Diagrams of premixed turbulent combustion based on direct simulation

Twenty-Third Symff.Jsium (International) on C o m b u s t i o n / T h e C o m b u s t i o n Institute, l,q~)/pp. 6 1 3 - 6 1 9

DIAGRAMS

OF PREMIXED TURBULENT COMBUSTION BASED ON DIRECT SIMULATION T. POINSOT Center for Turbulence Research Stanford University Stanford, CA .94305 U.S.A. AND

D. VEYNANTE AND S. CANDEL Laboratoire EM2C, CNRS Ecole Centrale de Paris 92295 Chatenay-Malabry Cedex, France

The structure and morphology of premixed turbulent flames is a problem of fundamental interest in combustion theory. Diagrams indicating the typical structure of a flame submitted to a given turbulent flow have been constructed on the basis of essentially intuitive and dimensional considerations. Knowing the turbulence integral scale and the turbulent kinetic energy, these diagrams indicate if the flow will feature flamelets, pockets or distributed reaction zones. A new approach to this problem is described in the present paper. The method is based on direct numerical simulations of flame/vortex interactions. The interaction of a laminar flame front with a vortex pair is computed using the full Navier-Stokes equations. The formulation includes non-unity Lewis number, non-constant viscosity and heat losses so that the effects of stretch, curvature, transient dynamics and viscous dissipation can be accounted for. As a result, flame quenching by vortices (which is one of the key-processes in premixed turbulent combustion) may be computed accurately. An elementary ('spectral') diagram giving the response of one flame to a vortex pair is constructed. This spectral diagram is then used, along with certain assumptions to establish a turbulent combustion diagram similar to those proposed by Borghi 2 or Williams. s Results show that flame fronts are more resistant to quenching by vortices than expected from the classical theories. A cut-off scale and a quenching scale are also obtained and compared to the characteristic scales proposed by Peters) Stretch is not the only important parameter determining flame/vortex interaction. Curvature, viscous dissipation and transient dynamics have large effects, especially for small scales and they strongly influence the boundaries of the combustion regimes. For example, the Kiimov-Williams criterion which has been advocated to limit the flamelet region, underestimates the size of this region by more than an order of magnitude.

Introduction

flame front curvature, dynamical features or viscous effects. This paper presents a new method leading to diagrams which take into account these effects. It is based on a more complete dimensional analysis combined with direct numerical simulations of typical v o r t e x / f l a m e interactions. This provides a "spectral' diagram describing the interaction between an isolated vortex pattern and a laminar flame front. This information is then used to infer the behavior of a complete turbulent reacting flow and establish a more quantitative diagram.

Models for premixed turbulent combustion are largely based on empirical ideas because of the complexity of flame/turbulence interactions. The first problem in the derivation of a turbulent combustion model is the determination of the structure of the reacting flow. For example, knowing if turbulence will induce local flame quenching and create holes in the flame surface is an important part of the modeling process. Diagrams defining combustion regimes in terms of length and velocity scale ratios have been proposed by Barrere, 1 Borghi, z Peters, 3 Bray4 and Williams. 5 The dimensional arguments used to derive these diagrams are essentially intuitive and neglect important effects such as

Turbulent Combustion Diagrams: Classical premixed turbulent combustion diagrams assume that a reacting flow may be para613

614

TURBULENTCOMBUSTION

metrized in terms of two non-dimensional numbers: the ratio of the turbulence integral scale l to the flame front thickness lF and the ratio of the rootmean-square velocity fluctuation u' to the laminar flame speed s~. Using the notations and assumptions of Peters, ~ different regime transitions may be associated with specific lines in this diagram (Fig. la). Of special interest is the Kiimov-Williams line which corresponds to the limit between flamelet regimes and distributed reaction zones. This transition occurs when the stretch 1/A dA/dt (A is the flame surface) imposed to the flame becomes larger than the critical stretch for extinction and induces local quenching. The critical stretch may be estimated z'4 by sL/l F. Defining the Karlovitz number by

The flame stretch 1/A dA/dt may be evaluated4 in terms of the Taylor microscale A and of u' as idA - --

-=- n ' / A .

