dynamical system analysis of ignition in reactive … seminars/mauro...kaust seminar, september 2012...

KAUST Seminar, September 2012

KAUST - September 2012

Dynamical System Analysis of Ignition in Reactive Systems

Mauro ValoraniDepartment of Mechanical and Aerospace Engineering


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Outline of the Talk

• CSP Basic Concepts• Applications of CSP diagnostics• A Toy Model for Ignition • Illustration of the new Stretching Rate diagnostics • Applications of Stretching Rate diagnostics in Ignition

T

t


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Auto-ignition of Hydrocarbon Mixtures

Time Histories Ignition Delay Time Sensitivity to Pressure, Equivalence Ratio

Propane/Air --- Petersen, et al. (118 species, 665 reactions)

N-Heptane/Air Auto-Ignition - Curran (561 specs/2600 reacs)

A detailed mathematical model of a batch reactor, can produce a number of “natural” outputs and observables such as:

Question: how can we “extract” more knowledge from the computed datasets ?

Answer: by approaching the evolution of the kinetics using concepts and tool developed for the analysis of nonlinear dynamical systems

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CSP Basic Concepts (D.A.Goussis, S.H.Lam)


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Computational Singular Perturbation (CSP) Method

CSP Projector Matrix P ≡ I-arbr = asb

s

dxdt

≈ as fs = as b

s ig(x)( ) = as bs ig(x)( ) = I - arbr( )g(x) = P g(x)

ar fr = ar b

r ig(x)( ) = I - P( )g(x) ≈ 0

dxdt

= I g(x) = arbr + asb

s( )g(x) = ar fr + as f

s

with mode amplitude defined as f j := b jg(x)

dxdt

= g(x)

Slow Dynamics (ODE): dxdt

≈ P g(x)

Equation of state (AE): I - P( )g(x) ≈ 0

Change of Frame of Reference (Ortho-normal Basis)I = aib

i; bi • a j = δ ji i, j = 1,N

f r e−λr t → t O(τ r = λr−1) → f r ≈ 0

dxdt

= ar fr + as f

s ≈ as fs

f r (x) ≡ br ig(x) ≈ 0

⎧⎨⎪

⎩⎪Exampleai right eigenvector of Jacobian Matrix J of g(x)bi left eigenvector

Fast/Slow DecompositionI = arb

r + asbs r = 1,M s = M +1,N

M (spectral gap) is the dimension of the fast subspaceIn general, it is a function of space and time

τM +1 ari

r=1,M∑ f r < ε i

Existence of a Spectral Gap between mode M and M+1


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CSP Slow/Fast Importance Indicesof the action of a Reaction to the evolution of a Species


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CSP Indices: Participation Indexof the action of a Reaction to the evolution of a Mode


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CSP Indices: Time Scale Participation Indexof the action of a Reaction to the development of a time scale


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convection

hi = ρ bni Lc

n

n=1,N∑ + bn

i Ldn

n=1,N∑ + b i •WSk

Rk

ρk=1,K∑⎡

⎣⎢

⎤

⎦⎥ i=1,N

Ac, ni = ρbn

i Lcn

Ad,ni = ρbn

i Ldn

As, ki = bi

•WSkRk

diffusion

chemistry

Ai = |ρbni Lc

n |n=1,N∑ + |ρbn

i Ldn |

n=1,N∑ + | bi

•WSkRk |k=1,K∑

Atrani Achem

i

How to extend the CSP Indices to PDEs

Ici = Ac,n

i / Ai

Idi = Ad ,n

i / Ai

Is,ki = As,k

i / Ai

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Applications of CSP Diagnostics


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)• Steady Detonations [Valorani, Goussis, ‘96]

• Transient Laminar Flames [Valorani, Goussis, Najm, ‘03]

• Premixed Steady Flames [Goussis, Skevis, ‘04], [Creta, ‘04]

• Ignitions [Lee, 2004], [Creta, ‘04] , [Donato, ‘06]

• Biological Systems [Goussis, ‘06]

• Edge Flames in Methane & n-Heptane [Najm, Valorani, ’07,’11]

• Auto-ignition in HCCI Systems [Gupta, Im, Valorani, ‘12]

• ....

