the interaction zone, Fr continues to increase, showing that the flow leads to a jet configuration.

Above this zone the Froude number increases significantly. However, this parameter has not a clear physical meaning when the density difference between the flow and the ambient tends towards to zero.

8

I

1 2 3 4 5 6 7 x /d

i

~g

0.5

0.4

0.3

0.2

0.1

0 0

1.0 ol O ~ t

0.8 "~ N~ N2

I

0.6

0.4

0.2 ~

0 . . . . ~ . . . .

0 1 2 3 4 5 6 7 x /d

Fig. 2a, b. Stratification. a Dimensionless density profile in the ambient medium; b dimensionless density difference on the axis of the flow velocity at the source: (A) uo=0.06 m/s; ( x ) 0.19 m/s; (IB) 0.45 m/s

1400

1200 I

1000

8o0

600

4o0

200 I

I

~ . . . . . . . . . . . . . . . . . 4

a x/d 5

I i , / 4 1 ~'

~ t 2

1

0 - .~-.-~-~ . . . . . . . i . . . . . . 0 1 2 3

b x / a

Fig. 3. a Local Reynolds number; b Froude number vs xld; velocity at the source: ([~) u0=0.06 m/s; (A) 0,19 m/s; ( x ) 0.45 m/s

3 Experimental results and discussion

t.1 Dynamic a s p e c t We investigated the mean and turbulent fields of the radial velocity for a better understanding of the phenomena due to the air entrainment and those caused by the interaction of the turbulent plume with a strong stratified layer.

The change of the mean radial velocity at different levels, is shown in Fig. 4 which represents the variations of ( - 9) only for the intermediate configuration (B); indeed, for the two others, the same change is observed. Owing to the choice made for the coordinates, (-a7) is positive when the mean flow is directed towards the axis, and here has an 'entrainment vel- ocity'-like behavior; ( - ~7) is negative when the flow is out- wards. Therefore, the sign of ( - -9) informs on entrainment occurrence. A general review of these results shows that the radial velocity is slow, exceeds rarely 0.10 m/s and becomes equal to zero at the flow axis in accordance with the flow symmetry.

Near the nozzle field, at 2=0.05, two symmetric extrema close to the edges of the nozzle may be assumed (Fig. 4a). These extrema are generated by the buoyancy forces which induce a strong acceleration of the flow, giving rise conse- quently to a significant horizontal air entrainment. Moreover, the area very close to the source may be influenced by vortex disturbance at the edge. This phenomenon can be clearly observed with the three configurations.

0.10

0.05

0

-0.05

f %%

1~ N i �9 /

i +

/ + - - . + . , , . _ +1 4p

*" e/r p / "

x "%

-0.I0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . -2.0 -1.5 -l.0 -0.5 0 "0.5 "1:O "115 2.0

a r / r l 1.0 ~

/4~x + ~

0.8 /" ~ x /x xex ~

x/ i ~

/ x +~x ! ~ I s x+t + . ~"

/ 'i t e l I I '~ l ~* I \

\ / " ~' +x * J \

. ' ~ [ \

0.6 �84 % )~ 0.4

/ 0.2 ,, } , , .t+ .~\

\ 4- �9

-2.5 -2.0 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0 2.5 b r / rm

Fig. 4. a Radial profiles of the mean horizontal velocity ( -v) : u0=0.19 m/s; (~) x/d=O.05; (+) 0.91; (0) 2.06; (x ) 3.25; b Normalized (--v) as a function of r/rM u0=0.19 m/s

From ~ = 0.91, the entrainment of the ambient air to the core of the plume decreases. More and more, the zones with strong air entrainment move away from the flow axis as the height

increases. This phenomenon seems to be linked to the decrease of the density difference between the flow and the ambient medium.

At the interface, close to 2 = 2.06, the plume impinges and the horizontal velocity changes in direction, which near the axis, is then directed away from the flow (Fig. 4a); moving away from the axis, the horizontal velocity first reaches a maximum value, then decreases and changes direction again. It becomes therefore an 'entrainment velocity' far from the plume core. According to Dehmani et al. (1989), this phenomenon could be directly related to the reduction of the vertical velocity near the axis, where the flow slowly breaks down while the height

increases and the horizontal velocity is directed towards the outer part of the flow. We can also notice that, inside this very stable zone, the increase of the injection flow rate leads to a greater radial velocity 9 and a better symmetry in its profiles. Lastly, 17 slows down in the neutral zone and the air en- trainment exists only at the edges of the flow.

