Prediction of axisymmetric turbulent diffusion flames and comparison with laser-doppler velocimetry data

' w

February 1987

PREDICTION OFAXISYMMETRIC TURBULENT DIFFUSION FLAMES

AND COMPARISON WITH LASER-DOPPLER VELOCIMETRY DATA

by

L.-Y. Jiang and J. P. Sislian

o

4

NOV.

1987

UTIAS Report No. 314

CN

'

ISSN 0082-5255

•

PREDICTION OF AXISYMMETRIC TURBULENT DIFFUSION FLAMES AND COMPARISON WITH LASER-DOPPLER VELOCIMETRY DATA

by

L.-Y. Jiang and J. P. Sislian

Subrnitted January 1987

© Institute for Aerospace Studies 1987

February 1987 UTIAS Report No.

314 CN ISSN 0082-5255

Acknowledgements

This research was financially supported by the Natural sciences and Engi neeri ng Research Council of Canada under Strategi c Grant G-0691. Thi s support is gratefully acknowledged •

Summary

Turbulent axisymmetric fuel jets issuing vertically upwards into stagnant air are calculated in non-combusting and combusting situations. The parabolic numerical prediction procedure is based on Reynolds averaged

conservation equations describing the investigated flow fields.

Experimentally measured initial values for the mean axial velocity component, the turbulence kinetic energy and length scale, are used to start the calculations. Turbulence transport properties are determined by means of a modified k-E two-equation model. Combustion is described by two models. In the first model it is assumed that chemical equilibrium prevails in the whole flow field; the second represents a relatively simple version of the eddy break-up concept. In the case without f1 ame good agreement has been obtained between predicted results and laser Doppler velocimetry measurements with the modified k-E turbulence model. Good agreement between the numerical results and measured values has also been obtained in the combusting flow situation when the same modified k-E turbulence model is combined with the eddy break-up model for turbulent combustion. This agreement is poorer when the equilibrium chemistry model is used to describe the combustion process. Further work with more elaborate turbulence and combustion models is in progress in order to improve the prediction capabilities of combusting systems.

TABLE OF CONTENTS I I Acknowledgements

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Summary

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Notation

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1 I NTRODUCTI ON 2 PHYSICAL MODELLING • • • 2.1 Governing Equations 2.2 The Turbul en ce Hodel

2.3 Combustion Hodels

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2.3.1 Equilibrium Chemistry Model. •• 2.3.2 The Eddy Break-Up Hodel

3 THE PREDICTION METHOD

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4 RESULTS AND DISCUSSION •

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4.1 4.2

Non-Combusting Jet Flow • • • • • • • • • • • • • • • • • • • • Predictions for the Combusting Jet Flow. Chemical Equilibrium. Mod e l . • • • • • • • • • • • • •

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4.3 Prediction for the Combusting Jet Flow. Eddy Break-Up r10del 5 CONCLUSIONS REFERENCES FIGURES

. .

.

. . .

. .

.

.

. .

.

. . . .

.

; i ; ; i v 1 2 2 3 4 4 6 7 8 8 9 10 11

..

H k L m· ₁ m'* ₁ p Pr r s T u v x y p Notation

specific heat at constant pressure

constant

specific static enthalpy

total (stagnation) enthalpy

turbulence kinetic energy

turbulence length scale species mass fract;on

non-dimensional species mass fraction fuel mass fraction

fuel mass fraction in unburned gas

fuel mass fraction in completely burned gas pressure

Prandtl number

radial distance stoichiometric ratio

temperature

axial mean velocity radial mean velocity

distance along axis of symmetry

radial distance density

laminar viscosity

turbulent viscosity

effective viscosity, Ileff

=

Il + Ilt

effective turbulent transport coefficients

w non-dimensional stream function

dissipation rate of turbulence kinetic energy effective PrandtljSchmidt number

1 INTRODUCTION

In most practical combustion systems the turbulent mixing process of the fuel and oxydant is slow compared to chemical reaction rates. Therefore, the burning rate in these systems is controlled, to a large extent, by the mixing process. Such mixing-controlled combustion systems are called diffusion flames. Turbulent diffusion flames have been studied from the beginning of this century up to the present time by many investigators (see, for example, Refs. 1-10). The understanding and prediction of turbulent combusting processes in such a relatively simple system as a fuel gas jet injected vertically upwards from a circular nozzle into stagnant air has been the subject of intensive research. There is little doubt th at the greatest impediment to developing viable prediction methods for turbulent diffusion flames is the inadequacy of the physical mode 1 s used to descri be turbul ence transport phenomena in the combust i ng environment and the turbulent combustion process. For isothermal flow situations, various investigators have advanced mathematical models of turbulence having varying degrees of complexity. The so-called k-E model is the most widely used. However, because of lack of relevant initial data for the turbulence kinetic energy k and the turbulence integral scale J. (from which the initial value for the dissipation of the kinetic energy of turbulence, E, can be derived), most researchers invoke assumed initial values of these quantities (see Refs. 23, 24), which greatly deteriorates the prediction capabilities of this turbulence model.

Recent advances in the non-intrusive laser Doppler velocimetry technique made it possible to measure the instantaneous fluctuating flow velocity in turbulent flames which was almost impossible by other measuring techniques. A number of papers were published where measured, experimental, data were used as initial values for the solution of the parabolic partial differential equation for k (Refs. 15, 19), while assumed, arbitrarily prescribed initial values were used for the solution of the parabolic partial differential equation for E. At the same time a variety of combustion models were used in computations of combusting flows (see, for example, Refs. 11-15).