Applications of CSP Diagnostics


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CH4/Air Stationary Planar Premixed Laminar

An automatic procedure for the simplification of chemical kinetic mechanisms based on CSPM Valorani, F Creta, DA Goussis, JC Lee, HN NajmCombustion and flame 146 (1), 29-51

http://scholar.google.it/citations?view_op=view_citation&hl=it&user=rW-uukoAAAAJ&pagesize=100&sortby=pubdate&citation_for_view=rW-uukoAAAAJ:u5HHmVD_uO8C

http://scholar.google.it/citations?view_op=view_citation&hl=it&user=rW-uukoAAAAJ&pagesize=100&sortby=pubdate&citation_for_view=rW-uukoAAAAJ:u5HHmVD_uO8C


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CH4/Air Edge Flame: CSP Time Scales and SIM Dimension

Analysis of methane–air edge flame structureHN Najm, M Valorani, DA Goussis, J PragerCombustion Theory and Modelling 14 (2), 257-294

http://scholar.google.it/citations?view_op=view_citation&hl=it&user=rW-uukoAAAAJ&pagesize=100&sortby=pubdate&citation_for_view=rW-uukoAAAAJ:hqOjcs7Dif8C

http://scholar.google.it/citations?view_op=view_citation&hl=it&user=rW-uukoAAAAJ&pagesize=100&sortby=pubdate&citation_for_view=rW-uukoAAAAJ:hqOjcs7Dif8C


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WSS/CI Paper, #07F-18, 2007

CH4/Air Edge Flame: HO2 Importance Indices along ξ=0.075

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Auto-Ignition and Time Scale Analysis

The role of eigenvalues with positive real part


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Positive Eigenvalues are characteristic of explosive systems

kstep

Temp

100 150 200 250 300

100

101

102

103

104

105

106

107

108

1000

1500

2000

2500

PosEvalR

PosEvalR_H

PosEvalR_T

NegEvalR_H

NegEvalR_T

Temp

NegEvalR

step

Temp

100 150 200 250 300 35010-1

101

103

105

107

109

1011

1000

1500

2000

2500

Ignition in Constant Volume Batch ReactorT0=750K p0=1 atm, stoichiometric, non-diluted air

Propane/Air Petersen et al. (118 spcs; 665 rcns) n-Heptane/Air Curran et al. (560 spcs; 2538 rcns)Methane/Air GRI 3.0(53 spcs; 325 rcns)

Heptane/Air Edge Flame in Partially-premixed Mixing Layer

00.1

0.20.3

0.4 0

0.1

0.2

0.3

0.40

2

4

6

y [cm]

x [cm]

log 10

(Rea

l par

t eig

enva

lue)

[1/s

]


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Three Goals

• We look for a simple model of ignition to extract the essence from the complexity of real fuel kinetics

• We want to apply CSP ideas to analyze both the simple and the real fuel kinetics

• We want to study ignition by blending CSP ideas with concepts developed in the geometrical analysis of nonlinear dynamical systems

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A Toy Model for Ignition

Initiation Reaction: R! C kiPropagation Reaction: R"C!# C"P kpTermination Reaction: C!P kt

R: reactantC: intermediate species !radical species"P: product#: branching factor# $ 1 , linear propagation#%1, branching propagation

Auxiliary parameters

&$ki

kp r0nondimensional rate constant of initiation reaction

'$kt

kp r0nondimensional rate constant of termination reaction

x1!!t"! "x1!t" " x1!t"x2!t"x2!!t"! 1

#!x1!t"$!%"1"x1!t"x2!t""& x2!t""


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A Toy Model of Isothermal Chain-Branching Ignition (Williams, 1985)

Non Dimensional Form (Creta, et al., 2007)

x1(0) = x1,0x2 (0) = 0

x1 !u1

1 " u12 " u22x2 !

u1

1 " u12 " u22u1 !

x1

1 " x12 " x22u2 !

x2

1 " x12 " x22


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Two-dimensional Poincarè Projection

Portraits of Toy Model TrajectoriesCartesian and Poincarè Projection Phase Spaces

Mapping from

0.0 0.5 1.0 1.5 2.0

0

2

4

6

8

(x1,x2)-plane =>

-1.0 -0.5 0.0 0.5 1.0-1.0

-0.5

0.0

0.5

1.0

Red Symbols are Stationary Points at Infinity

(u1,u2)-plane


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Poincarè Projected System at Infinity

α = 1± 2 ×10−3 ε = 0.01

Stationary Points at Infinity

Bifurcation of Stationary Points at Infinity

Stationary Condition at Infinity-1 1 2 3 4

-0.004

-0.002

0.002

0.004

-1.0 -0.5 0.5 1.0

-1.0

-0.5

0.5

1.0

0 0.000010.000020.000030.000040.00005-3

-2

-1

0

1

2

3

0 0.000010.000020.000030.000040.00005-3

-2

-1

0

1

2

3

ϑSS = 0,π2,π , 3π

2, 2α −1

ArcTan ε − 1− 2α +α 2 + ε 2( ), 2α −1

ArcTan ε + 1− 2α +α 2 + ε 2( )⎧⎨⎩

⎫⎬⎭

Cos(ϑ )Sin(ϑ ) α −1( )Cos(ϑ ) + εSin(ϑ )( ) = 0

α < 1P(ϑ = 0) is stable

P(ϑ =32π ) is unstable

α > 1P(ϑ = 0) is unstable

P(ϑ =32π ) is stable


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Sub and Super Critical Ignition Regimes

x2 äx1

x1 - x1 a + g

x2 = SIM(x1)

