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1. Introduction
Recent concerns over energy sustainability, energy security,
and global warming have presented grand challenges to develop
non-petroleum based low carbon fuels and advanced engines
to achieve improved energy conversion efficiency and reduced
emissions [1, 2]. Transportation contributes to one-third of the
global carbon emissions. Therefore, it is urgently needed to
develop alternative fuels and predictive combustion modeling
capabilities to optimize the design and operation of advanced
engines for transportation applications. To achieve this goal,
in 2009 DOE established an Energy Frontier Research Center
(EFRC) on Combustion Science at Princeton University [3]
to develop multi-scale predictive models for 21st century
transportation fuels.
Unfortunately, combustion involves various time and length
scales ranging from atomic excitation to turbulent transport
(Fig.1). Even with the availability of high-performance
supercomputing capability at the petascale and beyond, the multi
physics and multi timescale nature of combustion processes
remain to be the great challenges to direct numerical simulations
of practical reaction systems [4].
At first, the kinetic mechanisms for hydrocarbon fuels typically
consist of tens to hundreds of species and tens to thousands
of reactions. The large number of reactive species requires an
enormous computation and data storage power. For example, it
was shown that about 80% of the CPU time for a CFD simulation
with a simple and compact methane oxidation mechanism (36
species and 217 reactions) was spent on the computation of the
chemistry [5]. It is well known that the CPU time of a typical
matrix-decomposition based implicit algorithm (e.g. a stiff ordinary
differential equation (ODE) solver [6]) is proportional to the square
of the species number. The ratio of computing time for reactions
will become even larger as the size of the mechanism increases.
Secondly, the broad range of length and time scales of transport
and kinetic reactions creates problems of grid resolution and * Corresponding author. E-mail: [email protected]
■SERIAL LECTURE■
― New Aspects of Combustion Kinetic Modeling IV ―
日本燃焼学会誌 第 51巻 157号(2009年)200-208 Journal of the Combustion Society of JapanVol.51 No.157 (2009) 200-208
An On-Grid Dynamic Multi-Timescale Method with Path Flux Analysis for Multi-Physics Detailed Modeling of Combustion
JU, Yiguang1*, GOU, Xiaolong1,2, SUN, Wenting1, and CHEN, Zheng1,3
1 Princeton University, Princeton 08544, USA
2 Chongqing University, Chongqing 400030, China
3 Peking University, Beijing 100871, China
Abstract : Combustion is a kinetic and multi-physical-chemical process involving many time and length scales from atomic excitation to turbulent mixing. This paper presents an overview of the recent progress of numerical modeling using detailed and reduced chemical kinetic mechanisms for multi-scale combustion problems. A particular focus of this review is to introduce two new methods, an on-grid dynamic multi-time scale (MTS) method and a path flux analysis (PFA) to increase the computation efficiency involving multi-physical chemical processes using large kinetic mechanisms. Firstly, the methodology of the on-grid dynamic MTS method using the instantaneous time scales of different reaction groups is introduced. The definition of species timescales and the algorithm to generate species groups are presented. Secondly, a concept of hybrid multi-time scale (HMTS) method to selectively remove time histories of uninterested fast modes is introduced to increase the algorithm flexibility and efficiency. The efficiency and the robustness of the MTS and HMTS methods are demonstrated by comparing with the Euler and ODE solvers for ignition and flame propagation of hydrogen, methane, and n-decane-air mixtures. Thirdly, the PFA method to generate a comprehensive reduced mechanism by using the reaction path fluxes in multiple generations is summarized. The results are compared with that of the direct relation graph (DRG) method for n-decane and n-heptane-air ignition delay time. The PFA method shows a significant improvement in the accuracy of mechanism reduction over the DRG method in a broad range of species number and physical properties. Finally, the PFA method is integrated with MTS and HMTS methods to further increase the computation efficiency of combustion with large mechanisms. The new algorithm is used to predict unsteady flame propagation of n-decane-air mixtures in a spherical chamber. Excellent computation accuracy and efficiency are demonstrated.
