A method is presented to significantly improve the convergence behavior of batch nonpremixed counterflow flame simulations with finiterate chemistry. The method is applicable to simulations with varying pressure or strain rate, as it is, for example, necessary for the creation of flamelet tables or the computation of the extinction point. The improvement is achieved by estimating the solution beforehand. The underlying scaling rules are derived from theory, literature, and empirical observations. The estimate is used as an initialization for the actual solver. This enhancement leads to a significantly improved robustness and acceleration of batch simulations. The extinction point can be simulated without cumbersome code extensions. The method is demonstrated on two test cases and the impact is discussed.
Laminar steady nonpremixed counterflow flames have been investigated in numerous studies in the past [
From counterflow flame simulations, detailed insight is gained into the structure of nonpremixed flames, including temperature and species profiles [
Even though counterflow flame simulations usually run on a very small grid, the computation is time consuming because of the stiff equation system resulting from the detailed chemical mechanism. If the initialization is too far from the steady state solution, the solvers are prone to diverge.
A major aim of counterflow flame simulations is to compute extinction strain rates [
For both the computation of extinction strain rates and the creation of flamelet libraries, multiple simulations at varying strain rates are necessary. The operating pressure is often an investigated parameter as well. This can result in a large matrix of input parameters for which separate simulations are required. While the computational effort might still be acceptable for single simulations, it can be very large for the overall simulation campaign. A reduction of the computational effort is therefore favorable.
This paper develops a method to stabilize and accelerate batch counterflow flame simulations. These goals are achieved by improving the initialization for the iterative steps. Depending on the pressure and the strain rate changes, estimations for the change of the velocity, species, and temperature profiles are derived from the literature, theoretical considerations, and empirical observations.
To properly deduce the scaling rules, the setup of the counterflow flame and its definitions are briefly described first. The approach to improve the batch computations is then derived theoretically and finally put into an applicable form.
A schematic view of the considered counterflow flame configuration is shown in Figure
Schematic view of the counterflow flame configuration.
The wellknown governing partial differential equations are described by Kee et al. [
The flow field is fully characterized by the axial coordinate
Socalled plugflow boundary conditions are applied on both the oxidizer and the fuel inlet [
The counterflow flame is defined by setting the boundary conditions, the grid size, the operating pressure, and a chemical reaction mechanism. Counterflow flames are commonly characterized by their strain rate
In the plugflow configuration, the potential flow boundary conditions can be approximated by
With all parameters defined, counterflow flames can be computed with standard software like OPPDIFF [
As pointed out in the introduction, for the calculation of extinction strain rates or the generation of flamelet tables, a large number of counterflow flames have to be simulated. This is generally done by performing a batch simulation.
The idea of improving the convergence behavior of batch simulations is to estimate the solution beforehand. This guess is used as the initialization for the actual computation. The estimation is based on the assumption that the structure of flames with finiterate chemistry behaves similarly to the structure of flames with infinitely fast chemistry. In nonpremixed flames, the structure is mostly influenced by diffusion which is equally represented in flames with finiterate and infinitely fast chemistry.
Counterflow diffusion flames with infinitely fast chemistry are known to have a selfsimilar structure [
This formulation is equivalent to the postulation that the diffusion flame thickness (here, full width at half maximum (FWHM) of the temperature profile [
To apply this on batch simulations, an existing solution with the strain rate
However, instead of scaling the flame on a fixed grid, the grid size
In contrast to the temperature and species profiles, the magnitudes of the velocity profiles, the pressure curvature, and the boundary conditions have to be updated. Using the definition of the mean strain rate
The equations derived above are summarized in Table
Overview over the magnitude scaling parameters.
































The method described above is implemented on top of the Cantera environment (version 2.2a, revision r2777 [
The two benefits of the proposed method, the speedup and the robustness, are demonstrated using two test cases described in the following.
To indicate the speedup achieved by the proposed method, a series of counterflow flames at various pressures and strain rates is simulated. This could be the basis for the creation of a flamelet table. Since the computation of the underlying counterflow flames is the most computationally expensive step in the creation of a flamelet library, such a task is assumed to speed up accordingly. The acceleration is shown by comparing a batch simulation which incorporates the proposed method with a stateoftheart benchmark simulation. This is demonstrated for a hydrogenoxygen flame and an ethaneair flame. The overview over the simulation parameters is given in Table
The initial parameters of the example counterflow flame simulations.
Speedup H_{2}  Speedup C_{2}H_{6}  Extinction point  

