Computational Tools for Diagnostics and Reduction of ... Lecture... · Computational Tools for...
Transcript of Computational Tools for Diagnostics and Reduction of ... Lecture... · Computational Tools for...
Computational Tools for Diagnostics and Reduction of Detailed Chemical Kinetics
Tianfeng LuUniversity of Connecticut
Prepared for
2012 Princeton-CEFRC Summer School on CombustionPrinceton University
June 25–29, 2012
Outline Background
Need of detailed chemistry in flame studies Challenges and features of detailed chemistry:
large size, nonlinearity, stiffness, and sparse couplings
Computational tools for reduction and diagnostics Analytic vs. numerical Jacobian Directed relation graph (DRG) Analytic solution of quasi steady state approximation (QSSA) Diffusion reduction An integrated reduction method and reduced mechanisms Full Jacobian analysis for flame stability, ignition, and extinction Chemical explosive mode analysis (CEMA)
Applications in CFD
Need for detailed Kinetics: Example: H2-O2 ChemistrySpecies: H2, O2, H2O, (Major species); H, O, OH, HO2, H2O2 (Radicals)
No. Reactions
Detailed chemistry is important for:Ignition, extinction, instabilities …
p, a
tm
T, K
1st limit
2nd limit
3rd limit
Extended 2nd limit;
Weak chain branching
H+O2+M → HO2+M
HO2+H2 → H2O2+H
H2O2+M → OH+OH+M
Chain “termination”
H+O2+M → HO2+M
Strong chain branching
H+O2 → O+OH
O+H2 → H+OH
OH+H2 → H2O+H
k9
k1
Crossover temperature
k2
k3
k9
k17
k15
600 800 1000 12000.001
0.01
0.1
1
10
p, a
tm
T, K
1st limit
2nd limit
3rd limit
Extended 2nd limit;
Weak chain branching
H+O2+M → HO2+M
HO2+H2 → H2O2+H
H2O2+M → OH+OH+M
Chain “termination”
H+O2+M → HO2+M
Strong chain branching
H+O2 → O+OH
O+H2 → H+OH
OH+H2 → H2O+H
k9
k1
Crossover temperature
k2
k3
k9
k17
k15
600 800 1000 12000.001
0.01
0.1
1
10
Non-explosive Explosive
Temperature, K
Pre
ssur
e, a
tm
Ex1: Negative Temperature Coefficients (NTC)
0.0001
0.001
0.01
0.1
1
10
100
0.5 1 1.5 2
with low-T chemistry
without low-T chemistry
n-Heptane-Air
p = 1 atmφ = 1
1000/T (1/K)
Igni
tion
dela
y (s
ec)
10-8 10-6 10-4 10-2 100 102500
1000
1500
2000
2500
Tem
pera
ture
, KResidence time, s
Hydrogen
Methyl Decanoate
Propane p = 30 atmT0 = 700Kφ = 1
Perfectly Stirred Reactor
Ex2: Combustion “S”-curves
Accurate chemistry is needed to comprehensively predict flame behaviors under various flow conditions
Need for detailed Kinetics: Ignition & Extinction
Mechanism Size for Practical Fuels
Detailed mechanisms are large,
Size increases with Molecular size Time
Large transportation fuels >1000 species >10000 reactions
A useful correlation I ~ 5K
Updated from [Lu & Law, PECS 2009]
101 102 103 104
102
103
104
JetSURF 2.0
Ranzi mechanismcomlete, ver 1201
methyl palmitate (CNRS)
Gasoline (Raj et al)
2-methyl alkanes (LLNL)
Biodiesel (LLNL)
before 2000 2000-2004 2005-2009 since 2010
iso-octane (LLNL)iso-octane (ENSIC-CNRS)
n-butane (LLNL)
CH4 (Konnov)
neo-pentane (LLNL)
C2H4 (San Diego)CH4 (Leeds)
MD (LLNL)C16 (LLNL)
C14 (LLNL)C12 (LLNL)
C10 (LLNL)
USC C1-C4USC C2H4
PRF (LLNL)
n-heptane (LLNL)
skeletal iso-octane (Lu & Law)skeletal n-heptane (Lu & Law)
1,3-ButadieneDME (Curran)C1-C3 (Qin et al)
GRI3.0
Num
ber o
f rea
ctio
ns, I
Number of species, K
GRI1.2
I = 5K
Features of Typical Chemical Reactions (1/2)
A reaction in general formν1’S1 + ν2’S2 + … + νK’SK = ν1”S1 + ν2”S2 + … + νK”SK
Si is a species, νi’ & νi
” are the stoichiometric coefficients
The reaction rates are strongly nonlinear
An elementary reaction only involve a few species, e.g. SA + SB = SC + SD
i.e. each reaction can only couple a few pairs of species, say α<20 Directly coupled species pairs < αI = 5αK = O(K)
Detailed chemistry is sparse
∏=
=K
iiff
icTk1
')( νω
)()(
)(TKTk
TkC
fr =
∏=
=K
iirr
icTk1
")( νω
)exp()(RTEATTk n
f −=
Forward: Reverse:
Features of Typical Chemical Reactions (2/2)
Detailed chemistry is stiffSpecies lifetime ( / ): sub-nanoseconds to seconds
Modes with different time scales Invariant modes: e.g.
elements, energy conservation Slow modes: e.g.
NOx & Soot formation; CO → CO2 (often rate limiting)
Fast modes: reactions involving highly reactive radicals, e.g. H, O, OH, …HCO → CO;CH3O → CH2O
1000 2000 300010-15
10-12
10-9
10-6
Sho
rtest
Spe
cies
Tim
e S
cale
, Sec
Temperature
Ethylene,p = 1 atmT0 = 1000K
n-Heptane, p = 50 atmT0 = 800K
Typical flow time
Implications for Solving Combustion Problems
Nonlinearity Typically needs Newton Solver → Jacobian evaluation & factorization
(time consuming, no guarantee for convergence)
Sparse Couplings A Blessing From Above Most useful in improving computational efficiency
Stiffness Needs implicit integration solver → Newton Solver Can help to simplify ODE systems: Quasi Steady State Approximation
(QSSA), Partial Equilibrium Approximation (PEA), etc.
