Bifurcation diagrams for a

Paper 03F-52
Bifurcation diagrams for a discrete dynamical system model of
reduced-mechanism H2 {O2 combustion
J. C. Holloway , M. G. Chong and J. M. McDonough
Department of Mechanical Engineering
University of Kentucky, Lexington, KY 40506-0503
We present a turbulence modeling formalism similar to large-eddy simulation but distinct in the sense
that solutions, rather than equations, are ltered, and physical variables, instead of their statistics, are
modeled on the sub-grid scales. We argue that this approach yields specic advantages in the context of
combustion, but we also show that the associated discrete dynamical systems (DDSs) contain numerous
bifurcation parameters (hence, very high codimension) which must be calculated. It is thus useful
to understand details of the DDS bifurcation sequences in order to complete a subgrid-scale model
formalism. In the present work we provide 2-D regime maps (bifurcation diagrams) corresponding to
selected pairs of parameters appearing in a simple reduced mechanism for H2 {O2 combustion due to
Mauss et al. (in Reduced Kinetic Mechanisms for Applications in Combustion Systems, Peters and
Rogg, Eds., 1993). We discuss bifurcation sequences appearing in the regime maps and their relevance
to physical phenomena, as well as the ability of the DDS to model a wide range of H2 {O2 combustion
scenarios. In addition, we illustrate the prodigious complexity of the DDS model by presenting evidence
suggestive of fractal behavior.
Keywords: turbulent combustion, reduced mechanism, discrete dynamical system, bifurcation
1. Introduction
. [1] and Hylin and McDonough [2] used results
of modeled uctuating quantities to augment the
resolved-scale solution, unlike most approaches to
LES which simply discard them. LES involving
chemical kinetics has yet more challenges beyond
that of the usual closure problem and the associated subgrid-scale (SGS) models, particularly the
disparate time scales of the chemical kinetics and
velocity and temperature uctuations. This suggests the usefulness of employing SGS models as
outlined by McDonough and Huang [3].
McDonough and Zhang [4], [5] used the ideas
of [3] in constructing LES SGS models for six and
nine-step reduced mechanisms of H2 {O2 combustion. Results from these models compared qualitatively well with experiment. Furthermore, Zhang
et al. [6] were able to produce temporal uctuations of quantities which resulted in relatively good
Large-eddy simulation (LES) has begun to be
useful in simulating a wide range of uid ows for
engineering purposes. However, it is well known
that LES has deciencies, the rst and foremost
of which is the closure problem resulting from ltering of governing equations. An alternative approach to LES as explained by McDonough et al.
[1] alters the closure problem by ltering solutions
to the governing equations rather than the equations themselves. This ltering technique allows
modeling of uctuating quantities directly, rather
than their statistics, because no correlation quantities are produced in the process of ltering solutions. To further improve LES, McDonough et
Corresponding author. Tel: 859.269.1480
E-mail address:
quantitative agreement with experiments by Meier
et al. [7] by using an SGS model for a two-step reduced mechanism for H2 {O2 combustion similar
to those in [3], [4] and [5]. In all of these investigations, discrete dynamical systems (DDSs) were
employed to model uctuating quantities on the
sub-grid scales. But due to the large number of bifurcation parameters (and thus very high codimension) associated with these models, a suÆcient understanding of how to implement the models into a
complete LES of combustion requires further investigation. In the present work we continue analyses
begun of the two-step reduced mechanism for H2 {
O2 combustion in [6] in order to understand the
bifurcation sequences that can be observed in the
DDS model as the numerous bifurcation parameters are changed. Furthermore, we seek an understanding of limiting values of several key parameters that may be used in LES. This understanding
may then be employed to construct accurate SGS
models for LES involving turbulent H2 {O2 combustion.