Normalizing each negative and positive rlrl-domain of the ( - 9)-curve, by the local maximum or minimum ( - 9) value, and transforming r/rl into firm, gives rise to a single curve in Fig. 4b. Despite certain fluctuations, the axisymmetry assump- tion is justified. The relative radial velocity still reaches about 20% at twice the local extremum radius.

Scaling ( - 17) by the axis velocity tic we obtain the intensity of the entrainment; the dimensionless change ( -17)/~i~ is shown in Fig. 5.

For configurations (A) and (B) the entrainment inside the plume zone becomes strong only at a distance approximately equal to the radius of the nozzle where ( -9)/r~c reaches nearly 5%. On the contrary, configuration (C) is characterized by a weak air entrainment close to the axis. This is probably due to the strong density difference between the flow and the ambi- ent caused by the high injection flowrate. At the interface the radial velocity changes direction near the axis and its intensity increases, as described above. A similar development is observed within the neutral zone. The first part of the intense entrainment zone moves away from the axis when the

increases and the profiles of the horizontal velocity variations become irregular, especially for configuration (A). __The profile of the horizontal turbulent velocity variation

(v,2)in as a function of the axial abscissa, is shown in Fig. 6a. As for configuration (C), this property increases till it reaches the interaction zone, and then decreases. Going across the interface involves less modifications for the other two con- figurations, mainly the A-flow where the horizontal fluc- tuations decrease continually with the height. The greater the plume momentum at the source is, the more important will be the interface change of the rms (v '2)~.

On the axis, the intensity of the turbulence fluctuation of the radial velocity component, Iv, develops (Fig. 6b) in the same way as the centreline velocity: from a high value at the edge of the source, it decreases sharply, then reaches in the neutral zone a constant value of about 0.34 for the three configura- tions. It must be remarked that the intensity of turbulence Iv assumes a high value close to the source, which could mean there are severe perturbations in that area. May be that is why

x73

174

10

~ 0

~ - 5

-10

-15

-2G

a

~ 0

~-5

-10

-15

-20

0 0.5 1.0 1.5 r / r l

0 0.5 1.0 1.5 h r / r l

5 - *

+, /x

-;2I . . . . . . . . . .

0 0.5 1.0 1.5 e r / r l

2.0

2.0

2.0

Fig. 5a-c. Radial profiles of the dimensionless horizontal velocity ( - v)/uc, a: uo = 0.06 m/s: ( I ) x/d = 0.91; (~) 1.25; (+) 2.06; (A) 2.38; (• 3.25; b uo=0.19 m/s: (n)x/]d=0.91; (~) 1.25;(+) 2.06; (A) 2.38; ( • ) 3.25; (0) 4.81; c: Uo = 0.45 m/s: (n) x/d = 0.91; (~) 1.25; (+) 2.06; (A) 2.38; ( • ) 3.25

the literature provides few values for the mean horizontal velocity measurements. The works of Guillou (1984), Brahimi and Son (1986), Papanicolaou et al. (1988) and Brahimi et al. (1989), who found a weaker value of the intensity of turbulence L of about 0.16 for pure plumes could be mentioned. Accord- ing to these authors, the maximum values of the intensities of turbulence are located farther from the source, contrary to the case investigated above, where they appear at the origin of the flow. This apparent discrepancy could be due to the strong density difference between the flow and ambient, the edge effects near the source must be taken into account as already mentioned.

The intensity of turbulence L across the flow presents a maximum at the flow core axis for :~ values below 2, then tends

E

0.40

0.35

0.30

0.25

0.20

0.15

0.10 0

1.2

I I 1

I

�9 i

1 2 3 4 x/d

;>

b

1.0

0.8

0.6

0.4

0.2 0

n

I

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 x /d

Fig. 6a, b. Fluctuations of the horizontal velocity on the axis. a Standard deviation vs x/d; b v Turbulence intensity vs x/d; velocity at the source: ( x ) uo=O.06 m/s; (4') 0.19 m/s; (n) 0.45 m/s

to a large plateau for ~ values above 2, as shown in Fig. 7. For configuration (A) dealing with the weak flowrate, Iv decreases sharply at the axis between ~ = 0.91 and g = 1.25 contrary to the other two configurations where it does not vary much between these two height values. Inside this plume zone the weakest value of the intensity of turbulence is observed for configura- tion (C). This result is in agreement with the fact that a more intense flowrate creates a less disturbed flow.

