Verified Computations of Laminar Premixed Flames Ashraf N. - PowerPoint PPT Presentation

Verified Computations of Laminar Premixed Flames Ashraf N. Al-Khateeb Joseph M. Powers Samuel Paolucci Department of Aerospace and Mechanical Engineering University of Notre Dame, Notre Dame, Indiana 45 th AIAA Aerospace Science Meeting and Exhibit Reno, Nevada 8 January 2007

Objective To obtain an accurate a priori estimate for the finest length scale in a continuum model of reactive flow with detailed kinetics and multi-component transport of: • steady, • one-dimensional, • ideal gas mixture, • premixed laminar flame.

Mathematical Model Governing Equations ∂ρ − ∂ = x ( ρ ˜ u ) , ∂ ˜ ∂ ˜ t ∂ − ∂ u 2 + p − τ � � t ( ρ ˜ u ) = ρ ˜ , ∂ ˜ ∂ ˜ x u 2 u 2 � � e + ˜ �� � � e + ˜ � � ∂ − ∂ 2 + p ρ − τ + J q = ρ ˜ ρ u , ∂ ˜ 2 ∂ ˜ x ρ t ∂ − ∂ uY i + J m t ( ρY i ) = x ( ρ ˜ i ) + ˙ ω i M i , i = 1 , . . . , N − 1 . ∂ ˜ ∂ ˜

Constitutive Relations „ 1 N „ « 1 « 1 M i D ik Y k ∂χ k 1 − M k ∂p ∂T X J m − D T = x + ρ x , i i ∂ ˜ ∂ ˜ ∂ ˜ M χ k M p x T k =1 k � = i „ 1 N N D T „ « 1 « ∂χ i 1 − M i ∂p J q X J m X i = q + x + i h i − ℜ T , M i χ i ∂ ˜ M p ∂ ˜ x i =1 i =1 − k ∂T = q x , ∂ ˜ N ρY i X p = ℜ T M i , i =1 and others . . .

Dynamical System Formulation • PDEs − → ODEs d dx ( ρu ) = 0 , d dx ( ρuh + J q ) = 0 , d dx ( ρuY e l + J e l ) = 0 , l = 1 , . . . , L − 1 , d dx ( ρuY i + J m i ) = ω i M i , ˙ i = 1 , . . . , N − L. • ODEs − → 2 N + 2 DAEs A ( z ) · d z dx = f ( z ) .

A Posteriori Length Scale Analysis • Standard eigenvalue analysis is not applicable; A is singular. • The generalized eigenvalues can be calculated – from λ A ∗ · v B ∗ · v , = – and the length scales are given by 1 ℓ i = i = 1 , . . . , 2 N − L. | Re ( λ i ) | ,

Results Steady Laminar Premixed Hydrogen-Air Flame • N = 9 species, L = 3 atomic elements, and J = 19 reversible reactions, • Stoichiometric Hydrogen-Air: 2 H 2 + ( O 2 + 3 . 76 N 2 ) , • T unburned = 800 K , • p o = 1 atm , • CHEMKIN and IMSL are employed.

Mathematical Verification • Good agreement with Smooke et al., ’83 . 0.6 2500 0.6 HO 2 × 10 3 Temperature 0.5 0.5 2000 0.4 0.4 1500 [K] [K] χ i H 2 O χ i 0.3 0.3 H × 10 1 T 1000 0.2 0.2 O 2 OH × 10 1 500 0.1 0.1 H 2 0 0 0 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 x x [cm] [cm]

Experimental Validation • Good agreement with Dixon-Lewis, ’79. Experimental data 350 compiled by Dixon−Lewis, ’79. Present work 300 250 [cm/sec] 200 S 150 100 50 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 χ H 2

Fully Resolved Structure 0 10 N 2 H 2 O O 2 H 2 H −5 10 HO 2 H 2 O 2 Y i −10 10 OH O −15 10 −5 −4 −3 −2 −1 0 1 2 10 10 10 10 10 10 10 10 x [cm]

Predicted Length Scales 8 10 6 10 4 10 [cm] 2 10 ℓ i 0 10 −2 10 ℓ finest ∼ 10 − 4 cm −4 10 −5 −4 −3 −2 −1 0 1 2 10 10 10 10 10 10 10 10 x [cm]

Mean-Free-Path Estimate • The mixture mean-free-path scale is the cutoff minimum length scale associated with continuum theories. • A simple estimate for this scale is given by Vincenti and Kruger, ’65 : M √ ℓ mfp = 2 N πd 2 ρ.

• ℓ finest is well correlated with ℓ mfp . 4 10 2 10 ℓ reaction Length scales [cm] 0 10 −2 10 ℓ finest −4 10 ℓ mfp −6 10 −1 0 1 10 10 10 Pressure [atm]

Extensions • Two additional sets of calculations: – Variable fuel/air ratio, – Hydrocarbon mixtures (methane, ethane, ethylene, acetylene). • Two combustion regimes: – Freely propagating laminar fl ame, – Chapman-Jouguet detonation ( Powers and Paolucci, ’05 ).

