An integral model of turbulent non-premixed jet flames in a cross-flow

Twenty-Third Symposium (International) on Combustion/The Combustion Institute, 1990/pp. 653-660

AN INTEGRAL

MODEL OF TURBULENT NON-PREMIXED FLAMES IN A CROSS-FLOW

JET

D. K. COOK British Gas plc. Midlands Research Station Solihull B91 2JW, England

An integral model for calculating the structure of turbulent non-premixed jet flames in a cross-flow is presented. Entrainment rates are determined by summing contributions representing entrainment due to longitudinal and transverse shear. The turbulent non-premixed combustion process is modeled via the conserved scalar/prescribed probability density function approach and the laminar flamelet concept is used to specify the instantaneous thermochemical state of the combusting mixture. The model includes a representative profile for radial variations of mixture fraction, giving realistic predictions of fuel consumption rates, whereas previous integral models, based on top-hat profiles, have required tuning or additional empirical constants. Solutions of the model are compared with experimental data for reacting and non-reacting natural gas jets obtained from wind tunnel studies. For one particular cross-flow to jet velocity ratio, satisfactory agreement between theory and experiment is obtained for mean concentrations of the non-reacting jet and mean temperatures of the flame, providing the effects of flame lift-off and radiative heat loss are included in the latter case. Over a range of velocity ratios, predictions of flame trajectory and length are in good agreement with experimental data and with three-dimensional computations which provide a complete description of the flow but require significantly more computing resources. Introduction For routine hazard assessment work, the consequences of heat transfer from flames are usually estimated using simple empirical techniques 1-3 which have limited ranges of applicability and provide little insight into the physical and chemical processes involved. Although it has been demonstrated that finite-difference methods can successfully simulate flame structure 4, they require considerable computing resources and, therefore, are unsuitable for repetitive engineering calculations which must be performed relatively quickly. Integral modeling provides an intermediate approach, between empiricism and full three-dimensional simulations, which includes the concepts, but requires only a small fraction of the computing resources, of the latter. Previous integral models of flames~-7 have usually related flame structure to the time-averaged concentration of jet fluid or mixture fraction, completely ignoring turbulent fluctuations or unmixedness, such that complete combustion is considered to have occurred when the mixture fraction decays to the stoichiometric value. These models considerably underpredicted flame lengths, unless a constant fraction of the entrained oxygen is assumed to be inert or unreacted; this fraction being tuned em653

pirically to observed flame lengths. Tamanini s made a significant advance by developing an integral model in which unmixedness was allowed for using a probability density function (p.d.f.) of a conserved scalar and an integrated one-dimensional formulation of the k-~ turbulence model. However, Tamanini's turbulence model had to be adjusted to give realistic fuel consumption rates. A common simplification, applied to all the models discussed above, is to assume that the cross-stream variation of flow properties can be represented by a top-hat profile which neglects radial variations, p a r t i c u l a r l y in concentration. In a previous publication9, the author has shown that for axi-symmetric jet flames it is not only necessary to model temporal variations of concentration, with an assumed p.d.f., but also to model spatial variations with a representative radial profile. In the present paper, this work is extended to cover wind-blown jet flames. Description of the Model A detailed description of the model of jet flames in a quiescent environment can be found in the previous publication9; the emphasis here being placed on the additions required to extend the model

654

TURBULENT COMBUSTION

to wind-blown releases. The former model is essentially similar to that of Tamanini, s but has been developed to include radial variations of velocity, mean mixture fraction and mixture fraction variance, and also to include a more realistic combustion sub-model.

d

('V~/sin t9)

=

reeVe,

(4)

and

d

.

(W cos t9) = g(p= - (0))IrR ~.

(5)

The Conservation Equations: The conservation of mass and momentum are expressed in terms of the total mass and momentum fluxes, rh and W, passing through any cross-section normal to the mean trajectory of the jet flame. Then, continuity gives:-

ds

R

=

rhe.

