numerical investigation of turbulent plumes in both ambient and stratified surroundings

Indoor Air 1995, 5: 136146 Binzed in Denmark . all rights reserved

Copyrinht 0 Munksaaard 1995

Indoor Air ISSN 0905-6947

Numerical Investigation of Turbulent Plumes in both Ambient and Stratified Surroundings Vijay Shankar’, Lars Davidson’, Erik Olssonl

Abstract Kentilation displacement systems have, during the last ten years, become more and more popular. In these systems cool air is supplied to the room, and the air is heated by heat sources. The rising air above these heat sources is of paramount importance to the behavwur of the ventilation displacement systems. In the present work the turbuht flow in plumes is studied numerkaly, using finite volume methods. The standard k-E model was found to underpredict the spreading of the plumes, and it was thus modajied in two ways so as to predict spreading rates in agreement with experiments, We present a com- prehensive comparison between predictirms and experiments including spreading rates, velocity and temperature profiles, and turbulent shear stresses. The volumejlm rate versus the vertical distance j b m the plume is also presented. Good agreement between predictions and experiments is obtained.

KEY WORDS: Plume, Stratified, k-E model, NSA modification, Buoy- ant production

Manuscript received: 13 September 1993 Accepted for publication: 27 October 1994

Department of Thermo and Fluid Dynamics, Chalmers Uni- versity of Technology, Gothenburg, Sweden

Introduction Displacement Ventilation We spend about s i x t y to seventy percent of our life- time indoors. It is therefore important for us to make certain that our “indoor climate” is as comfortable as possible. An ideal ventilation flow system, apart fiom being “energy effective”, should also contribute to high quality air. This is where displacement ventilation flow systems can be applied to advantage.

Since displacement ventilation flow systems have become increasingly popular and are replacing the traditional mixing ventilation flow systems, it is of great interest to carry out a numerical investigation of the flow. A schematic diagram of a room ventilated by displacement is shown in Figure 1.

In displacement ventilation systems air is supplied to the room at low velocity, with a volume flow rate Kh near the floor, and is extracted at the ceiling. The temperature of the supplied air is slightly lower than that of the room. Air is heated by the objects in the room e.g., computer ter- minals, photocopy machines, etc. and it rises due to buoyancy.

Fig. 1 Displacement ventilation. Airflow patterns in a room with heat source, ventilated by displacement.

Shonkor et 0 1 . : Numerical Investigation of Turbulent Plumes in both Ambient and Strotified Surroundings 137

In order to design a displacement ventilation system that can synchronize with both thermal and comfort ventilation in rooms, it is important to predict the nature of flow over a heat source. The rising flow above the heat sources resembles a plume. The flow in the plumes rises to the ceiling. The volume flow rate in plumes for a given vertical distance from the heat source x, is t'pl,,,(x), and increases with x. At the ceiling the flow spreads out laterally. Air is extracted below the ceiling at a rate of c.n. The re- mainder of the flow, l&u,e(H) - ch (H is the height of the room) flows-downwards. The level where this downward flow has all been entrained in the plumes, occurs when vi,= VPlume(xxfront).

From the above discussion it is clear that knowledge of the volume flow rate in the plume is crucial for the efficient performance of displacement ventilation flow systems. Investigations have been carried out simulating the flow in a com- plete room. Here an elliptic solver was used (Dav- idson, 1989). In this approach the flow in the plume is not accurately resolved due to limitations in grid resolutions.

The aim of the present investigation is to study the flow in vertical plumes. The nature of flow in these configurations is parabolic in the flow direction (see Figure 2). This means that the flow at a given vertical level (plane z) is not influenced by the flow conditions at plane i+ 1. This allows us to use

a forward marching technique, i.e., a parabolic solver in a vertical direction. A solver of this type is extremely fast when compared to an elliptic solver (up to 100 times faster). This allows us to under- take parameter studies, and to ensure that the solutions are grid-independent.

Thermal Flows Thermal flows are an important type of natural convective flow, which relates to buoyancy-driven flows rising freely, without the constraining influence of a fixed boundary.

