J. M. McDonough
Departments of Mechanical Engineering and Mathematics
University of Kentucky, Lexington, KY 40506-0503
E-Mail: jmmcd@uky.edu
ABSTRACT
The basic mathematical nature of low-dimensional models hints at existence of interesting sequences of bifurcations that could result in drastic changes in model behavior as critical points in parameter space are crossed. In the present study an analysis technique based on construction of bifurcation diagrams is described and employed to analyze a specific class of such models for H
2
–O
2 combustion. These models consist of discrete dynamical systems
(DDSs) derived directly from the governing equations of fluid motion, thermal energy and species concentrations and provide good representations of physical behavior on small spatial scales, and for short times; they are employed as part of large-eddy simulation (LES) subgrid-scale (SGS) models of turbulent combustion in a framework not too different from that of linear eddy models and one-dimensional turbulence models. Bifurcation diagrams displaying changes in generic temporal behavior of temperature, species concentrations, various statistical correlations and scalar dissipation will be presented and cast in a physical context. These show the wide range of bifurcation sequences exhibited by the DDS, and at the same time the significant range of bifurcation parameter values over which nondivergent (and hence, useful for modeling) behavior can be achieved. It is concluded that a SGS modeling approach based on low-dimensional discrete dynamical systems holds significant potential in LES of combustion.
INTRODUCTION
Low-dimensional models of combusting systems have received considerable attention in recent years motivated by efficiency of calculation arising from their use and thus the improved ability to perform more realistic simulations of engineering importance, and many different approaches have been employed for their construction (see, e.g.
, Nafe and Maas [1], Blasco et al.
[2], Bhattacharjee et al.
[3]). Despite their low dimsionality, such systems are typically nonlinear, and this implies a potential for bifurcations as physical parameters are varied. Indeed, bifurcations are observed in physical combusting systems; so if a low-dimensional model is unable to replicate these, then it is important to employ such a model only far away from any bifurcation points. In light of the importance of this straightforward idea, it is somewhat surprising that little has previously been done along these lines, although it is noted in [3] that reduced mechanisms are generally valid only locally in space and time. In the present work we report results from such a study performed in the context of a specific low-dimensional model of H
2
–O
2 combustion consisting of a discrete dynamical system (DDS) derived directly from the physical governing equations and first introduced by McDonough and Zhang [4]. This is a particularly appropriate model for the present study because as will be evident below, it produces many different bifurcation sequences and thus poses significant difficulties in locating these and comparing the corresponding behaviors with actual physics. We will conduct this study by constructing bifurcation diagrams (also termed “regime maps” herein) similar to those introduced for a simpler
H
2
–O
2 mechanism by Holloway et al.
[5]. But here more emphasis will be placed on the nature of the bifurcation sequences and the relationship of these to physical parameters of the system.
The remainder of this paper consists of the following parts. In the next section we briefly recall the governing equations and the low-dimensional system derived from them for the specific nine-step H
2
–O
2 mechanism being considered. We then present a section of results focusing on bifurcation diagrams and statistical properties of the low-dimensional system. In a final section we summarize the work and draw some conclusions.
ANALYSIS
In this section we note the governing equations being considered, outline derivation of the discrete dynamical system that comprises the low-dimensional model that we study, and then introduce the form of this model corresponding to a particular reduced mechanism for H
2
–O
2 combustion. We then briefly describe the method employed to identify sequences of generic behaviors of this low-dimensional model.
The governing equations are, of course, a system of partial differential equations (PDEs) consisting of the Navier–
Stokes (N.–S.) equations of fluid motion, the thermal energy equation and equations for species transport. These are well known, so for the sake of brevity we refer the reader to any of a large number of references ( e.g.
, Libby and
Williams [6] and others) for details.
Derivation of the DDS has also been described in several earlier works (Refs. [4, 5] and McDonough and Huang [7]), so we will only outline the steps here and provide the final form of the equations. This begins with prescribing Fourier

representations for all dependent variables followed by substitution of these into the governing equations. Inner products of these equations with the basis functions of the Fourier representations are constructed, and the resulting
Galerkin ordinary differential equations (ODEs) are decimated to a single wavenumber. These ODEs are then formally integrated with Euler numerical methods, and in the case of the momentum equations two transformations
(one first introduced by May [8], and the other by McDonough and Huang [7, 9]) are performed to cast these into a form resembling coupled logistic maps. The result of all this, in the case of two space dimensions, is conversion of the original N.–S. PDE system to the following system of algebraic equations: a ( n +1) = β u a ( n ) 1 − a ( n ) − γ u a ( n ) b ( n ) , b ( n +1) = β v b ( n ) 1 − b ( n ) − γ v a ( n ) b ( n ) + α
T c ( n ) , c
( n +1)
=
" N s
X
α
T d i d
( n +1) i i =1
− γ uT a
( n +1)
− γ vT b
( n +1)
!
