7

CO2

Scaled molar heat capacity

6 H2O CO 5

4 H2 N2 500 1000 1500 2000

Ideal diatomic gas

2500

Temperature (K)

3000

2500

Mass heat capacity

H2O

2000

CO

CO2 1500

N2 1000 500 1000 1500 2000 2500

Temperature (K)

Stream 1: Fuel

Fuel + Air

Stream 2: oxidizer (a) Premixed flame (b) Diffusion flame

1.4 1.2

CO2

O2 1.0 Lewis number 0.8 0.6 0.4 H2 0.2 0.0 -2 -1 0 1 x [cm] 2 3 4 5 CH4 H2O OH

7 6 5 4 3 2 1 0 0.0 Abscissa (x) x=l Normalized fuel mass fraction Y / Y F1 Normalized temperature T / T 1

STATE 1: fresh gas

STATE 2: burnt gasReaction rate

1.00.0015

0.80.0010

HO2 H2O2

0.6

Y

0.40.0005

0.2

H2 O2 OH H2O O H

0.0 0.2 0.4 0.6 0.8 1.00.0000

Y

0.2

0.4

0.6

0.8

1.0

x [cm]4000.0 2e+12

x [cm]

2000.0

Heat release [erg/cm3]5e+11

3000.0

1e+12

Temperature [K]1000.0

0

0.0 0.2 0.4 0.6 0.8 1.0 0.2

5e+11

0.4

0.6

0.8

1.0

x [cm]

x [cm]

2400 2200 Maximum flame temperature (K) 2000 1800 1600 1400 1200 1000

0

1

2 3 Equivalence ratio

4

5

50

Laminar flame speed s L (cm/s)

40

0

30

20

10 Complex transport Lewis =1 for all species 0 0.6 0.8 1.0 Equivalence Ratio, 1.2 1.4

Adiabatic flame temperature (K)

2400

Full chemistry and variable Cp One-step chemistry and variable Cp One-step chemistry and constant Cp

2200

2000

1800

1600

1400 0.5 0.6 0.7

Equivalence ratio

0.8

0.9

1.0

1.1

1.2

-9

300x10

(1-)/ exp(1/)Reduced reaction rate

200

100

0 0.0 0.5 Reduced temperature = (T-T1) / (T2-T1)

1-1/(+)

1.0

Reduced reaction rate

= 0.75 = 0.75 = 0.75 = 0.75

=4 =8 =13 =20

0.2

0.4 0.6 Reduced temperature = (T-T1) / (T2-T1)

0.8

1.0

Reduced reaction rate

Zone 1: diffusion and convection: thickness = 1 in space

Zone 2: diffusion and reaction: thickness = 2/ in space

0.0

0.2

0.4 0.6 Reduced temperature = (T-T1) / (T2-T1)

0.8

1.0

700x10-9

600 Reduced reaction rate 500 400 300 200 100 0 0.0 0.2

Echekki Ferziger form

Arrhenius form

0.4 0.6 Reduced temperature = (1) / (21)

0.8

c = 1- 1/

1.0

1.0

0.8 Reduced temperature = c = 1-1/

0.6

0.4

0.2 Arrhenius form Echekki Ferziger model 0.0 -5 -4 -3 -2 -1 Reduced abscissa (x/ ) 0 1 2

0.7 0.6 0.5 Flame speed (m/s) 0.4 0.3 0.2 0.1 0.0 0.6 0.8 1.0 Equivalence ratio 1.2 1.4 Numerical solution Experiments Asymptotic solution

40x10

-3

n f =0.2 n f =0.4 n f =1

Reduced reaction rate

30

20

10

0 0.0 0.2 0.4 0.6 Reduced temperature (T-T1) / (T2-T1) 0.8 1.0

1.0 0.8 0.6 0.4 0.2 0.0 300 METHANE - AIR 350 400 450 Fresh gas temperature (K) 500= 1, P=1 atm = 1, P=5 atm = 1, P=10 atm

1.0 0.8 0.6 0.4 0.2 0.0 300

= 1, P=1 atm = 1, P=5 atm = 1, P=10 atm

Flame speed (m/s)

Flame speed (m/s)

PROPANE - AIR 350 400 450 Fresh gas temperature (K) 500

Reduced temperature

(T-T1)/(T-T2)

1

STATE 1: fresh gas

STATE 2: burnt gas

0

Lt

L o

Abscissa

n

w

m Burnt gases

p A(t)

