NASA Contractor Report 178266 

ICASE REPORT NO. 87-17 

ICASE 

AN ASYMPTOTIC ANALYSIS OF SUPERSONIC 

REACTING MIXING LAYERS 

T. L. Jackson 

M. Y. Hussaini 

(BASA-CR-178256) A81 ASYUE'IO'I3C AblALYSXS CP N8 7-2 C7 56 

SUPEBSGPIC BEAC'IILG BIXING LAYELS (NASA) 

17 P CSCL 12A 

Unclas 

63/59 45352 

Contract No. NAS1-18107 

March 1987 

INSTITUTE FOR COMPUTER APPLICATIONS IN SCIENCE AND ENGINEERING 

NASA Langley Research Center, Hampton, Virginia 23665 

Operated by the Universities Space Research Association 

National Aeronautics and 

Space Administration 

LnQky-- 

Hamptonvrginia 23665 
AN ASYMPTOTIC ANALYSIS OF SUPERSONIC 

REACTING MIXING LAYERS 

T. L. Jackson 

Department of Mathematical Sciences 

Old Dominion University 

Norfolk, Virginia 23508 

and 

Institute For Computer Applications In Science And Engineering 

NASA Langley Research Center 

Hampton, Virginia 23665 

M. Y. Hussaini 

Institute For Computer Applications In Science And Engineering 

NASA Langley Research Center 

Hampton, Virginia 23665 

Abstract. The purpose of this paper is to present an asymptotic analysis of the laminar mixing of and the simultaneous chemical reaction between parallel supersonic streams 

of two reacting species. The study is based on a one-step irreversible Arrhenius reaction 

and on large activation energy asymptotics. Essentially it extends the work of Linan and 

Crespo to include the effect of free shear and Mach number on the ignition regime, the 

deflagration regime and the diffusion flame regime. It is found that the effective parameter 

is the product of the characteristic Mach number and a shear parameter. 

This work was supported under the National Aeronautics and Space Administration under NASA Contract 

NAS1-18107 while the authors were in residence at the Institute for Computer Applications in Science and 

Engineering, NASA Langley Research Center, Hampton, VA 23665. 

i 
r 

1. Introduction. The study of supersonic combustion flames has become extremely 

important in view of the scramjet engine for the propulsion of hypersonic aircraft and 

cruise missile. The aerodynamic processes governing such reacting flows are exceedingly 

complex involving strong interaction between chemical and fluid dynamical effects. The 

simplest relevant problem of a combustion flame may be formulated by considering two 

uniform parallel streams of hydrogen and air, both supersonic and semi-infinite in extent, 

which begin to mix at some origin (Fig.1). The physical processes underlying this problem are 1) the evolution of the mixing flow-field, 2) the evolution of the temperature field 

through conduction and convection, and through the heat generated by viscous dissipation 

and chemical reaction, and 3) molecular diffusion of each gaseous component into the 

other together with the annihilation of the combustible gas and the consequent production 

of combustion products. Any realistic modelling of the complex kinetics will necessitate a 

full scale numerical solution (eg, Fem (1973), Drummond, Rogers and Hussaini (1986)). 

Then questions arise as to the reaction rates and their dependence on temperature and the 

relative importance of various reactions which entail thus rather large uncertainties. 

In a mathematical treatment of combustion, the kinetic model has to be necessarily 

simple. The one-step irreversible Arrhenius model has been extensively used with 

significant success in the study of combustion (Buckmaster and Ludford (1982)). It was 

employed by Marble and Adamson (1954) who considered the laminar mixing of two 

parallel streams, one of a cold combustible and the other of a hot inert gas. They discussed questions of ignition, location of the flame relative to the initial contact point, and 

of its detailed development in the isovelocity case where both the streams have the same 

uniform velocity. The method of solution involved expansions in terms of the streamwise 

coordinate and the classical von Karman integral technique for solving the resulting set of 

ordinary differential equations. A similar problem of the continuous evolution from nearly 

frozen flow to near equilibrium flow in unsteady diffusion flames was analyzed by Linan 

and Crespo (1976). Their analysis, which employs a two-reactant model, elucidated the 

existence of the ignition, deflagration and diffusion flame. regimes. Although their study 

was confined to the isovelocity case and one-dimensional situation, it captured surprisingly 

well the rather complex picture. 

