3.4 Effects of fuel composition and initial temperature RMS
3.4.2 Chemical explosive mode analysis
Chemical explosive mode (CEM) is adopted to further identify the ignition characteristics of PRF/air mixtures. Chemical explosive mode analysis (CEMA) has been developed for sys-
3.4 Effects of fuel composition and initial temperature RMS
Figure 3-8: Isocontours of the normalized heat release rate for Cases 1–3 (from left to right) at t/τig0 = 0.95, 0.82, and 0.56, respectively.
tematically detecting critical flame features such as ignition, extinction, and flame fronts and applied to DNSs of lifted flames in heated coflows [78–81], reacting jets in cross flows [82, 83], and ignition ofn-heptane/air mixture under HCCI conditions [53, 84].
The CEMA method is briefly introduced here. Readers may refer to [79] for more details about CEMA. The differential equations of a typical reacting flow can be described in discretized form as:
Dy
Dt =g(y) =ω(y) +s(y), (3.4.1)
where D/Dt is the material derivative, which can be replaced by d/dt in the Lagrangian co- ordinate, and y is the solution vector including species concentrations and temperature. Note that for spatially discretized flows, the chemical species concentration at different grid points corresponds to different entries iny. ω ands represent, respectively, the chemical source term and all non-chemical terms such as diffusion and homogeneous mixing.
The Jacobian matrix of the chemical source term,Jω(≡∂ω/∂y), can fully describe the local chemical information. As such, a chemical mode can be defined as an eigenmode ofJω, which is associated with an eigenvalue and a corresponding pair of the left and right eigenvectors. CEM is defined as a chemical mode of which the real part of the eigenvalue, λe, is positive [79]. By definition, CEM represents the reciprocal chemical time scale of a local mixture such that the existence of CEM implies that the corresponding mixture is explosive in nature. It is, therefore, likely to auto-ignite if the mixture is put in a lossless environment where the termsin Eq. 7.2.1 is negligible. Note that ignition may not actually occur in a mixture exhibiting CEM when significant loss in heat or radicals is present. Therefore, CEM remains an intrinsic chemical feature of ignitable mixtures.
In spatially inhomogeneous systems, CEMs interact with diffusion and other mixing pro- cesses. As such, ignition may not always result if the time scale of CEM is longer than those of the losses. The competition between CEMs and the losses can be approximately quantified by
a Damk¨ohler number defined as [79–81, 84]:
Dac=λe·χ−1, (3.4.2)
whereχ is a reciprocal characteristic timescale of the diffusion or loss terms; for instance, the scalar dissipation rate in turbulent flames. In this study,χ, is defined byχ= 2D|∇c|2, wherec and D are the progress variable and the thermal diffusivity of local mixture, respectively. c is defined asc≡Yc/YcEq, where Yc=YCO2+YCO andYcEq is the corresponding equilibrium value ofYc. Note that a mixture withDac1 indicates a dominant CEM which will likely result in actual ignition; otherwise, ignition may be suppressed by the losses.
Figure 3-9: Isocontours of (a) the timescale of the chemical explosive mode, (b)Dac, (c) temper- ature, and (d)χ for Case 1 (T0 = 15 K) att/τig0 = 0.95. The solid line denotes the reaction front (Dac = 1).
Figure 3-9 shows the isocontours of λe, Dac, temperature, and χ of Case 1 (PRF100 with T0 = 15 K) at t = τig. The sharp boundaries separating the burned and unburned mixtures are the reaction fronts that can be either spontaneous ignition or a deflagration wave. It is
3.4 Effects of fuel composition and initial temperature RMS
readily observed from Fig. 3-9b that the thin reaction fronts separate the whole domain into two bulk regions; i.e., the auto-igniting (red and yellow) region with large positive Dac where the chemical explosive process (large λe) overwhelms the mixing process (relatively small χ), and the post-ignition (blue) regions with large negative Dac where the mixing process (χ) is dominant compared with large negativeλe. It can also be observed that Dac upstream of the reaction fronts is much larger than unity (yellow region), suggesting that the chemical reaction (λe) is also much faster than the mixing process (χ). In deflagration waves, the reaction and mixing processes balance each other and, as such, large Dac( O(1)) ahead of the reaction fronts also verifies that the reaction fronts in the figure are attributed to spontaneous ignition rather than to deflagration.
Figure 3-10: Isocontours of (a) the timescale of the chemical explosive mode, (b)Dac, (c) temper- ature, and (d)χ for Case 3 (T0 = 60 K) att/τig0 = 0.56. The solid line denotes the reaction front (Dac = 1).
On the contrary, it can be observed from Fig. 3-10b for the highT0case that there exist three bulk regions: the auto-igniting region (red), the post-ignition region (blue), and the greenish region where mixing balances chemical explosion. Note thatDacupstream of the reaction fronts
(greenish region) becomes order of unity because the mixing process (χin Fig. 3-10d) shows the same order of magnitude as the chemical explosion process (λein Fig. 3-10a). These results also indicate that the reaction fronts of Case 3 are deflagrations rather than spontaneous ignitions.
It is also of interest to note from Fig. 3-10d that the reaction fronts mostly overlap with the strips with relatively large χ, a feature of deflagration waves propagating through unburned mixtures prior to the occurrence of auto-ignition.