Chapter III: Assessment of the Constant Non-Unity Lewis Number As-
3.3 Premixed hydrogen/air flame
3.3.2 Two-dimensional simulations
The laminar H2/air flame is simulated using the two-dimensional domain shown in Fig. 3.3, with the numerical parameters given in Table 1. The initial two- dimensional data file is generated from a one-dimensional unstretched flame, ob- tained using the FlameMaster code with mixture-averaged transport properties. The one-dimensional solution is then interpolated to build the two-dimensional initial data file. The two-dimensional grid is uniform in both directions, and the ratio of the laminar flame thickness to the spacing between nodes is approximately 15, which was found to be sufficient to represent the flame. Figure3.5 shows the di- mensionless source term of OH along the centerline of a data file corresponding to a two-dimensional simulation with mixture-average properties, at 10 ms. As can be seen, the grid spacing is such that there are about 20 grid points per OH layer, as suggested by Hawkeset al.[114].
The initial flame front is perturbed according to xf,0 = E+ AX
i=1,2
cos 2πkiy H
!
, (3.1)
where xf,0 is the initial flame position, E is the average flame position, k1 and k2 are two coprime modes, y is the vertical coordinate, and H is the height of the
3.3. Premixed hydrogen/air flame 30
640 N. Buraliet al.
Table 1 Parameters for the two-dimensional H2/air flame simulation.
Tu(K) P(atm) φ lF(mm) nx × ny x=y(mm) lF/SL(ms)
298 1 0.4 0.65 1888 × 472 0.0424 3
Figure 4. Normalised OH production term against grid spacing.
The initial flame front is perturbed according to
xf,0=E+A
i=1,2
cos
2πkiy H
, (11)
wherexf, 0 is the initial flame position,Eis the average flame position,k1andk2are two coprime modes,yis the vertical coordinate, andHis the height of the domain.Ais set to 10−4m, andk1andk2are 20 and 13, respectively. The two modes produce an asymmetric initial perturbation, which is intended to trigger the thermo-diffusive and Darrieus–Landau instabilities quickly. This flame was selected as its evolution is strongly dependent on the choice of the initial perturbation [50], and any deviations (even due to minute differences between the four cases) will undoubtedly lead to different flame evolutions.
An inflow velocity is provided to match the unstretched laminar flame speed of the benchmark case (i). However, due to the development of cellular structures, the flame burns faster and, hence, propagates to the left. Yet, the length of the domain is sufficient for the simulation time. A convective outflow condition is used for the right boundary, while periodic boundary conditions are used in the vertical direction.
4.2.2. Results
In this section, the four two-dimensional simulations corresponding to the four cases (i)–
(iv) are compared. All four simulations start from the same initial flow field (shown in Figure 3), and share the same grid (discussed in Section4.2.1).
Figure 5(a) shows the average flame position for the four simulations, which is calculated by averaging a progress variable in the vertical direction, and selecting the horizontal coordinate corresponding to a predetermined value. The progress variable is YH2O, and
Figure 3.5:Normalized OH production term against grid spacing.Combustion Theory and Modelling 641
Figure 5. Average flame position (a) and fuel consumption-based flame speed (b) for the four simulations of the two-dimensional lean H2/air flame. Snapshots corresponding to times B, C, and D are shown inFigure 6, while time A was shown inFigure 3.
the value corresponding to the flame is set to 0.04.Figure 5(b) shows the corresponding consumption-based flame speed,Sω˙H2(t), defined as
Sω˙H2(t)= 1 ρuYH2,uA
V−ω˙H2(t) dV, (12)
where ˙ωH2is the H2source term,Vis the computational domain,Ais the cross section, and ρuandYH2,uare the unburnt density and fuel mass fraction, respectively.Figure 6shows the snapshots marked B through D, which correspond to 50, 100, and 150 ms, respectively.
The initial snapshot, A, was shown inFigure 3.
In all cases, the initial perturbations disappear, leading to an almost smooth flame front.
