Prediction of buoyancy controlled turbulent wall diffusion flames

P R E D I C T I O N OF BUOYANCY C O N T R O L L E D T U R B U L E N T WALL DIFFUSION FLAMES LAWRENCE A. KENNEDY AND O. A. PLUMB Faculty of Engineering and Applied Sciences, State University of New York at Buffalo, Buffalo, New York 14214 A numerical model of a buoyancy dominated turbulent diffusion flame adjacent to an inert vertical surface is developed. The K-~-g model of turbulence was modified to account for the generation of turbulent kinetic energy and dissipation due to buoyancy and describes the fluctuations of a conserved scalar quantity. The combustion model assumes infinitely fast reactions and the local burning rate is obtained by solving for the source term in the fuel species equation. Due to the lack of experimental flame data, results are compared to non-reacting flow experimental data with good agreement. Sample results for a propane-air flame are presented.

b u o y a n c y . In the case of reacting flows, large local density differences can be generated and the generation of turbulence b y b u o y a n c y can be significant and should influence the energy exchange. F o r such combustion flows S p a l d i n g 6 suggested that in addition to the turbulent kinetic energy and dissipation rate, a third transport equation for the mean square temperature (or concentration) fluctuations be i n c l u d e d in the model, so called K-e-g model. T h e following work describes the application of this model to an idealized b u o y a n t diffusion flame adjacent to the inert surface.

Introduction At present, fire technology relies heavily on the use of full scale testing d u e to the incomplete u n d e r s t a n d i n g of the processes taking place. Therefore, in order to have a more thorough u n d e r s t a n d i n g of fire dynamics a n u m b e r of laboratories are a t t e m p t i n g to develop numerical and experimental models of the flow field and energy exchange. The present p a p e r describes a numerical model for calculating diffusion flames adjacent to vertical surfaces in a quiescent atmosphere. U p to the present, relatively little attention has been directed towards b u o y a n c y dominated t u r b u l e n t flows even for inert flows. Several investigators have calculated such flows using integral methods (Eckert a n d Jackson, 1 Bayley, 2 Kato et al. a) and these results agree well with experimental data on heat transfer. More recently the Spalding-Patankar m e t h o d has been a p p l i e d to these b u o y a n t flows assuming an e d d y diffusivity distribution, Mason and Seban 4 and P l u m b and Kennedy. 5 In the former, the turbulence was m o d e l l e d b y using Prandtl's mixing length (l) and the turbulent kinetic energy (K) while in Kennedy and P l u m b ' s formulation the t u r b u l e n c e was characterized b y the kinetic energy (K) and the d i s s i p a t i o n rate (~), e.g., the K-~ model. Both studies f o u n d an increase in the turbulence level due to the generation of t u r b u l e n c e b y

Formulation We examine the case of a t u r b u l e n t flame adjacent to an inert, adiabatic vertical surface. The fuel is provided b y a source at the base of the wall w h i c h is characterized by a flow rate Q. The ambient air is quiescent and ignition is considered to b e g i n i m m e d i a t e l y above the fuel source. The transition to turbulence occurs a few diameters downstream and the resulting flow can be d e s c r i b e d b y the b o u n d a r y layer approximation. Figure 1 depicts our configuration. T h e steady state b o u n d a r y layer flow for a m u l t i - c o m p o n e n t system can be written as:

1699

1700

TURBULENT FLAMES AND COMBUSTION and the volumetric heat release rate is 0. Radiation has been neglected even-though it has been shown to be important in turbulent flames. 7 However its effects could be approximated by modifying the heat of combustion. Species:

o~

pU - -

at,

+ pV-

Ox

Otj

WALL

=--

+

ay

ay j

+ m,

(4)

Here species "i" has mass concentration Yi and the volumetric mass generation rate #q. State:

pT= p~r~

(5)

Chemistry: v~(Fuel) + vo(Oxidant ) v~(Products) + Q(Heat)

it

FIG. 1. The schematic of buoyant fuel jet adjacent to inert vertical wall.

Here the v ~'s are the stoichiometric coefficients and Q is the net heat release by v) moles of fuel. The fuel and oxidizer are assumed to always unite in their stoichiometric proportions. Assuming equal exchange coefficients, the quantity

Continuity:

= Yox -

O(pU)

o(pv) +

= o

ax

(1)

Oy

(6)

s YF

(7a)

is a conserved property of the flow (s is the stoichiometric oxygen/fuel ratio). Similarly conserved is the quantity

Momentum: (7b)

la = h + QYF/V'F pU-+ pV-- =-Ox dy Oy

(~ + ~ r ) + a~(p~-

p)

(2)

Energy:

Equations (3) and (4) can then be written in terms of these quantities and the subsequent equation expressed as

ou ay ....