Key Words : Reduced Mechanism, Multi-timescale, Path flux analysis, Direct numerical simulation
Ju, Yiguang et al.: An On-Grid Dynamic Multi-Timescale Method with Path Flux Analysis for Multi-Physics Detailed Modeling of Combustion
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stiffness in direct numerical simulation. An explicit algorithm such
as the Euler scheme will have to use the smallest timescale based
on the finest grid and fastest reaction group, leading to a dramatic
increase of computation time. Thirdly, the multi physical non-
linear coupling between continuum regime (e.g. flow with small
concentration/temperature gradients) and non-continuum regime
(e.g. particle and molecular dynamics) and the statistical coupling
between mean values and sub-grid statistical values requires
accurate resolutions of the time histories of variables with different
timescales (Fig.1). For an example, a nanosecond non-equilibrium
plasma assisted combustion needs to calculate electron energy
distributions in nanoseconds and combustion in a few seconds.
In order to make numerical simulations of reactive flow
computationally affordable and comprehensively accurate,
various mechanism reduction methods have been developed to
generate reduced kinetic mechanisms [5]. The first approach is
the sensitivity and rate analysis [7-9]. Although these methods
are very effective in generating compact reduced mechanisms, it
provides neither the timescales of different reaction groups nor the
possible quasi-steady state species (QSS) and partial equilibrium
groups without human experience. The second approach is the
reaction Jacobian analysis which includes the Computational
Singular Perturbation (CSP) method [10-12] and the Intrinsic
Low Dimensional Manifold (ILDM) [13]. This approach can
effectively identify the timescales of different reaction groups,
QSS species, or low dimensional manifold. However, it requires
significant computation time to conduct Jacobian decomposition
and mode projection. This type of method is therefore not
suitable to be used in direct numerical simulation. To achieve
more efficient calculations of fast mode species, a third approach,
the parameterization method, was proposed. The In-Situ Adaptive
Tabulation (ISAT) [14], Piecewise Reusable Implementation
of Solution Mapping (PRISM) [15], High Dimensional Model
Representation (HDMR) [16], and polynomial parameterizations
[17] are the typical examples of this category. However, for
large mechanisms in non-premixed turbulent combustion with a
broad temperature distribution, the constraints for validity of low
dimensional manifolds, large time consumption in table buildup,
and the difficulties in data retrieval from table lookup lead to little
advantage of ISAT in comparison to direct integration [18].
In order to achieve efficient reaction model reduction, a fourth
method is presented to use the reaction-rate or reaction path
relations as a measure of the degree of interaction among species.
Bendsten et al. [19] adopts a reaction matrix (P) with each of
whose elements Pij defined as the net production rate of species i from all reactions involving j to establish a skeletal reaction path
way (or mechanism). The set of important species is selected by
going through the reaction matrix following the path that connects
one species to another that is most strongly coupled with it. The
skeletal mechanism is then completed by including a number
of reactions such that for each of the selected species, a certain
percentage of the total production or consumption rate (threshold
value) of that species is kept in the skeletal mechanism. Similar
selection procedures are used in Directed Relation Graph (DRG)
method [20] and Directed Relation Graph method with Error
Propagation (DRGEP) [21]. The DRG and DRGEP methods,
respectively, use the absolute or net reaction rates to construct the
species relation. However, the use of absolute reaction rates in
DRG makes the relation index not conservative, which may lead
to a less compact mechanism with the same accuracy. In order to
improve the accuracy of DRG method, a combination of DRG
with species sensitivity was proposed [22]. Unfortunately, the
addition of sensitivity analysis is very computationally expensive.
DRGEP [21] employed the absolute net reaction flux to include
error propagation across multi-generations. However, by using
the absolute net reaction flux, the reduced mechanism may fail to
identify a catalytic species, whose net reaction rate is near zero but
has a significant impact on ignition [23]. In addition, for indirect
relations, DRGEP only picks up the strongest reaction path which
may not be able to identify the species flux rigorously when the
intermediate species are more than one. Therefore, a more rigorous
and efficient reaction model reduction method is needed.