Basically, a thermal flow is a free convective flow, where the entrained air volume flow increases with the distance from the source.

Parameters The flow in a buoyant jet is influenced by the inertial, buoyant and viscous forces. Therefore, the local flow character is determined by the relative magnitude of these forces at each point. The following dimensionless numbers give the relationship between the inertial, buoyant and viscous forces.

1. Reynolds Number, Re, gives the relation between inertial and viscous forces.

Re = ~

2 . Grashof's Number, Gr, is the ratio between buoyant and viscous forces.

Gr=

3. Archimedes Number, Ar, gives the relation between the inertial and buoyant forces.

Ar =

Since the inlet velocity in pure plumes is zero, an alternate Grashof's number, based on the heat flux Q (Chen and Rodi, 1980), is applied and given by

uodo V

g(ex-eo)& -g@AT -

eov2 V 2

g(ex -@old0 - -~ eou: u:

Since j = O for plane plumes, j= 1 for round plumes

and Po = -, we have for round plumes 1

T,

4Qg Fig. 2 Vert coI plume %Q =

138 Shankar et al.: Numerical lnvestiqation of Turbulent Plumes in both Ambient and Stratified Surroundinas

Governing Equations of Motion Mean Flow Equations The continuity equation is written as

a - (@j) = 0. d X i

The momentum equation is expressed as

a aP a -((eU,v,) = - -+ - dX j axj ax,

T- T e f

T e f @r&i -*

where gl = -g, g2 = 0 (xl = x is vertically upwards, and x2 =y is the lateral direction, see Figure 2), and where the Boussinesq’s approximation is used for the gravitation term. The governing equation for temperature is given by

a (P a -((@U.T) = - eff-

dX j axj 0, axj J (3)

where the turbulent Prandtl number ot=0.7.

Turbulence Model The transport equations for k and E in the standard k-E model can be written in tensor notation as (Launder and Spalding, 1974)

and

where C,1 = 1.92, C,2 = 1.44, (3k = 1.0 and (3, = 1.3. Since the Boussinesq’s approximation is used, the density in Equations 1-5 is constant.

The generation terms can be expressed as

and the production due to buoyancy can be written as

where

@&j = - 2-* P dT 0, ax

The turbulent viscosity pt is calculated as

k2 Pc = ecp -*

The effective viscosity peff is obtained as

&

Clef= P+ ~ t -

(7)

The Code In this section the parabolic version of CALC-BFC, developed by Farhanieh and Davidson (1992), is presented. This is a derivative of the standard three- dimensional finite volume computer program CALC-BFC (Davidson and Farhanieh, 1992) (Boundary Fitted Coordinates) for three-dimensional complex geometries. The program uses Car- tesian velocity components, and the pressure-velocity coupling is handled with the SIMPLEC pro- cedure. In most finite volume programs, staggered grids for the velocities have been used (Patankar, 1980). In the present work collocated variables are used, which means that velocities are stored along with all scalar variables such as p, k , E at the center of the control volume. This concept was suggested by Rhie and Chow (1984). For more details on the code, see Shankar (1993).

The Implications of Parabolic Flow In parabolic finite volume methods, two basic as- sumptions are made (Patankar and Spalding, 1972). First, the streamwise diffusion term

is set at zero in all equations. Secondly, the streamwise pressure gradient dplat is set at zero.

It should be noted that in the literature, terms in the production term p k involving d/dt-derivate are usually neglected (Hussain, 1980; Leschziner, 1982; Ljuboja and Rodi, 1980). Furthermore, some approximations are also made when calculating the

Shonkar et 01.: Numerical Investigation of Turbulent Plumes in both Ambient and Stratifled Surroundings 139

non-orthogonal diffusion term (Leschziner, 1982). No such simplifications are used in the present parabolic solver.

