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

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

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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

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SLIDE 1

Spatial and Temporal Scales Coupling in Reactive Flows

Ashraf N. Al-Khateeb

REACTIVE FLOW MODELING LABORATORY KING ABDULLAH UNIVERSITY OF SCIENCE & TECHNOLOGY, SAUDI ARABIA

Joseph M. Powers and Samuel Paolucci

AEROSPACE & MECHANICAL ENGINEERING DEPARTMENT UNIVERSITY OF NOTRE DAME, NOTRE DAME, INDIANA, USA

The 3rd International Workshop on Model Reduction in Reacting Flows Corfu, Greece 27 April 2011

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SLIDE 2

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.

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SLIDE 3

Observed Physics Mathematical Model Computational algorithm Analysis Programming Validation Verification

  • Computations

should have fidelity with the underlying mathematics: verification.

  • The mathematical model needs to

represent observed physics: validation.

  • In computational studies, it is a neces-

sity to address these two issues.

  • Proper numerical resolution of all

scales is critical to draw correct con- clusions.

  • All relevant scales have to be brought

into simultaneous focus for DNS.

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SLIDE 4

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.

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SLIDE 5

Illustrative Model Problem

A linear one species model for reaction, advection, and diffusion:

∂ψ(x, t) ∂t + u∂ψ(x, t) ∂x = D∂2ψ(x, t) ∂x2 − aψ(x, t), ψ(0, t) = ψu, ∂ψ ∂x

  • x→L

= 0, ψ(x, 0) = ψu.

Time scale spectrum For the spatially homogenous version:

ψh(t) = ψu exp (−at) ,

reaction time constant:

τ = 1 a = ⇒ ∆t ≪ τ.

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SLIDE 6

Length Scale Spectrum

  • The steady structure:

ψs(x) = ψu

  • exp(µ1x) − exp(µ2x)

1 − µ1

µ2 exp(L(µ1 − µ2)) + exp(µ2x)

  • ,

µ1 = u 2D

  • 1 +
  • 1 + 4aD

u2

  • ,

µ2 = u 2D

  • 1 −
  • 1 + 4aD

u2

  • ,

ℓi =

  • 1

µi

  • .
  • For fast reaction (a ≫ u2/D):

ℓ1 = ℓ2 =

  • D

a = √ Dτ = ⇒ ∆x ≪ √ Dτ.

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SLIDE 7

Spatio-Temporal Spectrum ψ(x, t) = Ψ(t)eı

ikx

⇒ Ψ(t) = C exp

  • −a
  • 1 +

ı iku

a + Dk2 a

  • t
  • .
  • For fast reaction:

lim

k→0 τ = lim λ→∞ τ = 1

a,

  • For slow reaction: lim

k→∞ τ = lim λ→0 τ = λ2

4π2 1 D,          St =

  • 2π

λ

  • D

a 2 .

  • Balance between reaction and diffusion at k ≡ 2π

λ = a D = 1/ℓ,

  • Using Taylor expansion:

|τ| = 1 a

  • 1 −

D a λ

2π

2 − u2 2a2 λ

2π

2

  • + O

1 λ4

  • .
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SLIDE 8

10

−10

10

−11

10

−12

10

−9

10

−8

10

3

10

1

10

−1

10

−3

10

−5

1 2

|τ| [s]

1/a = τ

λ/(2π) [cm]

√ Dτ = ℓ

  • Similar to H2 − air : τ = 1/a = 10−8 s, D = 10 cm2/s,
  • ℓ =
  • D

a =

√ Dτ = 3.2 × 10−4 cm.

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SLIDE 9

Laminar Premixed Flames

Adopted Assumptions:

  • One-dimensional,
  • Low Mach number,
  • Neglect thermal diffusion effects and body forces.

Governing Equations:

∂ρ ∂t + ∂ ∂x (ρu) = 0, ρ∂h ∂t + ρu∂h ∂x + ∂Jq ∂x = 0, ρ∂yl ∂t + ρu∂yl ∂x + ∂jm

l

∂x = 0, l = 1, . . . , L − 1, ρ∂Yi ∂t + ρu∂Yi ∂x + ∂Jm

i

∂x = ˙ ωi ¯ mi, i = 1, . . . , N − L.

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SLIDE 10
  • Unsteady spatially homogeneous reactive system:

dz(t) dt = f (z(t)) , z(t) ∈ RN, f : RN → RN. 0 = (J − λI) · υ. St = τslowest τfastest , τi = 1 |Re(λi)|, i = 1, . . . , R ≤ N − L.

