Computational study of generic hypersonic vehicle flow fields

APPENDIX- A

AIAA-94-2311

Effect of Turbulence - Chemistry

Interactions In Compressible Reacting Flows

Johnny R. Narayan

MCAT Institute

Moffett Field

California

EFFECT OF TURBULENCE-CHEMISTRY INTERACTIONS IN COMPRESSIBLE REACTING FLOWS J.R,. Narayan" MCAT Institute, NASA ARC, Moffett Field, CA.

ABSTRACT At time step U dependent variable vector velocity vector The objective of this work is to investigate the ef- fect of the interactions between the turbulent scalar production rate of species n ;g streamwise coordinate field and chemistry in compressible, reacting turbu- lent flows. A model for these interactions has been j_h coordinate zj transverse coordinate proposed. In the thermochemistry model, the effects Y of temperature and species concentration fluctuations turbulence model coefficients turbulence model coefficients on the species mass production rates are decoupled.

Kronecker delta The effect of temperature fluctutations is modelled via a moment model and the effect of concentration turbulence energy dissipation rate _c compressible dissipation rate fluctuations is accounted for using an assumed fl-pdf dissipation rate in g-equation model. A two equation (k-w) model with compress- _g ibility correction is used to calculate the turbulent 77 compressibility correction coefficient velocity field. Preliminary results for the case of a 7 ratio of specific heats two-dimensional reacting mixing layer are presented. specific dissipation rate Computations are carried out using the Navier-Stokes laminar viscosity solver SPARK, with a finite rate chemistry model for turbulent viscosity hydrogen-air combustion. I/ kinematic viscosity P density turbulence model constants NOMENCLATURE O'd_O'k, 0"_ stress tensor rij flux vector in j_h direction A,b coefficients in Arrhenius rate equation Subscripts C1 ,C2 turbulence model constants t turbulent quantity E total internal energy /.

mass fraction of species n g scalar variance (enthalpy, mass fraction) INTRODUCTION H source vector H total enthalpy h Advanced air-breathing propulsion systems such as static enthalpy k the supersonic combustion ramjet (scramjet) have turbulent kinetic energy forward and backward reaction rate constants been studied for a long time as candidates for high- k/,kb L speed propulsion. Mixing between fuel and oxidizer, number of reaction steps heat release, presence of shocks and the chemical re- M.

Molecular weight of species n actions are some of the major aspects of such flows.

lVI, Turbulent Mach number N Computations and experiments of the flow fields in number of chemical species the combustors of such systems reveal a complex in- Pr,Prt laminar and turbulent Prandtl numbers terplay of these physical phenomena. The effect of P pressure turbulent mixing on chemical reaction is an example Re,ReT turbulence model coefficients Rk ,tL, of such an effect which must be addressed in design- turbulence model coefficients Sc,Sc_ ing such a propulsion system. One such effect, the laminar and turbulent Schmidt numbers T turbulence-chemistry interaction, is the subject of our temperature present study. Design parameters such as the mode of Activation Temperature t fuel injection, flame stabilization, ignition delay, com- time bustion efficiency etc. may be influenced by these in- *Senior Research Scientist, Senior Member AIAA. teractions.

Thereare manyfactorsthat affectthe successful

available in this area. There has also been a concerted

evaluation ofthecombustor flow.Theflowfieldin the

effort to generate validation data bases through direct

combustor is turbulentandcompressible. A further

numerical simulations. Even though there is a lot of

complication is introduced by the high heatrelease

publications in this area, this effort is still in its early

in the combustor. Today, anidealprocedure for the

stages.

studyof suchproblems (in theabsence ofexactana-

In a computational analysis of flow fields the choice

lyticalsolutions to thegoverning equations) will bea

of turbulence model(s) is dictated by two conflicting

combination of an accurate numerical solvercoupled

requirements. The model should (i) be reasonably ac-

with experimental datato (i) validate the numerical

curate with good physical justification; and, (ii) be

solver and(ii) toprovide valuable comparison datafor

computationally feasible for complex flow geometries.

theparticularcombustor configuration beingstudied.

The turbulence model used in the present instance is

The calculation method mustbe capable of address-

a two-equation (k- ¢v) model [6, 7], modified to ac-

inghigh-speed, turbulent, reacting, compressible fluid

count for compressibility. The effect of compressibil-

flowsinvolvinghighenergy release. Thereis a glut

ity in turbulent flows is an important aspect of high of useful numerical solvers applicable for a wide vari- speed turbulent flows. There has been a significant ef- ety of flows including all speed regimes. The capabili- fort aimed at developing models to account for these ties of the computational fluid dynamics (CFD) codes effects [8, 10]. These models have been used to predict depend upon the sophistication and accuracy of the shear layer growth rates with some success [11, 12].

models used to simulate the various physical processes The model used in the present computations is the involved. There has been a lot of progress made in the one developed by Zemen [10]. On the thermochemistry development of accurate turbulence models in the re- front, the effects of temperature and species fluctua- cent years. Finite rate chemistry including nonequilib- tions are decoupled. For the temperature-turbulence rium thermodynamics is another area where significant interaction, a moment model is used [1]. The chemical progress has been made. Computational algorithms species fluctations are accounted for using an assumed which are fast and accurate are being improved every- multivariate-_-pdf model [1, 2, 3]. The thermochem- day. Computers which can run these codes are also istry models used here are independent of the model being improved. However, every one of these areas for the turbulent velocity field. These models require is still under development and as a result there is no the solution of the evolution equations for all the mean single CFD code which includes the accurate models flow thermo-chemicai variables, turbulent kinetic en- and algorithms in all the areas. Limitations such as ergy (k), specific dissipation rate (w), enthalpy vari- the maximum grid size, temperature range of applica- ance and turbulent scalar energy (sum of all the species bility of thermodynamics models, number of steps in variances). The CFD code used is SPARK [13], devel- a finite rate reaction model, near-wall applicability of oped at NASA Langley Research Center.

the turbulence model etc. still affect the usefulness of Mixing plays a major role in high speed combustor these CFD codes. On the other hand, experimental study of such problems is not possible at present due flows. The reaction zone is mainly confined to mixing layers that exist between fuel and oxidizer streams.

to the lack of adequate facilities. Fortunately, many of Efficient mixing of fuel and oxidizer is very important the major issues can be addressed using current CFD tools.

for the design of combustor size. Mixing layers ranging from subsonic to supersonic speeds have been studied The focus of the present work is on the turbulence extensively over the years [1, 11, 14, 15, 16, 17, 18, 19, model(s) used for reacting, compressible flows. Specif- 20]. However, most of those studies use oversimplified ically, the interaction between the turbulence field and models for turbulence-chemistry interactions. In the the chemical kinetics is of interest. The effect of tur- present work, a two-dimensional, compressible, react- bulence on phenomena such as mixing between two ing, mixing layer (hydrogen-air) is computed with the streams, boundary layer etc. has been studied exten- aforementioned, more realistic turbulence-chemistry sively and it is an ongoing process. However, there interaction model. The reaction model (hydrogen-air) is not enough work done in the area of turbulence- used is described in Ref. [21] (9 species and 20 reac- chemistry interactions to either emphasize or neglect tion steps) and is given in Table 1. Experimental data the importance of such interactions. The main reason from compressible, reacting mixing layers is still scarce for this is the fact that it is extremely difficult to mea- which hinders the validation of the calculation proce- sure these interactions quantitatively. Recently, there dure. As a consequence, the comparison of results is have been efforts aimed at establishing pdf-based mod- confined to those obtained with and without the inter- els for coupling the turbulence field with the thermo- action model.

chemical field in numerical simulations. References [1], The governing and secondary equations used in the [2], [3J, [4] and [5] are just a few of the many reports

computations haveall beendescribed in detailin the with

references citedabove.Only an abreviated equation

_k

setwill begivenin the present paper.Thecomputa-

ReT = --= (10) wp

tionswereperformed on thesupercomputers of NAS

andNASAAmes Research Center (C-90).

and GOVERNING EQUATIONS P Oz i Ozi

-- _ (11)

The continuity equation, Navier-Stokes equations, The constant Re--6.

energy equation and species continuity equations gov- ern the instantaneous evolution of the flow vari- Closure of the averaged equations is achieved by ables [1, 15, 22]. Density-weighted averaging [23] is invoking the Boussinesq approximation which relates used to derive the mean flow equations from these the turbulent stresses to the mean strain rate. The equations. The dependent variables, with the excep- Reynolds stress tensor is written as, tion of density and pressure, are written as

- puTu_' = _ sq - ] _k6q

¢ = $ + ¢" (i)

where the ¢" is the fluctuating component of the vari- Sq - OUi OU 1 20U_, 6_i (12) Oz i + Ozi 30zk able under consideration and its Favre-mean ¢ is de- fined as where/_ is the turbulent/eddy viscosity defined as

_" = _ (2) ._k

fit = ot p-- (13) Cd In this equation, the overbar indicates conventional with time-averaging. Density and pressure are split in the _;+ conventional sense as, Or* -- Rk 1 + _ (14) R_ p = _ + p' and p = p + p' (3) and The averaged continuity and momentum equations are _; =Cd3, Rk =S, C_ =3 (15) O_" + Oz_ - 0 (4) The mean continuity equation (4) does not require any modeling. The modelled momentum equation is, " " Orli

#_u_ O_u_u_ O_ Opu_ ui + (5)

Y + Oz i - cgzi Ozj Oz i Ot + Oz i " Ox_ + ] 0_---/6q where • Oui Oui . 2 Ouk

- 0_--7 [ (_ + _') s,j 06)

(6)

ni = "_-_i + 8-;7*, ) - 3 _ s,i

There has been a considerable amount of activity in with repeated indices indicating summation.

the area of modeling the compressibility effects [8, 10].

The two turbulence variables for which we carry evo- In these models, the compressible dissipation terms are lution equations are the turbulent kinetic energy (k) expressed ms functions of the turbulent kinetic energy and the specific dissipation rate (w) [6] defined as dissipation rate and the local turbulent Mach number.

The compressibility effects are represented by a com- I! I!

k =_ pui ui ponent of the dissipation rate (¢_) given as

(71

_, = [i'ce

= _/(c:_) (s) Ko - 0F(M,)

2k where (17) M, -- --ft.

t_4 9 _8 +x Re ) where a is the local speed of sound and F(M,) is a r ___:_4 (g) C_- I00 I+_R= z function of the local turbulent Mach number (M_). As

mentioned before, the model used in thepresent work

where _ is a coefficient which, normally, is a constant.

wasproposed by Zemen [10] where F(M=) is given by

For ¢ = Jr, (n represents the species) , _¢_ = Sct, and for the static enthalpy, (¢ = h), _¢, = Prt.

-- Mto F(M,) = 1- ezp[-(M'0.6 )2], ;v/, > M,o Using the above definition, and omitting the body force contribution, the time-averaged and modelled en- = 0, Mt < M,o (18) ergy equation [1] is with Mto=0.1 and ,7=0.75. The modelled turbulent kinetic energy equation is [1, 9] Ot + 0_j - 0z_ (r-ZT- _ 6u - puTu_')u_ o_k o_kui 0--7" + Oaj - P_ - _k(l + Kc) o p ,a, ) O_ _,) _k..26 0 fit. Ok.

+ + (lo)

where ¢_ is a coefficient that appears in the turbulent kinetic energy equation. The modeled species continu- where ity equation is (20) d _ oz]. oz]. _ -.- o [ f, ) o]..

