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

SLIDE 1

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

45th AIAA Aerospace Science Meeting and Exhibit

Reno, Nevada 8 January 2007

SLIDE 2

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.

SLIDE 3

Mathematical Model

Governing Equations ∂ρ ∂˜ t = − ∂ ∂˜ x(ρ˜ u), ∂ ∂˜ t(ρ˜ u) = − ∂ ∂˜ x

ρ˜

u2 + p − τ

,

∂ ∂˜ t

ρ
e + ˜

u2 2

=

− ∂ ∂˜ x

ρ˜

u

e + ˜

u2 2 + p ρ − τ ρ

+ Jq
,

∂ ∂˜ t(ρYi) = − ∂ ∂˜ x(ρ˜ uYi + Jm

i ) + ˙

ωiMi, i = 1, . . . , N − 1.

SLIDE 4

Constitutive Relations

Jm

i

= ρ

N

X

k=1 k=i

MiDikYk M „ 1 χk ∂χk ∂˜ x + „ 1 − Mk M «1 p ∂p ∂˜ x « − DT

i

1 T ∂T ∂˜ x , Jq = q +

N

X

i=1

Jm

i hi − ℜT N

X

i=1

DT

i

Mi „ 1 χi ∂χi ∂˜ x + „ 1 − Mi M «1 p ∂p ∂˜ x « , q = −k ∂T ∂˜ x , p = ℜT

N

X

i=1

ρYi Mi ,

and others . . .

SLIDE 5

Dynamical System Formulation

PDEs −

→ ODEs d dx (ρu) = 0, d dx (ρuh + Jq) = 0, d dx (ρuY e

l + Je l )

= 0, l = 1, . . . , L − 1, d dx (ρuYi + Jm

i )

= ˙ ωiMi, i = 1, . . . , N − L.

ODEs −

→ 2N + 2 DAEs A(z) · dz dx = f(z).

SLIDE 6

A Posteriori Length Scale Analysis

Standard eigenvalue analysis is not applicable; A is singular.
The generalized eigenvalues can be calculated

– from λA∗ · v = B∗ · v, – and the length scales are given by ℓi = 1 |Re (λi)|, i = 1, . . . , 2N − L.

SLIDE 7

Results

Steady Laminar Premixed Hydrogen-Air Flame

N = 9 species, L = 3 atomic elements, and J = 19 reversible

reactions,

Stoichiometric Hydrogen-Air:

2H2 + (O2 + 3.76N2),

Tunburned = 800 K,
po = 1 atm,
CHEMKIN and IMSL are employed.

SLIDE 8

Mathematical Verification

Good agreement with Smooke et al., ’83.

0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.6

[cm]

HO2 × 103 H × 101 OH × 101

χi x

0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.6

[cm] [K] [K]

500 1000 1500 2000 2500 H2 O2 H2O Temperature

χi x T

SLIDE 9

Experimental Validation

Good agreement with Dixon-Lewis, ’79.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 50 100 150 200 250 300 350

[cm/sec]

Experimental data compiled by Dixon−Lewis, ’79. Present work

S χH2

SLIDE 10

Fully Resolved Structure

10

−5

10

−4

10

−3

10

−2

10

−1

10 10

1

10

2

10

−15

10

−10

10

−5

10

[cm]

N2 O2 H2 O H2O OH HO2 H2O2 H

Yi

x

SLIDE 11

Predicted Length Scales

10

−5

10

−4

10

−3

10

−2

10

−1

10 10

1

10

2

10

−4

10

−2

10 10

2

10

4

10

6

10

8

[cm] [cm]

ℓi x ℓfinest

∼ 10−4 cm

SLIDE 12

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:

ℓmfp = M √ 2Nπd2ρ.

SLIDE 13

ℓfinest is well correlated with ℓmfp.

10

−1

10 10

1

10

−6

10

−4

10

−2

10 10

2

10

4

Pressure [atm] Length scales [cm]

ℓmfp ℓreaction ℓfinest

SLIDE 14

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).

