Modeling of Lewis number dependence of scalar dissipation rate transport for Large Eddy Simulations of turbulent premixed combustion

Figures & data

Table 1. Initial values of simulation parameters and nondimensional numbers relevant to the DNS database considered for this analysis.

Case	Le	u′/SL	δL/δth	l/δth	τ	Ret	Da	Ka
A	0.34	7.5	2.17	2.45	4.5	47.0	0.33	13.0
B	0.6	7.5	1.40	2.45	4.5	47.0	0.33	13.0
C	0.8	7.5	1.15	2.45	4.5	47.0	0.33	13.0
D	1.0	7.5	1.0	2.45	4.5	47.0	0.33	13.0
E	1.2	7.5	0.90	2.45	4.5	47.0	0.33	13.0

For all cases τ = 4.5; β = 6.0; Pr = 0.7;  = 0.014159.

Table 2. The effects of Lewis number on normalized flame surface area A_T/A_L and normalized turbulent flame speed S_T/S_Lwhen the statistics were extracted (i.e., t = 1.75 α_T0/).

Case	AT/AL	ST/SL
A	3.93	13.70
B	2.66	4.58
C	2.11	2.53
D	1.84	1.83
E	1.76	1.50

Figure 1. Distributions of on x₁ − x₂ mid-plane for Δ = 0.8δ_th (1st column), 1.6δ_th (2nd column), 2.8δ_th (3rd column) for cases A–E (1st–5th row) when the statistics were extracted (i.e., t = 1.75 α_T0/).

Figure 2. Variations of T₁ ( —— ), T₂ (

), T₃ (

), T₄ (

), (−D₂) (

), and f(D) (

) conditionally averaged in bins of for Δ ≈ 0.4δ_th (1st column), 1.6δ_th (2nd column), and 2.8δ_th (3rd column) in cases A–E (1st–5th row). All the terms are normalized with respect to .

Table 3. Summary of the scaling estimates of the relevant quantities according to Gao et al. [28].

Quantities	Scaling estimates
T1	alternatively The above expressions can be combined as
T2
(T2)res (see Eq. 7ii)
T3	alternatively
(T3)res (see Eq. 11iii)
T4
(T4)res (see Eq. 14i)
(−D2)	alternatively
(−D2)res (see Eq. 14ii)
f(D)
{f(D)}res (see Eq. 14iii)

Figure 3. Variations of (

) conditionally averaged in bins of along with the predictions of Eqs. (6i) and (6ii) with Φ′ = 0.7 (

) and Eqs. (6i) and (6ii) with Φ′ according to Eq. (6iii) (

) for Δ ≈ 0.4δ_th (1st column), 1.6δ_th (2nd column), and 2.8δ_th (3rd column) in cases A–E (1st–5th row).

Figure 4. Variations of T₂ (

) and (T₂)_sg (

) conditionally averaged in bins of along with the predictions of Eqs. (8) (

) and (9) (

) for Δ ≈ 0.4δ_th (1st column), 1.6δ_th (2nd column), and 2.8δ_th (3rd column) in cases A–E (1st–5th row). All the terms are normalized with respect to .

Figure 5. Variations of T₃ (

) and (T₃)_res (

) conditionally averaged in bins of along with the predictions of Eqs. (12i) and (12iii) (

) and Eqs. (13i) and (13ii) (

) for Δ ≈ 0.4δ_th (1st column), Δ ≈ 1.6δ_th (2nd column), and Δ ≈ 2.8δ_th (3rd column) in cases A–E (1st–5th row). All the terms are normalized with respect to .

Figure 6. Variations of [T₄ + f(D) − D₂] (

) and [(T₄)_sg − (D₂)_sg + {f(D)}_sg] (

) conditionally averaged in bins of along with the predictions of Eqs. (15i) and (15ii) (

) and Eq. (16) (

) for Δ ≈ 0.4δ_th (1st column), 1.6δ_th (2nd column), and 2.8δ_th (3rd column) in cases A–E (1st–5th row). All the terms are normalized with respect to .

Table 4. Summary of the proposed models for the unclosed terms of the SDR transport equation (Eq. (3)) in this analysis.

Term	Model expression
	where γ1 = 1.8, γ2 = 4.9 − 3.2erf(0.15RetΔ), RetΔ = ρ0u′ΔΔ/μ0, Φ = 0.3(Le − 1) + 0.7 and CF = 0.11
T2	where KaΔ = (u′Δ/SL)^3/2(Δ/δth)^−1/2 is local sub-grid Karlovitz number and .
T3	where C3 = 7.5, C4 = 0.75(1.0 + KaΔ)^−0.4, and with p = 0.2 + 1.5(1 − Le)
[T4 − D2 + f(D)]	where fTD = exp[−0.27(SLΔ/αT0)^1.7], c* = 1.0 − 0.83erf[0.5(ΔSL/αt0) − 2.3] and β′3 = 5.7/Le^0.2