c
( n )
−
N s
X
H i
˙ i
#
/ (1 + β
T
) + c
0
, i =1 d
( n +1) i
= − β
Y i
+ γ uY i a
( n +1)
+ γ vY i b
( n +1) d
( n ) i
+ ˙ i
+ d i, 0
, i = 1 , 2 , . . . , N s
,
(1a)
(1b)
(1c)
(1d) with
ω i
=
N r
X
"
C f,ij j =1
N s
Y d
`
ν 0 j,`
` =1
− C b,ij
N s
Y d
`
ν 00 j,`
#
.
` =1
(2)
Here, superscripts ( n ) denote time step (or map iteration) index; a , b , c and the d i 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 of the scaled governing equations. For example,
Reynolds number (see [9]); α
T
β u is related to the Rayleigh or Grashof number; and β v
α
T d i are (the same) functions of the flow contains Schmidt and Lewis number information, and the H i are associated with specific enthalpies for each species i . The C f,ij
, C b,ij can be related to rates. The various γ s correspond to velocity, temperature and species concentration gradients (as would be available from resolved-scale results in a complete LES). For example, γ u
∼ u y
, γ uT
∼ T x
(subscripts x and y indicate partial differentiation), etc . The d i, 0 s and c
0 are high-pass filtered species concentrations and temperature, respectively (also obtained from the resolved-scale calculation), about which the subgrid-scale behavior fluctuates. In Eqs. (1), (2),
N r and N s are, respectively, number of reactions and number of species, and ν 0 j,` and ν 00 j,` are stoichiometric mole numbers for reactants and products corresponding to species ` in reaction j . Finally, we remark that in the present form of the model we assume constant pressure and calculate fluctuating density from the equation of state, rather than from a discrete form of the continuity equation.
We now present the reduced kinetic mechanism to be studied and specialize Eqs. (1), (2) to this case. We employ a nine-step mechanism for the H
2
–O
2 reaction consisting of the following:
H
2
+ O
2
→ HO
2
+ H
O
2
+ H → OH + O
H
2
+ O → OH + H
HO
2
+ H → OH + OH
OH + M → H + O + M
OH + O → O
2
+ H
OH + H + M → H
2
O + M
H
2
+ OH → H
2
O + H
H
2
O + H → H
2
+ OH
These reactions have been taken from among the 17 elementary H
2
–O
2 reactions listed by Peters and Rogg [10].
We have selected these according to the following constraints. First, we have chosen reaction (3a) as the initiation step following discussions in Glassman [11], 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 (H
2 and O
2
) and product (H
2
O). With regard to this, we note that this does not imply
(3a)
(3b)
(3c)
(3d)
(3e)
(3f)
(3g)
(3h)
(3i)

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; other approaches could be used. Next, because we have typically considered combustion of H
2 with N
2 dilution (and/or in air), the third body M is taken to be N
2
. 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 they can be used in subsequent elementary reactions.
To construct the DDS corresponding to this reduced mechanism we derive an iterated map for each product appearing in each elementary reaction. Each iterated map is of the general form of Eq. (1d) and the formula (2) for
˙ i following it. But now in this latter expression N r
= 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: d
1
∼ H
2
, d
2
∼ O
2
, d
3
∼ H
2
O , d
4
∼ OH , d
5
∼ H , d
6
∼ O , d
7
∼ HO
2
, d
8
∼ N
2
.
d
( n +1)
5
= − β
Y
5
+ γ uY
5 a
( n +1)
+ γ vY
5 b
( n +1) d
( n )
5
+ ˙
5
+ d
5 , 0
,
(4)
We now carry out the details for the initiation reaction Eq. (3a). We first observe that reaction (3a) yields two products; there must be an iterated map for each of these. The first product in reaction (3a) is HO
2
, so the corresponding DDS is d
( n +1)
7
= − β
Y
7
+ γ uY
7 a
( n +1)
+ γ vY
7 b
( n +1) d
( n )
7
+ ˙
7
+ d
7 , 0
, (5) with
7
= C f, 7 , 1 d
1 d
2
, and C f, 7 , 1
= ν 00
7 , 1
W
7
W
1
W
2 k f, 1
.