SLIDE 15

Equivalence ratio infl uence is negligible

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 10

−6

10

−4

10

−2

10 10

2

Length scales [cm] Φ

ℓmfp ℓfinest ℓreaction

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 10

−6

10

−4

10

−2

10 10

2

Length scales [cm] Φ

ℓmfp ℓfinest ℓinduction

(a) Laminar premixed fl ame (b) Chapman-Jouguet detonation

SLIDE 16

Defl agration

10 10

1

10

−6

10

−4

10

−2

10 10

2

Pressure [atm] Length scales [cm]

Methane−Air

ℓmfp ℓfinest ℓreaction

10 10

1

10

−6

10

−4

10

−2

10 10

2

Pressure [atm] Length scales [cm]

Ethane−Air

ℓmfp ℓfinest ℓreaction

10 10

1

10

−6

10

−4

10

−2

10 10

2

Pressure [atm] Length scales [cm]

Ethylene−Air

ℓmfp ℓfinest ℓreaction

10 10

1

10

−6

10

−4

10

−2

10 10

2

Pressure [atm] Length scales [cm]

Acetylene−Air

ℓmfp ℓfinest ℓreaction

SLIDE 17

Detonation

10 10

1

10

−6

10

−4

10

−2

10

Pressure [atm] Length scales [cm]

Methane−Air

ℓmfp ℓfinest ℓinduction

10 10

1

10

−6

10

−4

10

−2

10

Pressure [atm] Length scales [cm]

Ethane−Air

ℓmfp ℓfinest ℓinduction

10 10

1

10

−6

10

−4

10

−2

10

Pressure [atm] Length scales [cm]

Ethylene−Air

ℓmfp ℓfinest ℓinduction

10 10

1

10

−6

10

−4

10

−2

10

Pressure [atm] Length scales [cm]

Acetylene−Air

ℓmfp ℓfinest ℓinduction

SLIDE 18

Comparison with Published Results

Ref. Mixture molar ratio

∆x, (cm) ℓfinest, (cm) ℓmfp, (cm)

1 1.26H2 + O2 + 3.76N2 2.50 × 10−2 8.05 × 10−4 4.33 × 10−5

2 CH4 + 2O2 + 10N2

unknown

6.12 × 10−4 4.33 × 10−5

3 0.59H2 + O2 + 3.76N2 3.54 × 10−2 4.35 × 10−5 7.84 × 10−6

4 CH4 + 2O2 + 10N2 1.56 × 10−3 2.89 × 10−5 6.68 × 10−6

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.

SLIDE 19

Discussion

A lower bound for the grid resolution is desirable

Grid convergence, (Roache, ’98).

– Convergence rate must be consistent with truncation error

rder.

– 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 found in DNS of reacting fl

ws, ∆x = 4.30×10 −4 cm, (Chen

et al., ’06).

SLIDE 20

The modified equation for a model problem

∂ψ ∂t + a ∂ψ ∂x = ν ∂2ψ ∂x2 , ψn+1

i

− ψn

i

∆t + a ψn

i − ψn i−1

∆x = ν ψn

i+1 − 2ψn i + ψn i−1

∆x2 , ∂ψ ∂t + a ∂ψ ∂x = B B B B @ ν + a∆x 2 „ 1 − a∆t ∆x « | {z }

leading order numerical diffusion

1 C C C C A ∂2ψ ∂x2 + a∆x2 6 −1 + „ a∆t ∆x «2 + 6 ν∆t ∆x2 ! | {z }

leading order numerical dispersion

∂3ψ ∂x3 + . . .

Discretization-based terms alter the dynamics.
Numerical diffusion could suppress physical instability.

SLIDE 21

To solve for the steady structure

adψ dx = ν d2ψ dx2 ,

Exact solution ⇒ ψ

= C1 + C2 exp ax ν

.

– Analogous to what has been done in our work

λ = [0 a/ν], ⇒ ℓfinest = 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.

SLIDE 22

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.

SLIDE 23

Conclusions

To formally resolve the one-dimensional steady reactive fl
w,

micron-level resolution is needed.

Results will likely hold for multi-dimensional unsteady fl
ws.
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