Fuel  H_{2}  C_{2}H_{6}  H_{2} 
Oxidizer  O_{2}  Air  O_{2} 
Reaction mechanism 
Ó Conaire et al. [ 
GRIMECH 3.0 [ 
Li et al. [ 

1  1  1 

300  300  300 

3.6  20  18 

0.50  0.24  0.10 

3.00  0.72  0.60 
Starting from initial counterflow flames at a pressure of 1 bar, flames up to 100 bar should be calculated. The pressure is increased by approximately 15% in each step. Additionally, at 13 of the previously computed pressure levels, isobaric flames should be calculated at increasing strain rates. The strain rates are raised by 25% in each step until the flame is extinguished.
For each fueloxidizer configuration, the test case is run two times. The accelerated run makes use of the procedure described in this paper. In the corresponding benchmark run, the initialization procedure is not applied. The pressure and strain rate are set in this case by changing only the operating pressure and the inlet mass fluxes, respectively. Both cases are run sequentially on a standard desktop PC (housing an Intel i5 CPU at 3.2 GHz) under standard load conditions.
For the hydrogenoxygen flame, the benchmark computation fails to converge for pressures above 64 bar. At this point, the cumulated simulation time already amounts to 14:12 minutes. The accelerated run runs smoothly for all pressure levels; at 64 bar the cumulated simulation time is only 43 seconds. For the ethaneair flame, both variants are able to compute all pressure levels. The simulation time for the entire loop is accelerated from 2:47:35 hours to 15:15 minutes using the proposed method. The values are compared in Table
Simulation times for the pressure loops and the average strain rate loops with and without the scaling rules applied.
With scaling rules 
Without scaling rules (benchmark)  

Pressure loop H_{2}O_{2} (64 bar) 





Pressure loop C_{2}H_{6}air 





Strain rate loop H_{2}O_{2} 





Strain rate loop C_{2}H_{6}air 




The loops through the increasing strain rates profit even more from the preconditioning step. For the hydrogenoxygen flame, the average time for an isobaric loop decreases from 1:49 minutes to only 5 seconds. For the ethaneair flame, the corresponding times are 3:55:29 hours and 4:08 minutes. At pressures above 10 bar for the hydrogenoxygen flame and above 30 bar for the ethaneair flame, the solver diverges in the strain rate loop in the benchmark run. This problem does not occur for the accelerated run. This fact also indicates the improvement in robustness achieved by using the derived scaling rules.
Figures
Profiles of the normalized axial velocity
Profiles of the normalized axial velocity
A second independent test case is evaluated to demonstrate the robustness increase achieved by this method. This makes it possible to compute the extinction point of counterflow flames very accurately without having to apply a cumbersome method to circumvent the singularity at this point.
For multiple flames at constant pressure levels, the strain rate is increased gently until the flame is extinguished. From the last burning flame, the strain rate is increased in smaller steps to get closer to the extinction point. This iteration is repeated until the temperature change is below a threshold value. The strain rate and the maximum temperature of the last burning solutions are taken as the extinction strain rate and extinction temperature.
This is demonstrated for a hydrogenoxygen flame. The boundary and operating conditions of the initial simulation are given in Table
Maximum strain rates and maximum temperatures for H_{2}O_{2} counterflow flames just before the extinction point as a function of pressure. The reference data is taken from Wang et al. [
In this paper, a method is described which significantly improves the convergence behavior of batch counterflow flame simulations with varying pressure and strain rate. This is achieved by estimating the initial solutions from previously computed simulations. The scaling parameters for the solution vectors with respect to the pressure and the strain rate are derived from existing relations for flames with infinitely fast chemistry, dimensional analysis, and empirical observations.
The method is demonstrated using the Cantera software package [
The method is also used to compute the extinction point of a hydrogenoxygen counterflow flame. The result very accurately matches data from the literature. Using the described simple procedure, the extinction point can be calculated without having to use cumbersome continuation methods.
Diffusion coefficient [m^{2}/s]
Temperature [K]
Radial velocity potential [1/s]
Mass fraction [—]
Strain rate [1/s]
Isobaric heat capacity [J/(kg K)]
Grid size [m]
Mass flux density [kg/(s m^{2})]
Pressure [Pa]
Axial velocity [m/s]
Axial coordinate [m]
Pressure curvature [N/m^{4}]
Flame thickness [m]
Heat conductivity [W/(m K)]
Density [m^{3}/kg]
Fuel inlet
Oxidizer inlet.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The Deutsche Forschungsgemeinschaft (DFG) in the framework of Sonderforschungsbereich Transregio 40 provided financial support for this project. This work was supported by the German Research Foundation (DFG) and Technische Universität München within the funding programme Open Access Publishing.