(1)

In common with similar integral models of reacting6'7 and non-reacting l~ jets in a cross-flow, several entrainment mechanisms are considered independently, with the total entrainment given as the algebraic sum of these various components:

fo[O(

r -

(0) =

drh --

Equations (2) and (5) introduce a spatially-averaged density given by:-

p=]rdr

R

(6)

fo rdr and the jet radius R which is related to the prescribed radial profiles for velocity, mixture fraction and mixture fraction variance. Terminating cosine profiles 11 have been employed so that the velocity and all the scalar variables approach ambient values at r = a:-

,he = ck~r(~ + (r

fi(r) = v| sin t9 +

+ 2~r(a)p={(O)/o~} 1/2 af3~lv= cosOI.

+ p=,

ticS(r/R)

(7)

(2)

In the near field, where the jet momentum dominates, entrainment will be equivalent to that of a free jet, represented by the first term on the righthand side of Eq. (2), in which entrainment is related to the turbulent viscositY,12p~, following the analytical solution of Schlichting with the empirical constant, Ck, taking a value of 5.5 for momentum-dominated releases 13 . In the far field, where the cross-flow d o m i n a t e s , e n t r a i n m e n t is assumed ~'7'1~ to be proportional to the component of the crossflow velocity, v| normal to the jet trajectory as given by the second term on the righthand side of Eq. (2). The empirical entrainment constant, a, takes a value 6 "11 of 0.7 and a function, 13~, has been introduced to ensure that the second term does not dominate in the near field, resulting in excessive entrainment. This takes the following

f (r) = .(c[S(r/R) ]1/x

(8)

f"2(r) = (f"2)c[S(r/R)ll/X

(9)

where S(r/R) = 0.511 + cos(qrr/R)]

(10)

and h 2 is related to a turbulent Prandtl/Sehmidt number. In addition, an average jet radius (R) is more appropriate for the second term of Eq. (2) and is defined by the same relationship used by Tamanini 8 for top-hat profiles:rh (R) = ~ ~ "-:'~/2"

(11)

forln:-

- mo/v:]": 1

(3)

so that the second term of Eq. (2) is not fully introduced until the total momentum of entrained air exceeds the source momentum of the jet. A similar approach 1~ has been adopted in the past for nonreacting flows, The conservation of horizontal and vertical momentum is given by:-

The mixture fraction, f, is defined as the mass fraction of material in the mixture which originated in the fuel stream and, since f is invariant to combustion, the conservation of species is satisfied if the total flux of mixture fraction, F, is constant and equal to the source mass flux, #~o. Equations (1), (4) and (5) can be solved, in conjunction with suitable turbulence and combustion sub-models, to give the total mass and momentum fluxes as functions of s, with the vertical and horizontal components of momentum flux defining the jet trajectory. However, to ensure that the result-

JET FLAMES IN CROSS-FLOW i.ng structure calculation also conserves ~h, W and F, the following expressions must also be satisfied:-

655

(~,) = C.(0) ((~)~.

(19)

R

(12)

rh = 2~r fo P(r)u(r)rdr' R

(13)

"(V = 2~r fo O(r)[ft(r)]2rdr, R

= rho = 2~r fo O(r)a(r)j[r)rdr.

(14)

Apart from the evaluation of Pk, which includes the radial velocity profile, the turbulence model is essentially a top-hat model providing no information concerning the radial variations of k and ~. The values assigned to the empirical constants C~, C,I and C,2 are given in Table I and were obtained by Lockwood and Naguib 14 by tuning a two-dimensional parabolic model to axi-symmetric reacting and non-reacting jets.

