Analysis of Reaction Advection Diffusion Spectrum of Laminar Premixed Flames

Transcription

1 Analysis of Reaction Advection Diffusion Spectrum of Laminar Premixed Flames Ashraf N. Al-Khateeb Joseph M. Powers DEPARTMENT OF AEROSPACE AND MECHANICAL ENGINEERING UNIVERSITY OF NOTRE DAME, NOTRE DAME, INDIANA 48 th AIAA Aerospace Science Meeting Orlando, Florida 6 January 2010

2 Thermodynamics Diffusion Diffusion For a continuous random walk in 2-D, a particle must make (1) steps to travel a distance d, where l is the mean free path. The time required is then (2) where is the sound speed. Defining a diffusion coefficient (3) yields (4) Diffusion Coefficient, Diffusion Equation, Eddy Diffusion, Effusion, Graham's Law of DIffusion length vs. reaction time: l = Dτ

3 Outline Introduction Simple one species reaction advection diffusion problem. Simple two species reaction diffusion problem. Laminar premixed hydrogen air flame. Summary

4 Introduction Motivation and background Combustion is often unsteady and spatially inhomogeneous. Most realistic reactive flow systems have multi-scale character. Severe stiffness, temporal and spatial, arises in detailed gasphase kinetics modeling. As the scales range widens, more stringent demands arise to assure the accuracy of the results. Proper numerical resolution of all scales is critical to draw correct conclusions and achieve a mathematically verified solution.

5 Segregation of chemical dynamics from transport dynamics is a prevalent notion in combustion modeling, e.g. operator splitting. However, reaction, advection, and diffusion scales are coupled in reactive flows. The interplay between chemistry and transport needs to be captured for accurate modeling. Spectral analysis is a tool to understand the coupling between transport and chemistry. All relevant scales have to be brought into simultaneous focus a priori for DNS.

6 General objective To identify the scales associated with each Fourier mode of a variety of wavelengths for unsteady spatially inhomogenous reactive flow problems. Particular objective To calculate the time scale spectrum of a one-dimensional atmospheric pressure hydrogen air system.

7 Model problem I A linear one species model for reaction, advection, and diffusion: ψ(x, t) ψ(x, t) + u = D 2 ψ(x, t) t x x 2 aψ(x, t), ψ ψ(0, t) = ψ u, x = 0, ψ(x, 0) = ψ u. x=l Time scale spectrum For the spatially homogenous version: ψ(t) = ψ u exp( at), τ = 1 a t < 1 a.

8 Length scale spectrum The steady structure: ( ) exp(µ 1 x) exp(µ 2 x) ψ s (x) = ψ u 1 µ 1 µ 2 exp(l(µ 1 µ 2 )) + exp(µ 2x), µ 1 = u 2D ( aD u 2 ), µ 2 = u 2D l i = 1. µ i ( aD u 2 ), For fast reaction (a >> u 2 /D): l 1 = l 2 = D a x < D a.

9 Spatio-temporal spectrum 1) continuous spectrum: ψ(x, t) = Ψ(t)e ıikx Ψ(t) = C exp ( ( a 1 + ıiku ) ) a + Dk2 t. a long wavelength: lim k 0 τ = lim λ τ = 1 a, short wavelength: lim τ = lim τ = λ2 1 k λ 0 4π 2 D, S t = ( ) 2 2π D. λ a Balance between reaction and diffusion at k 2π λ = a D = 1/l, Using Taylor expansion: τ = 1 a ( 1 D a ( ) λ 2 2π u 2 2a 2 ( λ 2π ) 2 ) + O ( ) 1. λ 4

10 10 8 1/a τ 10 9 τ [s] l = D/a = Dτ λ/(2π) [cm] Similar to H 2 air : τ 1/a = 10 8 s,d = 10 cm 2 /s, D l = = 3.2 a 10 4 cm.

