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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.