Equivalence ratio infl uence is negligible 2 2 10 10 ℓ reaction 0 0 Length scales [cm] Length scales [cm] 10 10 ℓ induction −2 −2 10 10 ℓ finest ℓ finest −4 −4 10 10 ℓ mfp ℓ mfp −6 −6 10 10 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 Φ Φ (a) Laminar premixed fl ame (b) Chapman-Jouguet detonation

Defl agration Methane−Air Ethane−Air 2 2 10 10 ℓ reaction ℓ reaction Length scales [cm] 0 Length scales [cm] 0 10 10 −2 −2 10 10 ℓ finest −4 ℓ finest −4 10 10 ℓ mfp ℓ mfp −6 −6 10 10 0 1 0 1 10 10 10 10 Pressure [atm] Pressure [atm] Acetylene−Air Ethylene−Air 2 2 10 10 ℓ reaction ℓ reaction Length scales [cm] Length scales [cm] 0 0 10 10 −2 −2 10 10 ℓ finest ℓ finest −4 −4 10 10 ℓ mfp ℓ mfp −6 −6 10 10 0 1 0 1 10 10 10 10 Pressure [atm] Pressure [atm]

Detonation Methane−Air Ethane−Air ℓ induction ℓ induction 0 0 10 10 Length scales [cm] Length scales [cm] −2 −2 10 10 −4 −4 10 10 ℓ finest ℓ finest ℓ mfp ℓ mfp −6 −6 10 10 0 1 0 1 10 10 10 10 Pressure [atm] Pressure [atm] Acetylene−Air Ethylene−Air 0 ℓ induction 0 10 10 ℓ induction Length scales [cm] Length scales [cm] −2 −2 10 10 −4 −4 10 10 ℓ finest ℓ finest ℓ mfp ℓ mfp −6 −6 10 10 0 1 0 1 10 10 10 10 Pressure [atm] Pressure [atm]

Comparison with Published Results ∆ x, ( cm ) ℓ finest , ( cm ) ℓ mfp , ( cm ) Ref. Mixture molar ratio 2 . 50 × 10 − 2 8 . 05 × 10 − 4 4 . 33 × 10 − 5 1 . 26 H 2 + O 2 + 3 . 76 N 2 1 6 . 12 × 10 − 4 4 . 33 × 10 − 5 CH 4 + 2 O 2 + 10 N 2 2 unknown 3 . 54 × 10 − 2 4 . 35 × 10 − 5 7 . 84 × 10 − 6 0 . 59 H 2 + O 2 + 3 . 76 N 2 3 1 . 56 × 10 − 3 2 . 89 × 10 − 5 6 . 68 × 10 − 6 CH 4 + 2 O 2 + 10 N 2 4 1. Katta V. R. and Roquemore W. M., 1995, Combustion and Flame , 102 (1-2), pp. 21-40. 2. Najm H. N. and Wyckoff P . S., 1997, Combustion and Flame , 110 (1-2), pp. 92-112. 3. Patnaik G. and Kailasanath K., 1994, Combustion and Flame , 99 (2), pp. 247-253. 4. Knio O. M. and Najm H. N., 2000, Proc. Combustion Institute , 28 , pp. 1851-1857.

Discussion A lower bound for the grid resolution is desirable • Grid convergence, ( Roache, ’98 ). – Convergence rate must be consistent with truncation error order. – Grids coarser than the finest length scale could unphysically infl uence reaction dynamics. • Direct numerical simulation (DNS). – Our results are in rough agreement with independent estimates ows, ∆ x = 4 . 30 × 10 − 4 cm , ( Chen found in DNS of reacting fl et al., ’06 ).

The modified equation for a model problem ν ∂ 2 ψ ∂ψ ∂t + a ∂ψ = ∂x 2 , ∂x ψ n +1 ψ n i − ψ n ψ n i +1 − 2 ψ n i + ψ n − ψ n i − 1 i − 1 i i + a = ν , ∆ x 2 ∆ t ∆ x 0 1 B „ « C ∂ 2 ψ ∂ψ ∂t + a ∂ψ a ∆ x 1 − a ∆ t B C = ν + B C ∂x 2 B C ∂x 2 ∆ x @ A | {z } leading order numerical diffusion „ a ∆ t ! « 2 + a ∆ x 2 ∂ 3 ψ + 6 ν ∆ t − 1 + ∂x 3 + . . . ∆ x 2 6 ∆ x | {z } leading order numerical dispersion • Discretization-based terms alter the dynamics. • Numerical diffusion could suppress physical instability.

• To solve for the steady structure ν d 2 ψ adψ = dx 2 , dx � ax � Exact solution ⇒ ψ = C 1 + C 2 exp . ν – Analogous to what has been done in our work λ = [0 a/ν ] , ⇒ ℓ finest = ν/a. – The required grid resolution is ∆ x < ν/a . • This grid size guarantees that the steady parts of the dissipation and dispersion errors in the model problem are small.

Implications for combustion • Equilibrium quantities are insensitive to resolution of fine scales. • Due to non-linearity, errors at micro-scale level may alter the macro-scale behavior. • The sensitivity of results to fine scale structures is not known a priori . • Lack of resolution may explain some failures, e.g. DDT. • Linear stability analysis: – Requires the fully resolved steady state structure. – For one-step kinetics, Sharpe, ’03 shows failure to resolve steady structures leads to quantitative and qualitative errors in premixed laminar fl ame dynamics.

Conclusions • To formally resolve the one-dimensional steady reactive fl ow, micron-level resolution is needed. • Results will likely hold for multi-dimensional unsteady fl ows. • The finest length scales are fully refl ective of the underlying physics and not the particular mixture, chemical kinetics mech- anism, or numerical method. • The required grid resolution can be easily estimated a priori by a simple mean-free-path calculation. • Present steady results cannot show where unsteady models will fail, but accurate capture of bifucation dynamics will likely require capture of all scales.