an on-grid dynamic multi-timescale method with path flux...

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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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日本燃焼学会誌　第 51巻 157号（2009年）

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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

202

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.


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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.

203

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.


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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.


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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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Fig.8 Reduction of n-decane mechanism-Comparison of ignition delay times for various temperatures and pressures between skeletal mechanisms generated by PFA and DRG with detailed mechanism.

Fig.9 Comparison between reduced and detailed mechanisms with different sizes of reduced mechanisms.

Fig.10 Comparison of the transient trajectories of the outwardly propagating spherical n-decane flame by using HMTS and ODE methods with detailed chemistry.

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