SIM is an heteroclinic orbit connecting the stable fixed point and one of the unstable stationary point at infinity (theta=Pi/2)

0 < x1 < x1* = γ

α−1

x1* = γ

α−1 > x1 (0) sub-critical ignition

x1* = γ

α−1 < x1 (0) super-critical ignition

Ignition Regimes

x2¢@tD äx1@tD - g x2@tD + H-1 + aL x1@tD x2@tD

eä 0

x2 is QSSA

x1!!t"! "x1!t" " x1!t"x2!t"x2!!t"! 1

#!x1!t"$!%"1"x1!t"x2!t""& x2!t""

-1 - x2 -x11+x2 H-1+aL

e- x1-x1 a+g

e

- x1-x1 a+g+e+x2 e+ -4 H2 x1-x1 a+g+x2 gL e+Hx1-x1 a+g+e+x2 eL2

2 e

- x1-x1 a+g+e+x2 e- -4 H2 x1-x1 a+g+x2 gL e+Hx1-x1 a+g+e+x2 eL2

2 e

2 x1 - x1 a + g + x2 g

e

-x1 - x1 a + g + e + x2 e

e


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Time Scale Analysis of the Toy Model

Determinant of Jac

Jacobian

Eigenvalues of Jac

Trace of Jac


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Significant Loci of Ignition in Phase Space

0.0 0.5 1.0 1.5 2.00

10

20

30

40

50

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Locus of Zero Trace of Jac (Green Solid Line) SIM Approx by QSSA on x2 (Red Solid Line)

Loci of zero imaginary part of eigenvalues (Black Dashed Lines)

x1* = γ

α−1 > x1 (0) sub-critical ignition

x1* = γ

α−1 < x1 (0) super-critical ignition

Ignition Regimes


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α = 2 ε = 0.05 γ = 2 γα−1 α = 2 ε = 0.05 γ = 0.5 γ

α−1 α = 2 ε = 0.05 γ = γα−1

-1.0 -0.5 0.0 0.5 1.0-1.0

-0.5

0.0

0.5

1.0

-1.0 -0.5 0.0 0.5 1.0-1.0

-0.5

0.0

0.5

1.0

-1.0 -0.5 0.0 0.5 1.0-1.0

-0.5

0.0

0.5

1.0

0.0 0.5 1.0 1.5 2.0

0

2

4

6

8

0.0 0.5 1.0 1.5 2.0

0

2

4

6

8

0.0 0.5 1.0 1.5 2.0

0

2

4

6

8

Sub, Critical, and Super Critical Ignition Regimesx1

* = γα−1 > x1 (0) =1 sub-critical ignition regime

x1* = γ

α−1 < x1 (0)=1 super-critical ignition regime

Subcritical Critical Supercritical


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Eigenvalues Dynamics

0.00 0.05 0.10 0.15 0.20 0.25 0.30

-120

-100

-80

-60

-40

-20

0

20

Red lines: supercriticalBlue lines: subcritical

Toy model and complex kinetics feature similar

dynamics in term of eigvals !!!

kstep

Temp

100 150 200 250 300

100

101

102

103

104

105

106

107

108

1000

1500

2000

2500

PosEvalR

PosEvalR_H

PosEvalR_T

NegEvalR_H

NegEvalR_T

Temp

NegEvalR

Super-critical ignition regime -> crossing the complex eigvals region (from pos to cmplx to neg eigvals)Sub-critical ignition regime -> no crossing the complex eigvals region (no pos eigvals)

Equilibrium point always attained following the SIM (neg eigvals)

0.0 0.5 1.0 1.5 2.0

0

2

4

6

8

0.0 0.5 1.0 1.5 2.0

0

2

4

6

8

0.0 0.5 1.0 1.5 2.0

0

2

4

6

8Subcritical Critical Supercritical

PosNeg

Cmplx CmplxCmplx

NegPos

RealsReals

Reals

RealsReals


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0.0 0.2 0.4 0.6 0.8 1.0

0

10

20

30

40

0.0 0.1 0.2 0.3 0.4

!40

!20

0

20

40

0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.000

1

2

3

4

5

6

0.0 0.1 0.2 0.3 0.40

2

4

6

8

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

Eigenvector DynamicsSupercritical Ignition Regime

Real eigenvectors become co-linear with vector field while entering and leaving the region of complex eigenvalues, where no timescales gap exists

