Civil Engineering Reference
In-Depth Information
section on Stokes flow are applicable here; we now add the nonlinear terms aris-
ing from the convective inertia terms. From the first of Equation 8.50, we add a
term of the form
u
∂
d
A
[
N
]
d
A
u
v
∂
u
}
∂
[
N
]
∂
}
∂
[
N
]
∂
x
+
⇒
{
u
x
{
u
}+
[
N
]
{
v
y
{
u
}
∂
∂
y
A
(
e
)
A
(
e
)
(8.68)
and from the second equation of 8.50,
u
∂
d
A
[
N
]
d
A
v
v
∂
v
}
∂
[
N
]
∂
}
∂
[
N
]
∂
x
+
⇒
{
u
x
{
v
}+
[
N
]
{
v
y
{
v
}
∂
∂
y
A
(
e
)
A
(
e
)
(8.69)
As expressed, Equation 8.68 is not conformable to matrix multiplication, as in
being able to write the expression in the form
[
k
]
{
u
}
, and this is a direct result of
the nonlinearity of the equations. While a complete treatment of the nonlinear
equations governing viscous fluid flow is well beyond the scope of this text, we
discuss an iterative approximation for the problem.
Let us assume that for a particular two-dimensional geometry, we have
solved the Stokes (creeping) flow problem and have all the nodal velocities of the
Stokes flow finite element model available. For each element in the finite ele-
ment model, we denote the Stokes flow solution for the average velocity compo-
nents (evaluated at the centroid of each element) as
(
u
,
v
)
; then, we express the
approximation for the inertia terms (as exemplified by Equation 8.69) as
u
∂
d
A
d
A
u
v
∂
u
u
∂
[
N
]
∂
v
∂
[
N
]
∂
x
+
=
¯
+¯
{
u
}=
[
k
uv
]
{
u
}
∂
∂
y
x
y
A
(
e
)
A
(
e
)
(8.70)
Similarly, we find the
y
-momentum equation contribution to be
u
∂
d
A
d
A
v
v
∂
v
u
∂
[
N
]
∂
v
∂
[
N
]
∂
x
+
=
¯
+¯
{
v
}=
[
k
vu
]
{
v
}
∂
∂
y
x
y
A
(
e
)
A
(
e
)
(8.71)
Equations 8.70 and 8.71 refer to an individual element. The assembly procedures
are the same as discussed before; now we add additional terms to the stiffness
matrix as a result of inertia. These terms are readily identifiable in Equations 8.70
and 8.71. In the viscous inertia flow, the solution requires iteration to achieve sat-
isfactory results. The use of the Stokes flow velocities and pressures represent
only the first iteration (approximation). At each iteration, the newly computed
velocity components are used for the next iteration.