Another important fact is that no matter what kind of
mechanism reduction methods discussed above are used, the
reduced comprehensive mechanisms capable to reproduce low
temperature ignition delay time remain very large (typically
fifty to hundred species). Although the introduction of QSS
can significantly reduce computation time, the validity of QSS
assumption for a given species is questionable in the entire
simulation when the combustion environment such as temperature
and pressure are beyond the pre-specified conditions due to
turbulent transport. Although on-the-fly QSS assumption can be
used on different computation grids and time steps, the selection
and the solution of QSS remain as a difficult issue [4]. Moreover,
when the transient information of short timescales is needed
for the modeling of physical processes in larger timescales in a
multi-physics problem, the application of QSS is limited.
The question becomes: are there any simple methods which
can efficiently integrate a multi timescale problem and retain
the transient information of groups at different timescales.
A convenient way to obtain time accurate solution of various
reaction groups is the Euler method. However, an explicit Euler
method needs a time marching step smaller than the smallest
timescale in the combustion process. For stiff combustion
processes, the smallest timescale can be 10-12 second, which
makes the explicit Euler method only limited to small scale direct
numerical simulations. To resolve the stiffness of combustion
problems, an implicit approach such as the ordinary differential
equation (ODE) solver [6, 24-26] is often used. Curtiss and
Hirschfelder [25] provided the first pragmatic definition of
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stiff equations as those for which certain implicit methods,
particularly backward difference formulas (BDF), perform better
than explicit ones. The ODE solver [6] is popularly utilized
in the combustion community to handle detailed chemistry.
However, for an unsteady problem, an implicit solver for detailed
chemistry is often computationally expensive, even for the
simulation of one-dimensional propagating spherical methane/
air flames with detailed chemistry. The situation becomes even
worse for large hydrocarbons which have hundreds of species in
their oxidation mechanisms. More importantly, when an implicit
ODE solver is used, the time histories of species and variables
with fast timescales are lost. Therefore, in addition to the need of
development of a rigorous kinetic mechanism reduction method,
it is also necessary to develop an efficient, robust, and accurate
numerical method which can retain the time-accurate information
of variables of fast timescales for a stiff and multi-physics
reactive flow.
In this review, we introduce two new methods, an efficient
and time-accurate dynamic multi timescale (MTS) method
and hybrid multi-timescale method (HMTS) for the modeling
of unsteady reactive flow with detailed and reduced kinetic
mechanisms [27], and a Path Flux Analysis (PFA) method to
generate a comprehensive reduced chemical kinetic mechanism
for MTS [28]. In the following, first, the mathematical models of
MTS, HMTS, and PFA methods will be introduced. Second, the
efficiency and accuracy of the MTS and HMTS method will be
demonstrated through the comparisons with the Euler and ODE
solver to simulate the ignition process with detailed mechanisms
of methane and n-decane mixtures. Third, the PFA method is
compared with DRG to generate reduced mechanisms from
detailed n-decane and n-heptane mechanisms. Finally, the PFA
method is integrated with the HMTS method to simulate unsteady
spherical flame propagation of the n-decane-air mixtures with a
significant improvement of computation efficiency.
The Multi Timescale Method
In recent years, on-grid operator-splitting schemes have been
used in reactive flow simulations and the stiff source term due
to chemical reactions are treated by the so called point implicit
or fractional-step procedure [29, 30]. In this method, for a given
time step on each grid a non-reaction flow is solved in the first
fractional step and the stiff ODEs associated with detailed
chemical reactions are solved in the second fractional step for a
homogeneous reaction system. The present study is to develop an
efficient and accurate MTS and HMTS algorithm to integrate the
stiff ODEs on each computation grid with detailed and reduced
kinetic mechanisms.
To demonstrate the multi timescale nature of a reactive flow
with detailed chemistry, in this study, homogenous ignition of
methane/air and n-decane/air at a constant pressure is simulated.
