DRAFT
IECEC 2002 Paper No. xxxxx
DISCRETE DYNAMICAL SYSTEM MODELS OF TURBULENCE–
CHEMICAL KINETICS INTERACTIONS
J. M. McDonough
Departments of Mechanical Engineering and Mathematics
University of Kentucky, Lexington, KY 40506-0108
Phone: (859)257-5164
Fax: (859)257-3304
jmmcd@uky.edu
Sha Zhang
Department of Mechanical Engineering
University of Kentucky, Lexington, KY 40506-0108
Phone: (859)257-2368
szhan1@engr.uky.edul
ABSTRACT
A new approach to subgrid-scale (SGS) modeling
for large-eddy simulation of turbulent non-premixed
combustion is proposed and tested against
experimental data. The model is composed of three
specific factors: an amplitude, an anisotropy
correction and a temporal fluctuation to be evaluated
at each discrete point, during each time step, of
resolved-scale calculations. We employ discrete
dynamical systems (DDSs) for the third factor and in
the present work focus on construction of these for a
reduced kinetic mechanism and compare results with
experimental data from the Technische Universität
Darmstadt H2/N2−air jet diffusion flame H3. The DDS
model is derived as a single-mode Galerkin
approximation (with the mode left arbitrary) of the
governing partial differential equations, but with the
mode number and normalization(s) incorporated into
bifurcation parameters. Such algebraic systems are
capable of producing the full range of temporal
behaviors of the original differential equations (while
being very efficient to evaluate) and, in particular, can
exhibit the chaotic behavior of fractal (strange)
attractors that can be associated with turbulence.
Moreover, they are able to mimic specific reaction
pathways for any given kinetic mechanism on the
subgrid scales. We compare computed results from
the SGS model with the above mentioned data, both
qualitatively (appearance of the time series) and
quantitatively (rms fluctuation levels) and show
reasonable agreement, especially for the former.
INTRODUCTION
It is by now widely accepted that direct numerical
simulations (DNS) of turbulent combustion will be
viable only as a research tool in the immediate future,
and that large-eddy simulation (LES) in some form
presents a feasible direction for near-term
calculations. It is important, however, to recognize
that although much progress in LES has been made
in recent years, it is still far from being a completely
reliable tool in the context of turbulent combustion ―
especially if any but the simplest kinetic mechanisms
are used. Most recent research in LES has focused
on the subgrid-scale (SGS) models, and these are
especially problematic in the context of finite-rate
chemistry.
Various
approaches
have
been
implemented ranging from laminar flamelets, Cook et
al. (1997) through PDF models, e.g., Pope (1985),
and Cook and Riley (1994), and including extension
of scale similarity ideas to reactive scalars as in
Germano et al. (1997). Each has its strengths and
weaknesses as described in some detail by Peters
(2000), and for the sake of brevity we will avoid an
exhaustive review.
It is worth commenting that application to
combustion of notions from the theory of fractals has
declined rather considerably in recent years with a
few notable exceptions (e.g., Giacomazzi et al., 1999;
2000). Indeed, one of the main criticisms of such
theories has been their inability to produce reliable,
efficient predictive tools for turbulence in general, and
turbulent combustion in particular. They have mainly
provided ways to characterize experimental data of
premixed flames as in Gouldin et al. (1988), although
some success has been shown with modeling
premixed turbulent flame speeds in Gouldin (1987). In
the present paper we will revisit use of fractals in the
guise of attractors of discrete dynamical systems
(DDSs) to introduce a different approach to
constructing SGS models for LES of turbulent
combustion. These models will prove to be similar in
some respects to those proposed in Giacomazzi et al.
(1999; 2000), but they are rather different in detail. In
particular, the form of LES considered here differs in
three distinct ways from usual approaches.