R1

R2 Fresh gases

Burnt gases

Fresh gases Fresh gases

Burnt gases

(a) Cylindrical flame

(b) Spherical flame

Fresh gases

Burnt gases Steady flame front

u = (u1 ( x 2 ) , 0 ) x2

n Splitter plate

Flame front

x1

Flame front

Fresh gases (U j1 )

d

(a) The single flame: fresh gas against combustion products (steady flow)

Stagnation plane

Combustion products (U j2 )

Fresh gases (U j1 ) Stagnation plane (b) The twin flames: fresh gas against fresh gas (steady flow)

Flame fronts Fresh gases (U j2 )

Fresh gases Burnt gases (c) The spherical flame: unsteady stretched curved flame

Position of isolevel f at time t+dt s d n dt

FRESH GAS

w dt n Position of isolevel f at time t u dt BURNT GAS

8 Displacement speed of isolevel 6 Normalized speeds Flow velocity 4 Absolute speed of isolevel 2

0

-2 0 20 40 60 80 Distance along normal to flame front 100 120

1.0

0.8

0.6

0.4 Reduced temperature Velocity 0.0 -1.0 -0.8 -0.6 -0.4 Distance to stagnation plane -0.2 0.0

0.2

Consumption speed sc Adiabatic flame

LeF = 1

Markstein zone sc =sLo 0

Non adiabatic flame Stretch Critical stretch

Consumption speed sc Adiabatic flame

LeF < 1

Markstein zone sc =sLo-Lac 0

Non adiabatic flame Stretch Critical stretch

Consumption speed sc Adiabatic flame

LeF > 1

Markstein zone sc =sLo-Lac

Non adiabatic flame

Stretch Critical stretch

0

Flame tip: The flame moves at the flow speed u: sa = 0, sd= u, sc=sL . Fresh gas u=sd

Flame front

Fresh gas (state 1)

1.0

Normalized speeds

0.8 0.6 0.4 0.2 0.0 0 1

Reduced temperature Yp=Yp2

p

Burnt gas (state 2)R=0

r(t)L0

R

Products mass fraction Y

2

3

4

Normalized radius (R/r(t))

1.0 Displacement speed Normalized velocities u/(dr/dt) 0.8 Absolute speed 0.6

0.4 Flow speed 0.2

0.0 0 1 2 3 Normalized radius (R/r(t)) 4 5 6

Temperature Fuel mass fraction YF

YO=YO0

YF=YF0

Oxidizer mass fraction YO T=TO0

T=TF0

Heat release

Abscissa

Diffusion zone

Reaction zone

Diffusion zone

Fuel mass fraction Y F at t= t1

Temperature at t= t1 Temperature at t= t2 > t1

Fuel mass fraction Y F at t= t2 > t1

Abscissa x

(a) the unsteady unstretched diffusion flame Fuel mass fraction on the axis x2 Temperature on the axis

Fuel jet axis x1

Oxidizer jet

(b) the steady stretched diffusion flame

z gradient x2 Iso z surface x1 Tangent plane to iso z surface x3 y2 y3

Boundary conditions Initial conditions Geometry Flow field 1: Mixing problem determine the field of the mixture fraction Scalar dissipation rate field Assumptions on reversibility and chemistry speed

2: Flame structure problem specify: - species Yk for k = 1 to N - temperature T as functions of the mixture fraction z T(z) Yk (z)

z(x i ,t)

Solution of the full problem: T(xi,t), Yk (x i,t)

TO0

Temperature

TF0 0 Oxidizer mass fraction Fuel mass fraction 1 YF0 z

YO0

0

1

z

Tad Equilibrium lines Temperature : possible states TO0 Mixing line 0 zst Equilibrium lines Fuel mass fraction Mixing line 1 YF0 TF0 z

0 YO0 Oxidizer mass fraction

zst Mixing line Equilibrium lines

1

z

0

zst

1

z

Pure mixing

Fresh mixture T=zT F 0 +(1-z)T O 0 YF= YF0 z Y O = Y O 0 (1- z) Combustion

Pure mixing 1-z OXIDIZER

FUEL z

Burnt products T=zT F 0 +(1-z)T O 0 + Q YF 0 z /Cp

if zzs t

Stream 1: Temperature TF0 Fuel mass fraction YF0

Diffusion flame. Maximum temperature Tad

Stream 2: Temperature TO0 Oxidizer mass fractionYO0

Outlet plane. Mean temperature Tm

2000 Temperatures (K)

1500 Mean burner outlet temperature Maximum temperature in burner 1000

500

0.0

0.5

1.0 1.5 2.0 Global equivalence ratio g

2.5

3.0

z 1 Pure fuel

Ignited flame started at t=0 Pure oxidizer

x x=0

1.0 0.8

Fuel Oxidizer

2000 1600

Temperature

Mass fractions

0.6 1200 0.4 800 0.2 400 0.0 0.0 0.2 0.4 0.6 0.8 Mixture fraction z 1.0 0.0 0.2 0.4 0.6 0.8 Mixture fraction z 1.0