For methane/air mixtures, the GRI-MECH 3.0 mechanism [31] of
53 species and 325 reactions is used. For n-decane/air mixtures,
the high temperature mechanism for surrogate jet fuels [32]
which consists of 121 species and 866 reactions is utilized.
Figure 2 shows the timescales of species during homogeneous
ignition of the stoichiometric n-decane/air mixture initially at
T=1400 K and P=1 atm at t=0.1, 0.2, and 0.3 ms, respectively.
Before the thermal run-away, for example, at t=0.1 ms, the
characteristic time for most species is in the range of 10-4~10-10
s. However, after the thermal run-away, for example, at t=0.2 or
0.3 ms, the range of timescale distribution changes to 10-7~10-
13 s. Therefore, the reactive system is intrinsically dynamic and
multi timescale and the characteristic time of different species
spans a wide range, leading to the difficulty or invalidity of QSS
assumption for any pre-specified species.
In order to efficiently solve the multi timescale problem in a
reactive flow without introducing pre-specified QSS assumptions,
we have developed a multi timescale (MTS) method. The basic
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Fig.1 Multiscale nature of combustion process.
Fig.2 Characteristic species timescales for n-decane ignition at t=0.1, 0.2, and 0.3 ms.
Ju, Yiguang et al.: An On-Grid Dynamic Multi-Timescale Method with Path Flux Analysis for Multi-Physics Detailed Modeling of Combustion
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idea of the multi timescale (MTS) method is shown in Fig.3a.
First of all, in each simulation a base physical time scale, tbase,
is specified based on the interest of the physical problem and the
information needed at sub-time scale to construct models at the
physical time scale of interest. Then, based on the base time step
and the characteristic time of each species, all the species will
be divided into different groups according to their timescales. At
each time step of tbase, from the fastest group (ΔtF) to the slowest
group (ΔtS), all the groups are calculated with their own group
time steps by using the explicit Euler scheme until a converging
criterion is met. The Euler scheme is simple and can be easily
parallelized on each grid.
In some special cases, the transient time histories of certain
physical modes or reaction groups may not be needed. Therefore,
time accurate integration for these modes or groups by using
an explicit Euler scheme is not necessary and an implicit Euler
scheme can be applied more efficiently. As shown in Fig.3b, for
example, if the time histories of the fastest groups are not needed,
the integration of these groups will be done by using an implicit
Euler scheme. For the rest of groups, the explicit Euler scheme is
still applied. This algorithm is called the Hybrid Multi-Timescale
(HMTS) method. It combines the explicit Euler scheme with
the implicit Euler scheme for species integration. Note that,
the selection of the implicit Euler scheme for species groups is
not limited by the timescale, but by the interest of the need of
accurate time history of the reaction groups.
In the MTS method, the time step of integrating each group
has to be smaller than the characteristic time of any species in
this group to ensure the convergence of the explicit Euler method.
However, for HMTS, the groups which use the implicit Euler
method will be integrated by using the base time step (tbase).
In the following, we briefly summarize the implementation of
the on-grid dynamic MTS and HMTS method for combustion
modeling involving detailed or reduced kinetic mechanisms. For
an adiabatic and homogeneous system, the governing equations
for mass conservation and energy conservation in a constant
pressure process can be written as,
(1)
(2a)
or
(2b)
where t is the time, T the temperature, ρ the density, and Cp the
heat capacity. Yi, ωi, and hi are, respectively, the mass fraction,
the net production rate, and the enthalpy of species i. For a
constant volume process, the energy equation can be replaced by
the conservation of the internal energy,
(3)
where e0 is the initial internal energy per unit volume of the
mixture.
In the conventional Euler method, the conservation equations
are solved explicitly by using a time step smaller than the
minimum characteristic time of all species. Although the Euler
method is widely used in the direct numerical simulations,
because of the strong constraint in time step the computation
efficiency is very low. As shown in Fig.2, in order to calculate
the ignition of n-decane, the time step in the explicit Euler
method has to be smaller than 10-13 second. On the other hand,
in an implicit method such as the ODE solver [6], the governing
equation is solved implicitly by using a Jacobian decomposition.
