abstract

DECOUPLED SPECIES AND REACTION REDUCTION:
AN ERROR-CONTROLLED METHOD FOR DYNAMIC ADAPTIVE
CHEMISTRY SIMULATIONS
Oluwayemisi O. Oluwole1, Zhuyin Ren2, Christophe Petre3, Graham Goldin1
1
ANSYS, Inc, 10 Cavendish Court, Lebanon, NH 03766
Center for Combustion Energy and School of Aerospace, Tsinghua University, Beijing, 100084, China
3
ANSYS Belgium, Avenue Pasteur, 4, 1300 Wavre, Belgium
2
Introduction
Detailed kinetic mechanisms of hydrocarbon fuels
are developed to enable accurate computational prediction of combustion performance as well as emissions. Major difficulties are due to the large number
of chemical species and the wide range of timescales
involved in detailed chemistry [1]. Significant progress has been made in methodologies and algorithms
to reduce the computational cost imposed by the use
of detailed chemistry in reacting flow simulations [23]. Of the frequently used approaches, Adaptive
Chemistry methods have recently gained significant
interest, since most combustion solvers decouple the
solution of chemistry and transport terms.
Adaptive Chemistry methods are developed to exploit the time savings available through the use of
various locally-valid skeletal or reduced mechanisms
and have been successfully demonstrated using reduced-model storage and retrieval [4-5], as well as
on-the-fly mechanism reduction [3-6]. However, recent efforts have focused on the latter, thanks to the
success of the directed relation graph (DRG) approach [2-7], which allows fast mechanism reduction
and greatly simplifies the implementation of Adaptive
Chemistry. This paper addresses on-the-fly mechanism reduction, commonly referred to as “Dynamic
Adaptive Chemistry” (DAC) with particular emphasis
on improved accuracy control.
The basic approach of Adaptive Chemistry is to
avoid computations for the species and reactions that
have negligible impact on the kinetics, at each cell in
the computational domain, thus reducing the computational cost. In order to minimize computational
overhead costs, on-the-fly reduction requires a fast
method for mechanism reduction. In this study, we
present a new reduction method designed for DAC, in
which species and reaction reductions are decoupled.
Importantly, the method applies intuitive error controls, alleviating the empiricism typically required in
determining appropriate tolerances for mechanism reduction.
Methodolgy
We consider a reacting gas-phase mixture consisting of NS chemical species. The thermo-chemical state
1
Corresponding author: [email protected]
of the mixture is determined by the pressure p, the mixture temperature T, and the NS-sized vector Y of species
mass fractions. We are concerned with the efficient solution of the (adiabatic and isobaric) reaction step, in
which the composition
of each computational cell evolves according to Neq(= NS + 1) coupled
nonlinear stiff ordinary differential equations (ODEs)
resulting from chemical kinetics,
d k
 Rk ( )
dt
k  1,2 N eq
(1)
Here, Rk is the rate of change of species k due to
chemical reactions, with contributions from the Nrxn
elementary reactions ωi(ϕ) in the mechanism.
The task in the reaction fractional step is to solve
the chemical kinetics initial value problem in Equation
(1) over a flow time step Δt. Equation (1) is typically
stiff due the wide range of timescales and is most often
solved using an implicit ODE solver. The most expensive computations encountered in solving Equation (1)
are due to: 1) the high dimensionality (Neq ≈ NS) of the
ODE system and 2) evaluation of Nrxn >> NS reaction
rates. Our goal is to accelerate these expensive computations by reducing both NS and Nrxn.
Species reduction is performed by “freezing” species whose compositions are not expected to change
significantly over the ODE integration interval Δt. We
apply the criteria in Equation (2):
 d k   * max( 0,k , atol )
(2)

 
 dt

t
t0 

where  and atol are parameters that control accuracy
and precision, respectively. Effectively,  k
t  in Equa-
tion (1) is fixed at its initial value if a single-step forward Euler estimate of its final value is within a fraction  of  k , 0 (to precision atol).
Reactions are eliminated in specially defined
groups, with the constraint that the final reducedreaction mechanism must satisfy Equation (3):
 * max( 0, m , atol )
Rm ( )  Rm (~ ) 
m  1,2neq (3)
t
where neq is the number of species in the reduced
mechanism. By analogy to Equation (2) in the species
reduction step, the a priori estimated local error in
of Jacobian computation approach on overall speedup
due to mechanism reduction may be less pronounced.
~
 m (t+Δt) due to the reduced set of reactions ~ is required to be less than a fraction  of its initial value
 m,0 to a precision of atol.
Results
The mechanism reduction method developed in this
work has been tested on a few example applications including an isobaric auto-ignition of stoichiometric methane/air in 1D, a methane flame based on the co-flow
burner configuration of Bennett et al. [8], and a nheptane ignition and turbulent combustion occurring in
a generic Homogeneous Charge Compression Ignition
(HCCI) engine. For comparison, the DAC simulations
were repeated using the DRG approach of Lu et al. [2].
All the simulations were performed in ANSYS Fluent which (starting with release 15) uses analytic derivatives for computing the Jacobian matrices required in
the implicit kinetics ODE solver. This improves computational efficiency (faster evaluation and better Jacobian accuracy), but has implications for the expected
speedup of any mechanism reduction approach in
DAC. Mechanism reduction offers greater acceleration
for detailed chemistry simulations that apply finite differencing rather than analytic derivatives for Jacobian
computation.
The HCCI engine case related results are illustrated
on Figure 1. DSRR and DRG yielded speedup factors
that generally increased with the mechanism size.
However, in this case DSRR was both faster and more
accurate than DRG for the large mechanism simulation. This may be explained by Figure 1 which shows
that DRG clearly applied much larger models than
DSRR during the power stroke.
Conclusions
We have described a new method (DSRR) for dynamic mechanism reduction that emphasizes solution
error control and decouples species and reaction reductions. DSRR execution is similar in speed to DRG
(linear in the number of reactions in full mechanism),
making it an efficient approach for Adaptive Chemistry with on-the-fly reduction (so-called “Dynamic
Adaptive Chemistry”). We have demonstrated its application in combustion CFD with three examples. In
each case, DSRR yielded similar speedups as DRG,
with improved solution accuracy. We also highlighted
the dependence of the speedup obtainable by mechanism reduction on the Jacobian computation method
(analytic or numerical) in the implicit kinetics ODE
solver. Analytic Jacobian computation increases the
simulation speed (often significantly), but decreases
the opportunities for speedup by mechanism reduction. However, when the number of species is very
large (e.g thousands), linear algebra costs (e.g. matrix
factorization) become more significant and this effect
Figure 1: Average in-cylinder pressure traces (left)
and reduced-model sizes (right) for (top to bottom)
small [9], medium [10] and large [11] mechanism.
Acknowledgements
The authors gratefully acknowledge helpful interactions with Prof. Tianfeng Lu (University of Connecticut). The work by Z. Ren is supported by the startup
fund for the Center for Combustion Energy at Tsinghua
University and by the Young Thousand Talents Program from the Organization Department of the CPC.
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