1.0 0.8 0.6 0.4 0.2 500 0.0 -4 -2 0 2 Reduced abscissa 4 -4 -2 0 2 Reduced abscissa 4 Fuel YF Oxidizer Y O Mixture fraction z 2000 Temperature

1500

1000

Reaction rate Fuel mass fraction

Abscissa x fxf x f+

Oxidizer

x1

Flame (x1=xf)

x2

Fuel

Scalar dissipation (1/s)

10 8 6 4 2 0 -4 -2 0 2 4 Scalar dissipation at z=zst Flame position (z=z st )

Reduced abscissa eta

1.0

aeq / a0.8 0.6 0.4 0.2 0.0 -1 0 1 2 3 4 5

at

x2Oxidizer

z Flame2e0

Fuel

Lf

(a)

x2 YO z YF YF YO Lf

(b)

YF, YO, z T T

T z

x1

Equilibrium lines Temperature : possible states TO0 Mixing line 0 Fuel mass fraction zst Equilibrium lines Mixing line 1 YF0 TF0 z

0 Oxidizer mass fraction YO0

zst Mixing line Equilibrium lines

1

z

0

zst

1

z

Infinitely fast chemistry Temperature

Finite rate chemistry TO0 Mixing line TF0 0 Fuel mass fraction zst Mixing line 1 YF0 z

possible states with infinitely fast irreversible chemistry

Finite rate chemistry

Infinitely fast chemistry 0 Oxydizer mass fraction YO0 zst Mixing line Finite rate chemistry Infinitely fast chemistry 1 z

0

zst

1

z

Infinitely fast chemistry

Finite rate chemistry

Scalar dissipation at stoichiometric point ( st ) or Flame strain (a) or Inverse Damkohler (Da -1 )

Quenching

FAST CHEMISTRY (EQUILIBRIUM) Temperature Independent of strain

FINITE RATE CHEMISTRY (NON EQUILIBRIUM) Temperature Infinitely fast chemistry limit Finite rate chemistry (low strain) Finite rate chemistry (high strain)

IRREVERSIBLE REACTION TO0 Mixing line 0 zst TF0 TO0 Mixing line 0 zst

TF0 1

1

Temperature Independent of strain REVERSIBLE REACTION TO0 Mixing line 0 zst

Temperature

Infinitely fast chemistry limit Finite rate chemistry (low strain) Finite rate chemistry (high strain)

TO0 TF0 0 Mixing line zst 1 TF0

1

0.060 1.0 0.8 Y 0.6 0.4 0.2 0.0 1.0 0.5 0.0 0.5 x [cm] 1.0 0.000 1.0 0.5 0.0 0.5 x [cm] 1.0H2 O2 OH H2OH O

0.040 Y 0.020 Temperature [K] 1.0 3000 2000 1000 0 1.0

2e04HO2 H2O2

Y 1e04 0e+00 1.0

0.5

0.0 0.5 x [cm]

0.5

0.0 0.5 x [cm]

1.0

0.20 Scalar defined on O element (zO) Scalar defined on O2 and H2 (z1) 0.15

Normalized scalars

0.10

0.05

0.00 0.00 0.05 0.10 0.15 Mixture fraction defined on H element (zH) 0.20

1.0 Real Hydrogen Real Oxygen Ideal flame structure Hydrogen Ideal flame structure Oxygen

0.8

Mass fractions

0.6

0.4

0.2

0.0 0.00 0.05 0.10 0.15 0.20 Mixture fraction 0.25 0.30 0.35

4000

Method 1: 1D flame (full chemistry and transport) a = 300 s Method 2: Equilibrium calculation (with single step reaction) Method 3: Equilibrium calculation (with full chemistry)

-1

Temperature (K)

3000

2000

1000

0.0

0.2

0.4 0.6 Mixture fraction (zH)

