1
SIMPLIFIED FORMULATIONS FOR TURBULENT
COMBUSTION OF HYDROGEN
Forman A. Williams
UCSD, La Jolla, CA
General Basic Equations
Kolmogorov
Turbulence Scaling
Regimes of Turbulent Combustion
Approaches to Turbulent Combustion
ChemicalKinetic Approximations
TransportProperty Approximations
Simplified Conservation Equations
Turbulent Hydrogen Diffusion Flames
2
REACTING NAVIERSTOKES CONSERVATION
EQUATIONS
(
)
.
0
t
=
ρ
ν
⋅
∇
+
∂
ρ
∂
()
∑
=
+
ρ
Ρ
⋅
∇
−
=
ν
∇
⋅
ν
+
∂
ν
∂
N
1
i
i
i
f
Y
t
()
[]
()
∑
=
⋅
∇
−
+
⋅
+
⋅
Ρ
−
⋅
∇
+
∂
∂
=
⎟
⎠
⎞
⎜
⎝
⎛
+
∇
⋅
+
⎟
⎠
⎞
⎜
⎝
⎛
+
∂
∂
N
i
i
i
i
q
V
v
f
Y
v
pU
t
p
h
v
h
t
1
2
2
.
2
1
2
1
ρ
ν
ρ
ν
ρ
()
[
]
ρ
ρ
⋅
∇
−
ρ
=
∇
⋅
+
∂
∂
i
i
i
i
i
V
Y
w
Y
v
t
Y
3
(
)
()
()
[
]
T
v
v
U
v
3
2
p
∇
+
∇
µ
−
⎥
⎦
⎤
⎢
⎣
⎡
⋅
∇
⎟
⎠
⎞
⎜
⎝
⎛
κ
−
µ
+
=
Ρ
()
R
j
i
ij
i
Ti
j
N
1
j
N
1
i
N
1
i
0
i
i
i
q
V
V
D
W
D
X
T
R
V
Y
h
T
q
+
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
ρ
+
∇
λ
−
=
∑
∑
∑
=
=
=
()
()
()
N.
1,...,
i
,
1
1
1
=
∇
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
−
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∇
−
+
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
∇
∑
∑
∑
=
=
=
T
T
Y
D
Y
D
D
X
X
f
f
Y
Y
p
p
p
X
Y
V
V
D
X
X
X
i
Ti
j
Tj
ij
j
i
N
j
N
j
j
i
i
i
i
i
i
j
ij
j
i
N
j
i
ρ
ρ
()()
N
1,...,
i
,
T
R
p
X
e
T
B
W
w
k
,
v
0
j
N
1
j
T
R
/
E
k
k
'
ik
'
'
ik
M 1
k
i
i
'j
0
k
=
⎟
⎠
⎞
⎜
⎝
⎛
ν
−
ν
=
∑
∑
=
−
α
=
4
()
i
i
N
1
i
0
W
Y
T
R
p
∑
=
ρ
=
∑
=
=
N
1
i
i
i
Y
h
h
N
1,...,
i
,
dT
c
h
h
T
T
i
,
p
o
i
i
0
=
+
=
∫
(
)
()
N.
1,...,
i
,
W
Y
W
Y
X
N
1
j
j
j
i
i
i
=
=
∑
=
5
6
KOLMOGOROV SCALING
=
'
u
RMS velocity fluctuation
=
l
Integral scale ,
ν
/
'
lu
Rl
=
Turbulence Reynolds Number:
∈=
Rate of dissipation of turbulent kinetic energy
(
)
4
/
1
3
∈
=
ν
k
l
4
/
3
l
l
R
=
Kolmogorov
scale:
dk
k
E
dk
k
k
)
(
1
1
is
and
+
Kinetic energy in eddies of sizes between
(
)
3
/
5
3
/
2
k
k
E
−
=
∈
for the inertial range
Kolmogorov
scaling:
7
Regimes of turbulent combustion in a diagram of a length scale of the system and a representative
average velocity.
8
l
R
l
D
Regimes of turbulent combustion in a diagram of a turbulence Reynolds number and a Damköhler
number, both based on the integral scale of the turbulence.
,
R
D
D
c
k
K
l
l
=
τ
τ
=
9
Illustration of hydrocarbonair flamelet
structure as predicted by rateratio asymptotics.
10
Regimes of nonpremixed
turbulent comb
ustion in a diagram of a scalardissipation parameter related to a
Damköhler
number and the ra
tio of a representative mixturefraction fluctuation to the mixturefraction
range that spans a diffusion flamelet.
2
Z
2
∇
ν
=
χ
11
CATEGORIES OF APPROACHES TO ANALYSIS OF TURBULENT COMBUSTION
1.
Phenomenological
a.
Quasidimensional
b.
Age Theories
c.
Linear Eddy/OneDimensional Turbulence
2.