In jet flows, similar profiles have been commonly observed in a large self preserved field, by Chua and Antonia (1990), Wygnanski and Fiedler (1969); the depicted intensity of turbulence was about 0.2. It seems that the increase of the inertia forces results in a decrease of the turbulence.

Inside the interaction zone, the weaker the flow, the flatter will be the profiles of the intensity of turbulence. This is the case for configuration (A) from ~ = 2.06, which shows that the interface leads to homogenization of the turbulence. The impingement on the stratified interface could also provoke the increase of the properties (v'2) ~/2 and L when a high flow rate is experimented.

The neutral zone corresponds to a more homogeneous distribution of the fluctuations. The intensity of turbulence is greater than in the plume zone; the difference between the behavior in these two zones increases with the flow rate. For a greater value of the flowrate, a clear separation can be ob- served as shown in Fig. 7c. Great care is needed to measure the mean horizontal velocity. Although weak, this component is submitted to a significant fluctuations.

0.2

0.1

0 -2.0

0.5

0.4

0.3

0.4

. . . . , . . . . , . . . . , . . . . , . . . . , . . . . , . . . . , . . . . , . . . . , . . .

-2.0 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0 2.5 r / r l

~0 .3

0.2

0.1 -2.0

b 0.5

0.4

~0 .3

+ § l 'h l ' "~

x x x X x ) dr # # , - - +

§

o

g

-1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0 r / r l

§ ~,,,~o o ,o + �9 , + �9 x # .

: . . . . �9 x+ x �9

�9 # q>+x �9

o �9

�9 @

. . . . , . . . . i . . . . , . . . . . . . . , . . . . , . . . . , . . . .

- 1.5 - 1.0 -0 .5 0 0.5 1.0 1.5 2.0 r / r l

0.2

+ + + �9

g

# # #

#

0 . 1 , , , , I , , , �9 , . . . . i . . . .

-2.0 -1.5 -1.0 -0.5

+ + �9 A+

x x "~

g x •

# �9

# x

# A + A

�9 #

#

#

0 0.5 1.0 1.5 2.0 e r / r l (0,91)

Fig. 7a-c. v-intensity of turbulence as a function of r/r,. a: uo = 0.06 m/s: (m) x/d=0.91; (~) 1.25; ( + ) 2.06; (k) 2.38; ( • ) 3.25; b: uo=0.19 m/s: (m)x/d=0.91; (~) 1.25; ( + ) 2.06; (A) 2.38; ( • 3.25 (A) 4.81; c: uo = 0.45 m/s: ( I ) x /d =0.91; (O) 1.25; ( + ) 2.06; (A) 2.38; ( x ) 3.25

In the l i terature, Mor ton et al. (1956), Evans (1983) among other authors, applied the Gaussian dis t r ibut ion to describe the densi ty difference and the axial velocity profiles; this distr ibu- t ion relies on the self-similari ty deve lopment of these proper- ties which, sufficiently far f rom the source, is an experi- mental ly observed feature. These laws, despite stratification, are in good agreement with our results p rovided ~>~0.91, as shown in Fig. 8a and Fig. 8b. However , a top-hat shape profile is found close to the source of the densi ty difference (Fig. 8c). These laws can be writ ten:

~(x, r) = t~c(X) e -~/r,)~ (1)

A ~ (x , r) = fi (x , r) - Poo (x) = A Pc (x) e - ~r/r;)2 (2)

0.5 1.0

0.8

0.6

0.4

0.2

0 -2.5

1.0

0.8

i

t~ 0.6

~8 0.4 i&

0.2

0 -2.5 -2.0 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2,0 2.5

r / r ' l

1.0

0.8

i

I~ 0.6

0.4 I O . .

0.2

X~

x ~ m ~"~eA x X D A D x

x

\

A D

[]

x x

. . . . . . ~ 5,.,~ . . . . . . . . . . . . . . . . . . . . . . ~ . , . ~ . . . -2.0 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0

r / r ' l

Fig. 8a---c. Self-similar radial profiles, a: dimensionless mean vertical velocity; x /d >1 0.91; Gaussian fitted; b: dimensionless mean density difference; x/d>~0.91; Gaussian fitted; c: dimensionless mean density difference; source nearfield: x /d = 0.05; top-hat variation

3 . 2 .