  • Steady spatially inhomogeneous reactive system:

˜ B (˜ z(x))· d˜ z(x) dx = ˜ f (˜ z(x)) , ˜ z(x) ∈ R2N+2, ˜ f : R2N+2 → R2N+2. ˜ λ ˜ B · ˜ υ =

  • ˜

J − ˜ Ψ · d˜ z dx

  • · ˜

υ. Sx = ℓcoarsest ℓfinest , ℓi = 1 |Re(˜ λi)| , i = 1, . . . , 2N − L.

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SLIDE 11

Laminar Premixed Hydrogen–Air Flame

  • Standard detailed mechanisma; N = 9 species, L = 3 atomic elements,

and J = 19 reversible reactions,

  • stoichiometric hydrogen-air: 2H2 + (O2 + 3.76N2),
  • adiabatic and isobaric: Tu = 800 K, p = 1 atm,
  • calorically imperfect ideal gases mixture,
  • neglect Soret effect, Dufour effect, and body forces,
  • CHEMKIN and IMSL are employed.
  • aJ. A. Miller, R. E. Mitchell, M. D. Smooke, and R. J. Kee, Proc. Combust. Ins. 19, p. 181, 1982.
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SLIDE 12
  • Unsteady spatially homogeneous reactive system:

10

−20

10

−15

10

−10

10

−5

10 10

−25

Yi

HO

2

H O

2

H O

2

H

2

O

2

H OH O N

2

10

−30

10

−10 10 −8

10

−6

10

−4

10

−2

10

t [s]

10

2

2

600 1000 1400 1800 2200 2600 10

−10 10 −8

10

−6

10

−4

10

−2

10

t [s]

10

2

T [K]

10

−8

10

−6

10

−4

10

−2

10 10

2

10

4

10

−10 10 −8

10

−6

10

−4

10

−2

10

t [s]

10

2

slowest fastest

= 1.8×10 s

−2

= 1.0×10 s

−8 τi [s]

τ τ

St ∼ O ` 104´ .

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SLIDE 13
  • Steady spatially inhomogeneous reactive system:a

coarsest finest

= 2.6×10 cm = 2.4×10 cm

−4

10

−5

10

−4

10

−3

10

−2

10

−1

10 10

1

10

2

10

−4

10 10

4

10

8 i

[cm] 10

−15

10

−10

10

−5

10

Yi

10

−5

10

−4

10

−3

10

−2

10

−1

10 10

1

10

2

x [cm]

HO

2

H O

2 2

H O

2

O

2

H2 H OH O N

2

10

−5

10

−4

10

−3

10

−2

10

−1

10 10

1

10

2

800 1200 1600 2000 2400 2800

T K [ ] x [cm] x [cm]

ℓ ℓ

Sx ∼ O ` 104´ .

  • aA. N. Al-Khateeb, J. M. Powers, and S. Paolucci, Comm. Comp. Phys. 8(2): 304, 2010.
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SLIDE 14

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 − ze,

Ae · ∂z′ ∂t + Be · ∂z′ ∂x = Je · z′.

  • Spatially discretized spectrum,

Ae · dZ dt = (J e − Be) · Z, Z ∈ R2N (N+1).

  • The time scales of the generalized eigenvalue problem,

τi = 1 |Re (λi)|, i = 1, . . . , (N − 1)(N − 1).

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SLIDE 15
  • L = 1 cm and Dmix = 64 cm2/s,
  • modified wavelength:

λ = 4L/(2n − 1),

  • associated length scale: ℓ =

λ/(2π) ⇒ ℓ =

2L (2n−1)π ,

0.02 0.05 0.10 0.20 0.50 0.01 10

−6

10

−7

10

−8

10

−4

10

−5

τi [s]

= 1.0 × 10−8 s = 1.8 × 10−4 s

  • λ/(2π)

[cm]

ℓ1 = √Dmixτslowest

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SLIDE 16
  • Dmix =

1 N 2

N

i=1

N

j=1 Dij,

  • ℓ1 = √Dmixτslowest = 1.1 × 10−1 cm,
  • ℓ2 =
  • Dmixτfastest = 8.0 × 10−4 cm ≈ ℓfinest = 2.4 × 10−4 cm.

10

−6

10

−4

10

−2

10 10

2

10

−14

10

−12

10

−10

10

−8

10

−6

10

−4

1 2

τfundamental [s] 2L/π [cm]

τfastest = 1.8 × 10−4 s τslowest = 1.0 × 10−8 s

ℓ1 ℓ2

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SLIDE 17

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, H2 + air laminar flame, the length scale where fast

reaction balances diffusion is ∼ 2 µm, the necessary scale for a DNS.