The Combustion Sub-Model:

(15)

A widely-applied approach in the modeling of nonpremixed turbulent reacting flows15 is to assume fast chemistry so that the instantaneous scalar fields can be determined as unique non-linear functions of a conserved scalar, such as mixture fraction. Since these functions are highly non-linear, it is not possible to derive mean properties from temporal or spatial averages of the mixture fraction field. Temporal fluctuations are accounted for by prescribing a two-parameter 13-p.d.f.m for mixture fraction in terms of the density-weighted mean and variance of the mixture fraction. As demonstrated previously9, the only consistent way of formulating an integral model of flames is to solve for spatially-averaged mixture fraction and mixture fraction variance, prescribe radial profiles and apply local values to the p.d.f, in the combustion sub-model. This approach gives a realistic fuel consumption rate and the resulting density field can be integrated using Eq. (6) to give a consistent spatiaUy-averaged density, which leads to satisfactory predictions of the velocity field when substituting back into the vertical momentum equation, Eq. (5). ,.~e spatially-averaged mixture fraction variance, (f,,2), is evaluated by solving the following transport equation which is analogous to the integrated k-r equations:-

(10)

TABLE I Values of the empirical constants in the model

The Turbulence Sub-Model: A turbulence model is required for evaluating the turbulent viscosity, ~ , in the first entrainment term of Eq. (2) and also for evaluating the mixture fraction variance to define the p.d.f, in the combustion sub-model. The integrated one-dimensional k-e model presented with the previous axi-symmetric integral model9 has been retained without modification. Therefore, the influence of the cross-flow on fine-scale turbulence is ignored, with the wind-blown jet simply considered as a free axi-symmetric jet advected by the cross-flow. This approximation is justified for determining the first entrainment term of Eq. (2), which only dominates in the near field before the fully three-dimensional nature of the flow has been established. However, the mixture fraction variance is required over the whole of the reacting flow and it has been found that representative values are obtained in regions close to the flame tip, where the above approximation is less satisfactory. The model solves the following transport equations for (k) and (e):-

d

"7-((k)rh) = P k - Dk as d

(~)

Non-reacting

Reacting

Ck a h2 C~ C,1 C,~

5.5 0.7 1.35 0.09 1.44 1.84

5.5 0.7 0.75 0.09 1.44 1.79

c~1

2.8

2.8

where

P/~ = 2"tr(Iz~)

fo (

drr [a(r)]

rdr

(17)

and D/, = (15)(~) ~r(R)2.

(18)

(p~) is defined by tile usual algebraic expression:-

C~z •

1.84 --

1.79 0.15

TURBULENT COMBUSTION

656

-~d (( f~ '~)rhl = C,t[2~r(~)foa(d[f(r)])2rdr] -C.2[(O)~)(f""~)~r(R)'Z],

(20)

where C,,x and C,,z are empirical constants which again take values recommended by Lockwood and Naguib 14, given in Table I. Then, the centre-line mixture fraction variance, (f"2)c, is determined by satisfying the following relationship:-

21r (f"~) =

f,,2 (r)rdr (21)

~r(R) z

so that f"2(r), as specified in Eq. (9), is consistent with the solution of Eq. (20). ltaving defined f(r) and f"2(r), it is possible to construct the p.d.f., P(f, r), at any radial location within the flame. This p.d.f, is a density-weighted function which allows evaluation of both densityweighted and unweighted means from:t" x

r)df

(22)

+(f) P(f, r)df d~(r) = 15fo xp-~

(23)

= | +(f)P(f, Jo and

respectively, where ~b is any scalar quantity that can be uniquely related to f. The mean density is given by:[ f x p(f, r) f ] - t O(r) = [Jo P(f) d

(24)

The non-linear functions o f f which determine ~b(f) are specified using the laminar flamelet concept xT, with flamelet prescriptions derived from computations of counterflow diffusion flames with a global four step reaction scheme 18. The computations presented below are based on a methane flamelet prescription corresponding to a moderate strain rate of 100 s -1. A discussion of the suitability of this flamelet prescription, and the instantaneous densities mad temperatures used, is given elsewhere 4. For computations of the natural gas flames studied by Birch et al xg, considered further below, the flamelet prescription was adjusted to include radiative heat losses using the method suggested by Crauford et alz~ who modified instantaneous temperatures via:T(f) = Ta0a(f)[1 - •