Ot + 0zj w,-_ (_cc +_c_ -_iJ(27 ) The modelled w-equation used in the present anal- In the above, the mean species production rate due ysis [6] is given below. This equation does not include to chemical reaction (tb'--_) needs to be modelled. This any compressibility corrections.

term is a function of both the temperature and species 0_.,J concentrations. In the absence of a model for the O---i- + O_UJozj = c,-_Pk - C1_ _ + _a£D_,o interaction between the turbulent temperature and concentration fields, this term is evaluated using the 0 fit) Of - mean values of local temperature and species concen-

+ (21)

trations. In a finite-rate system involving L reaction where steps and N species, the instantaneous production rate Ok Ow of a species n can be represented (law of mass action) in the following general form: D_,_ - Ozj Ox i (22) L and iS. -" Mn Z_._, ,at- v,a_) x (28) 1 _o+ -&_ 1=t R_

- (23)

N f, , N , yl'r/, v,,',_ {kllP'_' H(M)"" - kbIP'_' J.J.'M " _' The turbulence constants are $=1 $ $=1 $ where c_0=0.1, R_ =2.2, ak = 1.0, o',o = 1.67, N N Pr=0.72, Prt = 1.0, Sc=0.22, Sc= = 1.0

E' E"

and _,----1 s=l o'a = 0.0, D_,,<0 In the above equations, u_t and u_] are the number of molecules of the scalar s involved in the l-th reaction = 0.3, D_,,,>0 step in the forward and backward directions, respec- tively. The forward and backward rate-constants of The mass-averaged total energy can be written in the reaction l are given by klt and kbt respectively.

terms of the total enthalpy as The reaction rates are usually strong functions of the temperature: _" = _r__ (24) kit = ArT _' ezp[ T=t] The correlations between the fluctuating velocity and ---_-_ (29) tI_e scalar fluctuations are modelled using a gradient- where At, bt and Ta, are numerical constants specific diffusion hypothesis. A typical model is of the form to the given reaction step l.

puT¢,, f, 0_"

- = --: (25)

Turbulence-chemistry interaction model The interactions between the temperature and con- Decoupling the effects of temperature and species centration fields (turbulent) may be of importance in concentrations, the mean species production rate can many applications. The following section describes a be written as, model for such interactions. Practical combustor flows L involve multiple scalar mixing and reactions. In order '/)_: Mr Z_.,< at - i%l) x (36) to calculate the mean species production rate using I----1 the pdf approach, a model for the joint pdf of temper- N N ature and various species is required. Such a joint pdf {gPm'( H M:V"')I;t- ;¢bZpa'(H MZ"")Ia,} is difficult to specify and as a preliminary step we effect s=1 s----I the simplification that temperature and species fluctu- where ations are uncorrelated. This permits us to deal with N N the temperature and species fluctuations separately.

b,-- <l--I s::'>, _,- <l-I s;;">

Equation (29) can be written as s=l s=l In the above, angular brackets represent conventional kf, = At(_' + T") b' ezp[- (_ + T,)] (30) time averaging. The modelled expression for/'fz using the multivariate 8-pdf is [2], Assuming that T" ---- < 1, the term (T+T") can be ex- t T panded in a series and the resultant modified reaction

b, = I-I l-I(_. + 4,- ,)I I]:(B + m,_ p) (an

rate term is written as, s----I r----i p=l Similarly, the expression for Ibl is, kl"_, = (1 + m) At:_ b_ ex.p[- Ta,.

-_-] (31) is 1_t r b'sl _l where

a,= I-[_(Z.+d_-,)/_(B+_,-p) (3s)

s----1 r=l p----I bl Ta,. i T_, )_ T"T" where

m = [(b,- i)(7 + -}-) + _(-_-] f,. (32)

B = 81 + 8._ + -" + 8_¢ (39) Terms of order higher than two in T" ---- are neglected T The production of turbulent scalar energy due to from the series expansion in the present analysis since chemical reaction can be decomposed as accurate models for these higher order terms do not ex- N N N ist. The factor m represents the effect of temperature

woJ" = w./o - F.. (40)

fluctuations on the mean reaction rate. The tempera- n=l n=l n=l ture variance is calculated from Only the firstterm on the right hand side of the above T,_T,, h'_, equation needs further modeling. This term can be = _ (33) written as, using the assumptions above, N N L in the present work. Other pdf-based models for the _"{I/' -- ' Z w'_f'_ -" Z M, A.._' "' u,_,)x (41) temperature effects exist [5] which will be explored in n=l n=l I=i the future.

N N For reacting flows involving multiple chemical

{Vp_" (rI M;':, )jr,. - _r,(rI M;<,)j_,. }

species, Girimaji suggests [2, 3] the use of a multivari- s=l s=l ate 8-pdf model to account for the effects of the scalar where fluctuations on the species production rates. This N N model is briefly outlined below. The parameters of

h,. = (A I-I ff:'>. J_,. = (A lI/::")

the multivariate 8-pdf for the N-scalar mixing process s=l s=l (81," • ", 8N) are functions of the mean mass fractions f,_ and turbulent scalar energy Q: The terms Jlt_ and Jbl. are also moments of the scalar joint-pdf. The modelled expression for these moments are [2], 8,=_(1_3 I) (34) where JI',, - b' ,6. + _'_l (42) B+mt N N

s= r o = (35) nl

r_=l n=1 B+n_ Substitution of equation (42) into equation (41) leads The term w* is taken from the most recent calcula- to the model for the source/sink of turbulent scalar tion step. The source terms of the w-equation are energy. The main advantage of this choice of the also manipulated in a similar manner. These nonlin- assumed-pdf model is that the chemistry related mod- ear turbulence source terms are treated in a pointwise- els are obtained analytically and no numerical inter- implicit manner while solving the turbulence equations gration in the species space is necessary.

by rewriting equation (46) as, The turbulence-thermochemistry interaction models (I OH 0+ 7 require the enthalpy variance and the turbulent scalar - At_U) (r-r "÷I - u") = - At [-_- - H"] (48) energy distributions. The modeled equations for en- The discretized equations are solved by means of a thalpy variance (h'h t') and turbulent scalar energy(Q) fourth-order compact scheme.

are of a form similar to that of the turbulent kinetic energy: RESULTS AND DISCUSSION Choosing a flow configuration to demonstrate the ef- + Ox I - 2pu_'G" Oz i 2fie_ fect of turbulence-chemistry interactions is a difficult task due to the lack of guidelines based on prior data

+ b 7 jC( + + +. (43)

(experimental or otherwise). Since the present work is part of a larger task of establishing a solution pro- For g = h"h', G = h, ffJ=0, _r = Pr and o'g = Pr_.

cedure for high speed propulsion systems (scramjets), Forg = L_n=ljnjn, 2 N • ,! the initial choice fell on a two-dimensional high speed v "_ t,,';7-,7,, G=.f,,,+= _,,__tw,,/_,o.= Sc and _g = Set. The dissipation term in the above reacting mixing layer. Non-reacting and reacting high equation is assumed to be speed mixing layers have already been computed us- ing the computational procedure described above [14].

Available experimental data were used to validate the (44) eg --- Cg kg -- CgC2wg prediction procedure. The fl-pdf model to account for the effect of species concentration fluctuations on the The model for _---n=IV'N _'_J,;';' r'', is given in equations (40)- mean production rate of species was also introduced in (42). The constants C_ are assigned a value of 0.5.

conjunction with the k- e turbulence model before [1].

The main aim of the present work is to introduce the Solution of the modeled governing equations results obtained by coupling the fl-pdf model with an improved version of the k- w model [6].

The equations are discretized and integrated in space and time to obtain steady state solutions using A schematic of the flow problem is given in Figure 1.

an elliptic solver SPARK [13]. The governing equa- The two streams are, air (U =1606 m/see, T =1600 K tions are written in vector form as follows.

with fH2=0.0, fo==0.267 and fN2=0.733) and hy- drogen (U =1250 m/see, T =254 K with fH_=l.O, bU b_ i fo_=0.0 and fY2=O.O). The two streams are super- _- + Oz-- 7 = H (45) sonic with the air stream Mach number of 2.07 and hydrogen stream Mach number of 1.03. The inlet mean where U is the vector of dependent variables, _i are velocity is assumed to have a hyperbolic tangent pro- flux vectors containing convective and diffusive terms file, thus imitating the flow that exists downstream of (repeated indices indicate summation), and H is the the splitter plate trailing edge. A constant turbulence source vector containing production/dissipation terms.

intensity level is used in the free stream for arriving The temporally discrete form of equation (45) is at the initial distribution of turbulent kinetic energy and the specific dissipation rate. The pressures are u "+1 = u" - _xt [0+;' _ H.+I ] (46) 0z i matched between the two streams (P =1 atm.). A 13- step, 8-species H2 - Air reaction model (Table 1) has where n is the old time level and n + 1 is the new time been used for the finite-rate chemistry system consid- level.

ered here. A 101 X 81 grid (101 points in flow direc- The source terms in the k- and w-equations are de- tion, 81 points in the transverse direction) was used coupled by suitable manipulation of the w term in the for the calculations. The length of the flow domain present analysis in order to alleviate the computational is 0.25 m and its width is 0.05 m. In all the figures shown in this report, y refers to the lateral distance stiffness introduced by these terms. For example, in the k-equation, the dissipation term is written as, measured from the outer edge of the lower stream. The title g refers to calculations that include the interac- tion model.

C_.fiwk = C2fik _" (47)

Asa first step,the difference made by thetemper-

as other similar computations (not shown) carried out

aturefluctuations model(28) is explored.Figure2

by the author, it is not possible to come up with a def-

shows the factor(1+ m) as a function of y for repre-

inite answer to this question. It is also not established sentative reaction steps at the exit plane of the flow whether the type of flow configuration (mixing layer, domain. The need for including the temperature fluc- jet, boundary layer) has any significance in this dis- tuations is evident in the figure. The forward reac- cussion. So far, all computations using the proposed tion rate changes by as much as 400 percent (reaction model have been done for reacting mixing layer (2- step 1) due to the effect of this model. The species D and coaxial jets) involving the mixing and reaction production rate is a strong function of the reaction between fuel and oxidizer streams. It is not certain rate (32) and hence will be affected by the inclusion whether a premixed reacting flow configuration will of this model.

show a amore pronounced effect of these interactions than the non-premixed cases studied so far. Also the The effect of the turbulence-chemistry interactions effect of the interactions near a solid wall is an area model on the species production rate is shown in the worth exploring.

next few figures. Figure 3 shows the production rates of the major species involved (H2, O,., H20, OH) An interesting aspect of the analysis is that the par- as functions of y at the exit plane. Computational ticular turbulence model chosen (k - e as opposed to results obtained with (k-w-g) and without (k-w) the k -w, for example) seems to have an effect on the effect of these interactions are shown in the figure. It overall predictions [1]. Since the g-equations, which is seen that the net change in the production rate in provide the variances needed to construct the interac- the shear layer/reaction zone is reduced by the effect tions model, are strongly coupled (via their produc- of these interactions for the species H2, 02 and H20.

tion terms) with the turbulence model equations it is For the species OH, the effect of these interactions also essential to differentiate between the effects of the seems to be to reverse the rate of production from the turbulence model and the turbulence-chemistry inter- case where the interactions are not included. While action model. This brings up the importance of the it is not conclusive whether the turbulence-chemistry accuracy of the turbulence model which was one of interactions increase or decrease the production rates the main reasons for carrying out this task using the of individual species, it is important to note that these k-w turbulence model. It is crucial to keep the short- interactions do have a significant effect on progress of comings of the model in perspective while attempting the chemical reactions.

to validate the results. The assumptions made in ar- riving at the model, such as the decoupling of temper- Figures 4 show the distribution of the streamwise ature and concentration effects, are as important as velocity and static temperature as a function of y at the model itself in some cases.

the exit plane. The effect of the interactions seems to be felt more at the edges of the mixing layer/reaction On the positive side, this effort represents a signifi- zone than anywhere else. In the Going back to fig- cant push in the right direction in this very important ure 3, the production rate seems to be sensitive to the area of turbulent reacting flows. The model is simple temperature gradients rather than the temperature it- to use and is easily adaptable to other types of tur- self. Turbulence affects the distribution of tempera- bulence closures such as the Reynolds stress models.

ture both directly (thermodynamic energy equation) More improvements can certainly be done in terms of and indirectly (via the shear layer effect). This effect using more realistic models for the temperature and is transmitted through the interactions model and is concentration fluctuations such as a pdf-based model seen in the chemical production rate profiles. Figure 5 for temperature effects (as opposed to the moment shows the effect of the interactions on the species dis- model employed here), a joint-pdf model to represent tributions. The effect seems to be more pronounced in the effects of both the temperature and the concen- the case of the products (H20, OH) than the primary tration fields etc. However, in order for the model to reactants (Hu, On). The changes are larger in regions be useful for practical applications, it must be kept in of higher gradients (on the fuel stream side for H2 and a form which promotes ease of use as well as compu- air stream side for 02, for example).

tational economy. This is the main reason for resort- ing to a two-equation level turbulence model in the One of the unknown yet crucial factors associated present work. A concerted validation effort is crucial with the present work is the lack of prior knowledge to ensure the success of the modelling efforts. One as to the effect of turbulence on chemical reactions debilitating factor in this regard is the nonexistence and the reverse effect of the chemical reactons (heat of useful experimental data. It is extremely difficult, release, concentration change) on the turbulent field.

if not almost impossible, to obtain such experimen- Which of these effects is dominant is a debatable is- tal data using present day equipment. An alternative sue. Based on the results shown in this report as well seems to be direct numerical simulations (DNS) which

hasshown promising signsin relatively simpler flow

[6]

Wilcox, D.C., "A Two-Equation Turbulence configurations. Inspite of the major advances made Model for Wall-Bounded and Free-Shear Flows", in the area of DNS in recent years, it still is a devel- AIAA Paper 93-2905, 1993.

oping field and is far from providing useful data for

[7]

Mentor, F. R., "Zonal Two Equation k-_,. Tur- validation purposes in cases such as the present work.

bulence Models for Aerodynamic Flows", AIAA CONCLUSIONS Paper 93-2906, 1993.