In order to provide relevant experimental data to check the validity of turbulence and combustion models used to predict diffusion flames, an experimental investigation into the turbulent structure of a jet diffusion flame were conducted, using the laser Doppler velocimetry technique, themocouple probes and instantaneous schlieren photography (Ref. 21). Measurements were made, in cold and combusting situations, at the initial nozzle exit section (2 mm above the nozzle) and several other locations in the jet diffusion flame flow, of the mean velocity components, all turbulence intensities (and hence of the turbulence kinetic energy) and the turbulence macroscale (from which initial experiment al values of E could be derived). The fuel jet which issued vertically upwards into still air, was a mixture of 70% methane and 30% argon, by volume. For cold flow experiments the mixture contained 78.6% air and 21.4% helium. Thus the density and molecular viscosity ratios for the cold fuel mixture and cold gas mixture were 1 and 1.3, respectively. The purpose of the present investigation is to compare results of a parabolic calculation procedure embodying a modified k-E turbulence model and two combustion models, viz., equilibrium chemistry and eddy break-up models, against experimental values

measured initial values were used for the solution of all the parabolic partial differential equations describing the investigated flow.

2 PHYSICAL MODELLING 2.1 Governing Equations

The Reynolds-averaged conservation equations, describing the steady axisymmetric turbulent combusting flow in an infinite still environment, with the turbulent Lewis number Let

=

1, are written in the following boundary layer (parabolic) form:

Cont i nu ity : o (pU) +

1

o( r pv)

= 0

ox r or (1) Axial Momentum: pu ~ + pv

ou = -

dp +

1

~ (J.lt r _OU) + 9 (p f - p) ox or d x r or or re ( 2) Species Conservation: ( 3)

Fuel Element Conservation:

oY f oY f 1 0 J.lt oY f pU - + pv - -

= - -- (--

r - ) ox or r or SCt or (4) Energy: pU _oH + pv oH =

1.

~ [~ r .Q _ (H _ k) + ~ (1 _ _1_) r o( u 2/2) ] ( 5 ) ox or r or Prt or Prt or

where all the dependent variables involved represent time-averaged flow quantities, u and v being the velocity components in the x and r directions, H the total enthalpy of the fluid element, and Prt and SCt the turbulent Prandtl and Schmidt numbers, respectively. The momentum equation (Eq. 2),

,

contains the buoyancy effect. The quantities YK represent mean species

mass fractions divided by their molecular weights: _YK

=

~/MK. In

the species conservation equation, Eq. (3), it is assumed that the mean rate

of production of each species, ~K' equals the instantaneous rate with

the mean quantities such as temperature, density, and species concentration replacing their instantaneous values.

The following initial and boundary conditions are imposed on the above

parabolic partial differential equations, Eqs. (1}-(5): at the initial

section, x = 0, velocity, temperature (or total enthalpy), species mass

fraction profiles should be given. These initial profile shapes should be

based on experiment al data for the particular configuration of interest. At the axis of symmetry the boundary conditions are:

r = 0, ~

=

'óH

=

'óf

=

'óY K

=

0

'ór 'ór or 'ór (6)

and at the outer edge of the mixing region the dependent variables should tend to their values in the outer (external) inviscid flow:

r -+ 0:>, (7)

2.2 The Turbulence Model

The eddy viscosity lJt in Eqs. (1}-(5) is determined by means of two

transport equations for the turbulence kinetic energy, k, and its

dissipation rate, E (two-equation model of turbulence) (Ref. 25). According

to this model the magnitude of the eddy viscosity depends only on the local

values of k, E and the fluid density in the following way:

=c

~.

I-It- ~ E ' ( 8)

The quantities k and E satisfy the following differential equations:

(9)

(la)

In this approach, several terms in the exact equations for k and E,

involving density fluctuations, are ignored for lack of more complete information. The justification for this step rests upon past success of the

model for a number of variable density flows (Ref. 19). Furthermore, the neglect of density fluctuations reduces the Reynolds averaged equations to a form identical to Favre (density) averaged equations. Buoyancy effects in Eqs. (9) and (10) are ignored. As pointed out in Ref. 19, including these effects in turbulence modelling equations has a relatively minor influence on the predictions of mean flow properties, including the temperature.

Computations performed with a modified version of Eqs. (9) and (l0) resulting from a slightly different modelling of the production terms in the exact equations for k and E, viz.

(11 )

pu ~ + pv ~ =

1.

~x ~r r (12)

were more stable. In the present investigation, Eqs. (11) and (12) were used together with measured i nit ial val ues to determi ne the turbul ence kinetic energy, k, and its dissipation rate, E. Initial values for E were

calculated from Eq. (8) by using measured values of k and ~.

2.3 Combustion Models

The combustion models, in essence, must provide a method of evaluating the rnean formation rate of each species present and, in addition, allow the evaluation of mean temperature and density of the gas mixture. In fact the determination of the mean reaction rates represents a major difficulty in the development of prediction methods for combusting flows. Fortunately, for diffusion flames of hydrocarbon fuels at high temperatures, the reacti ons usually have time scal es very short compared with the time scale of the turbul ent transport process. Therefore, "fast chemi s try 11 is assumed,

i.e., chemical equilibrium is reached at each point in the flow. The thermochemical state of the resulting mixture may be determined purely in terms of strictly conserved scalar variables, and the need to calculate mean reaction rates is thereby removed. With known pressure, enthalpy and element mass fractions of all elements present, the equilibrium composition, temperature and density of the gas mixture can be determined at each point of the fl ow.

2.3.1 Equilibrium Chemistry Model

The chemical equilibrium composition, temperature and density of the gas mixture are obtained by the method presented in Ref. 26, which is based on the minimization of the Gibbs free energy approach.

