c
TU Delft, The Netherlands, 2006
SOURCE TERM EVALUATION OF THE APE-RF SYSTEM
USING DNS DATA
Thanh Phong Bui∗, Wolfgang Schr¨oder∗, Matthias Meinke∗, Hemdan Shalaby†, Dominique Th´evenin†
∗ RWTH Aachen University, Institute of Aerodynamics,
W¨ullnerstr. zw. 5 u. 7, 52062 Aachen, Germany
e-mail: p.bui@aia.rwth-aachen.de web page: http://www.aia.rwth-aachen.de
†Otto-von-Guericke University Magdeburg, Institute of Fluid Dynamics & Thermodynamics,
Universit¨atsplatz 2, 39106 Magdeburg, Germany
web page: http://www.uni-magdeburg.de/isut/LSS
Key words: Combustion Noise, Acoustic Perturbation Equations for Reacting Flows, APE-RF, CAA, Direct Numerical Simulation
Abstract. Acoustic perturbation equations for reacting flows (APE-RF) in conjunction with direct numerical simulations (DNS) are used to investigate in detail the thermo-acoustic effects resulting from turbulent premixed flames. The basic procedure is a two-step DNS/APE-RF method, where the flow is simulated by direct numerical simulations and the acoustic analysis is performed using the APE-RF system. Based on the DNS data, the source terms of the APE-RF system can be thoroughly evaluated, since the full chemical reaction is taken into account in the DNS. The acoustic impact of several source mechanisms are investigated, such as the effect of unsteady heat release, that of heat flux, viscous effects, the effect of non-isomolar combustion, and that of species diffusion. The study shows the unsteady heat release to be the dominant source. All source terms but the heat diffusion term possess a monopole-like structure in the low frequency range. At high frequencies a multipole-like pattern is also determined for the sources due to species diffusion and viscous effects. This deviation from the monopole structure is caused by the chemical reaction time scales. It is shown in this study that the radiated acoustic energy is in good agreement comparing the impact of the total time derivative of the density as major source term with the unsteady heat release rate.
1 INTRODUCTION
unsteady heat release rate, which is well known to be the most relevant source as to acoustic radiation of open turbulent flames[13, 14, 15]. The comparison of the simulated radial intensity with experimental data shows a convincing agreement[3].
Using the energy equation for reacting flows, the total time derivative of the density can be separated into several terms which describe the unsteady heat release rate, species and heat diffusion, viscous effects, and a source, which represents the effect of non-isomolar combustion[19]. These source mechanisms are analyzed in this study. For this purpose, the data of a direct numerical simulation (DNS) of a premixed flame has been used. The DNS of the premixed flame is based on the complete chemical reaction mechanism. No look-up tables are used to reduce the reaction data so that the necessary information for computing all source terms of the APE-RF formulation is available. However, the required number of grid points and the computational costs induced by the full chemical reaction scheme restrict the DNS to a very focused spatial domain. Since the information on all species is directly provided in the DNS, the aforementioned source terms can be easily evaluated for the CAA analysis.
The purpose of the paper is to determine the major source terms, to evidence the direc-tivity patterns of the various sources over a wide frequency range and to relate them to the classical pole-like structure in acoustics. Moreover, it will be shown that using the total time derivative of the density as major source term is justified.
The paper is organized as follows. First, the governing equations for reacting flows are given. Second, the acoustic perturbation equations for reacting flows are discussed with special emphasis on the source terms. Then, the DNS and CAA approaches are outlined, followed by the discussion of the results. Finally, some conclusions are drawn.
2 GOVERNING EQUATIONS FOR REACTING FLOWS
Chemically reacting flows are described by a set of coupled partial differential equations, i.e., the conservation of mass, chemical species, momentum, and energy. These equations read ∂ρ ∂t + ∇ · (ρu) = 0 (1) ∂ρYk ∂t + ∇ · (ρYku) = −∇ · (ρYkVk) + Mkωk with k = 1, ..., Ns (2) ∂ρu ∂t + ∇ · (ρuu) = −∇p + ∇ · τ (3) ∂eτ ∂t + ∇ · ((eτ + p)u) = −∇q + ∇ · (τ · u) . (4)
The total energy eτ and the production rate ωk of species k are given by
and ωk = Nr X λ=1 kλ νkλ(p)− νkλ(r) Ns Y k=1 cν (r) kλ k . (6)
The rate coefficients kλ of the elementary reactions are given by a modified Arrhenius law
kλ = AλTbλexp −Eaλ RT , (7)
while the system of equations is closed by the ideal gas law
p = ρ
MRT. (8)
The parameters Aλ, bλand the activation energy Eaλare determined by a comparison with
experimental data. Multicomponent diffusion velocities and viscosity are computed using standard expressions[21]. Thermodynamical properties are determined by fifth-order fits of experminental data[21].
