Verified Calculation of Multiscale Combustion inGaseous Mixtures

Joseph M. PowersDEPARTMENT OF AEROSPACE & MECHANICAL ENGINEERING

DEPARTMENT OF APPLIED & COMPUTATIONAL MATHEMATICS & STATISTICS

UNIVERSITY OF NOTRE DAME

NOTRE DAME, INDIANA, USA

delivered to

Universidad Autónoma de San Luis Potosı́

26 October 2012

Acknowledgments

• Samuel Paolucci, U. Notre Dame

• Ashraf al-Khateeb, Khalifa U.

• Zach Zikoski, former Ph.D. and Post-doctoral student, U. Notre Dame

• Christopher Romick, Ph.D. student, U. Notre Dame

• Tariq Aslam, Los Alamos National Laborarory

• Stephen Voelkel, NSF-REU undergraduate student, U. Notre Dame; U. Texas

• Karel Matouš, U. Notre Dame

• NSF

• NASA

• DOE

Outline

• Part I: Preliminaries

• Part II: Fundamental linear analysis of length scales of reacting flows with de-tailed chemistry and multicomponent transport. (with al-Khateeb and Paolucci)

• Part III: Direct Numerical Simulation (DNS) of complex reacting and inert flowswith a) traditional methods, and b) a wavelet-based adaptive algorithm imple-

mented in a massively parallel computing architecture. (1D detonations with

Romick and Aslam; 2D detonations of Zikoski and Paolucci, inert implosions

with Voelkel and Romick using Zikoski’s algorithm)

Part I: Preliminaries

Some Semantics

• Verification: Solving the equa-tions right—a math exercise.

• Validation: Solving the rightequations—a physics exercise.

• DNS: a verified and validatedcomputation that resolves all

ranges of relevant continuum

physical scales present.

Direct numerical

simulation

Large eddy

simulation

Reynolds-averaged

simulation

Molecular

dynamics

Quantum

Mechanics

10-12

10-12

10-9

10-6

10-3

10-9

10-6

10-3

LENGTH SCALE (m)

TIM

E S

CA

LE

(s)

Kinetic

Monte Carlo

“Research needs for future internal

combustion engines,” Phys. Today,

2008.

Hypothesis

DNS of fundamental compressible reactive flow fields (thus, detailed kinetics,

viscous shocks, multi-component diffusion, etc. are represented, verified, and

validated) is on a trajectory toward realization via advances in

• adaptive refinement algorithms, and

• massively parallel architectures.

Corollary I

A variety of modeling compromises, e.g.

• shock-capturing (FCT, PPM, ENO, WENO, etc.),

• implicit chemistry with operator splitting,

• low Mach number approximations,

• turbulence modeling (RANS, k − ǫ, LES, etc.), or

• reduced/simplified kinetics, flamelet models,

need not be invoked when and if this difficult goal of DNS is realized; simple

low order explicit discretizations suffice if spatio-tempo ral grid resolution is

achieved.

Corollary II

Micro-device level DNS is feasible today; macro-device level DNS remains in the

distant future.

Corollary III

A variety of challenging fundamental unsteady multi-dimensional compressible

reacting flows are now becoming amenable to DNS, especially in the weakly

unstable regime; we would do well as a community to direct more of our

efforts towards unfiltered simulations so as to more starkly expose the

richness of unadulterated continuum scale physics.

[Example (only briefly shown today): ordinary WENO shock-capturing applied to

unstable detonations can dramatically corrupt the long time limit cycle behavior;

retention of physical viscosity allows relaxation to a unique dissipative structure in

the unstable regime.]

Part II: Fundamental Linear Analysis of Length Scales

Motivation

• To achieve DNS, the interplay between chemistry and transport needs to becaptured.

• The interplay between reaction and diffusion length and time scales is wellsummarized by the classical formula (see Al-Khateeb, Powers, and Paoucci,

CTM, 2012, to appear.)

ℓ ∼√D τ.

• Segregation of chemical dynamics from transport dynamics is a prevalentnotion in reduced kinetics combustion modeling.

• But, can one rigorously mathematically verify a Navier-Stokes model withoutresolving the small length scale induced by fast reaction? Answer: no.

• Do micro-scales play a role in macro-scale non-linear dynamics? Answer: insome cases, yes; see Romick, Aslam, & Powers, 2012, JFM.

Illustrative Linear Model Problem

A linear one-species, one-dimensional unsteady model for reaction, advection, and

diffusion:∂ψ

∂t+ u

∂ψ

∂x= D

∂2ψ

∂x2− aψ,

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.