The present study extends their analysis to the more general problem of the chemical 

reaction between two parallel streams with different velocities. It is assumed that the 

characteristic chemical time and the fluid dynamical time are of the same order. The 

kinetic model consists of a one-step two-species irreversible Arrhenius process with high 

activation energy. This model appears to cover the essential physics of the problem. In the 

last decade or so, the asymptotic studies of combustion based on this model have 

significantly enhanced our understanding of ignition, of flame stability and of diffusion 

flame structure in low subsonic flows. There is as yet no reason to believe that this model 

will not play an equally significant role in enhancing our understanding of supersonic 

combustion. In any event, this idealization makes the problem amenable to asymptotic 
2 

analysis and thus provides a semi-analytical solution. The results can then certainly verify 

and in tun be verified by full numerical simulations. The asymptotic results of the present 

study bring out the effect of free shear and Mach number. They indicate that the important 

parameter is the product of Mach number and shear. Further, they predict the existence of 

the ignition, deflagration and diffusion-flame regimes even in the supersonic case. The 

numerical solutions cited above for analogous problems appear to miss the ignition and the 

deflagration regimes. Perhaps, careful fine-tuning of the numerics will uncover these 

regimes; this will be the subject of future effort. 

2. Governing Equations. Consider two reactants initially separated by a finite length 

splitter plate coming in at different velocities and temperatures (see Fig. 1). The nondimensional shear-layer equations with zero pressure gradient which govern this mixing process are given by (Williams (1985)) 

(P4i + (pv>,fl = 0, 1 = pT, (2.la) 

p(uYi, + vYiy,) = (~LY~,,)~’ - aiR, i = 1,2, (2.ld) 

where p is the density, u the velocity in the x’-direction, v the velocity in the y’-direction 

and T the temperature. Yl and Y2 are the mass fractions above and below the plate, 

respectively, with reaction assumed to be irreversible and of Arrhenius-type. The viscosity 

p is assumed to be a function of temperature. The other nondimensional parameters 

appearing above are: 

a = QYIO/CPTIOVIWI Heat release number, 

e = EJRT,, Activation energy, 

a+b a+b-lv w 4 BPI0 YIO 1 1 D=- Damkohler number, 

ai = viWi/vl Wl Parameter involving stoichiometry, 

M = Uldalo Mach number, 
3 

Ir 

where a and b are the reaction orders of Y1 and Y2 , respectively; vi the stoichiometric 

coefficient for species j; W, the molecular weight of species j; E the dimensional activation 

energy; R the universal gas constant; B the preexponential constant in rate expression; alo 

the speed of sound referred to Tlo ; Q the chemical heat release; y the specific-heats ratio; 

and finally Cp the specific heat at constant pressure. The equations were nondimensionalized by selecting the freestream values Tlo, plo, ul0, Yl0 above the plate for the temperature, density, velocities and mass fractions, respectively. Lengths are referred to f, , some 

characteristic length scale of the flow. We have assumed unit Prandtl and Lewis numbers 

in writing down these equations. The assumption of unit Lewis number allows us to consider linear combinations of (2.1~) and (2.ld) to eliminate the source term, which then 

admits similarity-type solutions (see remarks after (2.10)). 

The boundary conditions consistent with (2.1) are 

T= u = Y1= 1, Y, = 0 X’ = 0, y’ > 0, and at x’ > 0, y’ + -, 

(2.2a) 

~=p,,~=p~,y~=o,Y~=Y~ x’=O,y’eO,andatx’>O,y’+-. 

For definiteness, we take the maximum initial temperature to occur above the plate, and so 

we require PT,S 1. We make no such restriction on where the maximum initial velocity 

can occur, we only require both streams to be supersonic; Le., M - 1 > 0 and 

- 1 > 0. In addition, matching the pressure across the mixing layer results in the 

compatibility equation 

(2.2b) -1 PU -v(x’,-), 1 

4- PT 

which reduces to that given by Ting (1959) for PT = 1. 

Equations (2.1) can be transformed into incompressible equations by the use of the 

Howarth-Dorodnitzyn transformation (S tewartson (1964)) 

>’ Y’ 

X’ 1 

x=- 9 = - I pdy‘, w = (pv + uj pi dy’), 1, &o 0 

(2.3) 

where 1, is a scaling factor based on a characteristic chemical length, to be chosen later. 