3 ACOUSTIC PERTURBATION EQUATIONS FOR REACTING FLOWS
The acoustic perturbation equations for reacting flows (APE-RF) are based on the homogeneous APE system developed by Ewert and Schr¨oder[5]. For combustion noise simulations this homogeneous system has been chosen to take advantage of its benign properties to simulate wave propagation, i.e., its validity for non-uniform mean flows, while instabilities are prevented. Using the decomposition of the density (ρ), the pressure (p), and the velocity (u) into a mean, denoted by an overbar, and a fluctuating part, denoted by a prime,
ρ = ρ + ρ0 (9)
p = p + p0 (10)
u = u + u0 (11)
the APE system[5] reads
∂ρ0 ∂t + ∇ · (ρ 0 u + ρu0) = qc,rf (12) ∂u0 ∂t + ∇ (u · u 0 ) + ∇ p 0 ρ ! = qm,rf (13) ∂p0 ∂t − c 2∂ρ 0 ∂t = qe,rf (14)
where the sources qc,rf, qm,rf, and qe,rf are zero when the homogeneous formulation is
rearranged such that the left-hand side describes the original homogeneous APE system, whereas the right-hand side (RHS) consists of all non-linear flow effects including the sources related to chemical reactions. After some algebra, which is described in detail by Bui et al.[3], the resulting source terms read
qc,rf = −∇ · (ρ0u0) 0 (15) qm,rf = − (ω × u) 0 − ∇(u 0)2 2 !0 + ∇ · τ ρ !0 + T0∇s − s0∇T + X n Ykfk !0 + Xµk Mk 0 · ∇Yk − X Yk0· ∇ µ k Mk ! (16) qe,rf = −c2 ρ ρ· α cp · N X n=1 ∂h ∂Yn ρ,p,Y m ρDYn Dt +∇ · q − ∂ui ∂xj τij ! −∇ · (uρe) − 1 c2 " 1 − ρc 2 ρc2 ! · Dp Dt − p − p ρ · Dρ Dt # + " −γ − 1 γ u · ∇ρ − p c2 · u ∇p p − ∇ρ ρ !# # . (17)
The total time derivative of the density Dρ Dt = 1 c2 Dp Dt + α cp · N X n=1 ∂h ∂Yn ρ,p,Y m ρDYn Dt + ∇ · q − ∂ui ∂xj τij (18)
can be decomposed into six parts. Since the density fluctuation includes the substantial time derivative of the pressure, the remaining five terms and the substantial time deriva-tive of the density are examined in this study. Note, all the following sources are included in the RHS of the pressure-density relation (14). Plugging in α/cp = (γ − 1)/c2 for an
ideal gas[4] the following relations constitute the subsources of the qe,rf term and describe
the effects of
unsteady heat release rate qe,rf,1 = −c2
ρ ρ γ − 1 c2 N X n=1 hnωn, (19)
non-isomolar combustion qe,rf,2 = −c2
ρ ρ γ − 1 c2 cpT W N X n=1 ωn Wn , (20)
species diffusion qe,rf,3 = −c2
ρ ρ γ − 1 c2 N X n=1 hn∇ · Jn− cpT W N X n=1 ∇ · Jn Wn ! ,(21)
heat diffusion qe,rf,4 = −c2
ρ ρ
γ − 1
c2 ∇ · (−λ∇T ) , (22)
and the effect of viscosity qe,rf,5 = −c2
4 DNS COMPUTATION
4.1 Flame configuration
The combustion configuration is a turbulent premixed CO/H2/Air flame. The
numer-ical domain is a square box of dimension 8×8 mm2 as shown in Fig.1. A fixed mesh of
251 points is used in each direction. This leads to a spatial resolution of 32 µm, necessary to resolve intermediate radicals. The left-hand boundary condition is a subsonic inlet at an imposed velocity of 1m/s, while on the right-hand boundary a non-reflecting subsonic outlet condition is formulated. A complete reaction scheme with 13 species (CO, HCO, CH2O, CO2, H2O, O2, O, H, OH, HO2, H2O2, H2, N2) and 67 individual reactions is
taken into account. First, the corresponding premixed laminar flame is computed for a one-dimensional flow along the x-direction and the inlet velocity is adapted to keep the flame front in the center of the domain. Then, the obtained steady solution is transposed to a 2-D flow with fresh gases on the left side and burnt gases on the right side. An isotropic 2-D turbulent velocity field is superposed using a von K´arm´an spectrum coupled with Pao correction for near-dissipation scales [18, 6, 7].
