Three-dimensional simulations - Premixed hydrogen/air flame

Chapter III: Assessment of the Constant Non-Unity Lewis Number As-

3.3 Premixed hydrogen/air flame

3.3.3 Three-dimensional simulations

3.3. Premixed hydrogen/air flame₆₄₂ _{N. Burali}_{et 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).

3.3. Premixed hydrogen/air flameCombustion Theory and Modelling 64333

Figure 7. Schematic of the three-dimensional H2/air simulation configuration (sketch adapted from Savardet al.[19]).

to the turbulent burning velocity, so that the statistically-planar flame is almost stationary.

The unburnt Karlovitz number for the benchmark test case is Ka = 149. The unburnt turbulent Reynolds number is Ret = ul/ν = 289. The ratiosl/lFandu/SL are 2 and 18, respectively, and the eddy turnover time, τ = k/, is about 500μs. The integral length scale,l, is computed as 16% of the domain width [54]. These conditions are close to case C from Aspdenet al.[51]. It should be noted that the definition of the Karlovitz number used in the current work differs from the definition used by Aspdenet al.[51] (in their work, Ka

= (u/SL,fp)^3/2(lF,fp/l)^1/2, where the subscript ‘fp’ stands for ‘freely-propagating’).

4.3.2. Results

(Figure 8(b)), which are shown against the number of eddy turnover times. As can be seen, both flame position and flame speed of test cases (i)–(iii) remain close for about 3τ. After this initial phase, the three test cases can be seen gradually drifting apart (Figure 8(a)), and the respective turbulent flame speeds fluctuate differently (Figure 8(b)).Table 2shows that test cases (ii) and (iii) have a slightly larger mean and RMSST fluctuation than the benchmark (i), while test case (iv) has a much reduced mean and RMSST. The unity Lewis

Figure 8. Average flame position (a) and fuel consumption-based flame speed normalised by the respective laminar unstretched flame speed (b), for the four simulations (i)–(iv) of the three-dimensional lean H₂/air flame.

Figure 3.8: Schematic of the three-dimensional H2/air simulation configuration (sketch adapted from Savardet al.[48]).

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 0.25L and 6.5L from the inlet [117–

119]. The average inflow velocity is set to a value close to the turbulent burning velocity, so that the statistically-planar flame is almost stationary. The unburnt Karlovitz number for the benchmark test case is Ka=149. The unburnt turbulent Reynolds number is Ret=u⁰l/ν=289. The ratiosl/lF andu⁰/SL are 2 and 18, respectively, and the eddy turnover time,τ=k/ε, is about 500µs. The integral length scale,l, is computed as 16% of the domain width [119]. These conditions are close to case C from Aspden et al. [116]. It should be noted that the definition of the Karlovitz number used in the current work differs from the definition used by As- pdenet al.[116] (in their work, Ka=(u⁰/SL,f p)^3/2(lF,f p/l)^1/2, where the subscript

‘f p’ stands for ‘freely-propagating’).

Results

Four three-dimensional DNS corresponding to cases (i)-(iv) are carried out. Each simulation is advanced for about 30 eddy turnover times. The statistics shown in this section correspond to the last 20, after any transient effects are gone. As was done for the two-dimensional case, two global parameters are shown: the average flame position (Fig.3.9a) and the normalized fuel consumption-based turbulent flame speed, ST (Fig. 3.9b), which are shown against the number of eddy turnover times. As can be seen, both flame position and flame speed of test cases (i)-(iii) remain close for about3τ. After this initial phase, the three test cases can be seen gradually drifting apart (Fig.3.9a), and the respective turbulent flame speeds fluctuate differently (Fig. 3.9b). Table3.10 shows that test cases (ii) and (iii) have a slightly larger mean and RMSST fluctuation than the benchmark (i), while test case

3.3. Premixed hydrogen/air flame 34

Combustion Theory and Modelling 643

Figure 7. Schematic of the three-dimensional H2/air simulation configuration (sketch adapted from Savardet al.[19]).

to the turbulent burning velocity, so that the statistically-planar flame is almost stationary.

The unburnt Karlovitz number for the benchmark test case is Ka = 149. The unburnt turbulent Reynolds number is Re_t = ul/ν = 289. The ratiosl/l_Fandu/S_Lare 2 and 18, respectively, and the eddy turnover time,τ = k/, is about 500 μs. The integral length scale,l, is computed as 16% of the domain width [54]. These conditions are close to case C from Aspdenet al.[51]. It should be noted that the definition of the Karlovitz number used in the current work differs from the definition used by Aspdenet al.[51] (in their work, Ka

= (u/S_L,_fp)^3/2(l_F,_fp/l)^1/2, where the subscript ‘_fp’ stands for ‘freely-propagating’).

