Numerical Simulation of Combustion Phenomena: Proceedings of the Symposium Held at INRIA Sophia-Antipolis, France May 21–24, 1985

402 Pages·1985·12.939 MB·English-French

by  K. N. C. Bray (auth.), Roland Glowinski, Bernard Larrouturou, Roger Temam (eds.)

INVITED LECTURES

LAMINAR FLAMELET MODELLING OF TURBULENT COMBUSTION
K. N. C. Bray
University Engineering Department,
Trumpington Street, Cambridge,
CB2 IPZ, England.
Abstract
The incorporation of a laminar flamelet description into the "presumed
pdf" approach to turbulent combustion modelling is reviewed. A sig-
nificant advantage of this approach is that it permits the calculation
of a library of strained laminar flame solutions, with realistic
chemical kinetics included, separately from subsequent turbulent flow-
field predictions. A unity of approach is identified between premixed
and nonpremixed combustion cases. Some outstanding problems are dis-
cussed which must be solved before such methods can be fully developed.
Introduction
There is a need for improved theoretical models of turbulent combustion
processes to assist designers of practical systems to develop more fuel
efficient, clean burning, and cost effective plant. Flow fields, even
in regions of combustors and furnaces where burning does not take place,
are often sufficiently complex to demand the use of a turbulence model.
Therefore the practical problem to be addressed by combustion modellers
is the adaption and extension of such models to include effects of
combustion. Progress towards this end is summarised in recent review
articles (],2)
The well known "closure problem" for the time or ensemble averaged
e~uations of turbulent flow is greatly accentuated in the presence of
combustion. Particular difficulty is found in the description of mean
chemical reaction rates while the large density changes generated by
COmbustion can lead to unexpected flow phenomena. Because of these and
other interactions between combustion and turbulence it is not obvious
that models of non-reacting turbulent flow can be adapted simply by
addition of appropriate mean species e~uations and chemical source
terms. Turbulent combustion models inevitably contain much empiricism
and it is only through comparison with relevant experiments that valida-
£ion can be obtained. Unfortunately suitable experiments are generally
difficult to perform.
Laminar flamelet modelling assumes that combustion and heat release in
a turbulent flame can be represented through the effects of one or more
moving laminar flames which are embedded in the turbulent flow. Klimov
)3( and Williams )4( have shown that laminar flames can survive in a
turbulent flow only if the laminar flame thickness is small in compari-
son with length scales of the turbulent velocity field. Laminar diffu-
sion flamelet models for nonpremixed turbulent combustion have recently
been reviewed by Peters )5( Similar methods are applicable to pre-
mixed flames.
In this paper we shall review some work in which laminar flamelets are
incorporated into both premixed and nonpremixed turbulent flame calcu-
lations by specifying the form of an appropriate probability density
function (pdf). Both effects of combustion on the turbulence and
effects of turbulence on mean chemical reaction rates are illustrated.
A significant computational advantage of this approach is that calcula-
tion of the structure and properties of the laminar flames, involving
detailed descriptions of complex chemical kinetic and molecular diffu-
sion mechanisms, is uncoupled from the turbulent flow field calculation
and can therefore be carried out separately. We can envisage the
building up of a "library" of laminar flame calculations covering a
range of initial conditions and flow field geometries corresponding to
different rates of strain imposed on the laminar flame. Chemical reac-
tion effects in a turbulent flow are to be modelled )6( as an ensemble
average of appropriate laminar flame solutions from this "library".
Our purpose in this paper is to explore the development of a coherent
strategy for modelling premixed and nonpremixed turbulent flames and to
identify some problems which remain to be solved. We shall assume that
an adeguate model of turbulent flow exists and concentrate on additions
and modifications to such a model to include combustion.
Premixed Flames
A formulation of the problem of premixed turbulent combustion which
has been used in several later studies is that of Bray and Moss )7(
As proposed, this analysis makes a number of simplifying assumptions,
namely: a single-step global reaction rate expression, constant mixture
specific heat and molecular weight, independent of the progress of the
combustion reaction, mixture Prandtl and Schmidt numbers of unity,
negligible pressure fluctuations, and adiabatic low Mach number flow.