In parabolic flow, the flow downstream (plane i+ 1 , see Figure 2) does not influence the flow up- stream (plane 2). This feature means that a space marching technique can be used, where the flow in each i-plane is calculated separately and only once. First, the flow is solved at i-plane i=2 (the inlet boundary conditions are set at i=l at x=O) and a convergent solution is obtained for this plane, then the flow at plane i=3 is calculated, and so on. The parabolic solver is up to two orders of magnitude faster than a standard elliptic SIMPLE solver, because in an elliptic solver the flow at each i-plane is not calculated once, but several hundred times.

Modification of the Standard &-E

Turbulence Model The Buoyancy Production Term (Plane Plume) It was found that the standard k-E model underesti- mates the spreading rate d61,2/& of plane plumes

denotes half-width of the plume). Since the volume flow rate is directly proportional to the the square of the plume width312, a correct prediction of this width is of importance. The standard production term due to buoyancy GB in Equations 7- 8 is almost negligible because the streamwise temperature gradient is very small. In order to enhance the importance of GB and thereby increase the spreading rate, a modification of the production term GB, used by Ince and Launder (1988) and Davidson (1990) is introduced. The idea is borrow- ed from the generalized gradient hypothesis of Daly and Harlow (1970) where (x and y denote streamwise and lateral direction, respectively)

(9)

and

Using Boussinesq’s assumption for UV in the second term in Equation 9, and substituting Equation 8 in the first part of Equation 9, we obtain

p dT ptkdTdU

eoc ax @ & dY 8Y In Ince and Launder (1988) and Davidson (1990) the constant C e l was taken as %; in the present study

+ c e - - - - . (10) = -

C e l = 1.7 was found to give the best agreement with experimental data.

In Equation 10 not only the streamwise temperature gradient d Tldx is present, but also the transversal gradient aTldy, which is an order of magnitude greater than dTldx. In a plume both dTldx and dUl dy are negative. Hence the prescribed modification will increase GB which will thus enhance the buoyant production of turbulence and increase the spreading rate.

Normal Stress Amplification (NSA) [hi- symmetric Plume] HanjaliC and Launder (1980) developed a modified k-c model, where the the production term in the dissipation equation was made more sensitive to the irrotational strains d Uldx, d Way. Recently some mi- nor alterations in HanjaliC and Launder’s model have been proposed by Malin ( 1989). The modified production term in the k and &-equation can be written (Hanjalic and Launder, 1980) as

and

where the shear stress UV is taken from Boussinesq’s approximation,

- k2 dU -uv = cp--

E dY

and

where ck = 0.35 and cE3 = 4.44 (Malin, 1989) were chosen in order to obtain a predicted spreading rate in agreement with experiments. The coefficient in the diffusion term in the &-equation was taken as O& = 1.1 (Malin, 1989).

The effect of the modified P, in Equation 11 for the axi-symmetric plume is that the decrease of P, in the outer region (where dUldx>O) outweighs the decrease of P, in the core region (where dUldx<O). Thus the overall effect is a decrease in E (increase in turbulent viscosity) and thus an increase in the spreading rate. Note that this is opposite to what happens in an axi-symmetric jet (Malin, 1989).

140 Shankar et 01.: Numerical Investigation o f Turbulent Plumes in both Ambient and Stratified Surroundinqs

Boundary Conditions Inlet All variables are prescribed according to experiments. The inlet is covered by half of the total number of nodes in the J-direction.

Table 1 Spreading rate for the plane plume

dS1/2/&

standard k-.z 0.068 modified k-.z 0.12 RSM (Haroutunian and Launder, 1988) 0.079 Experiments (Kotsovinos and List, 1977) 0.12

Symmetry line Iris set at zero, and zero gradient dWdy is set for the remaining variables. Table 2 Spreading rate for the axi-symmetric plume

Free Boundary The mass flux across the boundary is calculated from local continuity, and the variables are given by their free-stream values.

standard k-.z 0.10 k-.z with NSA 0.12 RSM (Haroutunian and Launder, 1988) 0.096 Exueriments (Chen and Rodi. 1980) 0.12

Results The flow in plane and axi-symmetric plumes is calculated. Sixty nodes are used in the transversal (r) direction, and 40-60 steps in the streamwise direction. The size of the forward steps was varied between 15% and 30% of the local width of the plume. The CPU time for one computation was about 1 minute on a DEC-3100 work station (approximately as fast as a SUN Sparc 1 +).