This initial transient (represented by the vertical section of the lines in Figure 5(b)) lasts for about 10 ms, after which test cases (i)–(iii) rapidly develop instabilities, and accelerate towards the inflow at a flame speed greater than the respectiveSL(Figure 5(b)). After about 5 ms, the unity Lewis number test case propagates towards the inlet with a flat flame front (up to about 70 ms), and at the unstretched laminar flame speed (Figure 5(b)). At 70 ms, there is a change of slope inFigure 5(a) of the unity Lewis number test case, corresponding to the appearance of the Darrieus–Landau instabilities, and a consequent increase of the average burning velocity. As can be seen fromFigure 5(b), despite the initial shift of about 4 ms, both cases (ii) and (iii) remain qualitatively and quantitatively very close to the mixture-averaged case (i) up to 40 ms. After that, the three cases deviate from each other, as is expected in very sensitive unstable dynamical systems. As expected, the unity Lewis number test case (iv), shows smaller fluctuations than the non-unity Lewis number cases.
In summary, while cases (ii) and (iii) are in reasonable agreement (both qualitatively and quantitatively) with the benchmark (i), the unity Lewis number simulation (iv) displays a radically different behaviour, resulting from the absence of the thermo-diffusive instabilities.
(a)Average flame position.
Combustion Theory and Modelling 641
Figure 5. Average flame position (a) and fuel consumption-based flame speed (b) for the four simulations of the two-dimensional lean H2/air flame. Snapshots corresponding to times B, C, and D are shown inFigure 6, while time A was shown inFigure 3.
the value corresponding to the flame is set to 0.04.Figure 5(b) shows the corresponding consumption-based flame speed,Sω˙H2(t), defined as
Sω˙H2(t)= 1 ρuYH2,uA
V−ω˙H2(t) dV, (12)
where ˙ωH2is the H2source term,Vis the computational domain,Ais the cross section, and ρuandYH2,u are the unburnt density and fuel mass fraction, respectively.Figure 6shows the snapshots marked B through D, which correspond to 50, 100, and 150 ms, respectively.
The initial snapshot, A, was shown inFigure 3.
In all cases, the initial perturbations disappear, leading to an almost smooth flame front.
This initial transient (represented by the vertical section of the lines inFigure 5(b)) lasts for about 10 ms, after which test cases (i)–(iii) rapidly develop instabilities, and accelerate towards the inflow at a flame speed greater than the respectiveSL(Figure 5(b)). After about 5 ms, the unity Lewis number test case propagates towards the inlet with a flat flame front (up to about 70 ms), and at the unstretched laminar flame speed (Figure 5(b)). At 70 ms, there is a change of slope inFigure 5(a) of the unity Lewis number test case, corresponding to the appearance of the Darrieus–Landau instabilities, and a consequent increase of the average burning velocity. As can be seen fromFigure 5(b), despite the initial shift of about 4 ms, both cases (ii) and (iii) remain qualitatively and quantitatively very close to the mixture-averaged case (i) up to 40 ms. After that, the three cases deviate from each other, as is expected in very sensitive unstable dynamical systems. As expected, the unity Lewis number test case (iv), shows smaller fluctuations than the non-unity Lewis number cases.
In summary, while cases (ii) and (iii) are in reasonable agreement (both qualitatively and quantitatively) with the benchmark (i), the unity Lewis number simulation (iv) displays a radically different behaviour, resulting from the absence of the thermo-diffusive instabilities.
(b)Consumption-based flame speed.
Figure 3.6: Average flame position (a) and fuel consumption-based flame speed (b) for the four simulations of the two-dimensional lean H2/air flame. Snapshots corresponding to times B, C, and D are shown in Fig.3.7, while time A was shown in Fig.3.3.
domain. A is set to 10−4m, and k1 and k2 are 20 and 13, respectively. The two modes produce an asymmetric initial perturbation, which is intended to trigger the thermo-diffusive and Darrieus-Landau instabilities quickly. This flame was selected as its evolution is strongly dependent on the choice of the initial perturbation [115], and any deviations (even due to minute differences between the four cases) will undoubtedly lead to different flame evolutions.