[8]

Sarkar, S., Erlebacher, G., Hussaini, M. Y., and A turbulence-chemistry interaction model has been Kreiss, FI. O., "The Analysis and Modeling of proposed in conjunction with a recent version of the Dilatational Terms in Compressible Turbulence", two-equation k- w model of turbulence for use in NASA CR 181959, 1989.

chemically reacting flows. Preliminary computations carried out with the model for the case of a two-

[9] Jones, W.P. and Launder, B.E., "The Prediction

dimesional, high speed, reacting mixing layer indicate of Laminarization with a Two-Equation Model of that these interactions have a significant effect on the Turbulence", Int. J. Heat Mass Transfer, Vol.15, flow predictions. The species production rates of in- 1972, pp 301-314.

dividual species seem to be affected by these interac- tions. It is not possible, yet, to quantify the effects

[lO]

Zeman, O., "Compressible Turbulence Subjected of these interactions based on available data. More to Shear and Rapid Compression", Eighth Sym- detailed analysis of the various aspects of the model posium on Turbulent Shear Flows, Munich, Ger- developmental process need to be done.

many, 1991.

ACKNOWLEDGMENTS

[11] Narayan, J. R., "A Two-Equation Turbulence

Model for Compressible Reacting Flows." AIAA- This work was supported by the Applied Computa- 91-0755, 1991.

tional Fluids Branch of the Fluid Dynamics Division at NASA Ames Research Center under Cooperative [12] Wilcox, D.C., "Progress in Hypersonic Turbu- agreement number NCC 2-715. The author would like lence Modeling", AIAA-91-1785, 1991.

to acknowledge the valuable role played by Dr.S.S. Gir- Carpenter, M. H., "Three-Dimensional Computa- imaji of ICASE, NASA Langley Research Center in the [13] tions of Cross-Flow Injection and Combustion in development of the model.

a Supersonic Flow", AIAA-89-1870, 1989.

[14] Narayan, J. R., Sekar, B., "Computation of Tur-

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ogy, Vol.78, 1991, pp 177-196.

[16] Drummond, J. P.,"A Two-Dimensional Numeri- [4] Pope, S. B., "Computations of Turbulent Com- cal Simulation of a Supersonic, Chemically Re- bustion: Progress and Challenges", Proc. 23rd acting ML'dng Layer", NASA TM 4055, 1988.

Symposium (Int.) on Combustion, The combus- [17] "Free Turbulent Shear Flows", NASA SP-321, tion Institute, Pittsburgh, PA, 1990, pp 591-612.

Vol.1, 1972.

[5] Frankel, S.H., Drummond, J.P. and Hassan, H.A., "A Hybrid Reynolds Averaged/pall Clo- [18] Brown, G. L. and Roshko, A., "On Density Effects sure Model for Supersonic "I_urbulent Combus- and Large Structure in Turbulent Mixing Layers", tion", AIAA-90-1573, 1990.

]. Fluid Mech., vol. 64, pt. 4, 1974, pp. 775-816.

[19]Papamoschou, D. andRoshko, A., "The Com-

Table 1. H2 - Air Reaction System

pressible TurbulentShearLayer: An Experi-

No. Reaction mentalStudy", J.Fluid Mechanics, v.197, 1988, pp 453-477.

1 H+O2=O+OH 2 OH + H2 _--- H20 + H [20] "Seventh Symposium on Turbulent Shear Flows", 3 O+H2=OH+H Vol.1 and 2, Stanford University, Stanford, Cali- 4 OH+ OH --- H_O + 0 fornia, 1989.

5 H+OH+M=H20+M [21] Oldenborg, 1%.et al., "Hypersonic Combustion Ki- 6 H+H+M_H2+M netics - Status Report of the Rate Constant Com- 7 O+O+M,=O2+M mittee, NASP High-Speed Propulsion Technology 8 H+O+M=OH+M Team", NASA TM 1107, 1990.

9 H+O2+M=HO2+M I0 OH + H02 = H_O + 02 [22] Williams, F. A., Combustion Theory. Addison- II H + H O_ = H_ + 02 Wesley Publishing Company, Inc., Reading, MA, 12 H + H02 = OH +OH pp. 358-429, 1965.

13 0 + H02 = OH + O_ 14 H02 + H02 _- H202 + 02 [23] Favre, A., "Statistical Equations of Turbulent 15 H + H_O_ = H2 + H02 Gases", Ins_itut de Mechanique Siatis_ique de la 16 OH + H_02 = H20 + H02 Turbulence, Marseille.

17 H + H20_ = H20 + OH 18 0 + H_02 = H02 + OH 19 OH + OH + M _- H202 + M 20 OH + OH _ H__ + 02 Species : H2, 02, H20, OH, H, O, HOz, H202 and N_(inert) M is a third body (all species included) U Air_-....- H 2 --- Fig. 1 Mixing Layer. schematic Ln -.,4 o

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APPENDIX- B

AIAA-94-Z313

Computation Of High Speed Turbulent

Reacting Flows Relevent To Scramjet

Combusters

Johnny Narayan and Gregory Molvik

MCAT Institute

Moffett Field, California

and

G.Wadawadigi

University of Texas at Arlington

Arlington, Texas

COMPUTATION OF HIGH-SPEED TURBULENT REACTING FLOWS RELEVANT TO SCRAM JET COMBUSTORS J.R. Narayan" MCAT Institute, Moffett Field, CA.

G. Wadawadigi?

U. of Texas at Arlington, Arlington , Texas.

and G. Molvik t MCAT Institute, Moffett Field, CA.

ABSTRACT j_h coordinate zj transverse coordinate Y Kronecker delta Computations are done on flow configurations that 61i ff resemble the reaction zone in the scramjet combus- turbulence energy dissipation rate tor flows. Compressible, reacting, turbulent flow so- % compressible dissipation rate rl compressibility correction coefficient lutions are obtained. A two equation (k-e) model with compressibility correction is used to calculate the 7 ratio of specific heats flow field. A finite rate (8-species, 13-reaction steps) specific dissipation rate chemistry model for hydrogen-air combustion has been laminar viscosity used. Computations are carried out using the Navier- turbulent viscosity V Stokes solver TUFF. Predictions are compared with kinematic viscosity available experimental data and also those obtained density P turbulence model constants by using the code UPS. O"k , 0"_ stress tensor rii NOMENCLATURE flux vector in j,h direction Subscripts A,b t coefficients in Arrhenius rate equation turbulent quantity turbulence model constants C1,C_.,C.

E total internal energy f. mass fraction of species n INTRODUCTION H total enthalpy h static enthalpy Hypersonic travel requires propulsion systems which k are different from the conventional ones used in most turbulent kinetic energy ]¢f ,kb forward and backward reaction rate constants of the modern aircraft. The supersonic combustion L number of reaction steps ramjet (scramjet) is a system considered to be suit- Mn Molecular weight of species n able for high speed applications. There has been a Turbulent Mach number M, tremendous amount of activity in the area of scramjet N number of chemical species research in recent years ( [1]- [11]). Some of the re- laminar and turbulent Prandtl numbers Pr,Pr_ lated topics include inlet configuration, mbdng layers, P pressure mixing enhancement, combustor configuration, finite Sc,Sct laminar and turbulent Schmidt numbers rate chemistry models and chemical kinetics. The fuel T temperature used in the scramjet varies depending upon the ap- T= Activation Temperature plication. For example, hypersonic waveriders using t time hydrocarbon fuels have been designed [12] for applica- velocity vector tions in the moderate hypersonic speed regimes. For Mach numbers of the order of 15 and above, hydro- production rate of species n streamwise coordinate gen is generally considered to be the fuel of choice.

In the present work, hydrogen is the fuel used in the * Senior Research Scientist, Senior Member AIAA.

computations.

t PostDoctorla Fellow, Member AIAA.

*' Senior Research Scientist, Member AIAA.

The present work represents a computational el-

fort in establishing a solution procedure for _hypersonic

upon factors such as the mixing between fuel and ox- propulsion applications. The entire task of establish- idizer streams, presence of shocks in the flow field, ing the solutions procedure must then be divided into boundary layer effects, flow separation, extent of chem- smaller tasks dealing with subsets such as turbulence ical reaction within the combustor and so on. The nu- modelling, chemical kinetics, geometry etc. One such merical solution procedure should have the capability task is the topic for the present study. Here, the rele- of addressing all of these factors while maintaining the vant flow features of the combustor, namely the mixing required accuracy and robustness. There is a glut of and chemical reaction between the fuel and oxidizer useful numerical solvers applicable for a wide variety streams, is addressed. The flow field in a scramjet is of flows including all speed regimes. Computational complex. It is turbulent and compressible involving algorithms which are fast and accurate are being im- high heat release. The solution procedure should ad- proved everyday.

dress all aspects of the flow field adequately. It should Even though there are a wide variety of sophisti- be capable of accurately modelling the turbulent field, cated and physically accurate thermodynamic, chem- taking into account the effects of compressibility, and ical kinetic and turbulence models available it is not addressing the changes associated with heat release.

always possible to use the most accurate and elabo- Also, the interactions between the distinct physical rate versions in a numerical simulation due to the lim- aspects of the flow such as the effect of heat release itations imposed by computer memory requirements, on turbulence, the interaction between turbulence and computational economy, ease of use and adaptability chemistry etc. must be properly addressed. Signifi- to practical problems. Solutions often are required, cant progress has been made in addressing these areas especially in ihe engineering industry which is the end via accurate and realistic modelling in recent years [6].

user for such solvers, in a short time using comput- Remarkable advances have been made in the area ers that may not be the fastest available. As a result, of turbulence modelling, accounting for a variety of compromises must be struck between physical accu- factors that affect the flow field. Compressibility cor- racy and computational feasibility and it is this aspect rection models to account for the effects of compress- which differentiates between various solvers that exist ibility, near-wall turbulence models to deal with the today.

transition from fully turbulent to zero turbulence, vis- In the present study, an attempt is made to es- cous dominated flow field near no-slip boundaries and tablish a solution procedure for scramjet combustor modifications to models to account for flow curvature flow predictions from the perspective of the discus- are some examples. A wide variety of turbulence mod- sion above. The models chosen to represent the tur- els, including algebraic (zero-equation), one-equation, bulent and chemistry fields reflect the compromise be- two-equation, Reynolds stress and large eddy simula- tween physical accuracy and computational economy tion models, are available (for example references [13] - mentioned above. The code chosen for the computa- [15]) depending upon the sophistication and accuracy tions is the TUFF [17] code and the solutions are com- desired and the limits imposed by numerical solution pared with those obtained with the UPS [18, 20] code.

procedures.

The turbulence model chosen is the two-equation k - e Thermodynamic and chemical kinetic models [16] turbulence model with low Reynolds number modifi- applicable to the scramjet flows have been undergoing cations [13]. However, the Baldwin-Lomax algebraic continuous improvements in recent years. Accurate model is also available as an option. The compress- modelling of thermodynamic variables as functions of ibility effects are included via the compressibility cor- temperature which are valid over a wide range of tem- rection model proposed by Zeman [19]. The fuel used peratures is an example. In flows such as the one is hydrogen although the numerical solver can easily associated with the scramjet, the time scales associ- be modified for hydrocarbon fuels. A 9-species, 20- ated with fluid dynamics and chemical reaction (not to reaction steps chemical kinetics model for hydrogen- mention the turbulence scales) require that the com- air combustion [16] is available. For the computations bustion process be modelled via a finite rate chem- presented in this report, an abbreviated version (8- istry mechanism. Such a mechanism should account species, 13-steps) of this model has been used.

not only for the major species (reactants and prod- Mixing plays a major role in high speed combus- ucts) involved in the chemical reactions but also the tor flows. The reaction zone is mainly confined to intermediate transient ones which play a vital role in mixing layers that exist between fuel and oxidizer the reaction progress process. Accurate models for the streams. Two flow configurations are chosen for the chemical reactions in the scramjet combustor, thus, is study. The first is the well known Burrows-Kurkov a crucial aspect of the solution procedure.

experiment [21] in which hydrogen and vitiated air The design of the combustor is strongly dependent streams (two-dimensional) mL_ and react. The second and

ease is that of anaxisymmetric configuration [22,23]

where twocoaxial jets (fuelandoxidizer) mixandre-

0U.'l 0t_zn

act. Experimental datafrom compressible, reacting

mixinglayers is still scarce whichhinders the valida-

= (8)

tion of the calculation procedure. Theavailable data

fromthe above twoexperiments areused to compare

Boussinesq approximation is used to obtain closure

with the predictions.The governing andsecondary

of the averaged equations. Here the Reynolds stress

equations usedin thecomputations haveallbeen de-

tensor is written as,

scribed in detailin thereferences citedabove. Onlyan

abreviated equation setwill begiven in thepresent pa-

. /"_"'_ 2

per. The computations wereperformed onthesuper-

pui u s = -_ Sis - "_"#k6#

computers of NASandNASAAmes Research Center

(C-90).