Here, the W i s are species molecular weights, and k f, 1 is the specific (forward) reaction rate for reaction (3a). Similarly, the second product of reaction (3a) is atomic hydrogen corresponding to d
5
. Thus, the DDS is
(6) with
5
= C f, 5 , 1 d
1 d
2
, and C f, 5 , 1
= ν 00
5 , 1
W
5
W
1
W
2 k f, 1
.
Of the nine elementary reactions listed above, seven lead to two products each, and only two result in a single product. In addition, two reactions contain the third body M whose concentration we assume to remain unchanged, and whose presence is taken into account via a reaction efficiency. Hence, there must be a total of 16 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.
It is important to consider the number of iterations undergone by each map comprising this overall low-dimensional system during one velocity time step. 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 each of these to the time scale of the hydrodynamic turbulence. This is done by calculating the Kolmogorov-scale
Damk¨ Da
K for each elementary reaction using its specific reaction rate to obtain a chemical time scale, and employing an input hydrodynamic turbulence time scale (which in practice would be automatically estimated from local resolved-scale flow field results). Then (in the case of Da
K
> 1 which we consider here) for each velocity time step (map iteration) a number of iterations proportional to Da
K 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-scale time step.
The system of 19 DDSs consisting of Eqs. (1a)–(1c) and the additional 16 equations of the form (5), (6) produces fluctuating, turbulent-like time series (see Ref. [4]). These must be analyzed as various bifurcation parameters are changed in order to characterize generic behaviors of the system. There are, of course, many possibilities for doing this. Because it is expected that at least some combinations of bifurcation parameters might lead to chaotic behaviors, one might consider employing a fractal dimension ( e.g.
, correlation dimension) or Lyapunov exponents to establish a characterization. But even in the context of parallel supercomputers, run times needed for these methods are far too great, and neither approach is completely automatic—both serious disadvantages when at least ∼ O (10 6 ) time series must be analyzed. An alternative is to employ power spectral analysis. As has been described in detail in
Ref. [9], the DDS of the form Eqs. (1a), (1b), termed the “poor man’s Navier–Stokes (PMNS) equation,” exhibits
14 distinct, well-defined characteristic behaviors that can be uniquely identified using their power spectral density
(PSD), so what is required is an automatic procedure for analyzing PSDs to establish a specific characterization for each time series corresponding to a unique combination of bifurcation parameters. It turns out that this is relatively

easy, and very efficient—it represents a quite simple image-processing (identification) problem, and this has been employed in numerous studies beginning with McDonough et al.
[12] to construct regime maps as will be presented below.
RESULTS and DISCUSSION
In this section we first describe the run conditions under which all of the bifurcation diagrams have been constructed.
We then present and discuss several different ones of these (there exist essentially an uncountable infinity) that are particularly illustrative. Following this we provide additional results associated with scalar dissipation and skewness of passive scalars to suggest the general capabilities of the DDS.
The results presented here have been obtained in a uniform way. All bifurcation parameters are held fixed (in this case at values that lead to a good qualitative match of data of Meier et al.
[13]—see Ref. [4]) except for two selected ones that in each separate case are simultaneously varied over intervals indicated in the regime maps that follow.
Each such interval is subdivided by 501 uniformly-spaced “grid” points. At each such point the DDS is iterated for
50,000 steps. The last 8192 of these are used to construct the PSD of a specified variable (see figure captions) using a standard radix-2 fast Fourier transform, and these digital results are then automatically analyzed to determine the type of time series according to 14 possible types listed below in the upper-left part of Fig. 1. These results are then plotted as in the remaining parts of this figure. The table to the far right-hand side provides values for all bifurcation parameters when they are not being varied. Finally, initial values of components of the DDS are: a = 0 .
51, b = 0 .
32, c = 0 .
05 and all d i s set to 6% of the equilibrium concentration computed with STANJAN at a temperature of 1205
◦ C corresponding to a location close to the axis of the H3 turbulent jet flame of Ref. [13].
