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X.S. Bai Modeling of TC
Lecture 12. Modeling of Turbulent Combustion
X.S. Bai Modeling of TC
• direct numerical simulation (DNS)
• Statistical approach (RANS)– Modeling of turbulent non-premixed flames– Modeling of turbulent premixed flames
• Large eddy simulation
Content
X.S. Bai Modeling of TC
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
X.S. Bai Modeling of TC
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
X.S. Bai Modeling of TC
Principles of DNS
• Fully resolving all flow scales
– Kolmogrov scales: length, time, velocity
– All flame scales: reaction zones
X.S. Bai Modeling of TC
Principles of DNS
• Fully resolving all flow scales
– Kolmogrov scales: length, time, velocity
– All flame scales: reaction zones
X.S. Bai Modeling of TC
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
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠∼ ∼
X.S. Bai Modeling of TC
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
X.S. Bai Modeling of TC
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
X.S. Bai Modeling of TC
Statistical methods (SM): Ensemble Averages and Modeling
(Reynolds averaged Navier-Stokes equations: RANS)
X.S. Bai Modeling of TC
Principles of ensemble averages
• Turbulent flame is a random process
• Only the statistical mean field is solved
X.S. Bai Modeling of TC
Ensemble average
u
1
1Reynolds decomposition: ,
Favre decomposition: ,
M
mm
u u u u uM
uu u u u ρρ
=
′= + =
′′= + =
∑
u
X.S. Bai Modeling of TC
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
X.S. Bai Modeling of TC
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
X.S. Bai Modeling of TC
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
X.S. Bai Modeling of TC
Modeling of Turbulent Non-premixed flames
• Flame sheet model• Flamelet models• Eddy dissipation concept model• Conditional moment closure models• Probability density function models
X.S. Bai Modeling of TC
Turbuelnt Combustionof a fuel jet
T Direct photo CH
X.S. Bai Modeling of TC
Presumed PDF Burke-Schumann model
X.S. Bai Modeling of TC
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
X.S. Bai Modeling of TC
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
X.S. Bai Modeling of TC
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
X.S. Bai Modeling of TC
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
X.S. Bai Modeling of TC
Presumed PDF flamelet model
X.S. Bai Modeling of TC
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
X.S. Bai Modeling of TC
Influence of the finite rate chemistryon maximum species mole fraction and T
• CH4/air diffusion flame, p=1 bar, Tu=300 K
X.S. Bai Modeling of TC
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!!
X.S. Bai Modeling of TC
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.
X.S. Bai Modeling of TC
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
X.S. Bai Modeling of TC
Other modeling approaches
X.S. Bai Modeling of TC
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
X.S. Bai Modeling of TC
Modeling of turbulent premixed flames
X.S. Bai Modeling of TC
Vo=0.45 m/s, phi=1.17; Vin=120m/s, phi=1.0
CH2O CHphoto
X.S. Bai Modeling of TC
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
X.S. Bai Modeling of TC
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
X.S. Bai Modeling of TC
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
X.S. Bai Modeling of TC
unburnedburned
Flamelet library approach
• Mean flame brush– ensemble of laminar flamelets
• global structure– Wrinkling and fluctuating laminar
flamelets
• local structure– stretched local laminar flamelet
X.S. Bai Modeling of TC
Stretched laminar flamelet library
X.S. Bai Modeling of TC
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
X.S. Bai Modeling of TC
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
X.S. Bai Modeling of TC
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
X.S. Bai Modeling of TC
A test case: Bluff-body stabilized premixed flames
VR-1 LDA data: u/SL = 10 - 14; l/δL = 40 - 200Thin reaction & flamelet regime (Peters) !
X.S. Bai Modeling of TC
Previous RANS: CO Simulation
EDC
X.S. Bai Modeling of TC
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
X.S. Bai Modeling of TC
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
X.S. Bai Modeling of TC
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
X.S. Bai Modeling of TC
Large Eddy Simulation of bluff-body flame
Streamwise vorticity 500 1/sFlame surface G=0
Flame fluctuations, large scale wrinkling are captured !
X.S. Bai Modeling of TC
LES of HCCI engine