- Source: G equation
In Combustion, G equation is a scalar
G
(
x
,
t
)
{\displaystyle G(\mathbf {x} ,t)}
field equation which describes the instantaneous flame position, introduced by Forman A. Williams in 1985 in the study of premixed turbulent combustion. The equation is derived based on the Level-set method. The equation was first studied by George H. Markstein, in a restrictive form for the burning velocity and not as a level set of a field.
Mathematical description
The G equation reads as
∂
G
∂
t
+
v
⋅
∇
G
=
S
T
|
∇
G
|
{\displaystyle {\frac {\partial G}{\partial t}}+\mathbf {v} \cdot \nabla G=S_{T}|\nabla G|}
where
v
{\displaystyle \mathbf {v} }
is the flow velocity field
S
T
{\displaystyle S_{T}}
is the local burning velocity with respect to the unburnt gas
The flame location is given by
G
(
x
,
t
)
=
G
o
{\displaystyle G(\mathbf {x} ,t)=G_{o}}
which can be defined arbitrarily such that
G
(
x
,
t
)
>
G
o
{\displaystyle G(\mathbf {x} ,t)>G_{o}}
is the region of burnt gas and
G
(
x
,
t
)
<
G
o
{\displaystyle G(\mathbf {x} ,t)
is the region of unburnt gas. The normal vector to the flame, pointing towards the burnt gas, is
n
=
∇
G
/
|
∇
G
|
{\displaystyle \mathbf {n} =\nabla G/|\nabla G|}
.
= Local burning velocity
=According to Matalon–Matkowsky–Clavin–Joulin theory, the burning velocity of the stretched flame, for small curvature and small strain, is given by
S
T
S
L
=
1
+
M
c
δ
L
∇
⋅
n
+
M
s
τ
L
n
n
:
∇
v
{\displaystyle {\frac {S_{T}}{S_{L}}}=1+{\mathcal {M}}_{c}\delta _{L}\nabla \cdot \mathbf {n} +{\mathcal {M}}_{s}\tau _{L}\mathbf {n} \mathbf {n} :\nabla \mathbf {v} }
where
S
L
{\displaystyle S_{L}}
is the burning velocity of unstretched flame with respect to the unburnt gas
M
c
{\displaystyle {\mathcal {M}}_{c}}
and
M
s
{\displaystyle {\mathcal {M}}_{s}}
are the two Markstein numbers, associated with the curvature term
∇
⋅
n
{\displaystyle \nabla \cdot \mathbf {n} }
and the term
n
n
:
∇
v
{\displaystyle \mathbf {n} \mathbf {n} :\nabla \mathbf {v} }
corresponding to flow strain imposed on the flame
δ
L
{\displaystyle \delta _{L}}
are the laminar burning speed and thickness of a planar flame
τ
L
=
δ
L
/
S
L
{\displaystyle \tau _{L}=\delta _{L}/S_{L}}
is the planar flame residence time.
A simple example - Slot burner
The G equation has an exact expression for a simple slot burner. Consider a two-dimensional planar slot burner of slot width
b
{\displaystyle b}
. The premixed reactant mixture is fed through the slot from the bottom with a constant velocity
v
=
(
0
,
U
)
{\displaystyle \mathbf {v} =(0,U)}
, where the coordinate
(
x
,
y
)
{\displaystyle (x,y)}
is chosen such that
x
=
0
{\displaystyle x=0}
lies at the center of the slot and
y
=
0
{\displaystyle y=0}
lies at the location of the mouth of the slot. When the mixture is ignited, a premixed flame develops from the mouth of the slot to a certain height
y
=
L
{\displaystyle y=L}
in the form of a two-dimensional wedge shape with a wedge angle
α
{\displaystyle \alpha }
. For simplicity, let us assume
S
T
=
S
L
{\displaystyle S_{T}=S_{L}}
, which is a good approximation except near the wedge corner where curvature effects will becomes important. In the steady case, the G equation reduces to
U
∂
G
∂
y
=
S
L
(
∂
G
∂
x
)
2
+
(
∂
G
∂
y
)
2
{\displaystyle U{\frac {\partial G}{\partial y}}=S_{L}{\sqrt {\left({\frac {\partial G}{\partial x}}\right)^{2}+\left({\frac {\partial G}{\partial y}}\right)^{2}}}}
If a separation of the form
G
(
x
,
y
)
=
y
+
f
(
x
)
{\displaystyle G(x,y)=y+f(x)}
is introduced, then the equation becomes
U
=
S
L
1
+
(
∂
f
∂
x
)
2
,
⇒
∂
f
∂
x
=
U
2
−
S
L
2
S
L
{\displaystyle U=S_{L}{\sqrt {1+\left({\frac {\partial f}{\partial x}}\right)^{2}}},\quad \Rightarrow \quad {\frac {\partial f}{\partial x}}={\frac {\sqrt {U^{2}-S_{L}^{2}}}{S_{L}}}}
which upon integration gives
f
(
x
)
=
(
U
2
−
S
L
2
)
1
/
2
S
L
|
x
|
+
C
,
⇒
G
(
x
,
y
)
=
(
U
2
−
S
L
2
)
1
/
2
S
L
|
x
|
+
y
+
C
{\displaystyle f(x)={\frac {\left(U^{2}-S_{L}^{2}\right)^{1/2}}{S_{L}}}|x|+C,\quad \Rightarrow \quad G(x,y)={\frac {\left(U^{2}-S_{L}^{2}\right)^{1/2}}{S_{L}}}|x|+y+C}
Without loss of generality choose the flame location to be at
G
(
x
,
y
)
=
G
o
=
0
{\displaystyle G(x,y)=G_{o}=0}
. Since the flame is attached to the mouth of the slot
|
x
|
=
b
/
2
,
y
=
0
{\displaystyle |x|=b/2,\ y=0}
, the boundary condition is
G
(
b
/
2
,
0
)
=
0
{\displaystyle G(b/2,0)=0}
, which can be used to evaluate the constant
C
{\displaystyle C}
. Thus the scalar field is
G
(
x
,
y
)
=
(
U
2
−
S
L
2
)
1
/
2
S
L
(
|
x
|
−
b
2
)
+
y
{\displaystyle G(x,y)={\frac {\left(U^{2}-S_{L}^{2}\right)^{1/2}}{S_{L}}}\left(|x|-{\frac {b}{2}}\right)+y}
At the flame tip, we have
x
=
0
,
y
=
L
,
G
=
0
{\displaystyle x=0,\ y=L,\ G=0}
, which enable us to determine the flame height
L
=
b
(
U
2
−
S
L
2
)
1
/
2
2
S
L
{\displaystyle L={\frac {b\left(U^{2}-S_{L}^{2}\right)^{1/2}}{2S_{L}}}}
and the flame angle
α
{\displaystyle \alpha }
,
tan
α
=
b
/
2
L
=
S
T
(
U
2
−
S
L
2
)
1
/
2
{\displaystyle \tan \alpha ={\frac {b/2}{L}}={\frac {S_{T}}{\left(U^{2}-S_{L}^{2}\right)^{1/2}}}}
Using the trigonometric identity
tan
2
α
=
sin
2
α
/
(
1
−
sin
2
α
)
{\displaystyle \tan ^{2}\alpha =\sin ^{2}\alpha /\left(1-\sin ^{2}\alpha \right)}
, we have
sin
α
=
S
L
U
.
{\displaystyle \sin \alpha ={\frac {S_{L}}{U}}.}
In fact, the above formula is often used to determine the planar burning speed
S
L
{\displaystyle S_{L}}
, by measuring the wedge angle.
References
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