The DNS computation is performed for a two-dimensional flow field. Since turbulence is in general a three-dimensional phenomenon, some of its aspects, like vortex stretching, cannot be captured with such a simulation. However, the purpose of this study is to most precisly describe the thermo-acoustic sources, such that complete chemical reaction schemes must be employed at a reasonable computational time. Today this is only pos-sible in two-dimensional DNS. Moreover, the coupling processes between flow, chemical reactions, and pressure waves are the same in two and three space dimensions, such that all qualitative features will be preserved. Quantitative values might nevertheless differ in a three-dimensional simulation, even if some studies tend to prove that three-dimensional flames show preferentially a cylindrical and thus, a locally two-dimensional structure.
4.2 Numerical methods
The simulation has been performed using the parcomb code developed by Th´evenin et al. [17, 18]. It is a finite-difference DNS code solving the compressible Navier-Stokes equations for multicomponent reacting flows. Derivatives are computed using sixth-order approximations except at boundaries where the discretization is fourth order accurate. The temporal integration is realized by a fourth-order 4-stage Runge-Kutta algorithm. Boundary conditions are formulated using the Navier-Stokes Characteristic Boundary Condition (NSCBC) technique [12] extended to take into account multicomponent ther-modynamic properties [1]. Transport coefficients and chemical kinetics are treated similar to the methods used in CHEMKIN II and TRANSPORT [9, 10]. The DNS code has been parallelized and used successfully to investigate turbulent flames[18, 6, 11]. Parcomb has already been validated for H2-O2-N2 premixed flame velocities of various compositions.
fresh gas burnt gas y x outlet inlet periodicity periodicity
Figure 1: Computational domain and boundary notation of the DNS
0 50 100 150 200 250 300 350 400 0,2 0,3 0,4 0,5 0,6 0,7 Jahn Bartholome Scholte and Vaags Burwasser and Pease Guenther and Janisch Andrews and Bradley
Flame velocity (cm/s)
X(H ) Miller, Evers and Skinner Gibbs and Calcote Smith and Pickering Edmondson and Heap Senior
calculation
2
Figure 2: Comparison of experimental and computed H2− O2− N2 flame velocities. References of the
experimental results are listed in [20]
to provide detailed information to determine the sources of the APE-RF system.
5 CAA COMPUTATION
The CAA code is based on the fourth-order dispersion-relation preserving (DRP) scheme of Tam and Webb[16] for the spatial discretization and the alternating low-dissipation low-dispersion Runge Kutta (LDDRK) method in the 5/6 mode for the tempo-ral integration by Hu et al.[8]. The CAA computation has been performed on a 13-block grid with approximately 3.5·105grid points. The inner block boundaries coincide with the
of 5 × 10−6s they have to be interpolated in time during the CAA computation. This is done via a quadratic interpolation algorithm. The DNS data are non-dimensionalized using following reference values
ρref = 1.1766 kg/m3 (24)
cref = 347.4 m/s (25)
Tref = 300 K (26)
Lref = 0.008 m. (27)
For the CAA computations mean flow effects are neglected.
6 RESULTS
The discussion of the results is organized as follows. First, the structure of the flame front is shown to evidence the unsteady behavior of the turbulent combustion. Then, the acoustical field generated by several source term compositions on the RHS of the pressure-density relation are discussed with emphasis on the presence of characteristic frequencies, the directivity behavior of each single source term in Eqs. (19)-(23), and the justification of using the total time derivative of the density as major source term instead of the unsteady heat release rate.
