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MODELING OF TURBULENT SUPERSONIC H2-AIR COMBUSTION /

WITH AN IMPROVED JOINT BETA PDF

• _ ._:,e  ̧ ̧ __

R. A. Baurle" and H. A. Hassan °*

North Carolina State University, Raleigh, NC 27695-7910

Abstract

Attempts at modeling recent experiments of Cheng et al. I indicated that discrepancies

between theory and experiment can be a result of the form of assumed PDF and/or the turbulence

model employed, 2 In this investigation, improvements in both the form of the assumed PDF and

the turbulence model are presented. The results are again used to compare with measurements.

Initial comparisons are encouraging.

Introduction

The successful design of scramjet engines requires the understanding of turbulent mixing

and its effect on combustion at supersonic speeds. The availability of detailed measurements of

composition and temperature and their variances in H2-Air supersonic reacting shear layers

provides the modeler with useful data to develop the codes that will eventually be used in the

design. The first attempt at predicting the data of Ref. 1 indicated needed improvements in the

form of the assumed PDF and the turbulence model.

The joint-Beta PDF used in Ref. 2 was developed by Girimaji 3 to study mixing.

According to the model, all variances and covariances involving mass fractions depend on their

mean values and an additional quantity which is the sum of their variances. Thus, using this PDF

requires one additional model equation. The newly developed PDF that is presented below

results in independent variances and covariances thus necessitating additional model equations.

Because of the above, the present effort limits the joint-Beta PDF to the elemental mass fractions

* Research Assistant, Mechanical and Aerospace Engineenng, Student Member AIAA.

"* Professor, Mechanical and Aerospac.e F.n_,ine.e.rine_As.w)cial_ Fellow AIAA.

(NASA-CR-191929} MODELING OF

TURBULENT SUP_:RSONIC H2-AIR

COMBUSTION WITH AN IMPROVED JOIN[

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N93-16389

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2

and employs a partial equilibrium assumption to relate the minor species to the elemental mass

fractions. To accomplish this, an approach similar to that employed by Janicka and Kollmann 4

was used in conjunction with Table 1 (Ref. 5) to determine combined variables whose production

rates are independent of the two-body reactions (the fhst fourteen reactions in Table 1). The

minor species, O, H, OH, H202 and HO2 are then expressed in terms of the elemental mass

fractions and one of the combined variables using an equilibrium assumption.

The New Joint-Beta PDF

The elerncntalmass fractionsobey therelation

s=x l+x2+x 3 -I (I)

where xjisthe mass fractionof element 'j'.For H2-aircombustion, the PDF isassumed tohave

the form (reflectingthe threeelements H2, 02, and N2)

1 - x3 )a-(Yl+_) _71"I ", _72"1 (5(1 - s)] (2)

where 8 istheDirac deltafunctionand c isa constantdetermined from a normalization

condition. The parameters a, 13,Yl, and ?2" can be expressed in terms of means, variances, and

covariances by taking appropriate moments of the joint PDF, i.e.,

f Ot 71

_-I = |Xl fdxl dx2 dx3 -

a+[_ ?I+Y2

xI f (x I -_I) 2 fdx I dx 2 dx 3 = Xl2 - (x-T)I2

with

f

XlX 2 =J(Xl-Xl)(X2-x2)fdxldx2dx3 = XlX2 - x 1 x 2

--'q- oc(a + I) YI(TI + I)

X 1 =

(a + 13)(a+ _ + I) (?I+ h)(?l+ ?2+ I)

(3)
a(a + 1) 7ff2

xlx2 = (4)

(a + _3)(_+ _+ 1) (YI + "Y2)('YI+ T2+ 1)

The parameters a, J3, Yl and Y2 follow from the expressions for x I + x2, x , x2,"2 XlX" "2 .

The Combined Variables

A general procedure has been developed which makes it possible to take a set of

reactions, such as those indicated in Table I, and determine the combined variables whose

production rates are independent of the rates of the two-body reactions. For the set shown in

Table 1, thecombined variablesare

YO2 YO2 YO YH

- + + + + WHO 2

YH2 YH2 YO YH I YOH 1 YHO2 YH202

_v.---_ - w._÷% ÷__ +_Woo.-_W.o_-W._o_

YH20 YH20 Yo YH YHO2 + 2 YH202

WH20- WH20 We- _ + WHO2 WH202

(5)

(6)

(7)

where W is the molecular weight.

The production rates of the combined variables axe given by

W_) 2 = - We 2 [V¢15 + _/16 + _/17 - _/lS]

Wry2=- 2w. 2[w_5+w_6+wt7- %s]

° °

WH20 =- 2WH20 ['_15 + W16 + W17- "_/lS]

(8)

(9)

(10)
The Proposed Model

S

The elemental mass fra_ons, Z O and Z H and YH2 O, are the thxe_ independent quami6¢s

for which co_adon of _._ys _uadons arc solved. The remaining mass fractions, YO, YH,

YOH, YHO 2, and YH202, arc detcrmine, d using an ¢_:luilibrium assumption in to'tins of Z O, Z H,

and YH20. The joint PDF describing temperature and concenu'arion fluctuations is chosen as

a

G(T, YH2 O, Z O, Z H, Z N) = f1(T) f2(YH20)/:(7-.0, ZH, ZN) (I I)

In the presentapproach, fl(T)isa Oaussian PDF, f2 isa singleBeta PDF while F isthejointBeta PDF indicatedin equation (2).