For a mixture of n spec i es, the Gi bbs free energy per ki logram of mixture is given by

n

G =

1.

gjnj j=1

(j

=

1, ••• , n) (13)

where nj is the number of kmole of of species j per kilogram of mixture, and 9j the chemical potential per kmole of species j defined as

g.

=~I

J ~. T .

J ,p ,ni *J

(14)

For gases it takes the form

n·

9j =

9J

+ RT ~ (n J ) + RT ~Patm (15 ) The numerical values of 9j are the chemical potentials in the standard state which are functions of temperature and are generally found in JANAF tables or calculated by means of polynomial functions. The condition for chemical equilibrium is the minimization of free ener9Y. This minimization should fit same mass balance constraints

n

L

j=1

o

a·· n· -_{lJ J} b· ₁ = 0 (i = 1, ••• , L) (16)

where the stoichiometric coefficients aij are the number of kilogram-at oms of element i per kmo 1 e of spec i es j; bp is the ass i gned number of kilogram-atoms of element i per kilogram of total reactants.

Defining W to be

L

W = G +

L

i=1

(17)

where Ài are Lagrangian multipliers, the condition for equilibrium becomes:

n L L 0

c5W =

L

(gi +

1.

~ aij )c5n_j +

L

(bi - bi) c5~ = 0

j=1 i=1 i=1

(18)

In the present case the thenmodynamic state is spec;fied by assigned enthalpy and pressure

h = (19)

where (H~)j is the standard state enthalpy for species j, a function of

temperature, and can be found in JANAF tables or calculated by polynomials.

The composition and temperature can be obtained through an iteration

procedure; then the density of the gas mixture can be detenmined.

Additional details are given in Refs. 27 and 28. In the calculations an

ideal gas mixture is assumed with variable specific heats. Dissociation is

also considered in the high temperature region of the flow. The specific

heat, enthalpy and entropy are given as follows:

o ST o HT

=

a +

~

T +

~

T2 +

~

T3 +

~

Tit +

~

RT 1 2 3 T 5 T

-

-R ( 20)

The coefficients are calculated from JA.NAF tables by using a modified

computer programme given in Ref. 29. In the calculation of the combusting

flow the following 15 species are considered (see Ref. 11):

CH_{It , H2CO, HCO, CO, CO2 , H20, H02' H2' O2, 0, CH 3, OH, H, N2, Ar} The inclusion of NO had no effect on the solution.

2.3.2 The Eddy Break-Up Model

The essential idea of this model is that the turbulent reacting mixture

consi sts of i nterspersed sheets and fi 1 aments of fully-reacted and

completely unreacted material , and that the rate of transformation of the gas from one state to the other depends on the rate of stretchi ng of the sheets and filaments. The latter is taken to be proportional to öu/êy in the

present version. The reaction rate (the term Ck in Eq. (3)) expression is:

( 21)

This expression takes no account of the chemical kinetics and relates to

entirely mixing controlled combustion. It may be hard to say whether the

predictions obtained in the present study, with this simple model are in quite good agreement with the measurements. The chemical reaction is characterized by a global single step reaction

CH 4+ _{2° 2 .. CO 2} + 2H 20

and the gas mixture is composed of CO 2' H20, CH4 , O~, N2 and Ar. The values

of mfu(m

CH ₄ ), the fuel mass fraction, are obtalned from Eq. (4). The molecular oxygen mass fraction, m0

2' can then be found from stoichiometric conditions. Finally, the combustion products, m

C02 and mH20, which are in proportion, can be obtained. The quantities mN and mAr are conserved

2

during combustion. For each species, the specific heat and enthalpy are given in the form of polynomials as in the chemical equilibrium model.

3 THE PREDICTION METHOD

The governing equations, Eqs. (1-5) and Eqs. (11-12), are of the form:

o( eu) ₊₁ o( r ev)

_{= 0}

_{( 22)}

ex

r or and

pu 04> + pv 04>

=

1 ~ _(r ~ff _{4> M) + S4>} ( 23) ox or r or ' or

Here 4> = _{u, Yk , Yf , H, k and} E. For boundary layer type (parabolic)

prob 1 ems, it is conveni ent to use the stream funct i on <\I as one of the independent variables. In the x, <\I coordinate system, the governing equations take the following general form:

04> _ 0 (2 04» 1 S - - - r pu reff "- - + -

"-oX 0<\1 ''f' 0<\1 pU 'f'

The continuity equation, Eq. (22), does not appear explicitly. This general form is further transformed to x, w coordi nate system, where wis the non-dimensional stream function

,

.

.

_ <\I - <\11 w

-<V[ - <\11

Subscri pts land E refer to the inner and outer boundari es of the flow, respectively. In this coordinate system, the resulting general form of the governing equations, is:

~~ + (a + bw) 04>

=

~ (c _04» + d

where

Here

mil

is the mass flow rate per unit area; a the mass flow rate through

the inner I boundary, b the mass flow rate through the outer E boundary, and

reff,~ the effective turbulent exchange coefficient for the corresponding

dependent variable~. S~ are the so-called source terms which can be

determined from Eqs. {l}-(5} and Eqs. {1l}-(12}. The computations were

performed using the modified GENMIX computer programme {Ref. 22} for cold and combusting flows with the modified turbulence model, Eqs. {1l}-{12}. The turbulence kinetic energy is neglected in the energy equation, Eq. {5}, as being small, and the turbulent exchange coefficients for all species

considered and enthalpy are assumed to be the same. Initially, calculations

were performed with 50 cross-stream grid points with the grid nodes 5 l11T1

apart. In subsequent computat i ons, 500 cross -stream gri d nodes were used

for cold flow calculations, and 200 for combusting flow calculations.