+ pv

Ox

oh oh pU+ pVOx

of

+

--

(s/

Oy

where the mixture fraction f at any point is defined by +

ay

+ (i (3) f = -

where the specific enthalpy is h =

=

O{t

f

T

Cp d T T~

-

~o-~

(9)

~o-a~

The subscripts o and ~ are respectively the value at the jet and the surrounding air. The mass fraction of the inerts is given by

TURBULENT WALL DIFFUSION FLAMES

YN -= YN,~ + (YN,o -- YN,~)f

(10)

The mass fractions and temperature (Yox, YF, Yr,"and T) may be obtained from f f o r a given combustion model.

Uc3~ p

Ox + P 0y = 0g L% 0y + C~KIXTkO~t /

(11)

where C~ is a constant of proportionality. 9 The assumptions of high Reynolds number where the flow tends toward isotropy is essential to the development of equation (11). In order to account for the contribution of buoyancy to the turbulent kinetic energy and dissipation rate a third transport equation for the mean square fluctuations of the mixture fraction must be included in the model. Thus, at high Reynolds where the dissipation process is essentially isotropic and the effects of molecular viscosity are negligible, the transport equations for the three turbulence parameters of interest can be written in the boundary layer approximation as follows: 9.

VOK + p

Ox

0 [ Ixt OK ] ~

Oy

-

-

-

-

Oy I_orr, cgy J

+ IXT

-

C2--

K

+ Ca pagl3 -K- (u?)

Numerous authors have proposed closure models for turbulent flows in an attempt to accurately predict the turbulent shear stresses. Prandtl and Kolmogorov proposed that the turbulent viscosity should be proportional to the square root of the turbulent kinetic energy and a length scale representative of the energy containing eddies. A transport equation for the turbulent kinetic energy can be derived from the Navier-Stokes equations through Reynolds decomposition, Various authors have developed transport equations for different variables in order to determine the length scale. In the K-e models characterized by that of Jones and Launder s the length scale is taken to be the dissipation length scale (e = Ka/2/L). Thus, the turbulent viscosity can be written

UOK

-

E

The Turbulence Model

IXT = C, p Ke /e

1701

-- I~ + Paal3 u-t (12)

UOg

VOg

0 [~.1,T Og ]

OX

Oy

Oy k(r ~ Oy J

--+p

+CgltXT \d~ /

(13)

-- C g 2 P K g (14)

An appropriate model for the correlation between temperature fluctuations and vertical velocity fluctuations must be determined in order to close the system of equations. Following the arguments presented by Launder and Spalding 10 & = c 4 ( g K ) ~/2

(15)

Equations 12-15 along with the equations of continuity, momentum and energy form a closed system of equations, describing a turbulent buoyancy driven flow at high turbulent Reynolds number. The r represent turbulent Prandtl numbers for the parameters in question and the C's are model "constants" preceding terms which are inexact in that their derivation relies heavily on dimensional arguments. This model for the contribution of buoyancy to the turbulent kinetic energy has also been proposed by Taminini 11 in calculating a turbulent diffusion flame. The model constants used in the calculations are given in Table I. Numerical values for C,, C1, C~, crK and ~, are those recommen~ted by Jones and LaunderY By assuming similar contributions from buoyancy and gradient production terms, C 3 can be shown to equal C r The value for C 4 was chosen based upon the experimental data of Smith. The constants Cgl , Cg,a and ~rg are those used by Tamini in turbulent diffusion studies. In order to predict accurately the behavior of the flow near the wall, particularly in the

TABLE 1 Model constants for high Reynolds number turbulence model C~

C1

C2

Ca

C4

Cg,

Cg~

%

%

r

.09

1.44

1.92

1.44

.5

2.8

1.7

1.0

1.3

.9

1702

TURBULENT FLAMES AND COMBUSTION

viscous sublayer, where the turbulent Reynolds n u m b e r is small, the effects of molecular viscosity and nonisotropic dissipation must be taken into account. T h e former can be easily a d d e d by its inclusion in the diffusive terms in all of the transport equations. The wall functions and wall terms w h i c h are a d d e d to account for the n o n i s o t r o p i c behavior near the walls are those of Jones a n d L a u n d e r s with one exception. They suggest a d d i n g an additional wall term to the equation for the dissipation rate (circled term in eq. 17) to give better agreement with the experimental results for the turbulent kinetic energy. However, in the present calculations, this term was d r o p p e d because it cannot be justified from a physical standpoint in b u o y a n t flows. The resulting equations depict the complete turbulence model.