FluidsBased
a.
Direct Numerical Simulation (DNS)
b.
LargeEddy Simulation (
LES)
c.
Moment Methods (RANS)
i.
Algebraic Closures
ii.
k
ε
Modeling
iii
ReynoldsStress Closure
3.
Turbulent Burning Velocity (ST)
a.
Perturbations for Low Intensities and Large Scales
b.
MomentMethod Modeling
of the G Equation
c.
Modeling
FlameSurface Evolution, such as CoherentFlamelet
Models (CFM)
d.
Fractals
e.
GEquati
ons Renormalization
f.
Pseudosolitons
4.
ProbabilityDensity Function (PDF)
a.
Flamelets
b.
Presumed PDF
i.
P(Z) for Diffusion Flames
ii
P(c) for Premixed Flames
iii
P(G) for Premixed Flames
c.
Conditi
onal Moment Closure (CMC)
d.
PDF Transport
i.
Linear MeanSquar
e Es
tim
at
ion (LMSE),
also call
ed Interaction by Exchange with the Mean (IEM)
ii
CoalescenceDispersion (CD)
iii.
Mapping Closure (MC)
iv.
Euclidean Minimum Spanning Tree (EMST)
12
FRONTTRACKING EQUATION
G
S
G
v
t
G
L
∇
=
∇
⋅
+
∂
∂
MIXTUREFRACTION EQUATION
(
)
ρ
∇
ρ
⋅
∇
=
∇
⋅
+
∂
∂
Z
D
Z
v
t
Z
th
13
TWOSTEP CHEMICALKINETIC DESCRIPTION
H
2
O
H
2
O
H
3
2
2
2
+
→
+
I.
2
H
H
2
→
II.
O
OH
O
H
2
+
→
+
1.
H
OH
H
O
2
+
→
+
2.
H
O
H
H
OH
2
2
+
→
+
3.
M
H
M
H
2
2
+
→
+
M
HO
M
O
H
2
2
+
→
+
+
OH
2
HO
H
2
→
+
4.
Not Reversible
5.
6.
2
2
2
O
H
HO
H
+
→
+
7.
14
(
)
5
f
7
f
7
6
f
7
ω
ω
=
ω
+
ω
ω
=
α
(
)
b
7
5
b
1
f
1
I
1
ω
+
ω
α
−
+
ω
−
ω
=
ω
5
4
II
ω
+
ω
=
ω
PARTIALEQUILIBRIUM APPROXIMATIONS
(
)
2
2
H
3
O
H
H
OH
X
K
X
X
X
=
(
)
2
H
3
2
O
H
2
H
O
2
2
X
K
K
X
X
X
=
O
H
2
/
1
O
2
/
3
H
3
2
/
1
2
2
/
1
1
H
2
2
2
X
X
X
K
K
K
X
=
2
/
1
O
2
/
1
H
2
/
1
2
2
/
1
1
OH
2
2
X
X
K
K
X
=
O
H
O
H
3
1
O
2
2
2
X
X
X
K
K
X
=
15
TRANSPORTPROPERTY APPROXIMATIONS
()
1
D
Sc
ij
ij
=
ρ
µ
=
1
c
Pr
p
=
λ
µ
=
=
=
p
pi
c
c
=
j
i
D
DTi
= 0,
constant
constant, µ
= constant or C = ρµ
= constant.
and
,
,
,
0
=
κ
16
0
1
2
3
4
5
012345
A
B
C
Schematic illustration of dependences of turbulent burning velocities on turbulence intensity.