2.1 Alternative Approach to LES
For simulation of chemically reacting ows, we
seek solutions to the conservation equations of
mass, momentum, energy, and species, which may
be represented as the general transport equation,
Qt + r F (Q) = r G(Q) + S (Q) :
Here, the subscript t denotes partial dierentiation
with respect to time, and r is the divergence operator. F and G are advective and diusive uxes,
respectively, and S is a general nonlinear source
term. To solve (1), we decompose dependent variables Q in the usual way to obtain
Q(x; t) = q (x; t) + q (x; t) ; x 2 Rd ; d = 2; 3 ;
with q and representing the large and sub-grid
scale contributions, respectively. After substitution of (2) into (1) leading to
(q + q )t + r F (q + q ) =
r G(q + q ) + S (q + q ) ;
The remainder of the paper proceeds as follows.
We begin with an analysis section outlining the
main ideas used in deriving the DDS SGS model
as proposed in [3]{[6]. Next, we describe the relationships of the DDS bifurcation parameters to
physical phenomena. We then present 2-D bifurcation diagrams illustrating the relationships between numerous pairs of these parameters, as well
as their limiting values and the observed bifurcation sequences of the DDS SGS model. Finally, we
discuss results and conclude the paper with a summary and a suggestion of plans for future work.
the large scale solution q may be calculated directly once q is set, much like in direct numerical
simulation (DNS).
As current hardware does not allow complete
resolution of all ow scales (as in DNS) for most
real-world engineering problems investigated, a
more eÆcient means of calculating q , and thus
a modeling approach, is required. We follow the
approach used in [8] and model q as
qi = Ai Mi ;
i = 1; 2; : : : ; Nv ;
where Nv is the number of dependent variables.
The Ai s are amplitude factors derived from Kolmogorov scalings [8], and the Mi s are algebraic
chaotic maps derived from governing equations associated with Q in (2), which compose a discrete
dynamical system. Detailed teatment has been
given the Ai s in [8] and [9]. In the present work we
investigate the chaotic behaviors associated with
the Mi s.
2. Analysis
For the sake of brevity we only outline the main
steps of the subgrid-scale model derivation. We
begin with an overview of the LES formalism and
present the general governing equations. We then
sketch the procedure that leads from these to a corresponding generalized DDS for reacting ows. We
conclude the section by introducing the two-step
mechanism that is the subject of the current inves- 2.2 Discrete Dynamical System
tigation and provide a specic DDS corresponding
To derive the Mi s we begin with the governto this mechanism.
ing equations for chemically reacting ows, written
here using a conventional notation:
crete dynamical system:
a(n+1) = u a(n) 1 a(n) u a(n) b(n) ;
0 = t + r (U ) ;
(5a) b(n+1) = b(n) 1 b(n)
v a(n) b(n) + T c(n) ;
= rp + r (rU ) + g ;
" N
T d d(in+1) uT a(n+1)
=r (rT ) +
cp Di Wi r
rT c(n+1) =
hi w_ i ;
vT b(n+1) c(n)
D(Yi )
=r (Di rYi) + w_ i ;
i = 1; : : : ; N s :
!_ i =
j =1
)wj ;
wj = kf;j
Y` s
Hi !_ i =(1 + T )
Thus the as, bs, cs and di s are quantities related to
the Fourier coeÆcients of the Fourier series representing the two velocity components, thermal energy, and chemical species concentrations, respectively. The superscripts (n) denote time steps (or
map iterations). Note that (6) will apply to any
chemically reacting ow; a DDS pertaining to a
particular mechanism simply requires a di for each
product species in that mechanism. The reader is
encouraged to refer to detailed derivations as found
in the extant literature, particularly [3] and [4].
Y` s
Y + uY a(n+1) + vY b(n+1) di
+ !_ i + di;0 ;
i = 1; 2; : : : ; Ns ; (6d)
j =1
+ c0 ;
Here, Ns is the total number of species and
w_ i = Wi
We assume dependent variables may be accurately represented as generalized Fourier series. To
simplify analysis, we further assume the basis functions f'k g used in constructing the series are i)
complete, ii) orthnormal, and iii) like complex exponentials with regard to dierentiation. After
substituting the Fourier representations into (5),
we then carry out the Galerkin procedure as in
[3]{[6], producing an innite number of ordinary
dierential equations for the Fourier coeÆcients.