(blue: slow eigenvector; red: fast eigenvector)

blue: slow eigenvalue; red: fast eigenvalue blue: slow eigenvector; red: fast eigenvector

8<

:

d(�x)d⌧

= g(x̄) + J(x̄)�x

�x(⌧ = 0) = 0

x(t̄ + ⌧) = x̄(t̄) + �x(⌧)


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α = 2 ε = 0.01 γ = 0.5γ cr

0.0 0.2 0.4 0.6 0.8 1.0

0

10

20

30

40

!3000 !2000 !1000 0

0

20000

40000

60000

80000

Linearized Dynamics from within cmplx region

Linearized dynamics is a spiraling motion aiming at infinity/fixed point

Pos/neg real part of complex eigval controls the growth rate of vector field module

Imaginary part control the rotation rate of the vector field direction

0.0 0.2 0.4 0.6 0.8 1.0

0

10

20

30

40

!6 !4 !2 0!1400

!1200

!1000

!800

!600

!400

!200

0


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CSP AnalysisCharacteristic time scale

0.0 0.1 0.2 0.3 0.4

1

1000

Log Re & Im Eigvals (Trace is cyan) Log Mode Amplitudes

The real part of the cmplx eigenvalue crossing the zero cannot be a proper measure of the local time scale

In the cmplx region, only one time scale is active and might be identified by taking the inverse of the MODULUS [!!] of the cmplx eigenvalue

The determinant is never zero ( Jacobian never singular ), and its max corresponds to the max of the cmplx eigenvalue modulus

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

0.0 0.1 0.2 0.3 0.4

-40

-20

0

20

40

0.0 0.1 0.2 0.3 0.4

0.1

0.51.0

5.010.0

50.0100.0

500.0

Log Timescale

The trajectory stays nearly tangent to the fast eigenvector until t=0.345, when the SIM is reached

There the trajectory becomes tangent to the slow eigenvector (blue)

CSP defines the local characteristic scale on the basis of the heuristic criterion

δyerr τM +1 ar f

r

r=1,M∑

At t = 0.32, the local characteristic scale abruptly switches from the fast to the slow scale, when the CSP fast mode amplitude (red) equals the slow mode amplitude (blue)


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A Few Intermediate Conclusions

1. Toy model and complex kinetics feature similar dynamics in term of eigvals !!!2. Super-critical ignition regime -> crossing the complex eigvals region (from pos to cmplx to neg eigvals); 3. Sub-critical ignition regime -> no crossing the complex eigvals region (no pos eigvals)4. Real eigenvectors become co-linear with vector field while entering and leaving the region of complex

eigenvalues, where no timescales gap exists5. Linearized dynamics starting from cmplx region is a spiraling motion aiming at infinity/fixed point6. SIM Approx by QSSA on x2 is an heteroclinic orbit connecting the stable fixed point and the unstable

stationary point at infinity (theta=Pi/2)7. In the complex region, only one time scale is active and might be identified by taking the inverse of the

MODULUS [!!] of the cmplx eigenvalue8. In the complex region, the max value of the cmplx eigenvalue modulus corresponds to the max value of

the determinant9. The determinant is never zero and thus the Jacobian is never singular, even when the real part of the

eigvals crosses the zero10.The CSP local characteristic time scale abruptly switches from the fast to the slow scale when the CSP

fast mode amplitude (red) equals the slow mode amplitude (blue)

Let us now take a different perspective to analyze the same dynamics,on the basis of the Stretching Rate Analysis

[Creta, et al. (2006) J. Phys. Chem. A, 110, 13447-13462]

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Stretching Rate Diagnostics

(M.Valorani and S.Paolucci)


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Stretching Rate AnalysisCreta, et al. (2006) J. Phys. Chem. A, 110, 13447-13462

dzdt

= g(z) z(0) = z0

State vector z: species concentration vectorVector field: g (z ) = S r (z ) with the species reaction rate vector S : stoichiometric coefficients matrix r : net reaction rates vectorz0 : initial concentrations vector

This stretching rate is NOT related to the flame stretch concept or any stretch rateassociated to the kinematic field

Consider a nearby trajectory: z1 = z0 + ε

Define a scaled vector distance between the two as: v := limε→0

z1 − z0

ε⎛⎝⎜

⎞⎠⎟

Vector Dynamics dvdt

= Jacg (z)v(t) v(0) =

1 Jacg := ∂g(z)

∂z

Vector Norm Dynamics dvdt

=vTJacg

vv 2

v v (0) = 1

Stretching Rate along any u ω u := uTJacgu u:=

v(t)v


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Stretching Rate Analysis combined with CSP decompositionValorani and Paolucci(2012), to appear

The stretching rate along the direction aj coincides with the eigenvalue λj corresponding to the eigen-direction aj, e.g., a unit vector along aj is modified by a factor λj by the action of the linearized dynamics represented by J.