Calculations with more grid nodes were similar to those with 50 grid nodes

except the boundary regions of the flow, where the flow variables varied

more smoothly. The last calculated section of the flow was 50 nozzle exit

diameters downstream for all the cases considered.

4 RESULTS AND DISCUSSION

4.1 . Non-Combusting Jet Flow

Computat i ons for the col d fl ow were carri ed out with 500 cross-stream

grid nodes. The following values of the constants in the k-E turbulence

model equations were used:

C~ = 0.12; _Ck = 0.30; Cl = 1.05; C₂ = 1.46; <\

=

1.0; cr

=

1.3

E

The initial conditions are illustrated in Fig. 1. The initial values for

the mean axial velocity U and the turbulence kinetic energy k were taken

from the relevant experimental data {see Ref. 21}. The initial values for

the turbulence kinetic energy dissipation rate, E, were calculated from the

relation E '" k3/2/.R. by using the experimental data at the initial section

for k and .R. - the turbulence macro-scale. The predicted {solid curves on

the figures} and measured radial distributions of mean axial velocity, turbul ence ki neti c energy together with the predi cted radi al di stri buti ons

of E at five downstream cross-sections: x

=

0, 10, 20, 30, 40 and 50 cm (50

, . . . . - - - -- - - - -- - - -- - - -- - - -- - - . ,

and measurements along the (axial) centreline of the cold jet flow are shown in Fi g. 7.

The agreement between predictions and measurements is, in general , reasonably good. The agreement is better for downstream sections. In Figs. 2 and 3 the discrepancy for k is larger than at downstream sections. In Fig. 7 the turbulence kinetic energy, k, in the potential core departs from experimental data. This may be due to the inability of the turbulence kinetic energy equation, Eq. (11), to model the flow in this region. According to this equation, if the radial gradients of k and U are zero (as in the potential core) , k should decrease. However, experimental data show that k increases in the potential core (Ref. 19,21). The exact theoretical equation for k is not well modelled, i.e., the effect of pressure fluctuations is not taken into account. The pressure fluctuations outside the nozzle may be stronger than in the flow inside the nozzle.

The empirical constants C , Ck' Cl and C₂ have different effects on the computed solution. Generally~speaking, the solution is most sensitive to the value of Ck• It has a st rong effect on the magnitude of k. With increasing Ck, k increases, so does~. This has a st rong decaying effect on axial velocity both in axial and radlal directions. When the value of Cl increases, k increases, ~ becomes larger and hence the velocity decreases. The effect of C₂ is opposite to Cl. The difference between Cl and C₂ should be appropriate. If this difference is large, k and ~ decrease rapidly, and the velocity decays slowly. In the extreme case when Cl

=

C_{2 ,} the differential equations for k and € become identical. The distributions of k

and € are proportional everywhere in the flow. They damp themselves and

always decrease, IJ.t becomes small and the velocity profiles decay slowly. At 50 diameters downstream the potential core still exists. Figure 8 shows the radial distributions of U, k and € at a section 40 diameters downstream,

with the same initial conditions, but with Cl

=

C_{2 •} The values of C\ and ar have relatively small effect on the computed solutions.

4.2 Predictions for the Combusting Jet Flow. Chemical Equilibrium Model Equations (1), (2), (4), (5), (11) and (12) are solved subject to the following initial and boundary conditions: experimentally measured data are used as initial values for U, k, ~ (or €) and T. Outside the jet flow U, k,

€ and element mass fractions are taken as zeros and standard pressure and

temperature prevail in this region. The initial values for the enthalpy can be found from JANAF tables or calculated by polynomial functions. The initial values for the carbon element mass fraction are derived from the known initial composition of methane/argon mixture. Mass fractions for all other chemical elements present in the mixture can be determined by writing them in the non-dimensional form:

y*

=

e

where the subscript e stands for C, H, 0, N, Ar. Here the subscript ic denotes the value at the centre of the nozzle exit section, and Erepresents the value at the outer boundary. These non-dimensional element mass fractions satisfy the partial differential equation (4) and have identical initial and boundary conditions. Hence the distributions of all these non-dimensional element mass fractions are identical. Thus the mass fractions Cl of atomic hydrogen, oxygen, nitrogen and argon can be obtained

from the following relations:

It is implied that the turbulent exchange coefficients of all species are the same.

Computations were performed with 200 cross-stream grid nodes and the following set of values for the empirical constants in the k,e: turbulence model equations:

CIJ.

=

0.10, _Ck

=

0.29, C_I

=

1.932, C₂ = 2.52, Pr = 0.89,

C\

=

0.8, ar

=

1.0

The initial (experimental) radial distributions of U, k and e: are shown in Fig. 9. Results for the case considered are presented in Figs. 10-15. The agreement between predictions and measurements are good for U and k. This agreement is better at upstream sections than at the downstream region of the flame. The predicted axial centreline distribution of k departs from experimental data in the potential core region of the flame and far downstream. The initial radial distributions of the mass fraction of the argon, methane and temperature are presented in Fig. 16. The predicted distributions of six main species at five downstream cross-sections are shown in Figs. 17-21. Figures 10-14 show that an obvious discrepancy exists between predicted and measured temperature distributions in the vicinity of the reaction zone. This is probably due to the fact that in this model the effect of turbulence on the combustion process is not taken into account. Also, the temperature in the near exit region is not high enough for equilibrium to prevail. Here a finite-rate reaction model would be more appropri ate.