For computational eÆciency, each series is decimated to retain a single term corresponding to an
arbitrary wavevector k. The resulting equations
can be integrated with rst-order Euler methods
and then rearranged to produce the following dis-
2.3 Bifurcation Parameters
Derivation of (6) leads to the construction of
numerous bifurcation parameters, all of which are
related to physical quantities associated with the
particular ow scenario investigated. It should be
observed that in a complete LES, these parameters are calculated automatically \on the y," using results from the resolved-scale solution. In the
current investigation, however, parameters are set
to xed values manually.
As discussed in [10], u and v in the momentum
equations are related to Reynolds number via
i = 4 1
i = u; v :
Here, is a time scale computed as the reciprocal
of the norm of the strain rate tensor, and k is a single wavevector. The components of k, kj , should
be treated as reciprocals of the Taylor micoscale
length in each separate coordinate direction. In
[11] it is shown that T is related to Peclet number,
and this is true for each Y as well [8],
T =
be modeled by using reduced mechanisms [4]|[6].
We follow such an approach and model concentrations with a two-step reduced mechanism. As illustrated in [12], under steady-state assumptions, the
17 elementary reactions comprising H2 {O2 combustion may be reduced to the two-step, global
3H2 + O2 ! 2H + 2H2 O
2H + M ! H2 + M:
i = 1; 2; : : : ; Ns :
P ei
The u , v , uT , vT , uY , and vY arise from the
In our investigations, the third body M is taken
to be H2 O , which has shown to be a reasonable
assumption in previous studies [4]{[6]. As mentioned above, a discrete dynamical system may be
Galerkin triple products [8],
used to model reacting species concentrations in
the form of (6). Each species concentration to be
j = j kj Ckkk ;
j = u; v; uT; vT; uYi ; vYi ;
(10) modeled, that is, each product species appearing
in (13), requires calculation of a corresponding di .
with Ckkk representing the Galerkin triple product
We set
of momentum Fourier basis functions with passive
scalar Fourier basis functions, and and kj comd1 H2 ; d2 O2 ; d3 H2 O ; d4 H : (14)
puted as above. We note, however, that kj is the
Fourier space representation of a partial derivative, Although d2 can only be a product in (13) if we
and in implementations we usually construct this assume the reverse reaction holds as well as the forin physical space. Each T is related to Galerkin ward, we nd in [12] that calculations of other di
require calculation of d2 (as will be shown below).
triple products similar to the s [8]:
In the current investigation we consider only the
T = jkj2 cdi Ckkk
i = 1; 2; : : : ; Ns ; (11) forward reaction in (13a); d2 , then, is set as constant for all time steps. All other product species
with the exception that the Ckkk
are computed in (13) are modeled in the form of di s in (6). Thus
from derivatives of basis functions. Finally, T is for H2 ,
T = 2 ;
+ !_ 1(n) + d1;0 ;
where Gr is a grid-scale Grashof number. In the
current investigation we set T = 0 to treat a nonwith
buoyant jet ame.
Y =
!_ 1(n) = Cf;1;2 d(4n+1) ;
2.4 Reduced Mechanism
Cf;1;2 = 100;2
k :
W4 f;2
According to Mauss et al. [12], combustion of For O2 we have simply
H2 {O2 involves eight chemically reacting species
d(2n+1) = d2;0 ;
and approximately 17 elementary reactions. Thus
an attempt to accurately model species concentrafor all time steps (n). We model H2 O with
tions involved in the phenomenon could prove to
be quite exhaustive for current hardware, at least d(n+1) =
in the context of 3-D LES or DNS. But there is
evidence that suggests reasonably good approx+ !_ 3(n) + d3;0 ;
imations of primary species concentrations may
!_ 3(n) = Cf;3;1 d(1n) d(2n) ;
Cf;3;1 = 300;1
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
Phase lock
Periodic w/ different fundamental
k :
W1 W2 f;1
Finally, H is modeled by
d(4n+1) = Y4 + uY4 a(n+1) + vY4 b(n+1) d(4n)
+ !_ 4(n) + d4;0 ;
!_ 4(n) = Cf;4;1 d(1n) d(2n) ;
Cf;4;1 = 400;1
k :
W1 W2 f;1
Thus to model H2 {O2 combustion primary
species concentrations with this two-step reduced
mechanism requires a total of only three iterated
maps, which may be calculated with ease. The entire DDS used to model momentum, thermal energy, and species concentrations is simply (6) with
the di s simplied as in (15){(18).