First, we choose to filter solutions rather than
equations. Much work still is in progress concerning
proper filtering of the governing equations in cases of
nonuniform gridding and complex geometry, while
filtering the solution reduces the problem to a
relatively simple (but not altogether trivial) one of
“signal processing.” At the same time it significantly
alters (simplifies, we contend) the requirements for
the SGS model. Second, we directly model fluctuating
primitive variables ― velocity components, temperature, species concentrations ― on the subgrid scales,
rather than modeling their statistics (correlations). The
ability to do this has a major impact on treatment of
advective terms in all governing equations, and also
on that of terms containing reaction rates in the
thermal energy and species concentration equations;
the closure problem becomes far more manageable
than is the case for all but the simplest Reynoldsaveraged Navier–Stokes (RANS) methods, and for
typical LES SGS closures. (There are many fewer
primitive variables than correlations, hence, much
less to model.) Finally, we directly employ the
modeled fluctuating quantities to augment the
computed large-scale solution, which by the nature of
LES is under resolved. Recall that in usual LES, and
in RANS approaches, the models are used only to
indirectly change the resolved part of the solution; but
they are never used to construct an approximation to
the complete solution. The modeling approach we are
proposing here does approximate the complete
solution and in that sense moves LES closer to DNS,
even when performed on relatively coarse grids.
We note that this approach was first studied by
McDonough et al. (1995), and presented in detail by
Hylin and McDonough (1999); Sagaut (2001) provides
a useful overview of the method. The basic idea is to
employ a typical LES-like decomposition of solution
variables, say
Q (x,t ) = q (x,t ) + q (x,t )
*
x∈R ,
d
d = 2,3,
(1)
and substitute this into the transport equation(s) for Q:
(q + q ) + ∇ ⋅ F (q + q ) = ∇ ⋅ G (q + q )+ S (q + q )
*
*
*
t
*
q = Aiζ i M i ,
i = 1,2,..., N v ,
).
T
Then Eqs. (2) can be directly
solved for q (followed by filtering to remove aliasing
*
due to under resolution, as needed), and q is added
to the result as suggested by the form of Eq. (1). In
Eqs. (3) the Ais are amplitudes derived from scaling
laws of Kolmogorov (see, e.g., Frisch, 1995), and the
ζis are anisotropy corrections calculated using scale
similarity as in the dynamic SGS models first
introduced by Germano et al. (1991). Details of these
constructions can be found in McDonough et al.
(1995), Hylin and McDonough (1999), Sagaut (2001).
Until only recently little attention was given to the
Mis. These represent the temporal fluctuations of the
SGS quantities, and they were initially constructed as
linear combinations of independent realizations of a
modified logistic map (a chaotic map with fractal
strange attractor described by May, 1976) for each qi*.
This turns out to be inadequate for treatment of
passive scalars, and we have recently introduced an
alternative procedure (described below) for obtaining
the Mis. This has previously been carried out for the 2D Navier–Stokes (N.–S.) equations in McDonough
and Huang (2001), for a Boussinesq approximation to
2-D thermal convection in McDonough and Joyce
(2002) and for a simple (and somewhat unrealistic)
kinetic mechanism for H2–O2 combustion in
McDonough and Huang (2000) and McDonough and
Zhang (2002). The purpose of the present paper is to
explore the behavior of such DDSs with a more
realistic kinetic mechanism for H2–air combustion, and
in particular provide comparisons with extant
experimental results.
The remainder of the paper consists of the
following sections. We begin by presenting governing
equations and assumptions, and then outline the
derivations of the corresponding DDS for a specific
reduced mechanism. We follow this with a section
containing computed results and discussions of these,
and end the paper with some conclusions.
ANALYSIS
In this section we introduce the governing
equations, and from these derive a general discrete
dynamical system that can be used to model any
desired kinetics (including a full mechanism). We then
provide a reduced mechanism for H2–air combustion
and give the specific DDS corresponding to this.
(2)
Here, the subscript t denotes partial differentiation
with respect to time, and ∇· is the divergence
operator. F and G are, respectively, advective and
diffusive fluxes, and S is a general nonlinear source
term.
We model q*, the fluctuating part of Eq. (1), as
*
i
(
q* = q1* , q2* ,..., q*N v
(3)
where Nv is the total number of dependent variables:
Governing Equations
The general equations describing fluid flow, heat
transfer and chemical reactions are well known and
can be found in any standard reference. We present
them here in the following form.