Axial change of the parameters

St ra t i f i ca t ion o f t h e a m b i e n t e n v i r o n m e n t

In Sect. 2 we studied the over-al l development of the dimen- sionless densi ty difference (f~ - f) as funct ion of the di- mensionless axial posi t ion x /d (Fig. 2b). It can be seen that these curves present the same general aspect. By normal iz ing the value (f~ - - f ) by its m a x i m u m value at the source, all the points can be assembled on one single curve (Fig. 9). The neutral zone is well defined by this curve despite a few per turbat ing effects close to the interface. A Gaussian fit is

175

176

1.0 I

08 , \ ~,

~ % 0.6 " %

'8 0.4

<~ 0.2

0 . . . . . . . . . . . . . . . . ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : . . 0 1 2 3 4 5

x / d

Fig. 9. Self-similar profile of dimensionless mean density difference on the axis; Gaussian fit attempted

10

nm)~-- ~ , . �9 �9

0 . 1 . . . . . . . . . ' . . . . . . . . . ' . . . . . . . . . ' . . . . . . . . . t . . . . . . . . . , . . . . . . . . .

0 20 40 60 80 1 O0 120 X/dm

Fig. 10. Correlation of the vertical velocity on the axis; determination of the dimensionless parameter A~

at tempted to describe the axial profile of the buoyancy force as follows:

(fo~ -- 0/(%0 -- Oo = exp ( - (x/1.58 d) 2) (3)

A significant variation of density occurs at x/d ~ 1.58 which corresponds to the beginning of the stable gradient zone (Fig. 2b). For this value of x/d, Eq. (3) attains a value of (l/e), Fig. 9.

Characteristic plume momentum lengthscale, ~m For vertical flows in plume or jet regime, the specific flux of buoyancy at the source, B, as well as the volume flux mo- mentum at the source, M0, play an important role in the organization of the flow (List et al. 1987; Papanicolaou et al. 1988). Equation (4) defines the characteristic lengthscale:

dm=M~ B -~ (4)

Moreover, the role played by dm is better expressed in dimen- sionless form in Eq. (5); particularly, Eq. (6) shows that using dm as reference lengthscale leads to consider a new dyna- mic variable thanks to the product by Fr~

~m/d : (n/4) ~ Fr ~ (5)

x/ f~ = (n/4) -0.2s Fro 0.5 (x/d) (6)

Axial velocity By predicting the mean axial velocity, uc =f (x ) , we can obtain the axial components of the mean dynamic field whose radial profiles are similar in rlrl as we have already shown in Sect. 2. We endeavoured to determine thanks to the Eq. (7), the di- mensionless parameter A~.

(x/B ) 1/3 -----At (7)

List et al. (1973) used this relationship in an equivalent form in a profile similarity in r/x. Figure 10 shows that A1 can be averaged to a constant, with, under our experimental con- ditions, A~ ~ 1.0, provided x/din >/4. This value of AI is close, while remaining inferior, to 1.16 which is the value we were able to deduce from the works on non-stratified plumes of Papanicolaou et al. (1988).

Plume angle coefficient, Cp The change of dimensionless (l/e)-lengths (F1]~m) as function of (x/dm) give rise to linear relations, whose negative virtual origins are (-xfldm)=22.2; 16.4 and 7.0 for configurations (A), (B) and (C) (Fig. l la ) . When (r,/d) is studied as a function of (x/d), the values above are relative to ( - x J d ) = 0 . 7 1 , 1.60 and 1.60, respectively, while the corresponding line slopes are 0.291, 0.183 and 0.183. Papanicolaou et el. (1988) found that expansion of buoyant jets in non-stratified media leads also to negative virtual origins; the authors find ( - x J d ) = 1.58, and a mean slope value of 0.104. These values mean that buoyant flows in stratified media spread wider than those in homogene- ous environment. For the conditions referenced above, the value of ( - x Jim) can be considered as negligible because of the high values of din. We therefore determined the angle coefficient from the virtual origin (Eq. (8)) for more detailed comparison.

Cp = (2n) '/2' (rlldm)l[(x +xv)ldm] (8)

Figure 1 lb proves that the development of the plume is linear in the domain, despite the stratification. The results show moreover a larger expansion for plumes in stratified media (Fig.1 lc) as mentioned above. The angle coefficient of pure plumes evolving in a homogeneous medium, is taken as reference.