17
SIMPLIFIED CONSERVATION EQUATIONS
,
v
t
Dt
D
∇
⋅
+
∂
∂
=
(
)
[
]
i
i
i
i
w
Y
S
Dt
DY
=
∇
µ
∇
−
ρ
i
N
1
i
c
i
T
Y
h
h
h
∑
=
−
=
()
[]
∑
=
⋅
∇
−
−
∂
∂
=
∇
µ
∇
−
ρ
N
1
i
R
i
o
i
T
T
q
w
h
t
p
h
Pr
Dt
Dh
(
)
(
)
[
]
II
'
iII
'
'
iII
I
'
iI
'
'
iI
i
i
v
v
v
v
W
w
ω
−
+
ω
−
=
18
(
)
(
)
[
]
i
i
i
i
i
Y
S
Dt
DY
Y
L
∇
µ
⋅
∇
−
ρ
=
(
)
(
)
[
]
T
T
T
T
h
Pr
Dt
Dh
h
L
∇
µ
⋅
∇
−
ρ
=
()
(
)
I
O
2
O
O
O
H
O
H
O
H
2
W
Y
L
2
W
Y
L
2
2
2
2
2
ω
=
−
=
(
)
I
II
H
H
H
3
W
Y
L
2
2
2
ω
−
ω
=
(
)
II
I
H
H
H
3
2
W
Y
L
ω
−
ω
=
19
(
)
II
II
I
I
T
T
Q
Q
h
L
ω
+
ω
=
H
o
H
O
H
o
O
H
I
W
h
2
W
h
2
Q
2
2
−
−
=
H
o
H
II
W
h
2
Q
=
()
(
)
(
)
II
O
O
O
H
H
H
O
H
O
H
O
H
2
W
Y
L
2
W
Y
L
W
Y
L
2
2
2
2
2
2
2
2
2
ω
=
−
=
−
=
(
)
(
)
,
Q
Q
h
L
II
II
I
T
T
ω
+
=
20
NONPREMIXED TURBULENT BURKESCHUMANN
HYDROGEN COMBUSTION
(
)
(
)
[
]
st
F
O
st
st
L
Z
1
L
L
Z
Z
Z
−
+
=
Pr
S
L
2
O
O
=
Pr
S
L
2
H
F
=
for Z < ZL; L = LF
for Z > ZL
(
)
Z
D
Dt
LDZ
th
∇
ρ
⋅
∇
=
ρ
(
)
Pr
µ
ρ
=
th
D
O
L
L
=
Everywhere:
21
(
)
()
L
L
H
II
I
T
H
F
Z
Z
H
Z
Y
Q
Q
h
W
L
2
2
2
+
+
=
∞
(
)
(
)
L
L
H
H
Z
1
Z
Z
Y
Y
2
2
−
−
=
∞
(
)(
)
(
)
[
]
L
L
H
II
I
T
H
F
Z
1
Z
1
H
Z
Y
Q
Q
h
W
L
2
2
2
−
−
+
+
=
∞
(
)
H
D
Dt
NDZ
Dt
DH
th
∇
ρ
⋅
∇
=
ρ
+
ρ
()
L
O
Z
L
1
N
−
=
for Z < ZL;
(
)
(
)
L
F
Z
1
1
L
N
−
−
=
for Z > ZL
(
)
L
O
O
Z
Z
1
Y
Y
2
2
−
=
∞
For Z < Z
L:
For Z > ZL:
Everywhere:
22
FINITERATE CHEMISTRY NONPREMIXED
TURBULENT HYDROGEN COMBUSTION (1)
H
2
O
H
2
O
H
3
2
2
2
+
→
+
temperature.
2
H
H
2
→
O
2H
O
2H
2
2
2
→
in a thin reaction zone at the highest
I.
in thicker recombination
+
and/or
II.
zones on each side.
Partial equilibrium of
O
OH
O
H
2
+
→
+
occurs at the reaction zone of I.
3
2
/
1
2
2
/
1
1
H
K
K
K
K
=
O
H
1/2
O
H
H
2
2
/X
X
K
X
=
2
/
1
2
2
/
1
1
OH
K
K
K
=
1/2
O
1/2
H
OH
OH
2
2
X
X
K
X
=
O
H
O
H
O
O
2
2
2
/X
X
X
K
X
=
KO
= K1K3
23
THE PARTIALEQUILIBRIUM QUANTITIES KH, KOH
and KO
AS FUNCTIONS OF TEMPERATURE
24
FINITERATE CHEMISTRY NONPREMIXED
TURBULENT HYDROGEN COMBUSTION (2)
(
)
()
2
2
2
2
2
2
O
O
O
O
H
O
H
O
H
W
Y
L
2
W
Y
L
−
=
()
(
)
(
)
2
2
2
2
2
2
H
H
H
O
O
O
H
H
H
W
Y
L
2
W
Y
L
4
W
Y
L
−
=
()
(
)
(
)
[]
(
)
2
2
2
2
2
2
2
2
2
O
O
O
I
O
O
O
H
H
H
II
T
T
W
Y
L
Q
W
Y
L
3
W
Y
L
Q
h
L
−
−
=
()
(
)
5
4
O
O
O
H
H
H
2
2
2
2
2
2
W
Y
L
3
W
Y
L
ω
+
ω
=
−
,
0
Y
2
O
=
0
Y
2
H
=
At the stoichiometric
surface of
H
2
O
H
2
O
H
3
2
2
2
+
→
+
(Z
st=0.036, not 0.024)
25
CONCLUSIONS
Turbulent hydrogen combustion obeys the reacting NavierStokes equations with
fluidmechanical turbulence.
There are four broad categories of approaches to the description
of turbulent
combustion, with optimal approaches differing in different regimes of regime
diagrams.
Of the flamelet
and distributedreaction limiting regimes, most applications lie
closer to the flamelet
regime.
A formulation specific to hydrogen can be based on twostep reduced chemistry
and account for finiterate recombination in nonpremixed
turbulent combustion.
It is important to account for Schmidt and Prandtl
numbers different from unity in
treating turbulent hydrogen combustion.
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