(4a)
ρt + ∇ ⋅ (ρU ) = 0 ,
ρ
DU
= −∇p + ∇ ⋅ (µ∇U ) + ρg ,
Dt
ρc p
(4b)
Ns
Ns
 ρY 
DT
= ∇ ⋅ (λ∇T ) + ∑ c pi DiWi∇ i  ⋅ ∇T − ∑ hiω! i ,
Dt
i =1
i =1
 Wi 
(4c)
IECEC 2002 Paper No. 20163 - 2
D(ρYi )
= ∇ ⋅ (ρDi∇Yi ) + ω! i
Dt
i = 1,⋅ ⋅ ⋅, N s .
(4d)
Here,
ω! i = Wi ∑ (ν i′′, j − ν i′, j )ω j ,
Nr
(5)
j =1
with
Ns
 ρY
ω j = k f , j ∏  l
l =1  Wl
ν l′, j




Ns
 ρY
− kb , j ∏  l
l =1  Wl
ν l′′, j




.
(6)
In these equations U is the velocity vector, (u, v)T in 2D; D/Dt is the substantial derivative; ∇ is the gradient
operator, and g is the body-force acceleration vector;
ρ and p are density and pressure, respectively, and T
denotes temperature. Yi is the mass fraction, and hi is
specific enthalpy of species i. The transport properties
are denoted as µ for (dynamic) viscosity, λ for thermal
conductivity and Di for the binary diffusion coefficient
of species i in the ambient background gas. The c pi
and Wi are, respectively, specific heat and molecular
are
weight of species i, and ν′i,j and ν″i,j
stoichiometric mole numbers for reactants and
products, respectively, corresponding to species i in
reaction j. Ns denotes the number of species, and Nr is
the number of reactions. Finally, kf,j and kb,j are the
forward and backward specific reaction rates for the jth
reaction, typically given in the form of an Arrhenius
expression,
 −E j
k j = B jT nj exp
 RoT

,


(7)
in which Bj is the pre-exponential factor, nj is the
temperature exponent, and Ej is the activation energy;
R0 is the universal gas constant. We would assume
for a complete LES that in addition to these
equations, sufficient initial and boundary data would
be provided to constitute a mathematically well-posed
problem for the chosen spatial domain. But because
the DDSs (and also the corresponding complete SGS
models) are local, this is not a particular concern in
the present study.
The Discrete Dynamical System
The approach we employ for modeling the Mis in
Eq. (3) was first introduced in McDonough and Huang
(2000) and studied in detail for the 2-D N.–S.
equations in McDonough and Huang (2001). It
provides a systematic technique for deriving DDSs
that are directly related to the partial differential
equations (PDEs) they are to model. The premise
employed to start the procedure is that all solution
variables possess (generalized) Fourier series
representations. For convenience we make the
following additional assumptions associated with the
basis sets {ϕk} used to construct the Fourier
representations: i) {ϕk} is complete (in an appropriate
norm); ii) the set is orthonormal; iii) each ϕk has
compact support; iv) the combination of ϕ k s used to
represent the velocity field is divergence free, and v)
the basis set “behaves like” complex exponentials
(which do not exhibit all of the desired properties) with
regard to differentiation.
We next substitute these into the governing
equations (4), and then construct the Galerkin
ordinary differential equations (ODEs) by forming the
inner product of each equation with every basis
function, utilizing the preceding simplifications (see
McDonough and Huang, 2001 for complete details).