(2)

Adt t

The Taylor microscale A'and the Kolmogorov scale 71 for isotropic homogeneous turbulence are given by:

A / l = RG 1/2

~/l = Re[ z/4,

(3)

where Ret = u'l /~. The Kolmogorov velocity scale is ur/u' = Rel 1/4. Using the relation sLIF/~' 1, four expressions for the Karlovitz number can be derived:

IdA Adt

Ka

(1)

=

s~llF'

one expects local quenching and distributed reaction zones for Ka > 1.

U~/sL

-f

~~ / 9 II('~'~

I'ama'ats I

I 1 Integralscate / flame thickness

,

I(b) Spectral dlagrem I

I... r OV IKOtmogO Iscale ~

: to-I. . / /

E

\ D D l/IF

BI~

j

~

/ / ~

/ /

/

X

I

I I

)

Integral scale ~;;,',,-Z./,~_ u. I ......

l/IF /

SL/Ir

=(/F) 2\~/ =

UK/~ 141

The Klimov-Williams (KW) criterion is derived from Eq. (4) by stating that, if the Kolmogorov scale ~/ is smaller than the flame thickness lF (Ka > 1), no flamelet should be observed because their internal structure is destroyed by stretching and quenching. The Ka = 1 limit is a line with a slope 1/3 in the diagram of Fig. la. The region lying below Ka = 1 is the flamelet region. Note that according to the last expression in Eq. (4), the critical stretch 1/A dA/dt is the strain rate at the Kolmogorov scale UK/~I. Therefore, the KW criterion involves a single turbulent scale, namely the Kolmogorov scale.

I(a) Standardcombustion diagram I I Distributed I [ reaction zones I

Ka=\

a~

"Turbulence line

!' Vortex scale / flar~ thickness

FIc. 1. Peters diagram for turbulent premixed combustion (a) and the spectral diagram principle (b).

A Spectral Diagram for Turbulent Combustion Regimes: The approach used in the previous section to derive the KW criterion has deficiencies. The most evident is that it assumes that the Kolmogorov fluctuations induce flame quenching because they generate the highest strain rates. This approach ignores three important points: 1. As indicated by Peters, z the Kolmogorov scales might be too small compared with the flame front thickness to stretch it. 2. The Kolmogorov scales (71 and ur,) are characterized by a unity Reynolds number OTuK/v = 1 where v is the kinematic viscosity). These structures might be dissipated by viscosity before they quench the flame front. 3. Scales smaller than the flame front thickness will induce high local curvatures which may counteract the influence of strain.

DIAGRAMS OF PREMIXED COMBUSTION Direct simulation is used here to derive criteria including viscous and curvature effects. This approach is adopted because we wish to reconsider the simplifications of previous studies and take a more basic point of view. Consider first a flame front interacting with a turbulent flow field. This turbulent flow features a complex combination of vortices with scales ranging from the Kolmogorov to the integral scale. To describe turbulence/combustion interaction, one has to take into account the existence of these various scales in the flow and, accordingly, define for each point of the combustion diagram, a spectral diagram (Fig. lb). Each point of the standard combustion diagram corresponds to a turbulent flow containing a spectrum of scales. Each of these scales corresponds to a point in the spectral diagram. In the spectral diagram, a turbulent flow field is represented by a straight line that may be designated as the 'turbulence line' bounded by the Kolmogorov and integral scales (It is assumed here that the turbulent reference quantities correspond to the fresh gases and that the turbulent spectrum in this part of the flow can be described by the Kolmogorov relation: u'(r)3/r = ~ where ~ is the dissipation rate). The Kolmogorov scale is located on the line Re n = u'(~)~l/v ~- U'(71)IsL ~IIF = 1. NOW, each scale of the turbulence line will have a different effect on the flame front. Some vortices may quench the flame front, others may be dissipated by viscous effects before they interact with the flame (Fig. 1). The interaction between one of these scales and the flame front may be calculated exactly and an accurate spectral diagram may be constructed. This article will focus on quenching limits and vortex decay mechanisms. The quenching limit is of special interest for modelling since it determines the validity domain of flamelet models. Vortex decay phenomena set the minimum scale which can affect the flame front. This scale is somewhat related to the Gibson scale3 and may be used as a cut-off scale in fractal studies of turbulent combustion. Direct Simulation of Vortex/flame Front Interactions:

Studies of vortex/flame interactions initiated by Marble s have been pursued by many authors, r-lz However, little is to be found on flame quenching in a vortex. Asymptotic studies 5 show that nonunity Lewis numbers and non-zero heat losses should be included in a computation aimed at the understanding of this effect. This is done here by solving the Navier-Stokes equations in a two-dimensional configuration under the following assumptions. Consider a compressible viscous reacting flow. The chemical reaction is represented by a single step mechanism B (reactants) ---> P (products). The re-

615

action rate tbR is expressed as tba = BpYR e x p ( - ~ )

[ -/3(1 - 0) '~ ApYR exPel : ~ l ' - ' @ i ] '

=

(~)

where @ = (T - T1)/(T2 - T1). T1 is the fresh gas temperature, T2 is the temperature of the burnt gases, 5 Toe is the activation temperature. The coefficients A, a and fl are, respectively , the reduced preexponential factor, the temperature factor and the reduced activation energy (see list of symbols). The mass fraction of the reactants Yn is nondimensionalized by its value Y~ in the fresh gases: = Ya/Y~ Radiative heat losses are introduced through a linear term h(T - T1) in the energy equation (7). Under these assumptions, the fluid dynamic equations are: Op

0

Ot

Oxi

-- + --

Oet

(m,,) = 0,

(6)

O

+ - - [(e, + p)uil #t Oxi

--

=

(ujTo) + - -

Ox~ \ -~xJ

Opu~

0 - -

Ot

0xj

a(p?) + 0t

(pu,uj)

Qw

-

Op +

- -

8xi

(pYu,) = ax~ \

h(T-

=

T1),

O~'#

-~x~'

a~,l - ~'

(7)

(8)

(9)

where 3

1~i~ et = 2 p uk 4 = w~/~.

P T

1'

(10)

(11)

We assume that the gas mixture is a perfect gas with constant molar mass and a ratio of specific heats y of 1.4. The viscosibty/~ is a function of temperature:/z = izl(T/T1) 9 The thermal conductivity A and the diffusion coefficient D are obtained from/z according to A = IzCp/P~ and D = /z/(pSc) where the Prandtl number Pr and the Schmidt number Sc are constant. As a consequence the Lewis number Le = Sc/P~ is also constant. The heat-loss coefficient h is expressed as in Williams: 5 h =

xs ~(sLl ~)%/ /3. The system (6) to (9) is solved using a finite-difference explicit scheme which is sixth-order accurate in space and third-order in time. 13A4 Typical grids contain 2.A(R~ points.

616

TURBULENT COMBUSTION

The initial configuration is sketched in Fig. 2a. Two counterrotating vortices are created at t = 0 on the upstream side of the laminar flame front. As the flow is symmetrical with respect to the x-axis, only the upper part is calculated and displayed. The inlet flow speed is equal to the laminar flame speed so that the flame does not move when it is not perturbed. The vortex-pair configuration allows an accurate evaluation of the flame stretch and speed on the axis. It also generates a high stretch and may be considered as one of the most efficient structures able to interact with the flame front because of its self-induced velocity. Like all 2D simulations,9'10'12 the present model fails to capture some mechanisms of real turbulence like vortex stretching. 18 However, because of the lack of knowledge of the exact structure of turbulence in 3D flows, it is reasonable to begin the investigation by a 2D computation even though we reckon that certain 3D effects might affect some of our conclusions. The present computation, which incorporates finite rate chemistry, variable density and viscosity, curvature, straining and transient effects is a significant improvement on classical approaches based on dimensional analysis and a first step towards a more realistic prediction of quenching in turbulent flames. The parameters adopted in the simulations are as follows:

Pr = '75, Le = 1.2,

/3 = 8,

sL/a=O.013,

b=0.76,

y~,

[ (a) Configuration

[

Periodic conditions

Vortex

Inflowat

a = .75,

pair

c = 10 -4 ,

I

[ Computation domain I

: :: : i ~:~

:! :

~ : ...........

i

I-axs --du',r,] i ::lllll: ::~"iHII

llii :

(12)

/

where a is the sound speed. With regards to these parameters, the final results are most sensitive to the heat-loss parameter c. The flame front thickness IF is 3.7 P/SL, The length scale r used to characterize the vortex pair is the sum of the vortex diameter d and the distance between vortex centers (Fig. 2a). The velocity scale u'(r) is the maximum velocity induced by the pair. Calculations have been performed for 0.81 < r/lF < 11 and 1 < u'(r)/sL < 100.

