5.2 Direct Numerical Simulations
5.2.2 Finite Difference Discretization Scheme
The computation of finite differences of order of six [53] (third-order at the boundaries) is particularly important in direct numerical simulations. The
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64 5 Direct Numerical Simulation
mesh can be equidistant or non-equidistant in space. The first and second derivatives are evaluated using the following relationships as summarized in [52]:
The second order derivative can be written in the same manner:
∂2f
The expression for first order derivative can be formulated in the core (inner) part of the domain:
∂f
∂ξ = (fi+3 −fi−3) + 9 (fi−2 −fi+2) + 45 (fi+1−fi−1)
60 . (5.4)
At the left boundary, we get:
∂f
while the right boundary is expressed as:
∂f
5.3 Computational Details 65
Derivatives of∂ξ/∂x appeared Eq. (5.2) can also be evaluated using the above presented finite difference schemes.
The second derivative in the domain is expressed as:
∂2f
∂ξ2 =�
2 (fi−3 −fi−2) + (fi+3 −fi+4) + 25 (fi−1−fi−2) + (fi+1 −fi+2) + 245 (fi−1−fi) + (fi+1 −fi)�
/180 . (5.11) The second order derivative at the left boundary is written as:
∂2f
similarly, the second order derivative at the right boundary is given by:
∂2f The derivatives of ∂2ξ/∂x2 appeared in Eq. (5.3) are also evaluated using the above presented finite difference schemes. An identical approach can be deduced for directions y and z.
5.3 Computational Details
Direct Numerical Simulations (DNS) have been conducted to study the re-sponse of initially laminar spherical premixed methane-air flame kernels to successively higher turbulence intensities at five different equivalence ratios.
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66 5 Direct Numerical Simulation
The numerical experiments include a 16-species/25-step skeletal mechanism for methane oxidation and a multicomponent molecular transport model. Highly turbulent conditions (with integral Reynolds numbers up to 4 513) have been accessed and the effect of turbulence on the physical structure of the flame, in particular the consumption speedSc (suitably scaled to quantify the turbulent flame speed ST), has been investigated [29, 28].
Initially perfectly spherical laminar premixed methane–air flames are con-sidered in all computations, within a cubic computational domain of sides L= 4.0 cm (Fig. 5.1) and a uniform grid spacing of 20 – 35 µm for the mild to the most intense turbulent cases, respectively, ensuring the full resolution of the smallest (Kolmogorov) scales, since their length decreases with increasing Ret. The three-dimensional computation shown in Fig. 5.1 has been realized on a grid with 400×400×400 finite difference nodes. The turbulent Reynolds number Ret is 1 400 and u� is 5 m/s. It has been shown [25, 26, 68] that the reaction zone is broad, if not completely homogeneous, making it possible to relax the constrains on the spatial resolution of the reaction zones, since there are no steep flame fronts [83]. The resolution is then purely dictated by the turbulent scales. The ignition and subsequent expansion/development of a premixed flame-kernel under the influence of a turbulent flow field is an excellent configuration, which allows turbulent flames to be studied well away from the influence of external perturbations such as walls and artificial bound-ary conditions. From a fundamental point of view, it offers the possibility to study highly complex multi-scale flows involving fully coupled physical pro-cesses. Simultaneously, it has direct practical relevance in a number of indus-trial cases including spark-ignition internal combustion engine and gas turbine re-ignition, as well as safety issues.
Methane oxidation is modeled by a 25-step skeletal scheme [12, 75], com-prised of 4 elements (H, C, O, N), 16 chemical species (CH4, O2, H2, H2O, CH2O, CO, CO2, HO2, OH, H, O, CH3, HCO, H2O2, CH3O, N2) and 50 el-ementary reactions (Appendix A). This reaction mechanism is retained here due to its simplicity compared to a full methane oxidation mechanism like the GRI-MECH [13] and provides sufficiently accurate results for lean up to stoichiometric conditions. It has been successfully used for large scale direct simulations of two-dimensional non-premixed methane jet flames [41] and most recently, highly turbulent premixed flames [26, 25]. However, it would be inad-equate for methane–rich flames due to the absence of C2 and higher carbon-chain reactions, the reason why Φ ≤ 1.0 for the present study.