This leads to a countable system of ODEs in place of
each of the original PDEs, and from the standpoint of
arithmetic complexity this is no more tractable than
would be any typical form of DNS. In order to produce
a model of minimal arithmetic complexity, we here
consider the extreme shell model (see, e.g., Bohr et
al., 1998 for a thorough treatment of shell models) ―
one consisting of only a single Fourier mode in each
original PDE. Then, as in McDonough and Huang
(2000, 2001), McDonough and Joyce (2002),
McDonough and Zhang (2002), we use very simple
numerical integrators (forward or backward Euler
methods) to perform the temporal discretization. After
possible rescaling (to account for a specific selected
wavenumber and the absence of those modes with
which it would strongly interact ― all included in a
bifurcation parameter) we obtain the following DDS
model of Eqs. (4):
(
)
( )
(1 − b )− γ a( )b( ) + α c( )
a (n +1) = β u a (n ) 1 − a (n ) − γ u a (n )b (n )
b
(n +1)
= β vb
(n )
n
n
n
n
v
T


c (n +1) =  ∑ α Td i d i(n +1) − γ uT a (n +1) − γ vT b (n +1) c (n )

 i =1
(8a)
(8b)
Ns

− ∑ H iω! i  (1 + βT ) + c0
i =1

Ns
(
)
d (n +1) = − βYi + γ uYi a (n +1) + γ vYi b (n +1) d i(n ) + ω! i + d i ,0
(8c)
(8d)
i = 1,2,⋅ ⋅ ⋅, N s ,
with
Ns
Ns
Nr

ν′
ν ′′ 
ω! i = ∑ C f ,ij ∏ d l j ,l − Cb ,ij ∏ d l j ,l  .
j =1 
l =1
l =1

Here, superscripts n denote time step (or map
iteration) index; a, b, c and the di s can be viewed
(heuristically) as Fourier coefficients of the two
velocity components, temperature and the species
concentrations, respectively; the subscripted α s ,
β s , etc., are bifurcation parameters of the DDS, all of
which are related to the various physical bifurcation
parameters. For example, βu and βv are (the same)
functions of the flow Reynolds number; αT is related to
the Rayleigh or Grashof number; α Td i contains
Schmidt and Lewis number information, and the Hi
are associated with specific enthalpies for each
species i; the Cf,ij, Cb,ij can be related to Kolmogorovscale Damköhler numbers. The various γ s
IECEC 2002 Paper No. 20163 - 3
correspond to velocity, temperature and species
concentration gradients (as would be available from
resolved-scale results). For example, γu ∼ uy, γuT ∼ Tx
(subscripts x and y indicate partial differentiation), etc.
The d i , 0 s and c0 are high-pass filtered species
concentrations
and
temperature,
respectively
(obtained from the resolved-scale calculation), about
which the subgrid-scale behavior fluctuates. Finally,
we remark that in the present form of the model we
calculate fluctuating density from the equation of
state, rather than from a discrete form of Eq. (4a).
each product appearing in each elementary reaction.
Each iterated map is of the general form of Eq. (8d)
and the formula for ω! i following it. But now in this
latter expression Nr = 1, and backward reactions are
treated separately (as forward reactions with the
backward reaction rate). Thus, the individual maps
are quite simple. To fix notation we make the
following identifications:
Reduced Mechanism for H2–Air Combustion
In this subsection we present the reduced kinetic
mechanism to be studied and specialize Eqs. (8) to
this case. We employ a nine-step mechanism for the
H2–air reaction consisting of the following:
We now carry out the details for the initiation
reaction Eq. (9a). We first observe that the reaction
(9a) yields two products. Thus, there must be an
iterated map for each of these. The first product in
reaction (9a) is HO2 so the corresponding DDS is
d1 ∼ H2 , d 2 ∼ O2 , d 3 ∼ H2O , d 4 ∼ OH , d 5 ∼ H ,
d 6 ∼ O , d 7 ∼ HO2 , d8 ∼ N2 ,
(
H2 + O2 → HO2 + H
(9a)
O2 + H → OH + O
(9b)
H2 + O → OH + H
(9c)
with
HO2 + H → OH + OH
(9d)
ω! 7 = C f , j ,1d1d 2 ,
OH +M → H +O + M
(9e)
OH + O → O2 + H
(9f)
OH + H + M → H2O + M
(9g)
H2 +OH → H2O + H
(9h)
H2O + H → H2 + OH
(9i)
These reactions have been selected from among
the 17 H2–O2 elementary reactions listed by Peters
(1993). We have chosen these according to the
following constraints. First, we have chosen reaction
(9a) as the initiation step following discussions in
Glassman (1996), and we have attempted to minimize
the number of reactions requiring a third body,
denoted as M. Second, to the extent possible within
the context of constructing a reasonable reaction
mechanism, we have tried to impose the constraint
that overall consumption nearly equals overall
production of species other than reactants (H2 and
O2) and product (H2O). With regard to this, we note
that this does not imply such a detailed balance
actually exists in the context of the resulting DDSs; it
is merely an heuristic leading to a well-defined
process for selecting the reactions, and other
approaches could be used. Next, because we are
considering combustion of H2 with N2 dilution, the
third body M is taken to be N2. Fourth, the order of
evaluation of the equations comprising the DDS
model is consistent with the ordering of the above
elementary reactions. In particular, we are able to at
least in part model a reaction pathway by requiring
any intermediate species to already be present before
it can be used in subsequent elementary reactions.
