Chapter VII: Analysis of effective Lewis numbers from the DNS of San-
7.1 Extracting effective Lewis numbers
7.1.3 Optimal flamelet parameters
Figure7.2shows a comparison of the conditional mean mass fractions of CO, H2O, and H2 from case RAD at x/d=30, against three flamelet solutions: the effective
7.1. Extracting effective Lewis numbers 129
2 10 100 260
10 20 30 40 50
Extinction limit
χstopt [s-1 ]
x/d RAD
Flame B Flame C Flame D Flame E
Extinction limit
(a) χoptst vs. x/d.
0.01 0.1 1 10 100
10 20 30 40 50
γopt
x/d 0.01
0.1 1 10 100
10 20 30 40 50
(b)γoptvs. x/d.
Figure 7.3: χoptst (left), and γopt (right) as a function of the downstream direction x/d.
Green with open symbols, RAD; green with full symbols, flame B; red, flame C; blue, flame D; black, flame E. See caption of Fig.4.4for a more complete description.
0.001 0.01 0.1
10 20 30 40 50
χstopt d / (Ujet - Upilot)
x/d
Flame B Flame C Flame D Flame E RAD
0.001 0.01 0.1
10 20 30 40 50
0.001 0.01 0.1
10 20 30 40 50
χstopt d / Ujet
x/d 0.001
0.01 0.1
10 20 30 40 50
Figure 7.4: χoptst normalized by (Ujet−Upilot)/d (left) and Ujet/d (right), as a function of downstream direction for flames B, C, D, and E and case (RAD).
Lewis number solution (γ=γopt), the unity Lewis number case (γ→ ∞), and the laminar Lewis number solution (γ=0). The χstvalue in all three flamelet solutions is set to χoptst . The optimal flamelet is found to be generally in good agreement with the DNS. For CO, all three flamelets are found to over-estimate its concentration on the lean side; moving towards the richer mixture, case RAD is found to move from the optimal flamelet, to the laminar Lewis number one. For both H2O and H2, the optimal flamelet is generally well representative of case RAD.
These results show that the optimal flamelet can be well representative of the data, especially given the assumptions associated with representing the conditional mean flame structure with a single steady-state, constant non-unity Lewis number flamelet.
Comparison of case RAD and the experiments
7.1. Extracting effective Lewis numbers 130 In this chapter, the results discussed in Ch.4 are compared to the DNS, with the goal of highlighting similarities and differences.
Figure7.3shows the optimal flamelet parameters from the experiments, compared to those from case RAD. The χoptst values from case RAD are found to be generally decreasing, similarly to the profiles extracted from the measurement data of flames B-E. The values extracted from case RAD are lower than the values extracted from the measurements of flame B. More specifically, at the two downstream locations where experiments are available, i.e., x/d=15 and x/d=30, the χoptst values ex- tracted from the experiments are 70s−1 and 60s−1, respectively, while the values found from case RAD are 40s−1 and 25s−1, respectively. It is interesting to note that, for x/d<30, the values of χoptst from case RAD and those extracted from flames C-E, are ordered according to their respective jet Reynolds numbers. To as- sess this further, Fig.7.4shows the χoptst values normalized by (Ujet−Upilot)/dand by Ujet/d. While being close, the profiles do not show a perfect collapse, with ei- ther normalization. For both normalizations, the χoptst values from the experiments of flame B are generally higher than those for flames C-E for x/d>30. The slope of χoptst from case RAD differs from flames C-E for x/d>30. That may be caused by the elongated shape of flame B compared to the higher Reynolds number flames, as was discussed in Ch.6.
Figure7.3bshows that theγoptst values from case RAD are generally increasing with downstream distance, similarly to the profiles extracted from the measurement data of flames B-E. The values extracted from case RAD are lower than those from the measurements of flame B. More specifically, theγoptvalues for flame B atx/d=15 and x/d=30 are 0.6 and 2.6, respectively, while the values computed from case RAD at the same locations are0.4and1.0, respectively. Interestingly, while flames C and D have practically the same γopt values for x/d<30, both case RAD and flame B are lower. As previously discussed, the stoichiometric flame tip for flames C-E is located at x/d≈45, and large differences are observed for the γopt values from those flames, at that location.