Our code is written for three-dimensional parabolic flows. In this investigation we have calculated only two-dimensional plumes. We do not need to solve the Vmomentum equation or the pressure correction equation, since Ir can be obtained from continuity. The equations that we solve are U, I; k and &-equations. In all calculations the modified k-E model is used as described earlier. For comparison, results obtained with the standard k-E are included in Tables 1 and 2.

Plane Plume The inlet boundary conditions Uin=0.076 m / s , Zn=3.68"C and T , = O (a denotes ambient) were used for the mean flow variables, inlet height D= 0.01 m, and k and E were estimated from mixing length theory as kin= (0.01 Ui,J2, ~ = k z ~ / 0 . 1 D. In the stagnant surroundings all variables were set at zero. The Archimedes number is calculated as

gDAT = 0.23. A?-= .rrr3

law&

We started to use the standard k-E model, but this gave too small a spreading rate compared with experiments (see Table 1). Note that not even a 1 1 1 Reynolds Stress Model gives a spreading rate ac-

cording to experiments (see Table 1). With the proposed modification of the heat flux & (see Equa- tions 9 and 10) we increased the predicted spreading rate to agree with the experiment.

In Figure 3 the predicted U, T and =profiles are compared with experiments and the agreement is good.

hi-symmetric Plume The boundary conditions were the same as those for the plane plume. The NSA-modification was used. The predicted spreading rates are compared with experimental data in Table 2, and the modified k-E model gives a spreading rate in agreement with experiments, which is further confirmed from Fig- ure 4.

Pure Axi-Symmetric Plume in Ambient Surroundings In the previous sections we have studied plumes with some initial momentum, i.e., inlet velocity dif- ferent from zero. In this section we will study pure axi-symmetric plumes in stagnant surroundings with no initial momentum, the flow being entirely dominated by buoyancy. A schematic representation of a pure plume is shown in Figure 5.

A pure axi-symmetric plume in ambient surroundings can be divided into three zones.

a. In zone one the transition from a boundary layer to a pure plume occurs. The laminar flow be- comes turbulent in this zone.

b. In zone two the plume is turbulent and axi- symmetric. The plume spreads nonlinearly and the

Shonkor et 0 1 . : Numeric01 Investigation of Turbulent Plumes in both Ambient and Strotified Surroundings 141

1 . 2 0 , I , . 1 . 1 . I . I .

AT TI -

0 . 8 0

0 4 0

0 03

0 0 2

0 0 2

0 01

0 0 0 1 0 8 1 2 1 6 2 . 0 2 4

Fig. 3 Plane plume. Predicted U, Tond Z -p ro f i l es compared with experiments of Chondrosekhorc 11 9 8 3 ) .

velocity and the temperature dismbution are of Gaussian type.

c. The plume is fully developed in zone three and it spreads linearly.

1 . 2 0

AT Tc -

0 . 8 0

0 . 4 0

0.00 0.0 0 . 4 0 . 0 1 . 2 1 . 6 2 . 0 2 4 . !I f

0 . 0 1 4 I .

!I - 0 0 0 4 0 1 1 2 1 6 2 0 2 b ,

!I t Fig. 4 Axi-symmetric plume. Predicted U/U,, T/T, ond Uv/Uz- profiles compored with experiments of Chondrosekhara 1 1 9 8 3 )

In ventilated rooms, the plume motion above a heat source arises simply because of density differ- ences, caused by the temperature difference. A part of the heat fiom a heat source is "lost", by means of radiative transport, and the remaining heat is