0.8

1.0

5 Max. Temp. Sens. Coeff. [%]

0

-5

-10 p = 1 bar a = 100 s 1 TF = TOx = 300 K

-15

-20 N2 O2 H OH O H2 Lewis Number H 2 O H O2 H 2 O 2

u(r)

ru(r)/ ult /

3

4u 1 uKK

3

1lt r

rK

lt

r/u(r) lt/ u

u(r)/r uK / K

33

22

u/l t K /uK K

rlt rK

lt

Turbulent fresh gases

Burnt gases

sT

Turbulent flame brush: speed sT thickness T

Ambiant air

dfuel

UFlame

Lf

Lf Laminar

Transition Turbulent Blow-off

Lift-off

Re

Temperature DNS RANS

LES

Time

E(k)

Modeled in RANS Computed in DNS Computed in LES Modeled in LES

kc

k

0.00

0.05

0.10

0.00

0.05

0.10

2 1 0 1 2 0 2 4 6 8 10 12

2 1 0 1 2 0 2 4 6 8 10 12

2 1 0 1 2 0 2 4 6 8 10 12

2 1 0 1 2 0 2 4 6 8 10 12

yFresh gases Limits of turbulent flame brush

AInstantaneous flame front

Burnt gases

CFlame holder

Burnt gases

Transverse cut axis

B

y

y

Laminar profiles

Turbulent profiles

Temperature

Reaction rates

Laminar rate profile () Reduced reaction rates

Turbulent rate profile ()

0 0.0 0.2 0.4 0.6 Reduced temperature = (T-T1) / (T2-T1) 0.8 1.0

t hot T2

T

T1 Time t cold t

Complex chemistry

2DPatnaik and Kailasanath 1988 Katta and Roquemore 1998 Baum et al 1994 Haworth et al 2000 Chen et al 1998 Tanahashi et al 1999

3D Tanahashi et al 1999 (TSFP) Constant density Variable density

2D One-step chemistry

3DTrouv and Poinsot 1994 Rutland and Cant 1994 Zhang and Rutland 1995 Swaminathan and Bilger 1999 Montgomery, Kosaly, Riley 1997

Rutland and Ferziger 1991 Poinsot, Veynante, Candel 1991

Leonard and Hill 1988 Ashurst, Peters, Smooke 1987 Rutland and Trouv 1990 Cant, Rutland and Trouv 1990 Laverdant and Candel 1989 ElTahry, Rutland, Ferziger 1991 Rutland and Trouve 1993

2DBarr 1990 Katta and Roquemore 1993

3DMcMurtry and Givi 1989

Ashurst and Barr 1983 Osher and Sethian 1987 Ashurst 1987

Ashurst, Kerstein, Kerr, Gibson 1987 Yeung, Pope 1990 Cattolica, Barr, Mansour 1989

NB: Non exhaustive list

Constant density Variable density

No chemistry

Constant density Variable density

1000

Re = 200

Da = 1

RMS velocity / Flame speed

100

DNS domain10

Aircraft engines

Re = 1

1

Piston engines

Infinitely thin flames100 1000

Integral scale / Flame thickness

1

10

F1

F1/

0

k kc

F

- / 2

0

/2

x

/ 2

0

/2

x

E(k)

Computed in LES

Modeled in LES

k

E(k)

Computed in LES

Modeled in LES

k

Volume V

Turbulent flame propagating against the mean flow at speed sT

Flame front Cross section A: premixed flow entering at speed sTx1

Fresh gas

Burnt gas

Locally laminar flame front propagating at speed sL

Turbulent flame speed (sT)

sL0

Turbulent RMS velocity in the fresh gas (u) Low turbulence zone s T = a u Bending zone Quenching limit

Velocity ub

Velocity

u uu uu t uu t

Fresh gases

A

C ub

Velocity ub

Burnt gases

B

t

1.0 0.8 Reynolds average 0.6 0.4 0.2 0.0 0.0 0.2

0 1 2 4 6 10

0.4 0.6 Favre (mass-weighted) average

0.8

1.0

0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0 0.2 0.4 0.6 0.8

1 2 4 6 10

1.0

x2

Propane + air

x1

A

B

RMS velocity / flame speed (u/s 0 )

Well stirred reactor Distributed reaction zones

L

D

a=

1

Re t= 1

Klim

ov

Wil

lia

lim ms

it:

Ka

=1

Corrugated flamelets 1 Laminar combustion Wrinkled flamelets

1

Integral length scale / flame thickness

(lt / )