Figure 1 lc shows that Cp grows rapidly when ones starts from Papanicolaou's results (N 2-- 0 s 2) to those obtained for a pure plume in the less stratified medium (N 2= 10.89 s -2). When the inertia forces become stronger Cp decreases while the stratification increases (N 2 = 31.18 s 2 and N 2 = 57.70 s - 2 ). The Brunt-V~iis~il~i frequency, N 2, can be scaled by (g/dm) on the basis of dimensional analysis. The characteristic dimensionless frequency can then be defined as:

N*=N(dm/g) in (9)

For a buoyant flow in a homogeneous medium, N* = 0. The dimensionless frequency allows to show (see Fig. 12) the influence of stratification on Cp. The curve obtained suggests there exists an extreme value for Cp. Let [Nm*, Cpm] be the coordinates of this maximum that we cannot determine with the required precision at this stage due to insufficient ex- perimental values. Two domains must be distinguished:

40

3O

20

10

00 . . . . . . . 2 0 . . . . . . . 4 o . . . . . . . 60 . . . . . . . 8 0 . . . . . . . i00 . . . . . . . i2o a X/~m

100

,,,.,,,

10

0.1

I I [ ]

I I ~L J L � 9

~ t , A

e,o.

azs A

1 10 0.01 0.1 100 1000

b (X+Xv)/ffm 1

A AS~ A ZS A A ~S

i

[ ] m

A ZSA ~A

A ~

0 . 1 , . . . . , , . , . . . . . . . . i . . . . , , , , r , . . . . . . .

0.1 1 10 100 1000 e (X+Xv)/tm

Fig. 11. a Negative virtual origin for the characteristic (l/e)-width vs X/~m: ( I ) N 2= 10.89 S -2, (--Xv)/fm=22.2; (A) 31.18 S 2, 16.4; ( • 57.70 S-2, 7.0; b Linear expansion of rill,, as a function of the dimensionless abscissa from the virtual origin: ( I ) , (A), ( I ) : Stratified medium, this paper. (Z~) homogeneous medium, Papanicolaou et al. (1988); c Angle coefficient Cp: ( I ) , (A), (@): Stratified medium, this paper. (A) homogeneous medium, Papanicolaou et al. (1988)

In the 0 ~< N*~< N~ domain the expansion angle increases with the stratification (in the case of a stable stratification). One domain starts from a flow in a homogeneous medium with C o = 0.27, i.e. roughly a 24 ~ angle, towards a plume flow which has more and more difficulty to pierce the interface because of the increase of N*. The increase of Cp could therefore express the tendency of the plume to collect under the interface and to evacuate more and more along the interface.

In the domain N'~>N*m the angle coefficient decreases significantly from Cpm towards the Cp values of flows in which the forces of inertia become more significant. Under these conditions an easy piercing of the interface without spreading

0.8

0.7

0.6

'T 0.5

0.4

o3

0.2 0

/ ,.., / -.,

/ \ / \

/ x x

/ x / \ I ~ '~ I i

0.05 0.10 0.15 0.20 0.25 0.30 N (#m/g) 0"5

Fig. 12. Influence of the dimensionless Brunt-V~iis/il~i frequency on Cp

under the interface is favored. The angle of the plume recovers then asymptotically values of the same order as those of little stratified media.

The critical value Cp ~ Cpm would correspond just to the beginning of piercing of the interface and to a flow through the interface. It is quite likely that a mixed mode of evacuation along the interface is superposed to the flow created by the piercing and this, during a certain time. If we accept an approximate value for Cpm ~ 0.80 an angle of 2 t a n - [270-~ Cpm] ~ 36 ~ can be deduced, which informs about the order of dimensions of the plume which would present just enough strength to elevate the interface and start off the piercing.

