Spatial and Temporal Scales Coupling in Reactive Flows Ashraf N. - PowerPoint PPT Presentation

Spatial and Temporal Scales Coupling in Reactive Flows Ashraf N. Al-Khateeb R EACTIVE F LOW M ODELING L ABORATORY K ING A BDULLAH U NIVERSITY OF S CIENCE & T ECHNOLOGY , S AUDI A RABIA Joseph M. Powers and Samuel Paolucci A EROSPACE & M ECHANICAL E NGINEERING D EPARTMENT U NIVERSITY OF N OTRE D AME , N OTRE D AME , I NDIANA , USA The 3 rd International Workshop on Model Reduction in Reacting Flows Corfu, Greece 27 April 2011

Motivation and Background • Severe stiffness, temporal and spatial, arises in detailed kinetics modeling. • Typical reactive flow systems admit multi-scale character. • To achieve DNS, the interplay between chemistry and transport needs to be captured. • The interplay between reaction and diffusion length and time scales is well √ summarized by the classical formula ℓ ∼ D τ. • Segregation of chemical dynamics from transport dynamics is a prevalent notion in reduced kinetics combustion modeling. Is this valid? • Spectral analysis is a tool to understand the coupling between chemistry’s and transport’s reaction and diffusion scales.

• Computations should have fidelity with the underlying mathematics: Observed verification. Physics Validation • The mathematical model needs to Analysis represent observed physics: validation. Mathematical • In computational studies, it is a neces- Model Programming Verification sity to address these two issues. • Proper numerical resolution of all scales is critical to draw correct con- Computational algorithm clusions. • All relevant scales have to be brought into simultaneous focus for DNS.

Objectives • To identify all the physical scales inherent in reacting systems with detailed kinetics and diffusive transport. • To illustrate the coupling of time and length scales in reactive flows. • To identify the scales associated with each Fourier mode of varying wavelength for unsteady spatially inhomogenous reactive flow problems.

Illustrative Model Problem A linear one species model for reaction, advection, and diffusion: = D∂ 2 ψ ( x, t ) ∂ψ ( x, t ) + u∂ψ ( x, t ) − aψ ( x, t ) , ∂x 2 ∂t ∂x � � ∂ψ � ψ (0 , t ) = ψ u , = 0 , ψ ( x, 0) = ψ u . � ∂x x → L Time scale spectrum ψ h ( t ) = ψ u exp ( − at ) , For the spatially homogenous version: τ = 1 = ⇒ ∆ t ≪ τ. reaction time constant: a

Length Scale Spectrum • The steady structure: � � exp( µ 1 x ) − exp( µ 2 x ) ψ s ( x ) = ψ u µ 2 exp( L ( µ 1 − µ 2 )) + exp( µ 2 x ) , 1 − µ 1 � � � � � � µ 1 = u 1 + 4 aD µ 2 = u 1 + 4 aD 1 − 1 + , , u 2 u 2 2 D 2 D � � � � 1 � � ℓ i = � . � µ i • For fast reaction ( a ≫ u 2 /D ): � √ √ D ℓ 1 = ℓ 2 = a = Dτ = ⇒ ∆ x ≪ Dτ.

Spatio-Temporal Spectrum � � � � a + Dk 2 i ku ı ψ ( x, t ) = Ψ( t ) e ı i kx ⇒ Ψ( t ) = C exp − a 1 + t . a  λ →∞ τ = 1  • For fast reaction:  k → 0 τ = lim lim a, � � 2 �   2 π D λ → 0 τ = λ 2 S t = . 1 λ a • For slow reaction: lim k →∞ τ = lim D,    4 π 2  λ = � a • Balance between reaction and diffusion at k ≡ 2 π D = 1 /ℓ, • Using Taylor expansion: � � � 1 � u 2 | τ | = 1 D 1 − � 2 − + O . � λ 2 a 2 � λ � 2 λ 4 a a 2 π 2 π

−8 1 /a = τ 10 −9 10 [ s ] 2 −10 10 1 √ | τ | −11 Dτ = ℓ 10 −12 10 −5 −3 −1 1 3 10 10 10 10 10 λ/ (2 π ) [ cm ] Similar to H 2 − air : τ = 1 /a = 10 − 8 s, D = 10 cm 2 /s , • � √ Dτ = 3 . 2 × 10 − 4 cm. D • ℓ = a =

Laminar Premixed Flames Adopted Assumptions: • One-dimensional, • Low Mach number, • Neglect thermal diffusion effects and body forces. Governing Equations: ∂ρ ∂t + ∂ ∂x ( ρu ) = 0 , ∂x + ∂ J q ρ∂h ∂t + ρu∂h ∂x = 0 , ∂x + ∂ j m ρ∂y l ∂t + ρu∂y l l ∂x = 0 , l = 1 , . . . , L − 1 , ∂x + ∂ J m ρ∂Y i ∂t + ρu∂Y i i ∂x = ˙ ω i ¯ i = 1 , . . . , N − L. m i ,