Thickened flame

KaRMS velocity /flame speed (u/s 0 )

00 = 1

Da= 1 Thickened-wrinkled flame

L

Re t= 1

Klim

ov

Wil

s liam

lim

it:

Ka

=1

1 Laminar combustion

Wrinkled flamelets

1

Integral length scale / flame thickness

(lt / )

FRESH GAS

BURNT GAS

FRESH GAS

BURNT GAS

T = 300 K T = 300 K (a) T = 300 K turbulent flame thickness (b) T = 300 K turbulent flame thickness

T = 2000 K

flamelet reaction zone

flamelet preheat zone

T = 2000 K

mean preheat zone

mean reaction zone

T = 2000 K

T = 2000 K

(c)

T = 300 K

T = 300 K turbulent flame thickness

T = 2000 K mean preheat zone mean reaction zone

T = 2000 K

Burnt gases Instantaneous flame front

Fresh gases

A

B

Laminar strained flamelet (curved and unsteady)

Fresh gases Curved flamelet

Burnt gases

Stretched flamelet

Vortex velocity / flame speed (u(r)/sL 0 ) uRMS

Po(r) = 1

Ka(r) = 1

Turbulence line u ( r ) 3 /r =

Vr(r) = 1

1 Kolmogorov line Re(r) = 1 Vortex size / flame thickness (r/ )

/Kolmogorov scale

1

l/ Integral scale

Computation domain

Vortex pair Max speed u'(r) Flame front (speed sL )

Symmetry axis r D

Vortex velocity / flame speed (u(r)/sL 0 ) uRMS

No effect zone Kolmogorov line A

Cutoff limit Quenching limit C

D B

1

Vortex size / flame thickness (r/ ) 1 Kolmogorov 100 Integral C Distributed reaction zones ? RMS velocity /flame speed (u/sL 0 ) A D B Corrugated flameletsKlim ov Wil =1

Cut off

im s l liam

it:

Ka

1

Pseudo laminar flames

Wrinkled flamelets

1

100

Integral length scale / flame thickness (l/ )

Arrhenius model () Mean reaction rates

EBU model ()

0 0.0 0.2 0.4 0.6 0.8 1.0

Reduced temperature = (T-T1) / (T2-T1)

pdf( )

0

1

1

0 time

T2 T T1 timeFresh gas Limits of turbulent flame brush u=s d

T2 T T1 time AFlame front

C

BBurnt gas

T2 T T1 time

1

0Time

tt

Fresh

Burnt

Fresh Flame

Burnt Flame LOCAL LEVEL (library): Flamelet consumption : wL

Flame stretch

GLOBAL LEVEL: Flame surface density

Mean reaction rate:

= 0 w L

Instantaneous flame front

Ly

mean surface

0.0 0.2 0.4 0.6 0.8 1.0

1.5

(I) (III) (V) (VII) (IX) unbalanced (II) (IV) (VI) (VIII) (X)

1.0

0.5

0

-0.5

-1.0

-3.0 0.0 0.2 0.4 0.6 mean progress variable 0.8 1.0

1000

k

100 10 1 0.1 0.01 0.001 100

u / sL0

10

1

10

2

10

3

10

4

lt / L0

14 12 10

sL0/

8 6 4 2 0 0 2 4

sT

u'

/

sL0

6

8

10

premixed reactants

u = sT

Flame

Fresh gases

Flame

Hot burnt gases

heat fluxes

flame front

LES computational mesh size (x)

Fresh gases =0 Burnt gases =1

Fresh gases LES computational mesh size (x)

real flame front

=1

sL=0 thickened flame front Burnt gases

G > G* LES mesh size Fresh gases

G < G*

flame front (G = G*)

sdBurnt gases

1.0

0.8

progress variable0.6 0.4 0.2 0.0 -2 -1

x/

0

1

2

0.20

0.10

0.00

0.10

0.20

0.30 0.0

0.2

0.4

0.6

0.8

1.0

0.15

0.10

0.05

0.00

0.05 0.0

0.2

0.4

0.6

0.8

1.0

0.020

0.010

0.000

0.010

0.020

0.030 0.015

0.005

0.005

0.015

_ =0 Fresh premixed gas + Turbulence N

_ =1 Burnt gas

n Instantaneous flame front

=0 =1 x Mean flame front (x) (x)

x

=0+

sd1

Burnt gases=0+

Burnt gases=1

sd 0

=1 sd

Fresh gases Fresh gases0