!⌧ := ⌧̃T J ⌧̃ where ⌧̃ :=g

|g|

Definition of Tangential Stretching Rate along g

!aj := aTj Jaj

Stretching Rate along (unit norm) eigen-directions

!aj = aTj Jaj = aT

j A⇤Baj = aTj

NX

i=1

ai�i

�bi.aj

�= aT

j

NX

i=1

ai�i�ij

!=�aT

j .aj

��j = �j |aj | 2 = �j

Easily proved: after CSP expansion of J = A Λ B, we obtain:

!aj = aTj Jaj = aT

j A⇤Baj = aTj

NX

i=1

ai�i

�bi.aj

�= aT

j

NX

i=1

ai�i�ij

!=�aT

j .aj

��j = �j |aj | 2 = �j

!⌧ :=⌧̃T J ⌧̃ where ⌧̃ =g

|g| =1

|g|

NX

i=1

aifi, with f i:=bi.g and g =

NX

i=1

aifi

!⌧ = ⌧̃T J ⌧̃ =1

|g|2�gT A⇤B g

�=

gT

|g|2NX

i=1

ai�i

�bi.g

�=

gT

|g|2NX

i=1

ai�i f i =1

|g|2NX

i=1

�gT · ai

��if

i

gT · ai =

NX

k=1

akfk

!T

· ai =NX

k=1

fk�aT

k · ai

�

where aTk · ai is the direction cosine between ai and ak (with |aT

k · ai| 1)


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of R

ome

- Mau

ro V

alor

ani (

2012

)

Stretching Rate and CSP decomposition

TSR can be recast after CSP expansion of J = A Λ B and g = Σi ai fi


Sap

ienz

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nive

rsity

of R

ome

- Mau

ro V

alor

ani (

2012

)

TSR is a weighted sum of the eigenvalues

!⌧ =1

|g|2NX

i=1

�gT · ai

��if

i =NX

i=1

1

|g|2NX

k=1

fk�aT

k · ai

�!

�i f i =NX

i=1

f i

|g|2NX

k=1

fk�aT

k · ai

�!

�i =NX

i=1

Wi �i

TSR can be expressed as a weighted sum of the local eigenvalues

Wi:=f i

|g|gT · ai

|g| =f i

|g|

NX

k=1

fk

|g|�aT

k · ai

�.

Weights depend on:

• mode amplitudes fi

• relative orientation of vector field g wrt eigen-directions ai

!⌧ =1

|g|2NX

i=1

�gT · ai

��if

i =NX

i=1

1

|g|2NX

k=1

fk�aT

k · ai

�!

�i f i =NX

i=1

f i

|g|2NX

k=1

fk�aT

k · ai

�!

�i =NX

i=1

Wi �i

where aTk · ai is the direction cosine between ai and ak (with |aT

k · ai| 1)

!⌧ =1

|g|2NX

i=1

�gT · ai

��if

i =NX

i=1

1

|g|2NX

k=1

fk�aT

k · ai

�!

�i f i =NX

i=1

f i

|g|2NX

k=1

fk�aT

k · ai

�!

�i =NX

i=1

Wi �i

Wi:=f i

|g|gT · ai

|g| =f i

|g|

NX

k=1

fk

|g|�aT

k · ai

�.

!⌧ can, at most, takes the value corresponding at the circumstance (never ver-

ified) of all ai being co-linear with g, that is:

gT · ai

|g| =

NX

k=1

fk

|g|�aT

k · ai

� f i

|g|

then

!⌧ NX

i=1

✓f i

|g|

◆2

�i

Note that because of the quadratic term, the sign of !⌧ relies on that of the

prevailing eigenvalues.


Sap

ienz

a U

nive

rsity

of R

ome

- Mau

ro V

alor

ani (

2012

)

TSR combines timescales with mode amplitudes !!

Let us consider three typical situations:

• let us consider the case of {fr ⇡ 0}r=1,M , that is, the state point lies on a (N-M)-dimensional SIM;

after ordering the terms in the summation by the descending value of the module of the eigenvalue,

we find

!⌧ =

NX

s=M+1

✓fs

|g|

◆2

�s +

MX

r=1

✓fr

|g|

◆2

�r ⇡NX

s=M+1

✓fs

|g|

◆2

�s

In this case, the tangential stretching rate is contributed only by slow scales.