• Unsteady spatially homogeneous reactive system: d z ( t ) f : R N → R N . z ( t ) ∈ R N , = f ( z ( t )) , dt 0 = ( J − λ I ) · υ . S t = τ slowest 1 i = 1 , . . . , R ≤ N − L. , τ i = | Re ( λ i ) | , τ fastest • Steady spatially inhomogeneous reactive system: z ( x )) · d ˜ z ( x ) f : R 2 N +2 → R 2 N +2 . ˜ = ˜ ˜ z ( x ) ∈ R 2 N +2 , B (˜ f (˜ z ( x )) , ˜ dx � � Ψ · d ˜ z λ ˜ ˜ J − ˜ ˜ B · ˜ υ = · ˜ υ . dx S x = ℓ coarsest 1 , ℓ i = , i = 1 , . . . , 2 N − L. | Re (˜ ℓ finest λ i ) |

Laminar Premixed Hydrogen–Air Flame • Standard detailed mechanism a ; N = 9 species, L = 3 atomic elements, and J = 19 reversible reactions, • stoichiometric hydrogen-air: 2 H 2 + ( O 2 + 3 . 76 N 2 ) , • adiabatic and isobaric: T u = 800 K, p = 1 atm , • calorically imperfect ideal gases mixture, • neglect Soret effect, Dufour effect, and body forces, • CHEMKIN and IMSL are employed. a J. A. Miller, R. E. Mitchell, M. D. Smooke, and R. J. Kee, Proc. Combust. Ins. 19 , p. 181, 1982.

• Unsteady spatially homogeneous reactive system: 0 10 2600 −5 10 2200 −10 10 T [ K ] 1800 −15 H 2 Y i 10 O 2 H O 2 1400 −20 H 10 O OH −25 1000 10 HO 2 H O 2 2 −30 N 2 10 600 −10 10 −10 10 −8 −6 −4 −2 0 2 −8 −6 −4 −2 0 2 10 10 10 10 10 10 10 10 10 10 10 10 t [ s ] t [ s ] 4 10 2 10 0 10 −2 [ s ] 10 −2 −4 τ = 1.8×10 s 10 slowest τi ` 10 4 ´ S t ∼ O . −6 10 −8 −8 = 1.0×10 s τ 10 fastest −10 10 −8 −6 −4 −2 0 2 10 10 10 10 10 10 t [ s ]

• Steady spatially inhomogeneous reactive system: a 0 2800 10 N 2 O H O 2 2 H 2 2400 H −5 10 [ ] HO 2 T K 2000 H O 2 2 Y i 1600 −10 10 1200 OH O −15 10 800 −5 −4 −3 −2 −1 0 1 2 −5 −4 −3 −2 −1 0 1 2 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 x [ cm ] x [ cm ] 8 10 4 10 [ cm ] 0 ℓ = 2.6×10 cm 0 i coarsest 10 ` 10 4 ´ S x ∼ O . −4 ℓ = 2.4×10 cm finest −4 10 −5 −4 −3 −2 −1 0 1 2 10 10 10 10 10 10 10 10 x [ cm ] a A. N. Al-Khateeb, J. M. Powers, and S. Paolucci, Comm. Comp. Phys. 8(2): 304, 2010.

Spatio-Temporal Spectrum • PDEs − → 2 N + 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 + B e · ∂ z ′ ∂x = J e · z ′ . • Spatially discretized spectrum, A e · d Z dt = ( J e − B e ) · Z , Z ∈ R 2 N ( N +1) . • The time scales of the generalized eigenvalue problem, 1 τ i = | Re ( λ i ) | , i = 1 , . . . , ( N − 1)( N − 1) .

• L = 1 cm and D mix = 64 cm 2 /s , • modified wavelength: � λ = 4 L / (2 n − 1) , • associated length scale: ℓ = � 2 L λ/ (2 π ) ⇒ ℓ = (2 n − 1) π , = 1 . 8 × 10 − 4 s −4 10 −5 10 [ s ] −6 ℓ 1 = √ D mix τ slowest 10 τ i −7 10 −8 = 1 . 0 × 10 − 8 s 10 0.01 0.02 0.05 0.10 0.20 0.50 � λ/ (2 π ) [ cm ]

� N � N 1 • D mix = j =1 D ij , N 2 i =1 • ℓ 1 = √ D mix τ slowest = 1 . 1 × 10 − 1 cm , � D mix τ fastest = 8 . 0 × 10 − 4 cm ≈ ℓ finest = 2 . 4 × 10 − 4 cm . • ℓ 2 = τ fastest = 1 . 8 × 10 − 4 s −4 10 [ s ] −6 10 τ fundamental τ slowest = 1 . 0 × 10 − 8 s −8 10 −10 1 10 2 −12 10 ℓ 2 ℓ 1 −14 10 −6 −4 −2 0 2 10 10 10 10 10 2 L /π [ cm ]

Conclusions • Time and length scales are coupled. • Coarse wavelength modes have time scales dominated by reaction. • Short wavelength modes have time scales dominated by diffusion. • Fourier modal analysis reveals a cutoff length scale for which time scales are dictated by a balance between transport and chemistry. • Fine scales, temporal and spatial, are essential to resolve reacting systems; √ the finest length scale is related to the finest time scale by ℓ ∼ Dτ . • For a p = 1 atm, H 2 + air laminar flame, the length scale where fast reaction balances diffusion is ∼ 2 µm , the necessary scale for a DNS.