The transformed equations in the (x,z) plane are 

ux 4- wz = 0, (2.4a) 

UU, + wu, = CU,, (2.4b) 
4 

a 

P 

UT, + wT, = CT,, + C(y-1)M2U: + lc-!2, (2.4~) 

(2.4d) 

where the constant C appearing above is given by Chapman’s (1950) linear viscosity law 

l=cT. (2.5) 

These equations may also be written in terms of the variables x and q , where 

?I-%== Z 

is the similarity variable for the chemically frozen heat conduction problem, and qo 

corresponds to a shift in the origin determined uniquely by (2.2b). (We note here that if 

the initial velocities were subsonic, then the compatability condition (2.2b) would be trivially satisfied, and thus qo would remain indeterminate (Klemp and Acrivos (1972)). By 

introducing the relationships 

(2.7) e 

G u=f(q) 7 w=-(rlf-J% 

we see that the continuity equation (2.4a) is automatically satisfied, while the q -momentum equation (2.4b) reduces to the well known Blasius equation for free shear 

f” + 2fs’ = 0, f(-) = 1, f(-) = pu, f(0) = 0, (2.8) 

which represents the mean flow profile of the wake velocity. Numerical solutions are 

given, for example, by Lock (1951). For our purposes, we shall consider the velocity field 

completely known. 

Once f is known, the temperature and mass fractions can be determined from the 

equations 

4x1,aSZ 

P 4dTx - 2g,, - T,,,, - (~l)M~(f‘)~ = 7 (2.9) 

4xlcaiSZ 

P 4XfYiX - 2fyii, - Yh = - , i = 1,2. (2.10) 

These do not have similarity solutions due to the presence of the nonlinear source term. 

However, the combinations T+aY, and T+-Y2 do satisfy the heat equation without 

source terms, and are given by the form 

a 

a2 
5 

Thus, a similarity solution exists (Stewartson (1964)) and is given by 

where g satisfies the differential equation 

g" + 2fg' = 0, 

with solution 

X 

c1 and c2 being arbritary constants. When the boundary conditions (2.2a) are taken into 

account, we find that 

T + aYl = 1 + a - (1 - PT + a)c + flM2(l-Pu)2c(l-c), (2.1 1) 2 

where 

(2.13) 

Note that as q + -, 5 + 0, and as q + -, 5 + 1. 

The relations (2.11) and (2.12) gives the mass fractions in terms of T and q , enabling us to reduce our problem to that of finding T from (2.9). If we choose for I, a nondimensional characteristic chemical length based on the maximum initial temperature Tlo 

and initial velocity ul0 , 

we obtain 

4x 4flTX - 2g,, - T,,,, - (')~l)M*cf')~ = -~~(~'6)Y~Y~e~p(0(~)}, 0 T (2.15) 

subject to the boundary conditions 

T + 1 as q + -, x > 0 and x = 0, q > 0, (2.16a) 
6 

T -+ PT q -+ -00, x > 0 and x = 0, q < 0. (2.16b) 

Eqn. (2.14) implies, in effect, that the characteristic chemical time and characteristic flow 

time are of the same order. 

For the special case Pu = 1 (no shear), (2.8) gives f = q corresponding to u=l, v=O 

everywhere. Then from (2.13), 

(2.17) 

and we see that (2.11)-(2.17) is equivalent to Linan and Crespo's (1976) equations (13)- 

(17); with x in our paper playing the role of the time-like variable t in theirs. Therefore, 

the analysis of Linan and Crespo (1976), hereafter referred to as I, can be considered a 

limiting case of the above system. The goal of this work is to assess the effect of shear 

on mixing and ignition, as well as the coupling effect of differing initial temperatures. 

3. Ignition Stage. In the absence of chemistry, equation (2.15) reduces to 

44% - Tqq - 2flq - (y-l)MZ(fy = 0 (3.1) 

which posesses the (inert) similarity solution 

which is also a solution of (2.15) for small x. We note here that for pu = 1, the Mach 

number M plays no role. Substituting (3.2) into (2.11) and (2.12) yields for the (inert) 

mass fractions 

Y,I = 1 - 5 , Y*1= Y& . (3.3) 

Since 5 increases as Pu increases (see (2.13)), we see from (3.3) that Y,I decreases while 

Y, increases. Thus, the effect of shear is to enhance the initial mixing process that would 

otherwise be present only by interdiffusion of the reactants. 