The Jacobian Jacobian is required in Newton solver For a chemically reacting flow:
The Jacobian:
Evaluation and factorization of Jacobian can be very time consuming
)()()( YgYsYωy =+=DtD
Chemical source term
Non-chemical terms, e.g. mixing/diffusion
sωg JJYgJ +=
∂∂=
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
=
n
nnn
n
n
yg
yg
yg
yg
yg
yg
yg
yg
yg
...
...
...
...
21
2
2
2
1
2
1
2
1
1
1
gJ
Evaluation of J through Numerical Perturbations (Numerical Jacobian)
Pseudo code to compute / :----------------------------------------Given y , compute (y)For each species, i = 1, Ky y , and y y/ ( ′) ( ) /End For----------------------------------------
Cost of numerical Jacobian: exponentiations Can be time consuming even for 0-D systems,
e.g. for auto-ignition of 2-methyl alkanes (LLNL) K ~ 8000, I ~ 30,000, # of Jacobian evaluations ~ 100cost ~ O(1010) (exponentiations), or several CPU hours
Analytic Chemical Jacobian
Si: Vector of the stoichiometric coefficients for the ith reactionRi: Rate of the ith reaction
: contribution of the ith reaction to the Jacobian
Ji is highly sparse
∑ is also sparse
YSJJ
YωJ ω ∂
∂==∂∂=
=
iii
Iii
R ,,1
Example of Sparse Jacobian Black pixel (i, j): a possible nontrivial entry (Ji,j) in Jω
K: number of species I: number of reactions
I ~ 5K Black pixels in Ji: ~16 Black pixels in J:N I 5K K Total pixels in J: K2
Fraction of non-trivial entries in Jω/K2= O(K−1)
Larger mechanisms are sparser100 200 300 400 500
50
100
150
200
250
300
350
400
450
500
550
Species
Spe
cies
n-heptane (LLNL, 561 species)
Analytical vs. Numerical Jacobian Cost
Analytic Jacobian: O(K)Numerical Jacobian: O(K2)
Analytic Jacobian only needs to be generated once for each mechanism
Analytic Jacobian is more accurate
Analytic Jacobian should always be used
1 2 3 40
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
CP
U T
ime,
s
Analytical JacobianNumerical Jacobian
φ = 1p = 1 atm
φ = 0.5p = 1 atm
φ = 1.5p = 1 atm
φ = 1p = 10 atm
Auto-ignition of CH4/air, T0 = 1000K
Luo et al, 2012
Skeletal Mechanism Reduction To eliminate unimportant species and reactions from a
detailed mechanism
What are unimportant species and reactions?
Quantification of the importance of species Jacobian/Sensitivity: , /
The entries in J can be arbitrarily large: difficult to chose a threshold
Relative error induced by elimination, , E.g. in directed relation graph (DRG) [Lu & Law, 2005]
Skeletal Reduction with DRG
Starts with pair-wise reduction errors
B is important to A iff , , a user-specified thresholdIn graph notation: → iff ,
Construction of DRG Vertex: species (A, B, C, …) Edges: species dependence, rAB>ε Starting vertices: species known to be important
e.g. H, fuel, oxidizer, products, a pollutant, …
( )( )iiAi
BiiiAiABr
ωνδων
,
,
maxmax
≡
=,0,1
Biδ If reaction i involves species B
otherwise
νA,i: stoichiometric coefficient of A in the ith reactionωi: net reaction rate of the ith reaction
A
B
C D
F
EA
B
C D
F
E
DRG and Sparse Chemical Couplings An alternative graph representation: adjacency matrix matrix E, where , 1 , Possible nontrivial entries are similar to that in the chemical
Jacobian,
DRG is a sparse graph
Many algorithms in graph theory can take advantage of the sparsitye.g. depth-first search (DFS)
100 200 300 400 500
50
100
150
200
250
300
350
400
450
500
550
Species
Spec
ies
n-heptane
Depth-First Search (DFS) A most basic algorithm in graph theory Recursively find all vertices reachable from a starting vertex Pseudo code:
-----------------------------------DFS (graph G, vertex U)
Mark U as discoveredFor each undiscovered node V, where there is an edge U → V,
DFS(G, V)End For
-----------------------------------
Properties of DFS Simple to implement Linear searching time, i.e. cost ~ number of edges/vertices
Reduction Curves of DRG Detailed mechanism
(LLNL 2010): 3329 species 10,806 reactions
Skeletal Mechanism 472 species 2337 reactions
Error ε/(1+ ε): ~30% (worst case)
Parameter range: p: 1-100 atm φ: 0.5 - 2.0 Ignition & extinction T0 >1000K for ignition
Biodiesel (MD+MD9D+C7) – Air
0.0 0.2 0.4 0.6 0.8 1.00
500
1000
1500
2000
2500
3000
detailed 2084 species 1034 species 472 species
Num
ber o
f spe