4.3.2. Results

(Figure 8(b)), which are shown against the number of eddy turnover times. As can be seen, both flame position and flame speed of test cases (i)–(iii) remain close for about 3τ. After this initial phase, the three test cases can be seen gradually drifting apart (Figure 8(a)), and the respective turbulent flame speeds fluctuate differently (Figure 8(b)).Table 2shows that test cases (ii) and (iii) have a slightly larger mean and RMSS_Tfluctuation than the benchmark (i), while test case (iv) has a much reduced mean and RMSS_T. The unity Lewis

(a)Average flame position.

Combustion Theory and Modelling 643

Figure 7. Schematic of the three-dimensional H₂/air simulation configuration (sketch adapted from Savardet al.[19]).

to the turbulent burning velocity, so that the statistically-planar flame is almost stationary.

= (u/SL,fp)^3/2(lF,fp/l)^1/2, where the subscript ‘fp’ stands for ‘freely-propagating’).

4.3.2. Results

Four three-dimensional direct numerical simulations corresponding to cases (i)–(iv) are carried out. Each simulation is advanced for about 30 eddy turnover times. The statistics shown in this section correspond to the last 20, after any transient effects are gone. As was done for the two-dimensional case, two global parameters are shown: the average flame position (Figure 8(a)) and the normalised fuel consumption-based turbulent flame speed,S_T (Figure 8(b)), which are shown against the number of eddy turnover times. As can be seen, both flame position and flame speed of test cases (i)–(iii) remain close for about 3τ. After this initial phase, the three test cases can be seen gradually drifting apart (Figure 8(a)), and the respective turbulent flame speeds fluctuate differently (Figure 8(b)).Table 2shows that test cases (ii) and (iii) have a slightly larger mean and RMSSTfluctuation than the benchmark (i), while test case (iv) has a much reduced mean and RMSS_T. The unity Lewis

(b)Consumption-based flame speed.

Figure 3.9:Average flame position (a) and fuel consumption-based flame speed normalized by the respective laminar unstretched flame speed (b), for the four simulations (i)–(iv) of the three-dimensional lean H2/air flame.

644 N. Buraliet al.

Table 2. Laminar unstretched flame speed,SL, mean and normalised mean turbulent flame speed, denoted bySTandST/SL, respectively (where each test case is normalised by the respectiveSL), and RMS turbulent flame speed,ST, for test cases (i)–(iv).

Test case SL(m/s) ST(m/s) ST/SL ST(m/s)

Mix.-av. (i) 0.22 6.5 29.5 1.0

Lei@Tmax(ii) 0.22 7.2 32.7 1.3

Lei@Ymax(iii) 0.21 7.7 36.7 1.8

Unity (iv) 0.41 4.0 9.8 1.2

Figure 9. Turbulence statistics of the four three-dimensional hydrogen–air flames corresponding to test cases (i)–(iv). The top row shows the area-weighted conditional means of the fuel mass fraction (a) and its normalised source term (b); the bottom row shows the PDF of the fuel mass fraction (c), and its normalised source term (d) at the temperature of the maximum mixture-averaged fuel source term (1191 K). The black line corresponds to case (i), while cases (i)–(iv) are shown by the blue, green, and red lines, respectively.

number test case (iv) presents an instantaneous drop, which reducesS_T/S_Lby a factor of about three.

A more quantitative comparison of the four test cases (i)–(iv) is made inFigure 9, which shows various turbulence statistics of the fuel mass fraction, and its (normalised) source term.Figure 9(a) and9(b) show a comparison of the area-weighted conditional means of the fuel mass fraction and its normalised source term, respectively, for the four cases (i)–(iv)

Figure 3.10: Laminar unstretched flame speed, S_L, mean and normalized mean turbulent flame speed, denoted byS_T andS_T/S_L, respectively (where each test case is normalized by the respectiveS_L), and RMS turbulent flame speed,S_T, for test cases (i)-(iv).

(iv) has a much reduced mean and RMSST. The unity Lewis number test case (iv) presents an instantaneous drop, which reducesST/SL by a factor of about three.

A more quantitative comparison of the four test cases (i)-(iv) is made in Fig.3.11, which shows various turbulence statistics of the fuel mass fraction, and its (normalized) source term. Figures3.11aand3.11bshow a comparison of the area-weighted conditional means of the fuel mass fraction and its normalized source term, respectively, for the four cases (i)-(iv) (the reader is referred to Lapointeet al.[43] for a description of the averaging procedure). These conditional means reveal that test cases (ii) and (iii) are both in good agreement with the benchmark. Test cases (ii) and (iii) have some minor differences with respect to the benchmark (i) in regions of high curvature, corresponding to the hot spots (T>1400K). As expected, the conditional mean fuel mass fraction and source term of the unity Lewis number test case are zero above 1400 K.

Finally, as predicting mean quantities is often not enough, Fig. 3.11c and 3.11d show a comparison of the probability density functions (PDFs) of the normalized fuel mass fraction and of its normalized source term, respectively, taken at the tem-

Dalam dokumen Towards a priori models for differential diffusion in turbulent non-premixed flames (Halaman 59-62)