Lecture 12. Modeling of Turbulent Combustion - LTH · X.S. Bai Modeling of TC Modeling of turbulent...

X.S. Bai Modeling of TC

Lecture 12. Modeling of Turbulent Combustion


• direct numerical simulation (DNS)

• Statistical approach (RANS)– Modeling of turbulent non-premixed flames– Modeling of turbulent premixed flames

• Large eddy simulation

Content


Direct Numerical Simulation: DNS

• Solve the entire set of governing equations

– Down to the smallest flow scales

– Down to the fine reaction zones

( ) 0,ρ ρvt

∂+∇ ⋅ =

∂

, 1j ii ii i

j j j

ρu YρY YρD ω i , Nt x x x

⎛ ⎞∂∂ ∂∂+ = + =⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

…

( ) , ( 1 2 3)v vv pI i , ,tρ ρ τ∂

+∇ ⋅ = ∇ ⋅ + =∂

1:

N

i i i ri

Dh Dp q YV h Q vDt Dt

ρ ρ τ=

⎛ ⎞− = −∇ ⋅ + + + ∇⎜ ⎟

⎝ ⎠∑

( ) ( ) ∑∑==

−+∇

−+−=∇N

jjijiiiij

N

j ij

jii ffYY

pppXYVV

DXX

X11

)(ρ

,1 TRW

p umix

ρ=

∑∑==

=⎟⎟⎠

⎞⎜⎜⎝

⎛++≡

N

iiicisi

N

ii hYpuuYh

11 ρ

( )∫ +=++=T

Trefifpcisii

ref

iThdTcpuuh ,

0

ρ


Principles of DNS

• Governing equations (N+5, N+4)– Continuity equation, 1– Momentum equations, 3– Species transport equations, N (number of species)– Enthalpy tranpsort equation, 1– Equation of state, 1– Calorific equation of state, 1– Transport coefficients, N+2

• Independent variables to be simulated (2N+9)– Density, pressure, temperature, 3– Velocity components, 3– Species mass fractions, N– Enthalpy, 1– Transport coefficients, N+2


Principles of DNS

• Fully resolving all flow scales

– Kolmogrov scales: length, time, velocity

– All flame scales: reaction zones


Principles of DNS

• Fully resolving all flow scales

– Kolmogrov scales: length, time, velocity

– All flame scales: reaction zones


Cost of DNS to resolve one large eddy

3/400

1/400

1/200

5/40 00

2

20 00

Re ;

Re ;

Re ;

Assuming the smallest grid is and smallest time step is

Computational cost for 1-D Re

Computational cost for 1-D Re

Computa

l

l

l

l

l

l

vv

l

l

η

η

η

η

η

η

ττ

η τ

τη τ

τη τ

∝

∝

∝

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

∼ ∼

∼ ∼

3

11/40 00tional cost for 1-D Rel

l

η

τη τ

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠∼ ∼


Cost of DNS to resolve one large eddy

Total number of spatial mesh points x time steps needed for resolving one large eddy scales of flames with different spatial dimensions and Reynolds numbers

0Re 1-D 2-D 3-D

1 1 1 110 17.8 100 562100 316 10,000 3162271000 5623 1000,000 177,827,90010,000 100,000 100,000,000 100,000,000,000

• DNS with detailed chemistry for an SI engine takes 30 years• DNS is used for 2D • DNS is used for low Reynolds number flames


DNS of hydrogen flame, Mizobuchi et al, 29th symp

H2 Jet flame: 9 species, 17 reactions, 30Dx30D, 22.8 million gridsN.F.I.: normalized flame index – square of concentration gradient

T=1000K

N.F.I iso-surfaces

D


Statistical methods (SM): Ensemble Averages and Modeling

(Reynolds averaged Navier-Stokes equations: RANS)


Principles of ensemble averages

• Turbulent flame is a random process

• Only the statistical mean field is solved


Ensemble average

u

1

1Reynolds decomposition: ,

Favre decomposition: ,

M

mm

u u u u uM

uu u u u ρρ

=

′= + =

′′= + =

∑

u


Cost of Statistical Methods to resolve one large eddy

Total number of spatial mesh points x time steps needed for resolving one large eddy scales of flames with different spatial dimensions and Reynolds numbers

0Re 1-D 2-D 3-D SM

1 1 1 1 110 17.8 100 562 1100 316 10,000 316227 11000 5623 1000,000 177,827,900 110,000 100,000 100,000,000 100,000,000,000 1


Governing equations for the mean flame

Momentum:

0xu~

j

j =+∂ρ∂

∂ρ∂t

j

ji

ij

jiix

uuxp

xuu

tu

∂

′′′′∂−

∂∂

−=∂

∂+

∂∂ ρρρ ~~~

iijjj

iji Yuxx

YutY

ωρ∂∂

∂ρ∂

∂ρ∂

+⎟⎠⎞

⎜⎝⎛−=+ ''''

~~~

Mass:

Species:

Energy equation: similar as above


Modeling issues

i ji ii i j i

j j j j

Y uY YD Y ut x x x x

ρρ ρ ρ ω⎛ ⎞∂∂ ∂ ∂ ∂ ⎛ ⎞′′ ′′+ = + − +⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ⎝ ⎠⎝ ⎠

Turbulent transport flux Turbulent reaction rate

Turbulence modelse.g. K-epsilon model

Combustionmodels


Modeling of Turbulent Non-premixed flames

• Flame sheet model• Flamelet models• Eddy dissipation concept model• Conditional moment closure models• Probability density function models


Turbuelnt Combustionof a fuel jet

T Direct photo CH


Presumed PDF Burke-Schumann model


Burke-Schumann flame sheet model

• In 1970s Bilger advocated - in diffusion flames there is such as ‘magic’ variable called mixture fraction (Z). All the species mass fractions, temperature, density etc, are uniquely related to Z …

• Burke-Schumann were the first one found this magic relationship

⎪⎩

⎪⎨⎧

<

≥−−

=

st

stst

st

FZZ

ZZZZZ

Y0

1⎩⎨⎧

>≤−

=st

ststO ZZ

ZZZZY

0)/1(233.0

2

⎪⎩

⎪⎨⎧

<

≥−−

=

stst

ststP

ZZZZ

ZZZZ

Y/

11

⎪⎪⎩

⎪⎪⎨

⎧

<+−

≥+−−−

=stOuOust

st

stFuFustst

ZZTTTZZ

ZZTTTZZ

T)(

)(11


Ensemble average of flame sheet in turbulent flows

Z Z

∫∑∑∑ ≡ΔΔ

≡≡====

1

0111

)()()(1)(1 ZdZZpZZZN

ZnZZnN

tZN

Z m

M

m

mm

M

mm

N

ii

∫∑∑∑ ≡ΔΔ

≡≡====

1

0111

)()()()()()(1))((1 dZZYZpZZYZN

ZnZYZnN

tZYN

Y m

M

m

mm

M

mm

N

ii

Probability density function: pdf

t n p

Z

ZΔ

mZ Measurementat a flow field point


How to obtain PDF? Presumed PDF approach

ZdZZ

ZZZpba

ba

∫ −

−= 1

0

)1(

)1()(

χρρ∂∂

∂ρ∂

∂ρ∂

−+⎟⎠⎞

⎜⎝⎛ ′′−=+ PZu

xxgu

tg

jjj

j 2'~

Presumed PDF:

Mixture fraction variance:

gZba ,, ⇔

( ) ( ) dZZpZZZZZgdZZpZZ )(',)(1

0

2221

0 ∫∫ −=−===

Two equations, two unknowns


Numerical implementation (flame sheet model)

Momentum:

0xu~

j

j =+∂ρ∂

∂ρ∂t

j

ji

ij

jiix

uuxp

xuu

tu

∂

′′′′∂−

∂∂

−=∂

∂+

∂∂ ρρρ ~~~

( ) equationgZuxx

ZutZ

jjj

j −′′−=+ ,~~~

ρ∂∂

∂ρ∂

∂ρ∂

Mass:

Mixture fraction:

Flame sheet relation: ,....)()(

1

0∫= dZZpZTT


Presumed PDF flamelet model


Influence of finite rate chemistry on flamelet structure

• Chemical kinetics does not affect the flame shape and flame height very much !!!

• Chemical reaction does affect the species and temperature distribution a lot !!!

CH4/air diffusion flame, p=1 bar, Tu=300 K

δZ

H


Influence of the finite rate chemistryon maximum species mole fraction and T

• CH4/air diffusion flame, p=1 bar, Tu=300 K


The flamelet library

• The flamelet equation can be derived using Crocco transformation

• Flamelet library

– How to get ?

ii w

dZYd

=2

2

21 χ

),(),,(),,( χρχχ ρ ZfZfTZfY Tii ===

Solve the above flamelet equation using detailed chemicalkinetic mechanisms!!


Numerical implementation

• Ensemble average

• Presumed PDF– How to get ?

dZdZfZfZY ii χχχχρ ρ ),(),(),(1~

1

0 0∫∫∞

℘=

),( χZ℘

dZdZfZ χχχρ ρ ),(),(1

0 0∫∫∞

℘=

Similar to flame sheet model. But here there are four unknownparameters. One needs 4 transport equations.


Numerical implementation

dZdZfZ χχχρ ρ ),(),(1

0 0∫∫∞

℘=

Continuity + momentum k-epsilon

equations

Transport equations for the mean and variance of mixture fraction, and scalar dissipation rate

Ensemble averages

....~iY


Other modeling approaches


Direct modeling of mean reaction rates: Eddy dissipation concept model

iijjj

iji Yuxx

YutY

ωρ∂∂

∂ρ∂

∂ρ∂

+⎟⎠⎞

⎜⎝⎛−=+ ''''

