Geoscience Reference
In-Depth Information
where
f
x
,
P
,
f
y
,
P
, and
f
z
,
P
are the interpolation factors, defined as
f
x
,
P
=
l
wW
/
(
l
Pw
+
l
wW
)
,
f
y
,
P
=
l
sS
/(
l
Ps
+
l
sS
)
, and
f
z
,
P
=
l
bB
/(
l
Pb
+
l
bB
)
, with
l
wW
,
l
Pw
,
l
sS
,
l
Ps
,
l
bB
, and
l
Pb
being the lengths of linear segments
wW
,
Pw
,
sS
,
Ps
,
bB
, and
Pb
, respectively.
4.3.3 Finite volume method for multidimensional
problems on moving grids
ξ
m
,
τ
The transport equation in the moving, curvilinear coordinate system (
) can be
written in conservative form as
ρ
∂
∂τ
(ρ
φ)
+
∂
∂ξ
j
∂φ
∂ξ
m
j
m
J
J
u
m
ˆ
φ
−
J
α
α
=
JS
(4.152)
m
m
where
u
m
(
m
ˆ
=
1, 2 or 1, 2, 3) are the velocity components in the
(ξ
m
,
τ)
system,
defined in Eq. (4.70).
Compared with Eq. (4.129) or (4.140) on fixed grids, Eq. (4.152) has additional
terms related to the moving grid. In particular,
t
that is
related to the gridmoving velocity. These terms can be eliminated for a steady problem,
but they should be considered for an unsteady problem.
Because the grid is moving, it needs to be generated repeatedly. Like the dis-
cretization of governing equations, the grid generation can be treated explicitly or
implicitly. In the explicit treatment, the grid is generated before the solution of govern-
ing equations at every time step, whereas in the implicit treatment, the grid generation
is coupled with the solution of governing equations.
The control volume in the moving grid system can still be arranged as Fig. 4.21
or 4.22 at every time step. For a 2-D case, integrating Eq. (4.152) over the control
volume, moving the cross-derivative diffusion terms into the source term, and then
using the numerical schemes described in Section 4.3.1.1 to determine the convection
and normal-derivative diffusion terms on cell faces yields the following discretized
equation (Wu, 1996a):
u
m
include the term
ˆ
∂ξ
/∂
m
n
+
1
A
n
+
1
P
n
+
1
n
P
A
P
φ
n
P
ρ
φ
−
ρ
P
P
n
+
1
n
+
1
n
+
1
=
a
W
φ
+
a
E
φ
+
a
S
φ
W
E
S
τ
n
+
1
n
+
1
+
a
N
φ
−
a
P
φ
+
b
(4.153)
N
P
A
P
varies with time. The
coefficients in Eq. (4.153) are evaluated in the same way as for the fixed grid in
Section 4.3.2.
Because of grid movement, the control volume area
4.4 NUMERICAL SOLUTION OF NAVIER-STOKES EQUATIONS
For incompressible flows, the momentum (Navier-Stokes) equations link the velocity
to the pressure gradient, while the continuity equation is just an additional constraint