4.3 Prediction for the Combusting Jet Flow. Eddy Break-Up Model

The initial and boundary conditions in this case are the same as in the chemical equilibrium model. Initial (experimental) distributions of U, k and E, as well as species mass fraction, are illustrated in Figs. 22-28. Figures 23-27 illustrate the radial distributions of U, k, E, ai and T.

Reasonable agreement is obtained for all the flow variables. In all sections, predicted temperatures are slightly higher than the measured values. This may be due to ---10% radiation heat loss (Ref. 19). The axial distributions of these variables along the centreline of the combusting jet are shown in Fig. 28. As in the case of the chemical equilibrium model, the predicted values of the turbulence kinetic energy depart from the measured in the potential core and at far downstream regions of the flow. In general, in this case, predictions agree very well with experimental data.

5 CONCLUSIONS

1. Predictions with measured initial data, the modified k,E turbulence model, the simple version of eddy break-up combustion model together with the commonly used values for the empirical constants in the turbulence model equations agree quite well with experimental data for cold, as we" as combusting jet flows.

2. The modified turbulence model used in the present investigation is stable and suitable for modelling turbulent flows. It must be improved to consider pressure fluctuation effects on turbulence kinetic energy production in the potential core.

3. The discrepancy between predictions and measurements in the case of the chemical equilibrium combustion model reveals the importance of turbulence on the combustion process.

4. More experimental data and comparison with predictions for a variety of combusting situations are needed to establish more realistic values for the empirical constants in the k,E model.

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26. Gordon, S., McBride, B. J., Computer Program for Calculation of Complex Chemical Equilibrium Compositions, Rocket Performance, Incident and Reflected Shocks and Chapman-Jouguet Detonations. NASA SP-273.

27. JANAF Thermochemical Tables, 1971.

28. Reynolds, W. C., Perkins, H. C., Engineering Thrmodynamics. McGraw-Hill, 1977.

29. ~1cBride, B. J., Gordon, S., Fortran IV Program for Calculation of Thermodynamic Data. NASA TN 0-4097, 1967.

2.0

N

-

_~

E

-

_>-

1.6

(!) Cl::: W

Z

w

_1.2

u

-.-

_W

Z

~

08

w

U

Z

w

0.4

-.J

=>

al Cl:::

=>

.-

0

20

100

U

/

o-u

/K

~-K

16

80

(\J

-

_~

0

-)(

É

12

₆₀

_z

>-

0

-.-

.-u

_~

9

8

0

40

ëi5

w

Cf)

>

-

_Cl

4

20

0

0

00

0.2

0.4

0.6

0.8

1.0

RADIAL DISTANCE

(cm)

FIG. 1 INITIAL RADIAL DISTRIBUTIONS OF MEAN AXIAL VELOCITY. TURBULENCE KINETIC ENERGY AND lTS DISSIPATION RATE. NON-COMBUSTING JET.

N

7.5

-

(/) ...

E

-

_>-

_6.0

l!>

Cl::: W Z W

4.5

U

-

I-w

z

S;Z 3.0

w

U Z

w

1.5

-.J

:::>

CD Cl:::

:::>

0

I-15

1500

a-u

.6.-K

12

₁₂₀₀

-

(/) ...

a

5 9

₉₀₀

_Q

z

>-

_~

I-U

a..

0

fJ) -.J

6

600

5Q

w

0

>

3

₃₀₀

0

₀

0.0

0.5

1.0

1.5

2.0

2.5

RADIAL DISTANCE

(cm)

FIG. 2 RADIAL DISTRIBUTIONS OF MEAN AXIAL VELOCITY, TURBULENCE KINETIC ENERGY AND lTS DISSIPATION RATE. NON-COMBUSTING JET. X/D=10.

2.5

15

N

-

Cl) ...

o-u

E

~-K

->-

2.0

6.0

(!)

a::

-w

Cl) 0 Z ... 0

w

1.5

_{- 4.5}

E

U

-

>-I-

I-W

-z

U ~

1.0

93.0

w

w

U

>

Z W ...J

0.5

1.5

=>

al

a::

=>

...

0

0

0

2

3

4

5

RADIAL DISTANCE

(cm)

FIG. 3 RADIAL DISTRIBUTIONS OF MEAN AXIAL VELOCITY, TURBULENCE KINETIC ENERGY AND lTS DISSIPATION RATE. NON-COMBUSTING JET. X/D=20.

150

120

z

90

_-

0

~

a:

en

60

en

0

30

0

1.0

4.0

N

-

(/)

o-u

""

E

~-K

-

_>-

_0.8

<.!)

et:

-W (/) Z ...

w

0.6

_-2.4

E

u

-

>-~ _~

w

_K

u

z

9

~0.4

1.6

w

w

_>

u

z

w

...J

0.2

0.8

::::>

al

et:

::::>

0

0

~

0

2

4

6

8

RADIAL DISTANCE

(cm)

FIG. 4 RADIAL DISTRIBUTIONS OF MEAN AXIAL VELOCITY, TURBULENCE KINETIC ENERGY AND lTS DISSIPATION RATE. NON-COMBUSTING JET. XjD=30.

20

16

z

12

0

-~

a.

en

8

en

0

4

0

10

0.40

2~5

10

N

-

_~

_o-u

E

6-K

...

>-

0.32

2.0

8

(!)

a::

-w

_~

z

5

1.5

w

0.24

6

z

0

-U

>--

_...

_I-

~

w

-

a.:

z

U

9

1.0

en

~

0.16

,

4

en

w

-w

_>

c

U Z

w

_0.08

_0.5

...J

2

:::J al

a::

:::J

0

...