OU_ OUs 20U_ ,hi (9) O:ci + Ozi 30zk GOVERNING EQUATIONS turbulent/eddy viscosity defined as The equations used for computations are described k 2 in detail in references [6, 7, 10] and [24]. Only the

#, = G#-- (1o)

forms of the modelled equations used in the present study are given here. Density-weighted averaging is with Cu=0.09.

used to derive the mean flow equations from the in- The modelled momentum equation, then, is, stantaneous coservation equations. The dependent variables, with the exception of density and pressure, are written as O-_U_ _r_u_uj _Y_ 20_k o--T + O_ - az_ + 3 az---T6_s ¢ = ¢ + ¢" (I) o

- a_:7 [ (# + _,) s,] (11)

where the ¢" is the fluctuating component of the vari- • able under consideration and its Favre-mean ¢ is de- fined as The effects of compressibility are included via the model proposed by Zemaa [19]. Here, the compress-

_-_

ible dissipation terms are expressed as functions of the

(2)

turbulent kinetic energy dissipation rate and the local In this equation, the overbar indicates conventional turbulent Mach number. The compressibility effects time-averaging. Density and pressure are split in the are represented by a component of the dissipation rate conventional sense as, (¢e) given as p = _ + p' and p = _ + p' (3) ¢, = Kce The averaged continuity and momentum equations are K_ = ,TF(M,) 2k

M, = a_ (12)

_- + 0_ - 0 (4)

where a is the local speed of sound and F(Mt) is a function of the local turbulent Mach number (Mr).

o_u, _u, uj

F(Mt) is given by T + oz i -

o_, a_s + _ (5)

where F(M,) = 1_ezp[_(Mt.o),],--M, M, > Mto

_,_ = _¢_-- +

= O, M_<M_o -_x_ )bus" 32 cgxkcgu_ 6ii (6) ozj with M,0=0.1 and 7/=0.75. The modelled turbulent with repeated indices indicating summation.

kinetic energy equation is [6, 13] In the two-equation turbulence model, the two tur- bulence variables are the turbulent kinetic energy (k) a#_ op_us and the dissipation rate (e) [13] defined as -- P_ - fie(1 + K¢) a--i- + Ozs k - pu_'u_' (13)

2_ (7)

+ _. [(_ + G _._. ]#, ) a_ .

where where N N (14) ml=Z ' Z " ---- -- pu i uj I/M, nl -- !./._I The modelled e-equation used (no compressibility ' and " are the number of molecules of the where, u_t u_t corrections) in the present analysis [13] is given below.

scalar s involved in the/-th reaction step in the forward and backward directions, respectively. The forward and backward rate-constants of the reaction l are given by k/_ and k_t respectively.

0 fi_) 0¢.

+ + 7, 1 (is)

k/_ = A_T _' ezp[ T_t.

---f-] (21) where P_ is the production term in the turbulent where A_, bt and Ta_ are numerical constants specific kinetic energy equation. The model constants used to the given reaction step I. k_ is determined from the in the analysis are C1-1.44, 6"2=1.92, _r,=l.0, equilibrium constant for the/-th reaction step and kit.

_=1.3, Pr=0.72, Pr,=l.0, Sc=0.22and Set=l.0.

The mass-averaged total energy can be written in Solution of the modeled equations terms of the total enthalpy as The equations are discretized and integrated in

P (16)

space and time to obtain steady state solutions using the finite-volume based numerical solver TUFF [17].

The correlations between the fluctuating velocity and The TUFF code contains many desirable features for the scalar fluctuations are modelled using a gradient- the computation of three-dimensional, hypersonic flow diffusion hypothesis..4 typical model is of the form fields. It has non-equilibrium, equilibrium and perfect gas capabilities along with an incompressible option.

- = A o8

It employs a finite-volume philosophy to ensure that 6r, (_zi) (17) the schemes are fully conservative. The upwind in- where _r_sis a coefficient which, normally, is a constant.

viscid fluxes are obtained by employing a new tem- For ¢ = fa (n represents the species) , _r_ = Sc,, and poral lZiemann solver that fully accounts for the gas for the static enthalpy, (_ = h), o'¢ = Prt. Using the model used. This property allows the flow field dis- continuities such as shocks and contact surfaces to be above definition, and omitting the body force contri- bution, the time-averaged and modelled energy equa- captured by the numerical scheme without smearing.

tion [6] is Total Variation Diminishing (TVD) techniques are in- cluded to allow extension of the schemes to higher or- ders of accuracy without introducing spurious oscilla- tions. The schemes employ a strong coupling between the fluid dynamic and species conservation equations o [ f, f,, ) o'_ f,,) ok 18 and are made fully implicit to eliminate the step-size restriction of explicit schemes. This is necessary since where _rk comes from the turbulent kinetic energy step-sizes in a viscous, chemically reacting calculation equation. The modeled species continuity equation is can be excessively small for an explicit scheme, and the resulting computer times prohibitively large. A fully conservative zonal scheme has been implemented

__ 0

+ - TM (19)

to allow solutions of very complex problems. The schemes are made implicit by fully linearizing all of The modelled form of the mean species production the fluxes and source terms and by employing a mod- rate due to chemical reaction (w"_n)is given, for a finite- ified Newton iteration to eliminate any linearization rate system involving L reaction steps and N species, and approximate factorization errors that might oc- in the following general form: cur. Approximate factorization is then employed to L avoid solving many enormous banded matrices. As UJr_ mentioned before, the options for turbulence models Z( It i M. v._ - v,.) x (20) I----1 include both zero and two equation models (both k - e and k - _). For more details about the solution pro- N , N cedure the reader is directed to the reference cited s=l s ,_----i above [1]].

RESULTS AND DISCUSSION (z/D=43.1 D) is good given the above mismatch be- tween the two data at the inlet. The development of the reaction zone after ignition is not predicted Two reacting flow configurations have been chosen well. The experimental data indicates that the reac- for the present study. As mentioned before, the Navier tion zone (depicted by the water mole fraction distri- Stokes solver TUFF has been used for the computa- bution) spreads more quickly than the predictions in- tions. The first one is the case of coaxial jets [22, 23] dicate. The predictions show the reaction zone to be where a hydrogen jet flows (inner jet) coaxially with off-center whereas the experimental data shows the re- an outer vitiated air (mass fractions: oxygen=0.246, cation zone to be closer to the axis of symmetry. How- water=0.209 and nitrogen=0.545) jet. A schematic of ever, there is very good qualitative agreement between the flow problem is given in Figure 1. The two streams the data with the peak values of the reaction prod- are, air (U=1380 m/sec, T=l180 K with p=107000 ucts predicted very well. The flow domain was seen N/m2) and hydrogen (U=1774 m/sec, T=545 K with to have a wave-like structure as shown by the pre- p=112000 N/m2)., The air stream is supersonic with a dicted profiles. The worst agreement seems to be for Mach number of 1.97 and the hydrogen stream Mach the case of oxygen. However, when the initial pro- number is 1.00. The inlet mean velocity is assumed files of oxygen are compared one finds that there too to have a step profile with the two jets having uni- is the worst agreement between computations and ex- form speeds at the specified values (no experimental periment which may be reason for the problem down- data available). The velocity in the lip region of the stream. Figure 3 shows the comparison of static tem- inner jet tube wall (finite wall thickness) is assumed perature data. The agreement between predictions to be zero. The inlet temperature profile is derived and experiment is good qualitatively displaying similar based on the experimental data given for a location trends. The uncertainty associated with the accuracy just downstream of it (shown later). The inlet species of the experimental data is unknown. There are con- mass fraction distributions are also chosen based on siderable differences between the data presented by the the experimental data provided at the same down- two references [22, 23], especially in the temperature stream location. A constant turbulence intensity level profiles. Overall, there is good qualitative agreement is used in the free stream for arriving at the initial between the predictions and experiment.

distribution of turbulent kinetic energy and the dis- The second test case considered is the Burrows- sipation rate. A 13-step, 8-species H2- Air reaction model (Table 1) has been used for the finite-rate chem- Kurkov experiment [21]. The flow configuration is two- dimensional. A schematic diagram of the configuration istry system considered here. A 81 X 91 grid (81 points is given in figure 4. No-slip walls bound both the upper in flow direction, 91 points in the radial direction) was and lower regions (y=0 and y=yma_). The lower wall used for the calculations. The inner jet/tube diameter is inclined to form an expansion surface. Hydrogen is (D=0.00236 m) is used as a reference length. The to- injected along this surface into a vitiated air stream.

tal length of the flow domain is equal to 43.1 D. The The two streams mix and react downstream of the outer boundary (radial) of the flow domain is taken to be at y=17 D. A more detailed description of the flow injection location (inlet). The hydrogen stream is in- parameters is given in Table 2. The region outside the jected at a velocity of 1216 m/sec and a temperature limits of the air jet is assumed to be still air at a tem- of 254 K. The airstream comes in at a speed of 1764 m/sec and a temperature of 1270 K. Full details about perature of 273 K. The two-equation (k-e) turbulence model is used along with the finite rate H2-Air chem- the flow parameters and geometry are given in Table 3.

In this case, the reference length used in the hydrogen istry model mentioned above. In all the figures shown in this report, y refers to the radial distance measured jet width at inlet, h (h=0.004 m). The models for tur- bulence and chemistry are identical to the ones used • from the axis of the coaxial system of jets.

for the coaxial jet case. The grid size is 81 X 121 Figures 2 - 3 show the results of the computations.

(81 grid points in the axial (z) direction and 121 grid Figure 2 shows the computed and experimental distri- points in the transverse direction). The total length butions of species mole fractions. The figure is de- of the solution domain is 0.356 m (z/h=89). Avail- signed in a two-column format. The left side col- able inlet data have been used for the first-plane pro- umn represents the inlet (first x-location) data and files which improved the predictions remarkably over the right side column is the data at the exit plane the solutions obtained with uniform profiles. The so- (z/D=43.1 D). As seen in these figures, the inlet lutions are compared with the available experimental data agreement between the computations and exper- data at this location (exit) in figures 5-7. The solu- iment is not perfect, especially around the jet edges, tions carried out with the space marching PNS code and this might affect the computed distributions at UPS [20] using the Baldwin-Lomax turbulence model downstream locations. The comparison between pre- are also given for comparison.

dictions and experiment at the downstream location

Figure5 shows the compariosn between the pre-

This work was supported by the Applied Computa-

dicteddistributions of thespecies molefractions and

tional Fluids Branch of the Fluid Dynamics Division at

the corresponding experimental data. As seenin

NASA Ames Research Center under the Cooperative

these figures, thereis excellent agreement between the

agreement number NCC 2-715.

TUFFpredictions andexperiment. Thepredictions by

theUPScode donot agree verywellbut still thereis

verygood qualitative agreement withtheexperimental References

data. Figure 6 compares the predicted profiles of exit

planetotal temperature andMuchnumber with the [1]

Marvin, J. G., "A CFD Validation Roadmap For

experimental data. Thereis goodqualitative agree-

Hypersonic Flows", NASA Technical Memoran-

mentin thecase oftemperature andverygood overall

dum 103935, 1992.

agreement in the case of the Muchnumber distribu-

tion. Figure7 shows thecomparison between thepre- [2]

Ebrahimi, H. B., "CFD Validation For Scramjet dictions andexperiment of the lowerwall (hydrogen Combustor and Nozzle Flows, Part I", AIAA-93- 1840, 1993.

jet side)pressure. Ignitioncauses thepressure risein

the profile. Ignitionseems to bedelayed in the case

Vitt, P. H., Riggins, D. W. and McClinton, C.

ofthepredictions accompanied by a more pronounced [3]

R., "The Validation and Application of Numeri- pressure rise.

cal Modelling to Supersonic Mixing and Reacting

Highspeed reacting flows such asthetwocases stud-

Flows", AIAA-92-0626, 1992.

iedherearecomplex inspiteof their simplegeome-

tries. The interactions between the different aspects [4] Pdggins, D. W. and McClinton, C. R., "A Com-

of theflowsuchasturbulence, chemical kinetics, heat

putational Investigation of Mixing and React- release etc. areverydifficultto understand and,to ing Flows in Supersonic Combustors", AIAA-92- a largeextent,impossible to modelaccurately. Mod- 0626, 1992.

erndayexperimental facilities still cannot make com-

Eklund, D. R. and Northam, G. B., "A Numeri-

pletemeasurements in suchflows.Onlymeanvalues [5]

cal Study of the Effects of Geometry on the Per-

of temperature, velocity, pressure, species concentra-

formance of a Supersonic Combustor." AIAA-92-

tionsetc. areavailable, if any. Eventhen,the un-

0624, 1992.

certainties associated with the dataforceoneto ac-

ceptthemonlywith certain reservations. Given that, [6]

Narayan, J. R. and Girimaji, S. S., "Tur- there is almost never a chance for perfect agreement bulent Reacting Flow Computations Including between predictions and experiment in all the areas.