- O~ + p a g ~ C 4 ( g K ) 1/2 + Ixr \ 311 / OK ~/2 12

- 2ix \ T / +p-3x

3!r

(16)

ix + tx--~ r cr /

3!t

"4- C1F1KId~T \011 ,1 -- C2F2~ K

flow. The assigned forms of Jones and L a u n d e r are used in the present calculation of these three functions, i.e., F 1 = 1.0 F 2 = 1.0 - .3 exp (--Re2T) F

= exp [ - 2 . 5 / ( i F 3 = 1.0

- - + p - 3x 311

311

~

(

1

7

y

)

+-trg E

+ Cg~TK311/

-2k

(3gl/212 \311

oK 2 ],LT = C F -

(18)

/ (19)

E

T h e wall functions F1, F 2 and F which are functions of the t u r b u l e n t Reynolds n u m b e r were chosen b y Jones a n d L a u n d e r b y a p p l y i n g the model to well d o c u m e n t e d turbulent forced

(20)

where the turbulence Reynolds n u m b e r is defined as, Re x = p K 2 / ~ . . The function F 3 was not evaluated b y Jones and L a u n d e r since they d i d not deal with b u o y a n t flows. In this study based u p o n d i m e n s i o n a l arguments F 3 was given a value of unity; however, further work may reveal that its o p t i m u m value is otherwise. The wall term a p p e a r i n g in equation (16) is again that p r o p o s e d b y Jones and Launder. Its necessity arises from the need to assign a b o u n d a r y condition to e at the wall. This b o u n d a r y condition was set equal to zero for lack of an exact prediction. Thus, ~, in equation (17) can be considered the isotropic part of the dissipation rather than the total dissipation. With this a s s u m p t i o n it can be shown that the wall term a p p e a r i n g in equation (16) will give the correct b e h a v i o r of the turbulent kinetic energy in the near wall region. A similar argument can be used to derive the wall term shown in equation (18) for the mean squared temperature fluctuations. Hoffman12 has shown that the wall term in equation (16) can be written alternately as 2p. OK

+

+ ReT/50) ]

(21)

311

to again get the correct behavior of K near the wall. Whether (21) or the term shown in (16) is used makes little difference in the final results. Calculations were m a d e with several different values of cr T. F o r p l a n e jets and plumes, ~ r = .5 has been shown to give results w h i c h agree well with experiment, thus, one might assume that this is an appropriate value, particularly outside of the velocity maximum. F o r non-reacting flows the experimental data of Smith 13 and the works of Blom and Pal and Whitelaw discussed in Ref. 10 indicate that a T should be s o m e w h a t higher inside the velocity maximum. In the related heat transfer p r o b l e m various constant values of ~rr were examined and the choice which gave best agreement with t h e heat transfer data was

TURBULENT WALL DIFFUSION FLAMES ~ r = 2.5 - 2.0

y/8

(22)

This value was used in the flame calculations. Time-Mean Mixture Fraction

The flame sheet approximation is invoked so that fuel and oxidant will c o m b i n e to yield a single product whenever they simultaneously exist at a point. Together with assuming equal exchange coefficients, the instantaneous values of the enthalpy and species may then be linearly related to f. W h e n calculating the time-mean values of the flow properties, their fluctuations should be taken into account. Since the governing equations provide only the time mean and mean square fluctuating values, it is necessary to assume a variation of f with time. Various authors have e m p l o y e d different models for this dependence. However this study uses the Gaussian distribution d e s c r i b e d b y Elghobashi and Pun 14 from w h i c h the mean value of a fluctuating quantity 0(f) can be evaluated by = J

104

l

df

0(f) p ( f )

(23)

o

Here p(f) is the p r o b a b i l i t y density function. The time averaged mass fractions, temperature and density can then be determined. Details of this procedure can be f o u n d in reference 14. Once the distribution of average fuel mass fraction has been determined, the local burning rate then may be calculated by solving the fuel species equation for its source term.

Discussion and Results The Patankar-Spalding finite difference p r o c e d u r e was used for the solution of the governing set of equations after performing a v o n Mises transformation. The advantage of this technique is that it allows the grid to expand with the b o u n d a r y layer and the resulting tridiagonal system is r a p i d l y inverted. The calculations reported were initiated in the laminar region and a small amount of t u r b u l e n t kinetic energy introduced at a point w h i c h yielded a numerical transition coincident with the experimental transition to turbulence. The model was also checked by calcnlating the heat transfer to a vertical wall for a non-reacting buoyant flow w h i c h agreed with experimental data (Fig. 2),

I

I

1703

I

i

_

10

Nu x

j

Nu x=.3"75

I

1 8 10

-

/fO

__

lo

j

2

Gr x

I 9

10

______

Nu x =, 022 Grx /s

(Eckert and Jackson)

__.__

NUx =,114 GrxI/s

(Bayley)

0

K-r

9

Cheesewright

I 10

Model (Experiment)

I C~ 1

lO

1

12 10

1

Gr x

FIG. 2. Comparison of predicted heat transfer results for a non-reacting buoyant flow adjacent to an isothermal wall. Local Nusselt number vs local Grashof number.