3.3 Entrainment coefficient It is essential to know the value of the entrainment coefficient to understand the flow organization. Indeed, this coefficient which, according to Evans (1983), List and Imberger (1973), and Brahimi (1987), is the main parameter for the numerical development, gives the lateral mass flow rate and the total increase in the vertical mass flowrate when the height in- creases. Table 2 lists the different values for this coefficient. Morton et al. (1956) defined the entrainment coefficient as the ratio between the rl value of the radial velocity, i.e. 9(rl), and tic, the vertical velocity at the axis:

= -- 9 (r,)l(~c (10)

Using the universal Gaussian formula for the vertical vel- ocity profiles, Morton et al. (1956) obtained a constant value for ~ satisfying the relation:

6 r l = - ~ x (11)

5

the authors suggested that

dm 1 (12)

dx 2~zp~ r,

where m is for the overall mass flow rate across the given cross section of the plume. Table 2 gives various values for ~. Ricou and Spalding (1961) estimated the entrainment rate by intro- ducing a turbulent gas jet into a reservoir at uniform pres- sure. The entrainment through the porous wall of the reservoir

177

178

was then measured. A coefficient K was defined as:

1 dm K - - - - (13)

(p09M) in dx

where M is the momentum flux, and K = 0.282. The authors noticed that there was better agreement between this value and the experimental results obtained with jets than with plumes. Many other authors used the coefficient K to process their results; the values they obtained are given in Table 3.

As for Eqs. (12) and (13), the integration of the equation of continuity in the case of strong density variations, leads to:

dm ~ - = - - l im (2n~rg)=--2n~091im (rg) (14)

r ~ 0 9 r ~ 0 9

To account for the change in the local entrainment coefficient, following the configurations and the zones of flows studied, the variations of the mass-flow-rate between two straight horizon- tal sections from the integration of the vertical velocity profiles were taken into consideration:

09

m = 2 n ~ f i~rdr=2npcucKl , rl=r/rl (15) 0

with

o Pc Uc

The difference of mass between two consecutive neighbouring sections was calculated to obtain the mean entrainment co- efficient at tr ibuted to the mean abscissa:

Am 1 o~ - - ( 1 6 )

Ax 2n [p09 rl ~ ] . . . .

Table 4 assembles the different calculation results. Between two configurations the same order of value is found. However, a significant variation as function of the abscissa xld can be remarked. Configurations (A) and (C) show a similar change. The maximum values of ~ occur at the same abscissa, just before the interface, whereas the lowest values occur just after. This observation concerning the results does not apply to the intermediate configuration (B), where the value of ~ is rather minimum.

As a first averaging approximation, the entrainment coeffi- cients relative to runs (A) and (B) can be considered to be constant. The parametric group X* defined by Eq. (17) which leads to this assumption takes into account the axial height of the flow scaled by the momentum length scale, the dimension- less axial density profile, and the local Froude number calculated with r~.

X*=(xl~m) \Poo/ (Fr)m (17)

A constant value of ~ is therefore proposed for X* >1 in Fig. 13. Moreover this figure shows that the increase in the momentum at the source leads to low values of the entrainment coefficient particularly in the sections located beyond x/d = 2.2. The ex- perimental values relative to this situation confirm a linear change as a function of the parametric group X*. The relation of linear change corresponds to the domain X*<~I. The ob- tention of lower values of an average value of ~ for the forced plume could be put in relation with the intensity of the impingement between the plume and the stratified interface. An impact like this could have the tendency to reduce the size of the large structures of the plume which start off the eddies responsible of the entrainment of the environing medium. The

Table 2. Coefficient a: Review Authors

Morton et al. (1956)

Morton (1958)

George et al. (1977) Evans (1983)

Guillou et al. (1986) Baines (1975)

Abraham et al. (1983) Zukoski et al. (1981)

Configurations c~

Atmospheric plume - - homogeneous ambient media or with stable stratification 0.093

�9 Homogeneous media 0.082 Hot jet:

�9 Stable stratification 0.116 Hot air jet 0.153 Plume with weak buoyancy arising from a flame 0.104 Pure buoyant plume 0.15 �9 Injection of salt water at the top of a

tank containing stratified water. 0.084 ~< ~ ~< 0.1 �9 Injection of fresh water in a uniform fluid 0.057 Pure buoyant plume 0.085 Plume arising from a flame 0.110

Table 3. Coefficient K: Review Authors Configurations K

Ricou and Spalding (1961)

Tamanini (1977)

Brahimi et al. (1988)

Isothermal jet or with weak temper- ature variations Hot jet with weak Froude number Isothermal jet A single thermal plume Two interacting thermal plumes

0.28 0.55 0.29 0.19 ~<K ~<0.23 0.21 ~<K ~<0.25

Table 4. Mean entrainment coefficient ct

xld Re0=27.31 Re0= 82.43 Re0=201.71

1.08 0.200 0.182 0.077 1.66 0.244 0.071 0.165 2.22 0.099 0.113 0.002 2.82 0.168 0.092 0.018

0.1

T 0.01

0.001 0.001 0.01 0.1 1 10

(X/dm)(~ /~-c)~ 0"5

Fig. 13. Correlation of the mean entrainment coefficient

�9 I t I - �9 " * A .

