Biomedical Engineering Reference
In-Depth Information
volume grid and it can be demonstrated that the discrete values of the velocity com-
ponent,
u
, from the
x
-momentum equation are evaluated and stored at the east,
e
,
and west,
w
, faces of the control volume.
By evaluating the other velocity components using the
y
-momentum and
z
-
momentum equations on the rest of the control volume faces, these velocities allow
a straightforward evaluation of the mass fluxes that are used in the pressure correc-
tion equation. This arrangement therefore provides a strong coupling between the
velocities and pressure, which helps to avoid some types of convergence problems
and oscillations in the pressure and velocity fields. The staggered grid is typically the
simplest strategy for incompressible flow calculations and it is used in this chapter
to demonstrate the pressure-velocity coupling.
Let us establish the staggered grid where scalar variables (e.g. pressure) are stored
at nodal points while the velocity components are stored at the control volume faces.
In the staggered arrangement the central node for pressure and other scalar variables
φ
are stored at nodal point
P
(
I, J
). This control volume is used for the pressure term
and is sometimes called the scalar control volume since other scalar variables are also
stored here. The velocity vector
u
, however is stored at nodal point
u(i, J)
and
v
is
stored at point
v
(
I, j
). This means that the vectors and scalars are stored in staggered
locations and there are three partially overlapping control volumes. The example
shown in Fig.
7.23
is a backward staggered grid, although a forward staggered grid
may also be used.
SIMPLE algorithm
The SIMPLE scheme is essentially a guess-and-correct pro-
cedure for the calculation of pressure through the solution of a pressure correction
equation. The method is illustrated by considering a two-dimensional steady laminar
flow problem in a structured grid with staggered arrangement as shown in Fig.
7.23
.
Firstly we discretise the momentum equations for a two-dimensional incompress-
ible flow using FV on a staggered mesh. The pressure gradient term for the
u
- and
v
- control volumes are
∂p
∂x
=
p
P
−
p
W
x
p
I
,
J
−
p
I
−
1,
J
x
∂p
∂y
=
p
P
−
p
S
p
I
−
1,
J
−
p
I
,
J
=
;
=
y
y
where
x
and
y
are the length and height of the control volume respectively.
Discretising the momentum equations we get
a
i
,
J
u
i
,
J
=
a
i
,
j
u
i
,
j
+
a
i
,
j
+
1
u
i
,
j
+
1
+
a
I
−
1,
J
u
I
−
1,
J
+
a
I
,
J
u
I
,
J
p
I
,
J
−
p
I
−
1,
J
x
b
u
V
−
V
+
a
I
,
j
v
I
,
j
a
I
,
J
−
1
v
I
,
J
−
1
+
a
I
,
J
v
I
,
J
+
a
i
,
j
v
i
,
j
+
a
i
+
1,
j
v
i
+
1,
j
=
p
I
−
1,
J
−
p
I
,
J
b
v
V
−
+
V
y
where
b
u
and
b
v
are source terms that may include gravity, and other effects. We
retain the subscript notation for the pressure term using the compass point notation
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