The initial mixture composition (Yi), prescribed burnt (Tb) and unburned (Tu) temperatures, laminar flame speedsL, thermal flame thicknessδth = (Tb− Tu)/max|∇T| and a Karlovitz number estimate Ka≈ [(u�/sL)3(lt/δth)]1/2 for
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5.3 Computational Details 67
Table 5.1: Initial flame and flow parameters
Φ Tb(K) Tu(K) YCH4 YO2 YCO2 YH2O sL(m/s) δth(mm) Ka 1.0 2 230 300 0.055 0.220 0.137 0.120 0.507 0.36 0.94–96.7 0.9 2 140 300 0.049 0.221 0.133 0.111 0.426 0.40 1.27–81.4 0.8 2 002 300 0.044 0.223 0.122 0.099 0.329 0.46 2.01–105.2 0.7 1 844 300 0.039 0.224 0.108 0.088 0.228 0.56 3.84–159.7 0.6 1 669 300 0.034 0.225 0.093 0.076 0.136 0.78 9.84–409.2
Table 5.2: Initial turbulence parameters
cases 1 2 3 4 5 6 7 8 9 10 11 12
u� (m/s) 1.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0 τ (ms) 3.20 1.60 0.80 0.53 0.40 0.32 0.27 0.23 0.20 0.18 0.16 0.15 Ret 205 410 821 1 231 1 641 2 051 2 462 2 872 3 282 3 668 4 103 4 513
the various mixture equivalence ratios (Φ) are given in Table 5.1. The given range for the Karlovitz number Ka corresponds to the different turbulence intensities, as defined below. The initial system is a hot (T = Tb) perfectly spherical laminar flame-kernel of initial radius ro = 5.0 mm, located at the center of the computational box and surrounded by a fresh premixed atmo-spheric mixture of methane and air at Tu. The initial mass fraction values of YCH4 and YO2 at Tu, and YCO2 and YH2O at Tb are prescribed outside and within the kernel, respectively. These initial values for any primitive variable φ are transformed into smooth profiles according to
φ =φo+ ∆φ where∆φis the difference between the initial values (φo) in the fresh and burnt gas mixture. The integer s is a measure of the stiffness at the fresh/burnt gas interface and is in the order of a few hundred. In this range, the influence of s is confined to the very early part of the simulation and therefore does not impact the analysis presented below at later times. In all cases, an appropriate nitrogen complement is added everywhere at start.
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Fig. 5.1: Exemplary view of the configuration showing the heavily wrinkled iso-surface of the mass fraction of CO2 and colored by the HO2 flame radical
Because of the extremely high computation efforts of the three-dimensional DNS computations, the following systematic study (Section 5.4) has been re-alized for a two-dimensional configuration.
To investigate systematically the influence of the integral Reynolds num-ber Ret on the fuel consumption/burning rate, the calculations for a given Φ were repeated with exactly the same initial composition, but with an ini-tial pseudo-turbulent velocity field at successively higher intensity. The rms velocity fluctuation u�, the eddy turn-over time τ = lt/u� and the turbulent Reynolds number Ret for the various cases are given in Table 5.2. The char-acteristic kinematic viscosity of the mixture, ν = 1.56 ·10−5 m2/s and the integral length scale lt = 3.2 mm are kept constant. Note also that not all the cases shown here are realized for every mixture composition given in Table 5.1, due to the higher sensitivity of the leaner mixtures to increasing turbulence intensity, as will be demonstrated later. For the mixtures with Φ = 0.6 & 0.7, only cases 1 – 7 are considered, while for Φ = 0.8, 0.9 & 1.0, only cases 1 – 8, 1 – 9 and 1 – 12 are performed, respectively.
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