An Example of Flame Quenching by a Vortex Pair: To illustrate these calculations, we will describe a case where the vortex pair size and speed are large enough to induce quenching of the flame front (r/1F = 4.8 and u'(r)/sL = 28). Figures 3 and 4 display the reaction rate (tb) and the temperature (O) fields at three instants. Time is normalized by the flame time IF/SL: t + = tsL/IF. The interaction is fast and ends after about two flame times. At t § = 0.65, the vortex pair has stretched and curved the flame but its inner structure is preserved. No quenching is observed (Fig. 3). The Karlovitz number on the symmetry axis is at that instant around one. The fact that the flame is still burning despite such a high Karlovitz illustrates the importance of transients. At t § = 1.3, quenching appears on the downstream side of the pocket of fresh gases formed by the vortex pair. These gases are pushed rapidly into regions where the burnt gases have been cooled because of heat losses (Fig. 4). This effect, combined with the high stretch generated by the vortices, causes a nearly complete extinction of the pocket after it has been separated from the bulk of the fresh gases. At t §

iigs~.~'t) l i!

i\ /illlll iiii!!!iii ii:i}

x

u,iow

R e d u c e d time = 0.65 Periodic conditions

Flame thickness IF

#

-*

I (b) Spectral diagram I

i-* R e d u c e d time = 1.30

~

i*

~,

i *

1

"-

-
0 -=

,o

~ ::

]

u

1"~

\.i.-"...............................

-:,

J"

101 t'" 10 "1

:

i in

:

\

..............

i

R e d u c e d time = 1.625

.....

\\

i ~'~

t \

1010" Length scale r / Flame front thickness I F

I 10 2

FIG. 2. Configuration for flame/vortex interactions (a) and final spectral diagram (b).

FIG. 3. Reaction rate fields at four instants for an interaction leading to quenching.

DIAGRAMS OF PREMIXED COMBUSTION Reduced time = 0.65

I

i

i

I

T

R e d u c e d t i m e = 1.30

I: R e d u c e d t i m e = 1.625

FIC. 4. Temperature fields at four instants for an interaction leading to quenching.

= 1.625, the pocket of fresh gases is convected through the burnt gases without burning except near its tail. in this case, the flame front is not only quenched locally by the vortex pair but some of the unburnt mixture is also able to cross the flame. This mechanism could be responsible for some of the pollutant formation in turbulent flames. The occurrence of quenching is a strong function of the structure size. Large vortices always lead to quenching if their characteristic strain is higher than the extinction strain rate. However, when the vortex-pair size diminishes, the thermodiffusive and viscous effects are felt. For Le = 1.2, simulations show that, when r/IF decreases, local quenching becomes more difficult because the flame speed is increased by curvature but also because vortices are rapidly dissipated by viscosity. The flame speed decrease related to stretch is compensated by the flame speed increase due to the stabilizing effect of the thermodiffusive mechanism. For small scales, viscous effects, stretch and curvature have to be considered together. All these effects result in the spectral diagram described in the next section.

The Spectral Diagram and the New Turbulent Combustion Diagram: The spectral diagram displayed in Fig. 2b is defined on the following basis. It is first observed that the outcome of a vortex/flame interaction depends on the scale r and on the vortex velocity u'(r) and leads to four typical configurations: (1) a local quenching of the front, (2) the formation of a pocket of fresh gases in the burnt gases without quenching, (3) a wrinkled flame front or (4) a negligible global effect without noticeable flame wrinkling or thickening. Two curves are plotted in Fig. 2b:

617

9 The quenching curve distinguishes vortices which locally quench the flame front. It is fitted to the data points for 0.81 < rile < 11 and extended for large scales rile > 11 to match the line Ka(r) = (u'(r)/r)/(sL/1F) = 1. (Large vortices stretch the flame front as in a stagnation point flow. The strain rate is sustained for long times and little curvature is induced. Therefore, quenching by these large structures is only determined by the ratio of vortexinduced stretch to critical flame stretch and occurs when Ka(r) = 1.) 9 The cut-off limit corresponds to vortices which induce a maximum modification of the total reaction rate of about 5 percent. From the diagram of Fig. 2b, it is possible to deduce a premixed turbulent combustion diagram under the following assumptions: (1) A single vortex structure interacts at a given time with the flame front, (2) Any turbulent structure located in the quenching zone of the spectral diagram will locally quench the flame front and induce a distributed reaction regime. These assumptions are rather crude. For example, it is clear that turbulent scales in the quenching zone will not quench the flame front if the energy density corresponding to these scales is too low. Therefore assumption (2) is probably not satisfied. However, these hypothesis lead to a 'worse case' quenching interaction diagram. An important limitation of the present approach is also found for small and energetic scales. In this case, the interaction between many small vortices and the flame front is difficult to assess from the behavior of a single vortex interacting with the front. Under the previous assumptions, the construction of the turbulent combustion diagram is straightforward. A turbulent field of type B (Fig. 5) will contain inefficient scales (dashed line) and scales able to corrugate the flame front but unable to quench it (solid line). Point B will therefore correspond to a flamelet regime. In the case of field A, even the integral scale will not be sufficiently energetic to interact with the flame and the latter will remain pseudo-laminar. Turbulent field C contains scales that may locally quench the flame (double-width solid line). These scales are larger and faster by orders of magnitude than the Kolmogorov scale. Type C turbulence will correspond to a distributed reaction zone. Comparing this diagram (Fig. 5b) with the classical diagram (Fig. la) reveals that the domain where distributed reaction zones are expected has moved at least an order of magnitude towards more intense fields. Taking into account the important value of heat losses used for this computation (Fig. 4), it is expected that the flamelet domain will be even larger in most practical cases.

618

TURBULENT COMBUSTION [ (a) Spectral diagram I

u'(r) I s L

uto.,Imi.,.

is: rill F = 1/e +1/4. Fig. 5a shows all relevant scales

"mlt i /

in a case without quenching where e+ > 1. The Gibson scale is lower than the Kolmogorov scale and does not constitute a physically meaningful quantity in this case.

102

101

Kolmoaorov

"~

L -- ~

I

-.a

~ | ,o~

ok u "FIllS~eL

I (b) Premlxed turbulent combustion diagram

9 The quenching scales are the cross points between the turbulence line and the quenching limit. ............... 1 They represent the sizes of the smallest and largest vortices which may locally quench the flame front (Fig. 5a). While the cut-off scale is always present, quenching scales do not exist in all turbulent fields. Turbulent field D in Fig. 5a corresponds to the minimum turbulence intensity for quenching. This ] leads to a simple quenching criterion: i

1

(u' /sL) 3

~+ =

J\

~ J

I

l /lF

> 103

or

u'/st.>4nt~

and

Rt = u'l/~, > 250.

10~ I ~,fCutoff

flames

limit

\ Integral length scale / Flame front thickness

I / IF

FIG. 5. Construction of the diagram for turbulent premixed combustion (b) using the spectral diagram

(14)

(15)

Equations (14) and (15) set minimum values for quenching on the reduced dissipation rate and on the integral Reynolds number respectively which may be compared to experimental correlations (see for example Abdel-Gayed and Bradleyl~).

(a). Conclusion

Characteristic Scales in Premixed Turbulent Combustion: Different characteristics scales may be extracted from the spectral diagram (Fig. 5a). 9 The cut-off scale lc~t-off is obtained at the intersection of the turbulence line with the cut-off limit. It corresponds to the smallest scale which may influence the reaction rate in a noticeable way. A best fit derived from our results for the cut-off scale is: /cut-off

IF

= 0.2 + 5 . 5 / ~ +1/6

This paper describes a new method for the construction of turbulent premixed combustion diagrams. It is based on a two-dimensional direct simulation allowing accurate calculations of the effect of vortices on a flame front. Results show that classical diagrams underestimate the resistance of flame fronts to vortices, mainly because they neglect viscous, transient and curvature effects. These effects are especially important when small scales are considered. Direct simulation appears as a powerful tool to improve our understanding of these mechanisms. Incorporating three-dimensional effects to the present work is a necessary and promising work.