As shown below, although this approach is robust for larger time
steps, the computation efficiency using this method for unsteady
flow simulations is very low because of the huge cost of matrix
inversion.
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Fig.3a Diagram of multi time scale scheme. ΔtF is the time step of the fastest group, ΔtM is the time step of an intermediate group, ΔtS is the time step of the slowest group, and Δtbase is the base time step.
Fig.3b Diagram of hybrid multi timescale (HMTS) scheme. Δt is the time step of the fastest group, ΔtM is the time step of an intermediate group, ΔtS is the time step of the slowest group, and Δtbase is the base time step.
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In the MTS method, as shown in Fig.3a, unlike the
conventional Euler method, the conservation equations for all
species are integrated with their own characteristic time scales.
In order to find the characteristic time of each species, the net
production rate, ωk, in the governing equation of each species can
be divided into two parts, a creation rate, Ck, and a destruction
rate, Dk,
(4)
From a linear analysis, a characteristic time, τk, for the destruction
of species k can be approximately defined as
(5)
In the numerical simulation process, the base time step tbase
is chosen from the interest of the physical phenomenon (e.g.
turbulent mixing, transient ignition, or plasma discharge). At the
same time, the characteristic time of all species can be estimated
from Eq.(5). Since many species may have the same or similar
timescales, these species can be grouped as one integration
group. By defining that each neighboring group has a difference
of timescale in one-order, the number of timescale groups, Nmx,
and the group index of the kth species Nk can be obtained as,
(6)
(7)
Therefore, the species in the nth group will be solved explicitly
using the characteristic time of nth group by using,
(8)
(9)
where Wi is the molecular weight of the ith species. The
superscript and subscript n means the nth time step, and m is the
mth iteration. Normally, for each group calculation, only a few
steps (e.g. 2-4 steps) are needed to meet a convergence criterion
(e.g. an absolute error and a relative error). Note that, MTS has
a similar form to the conventional explicit Euler method except
that each species in different groups is integrated with a different
time step. In fact, in the limiting case when all species are
integrated at the smallest time step, MTS method automatically
reduces to the conventional Euler method. However, MTS allows
integration of a much larger base time step and is more robust
and computationally efficient than the Euler method.
In the HMTS method, the species selected for implicit
integration are solved by using an iterative implicit Euler scheme.
In this method, by using Eqs. 2 and 4 the species iteration is
written as
(10)
where Ek=Dk/Yk. At a given time step of n, the species in Eq.10
are solved iteratively until a convergent criterion is met. After
the initial implicit steps for hybrid species, the MTS method
will continue to integrate the governing equation to the specified
based time step. Therefore, HMTS is a two-step approach. In one
limit, if no group is chosen to be solved implicitly, it reduces to
the MTS method. In the other limit, if all groups are chosen to be
solved implicitly, it becomes a fully implicit method. Therefore,
the HMTS is a general method which includes Euler, MTS, and
the fully implicit methods in three limiting cases, respectively.
Path Flux Analysis Method for Mechanism Reduction
We first review the DRG method and then introduce the PFA
method. The DRG method [5, 20] uses the direct interaction
coefficient for species reduction,
(11)
(12)
where νA,i is the stoichiometric coefficient of species A in the
ith reaction, ωf,i, ωb,i, and ωi, are the forward, backward, and net
reaction rates of the ith reaction, respectively. I is the number of
total reactions. The magnitude of rABDRG shows the dependence/
importance of species B to species A.
To initiate the selection process, a set of pre-selected species
(e.g. A) and a threshold value ε need to be specified. If rABDRG < ε,
the relation between B and A is considered to be negligible. Thus,
species B will only be selected when the index is larger than ε.
However, because the direct interaction coefficient is not
conservative, one drawback of this method is that only the
first generation (directed relation) of the pre-selected species
is considered. In the view point of reaction flux, both the
first generation and the second generation or even the higher
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generations are important to define the importance of relevant
reaction species.