An inflow velocity is provided to match the unstretched laminar flame speed of the benchmark case (i). However, due to the development of cellular structures, the flame burns faster and, hence, propagates to the left. Yet the length of the domain is sufficient for the simulation time. A convective outflow condition is used for the right boundary, while periodic boundary conditions are used in the vertical direction.
3.3. Premixed hydrogen/air flame 31 Results
In this section, the four two-dimensional simulations corresponding to the four cases (i)-(iv) are compared. All four simulations start from the same initial flow field (shown in Fig.3.3), and share the same grid.
Figure 3.6a shows the average flame position for the four simulations, which is calculated by averaging a progress variable in the vertical direction, and selecting the horizontal coordinate corresponding to a predetermined value. The progress variable is YH2O, and the value corresponding to the flame is set to 0.04. Figure3.6b shows the corresponding consumption-based flame speed,Sω˙H2(t), defined as
Sω˙H
2 = 1
ρuYH2,uA Z
V
−ω˙H2(t)dV, (3.2)
where ω˙H2 is the H2 source term, V is the volume of the computational domain, A is the cross section, and ρu and YH2,u are the unburnt density and fuel mass fraction, respectively. Figure3.7 shows the snapshots marked B through D, which correspond to 50, 100, and 150 ms, respectively. The initial snapshot, A, was shown in Fig.3.3.
In all cases, the initial perturbations disappear, leading to an almost smooth flame front. This initial transient (represented by the vertical section of the lines in Fig.3.6b) lasts for about 10 ms, after which test cases (i)-(iii) rapidly develop insta- bilities, and accelerate towards the inflow at a flame speed greater than the respec- tive SL (Fig. 3.6b). After about 5 ms, the unity Lewis number test case propagates towards the inlet with a flat flame front (up to about 70 ms), and at the unstretched laminar flame speed (Fig.3.6b). At 70 ms, there is a change of slope in Fig.3.6aof the unity Lewis number test case, corresponding to the appearance of the Darrieus- Landau instabilities, and a consequent increase of the average burning velocity. As can be seen from Fig.3.6b, despite the initial shift of about 4 ms, both cases (ii) and (iii) remain qualitatively and quantitatively very close to the mixture-averaged case (i) up to 40 ms. After that, the three cases deviate from each other, as is expected in very sensitive unstable dynamical systems. As expected, the unity Lewis number test case (iv), shows smaller fluctuations than the non-unity Lewis number cases.
In summary, while cases (ii) and (iii) are in reasonable agreement (both qualitatively and quantitatively) with the benchmark (i), the unity Lewis number simulation (iv) displays a radically different behavior, resulting from the absence of the thermo- diffusive instabilities.
3.3. Premixed hydrogen/air flame642 N. Buraliet al. 32
Figure 6. Snapshots of the temperature field corresponding tot = 50 ms (B),t = 100 ms (C), and t = 150 ms (D). The initial field is the same for all simulations (Figure 3), while B, C, and D, show a comparison of the four test cases (i)–(iv).
4.3. Three-dimensional simulations
Hydrogen flames under moderate levels of turbulence have been shown to present a stronger sensitivity to differential diffusion [51]. For this reason, the constant Lewis number as- sumption is also tested in a three-dimensional configuration with low-intensity turbulent conditions.
4.3.1. Configuration
A schematic of the three-dimensional configuration is shown in Figure 7. The reader is referred to Savardet al. [19,20,37,52] for more details on the configuration. Only a brief overview is given here. The domain has a square cross-section, where depth and width are of size L = 8.35 mm. The total length is 8L. The grid is uniform, with a cell size of 0.0424 mm, which is the same as that used for the previous two-dimensional laminar flames. The unburnt gas is injected with a low turbulent kinetic energy, and is generated through a separate homogeneous isotropic turbulence simulation. A velocity forcing is used to reach the desired level of turbulence intensity for each streamwise location between 0.25Land 6.5Lfrom the inlet [52–54]. The average inflow velocity is set to a value close Figure 3.7: Snapshots of the temperature field corresponding tot=50ms (B),t=100ms (C), andt=150ms (D). The initial field is the same for all simulations (Fig.3.3), while B, C, and D, show a comparison of the four test cases (i)-(iv).