TURBULENT FLAMES AND COMBUSTION

1704

The wall terms a n d wall functions act to damp the turbulent kinetic energy in the viscous sublayer where the molecular viscosity dominates the t u r b u l e n t transport. In inert forced flows, the importance of these wall terms decrease as one moves towards the free stream values. However, in buoyant flames, the maximum velocity occurs within the b o u n d a r y layer and these wall terms will also influence the d a m p i n g at the outer edge of the b o u n d a r y layer. Thus in order to overcome this difficulty the wall functions and wall

.4

i

terms were only applied out to the velocity maximum. The calculations presented utilized 80 cross stream grids. In order to check the adequacy of this n u m b e r runs were made with 40 grids in the turbulent region with no appreciable change in the results. The grid network was not evenly distributed across the b o u n d a r y layer. Roughly half of the grid points were inside of the velocity m a x i m u m where steep gradients of all of the flow variables occur. In addition it was essential to maintain several

I

I

1

I

i

I

I 1,,2

I

I 16

I

I 1.8

I

I 1.2

I

I 1,/B

I

I 1~8

,3 o

o,

,1

I 0

I .4

I

I .8 3a

,3

(t~ AT

l 0

I ,4

I

I ~8

Y/Yo.s 3b FIG. 3. (3a) Comparison of predicted values of K and experimental (fi,2)1/~ and (35) comparison of predicted and measured temperature fluctuations ({2)~/2 for a non-reacting buoyant flow adjacent to an isothermal wall. The distance y from the surface at which U~ U.... = 0.5 is indicated by Yo.5; O and 9 are Smith's data for Gr~ = 4.84 x l0 w and 6.81 x 10 l~ respectively. The cuives are values predicted by the K-~-g model, - - Gr x = 6.3 x 1 0 1 ~ - - - Gr x = 4.0 x 10 n.

TURBULENT WALL DIFFUSION FLAMES grids within the viscous sublayer in order to obtain satisfactory results. F o r w a r d step sizes of 4 or 5% of the total b o u n d a r y layer thickness proved to be effective; larger forward steps led to decreased accuracy. The entrainment rate at the outer edge of the b o u n d a r y layer was calculated from the m o m e n t u m equation w h i c h can be rewritten

P VE = - -

(~ + ~ r ) - - - ~

to

Oto

+ agf~(T- T~) U where to = ~ / ~ E ,

OU l 3U (24) Ox J Oto

here WE is the value of

14001

1705

the stream function at the outer edge of the b o u n d a r y layer. Equation (24) is solved at the second grid point from the outer edge. At this p o i n t the 0 U/Ox term and the b u o y a n c y term are negligible which simplifies the computation. Typical c o m p u t i n g time required to produce a solution was about 12 minutes on a CDC 6400. Since the authors know of no measurements of f and g in b u o y a n t diffusion flames, an alternate check was p e r f o r m e d on these timemean quantities by s h o w i n g that the method gives satisfactory predictions in non-reacting b u o y a n t flows. The calculated kinetic energy and mean square temperature fluctuations are c o m p a r e d with the experimental data of Smith

I

25

/

f

/ / 2O

1200

/ 1000

250-

15

I 200-

I I

o 150-

i-600

/ /

V

I 100-


It.

\

\

/

I

400

<

--10

I

Z

O I-, O

800

0

~^

"V'\

05 50-

200

/ //

/ /

I

I

I

4

6

8

0

Y/,r FIG. 4. The mean temperature, the rms temperature fluctuations, and the mass fraction of fuel and air as a function of distance from the surface, y/8, for an axial distance of x/Dia of fuel inlet of 100.