0

0

0

2

4

6

8

10

RADIAL DISTANCE

(cm)

FIG. 5 RADIAL DISTRIBUTIONS OF ~1EAN AXIAL VELOCITY • TURBLILENCE KINETIC ENERGY AND lTS DISSIPATION RATE. NON-COMBUSTING ,JET. X/D=40.

N

0.25

2.5

-

_~

o-u

E

A-K

-

_>-0.20

_2.0

C) Cl:: W

-

(J)

Z

...

w

0.15

5

_L5

U

>-i=

l-W

-U Z ~

0.10

9

1.0

W W

_>

U Z

w

...J

O.D5

0.5

::>

al Cl::

::>

0

0

I-0

2

5

7

10

RADIAL DISTANCE

(cm)

FIG. 6 RADIAL DISTRIBUTIONS OF MEAN AXIAL VELOCITY, TURBULENCE KINETIC ENERGY AND lTS DISSIPATION RATE. NON-COMBUSTING JET. X/D=50.

5

4

Z

3

0

-ti

a..

-Cl)

2

Cl)

-

0

0

12

7.5

20

2000

N

-

_~

_o-u

E

-

co

~-K

>-(!)

6.0

16

1600

a:

w

-

Cl) Z ...

Z

w

4.5

E

₁₂

1200Q

~

u

-~ W

z

~

w

U Z

w

...J

:::>

al

a:

:::>

~

>-

I--

_U

3.0

9

8

800

w

>

~

1.5

4

400

0

0

~

0

00

10

20

30

40

50

AXIAL DISTANCE

(cm)

FIG. 7 AXIAL CENTRELINE DISTRIBUTIONS OF MEAN AXIAL VELOCITY, TURBULENCE KINETIC ENERGY AND lTS DISSIPATION RATE. NON-COMBUSTING JET.

a:

en

en

4.8

C\J

-

,

Cl)

E 4.0

-

>-<.!)

a::

3.2

W Z W U

-I-

_2.4

W Z

-

_~ W

1.6

U

z

<l:

.

-l ::J

~

0.8

::J

I-0

24

20

0

-

,

Cl)

16

E

-

_r

I-

_-

12

U

9

W

>

8

4

0

0

4

8

12

16

20

RADIAL DISTANCE

(cm)

FIG. 8 RADIAL DISTRIBUTIONS OF MEAN AXIAL VELOCITY, TURBULENCE KINETIC ENERGY AND lTS DISSIPATION RATE. NON-COMBUSTING JET. X/D=40; C1=C2·

48

40

32

Z

0

-~

24

0-en

en

-0

16

8

0

4.0

C\I

-

_~

E

.

;: 3.2

(,!) Cl:: W

z

w

2.4

u

I-W

z

1.6

~

w

U Z

w

0.8

--1 :J CD Cl:: :J

I-

0

20

200

U

o-u

b-K

16

160

C\J

-

0

(/) ...

-

) (

512

120

z

>-

0

I-

-

~

U

g

8

80

a.:

Cf)

w

Cf)

>

-

0

4

40

0

0

0

.

0

0.2

0.4

0.6

0.8

1.0

RADIAL DISTANCE

(cm)

FIG. 9 INITIAL DISTRIBUTIONS OF MEAN AXIAL VELOCITY, TURBULENCE KINETIC ENERGY AND lTS DISSIPATION RATE. COMBUSTING JET. EQUILIBRIUM CHEMISTRY.

4.0 (\J

-

_~ E

>-

3.2 (.!) Cl:: W

z

w

2.4

u

I-W Z ~

_1.6

w

U Z W ...J _0.8 ::J al Cl:: ::J I- ₀ 20 100

o-u

ll.-K D-T 16 80

-

0 ~ )( EI2

-

60Z

>-

0 I-

_~

U

9

8 40-

a.:

w

0

en

>

en

₀ 0 4 20 0 0 0.0 0.4 0.8 1.2 1.6 2.0

RADIAL DISTANCE

(cm)

FIG. 10 INITIAL DISTRIBUTIONS OF MEAN AXIAL VELOCITY, TURBULENCE KINETI C ENERGY AND ITS DISSIPATION RATE. COMBUSTING JET. EQUILIBRIU~' CHEMISTRY. XjD=10. 2500 2000

w

Cl:: 1500

~

<:( Cl::

w

a.

1000 ~

w

I-500 0

4.0 N

-

Cl) ... E -3.2 ~ C)

a::

w

~2.4

u

~

w

Z _1.6 ~

w

U Z ~ 0.8 :::>

m

a::

:::> ~ 0 20 100 Q-U ~-K 16 o-T 80

--

0 ~ )( 512 60 _Z

>-

0 ~

-ti

()

g

8

40

a.

w

Cf) Cf)

>

₀

4

20 0 0 0 0.0 0.8 1.6 2.4 3.2 4.0

RADIAL DISTANCE

(cm)

FIG. 11 INITIAL DISTRIBUTIONS OF MEAN AXIAL VELOCITY, TURBULENCE KINETIC ENERGY AND lTS DISSIPATION RATE. COMBUSTING JET. EQUILIBRIUM CHEMISTRY. X/D=20. 2500 2000

w

a::

1500

:::>

~

a::

w

1000~

w

~ 500 0

5

C\I

-

_~

E

-

_>-4

<.!)

a:::

w

z

w3

u

I-w

z

~

2

w

U Z

w

...J

=>

CD

a:::

=>

1-0

15

400

o-u

L1-K c-T

₃₂₀

Cl) ...