4 4
(a)
3 3
2 2
1 1
0 0
−
1
-1
−
1 divergent broadband w/o fundamental broadband w/ different fundamental broadband w/ fundamental noisy quasiperiodic w/o fundamental noisy quasiperiodic w/ fundamental noisy phase lock noisy subharmonic quasiperiodic phase lock, N-periodic (N ≠ 2 ) subharmonic periodic w/ different fundamental periodic steady
2 2
(b)
1.5
1 1
0.5
0 0
-0.5
−
1
-1
-1.5
−
2
-2
0 0 0.5
1.5
β u
4 4
3
6 6
5
(c)
2 2
1
0
0
-1
−
2
-2
-3
-5
−
-4 4 -3
−
-2 2 -1
γ uY
3
0 0
2.5
1
3.5
3
Parameters
β u
β v
β
T
γ
γ
γ
γ
γ
γ
γ uT
γ uY
1
, uY
2
,
, uY
3
,
γ u
β
Y i
,
γ
γ
γ
γ v vT vY
1 vY
2
γ vY
3
γ uY
4 uY
5 uY
6
,
,
,
γ
γ uY
7
α
α
,
T
γ
Td i vY
4 vY
5 vY
6 vY
7 g
1 g
2 g
3 g
4 g
5 g
6 g
7
H
1
H
2
H
3
H
4
H
5
H
6
H
7
Value(s)
3.90, 3.45, 1.50
0.125
⁄
√ i
−
0.01, 0.60
−
4.2,
−
3.0
1.20, 1.20
−
1.30,
−
3.35
0.83,
−
1.595
1.00, 1.02
0.65,
−
0.65
0.65,
−
0.35
0.65,
−
0.65
g i
( γ uT
γ
0.0
uY i
+ γ vT
γ vY i
)
0.120
1.000
0.743
0.462
0.044
0.276
0.000
0.698
0.769
1.000
0.706
0.440
0.443
0.000
γ u
Figure 1: Regime maps of temperature and H
2
O mass fraction for various combinations of bifurcation parameters;
(a) temperature vs.
velocity strain rates, (b) H
2
O concentration vs.
strain rate and Re , (c) H
2
O concentration vs.
H
2
O concentration gradients.
Figure 1(a) displays behavior type of temperature time series as γ u
( ∼ u y
) and γ v
( ∼ v x
) are varied over the indicated range, with all other parameter values fixed. It is interesting that moderate positive values of both of these parameters lead to time series that are complex, but not always broadband, with intricate changes in behavior over a relatively small range. We remark, however, that γ u and γ v are scaled quantities, and the range of physical strain rates encompassed by the dimensionless parameter values shown here is very large. (Indeed, in LES of various flows with Re as high as O (10 6 ), computed values of γ u and γ v do not generally exceed O (10 − 1 ).) We also see in Fig. 1(a)

that with the set of parameters employed here, simultaneous negative γ u
, γ v produces divergence of the DDS; hence, this region of parameter space would be inaccessible by a model employing this system. Ramifications of this deserve further investigation. Finally, we see that increasingly large positive values of γ u
, with γ v fixed at moderate values, lead to an inverted bifurcation sequence resulting ultimately in steady behavior. Since γ u is associated with strain rate, and the temperatures computed here correspond to small-scale fluctuations, we expect this is an indication of extinction.
In Fig. 1(b) we display bifurcations of H
2
O mass fraction time series as β u
( ∼ Re
λ
) and γ u
( ∼ u y
) are changed. It is interesting to note that overall structure of this regime map is rather similar to those reported in [9] for the PMNS equations alone, thus indicating that concentration fluctuations are generally being driven by velocity fluctuations
(and not vice versa ). In general, the figure shows an increase in complexity of behavior as Re increases and, e.g.
with fixed and small | γ u
| , goes through the typical, rather simple, Feigenbaum subharmonic bifurcation sequence.
On the other hand, very complicated sequences arise if γ u and β u are varied simultaneously—as would occur from point to point in space and time in a physical flame.
Figure 1(c) presents effects of H
2
O concentration gradient variation on H
2
O concentration fluctuations. This regime map is quite simple (in comparison with the preceding ones) and displays only a few basic types of fluctuation.
Over a wide range of parameter space the behavior is completely turbulent, and for the set of physical parameters shown in the table no laminar small-scale H
2
O concentration behavior exists. That is, the DDS diverges without going through simpler periodic regimes. Nevertheless, we emphasize that other combinations of parameters lead to simpler, stable behaviors.
6 6 4 4
Figure 2 presents two key statistical properties of turbulent combustion, viz.