(13)

where r = eIF/s~ is the reduced dissipation rate. The cut-off scale is always larger than 0.2/F. It is also larger than the Kolmogorov scale even when the latter is larger than the flame front thickness. Kolmogorov scales do not carry enough energy to affect the flame front in any situation. Other expressions for cut-off scales may be found in the literature: Peters, 3 for example, proposed the Gibson scale lc which is given by: lc = SLa/~ or lG/lF = 1/e +. Note that the Kolmogorov scale

Nomenclature

Roman letters: a B c D h

Sound speed Preexponential factor Constant used in h Diffusion coefficient Radiative heat loss coefficient h = ASc2(sL/

Le

Lewis number

~,)~I~

DIAGRAMS OF PREMIXED COMBUSTION IF Pr Q

laminar unstretched flame thickness Prandtl number Heat of reaction per unit mass of fresh gases =

- AH~r~

r sL

Vortex pair size laminar unstretched flame speed (sLIF/v = 3.7) Sc Schmidt number To, Activation temperature Tl Fresh gas temperature Ta Burnt gas temperature for Lewis = 1 and no heat losses u'(r) Maximum velocity induced by the vortex pair ur Kolmogorov velocity scale tbn Reaction rate Yn Reactant mass fraction Y~ Initial reactant mass fraction 17 Reduced reactant mass fraction = Y R / ~ Greek letters: ot

t3

0 A A

Temperature factor = (T2 - T1)/T1 Reduced activation energy = aT~c/Tz Heat of reaction per unit mass of reactant Dissipation of kinetic turbulent energy Kolmogorov length scale Reduced temperature = (T - TI) /(T2 - TI) Thermal conductivity Reduced preexponential factor = B e x p ( - / 3 / a)

v P T

Kinematic viscosity Density Viscous tensor r 0 = bt(Oui/Oxj + auj/#x, 2 / 3 ~ Out/Oxk) Acknowledgments

The first author would like to thank Dr. Chris Rutland for many helpful discussions. This study was supported by the Center for Turbulence Research.

619

REFERENCES 1. BABRERE, M.: Revue Generale de Thermique, 148, 295-308, (1974). 2. BOBGrlI, R.: in Recent Advances in Aeronautical Science, (C. BRUNO, C. CASCI, Eds), Plenum N.Y., 1984. 3. PETERS, N.: Twenty-First Symposium (International) on Combustion, p. 1231, The Combustion Institute, 1988. 4. BRAY, K. N. C. : Topics in Applied Physics (P. A. LIBBY and F. A. WILLIAMS, Eds) Springer Verlag, 1980. 5. WILLIAMS, F. A.: Combustion Theory, 2nd ed. Benjamin Cummings, Menlo Park, 1985. 6. MARBLE, F. E.: Recent Advances in AeronauIdeal Science (C. BRUNO,C. CAscI, Eds), Plenum N.Y., 1985. 7. CETEGEN, B. AND SImGNANO, W.: 26th AIAA Aerospace Sciences Meeting, AIAA Paper 880730, 1988. 8. GHONIEM, A. AND Grvl, P.: 25th AIAA Aerospace Sciences Meeting, AIAA Paper 87-0225, 1987. 9. AStlUBST, W. T., PETERS, N. AND SMOOKE, M. D.: Comb. Sci. Tech., 53, 339 (1987). 10. RUTLAND, C. J. AND FERZmER, J.: 27th AIAA Aerospace Sciences Meeting, AIAA Paper 890127, 1989. 11. KARAC.OZlAN,A. AND MARBLE, F. E.: Comb. Sci. Tech., 45, 65 (1986). 12. LAVEBDANT, A. AND CANDEL, S.: J. PropuL Power, 5, 134 (1989). 13. POINSOT, T. AND LELE, S.: submitted to J. Comput. Phys., 1990. 14. LELE, S.: 27th AIAA Aerospace Sciences Meeting, AIAA Paper 89-0374, 1989. 15. ABDEL-GAYED,R. G. AND BRADLEY, n.: Comb. Flame, 76, 213 (1989). 16. ASHURST,W., KEBSTEIN, A., KERR, R. AND GIBSON, C.: Phys. Fluids, 30, 8, 2343 (1987).