Geoscience Reference
In-Depth Information
divergence-free field. Write the Navier-Stokes equations as
∂
u
1
ρ
∇
+
(
u
·∇
)
u
=
F
−
p
+
ν
u
(4.162)
∂
t
∇·
u
=
0
(4.163)
where
u
is the velocity vector;
F
is the external force;
∇
is the divergence or gradient
operator, defined as
z
, with
i
x
,
i
y
, and
i
z
being unit vectors
in the
x
-,
y
-, and
z
-axes in the Cartesian coordinate system; and
∇=
i
x
∂/∂
x
+
i
y
∂/∂
y
+
i
z
∂/∂
is the Laplace
z
2
.
The projection method consists of two steps in time. The first step computes the
intermediate velocity field
u
∗
by omitting the pressure term from the momentum
equation:
2
x
2
2
y
2
2
operator, defined as
=
∂
/∂
+
∂
/∂
+
∂
/∂
u
∗
−
u
n
u
n
u
n
u
∗
+
(
·∇
)
=
+
ν
F
(4.164)
t
The second step projects
u
∗
to the space of divergence-free field to obtain
u
n
+
1
:
⎨
u
∗
−
t
ρ
∇
u
n
+
1
p
n
+
1
=
(4.165)
⎩
u
n
+
1
∇·
=
0
Substituting the first equation into the second equation in (4.165) yields a Poisson
equation for the pressure:
t
ρ
p
n
+
1
u
∗
=∇·
(4.166)
To solve Eq. (4.166), the following boundary condition is often applied:
p
n
+
1
∂
∂
=
0
(4.167)
n
where
n
denotes the direction normal to the boundary.
In the projection method, Eqs. (4.162) and (4.163) are usually discretized on a
staggered grid (such as theMAC grid in Fig. 4.24). The convection terms are commonly
discretized using an upwind scheme, and the diffusion terms can be discretized using
the central difference scheme.
Various variants of the projection method have been proposed in the literature to
solve the shallow water equations and the Navier-Stokes equations. Some of them are
introduced in Chapters 6 and 7.