• let us consider the case of {fs ⇡ 0}s=K,N , that is, the state point is along a fast fiber (away from

a SIM), for which we obtain

!⌧ =

NX

s=K

✓fs

|g|

◆2

�s +

KX

r=1

✓fr

|g|

◆2

�r ⇡KX

r=1

✓fr

|g|

◆2

�r

In this case, the tangential stretching rate is contributed only by fast scales.

• let us consider the case of {fr ⇡ 0}r=1,M and {fs ⇡ 0}s=K,N , with M < K.

!⌧ =

NX

s=K

✓fs

|g|

◆2

�s +

K�1X

a=M+1

✓fa

|g|

◆2

�a +

MX

r=1

✓fr

|g|

◆2

�r ⇡K�1X

a=M+1

✓fa

|g|

◆2

�a

In this case, the tangential stretching rate is contributed only by the active scales, e.g., if some �a+

is positive, it is likely that the corresponding fa+will be the largest of all active mode amplitudes.

In this case !⌧ will be mostly a↵ected by �a+.


Sap

ienz

a U

nive

rsity

of R

ome

- Mau

ro V

alor

ani (

2012

)

TSR Decomposition in N-dim ProblemsThree Subspaces




we find

!⌧ =

NX

s=M+1

✓fs

|g|

◆2

�s +

MX

r=1

✓fr

|g|

◆2

�r ⇡NX

s=M+1

✓fs

|g|

◆2

�s




!⌧ =

NX

s=K

✓fs

|g|

◆2

�s +

KX

r=1

✓fr

|g|

◆2

�r ⇡KX

r=1

✓fr

|g|

◆2

�r



!⌧ =

NX

s=K

✓fs

|g|

◆2

�s +

K�1X

a=M+1

✓fa

|g|

◆2

�a +

MX

r=1

✓fr

|g|

◆2

�r ⇡K�1X

a=M+1

✓fa

|g|

◆2

�a







we find

!⌧ =

NX

s=M+1

✓fs

|g|

◆2

�s +

MX

r=1

✓fr

|g|

◆2

�r ⇡NX

s=M+1

✓fs

|g|

◆2

�s




!⌧ =

NX

s=K

✓fs

|g|

◆2

�s +

KX

r=1

✓fr

|g|

◆2

�r ⇡KX

r=1

✓fr

|g|

◆2

�r



!⌧ =

NX

s=K

✓fs

|g|

◆2

�s +

K�1X

a=M+1

✓fa

|g|

◆2

�a +

MX

r=1

✓fr

|g|

◆2

�r ⇡K�1X

a=M+1

✓fa

|g|

◆2

�a




Let us consider the case of {fr ⇡ 0}r=1,M and {fs ⇡ 0}s=K,N , with M < K.

!⌧ =

NX

s=K

✓fs

|g|

◆2

�s +

K�1X

a=M+1

✓fa

|g|

◆2

�a +

MX

r=1

✓fr

|g|

◆2

�r ⇡K�1X

a=M+1

✓fa

|g|

◆2

�a


Sap

ienz

a U

nive

rsity

of R

ome

- Mau

ro V

alor

ani (

2012

)

TSR DecompositionThree scales

ωai≡ λi

ωτ = Wiλi i=1,N∑ Wi := f i

ggT ⋅aig O f i

g⎛⎝⎜

⎞⎠⎟

2

f i := bi ⋅g

Three scales:1) ωτ most energy containing scale2) λK −1 slowest scale contributing to the making of ωτ

3) λM +1 fastest scale contributing to the making of ωτ

Scales contributing to the making of ωτ are in the range λK −1,λM +1{ }

The tangential stretching rate computed on the basis of the original definition !⌧ := ⌧̃T J ⌧̃ will be largely

inaccurate when complex conjugate eigenvalues occur.

It is expected that the driving scale will be a weighted average of the module of the N (possibly complex)

eigenvalues. This implies scaling the weights so to have unit norm.

The suggested formula for !⌧ at any state value x reads:

!⌧ [x] :=

NX

i=1

˜

Wi[x] Abs[�i[x]]

with

Wi:=f

i

|g|g

T · ai

|g|

and

˜

Wi:=WiP

j=1,Ns|Wj |

where it is intended that all complex conjugate eigenvectors pairs have been converted into real numbers

with the usual transformation.

Remark#1: ⌧chem := |1/!⌧ |, can identify the proper characteristic chemical time scale in the estimation

of the Damkoeler number for problems involving multi-steps chemical kinetic mechanisms.