The instantaneous thermochemical state is then characterised by a
progress variable c which can be defined either as a normalised
composition variable or as a normalised temperature
c = (T - T )r / (Tp - T )r )i(
where subscripts r and p refer to unburned reactant and fully
burned product, respectively , and by a constant heat release parameter
T = (Tp - Tr) r / T )2(
The mixture density is given by
p = pr/(l + T C) )3(
This instantaneous thermochemical state is averaged in a turbulent
flow through introduction of a pdf
P(c ; x) = e(x) 6(c) + 8(x) 6(1 - c)
+ [H(c) - H(c - )I ] 7(x) f(c ; x) )4(
where ~(c) and H(c) are the Dirac delta and Heaviside functions, res-
pectively,
Ii f(c;x)=l
and e(x) + S(x) + y(x) = i.
The delta functions at c = O and c = 1 may be identified (see Fig. )i
with packets of unburned and fully burned mixture, respectively, and
f(c ; x) with product distributions associated with burning. The
coefficients ~(x), 8(x) and ¥(x) describe the partitioning among these
three modes.
The theory was developed (7,8) to describe turbulent combustion situa-
tions characterised by a wide range of values of a Damk~hler number,
defined as the ratio of a turbulence time scale to a time characteris-
ing combustion chemistry. However, most later applications have con-
centrated on the most common case in practical systems namely one where
this Damk~hler number is large. The turbulent flame then consists
predominantly of regions of unburned mixture c( = O) and fully burned
product c( = )i separated by thin interfaces within which the combus-
tion occurs. A point measurement of temperature as a function of time
is a square wave with sharp transitions between two fixed levels at
r T and Tp. In these circumstances the pdf of Fig. 1 is bimodal with
large entries at c = O and c = 1 and a small probability for the inter-
mediate burning mode, i.e. y(x) << i. The occurrence of this thin
flame burning mode, which encompasses both wrinkled laminar flames and
the formation of isolated packets of reactants and/or products, is
amply demonstrated experimentally.
Although the thermochemical assumptions of the Bray-Moss formulation
)7(
are not excessively restrictive, they can in fact be considerably
relaxed for the thin flame burning regime with y << i. The assumed
single-step global reaction rate expression is replaced by the laminar
flamelet analysis described below which can include effects of realistic
chemical kinetic schemes as well as accurate values of Prandtl and
Schmidt numbers. Nor is it necessary to assume constant specific heats
and molecular weight for the mixture. Since the burning mode probabil-
ity is small the specific volume of intermediate states can be approxi-
mated as
v = (i - c) r v + c Vp
and the equation of state then becomes
p = pr/(l + ~ c) )5(
where ~ is a constant given by
Wr _
= T( + )i ~ 1 )6(
L-
Equation )5( reduces to Equation )3( if the reactant and product
molecular weights, r W and Wp respectively, are equal. However, so long
as y << i, Equations (I), (2), )5( and )6( can be applied also to cases
where specific heats and molecular weights are variable. The heat
release parameter T must then be obtained, via Equation (2), from an
adiabatic flame temperature Tp which has been calculated using variable
specific heats. Finally, situations have been analysed )9( where the
assumption of adiabatic flow can no longer be made.
The turbulent combustion model described below employs the simple state
equations of Equations )i( - )3( with y << i. It is based upon the
Fa~rre averaged balance equations of turbulent chemically reacting flow
)I( with turbulent transport terms described by a second order closure
scheme in which Reynolds stress components and turbulent mass transport
fluxes are described by additional balance equations. Gradient trans-
port assumptions for turbulent fluxes are avoided. A turbulence
Reynolds number is assumed to be large enough so that molecular trans-
port terms can be neglected.