. . . , . . . . . . . . , . . . . . . . .

100

variation of c~ from one section to the other could illustrate such an activity. The stretching and the flattening of these structures while crossing through the interface imply a signifi- cant reduction of the entrainment all the more so that during our experiments a reinforced stratification was associated to the forced plume. The same phenomenon acts as a conse- quence in the reduction of the integral lengthscale. Indeed Dehmani (1990), using a spectrum investigation, shows that the integral lengthscale reaches a min imum at the interface, then increases. Furthermore, this influence of the turbulence characteristics of the flow at the impinging zone of the in- terface was also remarked by Baines (1975).

The turbulent structure of the flow plays most likely an important role in the entrainment coefficient. Moreover, Papanicolaou and List (1987) showed that the intermittency factor reached unity further along the radius for jets than for plumes, this could imply that the entrained ambient fluid remains unmixed close to the axis for plumes than for jets. According to Guillou et al. (1986), the intermittency factor in the plume flow is smaUer than in the jet flow. We have not measured this factor in our paper. Which could justify the difficulty of expressing in a simple relation the results of c~.

4 Conclusion Experiments were performed on mass plumes, under various conditions of Reynolds, Froude numbers at the source. As these flows impinged a stratified medium, a homogenization of the turbulence occurred above this interface, even if a slight increase in the intensity of turbulence beyond the interface could be observed. Beyond the interface, the horizontal velocity component suddenly increased. This seemed to be quite significant since the stratification acted as a fluid obstacle with regard to the axial direction of the propagation; the

momentum, mass and buoyancy being continuously fed at the source, the plume needed a larger horizontal expansion to conserve the momentum. Correlatively, the radial Gprofile suddenly increased beyond the interface (~ >~2.06). The plume then spread out. The main change in the parameters did not occur just at the interface, but a little beyond. This could have been due to the dynamics of injection by entrainment of the environmental air. Indeed, the ambient air entrainment reached the core of the plume in the region near the source while it existed only at the edge of the flow far from the source. The maximum value of the horizontal velocity component was located on a radius when the expansion became radial. But the entrainment was subjected to considerable decay in the far field. To quantify this effect, the entrainment coefficient was experimentally evaluated. This value decreased considerably in the interaction zone, to increase in the neutral zone. The entrainment phenomenon is not only affected by the variations of the Reynolds or the Froude numbers, but also depends substantially on the turbulent characteristics of the flow which have not been taken into account in this paper. Larger inves- tigations including the turbulent structure of the flow will allow to establish a better correlation of the local entrainment co- efficient.

References Baines WD (1975) Entrainment by a plume or jet at a density interface

l Fluid Mech 68:309-320 Bejan A (1991) Thermodynamics of an isothermal flow: the two- dimensional turbulent jet. Int J Heat Mass Transfer 34:407 413 Brahimi M (1987) Structure turbulente des panaches thermiques -

Interaction. Th6se de Docteur de l'Universit6 de Poitiers, Poitiers, France

Brahimi M; Dehmani 14 Son DK (1989) Structure turbulente de l'6coulement d'interaction de deux panaches thermiques. Int I Heat Mass Transfer 32:1551-1559

Brahimi M; Son D-K (1986) Propagation of interacting turbulent plumes. Proceedings of the 3rd International Conference, Porto Carras, Greece

Brahimi M; Lamour M; Son DK (1988) Champs moyens et fluctuants des panaches thermiques isol6s ou en interaction. Rev G6n Therm Fr no. 315-316:236-243

Chen CJ; Rodi W (1980) Vertical turbulent buoyant jets, A review of experimental data. Pergamon Press. London

Chua LP; Antonia RA (1990) Turbulent Prandtl number in a circular jet. Int J. Heat Mass Transfer 33:331-339

Dehmani L (1990) Infuence d'une forte stratification de masse volumique sur la structure turbulente d'un panache/t sym6trie axiale. Th6se de Docteur de l'Universit6 de Poitiers, Poitiers, France