.... }a],

(25)

where • is a constant adjusted to account for radiative heat loss. In common with three-dimensional computations of these flames using the same flamelet prescription 4, a value of X of 0.15 was taken, resulting in predictions of peak mean temperatures which are in agreement with observations. Flamelet densities were then corrected using the equation of state, assuming constant pressure and molecular weights. In addition, flame lift-off was included using an approach based on the empirical finding of Chakravarty et a121 that, for flames in a quiescent environment, turbulence time scales are approximately invariant at the base of lifted natural gas diffusion flames. Therefore, combustion was introduced at the downstream location where (k)/(~) > 5 x 10 -3 s, with properties in the lift-off region being evaluated assuming isothermal mixing, giving a linear relationship between f and 1/p.

Computational Details: The model consists of six coupled ordinary differential equations which are solved using a fourth order Runge-Kutta technique. Initial conditions for the turbulence sub-model are typical of fully-developed pipe flow and (f,,z) is initially zero. The combustion sub-model is invoked to calculate the density field 15(r) in the cross-stream direction 'after each step, including internal steps of the RungeKutta routine. The centre-line mixture fraction, f~, and jet radius, R, are adjusted to satisfy Eqs. (12), (13) and (14) before the density, field is averaged using Eq. (6) to give (Ib) and the calculation can proceed to the next step. Typical run-times of about two minutes for combusting releases have been achieved on a VAX 8530, compared to three-dimensional computations 4 which required tens of hours. The prescribed radial profiles of Eqs. (7), (8) and (9) are not representative in the potential core of the jet, leading to velocities greater than (uo) and mixture fractions greater than unity. Therefore, values of velocity and mixture fraction are restricted to these physically meaningful limits. The pedbrmance of the turbulence sub-model was also improved by introducing an additional factor of Min[1,0.5 x (uo)/fi(0)] to reduce Pk in the developing flow region with the analogous factor of Min[1,0.5 • (fo)/J~0)] in the transport equation for (f,,2). The form of these factors was determined by comparing model predictions with experimental data and with predictions of a two-dimensional model for a free non-reacting jet. Results and Discussion

The experimental data 19 used to evaluate the model was obtained from a wind tunnel study of

JET FLAMES IN CROSS-FLOW reacting and non-reacting jets of natural gas in a uniform cross-flow with a low turbulence intensity o f - 0 . 5 % . Detailed measurements of concentration in the non-reacting jet and mean temperature in the reacting jet were made for a vertical release with an average velocity of 76 ms- t from a circular crosssection pipe with a diameter of 10.26 mm and a height of 298 mm into a 5 ms-l cross-flow. More global measurements were also made of flame length and trajectory over a range of jet to cross-flow veiocity ratios. To evaluate the turbulence sub-model and the various entrainment terms of Eq. 2, the model was applied to the non-reacting jet by assuming isothermal mixing and not invoking the combustion sub-model. Figure 1 shows a comparison of concentrations measured on two cross-stream planes, at downstream locations ofx = 0.175 m and 0.250 m, with predictions of the model. At these downstream locations, in a release with such a low jet to cross-flow velocity ratio, the influence of the vortex pair on the observed concentration field is selfevident. Although the radial symmetry of the prescribed profiles in the model cannot accommodate the presence of these vortices, it is encouraging to note that the position and spreading rate of the jet are predicted reasonably and that the centre-line peak concentrations given by the model are close to the maximum concentrations observed off-axis. The inclusion of ((0)/p~) 1/2 in the second entrainment term, as introduced by Escudier s, has a minimal effect on model performance for non-reacting releases of natural gas, in which the mean density of the jet quickly approaches that of the entrained air. However, this is not the case for reacting flows, as shown in Fig. 2 by the comparisons of predicted profiles of mean temperature on the symmetry plane of the flame with measurements at four downstream locations. If the ((15>/p=)l/~ factor is excluded the model underestimates jet penetration into the cross-flow and the peak temperature o ,o

--~

\\

E ' ~ o2b

o 2o

,k)J) \ V / J

ol5

ooe

o ~o

oos

o~

o.l~

oos

o.h

FIG. 1. Contours of mean concentration on two cross-stream planes of the non-reacting jet at downstream locations of x = 0.175 m and 0.250 m (contours correspond to volume %).