E

z

-

2400

>-

~

l-

_a:

U

en

g6

160

en

w

Cl

>

3

80

c

0

0

0.0

1.0

2.0

30

40

50

RADIAL DISTANCE

(cm)

FIG. 12 INITIAL DISTRIBUTIONS OF MEAN AXIAL VELOCITY. TURBULENCE KINETIC ENERGY AND lTS DISSIPATION RATE. COMBUSTING JET. EQUILIBRIUM CHEMISTRY. X/D=30.

2500

2000

w

et:

1500

~ <t

a:::

w

a...

IOOO~

w

I-500

0

2

.

5

(\J

-

_~

E

-

_>-

_2.0

(!) ct:: W Z

w

1.5

u

...

W

z

~

1.0

W U Z

w

-.J

0.5

::> CD ct:: ::>

...

₀

10

200

o-u

~-K

8

c-T

160

-

en "- _Z

5 6

1200

~

>-...

a...

U Cf)

g4

80

Cf)

w

0

>

2

40

0

0

0.0

2.0

4.0

6.0

8.0

10.0

RADIAL DISTANCE

(cm)

FIG. 13 INITIAL DISTRIBUTIONS OF MEAN AXIAL VELOCITY, TURBULENCE KINETIC ENERGY AND LTS DISSIPATION RATE. CDr1BUSTING JET. EQUILIBRIU~1

CHEMISTRY. X/D=40.

2500

2000

W 0::

1500~

<t 0::

w

a...

IOOO~

...

500

0

2.0

N

-

_~

E

->-

1.6

(!)

a::::

w

z

w

1.2

U

r-w

Z ~

0.8

w

U Z ~0.4 ~ CC

a::::

~

0

r-10

_{t. t.} b.

100

o-u

t.-K [ J - T

_.

₈₀

-

_~

56

60~

-

>-r-

b.

~

a.:

U

-94

40~

w

[J

C

>

_[J [J

2

[J

₂₀

[J [J [J [J [J [J

0

0

0

2.0

4.0

6.0

8.0

10.0

RADIAL DISTANCE

(cm)

FIG. 14 INITIAL DISTRIBUTIONS OF MEAN AXIAL VELOCITY, TURBULENCE KINETIC ENERGY AND ITS DISSIPATIDrI RATE. COMBUSTING JET. EQUILIBRIUM CHEMISTRY. X/D=50.

2500

2000

w

0:::

1500

::>

~

0::: W

1000

~

w

r-500

0

5

20

50

2000

N

-

Cf)

o-u

...

E

A-K

-

c-T

>-4

16

1600

~

-0:: Cf) 0 W ... ) (

w

~3

--12

E

CC c

305

_1200~ 0::

u

_>-

ce

_~

<:(

t-

...

0:: W

a:

w

~2

U 0

₈

oen

en

₈₀₀

a..

_~ ..J W Cl W W

_>

t-U A Z

w

4

10

400

..J ::::> al 0:: ::::>

t-O

0 0 0 0

10

20

30

40

50

AXIAL DISTANCE

(cm)

FIG. 15 AXIAL DISTRIBUTIONS OF MEAN AXIAL VELOCITY , TURBULENCE KINETIC ENERGY AND ITS DISSIPATION RATE. COMBUSTINr. JET. EQUILIBRIUM CHEMISTRY.

z

0

-....

u

«

0:::

LL Cl) Cl)

«

:!:

1.0

0.8

~

0.6

~

0.4

~

0.2

~

o

0.0

AR,

CH4

J

T7

I 1

"

~

0.2

0.4

0.6

0.8

RADIAL DISTANCE

(cm)

-2500

2000

W

0:::

1500~

«

0:::

W

a...

1000:!:

w

....

500

o

1.0

FIG. 16 INITIAL RADIAL DISTRIBUTIONS OF SPECIES MASS FRACTIONS AND TEMPERATURE.

1.0 r - - - . 2 5 0 0

0.8

N2\

2000

z

0

LU

-

_~

0::

U

0.6

AR

1500

~

<t

0::

<t

LL

_LU

0::

en

_Q4

0-en

1000

:E

<t

LU

~

I-0.2

500

o

0

0.0

0.4

0.8

1.2

1.6

2.0

RADIAL DISTANCE

(cm)

FIG. 17 RADIAL DISTRIBUTIONS OF SPECIES ~~ASS FRACTIONS AND TEMPERATURE. EQUILIBRIUM CHEMISTRY. X/D=10.

1 . o . - - - 2 5 o o

0

.

8

N2

2000

Z

0

LU

-

a::

~

u

0.6

1500

:::>

«

~

a::

a::

IJ..

LU

Cl)

0.4

a.

Cl)

1000 :E

«

LU

~ ~

02

0.2

₅₀₀

o~==~==~~~~~~--~o

0.0

0.8

1.6

2.4

3.2

4.0

RADIAL DISTANCE

(cm)

FIG. 18 RADIAL DISTRIBUTIONS OF SPECIES ~~SS FRACTIONS AND TEMPERATURE. EQUILIBRIUM CHEMISTRY. X/D=20.

1 . 0 , . . - - - , 2 5 0 0

0.8

N2

2000

Z

0

W

-

0:::

I-

_0.6

₁₅₀₀

::>

U

~

«

et:

a:::

lJ....

_W

en

_0.4

a..

en

1000

~

«

w

~

I-O

₂

0.2

500

o

0

0.0

1.0

2.0

3.0

4.0

5.0

RADIAL DISTANCE

(cm)

FIG. 19 RADIAL DISTRIBUTIONS OF SPECIES t'1ASS FRACTIONS AND TEMPERATURE. EQUILIBRIUM CHEMISTRY. X/D=30.