To construct the DDS corresponding to this
reduced mechanism we derive an iterated map for
(10)
)
d 7(n +1) = − βY7 + γ uY7 a (n +1) + γ vY7 b (n +1) d 7(n ) + ω! 7 + d 7 , 0 ,
C f , 7,1 = ν 7′′,1
(11)
W7
k f ,1 .
W1W2
We note that introduction of the species molecular
weights arises from the form of Eqs. (5,6).
Similarly, the second product of reaction (9a) is
atomic hydrogen corresponding to d5. Thus, the DDS
is
(
)
d5(n +1) = − βY5 + γ uY5 a (n +1) + γ vY5 b (n +1) d 5(n ) + ω! 5 + d 5, 0 ,
(12)
with
ω! 5 = C f ,5,1d1d 2 ,
C f ,5,1 = ν 5′′,1
W5
k f ,1 .
W1W2
One can check that of the nine elementary
reactions, Eqs. (9), seven lead to two products each,
and only two result in a single product. In addition,
two reactions contain the third body M whose
fluctuations also must be modeled. Hence, there must
be a total of 18 iterated maps in the complete DDS for
the chemical reactions. These are all obtained as just
demonstrated for the first reaction, and for the sake of
brevity we will not present details for the remaining
ones.
There are several items to mention with regard to
the overall DDS. First, for reactions involving a third
body M, as already noted we have taken that to
always be N2. Thus, we must use efficiencies
corresponding to this species. But once an efficiency
is found, there is still the question of how to
incorporate it in the present formulation. There appear
to be several possibilities, but the most
straightforward is simply to use it as a multiplier on
the corresponding specific reaction rate. This is the
approach followed for the results reported herein.
The second concern regarding the overall system
is evaluation of the terms in the thermal energy
IECEC 2002 Paper No. 20163 - 4
equation that arise from enthalpies of the chemical
species. We note that Hi appearing in Eq. (8c) is just
a scaled (due to normalization of Galerkin inner
products) instance of the physical enthalpy. Although
we include the possibility to input a scale factor in our
computer codes, we have always computed with that
factor set to unity.
Finally, it is important to consider the number of
iterations undergone by each map of the overall
system. One of the features of the present approach
is its ability to directly account for different reaction
rates of the individual elementary reactions
comprising a kinetic mechanism and relate this to the
time scale of the hydrodynamic turbulence. This is
done by calculating the Kolmogorov-scale Damköhler
number DaK for each elementary reaction using its
specific reaction rate to obtain a chemical time scale,
and employing an input hydrodynamic turbulence time
scale. Then (in the case of DaK >1 which we consider
here) for each velocity time step (map iteration) a
number of iterations proportional to DaK is performed
for each reaction (i.e., for the species maps
corresponding to products of that reaction). In fact, in
a complete LES employing the present approach, this
would be done at each discrete point of the large
scale, and for each resolved time step; moreover, the
local turbulence time scale would be estimated from
the resolved-scale calculations (see Hylin and
McDonough, 1999). Here, we will consider only a
single spatial location.
RESULTS AND DISCUSSION
In this section we present results of running the
DDSs discussed above. We begin by describing the
problem setup; we then present computed results and
compare these with a portion of the experimental coflow data, denoted H3, from Meier et al. (1996). We
will, for the sake of brevity, consider only one location
within the non-premixed 50% N2-diluted H2–air flame
of this experiment. This corresponds to an axial
location of x/D = 20 with D = 8mm being the H2/N2 jet
exit diameter; the measurement location is at a radial
distance of 10.5 mm from the jet centerline.