As x increases, more of the combustible mixes until, at some finite distance downstream of the plate, a thermal explosion occurs characterized by significant departure from 

the inert. The exponent of (2.15) suggests that, in the limit 8 -+ being considered here, 

this will occur in regions where T - 1 = O(e-'). Therefore, T = 1 is the switch-on temperature for the chemical reaction. 

To describe .the ignition stage, we follow I and set 

9 = e(T - Ti). (3.4) 
7 

The basic equation (2.15) becomes 

where 

from (2.1 1) and (2.12), respectively. 

The term 8(Tl - 1) in the exponent of (3.5) can be either positive or negative, 

depending on the amount of initial shear. For TI - 1 = 0(1), the rate term is either 

exponentially small or exponentially large, depending on its sign. The latter case shows 

that ignition takes place immediately at the trailing edge of the plate. A realistic model 

for this case should include the boundary layer effects at the trailing edge of the plate, 

which the present model neglects. Modifications of this model to allow for very short ignition distances will be the subject of a future paper. The former case shows that chemical 

activity is quenched. Thus, we confine our attention to regions where TI - 1 = O(e-'), 

consistent with (3.4) with T - 1 = O(e-'). 

We fist consider the important case of ignition for nearly equal initial temperatures 

(3.7) 

and velocities. To this end, we set 

pT = 1 - e-lp, , pu = 1 - e- 112 Pu, 

and seek an expansion for 8 + - about the inert, with PT, pu and M fixed and O(1) constants. In particular, we see that 

5 = 5, + o(e-ln), f= - e-lnD&, + o(e-'), (3.8) 

from (2.13) and (2.8), respectively, with 5, defined in (2.17). The inert solution can now 

be expanded as 

Y,, = 1 - 5, + o(e-1/2), Y, = y2&, + o(e-ln). (3.10) 

Substitution of (3.8)-(3.10) into (3.5) yields the following equation for the disturbance 

temperature: 
8 

subject to the boundary conditions 

$=O at x=O andatq+b. (3.12) 

The expansions are based on the assumption that e-' > where Re is the Reynolds 

number based on the free-stream values of the flow. (Typical values for 8 are on the 

order of twenty, and for Re, the order of several millions). Except for the last term in the 

exponent, eqn. (3.11) is identical to (33I). Thus, the important new parameter governing 

ignition is the product of Mach number M and shear parameter pu, which measures the 

deviation from the isovelocity case. 

The solution to this problem was computed numerically. The equation was discretized in q, and the resulting system of ordinary differential equations was solved via the 

Gear package. Computations were done for various values of the parameters & and 

and for reaction orders a = b = 1. Integration was terminated when $ became infinite, 

thus identifying the ignition point (x*,qr). Results of these computations are displayed in 

the table, which is discussed below. 

The first entry of the table (M&,= 0) corresponds to that of I. Clearly, as & 

increases, the igntion distance X. increases, while the ignition location (x*,q*) moves into 

the region of higher initial temperature. Thus, the effect of cooling the gas below the 

plate is to retard ignition. 

-pT xr rl* MDU XI rl* 

0 5.839 .O 0 5.839 0 

2 13.203 i3 2 4.899 0 

5 27.595 .62 5 1.912 0 

10 54.850 .84 10 0.057 0 

Mpu Xr q* MPU X* rl* 

0 13.203 .30 0 54.85 .84 

2 11.270 .28 2 51.01 .82 

5 4.666 .18 5 32.86 .70 

10 0.149 .08 10 2.92 .36 

Table. Ignition points for various values of PT and MPU, and for a = b = 1. 
I 

The other entries corresponds to fi ed values of BT. We note that these results do ot 

depend on the sign of pu; Le., does not depend on where the maximum initial velocities 

occur. As I pu( increases, with M fixed and 0(1), the ignition distance moves closer to 

the trailing edge of the splitter plate, while the ignition location (x*,q*) approaches the 

center line q = 0, where the maximum amount of shearing occurs. Thus, the effect of 

increase in I pul is to enhance mixing, and hence to enhance ignition. On the other 

hand, as M becomes large (Le., hypersonic speeds), for fixed I pul , ignition is found to 

occur rapidly even with a small amount of mixing! This is understandable since increasing 

the Mach number increases the inert temperature, which in turn feeds in more energy into 

the system causing it to ignite sooner. Consistent with the results of I, if we fix MI pu I 

and vary PT, we again see that ignition is retarded, moving away from the edge of the 

plate. 

