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on the velocity field without directly linking to the pressure. Because of such a weak
linkage, the convergence and stability of a numerical solution of the Navier-Stokes
equations depend largely on how the pressure gradient and velocity are evaluated.
Storing the variables at the geometric center of the control volume coupled with the
use of linear interpolation for internodal variation usually leads to non-physical node-
to-node (checkerboard) oscillations. One approach for eliminating such oscillations is
to use the staggered grid, as adopted in Harlow and Welch's (1965) MAC (Marker
and Cell) method, Chorin's (1968) projection method, and Patankar and Spalding's
(1972) SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm. The
other approach is to use the momentum interpolation technique proposed by Rhie and
Chow (1983) based on the non-staggered grid. In addition, the stream function and
vorticity approach is also useful for solving the 2-D Navier-Stokes equations.
4.4.1 Primitive variables: MAC formulation
on staggered grid
The 2-D Navier-Stokes equations (2.34) and (2.35) with a constant flow density are
written in the Cartesian coordinate system as
∂
u
x
∂
+
∂
u
y
∂
=
0
(4.154)
x
y
∂τ
∂
u
x
∂
u
x
∂
u
x
∂
u
y
∂
u
x
∂
1
ρ
1
ρ
∂
p
1
ρ
∂τ
1
ρ
xx
xy
t
+
x
+
y
=
F
x
−
x
+
x
+
(4.155)
∂
∂
∂
y
∂
u
y
∂
u
x
∂
u
y
∂
u
y
∂
u
y
∂
∂
∂τ
∂τ
1
ρ
1
ρ
p
1
ρ
1
ρ
yx
yy
+
+
=
F
y
−
y
+
+
(4.156)
t
x
y
∂
∂
x
∂
y
The MAC method first proposed by Harlow and Welch (1965) solves the Navier-
Stokes equations on the staggered rectangular grid, which stores the variables
u
x
,
u
y
,
and
p
at different grid points, as shown in Fig. 4.24. The continuity equation (4.154)
is discretized as
u
n
+
1
x
,
i
u
n
+
1
x
,
i
u
n
+
1
y
,
i
,
j
u
n
+
1
y
,
i
,
j
−
−
+
1
/
2,
j
−
1
/
2,
j
+
1
/
2
−
1
/
2
D
n
+
1
i
,
j
=
+
=
0
(4.157)
x
y
where
D
n
+
1
i
,
j
is the dilatation of the cell (
i
,
j
).
The momentum equations (4.155) and (4.156) are discretized as
F
i
+
1
/
2,
j
−
t
ρ
u
n
+
1
x
,
i
u
x
,
i
+
1
/
2,
j
+
p
n
+
1
i
p
n
+
1
i
,
j
=
x
(
−
)
(4.158)
+
1
/
2,
j
+
1,
j
G
i
,
j
+
1
/
2
−
t
ρ
u
n
+
1
y
,
i
,
j
p
n
+
1
i
,
j
p
n
+
1
i
,
j
u
y
,
i
,
j
+
1
/
2
+
=
y
(
−
)
(4.159)
+
1
/
2
+
1
where
F
i
+
1
/
2,
j
and
G
i
,
j
+
1
/
2
include the convection and diffusion terms in the momen-
tum equations (4.155) and (4.156) discretized by the finite difference schemes
introduced in Section 4.2.