6.1 Premixed flame
The typical temporal evolution of the premixed flame is indicated in Fig. 3. To visu-alize the structure the instantaneous heat relase rate has been chosen. The flame front
x (cm) y (c m ) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 heat release 4.2E+09 4E+09 3.8E+09 3.6E+09 3.4E+09 3.2E+09 3E+09 2.8E+09 2.6E+09 2.4E+09 2.2E+09 2E+09 1.8E+09 1.6E+09 1.4E+09 1.2E+09 x (cm) y (c m ) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 heat release 4.2E+09 4E+09 3.8E+09 3.6E+09 3.4E+09 3.2E+09 3E+09 2.8E+09 2.6E+09 2.4E+09 2.2E+09 2E+09 1.8E+09 1.6E+09 1.4E+09 1.2E+09
Figure 3: Instantaneous spatial distribution of the heat release in a premixed CO/H2/Air flame after
x (cm) y (c m ) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Y(H2O2) 3.8E-05 3.6E-05 3.4E-05 3.2E-05 3E-05 2.8E-05 2.6E-05 2.4E-05 2.2E-05 2E-05 1.8E-05 1.6E-05 1.4E-05 1.2E-05 1E-05 x (cm) y (c m ) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Y(HO2) 0.00019 0.00018 0.00017 0.00016 0.00015 0.00014 0.00013 0.00012 0.00011 1E-04 9E-05 8E-05 7E-05 6E-05
Figure 4: Instantaneous spatial distribution of H2O2and HO2mass fractions in a premixed CO/H2/Air
flame after interacting with a turbulent velocity field for t = 1.0 ms
Species Order of chemical Species Order of chemical
time scale [s] time scale [s]
CO 1.9D-7 HCO 5.5D-8 CH2O 1.0D-5 CO2 1.8D-6 H2O 1.5D-7 O2 1.6D-6 O 1.3D-7 H 1.1D-7 OH 1.5D-7 HO2 5.6D-6 H2O2 1.2D-7 H2 1.0D-5
Table 1: Comparison of the chemical time scales of each species
gets strongly stretched and curved forming locally cusp-like structures by interacting with the turbulence field, leading to considerable modifications of the heat release field up to almost local extinctions. In Fig. 4 the spatial distribution of the H2O2 and HO2 mass
fraction after an interaction duration of t = 1.0 ms are presented.
Note, the temporal evolution of the source term qe,rf,3, which represents the species
dif-fusion term, is highly influenced by species creation or species consumption along the reacting zone. Like mentioned above, a reaction scheme with 13 species and 67 individual reactions is taken into account. The maximum DNS time step is limited by the fastest reaction, which is of order O(10−8) seconds, while the slowest reaction time is still of order O(10−5) seconds. The range of chemical time scales is shown in Tab. 1. These chemical
f [Hz] S P L [d B ] 0 10000 20000 0 20 40 60 qe,rf,1 qe,rf,1-qe,rf,5 drhodt
Figure 5: Comparison of SPL spectra using different source term formulations
6.2 Acoustical field
The acoustical field is analyzed by directivity patterns and power spectral density plots. Figure 5 shows the sound pressure level (SPL) spectrum of three different compu-tations using three different source term compositions of the RHS of the pressure-density relation. The first computation was done using qe,rf,1 only, while the second and third
simulation were performed using all sources, i.e., qe,rf,1 through qe,rf,5 and the substantial
time derivative of the density, respectively. First of all, it is evident from Fig. 5 that the major source term is the unsteady heat release rate,i.e., qe,rf,1, since the contribution
of the other thermoacoustic sources in Eqs. (20)-(23) to the radiated acoustic field is on average less than 2dB.
Using the total time derivative of the density as source term of the pressudensity re-lation the spectrum distribution is in good agreement in comparison with the other two simulations. Especially in the frequency range close to 10kHz, the sound pressure level amplitude is underestimated by approx. 5dB to the SPL values caused by the source qe,rf,1.