As & and I pul increase, with M remaining 0(1), the expansions (3.7) break down. 

For the special case of no shear or zero Mach number, the analysis proceeds as in I. For 

0(1) shear, the location of the flame critically depends on the amount of initial tempertaure differences. If the two streams are nearly equal in temperature, then ignition ocurrs 

at the trailing edge of the plate, and a more delicate analysis is needed, as mentioned 

above. If, on the other hand, the two streams are sufficiently different in temperature, then 

ignition could occur away from the plate, and the analysis again proceeds as in I. We 

postpone the analysis for 0(1) velocity and temperature differences as a topic of future 

research. 

After ignition has occured, a pair of well-defined deflagration waves emerges according to classical thermal explosion theory. Since the effect of the Mach number and shear 

is to only quantitatively alter the direction and speed of the waves, we do not present any 

results here. We only note that they must exist, penetrating the mixing layer until all of 

the deficient reactant is consumed (Fig. 1). Just downstream of the deflagration wave a 

diffusion flame exist where the flow is in chemical equilibrium, and is described in the 

following section. 

4. Diffusion Flame. As in I, a thin diffusion flame exists behind the premixed flame 

and is characterized by near-equilibrium conditions; Y, = 0 on one side of the flame with 

Y2 = 0 on the other. From the relations (2.11) and (2.12). we see that the flow can be 

described by 

Y1 = 1 - (1 + 3)s , Y2 = 0, (4.la) 

a, 
10 

for q > q,, and 

(4.2a) yY Y1 = 0, Y, = -a,( 1 - (1 + -)!}, 

a2 

(4.2b) 

for q c qe, where qe gives the location of the flame sheet where both reactants vanish, 

and T takes the adiabatic flame value T,, given by 

and 

(4.3) 

defines the flame location. We remark here that the location of the diffusion flame 

depends not only on the initial concentration, but also on the product of Mach number and 

shear, a feature absent from I. Fig. 2 shows qe for various values of M(1-Pu). In particular, as pu decreases (or, equivalently, as M decreases), qe moves into the region of lower 

initial velocity, for fixed values of (1 + Y,$xz)-'. 

To study the structure of the diffusion flame, which appears as a discontinuity on the 

(4.5) 

q, or 6 scale, we introduce the variables 

6 = 5, + 8-92, T = T, - o-~T~T,, 

and expand for 8 + =. Here, B is a scaling factor defined by 

Following the notation of I, we introduce the relations 

Y, = qe-l(r - zy~, Y, = gT,e-l(r + Z~A, (4.6) 

with A defined by (831). By substituting (4.5)-(4.6) into (2.15), we obtain equation (84) 

of I (where a (-) minus sign is missing from the exponent), with I' given by (81I), except 

now the constant g of (851) is replaced by 
11 

Hence, the analysis of I for the structure of the diffusion flame also holds when shear is 

taken into account, with an appropiate interpretation of g. 

Conclusions. The laminar mixing of and the simultaneous chemical reaction 

between parallel supersonic streams of two reacting species has been discussed in the 

framework of high activation energy asymptotics. Attention is confined to the special but 

important case of low shear parameter, which is a measure of departure from the isovelocity case, and small temperature differences between the streams. It is found that the 

regimes of ignition, deflagration and diffusion flame uncovered by Linan and Crespo 

(1976) are found to exist in the supersonic case also. In particular, it is found that two 

parameters governs the ignition regime: a parameter giving the temperature difference 

across the plate (Linan and Crespo), and a second parameter which is the product of the 

characteristic Mach number and the shear parameter. The location of the ignition point 

for a given temperature difference is a function of the latter parameter. It is implied that 

the ignition point moves upstream as the value of this parameter increases. 

This work suggests resolution requirement for the numerical simulation of the problem to capture the ignition and deflagration regimes. The next improvement of this model, 

to be more realistic, should include the effect of the finite boundary layer thickness at the 

trailing edge. 