Turbulence-Chemistry Interactions." AIAA-92- While the advances made in measurement techniques 0342, 1992.

improve every day, the fruits of these advancements (ie. accurate measurements) are not realized imme- [7] Narayan, J. R., "A Two-Equation Turbulence diately. As a result, today's computations will have Model for Compressible Reacting Flows." AIAA- 91-0755, 1991.

only old data for validation (in the present case, the best data is already four years old) purposes which is Eklund, D. R., "Calculation of Supersonic Tur- certainly the case here. Unless more accurate experi- [8] bulent Reacting Coaxial Jets." AIAA Journal, mental data with minimum uncertainties are available, Vol.28, No.9, 1990, pp 1633-1641.

the best a computational effort can hope for, in terms of validation, is probably what is seen here. [9] Carpenter, M. H., "Three-Dimensional Computa- tions of Cross-Flow Injection and Combustion in CONCLUSIONS a Supersonic Flow", AIAA-89-1870, 1989.

Computation of the flow fields of two high-speed, [10] Drummond, J. P., Carpenter, M. I=I. and Riggins, turbulent, reacting flow configurations involving finite- D. W., "MLxing and MLxing Enhancement in Su- rate chemical kinetics for hydrogen-air combustion personic Reacting Flows", High Speed Propulsion have been carried out. A two-equation (k - e) turbu- Systems: Contributions to Thermodynamic Anal- lence model with compressibility corrections has been ysis, ed. E. T. Curran and S. N. B. Murthy, Amer- used. The predictions are compared with available ican Institute of Astronautics and Aeronautics, experimental data. Good qualitative agreement is Washington, D. C., 1990.

present between computations and experiment. More detailed experimental data is necessary. [11] Drummond, J. P.,"A Two-Dimensional Numeri- cal Simulation of a Supersonic, Chemically Re- ACKNOWLEDGMENTS acting Mixing Layer", NASA TM 4055, 1988.

[12]Molvik, G. A. Bowles, J.V.andHuynh,L. C., Validation." Proc. of the 25th JANNAF Combus-

"A Hypersonic Research Vehicle with Hydrocar-

tion Meeting, Huntsville, Alabama, 1988.

bonScramjet Propulsion: Design andAnalysis", [24] Williams, F. A., Combustion Theory. Addison- AIAA Paper 93-5097, 1993.

Wesley Publishing Company, Inc., Reading, MA,

[13]Jones, W.P.andLaunder, B.E.,"ThePrediction

pp. 358-429, 1965.

of Laminarization with a Two-Equation Model of

Turbulence", Int. J. Heat Mass Transfer, Vol.15, 1972, pp 301-314.

Table 1. H2 -- Air Reaction System [14] Launder, B.E., Reece, G.J.

No. Reaction and Rodi, W., "Progress in the Development of a Reynolds Stress Turbulence Closure", J. Fluid 1 H+O2,-_-O+OH Mech., Vol.68, 1975, pp 537-566.

2 OH + H2 = H20 + H [15] Wilcox, D.C., "A Two-Equation Turbulence 3 O+H_=OH+H Model for Wall-Bounded and Free-Shear Flows", 4 OH+ OH = H20 + 0 AIAA Paper 93-2905, 1993.

5 H+OH+Mr"---H20+M 6 H+H+M=_-H2+M [16] Oldenborg, R. et al., "Hypersonic Combustion Ki- 7 O+O+M,_--O2+M netics - Status Report of the Rate Constant Com- 8 H+O+M._-OH+M mittee, NASP High-Speed Propulsion Technology 9 H+O2+Mr'---HO2+M Team", NASA TM 1107, 1990.

10 OH + HO_ _ H20 + 02 [17] Molvik, G. A. and Merkle, C. L., "A Set 11 H + HO_ _-- H2 + 02 of Strongly Coupled Upwind Algorithms for 12 H + H02 _ OH + OH Computing Flows in Chemical Nonequilibrium", 13 0 + H02 _ OH + 02 AIAA Paper 89-0199, 1989.

14 H 02 + H 02 = H202 + 02 15 H + H202 = H2 + H02 [18] Lawrence, S. L., Chaussee, D. S. and Tannehill, 16 OH + H202 _ H20 + H02 J. C., "Application of an Upwind Algorithm 17 H + H20_ = H20 + OH to the Three-Dimensional Parabolized Navier- 18 0 + H202 _- H02 + OH Stokes Equations", AIAA 87-1112, 1987.

19 OH + OH + M ,_- H202 + M 20 OH + OH = H2 + 02 [19] Zeman, O., "Compressible Turbulence Subjected to Shear and Rapid Compression", Eighth Sym- Species : H2, 02, H20, OH, H, O, HO_., H202 and posium on Turbulent Shear Flows, Munich, Ger- N2(inert) many, 1991.

M is a third body (all species included) [20] Wadawadigi, G. Tannehill, J. C., Buelow, P. E.

and Lawrence, S. L., "A Three-Dimensional Up- wind PNS Code for Chemically Reacting Scram- jet Flowfields", AIAA 92-2898, 1992.

[21] Burrows, M. C. and Kurkov, A. P., "Analytical Table 2. Conditions for coaxial jet experiment and Experimental Study of Supersonic Combus- tion of Hydrogen in a Vitiated Airstream." NASA H2 Air TM X-2828, 1973.

Mach No. 1.0 1.97 Temperature 545 K 1180 K [22] Cheng, T. S., Wehrmeyer, J. A., Pitz, R. W., Jar- Pressure 0.112 MPa 0.107 MPa ret, O. and Northam, G. B., "UV Raman Scatter- Velocity 1774 m/s 1380 m/s ing Measurements in a Mach 2 H2-Air Flame for fH2 1.0 0.0 Assessment of CFD Models." Proc. of the Cen- fo2 0.0 0.246 tral States Meeting of the Combustion Institute,

fN, 0.0 0.545

Nashville, TN, 1991.

fH_o 0.0 0.209 [23] Jarret, O. Jr., Cutler, A. D., Antcliff, R. R., Chit- somboon, T., Daneey, C. L. and Wang, J. A., Fuel injector diameter=0.00236 m "Measurements of Temperature, Density, and Ve- Lip thickness=0.000725 m locity in Supersonic Reacting Flow for CFD Code Nozzle diameter(air flow)=0.01778 m

Table3. Conditions forBurrows-Kurkov

Experiment

H2 Air Mach No. 1.0 2.44 Temperature 254 K 1270 K Pressure 0.1 MPa 0.1 MPa Velocity 1216 m/s 1764 m/s fI_ 1.0 0.0 fo, 0.0 0.258 fN, 0.0 0.486 fH_o 0.0 0.256 Fuel injector height=0.004 m Duct height at inlet=0.0938 m Duct height at exit=0.1048 m

I I---" H2

Uai r --- 1380 m/s Tai r = 1180 K Pair = 107000 N/m 2 UH2 = 1774 m/s TH2 = 545 K PH2= 112000 N/m 2 Fig. 1 Coaxial jet case : schematic I.

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APPENDIX- C

j_ / J / r

AIAA-94-2948

Prediction Of Turbulent Reacting Flows

Related To Hypersonic Airbreathing

Propulsion Systems

Johnny Narayan

MCAT Institute

Moffett Field

California

PREDICTION OF TURBULENT REACTING FLOWS RELATED TO HYPERSONIC AIRBREATHING PROPULSION SYSTEMS J.R. Narayan ° MCAT Institute, Moffett Field, CA.

ABSTRACT 0

velocity vector production rate of species n streamwise coordinate Computation of scramjet combustor flows is the z jth coordinate main thrust of this work. Computations are done on xi transverse coordinate flow configurations that resemble the reaction zone in Y turbulence model coefficients the scramjet combustors. Compressible, reacting, tur- _,_" turbulence model coefficients bulent flow solutions are obtained. A two equation o_0,o_ turbulence model constants (k-e) model with compressibility correction is used to cd, _,, Kronecker delta calculate the flow field. A finite rate (9-species, 20- 61i turbulence energy dissipation rate reaction steps) chemistry model for hydrogen-air com- e compressible dissipation rate bustion has been used. Computations are carried out _c compressibility correction coefficient using the Navier-Stokes solver TUFF. Predictions are _?

ratio of specific heats compared with available experimental data and also 7 specific dissipation rate those obtained by using the codes UPS and SPARK.

laminar viscosity A turbulence-chemistry interaction model is included. # turbulent viscosity This model represents the first step in the effort to in- Pt kinematic viscosity clude these interactions in high-speed reacting flows, v density A two-dimensional reacting mixing layer has been used P for demonstration, o'k, o'_ turbulence model constants stress tensor r_j Subscripts NOMENCLATURE t turbulent quantity A,b coefficients in Arrhenius rate equation turbulence model constants Cl,C:,C.

INTRODUCTION Cs,C6 turbulence model constants E total internal energy Propulsion systems for hypersonic applications are f.

mass fraction of species n required to use the supersonic combustion ramjet g scalar variance (enthalpy, mass fraction) H (scramjet) due to the flight regime involved. Due to total enthalpy h the advent of the National Aerospace Plane (NASP) static enthalpy k proposed over a decade ago, there has been a consider- turbulent kinetic energy forward and backward reaction rate constants able amount of effort directed at developing a scram- kl ,kb L jet based propulsion system for a number of years.

number of reaction steps Even though interest in the NASP is waning at present M_ Molecular weight of species n due to various reasons, the research effort in the area Turbulent Mach number M, N of high speed propulsion systems such as the scram- number of chemical species jet still is going strong as evidenced by the enormous Pr,Pr_ laminar and turbulent Prandtl numbers number of reports published in the literature and na- P pressure turbulence model coefficients tional and international meetings. It is not within the Rc,ReT scope of the present report to list and describe all the R_ ,tL, turbulence model coefficients Sc,Sct work going on in the area of high speed propulsion.

laminar and turbulent Schmidt numbers T However, references [1] to [11] are some examples of temperature the vast amount of work in this area. The area of T_ Activation Temperature t high speed propulsion, especially for hypersonic appli- time cations, is a complex one encompassing a wide vari- *Senior Research Scientist, Senior Member AIAA.

ety of related topics such as inlet configuration, mix- ing layers,mixingenhancement, combustor configu- should have the capability of addressing all of these ration,finite rate chemistry models andchemical ki- factors while maintaining the required accuracy and netics.Both hydrocarbons andhydrogen areconsid- robustness. Computational algorithms which are fast

eredasfuelsdepending upontheapplication. Forex-

and accurate are being improved everyday and being ample, hypersonic waveriders using hydrocarbon fuels adapted in present day numerical solvers applicable for have been designed [1]forapplications in themoderate a wide variety of flows including all speed regimes. In- hypersonic speed regimes. ForMachnumbers of the spite of the existence of a wide variety of sophisticated orderof15andabove, hydrogen isgenerally considered and physically accurate models in many of the areas to bethefuelofchoice. Thepresent workrepresents a associated with high speed propulsion, it is not always computational effortin establishing a solution proce- possible to use the best models in a numerical sim- dureforhighspeed propulsion in general andscramjet ulation due to the limitations imposed by computer resources, computational economy and adaptability to flowsin paricular.

practical problems. Fast turnaround is a key factor

The flow field in a scramjetis complex. It is

in the success of any solver in practical problem solv-

turbulentand compressible involving high heatre-

ing situations. Compromises must be struck between

lease.The solutionprocedure shouldaddress all as-

physical accuracy and computational feasibility.

pectsof the flow field adequately. It shouldbeca-

pableofaccurately modelling theturbulent field,tak- The present work represents the effort made to es- ingintoaccount theeffects of compressibility, address tablish a solution procedure for high speed propulsion the changes associated with heat release and accuar- related flows. The models chosen to represent the turbulent and chemistry fields reflect the compromise rely describe the chemical kinetics mechanism which between physical accuracy and computational econ- is vital in high speed combustion. Also, the inter- actions between the distinct physical aspects of the omy mentioned above. The code chosen for the com- flow such as the effect of heat release on turbulence, putations are the TUFF [17] code and the SPARK the interaction between turbulence and chemistry etc.