657

x=O.125m v

2250

~ 2000 ~ 1750 '~ 1500 ~. E ~ ca

1250 1000 750 500

)

2250 2000 1750 1500 1250 1000 750 500

x=O.3OOm

0.4 x=O.5OOm v ~ ~ "~

2250 2000 1750 1500

~. ~ c o z~

1250 1000 750 500

0.4

2250

x=O.9OOm

2000: 1750 1500 1250 IOO0 75O 5OO 0.4 z/m

0.4 z /

m

FiG. 2. Profiles of mean temperature on the symmetry plane of the flame at four downwind locations (O measured, - predicted with ((tb)/p~)'/2 in the second entrainment term, - - - - - - predicted without ((lS)/p~)'/2 in the second entrainment term). at x = 0.9 m which is close to the observed luminous flame tip, quoted as x = 1.05 m. In fact, in this form the model predicts a greater penetration for the non-reacting jet than for the flame, which is not supported by the experimental data. These observations suggest that the rate of entrainment in the flame is excessive and the inclusion of ((15)/ p=)l/2 reduces the value of the second entrainment term so that the model satisfactorily predicts the penetration of both the reacting and non-reacting jet. Furthermore, the model gives a more representative prediction of peak temperatures, particularly at x = 0.9 m, provided that the flamelet prescription has been adjusted for radiative heat loss, as discussed previously. Both of the computations presented in Fig. 2 were performed for lifted flames, with the observed lift-off height being predicted accurately. If lift-off is not included, the model tends to overpredict jet penetration due to reduced entrainment in the development region, a trend which was also observed in three-dimensional computations of this flame4. Mean temperatures measured on two cross-stream planes of the flame, at downstream locations of x = 0.25 m and 0.5 m, are compared to model predictions in Fig. 3. The measured temperature con-

TURBULENT COMBUSTION

658

[PeO

c'ecl

05

04 "03 02 01

)) 02

r ~>:, I

01

o:1

02

01

02

T/m

FIG. 3. Contours of mean temperature on two cross-stream planes of the flame at downstream locations of x = 0.25 m and 0.50 m (contours range from 600 K, outer, to 1600 K, inner, in 200 K intervals). tours also indicate the influence of the vortex pair, whereas the model assumes radial symmetry. However, as for the non-reacting jet, the penetration and spreading rate of the flame are represented satisfactorily by the model. Figures 2 and 3 demonstrate that the model as formulated reasonably predicts the temperature field of a wind-blown jet flame without considering the three-dimensional nature of the flow. Finally, the overall performance of the model has been investigated by comparing measured and predieted flame trajectories over a range of jet to crossflow velocity ratios, as shown in Fig. 4. These flames were produced using the vertical pi~e described above with a jet velocity of 100 ms- , except for the lowest velocity ratio where the jet velocity was

1.B

u. // ~ , , I

/v~

=IOQ

I

1.4 1.2

l/ II

,50

o// oe

~

- -

meas~ea

- ~ --

DreOicted

I "/

02 Z7t ~ 0.0oo o~ 0.,A

30

~

~

.....

xlm

'0

,;

,',

FIG. 4. Flame trajectory over a range of jet to cross-flow velocity ratios.