Acknowledgments. The authors would like to thank A. Kapila for helpful comments 

in connection with this work. 
12 

REFERENCES 

Buckmaster, J. D. and Ludford, G. S. S. (1982). Theory of Laminar Flames, Cambridge University Press. 

Chapman, D. R. (1950). "Laminar mixing of a compressible fluid", NACA Rep. 958. 

Drummond, J. P., Rogers, R. Clayton and Hussaini, M. Y. (1986). "A detailed 

numerical model of a supersonic reacting mixing layer", AIAA Paper No. 86-1427. 

Fem, A. (1973). "Mixing-controlled supersonic combustion", Annual Review of Fluid 

Mechanics, Vol. 5, pp. 301-338. 

Klemp, J. B. and Acrivos, A. (1972). "A note on the laminar mixing of two uniform 

parallel semi-infinite streams", J. Fluid Mech., 55, pp. 25-30. 

Linan, A. and Crespo, A. (1976). "An asymptotic analysis of unsteady diffusion 

flames for large activation energies", Comb. Sci. Tech., 14, pp. 95-117. 

Lock, R. C. (1951). "The velocity distribution in the laminar boundary layer between 

parallel streams", Quart. J. Mech. and Appl. Math., 4, pp. 42-63. 

Marble, F. E. and Adamson, T. C. (1954). "Ignition and combustion in a laminar 

mixing zone", Jet Propulsion, 24, pp. 85-94. 

Stewamon, K. (1964). The Theory of Laminar Boundary Layers in Compressible 

Fluids, Oxford University Press, Great Britain. 

Ting, L. (1959). "On the mixing of two parallel streams", J. Math. Phys., 28, pp. 

15 3- 165. 

Williams, F. A. (1985). Combustion Theory, Benjamin Cummings Publishing Company. 
13 

Diffusion Flame 

Figure 1. Schematic showing the reacting mixing layer, with TI > 7'2. 
14 

Ye 

L 

Figure 2. Diffusion flame location q, as a function of initial concentrations. 
Standard Bibliographic Page 

ICASE Report No. 87-17 

. Title and Subtitle 

. Report No. NASA CR-178266 12. Government Accession No. 13. Recipient's Catalog No. 

5. Report Date 

and Engineering 

Mail Stop 132C, NASA Langley Research Center 

AN ASYMPTOTIC ANALYSIS OF SUPERSONIC 

REACTING MIXING LAYERS 

l. %it!!$ fS f6Yt No. 

6. Performing Organization Code 

Hampton, VA 23665-5225 

2. Sponsoring Agency Name and Address 

National Aeronautics and Space Admfnistration 

Washington, D.C. 20546 

'. Author(s) 

T. L. Jackson, M. Y. Hussaini 

13. Type of Report and Period Covered 

Contractor Report 

14. Sponsoring Agency Code 

505-90-21-01 

8. Performing Organization Report No. 

87-1 7 

20. ec ity C s'f.( f this page) 21. y6 of Pages I ancYissiYie8 cur'ty Clag 'f,(of this report) 19' ?fnciassi?!ied 

10. Work Unit No. 

I. YAfo i 

s M PFT%PT~%~v@ c a t i ons in s c i en c e 

22. Price 

A0 2 

Langley Technical Monitor: 

J. C. South 

Final Report 

Submitted to Combustion Science and 

Technology 

6. Abstract 

The purpose of this paper is to present an asymptotic analysis of the 

laminar mixing of the simultaneous chemical reaction between parallel supersonic 

streams of two reacting species. The study is based on a one-step irreversible 

Arrhenius reaction and on large activation energy asymptotics. Essentially it 

extends the work of Linan and Crespo to include the effect of free shear and 

Mach number on the ignition regime, the deflagration regime and the diffusion 

flame regime. It is found that the effective parameter is the product of the 

Characteristic Mach number and a shear parameter. 

17. Key Words (Suggested by Authors(s)) 

chemical mixing, supersonic flow, 

asymptotics 

18. Distribution Statement 

59 - Mathematical and Computer 

Sciences (General) 

Unclassified - unlimited 

For sale by the National Technical Information Service, Springfield, Virginia 22161 .. 

NASA Langley Form 63 (June 1985) NASA-Langley, 1987