code [6, 9]. The solutions are compared with those ob- must be properly addressed. Progress has been made tained with the UPS [18, 19] code and available exper- imental data. The turbulence model chosen is the two- in addressing these areas via accurate and realistic modelling in recent years [6]. However, a significant equation (k - e) turbulence model with low Reynolds amount of work remains to be done before any real- number modifications [12]. The compressibility effects istic claims for accurate models in these areas can be are included via the compressibility correction model made. There have been remarkable advances made proposed by Zeman [20]. The fuel used is hydrogen and in the area of turbulence modelling, accounting for a 9-species, 20-reaction steps chemical kinetics model a variety of factors that affect the flow field. Com- for hydrogen-air combustion [15] (Table 1) is used (in pressibility correction models to account for the ef- its abbreviated version).

fects of compressibility, near-wall turbulence models In the present task, major emphasis is on the turbu- to deal with the transition from fully turbulent to zero lence model(s) used for reacting, compressible flows.

turbulence, viscous dominated flow field near no-slip Specifically, the interaction between the turbulence boundaries and modifications to models to account for field and the chemical kinetics is of interest. The effect flow curvature are some examples. A wide variety of of turbulence on phenomena such as mixing between turbulence models, ranging from the algebraic (zero- two streams, boundary layer etc. has been studied ex- equation) models to Reynolds stress and large eddy tensively and it is an ongoing process. However, there simulation models are available (references [12]= [14] is not enough work done in the area of turbulence- are some examples) depending upon the sophistica- chemistry interactions in order to either emphasize or tion and accuracy desired and the limits imposed by neglect the importance of such interactions. The main numerical solution procedures.

reason for this is the fact that it is extremely difficult to Thermodynamic and chemical kinetic models [15, quantify these interactions experimentally. There have 16] applicable to the scramjet flows have been under- been efforts aimed at establishing pdf-based models for going continuous improvements in recent years. Ac- coupling the turbulence field with the thermochemi- curate modelling of thermodynamic variables as func- cai field in numerical simulations. References [6], [21] tions of temperature which are valid over a wide range and [22] are just a few of the many reports available in this area. There has also been a concerted effort to of temperatures is an example. The design of the com- bustor is strongly dependent upon factors such as the generate validation data bases through direct numeri- mixing between fuel and oxidizer streams, presence of cal simulations which is still in its early stages.

shocks in the flow field, boundary layer effects, flow In the present work, on the thermochemistry front, separation, extent of chemical reaction within the com- the effects of temperature and species the effects of bustor and so on. The numerical solution procedure temperature andspecies fluctuations aredecoupled.

a tilde represents Favre (density-weighted) averaging

Forthetemperature-turbulence interaction, amoment

and a simple overbar represents conventional time

modelis used[6]. The chemical species fluctations

averaging. The averaged continuity and momentum

areaccounted forusing anassumed multivariate-fl-pdf

equations are

model [6, 21,22].Thethermochemistry models used

hereareindependent of the modelfor the turbulent

velocityfield. Thesemodels requirethe solutionof 0-7 + 0z_ = 0 (1)

theevolution equations for allthe mean flowthermo-

chemical variables, turbulence variables (k andeither

w or e), enthalpy variance and turbulent scalar energy 0"_Ui _u_ ui _ Opui uj 0_-_ (sum of all the species variances). The CFD code used Ot + Ozi - Oz_ Oz_ + _ (2) is SPARK [9]. where Mixing plays a major role in high speed combustor flows. The reaction zone is mainly confined to mixing 2 0u_ &_ (3) layers that exist between fuel and oxidizer streams.

3 0z_ Efficient mixing of fuel and oxidizer is very impor- with repeated indices indicating summation.

tant for the design of combustor size. Mixing lay- ers ranging from subsonic to supersonic speeds have In the two-equation turbulence model, the two tur- been studied extensively over the years. In the present bulence variables are the turbulent kinetic energy (k) work, a two-dimensional and axisymmetric compress- and either the dissipation rate e [12] or the specific dis- ible, reacting, mixing layer (hydrogen-air) is computed sipation rate w [14]. The definitions of these variables with the aforementioned codes. Experimental data can be found in the cited references. Boussinesq ap- from compressible, reacting mixing layers is still scarce proximation is used to obtain closure of the averaged which hinders the validation of the calculation pro- equations. The Reynolds stress tensor is written as, cedure. Three flow configurations are chosen for the study. The first two are chosen to demonstrate the solution procedure without the turbulence-chemistry interaction model and the last case is used exclu- Ou_ OUj 20Uk (4) sively to demonstrate the effect of this model. The Sij = _ + Ozi 3 Ozk _<i first case is the well known Burrows-Kurkov experi- ment [23] in which hydrogen and vitiated air streams where/_t is the turbulent/eddy viscosity defined as (two-dimensional) mix and react. The second case is ]¢2 that of an axisymmetric configuration [24, 25] where d, = C.5-- (5) two coaxial jets (fuel and oxidizer) mix and react. The two-equation, (k-e) turbulence model is used for these with C_,=0.09, or, with w defined as, applications. Available data from the above two ex- periments are used to compare with the predictions.

(6) The last case considered is a two-dimensional, react- ing, mixing layer problem. Here, the two-equation, where (k-w) turbulence model [14] has been used. The gov- (/:_eT )4 9 _ +', Ro erning and secondary equations used in the computa- C5 = (7) tions have all been described in detail in the references 100 1+_ R° J cited above. Only an abreviated equation set will be with R¢=6 and given in the present paper. The computations were performed on the supereomputers of NAS and NASA _k Ames Research Center.

Re T = ---:_ (s) wp GOVERNING EQUATIONS /_t is given as _r, = _-Z_ The equations used for computations are described (9) Ld in detail in references [6, 7, 10] and [26]. Only the with forms of the modelled equations used in the present study are given here. Density-weighted averaging is used to derive the mean flow equations from the in- R_ {:]¢_ (10) I+K_: stantaneous coservation equations. In the following, R,.

and is given below.

=_=c6/3, R_=8, c6=_ (11) a} P_ - C6_ 2 + _£ D_ Or" + Ozj W The modelled momentum equation, then, is, a #,) ae

+ b-_jC(z + _ _] (18)

where Ok c3_ Dk,_ = (3zj Oz I (19) - a_---T [ (# + #') s,j (12) and The effects of compressibility are included via the 1 a0+ -_-z= model proposed by Zeman [20]. Here, the compress- c, - n_ (20) 2a* 1+ R--_ ible dissipation terms are expressed as functions of the R_ turbulent kinetic energy dissipation rate and the local The turbulence constants are c_0=0.1, /_:2.2, turbulent Mach number. The compressibility effects _=1.67, Ct=1.44, C__=1.92, _r_=l.0, _r_=l.3, are represented by a component of the dissipation rate Pr=0.72, Prr=l.0, Sc=0.22 and Sc_=l.0. and (e_) given as o'd = 0.0, D_,<0 ec = Kce = 0.3, D_ >0 K_ = rlF(Ms) 2k (13) The mass-averaged total energy can be written in M, = a--" 7 terms of the total enthalpy as where a is the local speed of sound and F(Ms) is a _=_ P function of the local turbulent Mach number (Mr).

(21) F(Mt) is given by The correlations between the fluctuating velocity and A/L the scalar fluctuations are modelled using a gradient- F(lVlt) = 1- ezp[-(M*_._"'°)e], Ms _M, o diffusion hypothesis. A typical model is of the form = 0, Mt < Mto (14) ,,,, #, o_) with Mso=0.1 and 0=0.75. The modelled turbulent - _,,__ = -- (-_-- (22) ¢7Z_ kinetic energy equation is [6, 12] where o'_ is a coefficient which, normally, is a constant.

O#k O#kUy For ¢ = fn (n represents the species) , _¢_ =Sct, and + - Pk - #_,(1+ Ko) Ot cgzi for the static enthalpy, (¢ = h), o'_ = Prt. Using the above definition, and omitting the body force contri- O #s) ak,

+ _-_C(_ + _ _-_J (15) bution, the time-averaged and modelled energy equa-

tion [6] is where _ is either e or wk and It It .

Ot + a_ - Ox---_. (_'_ - _ _ - pu, u¢ )u_ OU_ (16)

P_ = _ p_;,u_, _T_j

+ _ _+_ T;7_ +(_+_ _I_31

The modelled e-equation used (no compressibility corrections) in the present analysis [12] is given below.

where _r_ comes from the turbulent kinetic energy equation. The modeled species continuity equation is O#e O#eUy = (G Pk-C2Ze) "-_ + Ozj + -- _bn - "W:-'_ ] (24) c,zi a #, ) at .

+ b-_j[(# + 7, b_l (171

The modelled form of the mean species production where P_ is the production term in the turbulent ki- rate due to chemical reaction (_b,_) is given, for a finite- netic energy equation. The modelled w-equation [14] rate system involving L reaction steps and N species, in thefollowing general form: exist. The factor rn represents the effect of tempera- ture fluctuations on the mean reaction rate. The tem- L perature variance is calculated from the enthalpy vari- _"lu" ' (25) uL_ = M,_/__,t ,_l - v,_l) x ance [6].

For reacting flows involving multiple chemical species, a multivariate _-pdf model is used to ac- {kl,p",._._-:-. -k,,o"' I-I( )""}, count for the effects of the scalar fluctuations on the species production rates [6]. Relevant features of this where model are briefly outlined below. The parameters of N N the multivariate _-pdf for the N-scalar mixing process (ill,'", fiN) are functions of the mean mass fractions

Z' E" rnl -- lgsl, rll --" Psi

m s=l s=l f,_ and turbulent scalar energy Q: where, v_t and v_ are the number of molecules of the --- I) (30) scalar s involved in the l-th reaction step in the forward and backward directions, respectively. The forward where and backward rate-constants of the reaction I are given N N by kit and kbl respectively.

s= ]_E2, Q= _/z/z. (3_)

n=l n:l klt = ArT _'' ezp[ T_tl (26) ---_-j Decoupling the effects of temperature and species concentrations, the mean species production rate can where At, bt and Ta, are numerical constants specific be written as, to the given reaction step I. kbl is determined from the equilibrium constant for the/-th reaction step and kit. L --r- _-'_¢v" ' wn - Mn A.._ nt - v,zl) x (32) l---I Turbulence-chemistry interaction model N N Practical combustor flows involve multiple scalar {'_hpm'(H M;-":')II,_ kb"_pm(H M;"ld;)Ib,} mixing and reactions. In order to calculate the mean $=I $=i species production rate using the probability density where function (pdf) approach, a model for the joint pdf of N N temperature and various species is required. Such a

joint pdf is difficult to specify and as a preliminary #,- <I-I/5,>, r,, = <II.f::_>

s=i s=l step we effect the simplification that temperature and species fluctuations are uncorrelated. This permits us In the above, angular brackets represent conventional to deal with the temperature and species fluctuations time averaging. The modelled expression for I/r using separately. Equation (26) can be written as the multivariate _-pdf is [21],

To,_ ] (27/ N v',_ rnE

ki, = At(T + T") b' ezp[- (_ + T") Ij,= HH(il, +v:,-r)/H(B-Fmt-p ) (33) J:l r:l p=l Assuming that T" (_ < 1, the term +T") can be ex- T Similarly, the expression for [bt is, panded in a series and the resultant modified reaction rate term is written as, Ibl = HH(19, +v'i-rl/H(B+nt-p) (34) k]-'_ = (1 + m) A,T b' ezp[- T_,-

-_-I (281 $:I r:l p¼l

where where B = #t +f12 +...+#iv (35) b_ Ta,. I Ta, 2 T"T"

rn = [(b,- I)(_- + -_--) + _(T) ] _ (29) The production of turbulent scalar energy due to

chemical reaction can be decomposed as Terms of order higher than two in T" --_ are neglected T N N N from the series expansion in the present analysis since

= wo/o -E T.. (36)

accurate models for these higher order terms do not n=l n=l rt:[ Only the first term on the right hand side of the avoid solving many enormous banded matrices. As above equation needs further modeling and details mentioned before, the options for turbulence models can be found in reference [6]. The turbulence- include both zero and two equation models (both k -e thermochemistry interaction models require the en- and k - w). For more details about the solution pro- thaipy variance and the turbulent scalar energy distri- cedure the reader is directed to the reference cited butions. The modeled equations for enthalpy variance above [17]. The SPARK solver is a finite difference el- (h"h") and turbulent scalar energy(Q) are of a form liptic solver using a hybrid McCormack scheme. More similar to that of the turbulent kinetic energy: details about the solution procedure can be found in references [6] and [9].

ozg azg

0--Y + 0xi - _ 2/_e9 RESULTS AND DISCUSSION

+ [(g + + fi (37)

The computational effort is divided into two parts.

The first part deals with the demonstration and valida- For g = h"h", G = h, fi=0, o" = Pr and era = Prt.