cies
Error tolerance
Biodiesel surrogate - air
, ε
0 1000 2000 30000
50
100
150
200
250
Red
uctio
n Ti
me,
ms
Number of Species
methyl decanoate
iso-octanen-heptane
ethylenedi-methyl ether
More about DRG
Linear reduction time: cost ~ O(K) Error control at reduction time Fully automated
Other graph-based reduction methods DRG aided sensitivity analysis (DRGASA),
(Zheng et al, 2007; Sankaran et al 2007) DRG with error propagation (DRGEP),
(Pepiot-Desjardins & Pitsch 2008; Liang et al, 2009; Shi et al 2010) Path flux analysis (PFA): (Sun et al, 2009) DRGEP with sensitivity analysis (DRGEPSA): (Niemeyer et al 2010) Transport flux based DRG (on-the-fly reduction): (Tosatto et al, 2011) DRG with expert knowledge (DRGX): (Lu et al, 2011)
DRG with Error Propagation (DRGEP)(Pepiot & Pitsch 2005)
Assume that error decays geometrically along a graph path
or
Becomes a Shortest Path Problem that can be solved by the Dijkstra Algorithm (Niemeyer & Sung 2010)
A
B
C
rAB rBC
rAC
Example:
The Dijkstra Algorithm To solve the Shortest Path Problem (in GPS, etc)
e.g. how to find the shortest path between Friend’s center & Carnegie Lake? (slightly more complicated than DFS)
Time complexity: A sorting problem can be converted to the shortest path problem: cost ~ O log
(from http://maps.google.com)
A11
1X
Y Z
B
…
1
r1r2 … rn
Sort r1, r2, … rn → find shortest path from A to B
Example
B is a QSS species: consumption much faster than creation
QSS species stay in low concentrations
Reactions involving two QSS reactants are likely unimportant:→ :
QSSA are intrinsically a linear problem (Lu&Law, JPCA 2006)
Quasi Steady State Approximations (QSSA)
A B C1 1/ε
τcontrol ~ O(1)A B C1 1/ε
τcontrol ~ O(1)
0≈−=εBA
dtdB )(εε OAB =≈
Solving QSSA Equations
Traditional approach: algebraic iterations Slow convergence (inefficiency) Divergence (crashes, …)
Linear QSSA: analytic solution High accuracy High efficiency High robustness
0yygy
QSS == ),,;( Tpdt
dmajorQSS
QSS
Analytic Solution of LQSSA
Equation LQSSA:
0iik
kikii CxCxD +=≠
Destructionrate
Creation Rate involving
other QSS species
Creation Rate involving
major species
0 ,0 ,0 0 ≥≥> iiki CCD
Standard form: 0iij
jiji AxAx +=≠
• Gaussian elimination ~ N3
• The coefficient matrix A is sparse
0 ,0 0 ≥≥ iij AA
Decouple Species Couplings by Topological Sort
x1
x2 x3
x6
x4 x5
x1
x2 x3
x6
x4 x5
A B
C
A
B C12
3
A
B C12
3
Strongly Connected Component(SCC): coupled with cyclic path
Identification of SCC: DFS for G and GT
• Treat SCC as composite vertex• Acyclic graph obtained by
Topological Sort• Species groups can be solved
explicitly in topological order
Cyclic couplings: implicit; Acyclic couplings: explicit
QSS Graph (QSSG)
0iij
jiji AxAx +=≠
Each vertex is a QSS species xi → xj iff Aij>0
Example: ethylene
,
Solving Implicit Kernels
Paper & pencil: eliminate the most isolated variables first
Systematic: a spectral methodused by Google’s PageRank
cLc ⋅=T
Mccc ),...,,( 21=c
=
=M
kkjijij EEL
1/
c: Expansion cost vector, L: Averaging operatorE: the adjacency matrix
Measured Efficiency of the Analytic Solution
80 100 120 140 160 180 200 220 240 260 2800
50
100
150
200
Number of operations
Num
ber o
f ins
tanc
es
92
(*, /)
Ethylene/air, 9-species SCC, 10000 random sequences
The “pagerank” algorithm gives near-optimal efficiency
Diffusion Reduction Diffusion term: Time cost ~ K2, (quadratic speedup, but for
diffusion term only)
101 102 103102
103
104 PRFiso-octane
n-heptane
iso-octane, skeletal
n-heptane, skeletal1,3-Butadiene
DMEC1-C3
GRI3.0
Chemistry Diffusion
Num
ber o
f exp
func
tions
Number of species
GRI1.2
DiffusionChemistry
~K2
, K
~K
the crossing point:K~20
Mixture Average Model
Number of exp() ~ K2
Exact formulation of Di,j is complicated
Typically interpolated with polynomials inside exp()
iiii w
DtDY
+⋅−∇= )( Vρρ
i
iii X
XD
∇=V
≠
−≈ij ji
jii D
XYD
,
/)1(
,
Mixture average model:
≈ =
nN
njinji TapD )(lnexp
0,,,
Similarity in Species Diffusivities
Many species have similar diffusivities
Species with similar diffusivities can be lumped, their diffusivities evaluated as a group
10001
10
100
3000
CO2, C2H4
N2, O2
H2ODi,j/D
0
Temperature, K
H2
Lines: OSymbols: OH
300
Example: O and OH
