Geoscience Reference
In-Depth Information
)(
l
=
E
,
W
,
N
,
S
a
l
/
(
l
=
E
,
W
,
N
,
S
a
l
/
where
Q
i
,
w
=
Q
i
,
w
/
[
a
p
)
a
P
)
1
−
α
(
1
−
f
x
,
P
−
α
u
f
x
,
P
]
,
u
W
P
)(
l
=
E
,
W
,
N
,
S
a
l
/
(
l
=
E
,
W
,
N
,
S
a
l
/
and
Q
i
,
s
=
Q
i
,
s
/
[
a
P
)
a
P
)
.
The fluxes at cell faces are still determined by Eqs. (4.196) and (4.197), and the
pressure correction equation is still written as Eq. (4.199), with
Q
i
,
w
and
Q
i
,
s
replaced
by
Q
i
,
w
and
Q
i
,
s
.
1
−
α
(
1
−
f
y
,
P
−
α
u
f
y
,
P
]
u
S
P
4.4.5 Stream function and vorticity formulation
In the 2-D case, it is possible to avoid explicit appearance of the pressure in the Navier-
Stokes equations by introducing stream function and vorticity as dependent variables.
The voticity
is defined as
=
∂
u
y
−
∂
u
x
∂
(4.205)
∂
x
y
Cross-differentiating the
u
x
and
u
y
momentum equations (4.155) and (4.156)
with respect to
y
and
x
and then subtracting them yields the transport equation of
vorticity:
2
2
+
∂(
u
y
)
∂
∂
t
+
∂(
u
x
)
∂
+
∂
=
ν
(4.206)
∂
x
∂
y
∂
x
2
∂
y
2
Eq. (4.206) is for laminar flows. A similar equation can be derived for turbu-
lent flows.
The stream function
ψ
is defined by
=
∂ψ
∂
=−
∂ψ
∂
u
x
y
,
u
y
(4.207)
x
Substituting Eq. (4.207) into the continuity equation (4.154) leads to the following
Poisson equation for stream function:
2
2
∂
ψ
+
∂
ψ
=−
(4.208)
∂
x
2
∂
y
2
Eqs. (4.206) and (4.208) replace the continuity and Navier-Stokes equations
(4.154)-(4.156) and constitute the new governing equations. They can be solved
conveniently using the finite difference method,
finite volume method, or finite
element method.
Since the pressure does not appear in Eqs. (4.206) and (4.208) and the continuity
equation (4.154) is automatically satisfied, the stream function and vorticity method
is convenient in the 2-D case. However, extension of this method to the 3-D case is
not straightforward and loses the merits of the 2-D version.