Iv _, Jv tion process for the solution procedure by application fg f_:, Y.,fi= For g -- _-_,=1 G " " = 2_.=1 w./:_, _ = to high speed turbulent reacting flows involving finite Sc and u9 = Sct. The dissipation term in the above rate hydrogen-air chemistry. As mentioned before, the equation is assumed to be Navier Stokes solver TUFF has been used for the com- putations. Two reacting flow configurations have been eg = Cg_g = C_Czwg (38) chosen for this part. The first test case considered is the Burrows-Kurkov experiment [23]. The flow con- Solution of the modeled equations figuration is two-dimensional. A schematic diagram of the configuration is given in figure 1. No-slip walls The equations are discretized and integrated in bound both the upper and lower regions (y=O and space and time to obtain steady state solutions us- y--yrna_). The lower wall is inclined to form an expan- ing either the finite-volume based numerical solver sion surface. Hydrogen is injected into a vitiated air TUFF [17] (for the first two test cases) or the fi- stream. The two streams mix and react downstream of nite difference solver SPARK [9] (for the last case).

the injection location (inlet). The hydrogen stream is The TUFF code contains many desirable features for injected at a velocity of 1216 m/see and a temperature the computation of three-dimensional, hypersonic flow of 254 K. The airstream comes in at a speed of 1764 fields. It has non-equilibrium, equilibrium and perfect m/sec and a temperature of 1270 K. Full details about gas capabilities along with an incompressible option.

the flow parameters and geometry are given in Table 2.

It employs a finite-volume philosophy to ensure that In this case, the reference length used in the hydrogen the schemes are fully conservative. The upwind in- jet width at inlet, h (h=0.004 m). A constant turbu- viscid fluxes are obtained by employing a new tem- lence intensity level is used for arriving at the initial poral Riemann solver that fully accounts for the gas distribution of turbulent kinetic energy and the dis- model used. This property allows the flow field dis- sipation rate. A 13-step, 8-species H_. - Air reaction continuities such as shocks and contact surfaces to be model (Table 1) has been used for the finite-rate chem- captured by the numerical scheme without smearing.

istry system considered here. The grid size is 81 X 121 Total Variation Diminishing (TVD) techniques are in- (81 grid points in the axial (z) direction and 121 grid cluded to allow extension of the schemes to higher or- points in the transverse direction). The total length ders of accuracy without introducing spurious oscilla- of the solution domain is 0.356 m (z/h=89). Avail- tions. The schemes employ a strong coupling between able inlet data have been used for the first-plane pro- the fluid dynamic and species conservation equations files which improved the predictions remarkably over and are made fully implicit to eliminate the step-size the solutions obtained with uniform profiles. The so- restriction of explicit schemes. This is necessary since lutions are compared with the available experimental step-sizes in a viscous, chemically reacting calculation data at this location (exit) in figures 2-4. The solu- can be excessively small for an explicit scheme, and tions carried out with the space marching PNS code the resulting computer times prohibitively large. A UPS [19] using the Baldwin-Lomax turbulence model fully conservative zonal scheme has been implemented are also given for comparison.

to allow solutions of very complex problems. The schemes are made implicit by fully linearizing all of Figure 2 shows the comparison between the pre- the fluxes and source terms and by employing a mod- dicted distributions of the species mole fractions and ified Newton iteration to eliminate any linearization the corresponding experimental data. As seen in and approximate factorization errors that might oc- these figures, there is excellent agreement between the cur. Approximate factorization is then employed to TUFF predictions and experiment. The predictions by

theUPS andSPARK codes donot agree very well but

(z/D=43.1 D). As seen in these figures, the inlet still there is very good qualitative agreement with the data agreement between the computations and exper- experimental data. Figure 3 compares the predicted iment is not perfect, especially around the jet edges, profiles of exit plane total temperature with the ex- and this might affect the computed distributions at perimental data. There is good qualitative agreement downstream locations. The comparison between pre- here. Figure 4 shows the comparison between the pre- dictions and experiment at the downstream location dictions and experiment of the lower wall (hydrogen jet (_:/D=43.1 D) is good given the above mismatch be- side) pressure. Ignition causes the pressure rise in the tween the computational and experimental data at the profile. Ignition seems to be delayed in the case of the inlet. The development of the reaction zone after igni- TUFF predictions accompanied by a more pronounced tion is not predicted well by the TUFF code whereas pressure rise. A point to note here is the fact that the the SPARK code fares better. The experimental data SPARK, UPS and TUFF codes produced nearly iden- indicates that the reaction zone (depicted by the water tical results for nonreacting mixing layers which is not mole fraction distribution) spreads more quickly than the case in the present reacting flow simulations.

the predictions indicate. There seems to be a discrep- ancy between the locations of peak reaction activity The second case is that of coaxial jets [24, 25] where between the predictions (off the axis) and the exper- a hydrogen jet flows (inner jet) coaxially with an iment (closer to axis). However, there is very good outer vitiated air (mass fractions: oxygen=0.246, wa- qualitative agreement between the data with the peak ter=0.209 and nitrogen=0.545) jet. A schematic of the values of the reaction products predicted very well.

flow problem is given in Figure 5. The two streams The flow domain was seen to have a wave-like structure are, air (U=1380 m/sec, T=l180 K with p=107000 as shown by the predicted profiles. Figure 7 shows the N/m2) and hydrogen (U=1774 m/sec, T=545 K with comparison of static temperature data. The agreement p=l12000 N/m2)., The air stream is supersonic with a between predictions and experiment is good qualita- Much number of 1.97 and the hydrogen stream Much tively displaying similar trends. The uncertainty as- number is 1.00. The inlet mean velocity is assumed to sociated with the accuracy of the experimental data have a step profile with the two jets having uniform is unknown. There are considerable differences be- speeds at the specified values (no experimental data tween the data presented by the two references [24, 25], available). The velocity in the lip region of the inner especially in the temperature profiles. Overall, there jet tube wall (finite wall thickness) is assumed to be is good qualitative agreement between the predictions zero. The inlet temperature profile is derived based on and experiment.

the experimental data given for a location just down- stream of it (shown later). The inlet species mass frac- In the second part, the turbulence-chemistry inter- tion distributions are also chosen based on the exper- action model is demonstrated using a two-dimensional imental data provided at the same downstream loca- mixing layer configuration. A schematic of the flow tion. The models for turbulence and chemistry are problem is given in figure 8. The two streams identical to the ones used for the first test case. A are, air (U =1606 m/sec, T =1600 K with 81 X 91 grid (81 points in flow direction, 91 points f_z2=0.0, fo2=0.267 and f_¢_=0.733) and hydrogen in the radial direction) was used for the calculations.

(U =1250 m/sec, T =254 K with fH_=l.0, fo_=O.O The inner jet/tube diameter (D=0.00236 m) is used and f_2=O.O). The two streams are supersonic with as a reference length. The total length of the flow do- the air stream Much number of 2.07 and hydrogen main is equal to 43.1 D. The outer boundary (radial) stream Much number of 1.03. The inlet mean veloc- of the flow domain is taken to be at y=17 D. A more ity is assumed to have a hyperbolic tangent profile. A detailed description of the flow parameters is given in constant turbulence intensity level is used in the free Table 3. The region outside the limits of the air jet is stream for arriving at the initial distribution of tur- assumed to be still air at a temperature of 273 K. The bulent kinetic energy and the specific dissipation rate.

two-equation (k - e) turbulence model is used along The pressures are matched between the two streams with the finite rate H2-Air chemistry model mentioned (P =1 atm.). A 13-step, 8-species H2- Air reaction above. In all the figures shown for this case, y refers model (Table 1) has been used for the finite-rate chem- to the radial distance measured from the axis of the istry system considered here. A 101 X 81 grid (101 coaxial system of jets.

points in flow direction, 81 points in the transverse di- rection) was used for the calculations. The length of Figures 6 - 7 show the results of the computations.

the flow domain is 0.25 m and its width is 0.05 m. In Figure 6 shows the computed and experimental distri- the figures the title g refers to calculations that include • butions of species mole fractions. The figure is de- the interaction model.

signed in a two-column format. The left side col- umn represents the inlet (first x-location) data and The difference made by the temperature fluctuations the right side column is the data at the exit plane model is shown in figure 9. Here the factor (1 + m) is plotted as a function of y for the first three reaction A numerical solution procedure applicable to high steps at the exit plane of the flow domain. The need speed reacting flows has been demonstrated. In the for including the temperature fluctuations is evident first part, computation of the flow fields of two high- speed, turbulent, reacting flow configurations involv- in the figure. The forward reaction rate changes by as much as 400 percent (reaction step 1) due to the ing finite-rate chemical kinetics for hydrogen-air com- effect of this model. The species production rate is a bustion have been carried out. A two-equation (k - e) turbulence model with compressibility corrections has strong function of the reaction rate (29) and hence will be affected by the inclusion of this model. The been used. The predictions are compared with avail- effect of the turbulence-chemistry interactions model able experimental data. Good qualitative agreement is on the species production rate is shown in the next few present between computations and experiment. More detailed experimental data is necessary. In the sec- figures. Figure 10 shows the production rates of the ond part, a turbulence-chemistry interaction model is major species involved (H2, 02, H20, OH) as func- introduced and its effect is demonstrated via a two- tions of y at the exit plane. Computational results dimensional turbulent reacting flow problem. The obtained with (k-w-g) and without (k-w) the effect of these interactions are shown in the figure. It is seen interactions seem to have a significant effect on the that the interaction between turbulence and chemistry species production rate. More detailed analysis is nec- has a significant effect on the net production rate of essary.

chemical species. While it is not conclusive whether ACKNOWLEDGMENTS the turbulence-chemistry interactions increase or de- crease the production rates of individual species, it This work was supported by the Applied Computa- is important to note that these interactions do have tional Fluids Branch of the Fluid Dynamics Division at a significant effect on progress of the chemical reac- NASA Ames Research Center under the Cooperative tions. Figure 11 shows the effect of the interactions on agreement number NCC 2-715. The author wishes to the species distributions. The effect seems to be more acknowledge the contributions of Dr.Gregory Molvik pronounced in the case of the products (H20, OH) of MCAT Institute, NASA Ames Research Center, than the primary reactants (H2, 02). The changes California and Dr. Ganesh Wadawadigi of Univer- are larger in regions of higher gradients (on the fuel sity of Texas at Arlington, Texas. Also, the invaluable stream side for H2 and air stream side for 02, for ex- contribution to the interaction model by Dr. Sharath ample).

Girimaji of ICASE, NASA Langley Research Center is This effort represents a significant push in the right acknowledged and appreciated.

direction in this very important area of turbulent re- acting flows. The model is simple to use and is easily adaptable to other types of turbulence closures such References as the Reynolds stress models. More improvements can certainly be done in terms of using more realistic [1] Molvik, G. A. Bowles, J.V.and Huynh, L. C., models for the temperature and concentration fluctua- "A Hypersonic Research Vehicle with Hydrocar- tions such as a pdf-based model for temperature effects bon Scramjet Propulsion: Design and Analysis", (as opposed to the moment model employed here), a AIAA Paper 93-5097, 1993.

joint-pdf model to represent the effects of both the [2] Ebrahimi, H. B., "CFD Validation For Scramjet temperature and the concentration fields etc. How- Combustor and Nozzle Flows, Part I", AIAA-93- ever, in order for the model to be useful for practical 1840, 1993.

applications, it must be kept in a form which promotes ease of use as well as computational economy. This is [3] Vitt, P. H., Riggins, D. W. and McClinton, C.

the main reason for resorting to a two-equation level R., "The Validation and Application of Numeri- turbulence model in the present work. A concerted cal Modelling to Supersonic Mixing and Reacting validation effort is crucial to ensure the success of the Flows", AIAA-92-0626, 1992.

modelling efforts. One debilitating factor in this re- gard is the nonexistence of useful experimental data. [4] Riggins, D. W. and McClinton, C. R., "A Com- It is extremely difficult, if not almost impossible, to ob- putational Investigation of Mixing and React- tain such experimental data using present day equip- ing Flows in Supersonic Combustors", AIAA-92- meat. An alternative seems to be direct numerical 0626, 1992.

simulations (DNS) which has shown promising signs [5] Eklund, D. R. and Northam, G. B., "A Numeri- in relatively simpler flow configurations.

cal Study of the Effects of Geometry on the Per- formance of a Supersonic Combustor." AIAA-92- CONCLUSIONS 0624, 1992.

[6] Narayan, J. R. and Girimaji, S. S., "Tur- [18] Lawrence, S. L., Chaussee, D. S. and Tannehill, bulent Reacting Flow Computations Including J. C., "Application of an Upwind Algorithm to the Three-Dimensional Parabolized Navier- Turbulence-Chemistry Interactions." AIAA-92- 0342, 1992.

Stokes Equations", AIAA 87-1112, 1987.

[19] Wadawadigi, G. Tannehill, J. C., Buelow, P. E.

[7] Narayan, J. R., "A Two-Equation Turbulence

and Lawrence, S. L., "A Three-Dimensional Up- Model for Compressible Reacting Flows." AIAA- wind PNS Code for Chemically Reacting Scram- 91-0755, 1991.

jet Flowfields", AIAA 92-2898, 1992.