Quantification of Similarity in Species Diffusivities
Many species have similar molecular properties Molecular Weight Cross section parameters Molecular structure
How different are species i and j to everyone else:
=
<<=
kj
ki
TTTKkji D
D
,
,
,...,1, lnmaxmaxmin
ε
Formulation of Diffusive Species Bundling Strategy: divide species to least numbers of group for given
threshold error A Binary Integer Programming problem xi = 1: representative species
0: group member
Minimize: =
K
iix
1
Subject to: 11
, ≥=
K
jjji xA , i = 1, 2,…,K
}1 ,0{∈ix , i = 1,2,…,K
<
=otherwise ,0
if ,1 ,,
εε jijiA
=
<<=
kj
ki
TTTKkji D
D
,
,
,...,1, lnmaxmaxmin
ε
User specified error tolerance
Reduction Curve
0.01 0.1 1
0
5
10
15
20
global optimal solution greedy algorithm
Num
ber o
f gro
ups
Threshold value, ε
9 groups
Ethylene16-step reduced
0.01 0.1 10
50
100
global optimal solution greedy algorithm
Num
ber o
f gro
ups
Threshold value, ε
n-heptane188-species skeletal
9-groups
19-groups
Ethylene, 20 species Heptane, 188 species
Validation - Ethylene
0.6 0.8 1.0 1.2 1.4 1.60
20
40
60
80
30
Lam
inar
flam
e sp
eed,
cm
/s
Equivalence ratio
Ethylene/air
T0 = 300K
Lines: 16-step reducedSymbols: with bundling
p =1 atm
0.02 0.04 0.06 0.08 0.1010-6
10-4
10-2
100
C2H2
CH2O CH3OH
CO2
Mol
e fra
ctio
n
x, cm
Ethylene/air
p = 1 atmφ = 1.0T0 = 300K
Lines: 16-step reducedSymbols: with bundling
C2H4
16-step: 20 speciesWith bundling: 9-groups
Redu
ction Flow
Ch
art
Directed Relation Graph (DRG)
Detailed mechanismDetailed mechanism561 species, 2539 reactions561 species, 2539 reactions
DRG Aided Sensitivity Analysis
Reduced mechanismReduced mechanism52 species, 48 global52 species, 48 global--stepssteps
Analytic QSS solutionAnalytic QSS solution
Unimportant ReactionElimination
Isomer Lumping
Skeletal mechanismSkeletal mechanism78 species, 359 reactions78 species, 359 reactions
Skeletal mechanismSkeletal mechanism188 species, 939 reactions188 species, 939 reactions
Skeletal mechanismSkeletal mechanism78 species, 317 reactions78 species, 317 reactions
““SkeletalSkeletal”” mechanismmechanism68 species, 283 reactions68 species, 283 reactions
QSS ReductionReduced mechanismReduced mechanism
52 species, 48 global52 species, 48 global--stepssteps
QSS Graph
Diffusive Species Bundling
Reduced mechanismReduced mechanism52 species, 48 global52 species, 48 global--stepssteps
Analytic QSS solutionAnalytic QSS solution14 diffusive species14 diffusive species
On-the-fly Stiffness Removal
NonNon--stiff mechanismstiff mechanism52 species, 48 global52 species, 48 global--stepssteps
Analytic QSS solutionAnalytic QSS solution14 diffusive species14 diffusive species
Directed Relation Graph (DRG)
Detailed mechanismDetailed mechanism561 species, 2539 reactions561 species, 2539 reactions
DRG Aided Sensitivity Analysis
Reduced mechanismReduced mechanism52 species, 48 global52 species, 48 global--stepssteps
Analytic QSS solutionAnalytic QSS solution
Unimportant ReactionElimination
Isomer Lumping
Skeletal mechanismSkeletal mechanism78 species, 359 reactions78 species, 359 reactions
Skeletal mechanismSkeletal mechanism188 species, 939 reactions188 species, 939 reactions
Skeletal mechanismSkeletal mechanism78 species, 317 reactions78 species, 317 reactions
““SkeletalSkeletal”” mechanismmechanism68 species, 283 reactions68 species, 283 reactions
QSS ReductionReduced mechanismReduced mechanism
52 species, 48 global52 species, 48 global--stepssteps
QSS Graph
Diffusive Species Bundling
Reduced mechanismReduced mechanism52 species, 48 global52 species, 48 global--stepssteps
Analytic QSS solutionAnalytic QSS solution14 diffusive species14 diffusive species
On-the-fly Stiffness Removal
NonNon--stiff mechanismstiff mechanism52 species, 48 global52 species, 48 global--stepssteps
Analytic QSS solutionAnalytic QSS solution14 diffusive species14 diffusive species
Accuracy of Reduced Mechanisms: n-C7H16 (1/2)
0.6 0.8 1.0 1.2 1.41200
1400
1600
1800
2000
22000.6 0.8 1.0 1.2 1.4
10-5
10-4
10-3
10-2
(b)
50
10
Ext
inct
ion
tem
pera
ture
, K
Equivalence ratio
p = 1 atm Lines: detailedSymbols: reduced
50
10
Ext
inct
ion
resi
denc
e tim
e, s
p = 1 atm
PSRn-heptane - airT0 = 300K
(a)