, 5
12
(a)
1.5
(b)
3 3 scalar dissipation defined in terms of elemental hydrogen mixture fraction, ξ ≡
3
0 −
1.5
2 2 ( Y
H
− Y
H,o
) / ( Y
H,f
− Y
H,o
), and skewness of the temperature derivative ∂T /∂y . Part γ
2 2
1
γ
1 1 (a) of the figure shows the first of these computed as χ = h D ( ∇ ξ ) 2 i , where D
-1 is a mixture-averaged diffusion coefficient, and h · i denotes a time average. The spa-
−
2 -2
0 0 tial derivatives needed to compute ∇ ξ are obtained via Taylor’s “frozen flow” hy-
-3
-5
−
-4 4 -3
−
-2 2 -1
γ uY
3
1 3
−
1
-1
−
-1 1
γ u pothesis.
It is interesting to recall the experimental results of Nandula et al.
[14] for
H
2
–air jet flames collected in a Re range
Figure 2: Scalar dissipation and temperature derivative skewness; (a) scalar dissipation ξ , (b) skewness of ∂T /∂y .
of 5000–14000. These data show a range of values for χ between 0 and ∼ 15 from measurements taken at two axial locations and numerous corresponding radial locations. It is clear that our range of values 0 to 12 obtained with all bifurcation parameters fixed but two (and thus a limited range of accessible physics) demonstrates the ability of this low-dimensional model to at least qualitatively replicate scalar dissipation. It should also be noted that the computed results show the largest values of χ when one bifurcation parameter is relatively large and the other somewhat near zero. Locating these regions of high χ on the regime map Fig. 1(c) indicates that computed scalar dissipation is strongest in areas of broad-band behavior. In addition, if we note that γ uY
3
∼ ∂Y
H
2
O
/∂x and γ vY
3
∼ ∂Y
H
2
O
/∂y , we recognize that peak predicted values of χ occur in regions of the contour plot that would correspond to the reaction zone of a physical flame.
Skewness of passive scalars is one of the important statistical properties used to characterize turbulent flow and thus has been widely studied (see the review by Warhaft [15]). In Fig. 2(b) we present a contour plot of skewness of the temperature derivative, defined as s ≡ h ( ∂T /∂y ) 3 i / h ( ∂T /∂y ) 2 i 3 / 2 , where, again, the Taylor hypothesis has been used to obtain spatial derivatives from the time series for T produced by the DDS. Typical experimentally-observed values of s are in the range − 1 .
75 to 1 .
75, and the figure shows that for the conditions used for the calculations reported here computed values are well within this range. Furthermore, we see that the highest values correspond to a region of chaotic, but not completely broad-band behavior shown on Fig. 1(a). This too is in accord with both theory and experimental observation. In particular, it is well known that skewness is generally largest at relatively low Taylor microscale Reynolds numbers, Re
λ
, and decays (in magnitude) to a value of ∼ 0 .
8 as Re
λ increases (see
Sreenivasan and Antonia [16]). But in addition, theory indicates that s → 0 as Re
λ
→ ∞ , and this is indeed observed experimentally for high-pass filtered data. Figure 2(b) displays such a decay in magnitude and approach to zero as
γ u and/or γ v become large. We note that a detailed interpretation (in Fourier space) of these bifurcation parameters
(see [12]) includes a factor of wavenumber, and hence the observed effect appearing in high-pass filtered data.

SUMMARY and CONCLUSIONS
In this paper we have introduced a low-dimensional model of turbulent combustion of H
2 and O
2 consisting of a discrete dynamical system derived directly from the governing equations. We then described a technique for efficiently identifying bifurcation sequences of DDSs via analysis of the PSDs of their time series, and we presented several interesting cases demonstrating its use. We then provided limited results comparing statistical quantities derived from time series of the DDS using Taylor’s hypothesis to obtain approximations of spatial derivatives. These results are in at least general qualitative agreement with both experiment and theory.
There are two main conclusions to be drawn from the results of this study. First, the analysis technique introduced provides an efficient, automatic characterization of even very complicated bifurcation sequences (more than threequarters of a million separate cases were analyzed to obtain the presented results) leading to a detailed, if not complete, understanding of behavior of the DDS in response to changes in its physically-based bifurcation parameters. Second, statistical results obtained from the DDS time series are in quite good agreement with both theory and experiment suggesting that this modeling approach could potentially be very effective in the context of constructing LES subgridscale models. Moreover, these results seem sufficiently faithful to begin considering their use in real-time control of turbulent combusting systems.
Acknowledgements
The author expresses his gratitude for support provided by NASA/EPSCoR Grants WKURF 515107-04-14 and
596217-05-09.
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