The skeletal and reduced mechanisms were originally validated based on reaction states sampled from auto-ignition and perfectly stirred reactors (PSR) covering the parameter range of pressures from 1 to 100 atm, equivalence ratios from 0.5 to 2.0, and initial temperature from 700 to 1800 K for auto-ignition [81, 113]. Extended validation of the reduced mechanism relevant to the present DNSs is shown in Fig. 4-1 for biodiesel/air at equivalence ratio of 0.3 and pressure of 40 ∼ 100 atm. The accuracy of the reduced mechanism is mostly identical to that of the skeletal mechanism over the entire parameter range of the reduction. Although it is significantly reduced from the original detailed chemistry, the reduced mechanism shows a good agreement with the detailed mechanism and experimental results in terms of ignition delays, flame propagation speeds, and extinction residence times. Details of the skeletal and reduced mechanisms of biodiesel oxidation can be found in [81,113]. In addition, the method of dynamic stiffness removal [53, 54, 65, 91] is employed to eliminate chemical time scales shorter than 10 ns such that explicit time integration could be applied in DNSs.
For all DNSs in the present study, the initial mean temperature, T0, mean equivalence ratio, φ0, and the initial uniform pressure, p0, are 850 K, 0.45, and 40 atm, respectively. The initial conditions are chosen to employ the low temperature heat release of two-stage ignition of biodiesel/air mixture relevant to high-load naturally aspired HCCI engines such that the initial pressure of 40 atm is relatively low compared to that in modern boosted engines at the TDC. Twelve different DNSs are performed by changing two key parameters: temperature fluctuations,T0, of 15 K and 60 K, and equivalence ratio fluctuations,φ0, of 0.05 and 0.10. Three additional DNSs are carried out to elucidate the effect of turbulence on HCCI combustion by varying turbulence intensity,u0, from 1.0 to 5.0 m/s.
Note that φ0 and φ0 are carefully selected such that local φ is maintained precisely below unity, thereby preventing locally-high temperature that may cause excessive NOxformation [17].
Furthermore, the initial conditions are representative of the TDC under high-load conditions in air-diluted HCCI combustion [90, 116]. From 0-D simulations, the homogeneous ignition delay of a biodiesel/air mixture withφ0 = 0.45 is found to be τig0 = 1.0 ms at the initial mean temperature and pressure. Henceforth, τig represents the time at which the maximum mean heat release rate (HRR) occurs for all 0-D and 2-D simulations. The superscript 0 denotes the zero-dimensional simulation at a constant volume.
The initial turbulent flow field is prescribed by an isotropic kinetic energy spectrum function by Passot-Pouquet [38] as in [49–51, 53, 54, 75–77, 91]. For the first fifteen DNS cases, the turbulence intensity, u0, and length scale, le, are specified as 1.0 m/s and 1.0 mm respectively.
As such, the turbulence time scale,τt, is 1.0 ms, which is comparable to the ignition time scale.
Note that turbulence time scale in real HCCI engines is ∼ O (1 ms) such that the turbulence time scale in the present DNS study is representative of HCCI combustion. It is also of interest to note that the evolution of 2-D turbulence without 3-D vortex stretching may be different
4.2 Initial conditions
from that of 3-D turbulence. However, investigation of HCCI combustion using DNSs with 2-D random turbulence exhibiting a wide range of spectrum of length and time scales is still of value because the effect of turbulent mixing on HCCI combustion plays a secondary role compared with mixture stratifications [49–54, 91]. Therefore, it is reasonable to expect that overall HCCI combustion characteristics from 2-D DNSs may not differ significantly from those of 3-D DNS which is extremely expensive [106].
In addition to the velocity fluctuation, temperature and concentration fluctuations are also superimposed on the corresponding mean fields to investigate the effects of initial hot/cold and/or relatively fuel-rich/lean spots on the ignition characteristics of the biodiesel/air mixture.
The scalar fluctuations are generated from the same energy spectrum as turbulence with different random numbers. The most energetic length scales of the temperature fluctuations, lT e, and composition fluctuations, lφe, are 1.0 mm in all cases. Identical characteristic length and time scales are specified for the ignition delay and all fluctuation fields, so allowing most effective turbulent mixing of initial mixtures to be elucidated. Note that turbulence and other scalar fields such as temperature and concentration are not correlated. Details of the physical and numerical parameters for each case are listed in Table 4-1.