Perfectly Stirred Reactor
Auto-ignition
0.6 0.8 1.0 1.2 1.4 1.6 1.8
10-5
10-3
10-1
10-5
10-3
10-1
10-5
10-3
10-1
1
5
50
φ = 1.5 Detailed Reduced
1000/T, 1/K
1
5
50
φ = 1.0
Igni
tion
Del
ay, S
ec
φ = 0.5
p = 1 atm5
50
n-heptane
Detailed (LLNL): 561 species
Reduced: 58 species
Accuracy of Reduced Mechanisms: n-C7H16 (2/2)
0.00 0.05 0.10 0.1510-6
10-5
10-4
10-3
10-2
0.00 0.05 0.10 0.1510-4
10-3
10-2
10-1
100
500
1000
1500
2000
Mas
s Fr
actio
n
x, cm
h
oh
ch4ho2
c2h4ch2o
(b)
Lines: detailedSymbols: 55-species
Mas
s Fr
actio
n
φ = 1.0p = 1 atmT0 = 300K
Tem
pera
ture
, K
To2
nc7h16
h2o
co
co2
(a)
Premixed Flame StructureOther reduced mechanisms
(All suitable for DNS) CH4 (GRI3.0): 19 species
C2H4 (USC Mech II): 22 species
DME (Zhao et al): 30 species
nC7H16 (LLNL): 58 species
Biodiesel (LLNL): 73 species
…
More reduced mechanisms:http://www.engr.uconn.edu/~tlu
Sample Flame Simulations: RANS
Liu et al, AIAA/ASM 2006
3-D cavity stabilized ethylene flame at scramjet conditions C2H4, 19 species
(from Qin et al 2000, 70 species) RANS with VULCAN
Sample Flame Simulations: RANS
Biodiesel jet flame 89-species with low-T chemistry (Luo et al, from LLNL MD+MD9D+C7, 3329 species) Updated C7 subcomponent Simulated with CONVERGE
Experiment: Pickett et alSimulation: S. Som
3000 μs ASI, Ta = 1000K
Parameter Quantity
Injection System Bosch Common Rail Nozzle Description Single-hole, mini-sac
Duration of Injection [ms] 7.5 Orifice Diameter [µm] 90 Injection Pressure [Bar] 1400
Fill Gas Composition (mole-fraction) N2=0.7515, O2=0.15, CO2=0.0622, H2O=0.0363
Chamber Density [kg/m3] 22.8 Chamber Temperature [K] 900, 1000
Fuel Density @ 40˚C [kg/m3] 877 (Soy-biodiesel) Fuel Injection Temperature [K] 373
Sample Flame Simulations: LES 3-D premixed sooting flame C2H4, 19-species (from Qin et al 2000) LEM + Simplified soot model (Lindstedt 1994) C2H2 as soot precursor
El-Asrag et al, CNF 2007
φ=1
Sample Flame Simulations: DNS 2-D & 3-D non-premixed jet flame with soot C2H4, 19 species, non-stiff (from Qin et al 2000) Simplified soot model (Leung & Lindstedt 1991) C2H2 as soot precursor
Lignell et al, CNF 2007
Sample Flame Simulations: DNS 3-D premixed Bunsen flame CH4-air (lean): 13 species, (from GRI1.2) Re: 800 Grids: 50 millionTime steps: 1.3 million CPU hours: 2.5 million (50Tflops Cray)
Sankaran et al, PCI 2007
Sample Flame Simulations: DNS
Yoo et al, CNF 2011
2-D HCCI nC7H16-air, 58 species, non-stiff (from LLNL C7, 561 species) φ = 0.3, p = 40atm Tmean = 900K, T’ = 100K (RMS) u’ = 5m/s (isotropic turbulence)
“S”-Curve for Steady State Combustion
The canonical “S”-curve
J is singular at turning points. i.e. turnings are bifurcation points
Tem
pera
ture
or b
urni
ng ra
te
Residence time or Damköhler number
upper branch: strong flames
lower branch: weakly reacting
middle branch: physically unstable
ExtinctionState, E
Ignition state, I
0),( =τygGoverning equations:
Expansion at a turning point:
( ) ( )00
00
00
),(),(
τττ
ττ
ττ
−
∂∂+−
∂∂
+≈
==
gyyyg
ygyg
yy
( )( ) ∞=
∂∂−=
−−≈
=
−
0
1
00
0
τττττdτd yJyyy
An Example of Steady State Reactors:Perfectly Stirred Reactor (PSR)
Governing equations:
)()()( ysyωygy +==dtd
(from CHEMKIN manual)
ω: chemical sources: mixing term
sωg JJys
yω
ygJ +=
∂∂+
∂∂=
∂∂=
The Jacobian:
“S”-Curves for Practical Fuels in PSR
Residence time, s
Tem
pera
ture
,K
10-6 10-5 10-4 10-3 10-2 10-1 1001000
1400
1800
2200
2600
3000
Crossover point (λ1=0)
(a) CH4 - Air
CH4-airp = 1atmφ = 1.0Tin = 1200K
0.01 0.1 1600
800
1000
1200
1400
Tem
pera
ture
, K
Residence time, τ, ms
DME-airp = 30 atmφ = 5.0Tin = 700K
DME-air, with NTC(Zhao et al, 2008)
CH4-air, no NTCGRI-Mech 3.0
Fuels with NTC can feature multiple criticalities
Are the turning points physical ignition/extinction states?
Effect of Eigenvalue λ on Stability: Real λ
time
Real(λ) < 0
0
δf
Stable
δf
Real(λ) > 0
0
time
Unstable
yb δδ ⋅=fteff λδδ ⋅= 0
yJyygygyygyyy
sss δδδδδ ⋅=⋅+≈+=+=
dd
dtd
dtd )()()()(yyy δ+= s
δy is a small perturbation on the steady state solution, ys, for
where , b is a left eigenvector of J
0)( == ygydtd
Effect of Eigenvalue λ on Stability: Complex λλ : Eigenvalue of Jacobian matrix Jg, Complex numberλ : Eigenvalue of Jacobian matrix Jg, Complex number