Length Scale Spectrum

• The steady structure:

ψs(x) = ψu

(

exp(µ1x) − exp(µ2x)1 − µ1

µ2exp(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.

Spatio-Temporal Spectrum

ψ(x, t) = Ψ(t)eıikx ⇒ dΨdt

=

(

−a(

1 +ıiku

a+Dk2

a

)

t

)

Ψ.

Ψ(t) = C exp

(

−a(

1 +ıiku

a+Dk2

a

)

t

)

.

• For long length scales: limk→0

τ = limλ→∞

τ =1

a,

• For fine length scales: limk→∞

τ = limλ→0

τ =λ2

4π21

D,

St =(

2π

λ

√

D

a

)2

.

• Balance between reaction and diffusion at k ≡ 2πλ

=√

aD

= 1/ℓ,

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 =

√

Da

=√Dτ = 3.2 × 10−4 cm.

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

+∂jml∂x

= 0, l = 1, . . . , L − 1,

ρ∂Yi∂t

+ ρu∂Yi∂x

+∂jmi∂x

= ω̇im̄i, i = 1, . . . , N − L.

• Unsteady spatially homogeneous reactive system:dz(t)

dt= f (z(t)) , z(t) ∈ RN , f : RN → RN .

0 = (J − λI) · υ.

St =τslowestτfastest

, τi =1

|Re(λi)|, i = 1, . . . , R ≤ N − L.

• Steady spatially inhomogeneous reactive system:

B̃ (z̃(x))· dz̃(x)dx

= f̃ (z̃(x)) , z̃(x) ∈ R2N+2, f̃ : R2N+2 → R2N+2.

λ̃B̃ · υ̃ = J̃ · υ̃.

Sx =ℓcoarsestℓfinest

, ℓi =1

|Re(λ̃i)|, i = 1, . . . , 2N − L.

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 = 800K, 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.

• Unsteady spatially homogeneous reactive system:

10−20

10−15

10−10

10−5

100

10−25

Yi

HO 2

H O 2

H O2

H 2

O 2

H

OH

O

N 210−30

10−10

10−8

10−6

10−4

10−2

100

t [s]10

2

2

600

1000

1400

1800

2200

2600

10−10

10−8

10−6

10−4

10−2

100

t [s]10

2

T [K

]

10−8

10−6

10−4

10−2

100

102

104

10−10

10−8

10−6

10−4

10−2

100

t [s]10

2

slowest

fastest

= 1.8×10 s−2

= 1.0×10 s−8

τi

[s]

τ

τ

St ∼ O`

104´

.

• Steady spatially inhomogeneous reactive system:a

coarsest

finest

= 2.6×10 cm0

= 2.4×10 cm−4

10−5

10−4

10−3

10−2

10−1

10 0

10 1

10 2

10−4

10 0

10 4

10 8

i[c

m]

10−15

10−10

10−5

10 0

Yi

10−5

10−4

10−3

10−2

10−1

10 0

10 1

10 2

x [cm]

HO 2

H O 22

H O 2O 2H2

H

OH

O

N 2

10−5

10−4

10−3

10−2

10−1

100

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.

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 spatiallyinhomogeneous 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).

• Dmix = 1N2∑N

i=1

∑N

j=1 Dij ,

• ℓ1 =√Dmixτs = 1.1 × 10−1 cm,

• ℓ2 =√

Dmixτf = 8.0 × 10−4 cm ≈ ℓfinest = 2.4 × 10−4 cm.

10−6

10−4

10−2

100

102

10−14

10−12

10−10

10−8

10−6

10−4

1

2

τ(s

)

2L/π (cm)

τs = 1.8 × 10−4 s

τf = 1.0 × 10−8 s

ℓ1ℓ2

Conclusions: Part II

• 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 aredictated 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 fastreaction balances diffusion is ∼ 2 µm, the necessary scale for a DNS.

Part III: DNS of Complex Reacting and Inert Flows

Effect of Diffusion on Detonation Dynamics: 1D, 1-Step

4

6

8

10

Pm

ax

(MP

a)26 27 28 29 30 31 32

E

4

6

8

10

Pm

ax

(MP

a)

26 27 28E

inviscid viscous

• Small physical diffusion significantly delays transition to instability.

• In the unstable regime, small diffusion has a large role in determining role forthe long time dynamics.

• Romick, Aslam, Powers, JFM, 2012.