Remark#2: The time scale associated with the fastest of the energy energy containing scales ⌧M+1 :=

|1/�M+1|, can be rightly adopted to define the proper integration time step during the numerical in-

tegration of the sti↵ problem. It does not seem the right choice in the estimation of the Damkoeler

number.


Sap

ienz

a U

nive

rsity

of R

ome

- Mau

ro V

alor

ani (

2012

)

A modified definition of the tangential stretching rate


Sap

ienz

a U

nive

rsity

of R

ome

- Mau

ro V

alor

ani (

2012

)

TSR vs CEM

Lu, et al., introduced a chemical explosive mode (CEM) scale defined as

where λa+ is the fastest eigenvalue with positive real part in the system.

Our analysis indicates that the proper characteristic chemical time scale is

This scale (i) is the most relevant one during both the explosive and relaxation regimes, (ii) is always automatically identified without the need of any ad-hoc assumption(iii) recovers the CEM scale when:

τ chem =1λa+

τ chem =1ωτ

τ chem =1ωτ

=1Wiλ

i

i∑

=1λa+

if Wa+ = 1 and Wi≠a+ = 0

when all modes but i = a+ have zero amplitude and aa+ is co-linear with the vector field.


Sap

ienz

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nive

rsity

of R

ome

- Mau

ro V

alor

ani (

2012

)

Comparing CSP vs TSR vs CVODEAuto-ignition (GRI 3.0)

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úúúúú

0 20 40 60 80 100

0.001

10

105

CSP fast/slow decomposition (orange/green/black)Pos eigvals (red), neg evals (light gray)

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ÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛ

ı

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ı

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ı

0 20 40 60 80 100

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100

104

106

108

TSR (black), Active subspace (upper/lower triangles)

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0 20 40 60 80 100

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CSP fast/slow decomposition (orange/green/black)TSR (black), Active subspace (upper/lower triangles)

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0 20 40 60 80 100

10-710-610-510-4

0.001

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CVODE integration time step (blue)TSR (black), Active subspace (upper/lower triangles)

1.The fastest of the modes contributing to TSR coincides with the fastest of the slow CSP modes2.TSR coincides first with the fastest of the pos eigenvalues (the most energy containing mode) and

later with the fastest of the slow CSP modes 3.CVODE’s integration time step envelopes the fastest of the modes contributing to TSR


Sap

ienz

a U

nive

rsity

of R

ome

- Mau

ro V

alor

ani (

2012

)

Participation indices related to tangential stretching rate

Modes with a large Piωτ are the most contributing to the ωτ scale (energy containing)

Reactions with a large Pki are the most contributing to the i-th mode

Reactions with a large Pkωτ = Pi

ωτ Pki are the most contributing to the ωτ scale

f i = bi ⋅g = (bi ⋅Sk )rkk=1,Nr∑

Pki =

(bi ⋅Sk )rk

(bi ⋅Sk ' )rk '

k '=1,Nr∑

CSP PI index between reaction & mode PI index between mode & TSR

ωτ = Wi λii=1,N∑

Piωτ =

Wi λi iWj λ j

j=1,N∑

Pkωτ = Pi

ωτ Pki

PI index between reaction & TSR


Sap

ienz

a U

nive

rsity

of R

ome

- Mau

ro V

alor

ani (

2012

)

TSR Break-downParticipation Index

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npick = 54;

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mode amplitude direction cosine weight

log(eigval)

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Sap

ienz

a U

nive

rsity

of R

ome

- Mau

ro V

alor

ani (

2012

)

TSR analysis of Gri-1.2, Propane, and n-Heptane AutoignitionAutoignition iso-choric batch reactor:T0=1000K p0=1 atm, stoichiometric, non-diluted air.

n-Heptane

gray: neg eigvalsblack: pos eigvals blue: first non-exhausted modered: TSR green: active subspace

Gri-1.2.

gray: neg eigvalsblack: pos eigvals blue: first non-exhausted mode red: TSRgreen: active subspace

Propane

gray: neg eigvalsblack: pos eigvalsFirst non-exhausted mode: purple, blue, orange Tol = 10^(-4), 10^(-3), 10^(-2) red: TSRgreen: active subspace

Following Bisetti’s conjecture, we observe that the most energetic scale in 3 fuels takes a max value of about the same magnitude [tau ~1/(2 x 105) s ~ 50 x 10-6 s = 50 microsec]

iter

|λi|

50 100 150 200 250100

102

104

106

108

1010

1012

1014

1016

Max |λ|, methaneλ+, methaneMax |λ|, propaneλ+, propaneMax |λ|, n-heptaneλ+, n-heptane


Sap

ienz

a U

nive

rsity

of R

ome

- Mau

ro V

alor

ani (

2012

)

TSR analysis of Gri-1.2, Propane, and n-Heptane Autoignition

Autoignition iso-choric batch reactor:T0=1000K p0=1 atm, stoichiometric, non-diluted air.

