Biomedical Engineering Reference
In-Depth Information
maximum difference
φ
(
k
j
φ
(
k
+
1)
j
is chosen as the condition for the termination
process, the accuracy of the solution depends on the targeted number of significant
figures you wish to obtain for the temperatures. The smaller the acceptable error, the
higher the number of iterations but this will achieve greater accuracy.
−
7.3.4
Pressure-velocity Coupling
The discretisation methods that have been presented are for a single equation. If we
consider the governing equations for a 2D incompressible steady laminar flow we
have
∂u
∂x
+
∂v
∂y
0
=
continuity
ν
∂
2
u
ν
∂
2
u
∂y
2
diffusion
∂u
∂t
u
∂u
v
∂u
∂y
convection
∂p
∂
x
pressure gradient
1
ρ
+
∂x
+
=−
+
∂x
2
+
local acceleration
F
B
body force
+
x
-
momentum
∂v
∂t
u
∂v
v
∂v
∂y
convection
∂p
∂
y
pressure gradient
1
ρ
ν
∂
2
v
ν
∂
2
v
∂y
2
diffusion
+
∂x
+
=−
+
∂x
2
+
local acceleration
F
B
body force
+
y
-
momentum
where there are three unknowns,
u
,
v
and
p
with the other variables,
ν
,
ρ
determined
from the material property of the fluid. The velocity field
u
and
v
can be obtained
after discretisation of its transport equation (i.e. the
x
- and
y
- momentum equations
above) if we know the pressure field, (because of the pressure gradient term in the mo-
mentum equation). However for an incompressible flow assumption, the governing
equations lack an independent equation for pressure. With the additional continuity
equation, this system of equations is self-contained; there are three equations for
three dependents
u
,
v
and
p
, however we cannot use the continuity equation directly
to obtain
p
. In order to link the pressure with the velocity for an incompressible flow,
a correction to the pressure field is made to ensure that the continuity and momentum
equations are satisfied.
Within this section, we describe the basic philosophy behind one of the most
popular schemes of pressure-velocity coupling for an incompressible flow. It belongs
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