DDS Model Problem Setup
The full system of Eqs. (8) has been coded in
Fortran 90 and for the present study was specialized
to equations of the form (11) and (12) corresponding
to the nine-step mechanism (9). We have made the
reported runs with constant density (i.e., ρ* = 0) to
more simply elucidate the ability of the DDS to model
effects of turbulence on combustion (and not vice
versa). The model is intended to provide
instantaneous fluctuations about local in time and
space mean temperature and species concentrations;
the reported mean quantities from the H3 data set
have been employed for this, with some modification.
In particular, no minor species concentrations are
reported in Meier et al. (1996), so we have slightly
altered the mean values of those reported data to
allow inclusion of mass fractions of minor species
while maintaining the overall average (in time) sum of
mass fractions equal to unity.
The velocity time scale used in the code to set the
Damköhler number was equated to the data sampling
4
frequency of the H3 experiment (10 kHz), and 10
velocity time steps were computed using Eqs. (8a,b).
During each such time step a number (≥1) of
chemical time steps was computed for each reaction
with the number being calculated during each velocity
time step using local in time temperature results to set
the reaction rate, and thus DaK, for each reaction.
Following calculation of all fluctuating species, the
fluctuating temperature is updated via Eq. (8c).
Clearly, this is not the only possibility for managing
the time stepping procedure.
Computed Results, Comparisons with Data
Figures 1 (experimental, Meier et al., 1996) and 2
(computational, present model) provide a qualitative
comparison of our results with those from a wellestablished data base that was selected as a
“standard flame” at the International Workshop on
Measurements and Computation of Turbulent
Nonpremixed Flames, Naples, 1996. We first observe
that the qualitative appearance is very similar
between corresponding parts of these figures. This
demonstrates the general ability of the DDS model to
provide physically realistic turbulent fluctuations in a
nontrivial situation. We note, however, that with the
exception of temperature, the fluctuation amplitudes
are not extremely accurate. Considerable effort was
made to adjust the bifurcation parameters for
temperature because of its direct influence on
reaction rates, but this was done only to a lesser
extent for the species mass fractions for several
reasons. First, our main objective in the present study
was to demonstrate the model's ability to mimic
physical qualitative behavior; the fluctuation
amplitudes in a complete LES would be set by the
factors Ai in Eq. (3) described above, while the DDS
results would be normalized to unity. Second, the
velocity field for the experiments was not available in
detail beyond the fact that it corresponds to a
4
Reynolds number of 10 ; so a somewhat arbitrary
fluctuating velocity was generated via Eqs. (8a,b).
This enters in the advection terms of all remaining
equations and clearly has a direct effect as shown in
McDonough and Huang (2000), McDonough and
Zhang (2002). Finally, we had no data for the minor
species H, O, OH, and HO2 about which to center the
fluctuations. As noted above, somewhat arbitrary (but
small) values for their mass fractions were employed;
moreover, it can be expected that a different reduced
mechanism might also alter results somewhat. All of
these factors can lead to inaccuracies, but if the DDS
behavior is stable they should have only relatively
minor effects on qualitative computed results.
Comparison of Figs. 1 and 2 suggests this is the
case. In particular, it can be seen that the general
features of the physical turbulent fluctuations are
preserved in each of the cases of model results
IECEC 2002 Paper No. 20163 - 5
although there appears to be a mild discrepancy in
oscillation frequency of H2O concentration.
FIGURE 1: EXPERIMENTAL DATA, TEMPERATURE
AND MASS FRACTIONS; (a) TEMPERATURE, K, (b)
H2, (c) O2, (d) H2O.
are considering only a single location of the
experimental flow field data in this brief
communication we cannot establish whether the
current model exhibits improved behavior in this
respect, but we note that the data location was
chosen a priori, and results presented herein are not
a “best case” (nor are they necessarily a worst case);
they are, we believe, representative. As already
indicated, we have employed the mean quantities
provided with data set H3 as reference levels about
which fluctuations produced by the DDS model would
take place. However, because of the spatial
sparseness of these data it was not possible to
perform the usual high-pass filtering (as would be
done in our complete LES implementations) in order
to directly obtain the di,0s of Eqs. (8d). In fact, as can
be seen from Fig. 2 and Table 1 we did not
accurately match the mean values for H2 and O2
mass fractions. On the other hand, the computed
mean values for temperature and H2O concentration
are reasonably accurate ― probably to within
experimental discrepancies. Table 1 summarizes
these results. As can be seen, the amplitudes for rms
fluctuations produced by the model are generally low
compared with the experimental observations; but as
already emphasized, it is the temporal qualitative
behavior that is more important for the factors Mi of
the overall model.