Two aspects of the model may be identified in this thin flame regime
with 7 << i. The first is the fluid mechanics of a two fluid flow
consisting of packets or regions of gas with two different densities:
for this purpose the interfaces may be replaced by discontinuities.
The second aspect, which is concerned with mean reaction rates, depends
upon the detailed structure of these interfaces.
To understand the fluid mechanical problem it is helpful to introduce
a joint pdf of velocity and thermodynamic state. For y << 1 the pdf
P(u a, c ; x) will have the form illustrated in Fig. 2, where it is
seen to consist mainly of entries at c = 0 (representing unburned gas
packets) and at c = 1 (representing burned gas packets). The quantities
P(u , O ; x) and P(u , 1 ; x) describe conditional velocity distribu-
tions in these reactant and product packets, respectively, at location
within the turbulent flame. They may be used to define conditional
mean velocities and turbulence intensities:
5r(X) _ = du u P (u, 0 ; _ ~) (7)
rm2'u _)x( = du e (u- ~ )2 r P(u , O ; )x_ )8(
for reactants, with similar definitions of the analogous quantities
uep and u 2, for products The unconditional Favre mean values are
related to-the conditional means through
u = i( - ~)Umr + c Uap + O(¥) )9(
u "2~ a = i( - ~) u ra2' + ~ p~2'u + ~(i - ~)(u p - u )2 + r O(¥)
(10)
where ~ = pc/p is the FaVre mean of the progress variable c. Mean
turbulent fluxes may be expressed similarly, for example
0u"c" a = ~ c - ~ 1( - ~) (~ap - u ) + r 0(¥) (ll)
and can thus be interpreted in terms of the conditional statistics.
If a simple one dimensional configuration is assumed for the turbulent
flame, with averaged flow variables dependent only on a single spatial
coordinate, x, a transformation from x to ~ is found to lead to elimina-
tion of all terms dependent on mean reaction rates. This result is
independent of modellinq details (I0,11,12) A consequence is that the
two fluid flow aspects of the problem can be studied in isolation from
consideration of mean reaction rates. The one dimensional geometry is
however an idealisation of real turbulent flame configurations. It may
be regarded, Fig. 3, as an approximate description of an unconfined
oblique turbulent flame viewed from a coordinate system moving with
velocity 9. See Cheng and Shepherd (13) for an experimental interpreta-
tion of the one dimensional approximation.
Studies of turbulent transport (]4) and turbulent production (11) in a
one dimensional configuration are based on a second order closure model
of the turbulent transport equations. Unknown covariances are related
to conditional velocities throuah eauations similar to Esuation (ii)
while dissipation terms are modelled (15) on the assumption that the
largest gradients occur in laminar flamelets. Pressure fluctuation
terms are neglected. To obtain a closed set of model equations, with-
out recourse to gradient transport assumptions, one additional empiri-
cal relationship must be assumed between conditional velocities and
intensities. This is taken (l]) to be
u 2' - u 2' 3 = K (Up - ~ 2) (12)
p r r
where K3 is supposed to be a constant. However, subsequent experiments
(]3) have shown it to take a wide range of values. We shall return to
this problem later.
Results of these calculations show (l]) that at realistic values of the
heat release parameter T of Equation (2), the conditional mean product
velocity, Up, is greater than the corresponding mean velocity for
reactant packets, ~r' everywhere except at the leading edge of the
turbulent flame where c = O. Behind this small leading edge region
Equation (Ii) then demonstrates that the turbulent mass flux p U"c" is
positive corresponding to transport in the opposite direction to that
predicted by the usual gradient transport assumptions. This counter-
gradient diffusion is also observed experimentally (16,17) . The mechan-
ism leading to relative motion between burned and unburned gas packets
is found (14) to involve the action of the self induced mean pressure
gradient dp/dx acting preferentially on low density unburned gas
packets. A result of the relative motion is the generation of substan-
tial additional turbulence within the flame . (II)
It is noteworthy that the model equations (II,14) do not predict a tur-
bulent burning velocity and this velocity must be provided as an input
to the one dimensional flame predictions described above. However, the
averaged equations describing the analogous one dimensional propagating
flame problem are found to be hyperbolic and the burning velocity does
emerge (18) as one of the characteristic directions for this set of
equations.