reduced to 60 ms -1 to avoid flame blow-off. The measured trajectories have been chosen to be representative of the length and position of observed flame envelopes obtained from photographic data averaged over a period of 10 s and, as such, are relatively qualitative. Consequently, apart from the highest velocity ratio release, where the model underpredicts the observed flame deflection, the comparisons can be considered to be satisfactury, exhibiting similar agreement to that obtained with the three-dimensional model4. Similar agreement is observed for the trajectories of non-reacting releases over the same range of velocity ratios. The predicted trajectories in Fig. 4 have been terminated at the observed flame length since the association of a temperature with the flame tip is not straightforward. However, as shown in Fig. 2, the model closely predicts the peak temperature on the symmetry plane of a low velocity ratio flame in the region of the observed flame tip. In addition, the predicted distance along the trajectory to a given temperature increases with velocity ratio, due to reduced entrainment, and is in qualitative agreement with the three-dimensional computations 4. These observations suggest that the model predicts a realistic fuel consumption rate and flame length over a range of velocity ratios.

Conclusions An integral model has been developed for calculating the structure of turbulent non-premixed jet flames in a uniform cross-flow. The model includes a representative profile for radial variations of mixture fraction, giving realistic predictions of fuel consumption rates, whereas previous integral models, based on top-hat profiles, have required tuning or additional empirical constants. Predictions of the model have been compared with experimental data for reacting and non-reacting natural gas jets obtained from wind tunnel studies. For one particular jet to cross-flow velocity ratio, satisfactory agreement between theory and experiment has been demonstrated for mean concentrations of the non-reacting jet and mean temperatures of the flame. Over a range of velocity ratios, predictions of flame trajectory and length were also found to be in good agreement with experimental data. Although the model does not accommodate the three-dimensional nature of jets in a cross-flow, it has been shown that predictions of mean temperature fields are reasonably representative. The model also gives predictions of gas composition and, therefore, could be combined with a suitable radiation sub-model to calculate the incident thermal radiation from wind-blown flames, providing a more fundamental alternative to the usual empirical methods.

JET FLAMES IN CROSS-FLOW Nomenclature

Superscripts unweighted time-average density-weighted time-average fluctuation with respect to densityweighted time-average spatial-average over jet cross-section

first entrainment constant in Eq. (2) Ck Ctl , Cd, C~ empirical constants of the k-t turC~rl, Cg2

Dk

f g k rh

the e(f, r) Pk R

(R) r

S(r/R) 8 U 1)

W X

Y

bUlence model empirical constants of the variance transport equation, Eq. (20) integrated sink term representing dissipation of (k) total mixture fraction flux through normal jet cross-section mixture fraction or conserved scalar acceleration due to gravity turbulence kinetic energy total mass flux through normal jet cross-section total mass of air entrained per unit jet length per unit time density-weighted probability density function o f f integrated source term representing production of (k) jet radius associated with prescribed radial profiles jet radius associated with spatiallyaveraged variables radial distance from jet axis similarity function for prescribed radial profiles curvi-linear distance along jet trajectory axial velocity component horizontal velocity component total momentum flux through normal jet cross-section downwind distance from release point cross-wind distance from release point vertical distance from release point

Greek Ot

second entrainment constant in Eq.

Subscripts adi C

max 0 o0

0

P •

adiabatic value centre-line value at r = 0 maximum value release point value at s = 0 ambient free stream value

Acknowledgment The author would like to thank A. D. Birch, R. P. Cleaver and M. Fairweather for helpful discussions. This work is published with the permission of British Gas plc.

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(2) additional function for second entrainment term in Eq. (2) dissipation rate of turbulence kinetic energy angle made by jet trajectory to the vertical dynamic laminar viscosity dynamic turbulent viscosity gas density general scalar variable adjustable constant in Eq. (25) for correcting flamelet temperatures for radiative heat loss constant related to turbulent Prandtl/ Schmidt number

659

9. 10. 11.

12. 13. 14. 15. 16.

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660

TURBULENT COMBUSTION

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