Dehmani 14 Rey C; Rongere FX; Son DK (1989) Etude de la p~n&ration d'un panache thermique dans une zone stratifi6e stable. 96me Congr6s Franqais de M6canique Metz

Douglas G Fox (1970) Forced plume in a stratified fluid. ] Geophys Res 75:6818-6835

Evans DD (1983) Calculating fire plume characteristics in a two layer environment, National Bureau of Standards, Department of Com- merce, Washington D.C. 20234-NBSIR 83-2670

Fukui K; Nakajima M; Ueda H (1991) Coherent structure of turbulent longitudinal vortices in unstably-stratified turbulent flow. Int I Heat Mass Transfer 34:2373-2385

Gebhart B; Hilder DS; Kelleher M (1984) The diffusion of turbulent buoyant jets. Copyright by Academic Press. Inc. ISBN 0-12-020016- 3, pp. 1 57

George WK; Alpert RL; Tamanini F (1977) Turbulence measurements in an axisymmetric buoyant plume. Int l Heat Mass Trans 20: 1145-1154

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18o

GuiUou B (1984) Etude num~rique et exp~rimentale de la structure turbulente d'un panache pur ~ sym&rie axiale. Th&e de Docteur Ing~nieur, Universit~ de Poitiers, Poitiers, France

Guillou B; Brahimi M; Son DK (1986) Structure turbulente d'un panache thermique. Aspect dynamique. Journal de M&anique th~orique et appliqu& 5:371-401

Kapoor K; Jaluria Y (1993) Penetrative convection of a plane turbulent wall jet in a two-layer thermally stable environment: a problem in enclosure fires. Int J Heat Mass Transfer 36:155-167

Kotsovinos NE (1990) Dissipation of turbulent kinetic energy in buoyant free shear flows. Int J Heat Mass Transfer 33:393~400

Kotsovinos NE; List El (1977) Plane turbulent buoyant jets. Part I. Integral properties. J Fluid Mech 81:25~44

List EJ; Imberger J (1973) Turbulent entrainment in buoyant jets and plumes. J Hydr Div A.S.C.E. 99:1461-1474

Morton BR (1958) Forced plumes. J Fluid Mech 5:151 163 Morton BR; Taylor FRS; Turner JS (1956) Turbulent gravitational

convection from maintained and instantaeous sources. Proc Roy Soc A 234:1 23

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axisymetrical turbulent jets. J Fluid Mech 11:21 32 Srinivasan J; Angirasa D (1990) Laminar axisymmetric muiticompo-

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Taylor GI (1945) Dynamics of a mass of hot gas rising in air. U.S. Atomic Energy Commission MDDC-919 LADC-276

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renouncement!

Call for papers 37th Israel Annual Conference on Aerospace Sciences

TeI-Aviv and Haifa, February 26-27, 1997

The 37th Israel Annual Conference on Aerospace Sciences will be held on February 26-27, 1997. The Conference constitutes a forum for the presentation and discussion of recent advances in the following areas:

Aerodynamics and Ballistics - Aeronautical Design, CAD/CAM, Manufacturing and

Maintenance - Aeronautical Systems and System Engineering - Materials, Aeronautical Structures and Aeroelasticity - Propulsion and Combustion

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Papers on recent advances is basic research and technology applications in the above-mentioned areas, as well as other aerospace-related fields, are invited including student papers. The accepted manuscripts will appear in the conference pro- ceedings. The conference will include invited lectures.

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- Three copies of drafts of complete papers (or extended abstracts of 1000 to 1500 words) should reach the Program Committee Chairman by September 13, 1996 (for extended

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- Extended abstracts and papers will be sent for review. Scientific/technical content, importance to the field, rel- evance to the scope of the conference, and originality, are the principal criteria for the selection of papers. Authors will be notified of the acceptance of their papers by November 30, 1996. Final versions of accepted papers should be submitted to the Chairman of the Program Committee. The deadline for submission is January 1, 1997. It is the responsibility of each author to secure any and all necessary approvals from their own or sponsoring organiza- tions.

Abstracts and complete papers should be sent to: Prof. Omri Rand Chairman, Program Committee 37th Israel Annual Conference on Aerospace Sciences Faculty of Aerospace Engineering Technion-Israel Institute of Technology Haifa 32000, Israel FAX: 972-4-8231848, TEL: 972-4-829713 email: AERCONF@Technion.Technion.Bitnet