Cefrc 2012 Tools

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.

Analytic vs. Numerical Jacobianfor Sparse Chemistry

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 Reduction ofSparse Chemistry with Graph-Based Methods

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

Analytic Solution of QSSA:Sparse Coupling between Fast Species

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

Binary Integer Programming forMixture-Averaged Diffusion Reduction

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

A Suite of Algorithms forSystematic Mechanism Reduction

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



““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


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


Directed Relation Graph (DRG)

Detailed mechanismDetailed mechanism561 species, 2539 reactions561 species, 2539 reactions

DRG Aided Sensitivity Analysis


Analytic QSS solutionAnalytic QSS solution

Unimportant ReactionElimination

Isomer Lumping




““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



On-the-fly Stiffness Removal

NonNon--stiff mechanismstiff mechanism52 species, 48 global52 species, 48 global--stepssteps


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

Applications of Reduced Mechanisms in CFD Simulations

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)

Jacobian Analysis:Flame Stability, Ignition, and Extinction

“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)

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)

Applications of CEMA in CFD Simulations

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

Explosive Mode Analysis vs. Conventional Scalars

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

Chemical Structure of the Lifted H2 Jet Flame

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

Special Thanks

My students, Ruiqin Shan, Zhaoyu Luo, and Max Plomer

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

Documents

Cefrc 2012 Tools