An analysis by Champion and Libby (19) of premixed turbulent combustion
in a boundary layer uses a similar theoretical model to that outlined
above. In this case the mean pressure gradient in the direction per-
pendicular to the boundary layer is predicted (19) to cause a substan-
tial reduction in the normal gradient transport flux but not to
reverse it. Generalised model equations have recently been proposed
(12) for two and three dimensional flows.
We now turn to the mean reaction rate aspect of the problem. The
instantaneous balance equation )8( for progress variable c contains a
term ~(x, t) representing the rate of production of product per unit
volume and time due to chemical reactions. The Favre averaged version
of this eguation, which is a balance equation for ~(~), therefore
includes the mean chemical source term ~(x), which must be modelled.
Terms similar to ~ occurring in other balance equations must also be
represented.
In the thin flame case y << 1 it may be shown )8( that ~(x) is propor-
tional to the mean scalar dissipation rate ~c(~). Since this latter
quantity is the rate at which fluctuations in c are reduced, as a
result of molecular transport processes, we have an example of mixing
limited reaction as might be anticipated at large Damk~hler numbers.
Unfortunately, however ~c is as difficult to model as is ~. A con-
ventional turbulence model of ~c (I0) and a heuristic description of
based (]5) on thin laminar flamelets both give ~ expressions similar to
Spalding's intuitive eddy break up model (~0). Neither of these
approaches provides guidance as to the variation of the scalar length
or time scales which the models contain. Such scales will generally
not be the same as those of the velocity field (21)
Recent work (22) attempts to provide a rational basis for the prediction
of ~. If y << 1 the progress variable c{~, )t is a square wave func-
tion of time, with sharp transitions occurring between zero and unity
whenever a flame interface crosses location x. The corresponding time
variation of e(x, )t is a series of narrow spikes each of which corres-
ponds to a flame crossing. This leads us to write
~(£) = ~F(X ) ~(~) (13)
01
where ~(~) is the mean frequency of flame crossings and ~F(~) is the
mean conversion of reactants to products per crossing.
Progress is reported (22) in predicting the crossing frequency ~. A
balance eauation is formulated for a two point two time covariance
P]I(~I, ~2, )~ = Cl (~l, )t C2(X2 , t + )~ (14)
where cl lx( , )t refers to location xl while c2(x2 , t + )~ concerns
location ~2 and a time advance .T The quantity P]] thus contains
information about both length and time scales of the scalar field.
Closure of the balance equation for Pl] is greatly facilitated when
y << 1 because cl and 2 c are then 0-i square wave functions. The
analysis is applied (22) to prediction of scalar time scales at a
single fixed location x. It is assumed that the passage times, within
which individual unburned and burned gas packets move past station x
t( r and tp, respectively, as illustrated in Fig. )4 are exponentially
distributed. This is equivalent to postulating that the duration of
each event is statistically independent of previous events. The
modelled balance equation is solved analytically leading to the follow-
ing results.
The mean flame crossing frequency is found to be
= 2 c(l - ~) / T (15)
where ~ is the Reynolds average of progress variable c and T is the
integral time scale obtained from PI]- Mean values of the passage
times for unburned and burned gas packets, ~r and tp, respectively,
are predicted to be
r = i~_ ; ~j_ = 1 (16)
where [m is the constant value of both r Z and [p when c = 9. Finally
} is found to be qiven by
} = ~/2 )71(
and is therefore a constant. The time scale ~m which is not at present
predicted is the goal of future analysis. It is determined experi-
mentally in one flame configuration (23)
Experimental data from several sources (22,23,2~) lends strong support
to Equations (16) while power spectral densities deduced from Pll are
also similar to results from these experiments. Finally the assumed