The most energetic scale in 3 fuels takes a max value of about the same magnitude [tau ~1/(2 x 105) s ~ 50 x 10-6 s = 50 microsec]

TSR (red symbols) captures the most energetic scale at all times The modes mostly contributing to TSR are in green

Methane/Air Ignition


Sap

ienz

a U

nive

rsity

of R

ome

- Mau

ro V

alor

ani (

2012

)

Comparison CSP-TSR: Propane

PI index between mode & TSR

ωτ = Wi λii=1,N∑

Piωτ =

Wi λi iWj λ j

j=1,N∑

The first non exhausted mode indicated by CSP has a very low PIModes with the highest PI are in the “middle” of the active subspace


Sap

ienz

a U

nive

rsity

of R

ome

- Mau

ro V

alor

ani (

2012

)

Reaction Index: Propane

Pkωτ = Pi

ωτ Pki

PI index between reaction & TSR

The reactions contributing to the slowest positive eigvals do not contribute to TSRThe reactions contributing to the fastest positive eigvals do contribute to TSRThere are reactions contributing to TSR which do not contribute to positive eigvals

slow positive eigval

fast positive eigval

Table 1List of important reactions at iter = 30 (omega tau = 5.52714)O2+CH2O<=>HO2+HCO 32 0.040129H+O2<=>O+OH 38 0.0202577H+CH2O(+M)<=>CH3O(+M) 57 0.01242242OH(+M)<=>H2O2(+M) 85 0.111698OH+H2O2<=>HO2+H2O 89 0.0349801OH+CH4<=>CH3+H2O 98 0.058769OH+CH2O<=>HCO+H2O 101 0.01683092HO2<=>O2+H2O2 115 0.01653242HO2<=>O2+H2O2 116 0.104433HO2+CH3<=>O2+CH4 118 0.0173129HO2+CH3<=>OH+CH3O 119 0.107882HO2+CO<=>OH+CO2 120 0.0808342HO2+CH2O<=>HCO+H2O2 121 0.0261494CH3+O2<=>OH+CH2O 156 0.0221534CH3+H2O2<=>HO2+CH4 157 0.0960371CH3+CH2O<=>HCO+CH4 161 0.0252745HCO+M<=>H+CO+M 167 0.0151391HCO+O2<=>HO2+CO 168 0.0179481CH3O+O2<=>HO2+CH2O 170 0.0173865

Table 2;List of important reactions at iter = 50 (omega tau = 212718.)O+H2<=>H+OH 3 0.0309192O+CH3<=>H+CH2O 10 0.0251784H+O2<=>O+OH 38 0.201172H+CH4<=>CH3+H2 53 0.0554681H+HCO<=>H2+CO 55 0.0171357H+CH2O<=>HCO+H2 58 0.0100996OH+H2<=>H+H2O 84 0.05221062OH<=>O+H2O 86 0.0107618OH+CH3<=>CH2(S)+H2O 97 0.0204665OH+CH4<=>CH3+H2O 98 0.0253432OH+CO<=>H+CO2 99 0.0148563CH2+O2<=>OH+HCO 135 0.0223359CH2(S)+O2<=>H+OH+CO 144 0.0127997HCO+H2O<=>H+CO+H2O 166 0.0338651HCO+M<=>H+CO+M 167 0.0208143HCO+O2<=>HO2+CO 168 0.0180102


Sap

ienz

a U

nive

rsity

of R

ome

- Mau

ro V

alor

ani (

2012

)

Mechanism Simplification based on TSR Participation Index


Sap

ienz

a U

nive

rsity

of R

ome

- Mau

ro V

alor

ani (

2012

)

Conclusions

Tangential Stretching Rate diagnostics seems a promising tool of analysis

It retrieves the same results found by CSP in terms of fast/slow decomposition

It also provides additional information about (i) the most energetic scale(ii) the range of active scales, which are not uniquely identified in CSP

It can be used to define (i) the Damkoeler number in a reactive flow (ii) the proper integration time step during the numerical integration of the stiff problem.

TSR and CSP participation indices can also be used in mechanism simplification procedures.


Sap

ienz

a U

nive

rsity

of R

ome

- Mau

ro V

alor

ani (

2012

)

Questions ...

dynamical system analysis of ignition in reactive … seminars/mauro...kaust seminar, september 2012...

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