142 Shankar et al.: Numer ica l Investigation of Turbulent Plumes in both Ambient a n d Stratified Surroundinas

h J Zone 3

--

Zone 2

Fig. 5 Pure axi-symmetric plume in ombient surroundings

. E x p e r i m e n t 0 = 2 1 5 w . 0 = 5 c m -

v) -. m ; 0 . 1 6

E Y

z 0 . 1 2

g 0 . 0 8 - 0 w

0 . 0 4

0 . 0 0 1 . o o 2 . 0 0 3 . 0 0 x [.I

Fig. 6 Axi-symmetric plume. Predicted volume flow rate in an axi-symmetric plume for heat source, Q=2 15 W (convective heat) a n d diameter D = 5 cm, compared with experiments of Ko- foed ( 1991 ) .

0 . 2 4

0 . 2 0 - VI --.

rc)

0 . 1 6

E Y

g o . 1 2 0

0.08 3 - 0 >

0 . 0 4

0 . 0 0 1 . o o 2 . 0 0 3 . 0 0 x [.I

Fig. 7 Axi-symmetric plume. Predicted volume flow rate in a n axi-symmetric plume far heat source, Q= 343 W (convective heat) a n d diameter 0=5 cm, compared with experiments of Kofoed ( 1 99 1 ) .

0 . 3 2 I I 1

0 . 2 8

- 0 . 2 4

c) * * 0 . 2 0 E

3 0 . 1 6 Y

- LL

~ 0 . 1 2

5 - 0 0 . 0 8 >

0 . 0 4

0 . 0 0 1 . o o 2 . 0 0 3 . 0 0 x [.I

Fig. 8 Axi-symmetric plume. Predicted volume flow rate in a n axi-symmetric plume for heat source, Q=729 W (convective heat) a n d diameter 0=5 cm, compared with experiments of Kofoed ( 199 1 ) .

transported by natural convection. Only the convective part of the heat source is taken into consider- ation in this paper.

The boundary conditions at the inlet are Ujn = 0, zh = T, (subscript a denotes ambient) and a heat source Q is prescribed for temperature equations (actually Q/cp, since we are not solving for enthalpy but for temperature). This gives an Archimedes number Ar= 0. The volume flow rate v(x) for dif-

ferent heat sources and for the diameter, D = 5 cm is shown in Figures 6-8. We can thus conclude that the volume flow rate v(x) increases with the distance from the source and with an increase in the convective heat of the source.

It can be seen that v(x) increases with x and Q as expected. But an increase in Q with a factor of

Shankar et 01.: Numerical Investigation of Turbuleni Plumes in both Ambient and Stratified Surroundings 143

two, does not give rise to a large increase in P a s indicated in Figures 9-10. The reason for this can be explained as follows. A fully developed turbulent plume is characterized by its initial weight deficit Win (Chen and Rodi, 1980).

and from dimensional analysis the characteristic velocity can be written as

As the volume flow rate in the plume can be estimated as

v cx v d / 2 , (14)

it can be seen from Equations 12-14, that an increase in Q with a factor of two only increases the volume flow rate with a factor of P3.

Comparing Figures 9-1 0, we find that the volume flow rate v ( x ) increases as the diameter D is increased. We know that the spreading rate for a fully developed axi-symmetric plume is related to x as

YllZ x* (15)

The volume flow rate P(x) in a fully developed plume is independent of its initial diameter 0, but the predicted volume flow rates in Figures 9-10, increase with D. This can be clarified by the following.

0 3 0 ~ I I 1 0 2 8

0 2 6

0 2 4

- - - Q=JOOw. D=lOcm

Q=200w. D=lOcm - : ....

- - - -

-

- -

- -

- - -

-

0 00 1 00 2 00 3 00 x 1.1

Fig. 9 Axi symmetric plume Volume flow rate in an 0x1-sym metric plume with inlet diameter D= 10 cm for heat sources Q= 300 W o a d Q=200 W

0.30 c 1 I

2 0.20

- 0 . 1 6

- a=300w. 0 = 2 O c m

.... Q=200w. 0=20cm

0.00 1 . o o 2 . 0 0 3.00 x I m l

Fig. 10 Axi-symmetric plume. Volume f low rate in on axi-symmetric plume with inlet diameter 0=20 cm for heot sources Q= 300 W a n d Q=200 W.