1.0

r---~---

___

2500

"

0.8

N2

2000

Z

0

W

-

_n:::

~

₀₆

U

1500

::::>

<t

_~

n:::

I..L.

n:::

W

en

0.4

₁₀₀₀

~

en

<t

_w

~ _~

0

0.2

₅₀₀

o

0

0.0

2.0

4.0

6.0

8.0

10.0

RADIAL DISTANCE

(cm)

FIG. 20 R.l\DIAL DISTRIBUTIONS OF SPECIES MASS FRACTIONS AND TEMPERATURE. EQUILIBRIUM CHEMISTRY. XjD=40.

1 . 0 , . . . - - - , 2 5 0 0

0.8

N2

2000

Z

w

0

-

0:

...

_0.6

₁₅₀₀

_~

U

<t

_cd:

a::

_0:

LL

W

~

0.4

₁₀₀₀

~

<t

_w

::E

I-CH4

02

Q2

_H20

₅₀₀

T

0

0

0.0

2.0

4.0

6.0

8.0

10.0

RADIAL DISTANCE

(cm)

FIG. 21 RADIAL DISTRIBUTIONS OF SPECIES MASS FRACTIONS AND TEMPERATURE. EQUILIBRIUM CHEMISTRY. X/D=50.

200r

40

20

2500

N

o-u

-

_~

_~-K

E

-

o-T

1601-

~

3.2

16

r"\fIl.

-i

2000

N

a::

-

/ K

0

lLJ Cl)

-

₎₍

Z

,

lLJ

E

0::

z

120

w

2.4

-12

I -

1500

~

U

>-

AR7

~

0

_-

-

_~ ~

_a::

lLJ lLJ

~

Z

U

CH4~

~

-g 8

1000

~

ü) 80

~

1.6

I

-en

-

₀

w

U

w

>

~

,

Z

lLJ

401-

~

O.Sr

_{4r T7}

""-11

Ub

\

--1500

a::

~

OL

~

oL

ol

o

I

LIl

~~

10

0

0.2

0.4

0.6

0.8

1.0

RADIAL DISTANCE

(cm)

FIG. 22 INITIAL RADIAL DISTRIBUTIONS OF MEAN AXIAL VELOCITY TURBULENCE

KINETIC ENERGY AND lTS DISSIPATION RATE, SPECIES MASS FRACTION AND

w

~

-d.O

-iO

.

S

Z

0

-0.6

t;

«

a::

1.L

0

.

4

~

«

~

-i 0.2

-.JO

1000r

N

4.0

20

2500

-

o-u

~

U

A-0

A-K

E

-

o-T

8001-

~

3.2

NI')

~2000

a::

LU

-Z

W

Z

_w

a::

0600

(.)2.4

1500

::>

-

_~

~

-

_...

>-a.

w

...

-

a::

_w

-

Z

(.) Cl)

a.

Cl)

400

~

1.6

9 8

1000

::E

-0

w

w

(.)

>

Z

W

02

2001-

5

08

-1500

al

a::

:::>

OL ...

OL

01~1 ~ ~~

la

0

0.5

1.0

1.5

2.0

2.5

RADIAL DISTANCE

(cm)

w

...

-,1.0

~0.8

Z

0

-0

.

6

ti

«

a::

IJ..

0

.

4

~

«

::E

-10.2

-.JO

IOOOr

N

4.0

20

2500

-,1.0

-

_~

o-u

E

ot.

ot.-K

-

_/'\vK

o-T

8001-

b

3.2

16

- N2

~2000 ~0.8

a::

~

-W

_~

W

Z

Z

₀

E

a::

-~

600

w

U

2.4

-12

₁₅₀₀

~

0.6

t;

-

>-I-

_I-

-

_I-

_<t

<t

~

w

-

a::

a::

z

U

_W

LL

-98

₁₀₀₀

~

_10.4

~

~

400

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~

1.6

0

w

w

W

<t

U

>

_I-

_:E

Z

W

02

200

"

r-

~

0.8

-1500

al

a::

:::>

OL

I-" I L,.../" . / '

"'''

_~~""- 0' 0

OL

Ol

~I ~--,... _~

₁₀

0

0.8

1.6

2.4

3.2

4.0

RADIAL DISTANCE

(cm)

FIG. 24 RADIAL DISTRIBUTIONS OF MEAN AXIAL VELOCITY, TURBULENCE KINETIC

ENERGY AND lTS DISSIPATION RATE, SPECIES MASS FRACTIONS AND

-10.2

•

4

400r

N

4.0

20

2500

-,1.0

-

o-u

~

E

à-K

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o-T

3201-

b

3.2

16

₀

I<

N

2

-i2000

-i 0.8

a::

LaJ

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Z

Z

~

LaJ

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l&.J

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a::

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....

0240

U

2.4

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1500

~

0.6

U

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>-~

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....

_...

<t

«

l&.J

Cl::

Cl::

Z

U

l&.J

LL

en

~

1.6

0

₁₀₀₀

~

₁₀₄

.

~

en

160

...Ja

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l&.J

Cl

l&.J

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l&.J

«

U

....

:E

z

l&.J

ao~

5o.ar-

₄

r-

_{CH4 /""}

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~\

\.

~

..,500

m

a::

:::l

OL ... OL

0'

,

---I

--- r::

~

T

'0

0

.

0

1.0

2.0

3.0

4.0

5.0

RADIAL DISTANCE

(cm)

..,0.2

.-JO

200r '"

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FIG. 26 RADIAL DISTRIBUTIONS OF MEAN AXIAL VELOCITY, TURBULENCE KINETIC

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