1706

TURBULENT FLAMES AND COMBUSTION

in Fig. 3. Both this data and the calculation indicates a m a x i m u m turbulent intensity slightly less than 0.3. Vliet and Liu a5 experiments yields a value of approximately 0.3 while Kutateladze's 16 data for ethyl alcohol gives a maximum intensity of 0.36. The model results and the experimental data agree closely w i t h the only d i s c r e p a n c y arising in the kinetic energy very close to the wall where either the data a n d / o r the m o d e l could be suspect. The mean temperature, RMS temperature fluctuation and the mass fractions were calculated for a p r o p a n e b u o y a n t jet adjacent to a wall. Representative profiles are given in Fig. 4. With increasing distance from the surface, the temperature fluctuations exhibits a maximum. This m a x i m u m lies in a region where the air a n d fuel vortices are b o t h numerous and large. T h e m a x i m u m value of these fluctuations are of the order of 200~ w h i c h is 15-20 percent of the m a x i m u m mean temperature. Relative to the fluctuating temperature maximum, the peak in the mean temperature curve lies slightly nearer the wall where reacted fluid predominates. Conclusions It has been d e m o n s t r a t e d that a mathematical model can b e d e v e l o p e d which accounts for fluctuating scalar quantities and their effect on the time-mean quantities in b u o y a n c y dominated flames. T h e K-e-g turbulence m o d e l was m o d i f i e d to i n c l u d e the effects of b u o y a n c y on the generation of turbulent kinetic energy and dissipation a n d in the absence of flame data, the results were c o m p a r e d with nonreacting flow measurements. The model is still b e i n g developed and must be expanded to i n c l u d e an explicit radiation model and to relax the assumption of an inert wall to a more realistic fire condition. Nomenclature Cv C1, C2, Ca, C ,

specific heat

Cql, C qz, C~

m o d e l constant wall functions mixture fraction m e a n square fluctuation of the mixture fraction enthalpy kinetic energy of t u r b u l e n c e (T.K.E.) thermal conductivity Lewis number

F1, F2, F3, F~ f g h K k Le

M rh~ Q T t U u V VE Y~ 13 9 Ix Ix'r p q~

molecular weight mass source per unit v o l u m e of species i net heat release b y v ~ m o l e s of fuel m e a n temperature fluctuating temperature m e a n vertical velocity fluctuating vertical velocity m e a n horizontal velocity entrainment velocity mass fraction of species i coefficient of thermal expansion d i s s i p a t i o n rate of T.K.E. t u r b u l e n t viscosity t u r b u l e n t viscosity density Prandtl and Schmidt n u m b e r s stream function

Subscripts F OX P T i O

fuel oxidizer products turbulent species c o n d i t i o n in environment c o n d i t i o n in fuel REFERENCES

1. ECKERT, E. R. G. ~'~D JACKSON,T. W., NACA TN-2207, 1950. 2. BARLEY,F. J., Proe. Inst. Mech. Engrs. (London), 169, 361 (1955). 3. K,TO, H., NlSmWAK1,M. ANDHIaaTA,M., Intern. J. Heat Mass Transfer, 11, 1117 (1968). 4. MASON,H. B. AnD SEBAN,R. A., Intern. J. Heat Mass Transfer, 17, 1329 (1974). 5. PLUMB,O. A. AND KENNEDY, L. A., "Application of a K-e Turbulence Model to Natural Convection from a Vertical Isothermal Surface" to be presented ASME-AIChE National Heat Transfer Conference, St. Louis, Mo., Aug. 1976. 6. SPALDING,D. B., Chem. Engr. Sci., 26, 95 (1971). 7. MAnKSTEIN,G. H., Fifteenth Symposium (International) on Combustion, p. 1285, The Combustion Institute, 1974. 8. JoNEs, W. P. ANDLAUNDEa,B. E., Intern. J. Heat Transfer, 15, 301 (1972). 9. RODI,W., Trans. Am. Soc. Mech. Engrs., Series I, 97, 386 (1975). 10. LAUNDER,B. E. ANDSPALDING,D. B., Mathematical Models of Turbulence, p. 386, Academic Press, 1972. i1. TAMININI, F., " O n the Numerical Prediction of Turbulent Diffusion Flames," Paper presented

TURBULENT WALL DIFFUSION FLAMES at the Central and Western State Section of the Combustion Institute, San Antonio, Texas, April 1975. 12. HowMa~, G. H., Phys. Fluids, 18, 309 (1975). 13. SMITH,R. R., "Characteristics of Turbulence in Free Convection Flows Past a Vertical Plate," Ph.D. Thesis, Queen Mary's College, University "of London, 1972.

1707

14. ELGHOBASH1,S. E. ANn Pu~, W. M., Fifteenth Symposium (International) on Combustion, p. 1353, The Combustion Institute 1974. 15. VL1ET, G. C. AND LIU, C. K., J. of Heat Transfer, 91,517 (1969). 16. KUTALELADZE,S. S., KI~DYASHKIN,A. G. ANDIVAKIN, V. P., Intern. J. Heat Mass Transfer, 15, 193 (1972).