TABLE 1. MEAN AND RMS VALUES FOR
TEMPERATURE AND MAJOR SPECIES MASS
FRACTIONS
T,K
YH 2
YO2
YH 2 O
YN 2
FIGURE 2: DDS MODEL RESULTS; (a) THROUGH
(d), SAME AS IN FIG. 1.
One of the main requirements of any turbulence
model is to produce reasonably accurate statistics.
Indeed, most models are not actually able to do this
globally throughout a flow field; they may be accurate
for some quantities at some locations, but never
accurate for all quantities at all locations. Because we
Experi- Mean
ment rms
1639.5
0.0054
0.0376 0.1372 0.8198
288.0
0.0061
0.0500 0.0330 0.0220
DDS Mean
Model rms
1548.1
0.0028
0.0213 0.1349 0.8190
252.0
0.0026
0.0221 0.0083 0.0033
One additional output from the model that we have
monitored is the sum of mass fractions. We do not
force this to be unity by calculating all but one species
and then setting it to satisfy the required unity value.
Rather, we directly calculate all species and then
utilize the sum of their mass fractions as a measure of
the success of the model simulations. For the runs
presented here the time average of the sum of the
mass fractions was 0.983, showing less than 2%
discrepancy from the required value. Moreover, the
instantaneous deviation from unity over the run of
4
10 time steps never exceeded ±6%.
FURTHER DISCUSSION
The results we have presented and the manner in
which they have been calculated are somewhat
different from those of usual LES SGS modeling
techniques. There are some current models that are
at least somewhat similar, and these should be briefly
IECEC 2002 Paper No. 20163 - 6
compared. The two approaches that seem closest to
the present one are those of Kerstein and various coworkers (see e.g., Kerstein, 1992 and Echekki et al.,
2001), viz., the linear eddy models (LEMs) and the
more recent “one-dimensional turbulence” (ODT)
models, and those of Giacomazzi et al. (1999, 2000).
But it should be noted that the former of these is
mainly phenomenological and ultimately driven by a
random number generator; moreover, physical
turbulence is not one dimensional. The latter
approach, while incorporating some of the ideas we
employ to evaluate Ai and ζi (not carried out in the
present study) of Eq. (3), nevertheless utilizes
averaged or filtered equations of motion, and an eddy
viscosity.
The method we have presented, although two
dimensional as given herein, extends to three
dimensions in a straightforward way. It employs
approximations of the physical equations themselves
on the subgrid scales to produce turbulent fluctuations
that are completely deterministic, just as would be the
case in a DNS. Moreover, the equations are neither
averaged nor filtered; problems associated with
closure of nonlinear terms are thus greatly reduced.
CONCLUSIONS
In this paper we have presented a discrete
dynamical system model of a nine-step reduced
mechanism for H2–air combustion derived directly
from the governing equations which we propose for
use as the temporal fluctuations in subgrid-scale
models for large-eddy simulation of the type studied in
McDonough et al. (1995), Hylin and McDonough
(1999), Sagaut (2001). We have demonstrated that
the fluctuations produced by this DDS are very similar
to those observed in well-known experimental data
(Meier, 1996) in a qualitative sense, but that
turbulence
statistics,
especially
second-order
statistics, are not very accurate. We have, however,
noted reasons for which this might be expected; viz.,
the conditions under which the DDS was run for the
present study were lacking information that would
generally be available in a complete LES.
It is thus difficult to offer firm conclusions beyond
noting that the qualitative behavior of the DDS is
strikingly similar to that of all aspects of the
experimental data for an arbitrarily chosen
measurement location. We believe this is ample
justification for more thorough studies of this
formalism for constructing SGS models for LES.