Figure 1: Regime map color legend. Each color
represents a distinctive classication of temporal
uctuations that may be observed in the time series of components in (6).
3. Results
here are not meant to be representative of all possible behaviors of the components of (6). Each
map represents time series behavior for only one
combination of bifurcation parameters (except, of
course the two parameters varied within the regime
map), the slightest perturbation of which can potentially alter the map signicantly. In most cases,
we will omit reporting specic values of bifurcation
parameters employed as such is not neccessary to
illustrate the potential of the DDS to model H2 {
O2 combustion or to comprehend its inherent complexity. Finally, recall in a complete LES, all bifurcation parameters are calculated automatically
by using results from the resolved-scale solution.
In the current investigation we simply seek an understanding of the possible behaviors that could
be observed in such an LES.
We present 2-D bifurcation diagrams corresponding to various pairs of bifurcation parameters
in (6) to illustrate the wide range of temporal behaviors that can be observed in the components of
(6). These diagrams, which we call regime maps,
are produced by a Fortran 77 program that iterates
(6) for a specied number of time steps (producing a time series for each component in the DDS),
calculates power spectral densities (PSDs) based
on these time series, and then essentially \looks"
at each PSD to classify the temporal behavior produced by the particular combination of bifurcation
parameters employed. Temporal behavior of the
components of (6) can normally be classied into
one of several generalized groups of temporal uctuations, which to each we assign a color as shown
in Fig. 1. On a regime map for a particular component of (6), each point corresponds to a unique
combination of bifurcation parameters. Thus for
each point on a map the color represents the type
of temporal uctuations observed in the time series for the component of (6) portrayed in the map
when this combination of bifurcation parameters is
Regarding discussions that follow, it is important to understand that the regime maps presented
3.1 Regime Maps for
H2 O
In this section we investigate the temporal behavior of the uctuating H2 O concentration calculated from (17) as the parameters v and uY3 of
(6) are changed. As the main product of H2 {O2
combustion, H2 O is of particular interest in our
studies. Recall that uY3 is related to strain rate
and a spatial derivative for H2 O concentration via
(10), and v is directly related to Re (7). Thus
increasing v in (6) is analagous to physically increasing the Re in the vertical direction. In Fig.
2 we can see a common bifurcation sequence leading to turbulence as v is increased and uY3 is
held constant. Note the transition from subharmonic to phase lock (v 2:8), which then bifurcates to noisy quasiperiodic with a distinctive
fundamental mode. We also observe reverse bifurcation sequences, such as where the quasiperiodic
uctuations bifurcate back into phase lock as v is
further increased (v 3:2). Note as v ! 4, the
time series diverges, symbolizing a mathematical
irregularity and a non-physical solution.
If instead v is held constant as uY3 is changed,
we notice little change in time series behavior,
and few clear bifurcation sequences. The main
visible transitions as uY3 is varied occur around
uY3 1:5, where subharmonic H2 O uctuations
become phase locked as uY3 is increased before
bifurcating back into subharmonic. Other transitions are less distinctive, mainly comprised of
changes between broadband with, without, or with
dierent fundamentals.