Unstable
time
0
δf Real(λ) > 0
• Real(λ) : Stability• Imag(λ) : Oscillation frequency
time
Real(λ) < 0
0
δf
Stable
Real(λ) < 0
Ignition Points I1 & I2
Time, ms
Tem
pera
ture
,K
0 0.5 1 1.5 2 2.5 3600
800
1000
1200
1400
1600
I1
I2
DME - Airτ′ = 0.05ms
P = 30atmT0 = 700Kφ = 5.0
Residence time, ms
Tem
pera
ture
,K
10-2 10-1 100 101600
800
1000
1200
1400
1600(a)
I1
E1E1
I2
E2 E2′
′
DME - AirP = 30atmT0 = 700Kφ = 5.0
• I1: Cool flame ignition• I2: Strong burning ignition
Steady state PSR Unsteady PSR
Points P1 & P2 on Upper Branch: Re(λ1)<0, Stable
• P1 and P2 are both stable
τ = 4.9ms, λ1 = -2.1E2 s-1
Time, ms
T-T ∝
,K
0 10 20 30 40-0.1
-0.05
0
0.05
0.1
(a)Point P1
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
τ, ms
T,K
10-2 10-1 100 101
1200
1400
1600
E2
P1
P2
E2′
100.01 0.1 1
Time, ms
T-T ∝
,K
0 0.2 0.4 0.6 0.8-0.2
-0.1
0
0.1
0.2
T = 0.1KT = -0.1K
(b)
′′
Point P2
τ = 0.1ms, λ1 = -9.5E3+4.2E4i s-1
Point P3 & P4 on Upper Branch: Re(λ1)>0, Unstable
• P3 and P4 are not stable• Extinction occurs prior to the turning point
Time, ms
Tem
pera
ture
,K
0 0.5 1 1.5 2700
900
1100
1300
1500
T = 0.1KT = -0.1K
(a)′′
Point P3
Time, ms
Tem
pera
ture
,K
0 0.2 0.4 0.6 0.8700
900
1100
1300
1500
(b)
Point P4
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
τ, ms
T,K
10-2 10-1 100 101
1150
1250
1350
P4 (E2)
P3
100.01 0.1 1
τ = 0.07ms, λ1=7.8E3 + 3.5E4i s-1 τ = 0.06ms, λ1=5.0E4 s-1, λ2=0
Points P5 & P6 on Cool Flame Branch
• P5 is stable• P6 is unstable, perturbation leads to extinction
τ = 0.1ms, λ1=-8.5E3 + 3.5E4i s-1 τ = 0.04ms, λ1=1.8E3 + 3.5E4i s-1
Time, ms
T-T ∝
,K
0 0.2 0.4 0.6 0.8-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15(a)
Point P5
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
τ, ms
T,K
10-2 10-1 100800
900
1000 I2P5P6E1
10.01 0.1
E1′
Time, ms
Tem
pera
ture
,K
0 0.3 0.6 0.9700
750
800
850
900
950
T = 0.1KT = -0.1K
(b)
′′
Point P6
Point P7 on Cool Flame Branch
• Perturbation evolved to limit-cycle oscillation
τ = 0.07ms, λ1=3.0E3 + 5.6E4i s-1
Time, ms
T-T ∝
,K
0 0.5 1 1.5 2 2.5 3-150
-100
-50
0
50
100
150
T = 0.1KT = -0.1K
DME - AirPoint P7
′′
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
τ, ms
T,K
10-2 10-1 100800
900
1000
E1
I2P7E1 ′
10.01 0.1
Flame Stability for PSR: DME (1/2)
0.01 0.1 1600
800
1000
1200
1400
stable unstable
Tem
pera
ture
, K
Residence time, τ, ms
DME-airp = 30 atmφ = 5.0Tin = 700K
SE1SE2
+A
Turning points may not be physical extinction or ignition states
Full Jacobian analysis is difficult for (multi-dimensional) diffusive flames
Chemical Explosive Mode Analysis (CEMA)• Governing equations for a chemically reacting flow
• The chemical Jacobian
• Chemical explosive mode (CEM): Re(λe)>0 for Jω
• CEM indicates the propensity of a mixture to explode when isolated
• The mixing timescale competing with the CEM:• be, ae: right and left eigenvectors of the CEM, respectively
• A Damköhler number
)()( ysyωy +=DtD
yωJ
∂∂=ω y
sJs ∂∂=
ees aJb s ⋅⋅
−= 1τ
seDa τλ ⋅= )( Re
CEMA for Auto-Ignition (no mixing)
10-6
10-5
10-4
10-3
10-2
10-1
1000
1500
2000
2500
3000
Residence time, sec
Tem
pera
ture
, K
10
-610
-510
-410
-310
-210
-1
1000
1500
2000
2500
Residence time, sec
-6
-4
-2
0
2
4
6
C2H4-airφ = 1.0p = 1 atm
nC7H16-airφ = 0.5p = 10 atm
(a) (b)
CEM indicates the propensity of a mixture to explode if isolated
CEM is present in pre-ignition mixtures; absent in post-ignition mixtures
Color indicates sign(Re(λe))×log10(1+|Re(λe)|).
CEMA for 1-D Premixed Flames
0 0.05 0.1 0.15 0.2 0.250
500
1000
1500
2000
2500
x, cm
Tem
pera
ture
, K
C2H4-airp = 1 atmT0 = 300K
φ = 1.0
2.0
0.5
0 0.05 0.1 0.15 0.2 0.250
500
1000
1500
2000
2500
x, cm
-6
-4
-2
0
2
4
6
φ = 1.0
1.6
0.5
nC7H16-airp = 1 atmT0 = 300K
(a) (b)
ethylene n-heptane
CEM present for pre-ignition mixtures, absent for post-ignition mixtures (similar to auto-ignition)
CEM zero-crossing indicates locations of premixed flame fronts
cool flame
strong flame
CEMA for Ignition/Extinction in Steady State PSR
Da = 1: CEM balances mixing at ignition/extinction Competition between CEM and mixing is the key
reason for ignition/extinction in steady state flames
Color indicates sign(Re(λe)-1/τ)×log10(1+|Re(λe)-1/τ|).