11 2) Spatially discretized spectrum: ψ(x,t) ψ i (t), i = 1,...,N. Original boundary conditions: A dψ dt = B ψ (µa B) υ = 0. Dirichlet boundary condition modification: τ j = a + 2D(N+1)2 L 2 (1 1 u 1 2 L 2 4D 2 (N+1) cos 2 Effects of advection and diffusion: τ 1 1 a ( )), j = 1,..., N 2, jπ N 1 ( 1 D a(l/π) ) u 2 ad, For small N : lim x τ j 1/a, For large N : lim x 0 τ j L 2 (4D(N + 1) 2 ), S t = ( ) 2 2(N + 1) D. L a

12 Model problem II An uncoupled reaction-diffusion system with chemical stiffness: ψ i (x, t) t = D 2 ψ i (x, t) x 2 a i ψ i (x, t), ψ i (0, t) = ψ iu, ψ i x (L, t) = 0, ψ i(x, 0) = ψ iu. Time scale spectrum For the spatially homogenous version: ψ i (t) = ψ iu exp( a i t), τ i = 1 a i S t = a largest a smallest t < 1 a largest.

13 Length scale spectrum The steady structure: ψ is (x) = l i = D a i S t = Spatio-temporal spectrum ψ iu cosh L/ q D ai cosh alargest a smallest, x < ( L x q D ai ) D a largest. 1) Continuous spectrum: ) ψ i (x, t) = Ψ i (t)e ıikx Ψ i (t) = C exp ( a i (1 + Dk2 a i, ) t. 2) Discrete spectrum: " X ψ i = ψ is + A κ exp a i 1 + κ=1 (2κ 1)π 2L r! 2 #! D (2κ 1)π t sin 2L a i «x.

14 3) Spatially discretized spectrum: for a 1 = 10 4 s, a 2 = 10 2 s, D = 10 cm 2 /s, and L = 10 cm, modified wavelength: λ = 4L/(2n 1), associated length scale: l = λ/(2π) l = prediction from length scale spectrum: l i = D/a i, 2L (2n 1)π, τi [s] l1 = cm o o oooooooooooo o o o l2 = cm ooooooooooooo o o o o o o o o o o o o o o Analytical r Numerical o = 1/a 2 = 1/a oo oooo oooooooooooooo ooooooo λ/(2π) 1 [cm] 10

15 Prediction from length scale spectrum: l i = D/a i, τf undamental [s] l1 = cm l2 = cm L/π [cm] = 1/a 2 = 1/a 1

16 Laminar Premixed Hydrogen Air Flame N = 9 species,l = 3 atomic elements, and J = 19 reversible reactions, Y u = stoichiometric Hydrogen-Air: 2H 2 + (O N 2 ), T u = 800 K, p o = 1 atm, neglect Soret effect, Dufour effect, and body forces, CHEMKIN and IMSL are employed.

17 TIme evolution of the spatially homogenous version Y i t [s] H 2 O 2 H 2 O H O OH HO 2 H 2O N

18 Time scale spectrum S t O (10 4 ), t < τ fastest = s, τi [s] t [s] 10 2 = s = s

19 Fully resolved steady structure a 10 0 H 2 O 2 N 2 H O H HO 2 H 2 O 2 Y i OH O x [cm] a Al-Khateeb, Powers, and Paolucci, Communications in Computational Physics, to appear.

20 Length scale spectrum S x O (10 4 ), x < l finest = cm, 10 8 [cm] 10 4 i x [cm]

21 Spatio-temporal spectrum PDEs 2N + 2 PDAEs, A(z) z t + B(z) z x = f(z). Spatially homogeneous system at chemical equilibrium subjected to a spatially inhomogeneous perturbation, z = z z e, A e z t + Be z x = Je z. Spatially discretized spectrum, A e dz dt = (J e B e ) Z, A e and (J e B e ) are singular matrices.

22 L = 1 cm and D mix = 64 cm 2 /s, modified wavelength: λ = 4L/(2n 1), associated length scale: l = λ/(2π) l = 2L (2n 1)π, 10 4 = s τi [s] l s = D mix τ slowest λ/(2π) [cm] = s

23 l finest = cm, l f = D mix τ fastest = cm, l s = D mix τ slowest = cm, 10 4 = s τf undamental [s] lf ls = s L/π [cm]

24 Summary Time and length scales are coupled. Short wavelength modes are dominated by diffusion, and coarse wavelength modes have time scales dominated by reaction. For a resolved diffusive structure, Fourier modes of sufficiently fine wavelength must be considered so that their associated time scale is of similar magnitude to the fastest chemical time scale. For a p = 1 atm,h 2 +air laminar flame, the length scale where fast reaction balances diffusion is 2 µm; the associated fast time scale is 10 ns.