6.2.1 Characteristic frequencies
Each single source term is analyzed by Fourier transforming the perburbation pressure at a point with a radial distance to the center of the domain of R/L = 4. In Fig. 6(a) it can be observed that the maximum acoustic energy generated by the unsteady heat release rate qe,rf,1, can be found at approximately 1kHz. From the sound pressure
spectrum of the source describing the effect of non-isomolar combustion qe,rf,2 in Fig.
(a) f [Hz] P S D [-] 0 5000 10000 15000 20000 0.00E+00 5.00E-05 1.00E-04 1.50E-04 qe,rf,1 (b) f [Hz] P S D [-] 0 5000 10000 15000 20000 0 5E-06 1E-05 1.5E-05 2E-05 qe,rf,2 qe,rf,3 qe,rf,4 (c) f [Hz] P S D [-] 0 5000 10000 15000 20000 0.00E+00 6.00E-11 1.20E-10 1.80E-10 qe,rf,5
decay. The remaining three spectra shown in Figs. 6(b) and 6(c) definitely evidence the existence of characteristic frequencies. The characteristic frequencies of the source qe,rf,3
are approx. 550 Hz, 5.3 kHz and 9.5 kHz, while the source terms qe,rf,4 and qe,rf,5 show a
peak value at approx. 5 kHz and 4.3 kHz, respectively. 6.2.2 Directivity patterns
From the directivity patterns in Fig. 7 it is apparent that the frequencies containing the major acoustic energy content are of monopole type. In general all source terms show a monopole behavior in the low frequency range but the heat diffusion source term qe,rf,4,
which includes a non-monopole-like sound field even in the frequency range 1.6kHz < f < 2.6kHz. The acoustic field generated by the sources qe,rf,3 and qe,rf,5 illustrated in Figs.
7(c), (d), (e), (h), (i), and (j) show a multipole behavior in the very high frequency range only, i.e., at 15kHz and higher. That frequency range correlates highly with the chemical time scales which were mentioned above. The species diffusion term shows monopole-like directivity patterns in the low frequency range, while the multipole behavior is dominated by the effect of species creation and consumption along the reaction zone. Since the chemical reaction time scales are extremely small, i.e., 10−8s < t < 10−5s, their acoustic impact can be found in the very high frequency range starting from approx. 21kHz.
7 CONCLUSIONS
The comparison of the power spectral density of three acoustic simulations using dif-ferent source terms shows the total time derivative of the density, which is the major source term as far as the acoustic radiation of reacting flows is concerned, to be in good agreement with the acoustic impact of the unsteady heat release rate or the summation of all sources examined in this study. Moreover, all source terms but the heat diffusion source term possess a monopole character in the low frequency band, whereas the latter has a multipole behavior in the frequency range of 1.6kHz < f < 2.6kHz. Since the time scale of the classical species diffusion is very large compared to the chemical reaction time scale, it can be concluded, that in this case “ordinary” species diffusion shows a monopole behavior, while the effect of species creation and species consumption is of multipole type.
8 ACKNOWLEDGMENTS
(a) qe,rf,1 0 30 60 90 120 150 180 210 240 270 300 330 0 2E-05 4E-05 6E-05
1 kHz 3 kHz 5 kHz 7 kHz 9 kHz (b) qe,rf,2 0 30 60 90 120 150 180 210 240 270 300 330 0 3E-06 6E-06 9E-06
1 kHz 3 kHz 5 kHz 7 kHz 9 kHz (c) qe,rf,3 0 30 60 90 120 150 180 210 240 270 300 330 0 3E-066E-06 9E-06
1 kHz 3 kHz 5 kHz 7 kHz 9 kHz (d) qe,rf,3 0 30 60 90 120 150 180 210 240 270 300 330 0 4E-088E-08 1.2E-07
21 kHz 22 kHz (e) qe,rf,3 0 30 60 90 120 150 180 210 240 270 300 330 0 2E-08 4E-08 6E-08
25 kHz 29 kHz (f) qe,rf,4 0 30 60 90 120 150 180 210 240 270 300 330 0 4E-07 8E-07 1.2E-06
1.0 kHz 1.6 kHz (g) qe,rf,4 0 30 60 90 120 150 180 210 240 270 300 330 0 4E-07 8E-07 2.1 kHz 2.6 kHz (h) qe,rf,5 0 30 60 90 120 150 180 210 240 270 300 330 0 8E-13 1.6E-12 15.3 kHz 16.4 kHz (i) qe,rf,5 0 30 60 90 120 150 180 210 240 270 300 330 0 1.3E-12 2.6E-12 3.9E-12
23.3 kHz 27.0 kHz (j) qe,rf,5 0 30 60 90 120 150 180 210 240 270 300 330 0 1E-12 2E-12 3E-12
29.7 kHz 38.7 kHz
REFERENCES
[1] M. Baum, T. Poinsot, and D. Th´evenin. Accurate boundary conditions for multi-component reactive flows. J. Comput. Physics, 116:247–261, 1994.