In addition to understanding the possible behavior of H2 O uctuations permitted by (6), we take
interest in the temporal uctuations of products H
and H2 . In Fig. 4 we choose (arbitrarily) to present
a regime map for H concentration as the parameters uY4 and vY4 are varied. Recall that uY4 and
vY4 are related to products of momentum Fourier
basis functions with the Fourier basis functions for
H as in (10). Here we see the wide range of uY4
and vY4 values that are permissible to produce
non-divergent time series, a range which is much
larger than allowable for many other parameters
appearing in (6). This large range of permissible
values has shown to be characteristic of most s in
(6) for our investigations thus far. Also in Fig. 4
it is interesting to observe bifurcation sequences as
uY4 or vY4 increase in magnitude. In particular,
beginning one sequence we see that for small magnitudes of uY4 and vY4 (near the center of the
regime in Fig. 4), there are quasiperiodic uctuations of H concentration. As uY4 and vY4 are both
that as we increase grid resolution of the regime
map there are more visible bifurcations than at the
coarser resolution. This illustrates the complexity
of a DDS with such high codimension as (6).
3.2 Regime Map for
Figure 3: Regime map for H2 O; a magnication of
Fig. 2, illustrating the possible fractal behavior of
Figure 2: Regime map for H2 O illustrating relationships between H2 O concentration uctuations
and changes in v and uY3 .
It is also interesting to observe what appears to
be fractal behavior in certain regions of Fig. 2. In
Fig. 3 we magnify one of these regions. We see
when T < 0. This suggests the mathematics of
(6) is, perhaps, more complex in nature than the
actual physics described by it.
Figure 4: Regime map for H displaying changes in
H uctuations as uY4 and vY4 are varied.
increased in (positive) magnitude, the H uctua- Figure 5: Regime map for thermal energy showing
tions bifurcate to noisy subharmonic, then to noisy relationships between thermal energy uctuations
quasiperiodic, to eventually broadband without a and changes in and .
fundamental mode before nally diverging.
In Fig. 6 we present regime maps for the thermal energy component of (6) where u and v
are varied. For the combination of bifurcation
parameters employed here, thermal energy uctuations are extremely sensitive to perturbations
in either u or v . Observe, for example, in the
lower middle region of the regime of Fig. 6(a)
(v < 0:75; u 0), thermal energy uctuations
bifurcate from periodic to subharmonic, and then
further to noisy quasiperiodic with changes in u
over a range as small as 0:2. In addition, notice
the bifurcations to phase-locked uctuations littered throughout the quasiperiodic region (in the
lower left quadrant of the regime) as well as bifurcations to noisy quasiperiodic with a fundamental
mode in the noisy quasiperiodic region without a
fundamental mode (in lower right quadrant of the
regime). Such regions illustrate the extreme sensitivity to initial conditions in (6), which is a common characteristic of turbulence in general. Unlike
in other gures presented thus far, in Fig. 6(a) we
observe regions in the regime which produce periodic uctuations and even steady state behavior.
This can be attributed to having set u = 3:2,
3.3 Regime Maps for Thermal Energy
In this section we wish to demostrate the ability
of (6) to model temporal uctuations of thermal
energy as are often observed in H2 {O2 combustion
systems. To do so we present regime maps for the
c s in (6), which represent the thermal energy of
the system, as u and T in (6) are varied. Recall
from (8) that T is directly related to Peclet number; thus changing T is analagous to altering the
thermal diusivity of the system. Also, from (10),
u is related to products of Fourier basis functions
for momentum, which has a less intutive relation to
physical space than that of T , but it carries spectral information on velocity derivatives. An important characterisic of Fig. 5 is that as T is varied,
there is little change to the temperature uctuation behavior, except at low values of T . Clearly,
at least in the case shown in Fig. 5, temperature
uctuations have a much larger dependence on the
value of u . Finally, it should be noted that due to
the nature of (8) we consider only positive values
of T , although regime maps can be used to show
that there exists non-divergent behavior for cases
which in our investigations normally produces less
chaotic uctuations than s > 3:5. For example,
in Fig. 6(b) we employ the same combination of
bifurcation parameters as in Fig. 6(a), only we increase u to 3:6. Notice here the steady and periodic regions in Fig. 6(a) have now bifurcated to the
more turbulent uctuations of noisy quasiperiodic
and broadband.