S p r e a d i n g R o l e

I I I I i n 1 7 I-

d y l , ,

4 t I I I I 1

X l d

I

0 20 40 60 80 1 0 0 0 001

Fig. 1 1 Axi-symmetric plume. The spreading rote o f o pure axi- symmetric plume in ambient surroundings.

The vertical volume flux for a free plume is given by (Kofoed, 199 1)

V = 0.005 1 @'3(~-~,)5'3, (16)

where x, is the position of the virtual origin of the plume and a is the convective heat. Kofoed (199 1) finds that the virtual origin is located below the source, i.e. x,<O. The explanation as to why the predicted volume flow rate increases with increasing initial diameter can thus be that x, decreases (be- comes more negative) with increasing D.

The spreading rate for a pure axi-symmetric plume in ambient surroundings is shown in Figure 11. From this figure, we can conclude that the axi- symmemc plume spreads non-linearly in zone 2

144 Shonkor et 0 1 . : Numer ica l Investigation of Turbulent Plumes in both Ambient and Stratified Surroundings

k v r l uf ~ir~ill ive huoyalcy I - I

Fig. 12 Buoyont jet in stratified surroundings

0 . 2 4

r---

ln .. vl * *

0 . 1 6 E u

3 0

LL

RJ

-

5 0.08 - 0 >

I I

( s t r o t i f i e d , O . 6'/rn) ( s t r o t i f i e d , 1 .O' /m)

- A m b i e n t _ _ _ ~ - -

A

0.00 1 . o o 2 . 0 0 3 . 0 0 x [ m l

Fig. 13 Axi-symmetric plume in stratified surroundings. The volume flow rate in a n axi-symmetric plume for heat source Q=2 15 W(convec t i ve heat) a n d diameter D=5 cm.

(see Figure 5 ) and corresponds with the fully developed plume in zone 3 where it spreads linearly. We can also infer that zone 1 extends up to x/d= 10,

0 . 2 4

T 0 . 2 0 --. vl * *

0. 1 6 E Y

i= - O 0 . 1 2 LL

a,

E 2 0.08 - 0 >

0 . 0 4

I - ' ( ornbli e n t __.- ( s t r o t i f i e d , 0 . 6 ' / r n ) _ _ ( s t r o t i f i e d , l . O ' / m )

'

0 . 0 0 1 . o o 2 . 0 0 3 . 0 0 x [ m l

Fig. 14 Axi-symmetric plume in stratified surroundings. The volume f l ow rote in o n axi-symmetric plume for heat source Q=343 W(convec t i ve heat) a n d diameter D=5 cm.

0 . 2 4

';;; 0 . 2 0 .. vl * *

0 . 1 6 E u

3 - O 0 . 1 2

5 0.08

L L

W

- 0 >

0 . 0 4

0.00 1 . o o 2.00 x [ m l

3 . 0 0

Fig. 15 Axi-symmetric plume in stratified surroundings. The volume flow rote in o n oxi-symmetric plume for heat source Q=729 W(convec t i ve heat) o n d diameter D=5 cm.

zone two lies between x/d= 10 to 40 and zone 3, stretches beyond x/d= 40.

Pure hi-Symmetric Plume in Stratified Surroundings The fluid density in a stratified environment is non- uniform and varies with height, which is true for

Shankar et a / : Numerical Investigation of Turbulent Plumes in both Ambient and Stratified Surroundings 145

several natural convective flows of practical interest. The environment in this case is said to be stratified, i.e., the natural convective flow arising in such stratified surroundings is affected to a considerable ex- tent by the density variations. This situation is called thermal stratification, and occurs in ventilated rooms. Figure 12 shows a schematic representation of a buoyant plume in stratified surroundings.