10-6
10-5
10-4
10-3
10-2
800
1200
1600
2000
2400
2800
Residence time, sec
Tem
pera
ture
, K
C2H4-airφ = 1.0p = 1 atm
10-5
10-4
10-3
10-2
10-1
600
800
1000
1200
1400
1600
1800
2000
Residence time, sec
-6
-4
-2
0
2
4
6
nC7H16-airφ = 0.5p =10 atm
λ2-1/τ
E1
I2
I1
E2cool flame
strong flame
E2’
(a) (b)
ethylene n-heptane
+
+ + +
+
+
+: Da = 1
Flame Stability Based on CEMA
0 0.01 0.02 0.03 0.04 0.05700
800
900
Time, s
Tem
pera
ture
, K
0 0.005 0.01 0.015 0.02 0.025700
750
800
850
Time, sTe
mpe
ratu
re, K
(a)
(b)
T = 799.3 K, τ = 4.874e-4 s
T = 795.5K, τ = 4.138e-4 s
nC7H16-airφ = 0.5p =10 atm
10-5
10-4
10-3
10-2
10-1
600
800
1000
1200
1400
1600
1800
2000
Residence time, sec
Tem
pera
ture
, K
-6
-4
-2
0
2
4
6
nC7H16-airP=10 atmφ= 0.5
ab
n-heptane
CEMA can capture non-turning extinction states
CEMA vs. Full Jacobian Analysis
10-5
10-4
10-3
10-2
10-1
600
800
1000
1200
1400
1600
1800
2000
Residence time, sec
Tem
pera
ture
, K
-6
-4
-2
0
2
4
6
10-5
10-4
10-3
10-2
10-1
600
800
1000
1200
1400
1600
1800
2000
Residence time, s
Tem
pera
ture
, K
stableunstable
nC7H16-airp=10 atmφ= 0.5
Full Jacobian Analysis CEMA
CEMA gives mostly identical result to that of the full Jacobian analysisCEM is the major chemical process in determining ignition, extinction and flame stability
CEMA for Unsteady Flames in PSR: Extinction and Re-ignition
sign
(λe)
×log
10(1
+| λ
e|)
“+”: Da = 1 (or λe = 1/τs)
Da = 1: onsets/ends of extinction and ignition
λe = 0 near start of extinction or end of ignition
+
+
+
+
Summary of CEMA
CEM is an important chemical property
are important criteria for limit phenomena detection
CEMA can systematically detect Auto-ignition Premixed flame fronts Ignition/extinction/Re-ignition in steady and unsteady systems Diffusive flame kernels (Da << -1, fast chemistry, no CEM)
Example of CEMA for DNS:2-D n-heptane-air at HCCI Condition
2-D lifted ethylene jet flame(Yoo et al, CNF, 2011)
58-species non-stiff reduced mechanism Domain size: 3.2mm x 3.2mm Grid size: 2.5μm, uniform
Initial conditions φ = 0.3 p = 40 atm Tmean = 934K, T’ = 100K (RMS) Isotropic turbulence, u’ = 5m/s
Autoignition for n-heptane-air (constant volume)
y, m
m
sign(Re(λe)) × log10(1 + Re(λe), 1/s)
0 1 2 30
1
2
3
-6
-4
-2
0
2
4
6
y, m
m
YOH/max(YOH)
0 1 2 30
1
2
3
0
0.2
0.4
0.6
0.8
1
y, m
m
T, 1000K
0 1 2 30
1
2
3
0.811.21.41.61.822.2
y, m
m
0 1 2 30
1
2
3
-6
-4
-2
0
2
4
6
y, m
m
0 1 2 30
1
2
3
0
0.2
0.4
0.6
0.8
1
y, m
m
0 1 2 30
1
2
3
0.811.21.41.61.822.2
y, m
m
0 1 2 30
1
2
3
-6
-4
-2
0
2
4
6y,
mm
0 1 2 30
1
2
3
0
0.2
0.4
0.6
0.8
1
y, m
m
0 1 2 30
1
2
3
0.811.21.41.61.822.2
x, mm
y, m
m
0 1 2 30
1
2
3
-6
-4
-2
0
2
4
6
x, mm
y, m
m
0 1 2 30
1
2
3
0
0.2
0.4
0.6
0.8
1
x, mm
y, m
m
0 1 2 30
1
2
3
0.811.21.41.61.822.2
0 ms
0.35ms
1 ms
2 ms
Play movie here
Example of CEMA for DNS: 2-D Non-Premixed Flame with Local Extinction/Re-Ignition
2-D temporally evolving sooting jet flame
C2H4-air+PAH, 60 species, non-stiff (Luo et al, from the ABF-mech, 100 species)
Detailed soot model (Roy & Haworth)
Pyrene as soot precursor
Arias et al, Paper# C-05
Detection of Local Extinction and Re-ignition with CEMA
x, cm
y, c
m
T, 1000K
-1 -0.5 0 0.5 10
0.2
0.4
0.6
0.8
1
1.2
0.30.50.70.91.11.31.51.71.92.1
x, cm
y, c
m
sign(λexp) × log10(max(1, λexp (s-1)) )
-1.2 -0.8 -0.4 0 0.4 0.8 1.20
0.2
0.4
0.6
0.8
1
1.2
1.4
-6
-4
-2
0
2
4
6
(a)
local extinction
(b)
diffusionflame
DNS by Lecoustre et al
Example of CEMA for DNS: Lifted H2 Jet into Heated Coflowing Air A 3-D lifted hydrogen jet flame
1 billion grid points
3.5 million CPU hours
>30TB output
Systematic algorithms required to extract hidden information:Where are the flame fronts, ignition, or extinction spots?
Courtesy of Ma&Yu SciDAC UltraViz Inst.
Yoo et al, Sandia, 2008
Lifted Hydrogen Flames Visualized with Conventional Scalars
y, m
m
Temperature, 10 3 K
-5
0
5
0.5
1
1.5
2
y, m
m
Mixture Fraction
-5
0
5
0
0.5
1
x, mm
HO2 Mass Fraction, x10 4
0 5 10 15 200
1
2
OH Mass Fraction, x10 2
0
0.5
1
1.5
H Mass Fraction, x10 2
0
0.2
x, mm
y, m
m
Heat Release Rate, 1010 J/(m3.s)
0 5 10 15 20
-5
0
5
0
1
2
Flame Fronts Revealed by Explosive Mode Analysis
Lean front
Rich front
Lifting points
Sharp boundaries: premixed flame fronts
x/H
y/H
log10(Da)
0 5 10-4
-2
0
2
4
-4
-2
0
2
4
Damköhler Number Defined with Explosive Mode
1exp
−⋅= χλDa
Da >> 1, Auto-igniting
Da ~ 1, Premixed flames
Composition of a Chemical Mode:Species & Reactions
Explosion Index for Species
Participation Index for Reactions
|)|(||
expexp
expexp
baba
EIdiagsum
diag= a: the right eigenvector
The correlation of the species with the chemical explosive mode
( )( ) |)(| exp
exp
RSbRSb
PI⊗⋅
⊗⋅=
sumS: the stoichiometric coefficient matrixR: the vector of net rates for the reactions⊗: element-wise multiplication
The contribution of the reactions to the chemical explosive mode
Example of CEMA for DNS:Lifted Ethylene Jet Flame
3-D lifted ethylene jet flame(Yoo et al, POCI, 2011)
22-species reduced mechanism 1.3 billion grid points 14 million CPU hours 240 TB output
Volume rendering by H. Yu at Sandia
Fuel: 18% C2H4+82%N2, 550K, 204m/s Air: 1550K, 20m/s Re: 10000 Domain size: 30mm x 40mm x 6mm
HO2 CH3 CH2Oηlogχ
CEMA of the Lifted Ethylene Flame
y/H
x/H
sign(λexp) × log10(max(1, |λexp|), 1/s)
-4 -2 0 2 40
5
10
15
-4
-2
0
2
4(a)
Time scale
(c)
2. OH1. O
3. HO2
4. T
5. CO
6. CH3CHO
Chemical Structure
2-D center cut at z = 0 t = 5 flow-through-time |)|(
||
expexp
expexp
baba
EIdiagsum
diag=
CEMA vs. Conventional Scalars
y/H
x/H
Temperature
-4 -2 0 2 40
5
10
15
y/H
YOH
-4 -2 0 2 40
5
10
15
y/H
YHO2
-4 -2 0 2 40
5
10
15
y/H
YH2O
-4 -2 0 2 40
5
10
15
0
0.2
0.4
0.6
0.8
1
Summary Detailed chemistry is nonlinear, stiff, and sparse
Algorithms in discrete mathematics (graph theory, integer programming, trees, etc) can be very useful for combustion study
Flame responses to mixing can be complex for practical fuels
CEM is a very important chemical property
To detect limit flame phenomena in complex flow fields, try CEMA
Growth of Supercomputer Power
Jaguar XT-5 224,256 cores
(picture from NCCS)1990 2000 2010 2020
109
1012
1015
1018
FLO
PS
Year
CM-5
Num. Wind TunnelSR2201
CP-PCASASCI Red ASCI Red
ASCI WhiteEarthe Sim.