Eklund, D. R., "Calculation of Supersonic Tur-

[8]

[20] Zeman, O., "Compressible Turbulence Subjected bulent Reacting Coaxial Jets." AIAA Journal, to Shear and Rapid Compression", Eighth Sym- Vol.28, No.9, 1990, pp 1633-1641.

posium on Turbulent Shear Flows, Munich, Ger- many, 1991.

Carpenter, M. H., "Three-Dimensional Computa-

[9]

tions of Cross-Flow Injection and Combustion in [21] Girimaji, S. S., "A Simple Recipe for Mod- a Supersonic Flow", AIAA-89-1870, 1989.

eling Reaction-rates in Flows with Turbulent- Combustion", AIAA-91-1792, 1991.

[10] Drummond, J. P., Carpenter, M. H. and Riggins, D. W., "Mixing and Mixing Enhancement in Su- [22] Girimaji, S. S., "Assumed fl-pdf model for turbu- personic Reacting Flows", High Speed Propulsion lent mixing: Validation and Extension to Multiple Systems: Contributions to Thermodynamic Anal- Scalar Mixing", Combustion Science _ Technol- ysis, ed. E. T. Curran and S. N. B. Murthy, Amer- ogy, Vol.78, 1991, pp 177-196.

ican Institute of Astronautics and Aeronautics, [23] Burrows, M. C. and Kurkov, A. P., "Analytical Washington, D. C., 1990.

and Experimental Study of Supersonic Combus- tion of Hydrogen in a Vitiated Airstream." NASA Drummond, J. P.,"A Two-Dimensional Numeri- [11] TM X-2828, 1973.

cal Simulation of a Supersonic, Chemically Re- acting Mixing Layer", NASA TM 4055, 1988.

[24] Cheng, T. S., Wehrmeyer, J. A., Pitz, R. W., Jar- ret, O. and Northam, G. B., "UV Roman Scatter-

[12] Jones, W.P. and Launder, B.E., "The Prediction

ing Measurements in a Much 2 H2-Air Flame for of Laminarization with a Two-Equation Model of Assessment of CFD Models." Proc. of the Cen- Turbulence", Int. J. Heat Mass Transfer, Vol.15, tral States Meeting of _he Combustion Institute, 1972, pp 301-314.

Nashville, TN, 1991.

[13] Launder, B.E., Reece, G.J.

[25] Jarret, O. Jr., Cutler, A. D., Antcliff, R. R., Chit- and Rodi, W., "Progress in the Development of somboon, T., Dancey, C. L. and Wang, J. A., a Reynolds Stress Turbulence Closure", 3". Fluid "Measurements of Temperature, Density, and Ve- Mech., Vol.68, 1975, pp 537-566.

locity in Supersonic Reacting Flow for CFD Code Validation." Proe. of the 25th JANNAF Combus- [14] Wilcox, D.C., "A Two-Equation Turbulence tion Meeting, Huntsville, Alabama, 1988.

Model for Wall-Bounded and Free-Shear Flows", AIAA Paper 93-2905, 1993.

[26] Williams, F. A., Combustion Theory. Addison- Wesley Publishing Company, Inc., Reading, MA,

[15] Oldenborg, R. et al., "Hypersonic Combustion Ki-

pp. 358-429, 1965.

netics - Status Report of the Rate Constant Com- mittee, NASP High-Speed Propulsion Technology Team", NASA TM 1107, 1990.

'[16] Jachimowski, C. J., "An Analytical Study of the Hydrogen-Air Reaction Mechanism with Applica- tion to Scarmjet Combustion", NASA TP 2791, 1988.

[17] Molvik, G. A. and Merkle, C. L., "A Set of Strongly Coupled Upwind Algorithms for Computing Flows in Chemical Nonequilibrium", AIAA Paper 89-0199, 1989.

Table 1. H2 - Air Reaction System Table 3. Conditions for coaxial jet experiment

No. Reaction H2 Air Mach No. 1.0 1.97 1 H+O2_O+OH Temperature 545 K 1180 K 2 OH+H2_---H20+H Pressure 0.112 MPa 0.107 MPa 3 O+H2-----OH+H Velocity 1774 m/s 1380 m/s 4 OH + OH _ H20 + 0 fH_ 1.0 0.0 5 H+OH+M_-_-H20+M fo, 0.0 0.246 6 H+H+M=H2+M fN, 0.0 0.545 7 O+O+M_O2+M faqo 0.0 0.209 8 H+O+M_--OH+M 9 H+O2+M_HO2+M i0 OH + H02 = H20 + 02 Fuel injector diameter=0.00236 m ii H + H02 = H2 + 02 Lip thickness=0.000725 m 12 H + H02 = OH + OH Nozzle diameter(air flow)=0.01778 m 13 0 + H02 _ OH + 02 14 H 02 + H 02 = H202 + 02 15 H + H202 _- H2 + H02 16 OH+ H202 = H20 + H02 17 H + H202 _- H20 + OH 18 0 + H202 = H02 + OH 19 OH + OH + M = H202 + M 20 0 K+ OH = H2 + 02 Species : H2, 02, H20, OH, H, O, H02, H202 and N2(inert) M is a third body (all species included) Uai r = 1764 m/s Air (vitiated) Tai r = 1270 K i UH2= 1216 m/s Table 2. Conditions for Burrows-Kurkov TH = 254 K Experiment X H2 Air Mach No. 1.0 2.44 Temperature 254 K 1270 K Fig.1 Burrows-Kurkov experiment : schematic Pressure 0.1 MPa 0.1 MPa Velocity 1216 m/s 1764 m/s fH= 1.0 0.0 fo_ 0.0 0.258 fN: 0.0 0.486 fargo 0.0 0.256 Fuel injector height=0.004 m Duct height at inlet=0.0938 m Duct height at exit=0.1048 m I • : J ! • : o o e- °_ , i i °° II e_

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APPENDIX- D

Two-Equa!ionTurbu!enceModel for

Compressible Reacting Flows

J. R. Narayan

Reprinted from

AIAA Joumal

Volume31, Number 2, February 1993,Pages 398-401

IIA/AA_

A publication of the American Institute of Aeronautics and Astronautics, Inc.

The AerospaceCenter, 370 L'Enfant Promenade,SW Washington, DC20024-2518 398 AIAA JOURNAL, VOL. 31, NO. 2: TECHNICAL NOTES lieGE BI.ANK NOT FtLME_._

Two-Equation Turbulence Model for

Compressible Reacting Flows

J. R. Narayan* NASA Ames Research Center, Moffett Field, California 94035 Introduction NE of the major areas of current interest where compu- tational fluid dynamics (CFD) is used extensively is the development of advanced air-breathing propulsion systems for hypersonic vehicles. A hydrogen-fueled supersonic combus- Received Jan. 13, 1992; revision received July 9, 1992; accepted for publication July 10, 1992. Copyright © 1992 by the American Insti- tute of Aeronautics and Astronautics, Inc. All rights reserved.

*Research Scientist, MCAT Institute. Member AIAA.

400 AIAA JOURNAL, VOL. 31, NO. 2: TECHNICAL NOTES

Evans et al. s In the experiment, hydrogen at a temperature of comparison between the predictions and representative experi- mental data. 6,7 Here, C_ is defined as 251 K was injected (Much number = 2.0 and velocity = 2418 m/s) along the axis of a supersonic jet (Much number 1.9, d_ U_ + UE velocity= 1510 m/s) of vitiated air (temperature= 1495 K).

C,-

The species mass fractions in the alrstream are nitrogen = dx UI - U 2 0.478, oxygen = 0.241, and water = 0.281. The fuel injector has an inner diameter of 0.6525 cm and outer diameter of 0.9525 and C_ 0 is its value for incompressible flow. KEPSI refers to the predictions with compressibility correction included, and cm. The outer jet (air) diameter is 6.53 cm. The predicted KEPS2 refers to the predictions without the correction. RS centerline distribution of hydrogen and the profiles of the refers to the Reynolds stress closure predictions) The scatter major species mass fractions (H 2, O2, HEO, NE) at three axial among the experimental data is indicative of the nature of locations are compared with experimental data in Figs. 3. In available data. The effect of compressibility on the mixing- these figures, D is the fuel injector outer diameter. The predic- layer growth rate is to reduce the growth rate with increasing tions agree well with experiment in the region where the mixing convective Much number. The two-equation model without effects dominate. The discrepancy between the predictions and the compressibility correction does not predict the reduced experiment immediately downstream of the injector exit may be due to the fact that the conditions set at the inlet location growth rate with increasing compressibility. However, the compressible two-equation model does predict this trend very (boundary condition for computations) in the calculations well. The close comparison between the Reynolds stress clo- may not match the exact experimental conditions. These con- sure and the k-¢ model is important because for complicated ditions are unknown and, hence, could not be used for the flowfields, such as that in the scramjet combustor, the higher- calculations. The predicted profiles of water vapor, which is an order Reynolds stress closure may be very expensive to use. indication of the extent of reaction, agree reasonably well with The predictions agree well with experiment within the scatter in the experiment. The location of the peak in the profile is the available data. It must be pointed out that there are other farther into the airstream for the predictions than the experi- compressibility correction models, similar to the one used ment indicates. This may be because the interaction between here, available today. However, a detailed comparison be- turbulence and chemistry is not fully accounted for in the tween such models is beyond the scope of this report. calculations. As mentioned earlier, the calculations account Mixing between fuel and air and the ensuing chemical reac- for only the effect of temperature fluctuations on the reaction tion is the main focus of study for a configuration such as the rate. In addition, correlations of order higher than 2 were scramjet combustor. The presence of turbulence and its effect dropped from the model. Also, the experimental data itself on the flowfield, especially the mixing aspect of it, is an impor- may not be very accurate. Given that, the compressible turbu- tant part of such studies. A representative mixing dominated lence model used here performs very well in predicting the reacting flow (hydrogen and air) is considered here. A seven- turbulent reacting flow.

step, seven-species HE-Air reaction model (Table 1) has been

used for the finite rate chemistry system considered here. Conclusions

The test case studied is the reacting coaxial jet problem A two-equation turbulence model (k-O has been modified to (Figs. 3) for which the experimental data was obtained by be applicable for compressible flows by adapting a compress- ibility correction model. Computation of compressible mixing layers indicate that the decrease in the growth rate of the mixing layer with increasing convective Mach number is well predicted by the model. Comparisons of the predictions agree very well with available experimental data and the predictions • , , G: , - G: -" , 0.0 0.5 1.0 0.0 0.5 0.0 0.5 of a compressible Reynolds stress closure. Preliminary studies /s, of reacting mixing problems involving dissimilar gases indicate that the model is well suited for application to such flows. The computations use a model for the effect of temperature fluctu- ations on the reaction rate in the finite rate chemistry model.

0_' 0 Further improvement to the model, especially the effect of o.o s_, o15 o o.o 0.5 oo o15 turbulent fluctuations in species concentrations on the produc- tion rate, is necessary.

V/D 1 • Acknowledgments This work was supported by Theoretical Flow Physics 0.0 ..t.H_ ° 0.5 0,0 0.5 0.0 0.5 Branch, Fluid Mechanics Division, NASA Langley Research Center under Contract NAS1-18599. The Technical Monitor was J. P. Drummond.

References

0_ r I , , , ICarpenter, M. H., "Three-Dimensional Computations of Cross- o.o & 0.5 o.o o.5 o.o o15 Flow Injection and Combustion in a Supersonic Flow," AIAA Paper a) Species mass fractions Hi, 02, HEO, and N2 89-1870, June 1989.

2Narayan, J.R., "A Two-Equation Turbulence Model for Com- 1.50 pressible Reacting Flows," AIAA Paper 91-0755, 29th Aerospace Sci- -- Pr*dictions ences Meeting, Reno, NV, Jan. 1991.

1.25 • Expermmtt 3Narayan, J. R., and Sekar, B., "Computation of Turbulent High 1.00 _..._ m, Speed Mixing Layers Using a Two-Equation Turbulence Model," /'8, 0.75 Proceedings of the CFD Symposium on Aeropropulsion, NASA Lewis 0,50 Research Center, Cleveland, OH, April 1990.

4Sarkar, S., Erlebacher, G., Hussalni, M. Y., and Kreiss, H. O., 0.25 "The Analysis and Modeling of Dilatational Terms in Compressible 0.00 10 20 30 Turbulence," NASA CR 181959, Dec. 1989.

z/D 5Jachimowski, C. J., "An Analytical Study of the Hydrogen-Air b) Centeriine Hi distribution Reaction Mechanism with Application to Scramjet Combustion," NASA TP 2791, 1988.

Fig. 3 Reacting flow: co_Aal Jets.