[2] T. P. Bui, M. Meinke, W. Schr¨oder, F. Flemming, A. Sadiki, and J. Janicka. A hybrid method for combustion noise based on LES and APE. In AIAA Paper 2005-3014, Monterey, CA, May 2005. AIAA.
[3] T. P. Bui, W. Schr¨oder, and M. Meinke. Acoustic perturbation equations for reacting flows to compute combustion noise. submitted to Int. J. Aeroacoustics, 2005.
[4] D. G. Crighton, A. P. Dowling, J. E. Ffowcs Williams, M. Heckl, and F. G. Lepping-ton. Modern Methods in Analytical Acoustics. Springer, 1992.
[5] R. Ewert and W. Schr¨oder. Acoustic perturbation equations based on flow decom-position via source filtering. J. Comput. Physics, 188:365–398, 2003.
[6] R. Hilbert and D. Th´evenin. Autoignition of turbulent non-premixed flames investi-gated using direct numerical simulations. Combust. Flame, 128:22–37, 2002.
[7] J. O. Hinze. Turbulence. 2nd edition, McGraw-Hill, 1975.
[8] F. Q. Hu, M. Y. Hussaini, and J. L. Manthey. Low-dissipation and low-dispersion runge-kutta schemes for computational acoustics. J. Comput. Physics, 124(1):177– 191, 1996.
[9] R. J. Kee, J. A. Miller, and T. H. Jefferson. CHEMKIN, A general purpose problem-independent transportable FORTRAN chemical kinetics code package, SAND80-8003. Sandia National Laboratories Report, 1980.
[10] R. J. Kee, J. Warnatz, and J. A. Miller. A FORTRAN Computer Code Package for the Evaluation of Gas-Phase Viscosities, Conductivities, and Diffusion Coefficients, SAND83-8209. Sandia National Laboratories Report, 1983.
[11] U. Maas and D. Th´evenin. Correlation analysis of direct numerical analysis data of turbulent non-premixed flames. Proc. Combust. Inst., 27:1183–1189, 1998.
[12] T. Poinsot and S. Lele. Boundary conditions for direct simulations of compressible viscous flows. J. Comput. Physics, 101:104–129, 1992.
[14] T. J. B. Smith and J. S. Kilham. Noise generation by open turbulent flames. J. Acoust. Soc. Am., 35:715–724, 1963.
[15] W. C. Strahle. Some results in combustion generated noise. J. Sound Vibration, 23(1):113–125, 1972.
[16] C. K. W. Tam and J. C. Webb. Dispersion-relation-preserving finite difference schemes for computational acoustics. J. Comput. Physics, 107(2):262–181, 1993. [17] D. Th´evenin, F. Behrendt, U. Maas, B. Przywara, and J. Warnatz. Development
of a parallel direct simulation code to investigate reactive flows. Comput. & Fluids, 25:485–496, 1996.
[18] D. Th´evenin, E. van Kalmthout, and S. Candel. Two-dimensional direct numerical simulations of turbulent diffusion flames using detailed chemistry. In Direct and Large-Eddy Simulation II: (Chollet, J.P., Voke, P.R. and Kleiser, L., Eds.), pages 343–354. Kluwer Academic Publishers, 1997.
[19] J. M. Truffaut, G. Searby, and L. Boyer. Sound emission by non-isomolar combustion at low mach numbers. Combust. Theory Modelling, 2:423–428, 1998.
[20] J. Warnatz. Calculation of the structure of laminar flat flames II:flame velocity and structure of freely propagating hydrogen-oxygen and hydrogen-air flames. Ber. Bunsenges. Phys. Chem., 82:642–649, 1978.