Effect of Diffusion on Detonation Dynamics: 1D, N -Step

Case Examined

• Romick, Aslam, Powers, AIAA ASM, 2012

• Overdriven detonations with ambient conditions of 0.421 atm and 293.15 K

• Initial stoichiometric mixture of 2H2 + O2 + 3.76N2

• Detailed kinetics: 9 species, 19 reversible reactions

• a) Inviscid via shock-fitting, and b) viscous (multicomponent diffusion of a viscous, heat

conducting fluids) via WAMR studied

• DCJ ∼ 1961 m/s

• Overdrive is defined as f = D2o/D2

CJ

• Overdrives of 1.025 < f < 1.150 were examined

Continuum Scales

• The mean-free path scale is the cut-off minimum length scale associated with contin-

uum theories.

• A simple estimate for this scale is given by Vincenti and Kruger (1967):

λ =M

√2πNAρd2

∼ O`

10−6 cm

´

. (1)

• The finest reaction length scale is Lr ∼ O`

10−4 cm´

.

• A simple estimate of a viscous length scale is:

Lµ =ν

c=

6 × 10−1 cm2/s

9 × 104 cm/s∼ O

`

10−5 cm

´

. (2)

• λ < Lµ < Lr

Inviscid Steady-State: Mass Fractionsf = 1.15

10-8

10-6

10-4

10-2

100

10-4 10-3 10-2 10-1 100 101

Ma

ss F

ractio

n

Distance from Front (cm)

O

H

HO2

H2O

2

OH

H2O

N2

O2

H2

Inviscid Steady-State: Pressuref = 1.15

10

11

12

13

10-4 10-310-2 10-1 100 101

P (

atm

)

Distance from Front (cm)

Inviscid Transient Behavior: Stable Detonationf = 1.15

13.2827

13.2828

13.2829

13.2830

0 10 20 30

De

ton

atio

n P

ressu

re (

atm

)

t (µs)

unsteadysteady

Near Neutral Stability

0.999

1.000

1.001

50 60 70 80 90 100

Pfr

on

t/PZ

ND

t (µs)

f=1.125f=1.130f=1.135

Inviscid Transient Behavior: Unstable Detonationf = 1.10

12.5

13.0

13.5

50 75 100 125

De

ton

atio

n P

ressu

re (

atm

)

t (µs)

frontmax

12.5

13.0

13.5

110 115 120

• Frequency of 0.97 MHz agrees well with both the frequency, 1.04 MHz, observed

by Lehr (Astro. Acta, 1972) in experiments and the frequency, 1.06 MHz, predicted

by Yungster and Radhakrishan.

• The maximum detonation front pressure predicted, 13.5 atm, is similar to the value

of 14.0 atm found by Daimon and Matsuo.

Validation: Recovering Lehr’s High Frequency Instability

Lehr, Astro. Acta, 1972

• Experiment of shock-induced combus-

tion in flow around a projectile in

an ambient stoichiometric mixture of

2H2 +O2 +3.76N2 at 0.421 atm.

• Projectile velocity yields an equivalent

overdrive of f ≈ 1.1

• The observed frequency was approxi-

mately 1.04 MHz

• Compare to 1D computation’ predic-

tion: 0.97 MHz

Unstable, Inviscid Detonation: x-t Diagramf = 1.10

106

107 Shock Front

Induction Zone

Compression Wave

Contact Discontinuity

0.00 0.01-0.01-0.02-0.03-0.04-0.05

108

105

t (µ

s)

X - DZND

t (cm)

A x-t diagram of density in a Galilean reference frame traveling at 2057 m/s.

Inviscid Transient Behavior: Various Overdrives

Pfr

on

t/PZ

ND

Pfr

on

t/PZ

ND

t (µs)

Pfr

on

t/PZ

ND

t (µs)

0.8

1.0

1.2

1.4

0 50 100 150 200

(d) f=1.025

Pfr

on

t/PZ

ND

0 50 100 150 200

0.96

1.00

1.04

1.08

1.12(b) f=1.100

t (µs)

0.96

1.00

1.04

1.08

1.12(c) f=1.050

0 50 100 150 200

(a) f=1.150

0.96

1.00

1.04

1.08

1.12

0 50 100 150 200

t (µs)

Inviscid Phase Portraits: Various Overdrives

dP

fro

nt/d

t (a

tm/µ

s)

Pfront

/PZND

dP

fro

nt/d

t (a

tm/µ

s)

dP

fro

nt/d

t (a

tm/µ

s)