Moreover, as is the case for the fractal modeling
procedure reported in Giacomazzi et al. (1999, 2000),
it would in principle be possible to incorporate the
present approach into RANS modeling methods,
although we have not as yet attempted this.
ACKNOWLEDGMENTS
The authors gratefully acknowledge the support for
these studies provided in part by AFOSR Grant
#F49620-00-1-0258 and NASA/EPSCoR Grant
#WKU 522635-00-10.
REFERENCES
Bohr, T., Jensen, M. H., Paladin, G. and Vulpiani,
A., Dynamical Systems Approach to Turbulence,
Cambridge University Press, Cambridge, 1998, pp.
78–88.
Cook, A. W., Riley, J. J. and Kosály, G., Combust.
Flame 109:332–341 (1997).
Cook, A. W. and Riley, J. J., Phys. Fluids 6:2868–
2870 (1994).
Echekki, T., Kerstein, A. R. and Dreeben, T. D.,
Combustion and Flame 125:1083–1105 (2001).
Frisch, U., TURBULENCE The Legacy of A. N.
Kolmogorov,
Cambridge
University
Press,
Cambridge, 1995, pp. 89–92.
Germano, M., Maffio, A., Sello, S. and Mariotti, G.,
in Direct and Large Eddy Simulation II (Collet et al.,
eds), Kluwer Academic Publishers, Amsterdam, 1997,
pp. 291–300.
Germano, M., Piomelli, U., Moin, P. and Cabot, W.
H., Phys. Fluids A3:1760–1765 (1991).
Giacomazzi, E., Bruno, C. and Favini, B.,
Combust. Theory Modelling 3:637–655 (1999).
Giacomazzi, E., Bruno, C. and Favini, B.,
Combust. Theory Modelling 4:391–412 (2000).
Glassman, I., Combustion, Academic Press, San
Diego, 1996, pp. 65–72.
Gouldin, F. C., Hilton, S. M. and Lamb, T., Proc.
Combust. Inst. 22:541–550 (1988).
Gouldin, F. C., Combust. Flame 68:249–266
(1987).
Hylin, E. C. and McDonough, J. M., Int. J. Fluid
Mech. Res. 26:539–567 (1999).
Kerstein, A. R., Comb. Sci. and Tech. 81:57–96
(1992).
May, R. M., Nature 261:459–467 (1976).
Meier, W., Prucker, S., Cao, M.-H. and Stricker,
W., Combust. Sci. Technol. 118:293, (1996).
McDonough, J. M., Yang, Y. and Hylin, E. C., in
Proc. First Asian Comput. Fluid Dyn. Conf. (Hui et al.,
eds), Hong Kong University of Science and
Technology: Hong Kong, 1995, pp. 747–752.
McDonough, J. M. and Huang, M. T., submitted to
Int. J. Numer. Meth. Fluids (2001).
McDonough, J. M. and Huang, M. T., paper
ISSM3-E8 in Proceedings of Third Int. Symp. on
Scale Modeling, Nagoya, Japan, Sept. 10–13, 2000.
McDonough, J. M. and Joyce, D. L., to be
th
presented at 8 AIAA Thermophysics and Heat
Transfer Conference, St. Louis, June 24–27, 2002.
McDonough, J. M. and Zhang, Sha, to be
nd
presented at 32 AIAA Fluid Dynamics Conference,
St. Louis, June 24–27, 2002.
Peters, N., in Reduced Kinetic Mechanisms for
Applications in Combustion Systems (Peters and
Rogg, eds), Lecture Notes in Physics m15, SpringerVerlag, Berlin, 1993, pp. 8–12.
Peters, N., Turbulent Combustion, Cambridge
University Press, Cambridge, 2000, pp. 57–65.
Pope, S. B., Prog. Energy Combust. Sci. 11:119–
192 (1985).
Sagaut, P., Large Eddy Simulation for
Incompressible Flows, Springer, Berlin, 2001, pp.
191–194.
IECEC 2002 Paper No. 20163 - 7