~~~Species:

⎟⎟⎠

⎞⎜⎜⎝

⎛=

γω O

FEDCiYY

tC ,min1

0

‘Mixed is burned’model

Fuel air

Mixing and reaction zone

l0

u0


Modeling of turbulent premixed flames


Vo=0.45 m/s, phi=1.17; Vin=120m/s, phi=1.0

CH2O CHphoto


Modeling of turbulent premixed flames

• Desirable Models– taking into account the basic features of turbulent

premixed flames• wrinkling • stretch• local extinction, re-ignition• local flame structure • ...

– Computationally inexpensive– Valid for wide parameter range

with reasonably detailed chemistry


Modeling of turbulent premixed flamesa unified model does not exist

• Examples of models• k-ε model • global chemistry +

EDC/EBU ...• detailed chemistry +

G-equation + presumed PDF + flamelet library

• BML ...• Flame surface density

models

• Resolved issues– Mean flame position– Mean major species

• CO2, O2, UHC, …– Mean temperature

• Unresolved issues– intermediate species

• CO• NOx• soot

– flame dynamics


Direct modeling of mean reaction rates: flame surface density model

iijjj

iji Yuxx

YutY

ωρ∂∂

∂ρ∂

∂ρ∂

+⎟⎠⎞

⎜⎝⎛−=+ ''''

~~~Species:

unburned burned

mean reaction zone

l0

VsL

Σ

,

,

,

u L L F uF

Lu L F u

u L F u

A S YV

AS YV

S Y

ρω

ρ

ρ

⎛ ⎞= ⎜ ⎟⎝ ⎠

= Σ


unburnedburned

Flamelet library approach

• Mean flame brush– ensemble of laminar flamelets

• global structure– Wrinkling and fluctuating laminar

flamelets

• local structure– stretched local laminar flamelet


Stretched laminar flamelet library


Influence of flame stretch on Laminar flames

• 1-D geometry• Counterflow fresh-to-

burned configuration• Counterflow fresh-to-

fresh twin-flame configuration

• Detailed chemical kinetic mechanisms (up to C3)

• Peters’ group (Lecture notes in physics m15)

• Numerical code • Chemkin• Cantera


Level-set Based Flamelet Library Approach

Structures of laminar flamelet (quenching & species distributions)

Statistics of flamelets(fluctuations and wrinkling)

Level-set G formulationCounterflow DNMwith detailed chemistry

Ensemble average based on presumed PDF

Mean Turbulent Flame


Mean Flame Position – Level-set G-equation

jjT

ii

iiT

ii

Tiii

jj

ii

i

ii

xG

xGs

xGu

tGUse

xGns

xGu

tGinInsert

snudtdx

xG

xG

xG

n

tx

xG

tGGtxG

∂∂

∂∂

=∂∂

+∂∂

⇒

∂∂

−=∂∂

+∂∂

⇒

+=

∂∂

∂∂

∂∂

−=

=∂∂

∂∂

+∂∂

⇒==

~~~~

~)2(

~~~

~)1()3(

)3(~

)2(~~

~)1(0

~~0~),(~

0


A test case: Bluff-body stabilized premixed flames

VR-1 LDA data: u/SL = 10 - 14; l/δL = 40 - 200Thin reaction & flamelet regime (Peters) !


Previous RANS: CO Simulation

EDC


RANS with new FLA: profiles at x=150 mm

Nilsson & Bai 29th symp(1)no stretch & wrinkling; (2)with stretch, no wrinkling; (3) with stretch & wrinkling


RANS with new FLA: profiles at x=350 mm

Nilsson & Bai 29th symp

(1)no stretch & wrinkling; (2)with stretch, no wrinkling; (3) with stretch & wrinkling


Large eddy simulations: LES

• Filter away the small scales

• Retain the large eddies (larger than Taylor micro scales)

• Large scale unsteady motion is resolved

• Eddies smaller than the filter size need to be modeled

• Flame thickness is typically thinner than the LES grid size

• Models are needed to account for the unresolved scales

• Models are similar to the RANS models

• Computational cheaper than DNS, but more expensive than RANS


Large Eddy Simulation of bluff-body flame

Streamwise vorticity 500 1/sFlame surface G=0

Flame fluctuations, large scale wrinkling are captured !


LES of HCCI engine

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Lecture 12. Modeling of Turbulent Combustion - LTH · X.S. Bai Modeling of TC Modeling of turbulent...

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