Case Type T0 T0 φ0 φ0 le lT e lφe u0 τt τig0 N
(K) (K) (mm) (mm) (mm) (m/s) (ms) (ms)
1 BL 850 15 0.45 - 1.0 1.0 1.0 1.0 1.0 1.0 640
2 BL 850 60 0.45 - 1.0 1.0 1.0 1.0 1.0 1.0 640
3 BL 850 - 0.45 0.05 1.0 1.0 1.0 1.0 1.0 1.0 640
4 BL 850 - 0.45 0.10 1.0 1.0 1.0 1.0 1.0 1.0 640
5 UC 850 15 0.45 0.05 1.0 1.0 1.0 1.0 1.0 1.0 640
6 UC 850 15 0.45 0.10 1.0 1.0 1.0 1.0 1.0 1.0 640
7 UC 850 60 0.45 0.05 1.0 1.0 1.0 1.0 1.0 1.0 640
8 UC 850 60 0.45 0.10 1.0 1.0 1.0 1.0 1.0 1.0 1280
9 NC 850 15 0.45 0.05 1.0 1.0 1.0 1.0 1.0 1.0 640
10 NC 850 15 0.45 0.10 1.0 1.0 1.0 1.0 1.0 1.0 640
11 NC 850 60 0.45 0.05 1.0 1.0 1.0 1.0 1.0 1.0 640
12 NC 850 60 0.45 0.10 1.0 1.0 1.0 1.0 1.0 1.0 640
13 BL 850 - 0.45 0.10 1.0 1.0 1.0 5.0 0.2 1.0 640
14 UC 850 60 0.45 0.10 1.0 1.0 1.0 5.0 0.2 1.0 1280
15 NC 850 60 0.45 0.10 1.0 1.0 1.0 5.0 0.2 1.0 640
Table 4-1: Physical and numerical parameters of the DNS cases. BL, UC, and NC represent baseline, uncorrelatedT−φ, and negatively-correlatedT−φdistribution, respectively.
Depending on factors including fuel delivery strategies, injection timings, amount of EGR, intake charge heating, and wall heat loss, different T −φ distributions may occur at the TDC prior to the main auto-ignition event. However, only two most-probable scenarios are considered in the present study: (1) early direct injection (one stage injection) combined with EGR may
produce ‘uncorrelatedT−φ’ fields due mostly to turbulent mixing, and (2) two-stage injection strategy with late second direct injection may result in ‘negatively-correlated T −φ’ fields because of the evaporative cooling of injected fuel and incomplete mixing [52, 117]. On this basis, three distinct cases of initial T −φ correlations are elucidated: (1) baseline cases with fluctuations either in temperature (Cases 1 and 2) or equivalence ratio (Cases 3 and 4), (2) uncorrelated T −φ distribution (Cases 5–8), and (3) negatively-correlated T −φ distribution (Cases 9–12). Figure 4-2 shows various initialT−φdistributions together with a representative isocontour of the initial equivalence ratio field for Case 12.
Figure 4-2: Initial T−φdistribution for (a) Cases 2, 4, 12 and (b) Case 8, and (c) initial φfield for Case 12.
All of the cases are simulated in a 2-D computational domain of a 3.2 mm×3.2 mm square box. Each direction is discretized with grid points, N, of 640 or 1280 and the corresponding uniform grid sizes are 5 or 2.5 µm. Note that based on the initial conditions and the integral length scale, L11, the turbulent Reynolds numbers for cases with u0 = 1.0 and 5.0 m/s are 145 and 730, respectively. The corresponding Kolmogorov length scales, ηK, are approximately 8.2 and 2.5 µm, respectively. For all DNS cases, therefore, at least half grid point is located within the Kolmogorov length scale as suggested in [118]. Moreover, the thinnest reaction layers in 2D DNSs were resolved with at least 12∼ 16 grid points and as such, turbulence and scalar fields are well resolved in the DNSs. The DNSs were performed on the IBM Blue Gene/P at King Abdullah University of Science and Technology (KAUST).