Resultsand Discussion

A schematic of the experimentalsetupisshown inFigure I,and Table 2 givesa summary

of the burner exitconditions.Measurements wcrc taken ataxiallocationsof 0.85, 10.8,32.3,

43.l,64.7,and 86.1 innerdiametersdownstream of the burner.

A complex model of the type described above has tobc developed step by step. Initial

calculationscompared theresultsof a fullfiniteram kineticmodel calculationwith those

employing a partialequilibriumassumption. These calculationswere tamed out using a 91 x 91

gridon a domain of 4 x 4 inches.As a result,only comparisons for the firstfrye axialstationsarc

given. Figures 2-4 compare mcasur_ and calculatedmole fractionsof H20 and OH and the

tcmpcraturc distribution.As issccn from the figures,thepartialequilibrium assumption isan

acceptableassumption esgw.ciallyatthedownstream stations. -

The paper willimplement thefullmodel and willpresentdetailedcomparisons with

cxpcrimcnts of Rcf. I.
5

Acknowledgement

/

Y

This work is supported in part by the following grants: NASA Grant NAG-I-244, and the

Mars Mission Research Center funded by NASA Grant NAGW- 1331.

References

.

*

o

o

o

Cheng, T. S., Wehrmeyer, J. A., Pitz, R. W., Jarrett, O., Jr., and Northam, G. B., "Finite-Rate

Chemistry Effects in a Math 2 Reacting Flow," AIAA Paper 91-2320, June 1991.

Baurle, R. A., Alexopoulos, G. A., Hassan, H. A., and Drummond, J. P., "An Assumed JointBeta PDF Approach for Supersonic Turbulent Combustion," AIAA Paper 92-3844, July

1922.

Oirimaji, S. S., "A Simple Recipe for Modeling Reaction-Rates in Flows with TurbulentCombustion," AIAA Paper 91-1792, June 1991.

Janicka, J. and Kollmarm, W., "A Two-Variables Formalism for the Treatment of Chemical

Reactions in Turbulent Diffusion Flames," Seventeenth Symposium (International) on

Combustion, The Combustion Institute, Pittsburgh, 1979, pp. 421-430.

Jachimowski, C. J., "An Analytic Study of the Hydrogen-Air Reaction Mechanism with

Application to Scramjet Combustion," NASA Technical Paper 2791, Feb. 1988.
Tables

Table 1. Chemistry Model

Reaction

Number

5

6

7

8

9

i0

ii

12

13

14

15

16

17

18

R_axtion

H_ +0_, -- OH + OH

A _ b T_

.170E + 14 0.00 24157.0

.120E+ 18 -0.91 8310.5

.220E + 14 020 2591.8

.506E + 05 2.67 3165.6

.630E + 13 0.00 548.6

.130E + 14 0.00 0.0

.150E+ 15 _ 0.00 503.3

.200E + 14 I 0.00 0.0

.200E + 14 0.00 0.0

.301E + 12 0.00 9411.2

.200E + 13 , 0.00 0.0

.100E + 14 0.00 1801.7

.280E + 14 0.00 3220.9

.700E + 13 020 722.2

.221E + 23 -2.00 0.0

.730E + 18 -120 0.0

.230E + 19 -1.00 0.0

.121E + 18 0.00 22898.8

H+02 ---,OH+O

OH + H2 .-, H20 + H

O+H_ --OH+H

OH +OH _ H20 +0

HO_ + H -- H_ + 02

HO2 + H ---. OH + OH

H02 + 0 -- O_ + OH

H02 + OH ---*H20 + 02

H02 + H2 --* H202 + H

H02 + H02 ---, H202 + 02

H + H202 -- H20 + OH

0 + H202 --- H02 + OH

OH + H202 -- H20 + H02

H +OH + M -- H_O + M

H+H+M --H2+M

H + 02 + M ---, H02 + M

H202 + M -- OH + OH + M

Units of A are a multiple of cm s . mote -t . s -1

Table 2. Exit Conditions

Exit Conditions Hydrogen Jet

Math Number 1.0

Outer Jet Ambient Air

2.0 0.0

Temperature, K 545.0 1250.0 300.0

Velocity, m/s 1780.0 1417.0 0.0

Pressure, MPa 0.112 0.107 0.101

Mass Fraction

YH_ 1.0 0.0 0.0

Yo_ 0.0 0.245 0.233

YN_ 0.0 0.580 0.757

}_,o 0.0 0.175 0.010

J

Figures

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Figure 2: Mean H_.O Mole Fraction Comparisons Between Full Finite Rate Chemistry and

Assuming Partial Equilibrium
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Figure 3: Mean OH Mole Fraction Comparisons Between Full Finite Rate Chemistry and

Assuming Partial Equilibrium
?5

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°°of ::-.

500 I_O !500 20_ 2,500 500 _0(_ 1500 2X)O 2500

Figure 4: Mean Temperature Comparisons Between Full Finite Rate Chemistry and Assuming

Partial Equilibrium