Here the density of the surroundings decreases with the increase in x. In the beginning, the plume behaves like a buoyant plume. The density difference between the plume and the surroundings decreases with the increase in the distance from the heat source, due to stratification. At a certain distance from the heat source a point of neutral buoyancy is reached. At this height, the density difference between the plume and the surroundings is zero. From this height, the plume behaves like a negative buoyant jet.

The Effect of Stratification In the case of axi-symmetric plumes in ambient surroundings, the entrainment rate is higher than that for axi-symmetric plumes in a stratified environment because the entrained air has a “cooling effect” on the plumes which decreases the driving buoyancy force.

The effect of stratification on the volume flow is shown in Figures 13-15. In general, the volume flow is bound to decrease when the density difference between the axi-symmetric plume and its surroundings decreases. As the intensity of stratification is increased from O”C/m to 0.6”C/m and 0.6”C/m to 1 .O”C/m, the volume flow rate decreases by approximately 25 and 20 percent respectively. This is due to the fact that the density difference between the axi-symmetric plume and its surroundings decreases as the “intensity” of stratification is increased.

An interesting feature that can be noticed in Fig- ures 13-15, is that the effect due to stratification occurs at a certain distance from the convective heat source. The smaller the “stratification intensity”, the lesser is its effect on the flow. For instance, in Figure 14, the effect due to stratification, when the intensity is 0.6”C/m, can be seen to occur at approximately 30 “diameter height” from the heat source. When intensity of stratification is increased from 0.6”C/m to 1 .O”C/m, its effect on the flow can already be observed at nearly 20 “diameter height”, from the heat source.

Conclusions In ventilation displacement systems plumes are formed above objects which heat the cool supplied air. The volume flow rate in the plumes is crucial for the performance of the ventilation system, and it is thus a parameter of great interest to a ventilation engineer. When computing displacement ventilation flows, elliptic solvers have traditionally been used. However, this approach can be inaccurate due to poor grid resolution.

In the present work, a parabolic solver has been used for computing the turbulent flow in plane and axi-symmetric plumes. In this approach a space marching technique is utilized which is up to 100 times faster than elliptic solvers. A sufficiently fine grid can thus be afforded, and parametrical studies can be carried out.

In this paper, the behaviour of axi-symmetric plumes in both ambient and stratified surroundings is studied. For accurate numerical simulation of the above-mentioned flow, the k-E model was modified. By the application of the modified k-E model, con- solidation of the “numerical spreading rate”, with the “experimental spreading rate”, was achieved.

The volume flow rate for axi-symmetric plumes in an ambient environment with convective heat strength Q=215 K Q=343 II;: Q=729 Wand diameter D= 5 cm was found to agree with the experiments. Also, as expected, the volume flow increased with the increase in the strength of the heat source.

In the case of axi-symmetric plumes in stratified surroundings, plumes with convective heat sources Q=215 K Q=343 K diameter 0 = 5 cm and Q= 729 W with D=5 cm, were numerically simulated. The volume flow rate decreased in the case of axi- symmetric plumes in stratified surroundings compared to that of ambient surroundings. When the “intensity”, of stratification was increased from 0.6”C/m to l.O”C/m, the volume flow rate decreased by approximately 20 percent. This is due to the fact that the density difference between the plume and its surroundings decreases with an increase in the “strength” of stratification.

References Chandrasekhara, M.S. (1983) Study of Rrtical Plane Turbulent

Jets and Plumes, Ph.D. thesis, University of Iowa, Iowa. Chen, C.J. and Rodi, W. (1980) “Vertical turbulent jets: a review

of experimental data”. In: The Science and Applications of Heat and Mass Tramfg, Oxford Pergamon Press, Vol. 4.

146 Shankar et 01.: Numerical Investiqation of Turbulent Plumes in both Ambient and Stratified Surroundinas

Daly, B.J. and Harlow, F.H. (1970) “Transport equations of turbulence”, Physics of Fluids, 13, 2634-2649.

Davidson, L. (1989) “Ventilation by displacement in a three- dimensional room - a numerical study”, Building and Environ- ment, 24, 363-392.