Pfront

/PZND

Pfront

/PZND

dP

fro

nt/d

t (a

tm/µ

s)

-5

0

5

10

0.96 1.00 1.04 1.08 1.12

(c) f=1.050Limit Cycle

(b) f=1.100

-5

0

5

10

0.96 1.00 1.04 1.08 1.12

Limit Cycle

(a) f=1.150

-5

0

5

10

0.96 1.00 1.04 1.08 1.12

Stable Point

-150

-100

-50

0

50

100

150

0.8 1.0 1.2 1.4

(d) f=1.025

Stable, Viscous Detonation: Long Time Structuref = 1.15

0.0

0.1

0.2

2 3 4 5 6

YO

H

x (cm)

InviscidViscous

10-8

10-6

10-4

10-2

100

3.96 3.98 4.00

0

4

8

12

2 3 4 5 6

P (

atm

)

x (cm)

InviscidViscous

0

4

8

12

3.96 3.98 4.00

Stable, Viscous Detonation: Transient Behaviorf = 1.15

0.9

1.0

1.1

0 10 20 30 40

Pm

ax/P

ZN

D

t (µs)

InviscidViscous

Unstable, Viscous Detonation: Long Time Structuref = 1.10

0.0

0.1

0.2

2 3 4 5 6

YO

H

x (cm)

InviscidViscous

3.97 3.98 3.99 4.00

10-8

10-6

10-4

10-2

100

0

4

8

12

2 3 4 5 6

P (

atm

)

x (cm)

Inviscid

Viscous

0

4

8

12

3.97 3.98 3.99 4.00

Unstable, Viscous Detonation: Transient Behaviorf = 1.10

0.96

1.00

1.04

1.08

20 30 40

Pm

ax/P

ZN

D

t (µs)

InviscidViscous

The addition of viscous effects have a stabilizing effect, decreasing the amplitude of the

oscillations by ∼ 25%.

Unstable, Viscous Detonation: x-t Diagramf = 1.10

Smooth Shock Front

Induction Zone

Compression Wave

Contact Discontinuity

X - DZND

t (cm)

0.00 0.01-0.01-0.02-0.03-0.04-0.05

42

t (µ

s)

43

44

41

A x-t diagram of density in a Galilean reference frame traveling at 2057 m/s.

2D Viscous Detonation in Hydrogen-Air

• Wavelet Adaptive Multilevel Representation (WAMR, Zikoski and Paolucci),resolves multi-scale solutions in an adaptive fashion.

• User-defined error control guarantees a verified solution.

• The algorithm has been implemented with an MPI-based domain decomposi-tion in a massively parallel computational architecture with linear scaling to at

least 103 processors.

2-D Viscous Detonation

Initial Conditions:

Domain: [0, 60]× [0, 6] cmFront: x = 15.0 cmUnreacted pocket:

[1.05× 1.43] cmat x = 14.7 cmP = 4.7× 105 dyne/cm2T = 2100 K

128 cores391 hrs runtime

2H2 : O2 : 7Ar mixture9 species, 37 reactions

Wavelet parameters:� = 1× 10−3p = 6, n = 5[Nx ×Ny]j0 = [600× 60]J − j0 = 10

12

2-D Viscous Detonation (cont.)

100 µs 120 µs 140 µs 160 µs

13

2-D Viscous Detonation (cont.)

240 µs 250 µs 260 µs 270 µs

14

Inert Viscous Cylindrical Implosion

• WAMR algorithm employed

• 100 µm× 100 µm square domain,

• Pure argon,

• Initial uniform temperature, T = 300K ,

• Initial pressure ratio is 4 atm : 0.2 atm between argon on either side of anoctagonal diaphragm,

• Tmax(r = 0, t ∼ 40 ns) ∼ 2400K .

Conclusions

• Verified 1D and 2D combustion physics spanning over five orders of magnitude–from near mean-free path scales (10−4 cm) to small scale device scales

(10 cm)–can be calculated today with modern adaptive algorithms working

within a massively parallel computing architecture.

• Micro-scale viscous shock dynamics can influence oscillatory detonation dy-namics on the macro-scale.

• Some 1D detonations can be validated; others await 3D extension.

• Realization of verified and validated DNS would remove the need for common,but problematic, modeling assumptions (shock-capturing, turbulence model-

ing, implicit chemistry with operator splitting, reduced kinetics/flamelets).

• Such 3D V&V could be viable in an exascale environment; however, routinedesktop DNS calculations remain difficult to envision at macro-device scales.