5. Discussion
In the present work we have presented results
from preliminary analysis of the complex behaviors associated with (6). Complete analysis for a
DDS with such high codimension is slowed by the
vizualization limitations of 3-D space. As we have
shown, 2-D bifurcation diagrams can be used to
deduce relationships between pairs of bifurcation
parameters appearing in (6). Likewise, 3-D bifurcation diagrams would be capable of illustrating
such relationships among three parameters at one
time. Thus to predict all possible changes in temporal uctuations for a DDS with codimension of
n requires an n-dimensional space in which to visualize it. Under these circumstances, an attempt
at understanding all possible behaviors and bifurcations associated with such a system would prove
to be an insurmountable task, indeed. Rather, a
more realistic goal is to implement such a DDS
SGS model into a complete LES to observe its
strengths and weaknesses in simulations by simply comparing the results to those of experiment
and DNS.
6. Conclusion
It has been the goal of this work to present a possible
solution to the computational diÆculties as4
sociated with accurately modeling turbulent ows
involving chemical kinetics. We propose a simulation formalism similar to LES, but distinct in
that solutions are ltered rather than equations,
dependent variables are modeled directly rather
than their statistics, and subgrid-scale solutions
are used to augment the resolved-scale solutions.
In this context we have presented a subgrid-scale
model for H2 {O2 combustion in the form of a disb)
crete dynamical system, which was derived based
on a two-step reduced mechanism constucted by
Mauss et al. [12]. 2-D regime maps, or bifurcation diagrams, were used to illustrate relationFigure 6: Regime maps for thermal energy illus- ships between the numerous bifurcation parametrating relationships between thermal energy uc- ters in (6) and were able to demonstrate the comtuations and changes in u and v .
plex temporal uctuations that can be produced in
the components of this DDS. Additionaly, regime
maps were able to model bifurcation sequences
commonly observed in turbulent combustion phenomena. We conclude by calling attention to the
potential for this approach to accurately model
H2 {O2 combustion in a complete LES. For future
work, it will be worthwhile to implement (6) and
compare results to those obtained by employing
DDS SGS models as in [4] and [5], which were
based on six and nine-step reduced mechanisms.
[12] F. Mauss, N. Peters, B. Rogg and F. A. Williams,
in Reduced Kinetic Mechanisms for Applications
in Combustion Systems (Peters and Rogg, eds),
Lecture Notes in Physics m15, Springer-Verlag,
Berlin, 1993.
7. Acknowledgements
The rst author would like to express his gratitude to the NASA Kentucky Space Grant Consortium for nancial support of this work.
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[4] J. M. McDonough and S. Zhang, presented at 32nd
AIAA Fluid Dynamics Conference, St. Louis, MO,
June 24{27, 2002.
[5] J. M. McDonough and S. Zhang, presented at 37th
Intersociety Energy Conversion Engineering Conference, Washington, D. C. July 29{31, 2002.
[6] S. Zhang, J. D. Slade and J. M. McDonough, Proc.
of Technical Meeting of Central States Section of
the Combustion Institute, Knoxville, TN, April 7{
9, 2002.
[7] W. Meier, S. Prucker, M. H. Cao and W. Stricker,
Combust. Sci. Technol. 118:293, 1996.
[8] J. M. McDonough and J. C. Holloway, Proc. of
Third Joint Meeting of U. S. Sections of the Combustion Institute, Chicago, IL, March 16{19, 2003.
[9] J. M. McDonough, paper c3.4, Proc. of Technical
Meeting of Central States Section of the Combustion Institute, Knoxville, TN, April 7{9, 2002.
[10] J. M. McDonough and M. T. Huang, Int. J. Numer. Meth. Fluids, accepted 2003.
[11] J. M. McDonough and D. L. Joyce, presented at
th AIAA Thermophysics and Heat Transfer Con-
ference, St. Louis, MO, June 24{27, 2002.