Columbia
SX-8 ASCI PurpleBlue Gene/L
MDGrape 3Roadrunner Jaguar XT-5
Tianhe-1A K Computer
Source: Top500
Partial Equilibrium Assumptions An example:
Forward and backward rates are much faster than the net rate
Reaction B↔C is in PE:
Question: How to apply PE assumptions?
A B C1 1/ε
τcontrol ~ O(1)
0≈−εεCB
CB ≈
Properties of QSS & PE
QSS Species PE involved species
Concentration ~ O(ε) O(1)
Can hide from governing equations
Has to be retained in governing equations
Can be directly applied back for rate computation
Should not be directly applied back for rate computation
Both are fast to apply
QSS and PE species need to be treated differently
Linearized QSSA (LQSSA)
QSS species are in low concentrations, say O(ε) Reactions with more than one QSS reactant are mostly
unimportant; reaction rate: O(ε2)
0 0.2 0.4 0.6 0.8 10
500
1000
1500
Maximum normalized contribution of nonlinear terms
Num
ber o
f ins
tanc
es
creation ratedestruction rate
(b)
33 species skeletal
Example: ethylene>1000 sampled instances, 12 QSS Species
Example: n-heptane
0.6 0.8 1.0 1.2 1.40
20
40
60
188-species skeletal 19-group 9-group 3-group
Lam
inar
flam
e sp
eed,
cm
/s
Equivalence ratio
n-heptane/air
p = 1 atmT0 = 300K
Skeletal: 188 speciesWith bundling: 19, 9, and 3 -groups
0.02 0.04 0.06 0.08 0.10 0.1210-4
10-3
10-2
10-1
100
188-species skeletal 19-group 9-group 3-group
H2
nC7H16
CO2
CH4
OHMol
e fra
ctio
n
x, cm
n-heptane/air
p = 1 atmφ = 1.0T0 = 300K
(Measured with S3D)
Dynamic Chemical Stiffness Removal (DCSR) (Lu et al, CNF 2009)
Mechanisms are still stiff after skeletal reduction & global QSSA
Implicit solvers needed for stiff chemistry Cost in evaluation of Jacobian ~ O(K2)
Cost in factorization of Jacobian ~ O(K3)
Idea of DCSR Chemical stiffness induced by fast reactions
Fast reactions results in either QSSA or PEA, Classified a prioriAnalytically solved on-the-fly
Explicit solver can be used after DCSR Time step limited by CFL condition Cost of DNS: ~ O(K)
1000 2000 300010-15
10-12
10-9
10-6
Sho
rtest
Spe
cies
Tim
e S
cale
, Sec
Temperature
Ethylene,p = 1 atmT0 = 1000K
n-Heptane, p = 50 atmT0 = 800K
Typical flow time
φ/(1+φ)
Tem
pera
ture
,K
Res
iden
cetim
e,s
0.1 0.3 0.5 0.7 0.91000
1200
1400
1600
1800
2000
2200
2400
10-7
10-6
10-5
10-4
10-3
10-2
10-1
DME - AirP = 30 atmTin = 700 K
(a)
φ/(1+φ)
Tem
pera
ture
,K
Res
iden
cetim
e,s
0.1 0.3 0.5 0.7 0.9750
800
850
900
950
10-5
10-4
10-3
Solid line: Re(λ1)=0Dashed line: Turning point
(b)
strong flame extinction cool flame extinction
Differences observed for extinction for Rich strong flames; Lean and rich cool flames
No difference observed for ignition
Flame Stability for PSR: DME (2/2)
Time Complexity ofTypical Combustion Simulations Time complexity of major components in CFD:
Chemistry: ~ I ~ 5K Jacobian evaluation (numerical): ~ KI ~ 5K2
Jacobian evaluation (analytic): ~ I ~ 5K Jacobian factorization: ~ O(K3) Diffusion (mixture average): ~ K2/2
Implicit solvers (Jacobian, chemistry, diffusion) Numerical J: ~ O(K3) + 10K2 + 10K + K2
Analytic J: ~ O(K3) + 20K + K2
Explicit solvers (chemistry, diffusion) ~ I + K2/2 ~ 10K + K2
Selection of solvers: Small to large mechanisms: implicit solver + large time steps Extremely large mechanisms: explicit solver + small time steps