Instead of using the absolution reaction rates, we propose to
use the production and consumption reaction fluxes to construct
reduced mechanisms. The production and consumption reaction
fluxes of one species A can be give as:
(13)
(14)
where PA and CA, respectively, denote the total production rate
and consumption rate of species A. Therefore, the production
and consumption fluxes of species A through species B can be
calculated as,
(15)
(16)
Here PAB and CAB mean the production rate and consumption rate
of species A through species B. δB,i is unity if B is involved in the
production or consumption of A and 0 otherwise.
In order to consider the conservative flux information, we
introduce a different definition of the flux interaction coefficients
which contain the flux information for both the first and second
generations.
The flux interaction coefficients for production and
consumption of the first generation are defined as:
(17)
(18)
The flux interaction coefficients which are the measures of flux
ratios between A and B via a third reactant (Mi) for the second
generation are defined as:
(19)
(20)
The summation here includes all possible reaction paths (flux)
relating A and B.
Different threshold values can be set for different interaction
coefficients. For simplicity, we can lump all the interaction
coefficients together and set only one threshold value,
(21)
A schematic of flux transfer between A and B via other
reactants (Mi) is depicted in Fig.4. Assume A is a selected
reactant, and M1 to M5 are reactants which are the direct products
of species A. Species B to G are the indirect products of A via
M1 to M5. The values on the arrows are the fractions of the flux
of species A to each product. Let us assume that the specified
threshold value is ε=0.16. Based on the methodology of DRG,
two species, M5 and G, will be selected from the reaction path
of species A. With the same threshold value, however, PFA will
select three species, M5, B and G. Simple algebra calculations
will show that the selection of DRG captures 15% of the flux
of species A. However, the selection of PFA can capture 64%
of the target flux. If we increase the threshold value so that PFA
also only selects two species, we can see that M5 and B will be
chosen. In this case, PFA can still capture 49% of the target flux
of species A. Note that it is also easy to see that DRGEP cannot
capture the flux of species A correctly either (if two species are
chosen, DRGEP will point to M5 and G, which are the same as
DRG). From the above schematic of reaction path flux, it is seen
that PFA can identify a better reaction path and capture a higher
total flux of selected species than DRG and DRGEP.
Another issue needed to be examined is the catalytic reaction
cycle such as,
(22)
(23)
Both the production and consumption rates are fast and the
net reaction rates NO (species A) and NO2 (species B) are almost
zero. In this case, the interaction coefficient between NO and
NO2 given by DRGEP is nearly zero and the catalytic species
which is very important in low temperature methane ignition will
not be identified by DRGEP. However, based on the consumption
and production fluxes, the present PFA method will identify the
catalytic species effectively.
Therefore, compared to DRG and DRGEP, the present
PFA method is more accurate to capture the reaction fluxes of
205
Fig.4 Schematic of flux relation between different species.
日本燃焼学会誌 第 51巻 157号(2009年)
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important species to generate a more comprehensive reduced
mechanism. In addition, the integration of the PFA algorithm
with MTS can significantly increase the computation efficiency.
Results and Discussions
The MTS/HMTS and PFA methods described above were
tested and compared with implicit ODE solver [6] and DRG
method for homogeneous ignition and unsteady spherical flame
propagation for hydrogen, methane, n-heptane, and n-decane-
air mixtures. In the calculations of the HMTS method, only the
fastest group was chosen for the implicit method.
Figure 5 shows the comparison of the major species
concentrations predicted by MTS, HMTS, and ODE solver for
ignition of stoichiometric n-decane-air mixture at initial pressure
of P=1 atm and initial temperature of T=1400 K. It is seen that
both MTS and HMTS agree well with the VODE method.
Figure 6 shows the comparison of the dependence of ignition
delay time on temperature for the stoichiometric n-decane-
air mixture at two different pressures. The comparison also
demonstrates that MTS and HMTS are accurate to reproduce
ignition delay time in a broad range.