Multiple reasons could explain the discrepancies between the values of χoptst from case RAD, and those from measurements of flame B, including experimental uncer- tainties, and biases introduced by the models used (e.g., the chemical model). The fact that both the χoptst values and theγopt values for case RAD are lower than those extracted from the measurements of flame B is consistent with the shape of the er- ror maps shown in Fig. 7.1. The shape of the error contours around the optimal
7.1. Extracting effective Lewis numbers 131
10 100 260
10 20 30 40 50
Extinction limit
χstopt [s-1 ]
x/d RAD
BL UL
Extinction limit
(a) χoptst vs. x/d.
0.1 1 10 100
10 20 30 40 50
γopt
x/d 0.1
1 10 100
10 20 30 40 50
(b)γopt vs. x/d.
Figure 7.5: χoptst (left), and γopt (right) as a function of the downstream direction x/d.
Green, RAD; blue, BL; red, UL. The shaded region corresponds to values of χstabove the laminar Lewis number extinction limit.
solution is approximately oriented at 45o. In other words, small error differences are associated with the simultaneous increase or decrease of χoptst and γopt. That means that χst and γ generally have opposing effects, which tend to offset each other. For example, for the three species shown in Fig.7.2, an increase inγ will result in higher mass fractions (the red lines are always above the blue lines). On the other hand, increasing χst values will cause the mass fractions to decrease.
Comparison of cases BL, RAD, and UL
Figure7.5shows a comparison of optimal flamelet parameters from cases BL, UL and RAD. Figure 7.5a shows that, for all three cases, the χoptst profiles are gener- ally decreasing with axial distance, i.e., the qualitative behavior is the same. For x/d<10, cases RAD and BL are partly overlapping, and lower than case UL. All three profiles remain within a factor of 2 of each other. Figure7.5bshows that the γoptprofiles for cases BL and RAD remain close for the entire range7.5<x/d<45.
This suggest that not including radiation heat loss, and not considering the condi- tional temperature profiles in the error map analysis, has a small impact onγopt. As expected, the γopt values for case UL are much higher than the other two cases for the entire range 7.5<x/d<45. As previously discussed for the optimization analysis presented in Ch.4, the chosen range for γ was[0,120], whereγ=120is considered a proxy forγ→ ∞. While theγopt values for case UL remain high for the entire range 7.5<x/d<45, they are not always equal to 120. That is not sur- prising, since the expression for the effective Lewis numbers, Eq. (7.1), presents a small sensitivity toγ for large values of this parameter. For example, the effective
7.1. Extracting effective Lewis numbers 132
0 2 4 6
0 0.2Zp 0.4 0.6 0.8 1
effect of pilot
〈χ | Z〉/〈χ | Zst〉
Z
x/d=1 x/d=5 x/d=10 x/d=30 mixing layer
Figure 7.6:Comparison ofhχ(Z)|Zi/hχ|Zsticomputed from case RAD (no symbols), and Eq. (4.5) with Eq. (4.6) (symbols). The vertical line represents the stoichiometric mixture fraction. “Zp” is the pilot feed mixture fraction.
Lewis number of H2is 0.98 forγ=120and 0.93 forγ=30. The values ofγoptclose to the burner exit plane are likely affected by the pilot flame. The pilot strongly af- fects the scalar dissipation rate at the base of the jet, and acts as an additional feed (in the flamelet framework discussed in Ch.4, non-premixed jet flames are modeled as a two-feed system). The effect of the pilot flame can be seen in Fig.7.6, where thehχ(Z)|Zi/hχ|Zstiprofiles from case RAD at several downstream locations, are compared to the shape of χ(Z) used to solve the flamelet equations (Eq. (4.6)).
One approach to include the effect of the pilot, could be to use the following closure for the scalar dissipation rate in the flamelet equations [23]
χ(Z,x) =hχ|Zi(x), (7.2)
where hχ|Zi(x) is computed from available data. This approach is not pursued here.
Comparison of optimal parameters using Z and ZTNF
In Ch. 4, the optimal parameters were extracted from the data through an error map analysis based on six measured species (including O2, CH4, H2O, H2, CO2, and CO), as well as a measured mixture fraction, ZTNF. Moreover, to take into account interference in the measured CH4 signal, a correction was considered for this species (see Sec. 4.2). Figure 7.7 compares the optimal parameters χoptst and γopt extracted by performing the error map analysis with either Z or ZTNF. The