Davidson, L. (1990) “Second-order correction of the k-E model to account for non-isotropic effects due to buoyancy”, Interna- tionalJacrnal of Heat and Mass Transfer, 33(12), 2599-2608.

Davidson, L. and Farhanieh, B. (1992) CALC-BFC: A Finite- volume Code employing CoUocated Va’ariable Arrangement and Car- tes& Elocity Components for Computation of Fluid Flow and Heat Transfer in Complex Three-dimensional Geometries, Gothen- burg, Sweden, Dept. of Applied Thermodynamics and Fluid Mechanics, Chalmers University of Technology (Report 92/ 4).

Farhanieh, B. and Davidson, L. (1 992) A Finite-volume Code em- p b n g CoUocated Variable Arrangement and Rhie-Chm Inter- polation Method for Computation of Three-dimensional Parabolic E;lau in Non-orthgonal Corndinam, Gothenburg, Sweden, Dept. of Applied Thermodynamics and Fluid Mechanics, Chalmers University of Technology, (Report 92/5).

Hanjalik, K. and Launder, B.E. (1980) “Sensitizing the dissipation equation to irrotational strains”, Journal of Fluids Engin- eering, 102 34-39.

Haroutunian, V. and Launder, B.E. (1988) “Second-moment modeling of free buoyant shear flows: a comparison of parabolic and elliptical solutions”. In: Puttock, J.S. (ed.) Stably Stratifid Fzav and Dense Gas Dispersion, Oxford, Claredon Press, pp. 409-430.

Hussain, M.D. (1980) Mathemattiche Modelierung von turbulenten Aufmebsstriimungen, PhD Thesis, University of Karlsruhe, Karlsruhe.

Ince, N.Z. and Launder, B.E. (1988) “Computation of turbulent

natural convection in closed rectangular cavities”. In: Proceed- ings of 2nd UK. National Conference on Heat Transfer, Glasgow, University of Strathclyde, Vol. 2, pp. 1389-1400.

Kofoed, P. (1 99 1) Thermal Plumes zn Ventilated Rooms, Ph.D. thesis, Aalborg, Danmark, Instituttet for Bygningsteknik, Aalborg Universitetscenter, AUC.

Kotsovinos, N.E. and List, E.J. (1977) “Turbulent buoyant jets. Part 1: Integral properties”, Journal of Fluid Mechanics, 81, pp. 25-44.

Launder, B.E. and Spalding, D.B. (1 974) “The numerical computation of turbulent flow”, Computational Methods in Applied Mechanics and Engineering, 3, pp. 269-289.

Leschziner, M. (1982) A n Introduction and Gut& w PASSABLE: A jinite-volume-based code for mo-dimensional boundary-layer flows, Manchester, UMIST

Ljuboja, M. and Rodi, W. (1980) “Calculation of turbulent wall jets with an algebraic Reynolds stress model”, Journal of Fluids Engineering, 102, 350-356.

Malin, M.R. (1989) “Modeling the effects of lateral divergence on radially spreading turbulent jets”, Computers and Fluids, 17(3), 453465.

Patankar, S.V. (1980) Numerical Heat Transfm and Fluid Flow, Washington, McGraw-Hill.

Patankar, S.V. and Spaldmg, D.B. (1972) “A calculation pro- cedure for heat, mass and momentum transfer in three-dimensional parabolic flows”, Internatkmal Journal of Heat Mass Transfer, IS, 1787-1806.

Rhie, C.M. and Chow, W.L. (1984) “Numerical study of the turbulent flow past an airfoil with trailing edge separation”, AIRA Journal, 21, 1527-1532.

Shankar, V. (1 993) Numerical Investigation of Turbulent Plumes in both Ambient and Stratijied Surroundings”, Thesis of Lic. of Engng., Goteborg, Dept. of Thermo and Fluid Dynamics, Chalmers University of Technology (Report 93/1).

numerical investigation of turbulent plumes in both ambient and stratified surroundings

Documents