Figure 7 shows the comparisons of the normalized CPU
time between MTS, HMTS, and ODE solver for ignition of
the stoichiometric n-decane-air mixture at 1 atm and different
temperatures. It is seen that both MTS and HMTS dramatically
reduce the computation time in the entire temperature range.
Similar results were also obtained for n-decane ignition at 20
atm. In addition, it is also seen that the computation efficiencies
of MTS and HMTS are comparable. However, additional tests
show that MTS is more robust at a larger base time step.
The comparison between reduced mechanisms generated by
PFA and DRG for n-decane-air stoichiometric mixture is shown
in Fig.8. The reduced mechanism generated by PFA has 48
species and that generated by DRG has 50 species. Fig.8 shows
that for less number of species, the reduced mechanism generated
by PFA well reproduced the results of detailed mechanism in the
entire temperature range. By reducing the species number from
121 to 50, DRG resulted in a significant discrepancy.
Figure 9 shows the comparison of ignition delay times of
n-decane-air mixtures at 1 and 20 atm produced by detailed
and different sized reduced mechanisms from PFA and
DRG methods. It is seen that at both 1 and 20 atm, PFA has a
significant improvement over DRG in reproducing the ignition
delay time. For species number larger than 73, both PFA and
DRG gave good results. However, for species number less than
70, DRG generated much larger discrepancy than PFA. Similar
results were observed for ignition of n-heptane/air mixtures.
Comparisons of flame speeds and flame structures also showed
that PFA improved the prediction.
The PFA based mechanism reduction can be easily integrated
with the MTS/HMTS method. By using the reduced n-decane
mechanism with 54 species generated by PFA, the unsteady
outwardly propagating spherical flame of the stoichiometric
n-decane mixture was simulated by using HMTS and VODE
with the adaptive simulation of unsteady reactive flow (ASURF)
206
Fig.5 History of temperature and species mass fractions during ignition predicted by different schemes.
Fig.6 Ignition delay time for different pressures and different initial temperatures (n-decane-air).
Fig.7 Comparison of CPU time for n-decane ignition for different initial temperatures at 1atm.
Ju, Yiguang et al.: An On-Grid Dynamic Multi-Timescale Method with Path Flux Analysis for Multi-Physics Detailed Modeling of Combustion
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code [33]. Figure 10 shows the evolution of flame trajectories. It
is seen that there is excellent agreement between the HMTS and
VODE solver in the simulation of flame trajectory. Similar results
were also obtained for detailed n-decane mechanism. However,
the integration of MTS/HMTS with PFA generated reduced
mechanism significantly reduced the computation time.
Conclusions
Simulation of multi-physical combustion problem requires
an efficient multi timescale modeling approach as well as
comprehensive and compact reduced mechanisms. A dynamic
multi time scale (MTS) and hybrid multi time scale (HMTS)
methods were developed to achieve efficient and time accurate
computations of combustion with detailed and reduced
mechanisms. A reaction path based path flux analysis (PFA)
method was also developed to generate computationally efficient
reduced mechanism. Comparisons between the MTS and
HMTS methods with the conventional Euler and ODE solvers
demonstrated that the MTS/HTMS methods are efficient and
accurate. Comparison between the PFA and DRG methods for
n-decane ignition revealed that PFA method had a significant
improvement in generating compact and accurate reduced
mechanisms. Computation of unsteady spherical propagating
flames also showed that the present method is able to accurately
calculate transient flame evolutions. The integration of the MTS/
HMTS methods with the path flux analysis based mechanism
reduction further increases the computation efficiency
dramatically. The present method is general and can be used
directly in DNS and large eddy simulations.
Acknowledgments
This research is jointly supported by the Air Force Office of
Scientific Research (AFOSR) under the guidance of Dr. Julian
Tishkoff and the DoE gas turbine research grant. Yiguang Ju
would like to thank Weinan E and Yannis G. Kevrekidis at
Princeton University and Dr. Viswanath R. Katta (Innovative
Scientific Solutions, Inc.) for many helpful discussions.
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