Civil Engineering Reference
In-Depth Information
Example solutions to steady state problems described by (2.122) are given in Chapter 7.
Three-dimensional problems are also solved in Chapter 12.
2.17.2 Transient state
Transient conditions must be analysed in many physical situations, for example in the case
of Terzaghi “consolidation” in soil mechanics or transient heat conduction. The governing
consolidation diffusion equation for excess pore pressure
u
w
in 2D, takes the form
2
2
c
x
∂
u
w
∂x
+
c
y
∂
u
w
∂y
∂u
w
∂t
=
(2.130)
2
2
where
c
x
and
c
y
are the coefficients of consolidation in the
x
-and
y
-directions. Discretisa-
tion of the left hand side of (2.130) clearly follows that of (2.122) while the time derivative
will be associated with a matrix of the “mass matrix” type from (2.71), without the multiple
ρ
. Hence, the discretised system is,
[
m
m
]
d
u
w
d
t
[
k
c
]
{
u
w
} +
= {
0
}
(2.131)
are the nodal values of
u
w
.
This set of first order, ordinary differential equations can be solved by many methods,
the simplest of which discretise the time derivative by finite differences. The algorithms
are described in Chapter 3 with example solutions in Chapters 8 and 12.
where
{
u
w
}
2.17.3 Advection
If pollutants, sediments, tracers, etc, are transported by a laminar flow system they are at
the same time translated or “advected” by the flow and diffused within it. The governing
differential equation for the two-dimensional case is (Smith
et al
., 1973),
2
2
c
x
∂
φ
∂x
+
c
y
∂
φ
∂y
−
u
∂φ
∂x
−
v
∂φ
∂φ
∂t
=
(2.132)
2
2
∂y
where
φ
can be interpreted as a “concentration” and
u
and
v
are the fluid velocity compo-
nents in the
x
-and
y
-directions (compare equation 2.117).
The extra advection terms
−
v∂φ/∂y
compared with (2.130) lead, as
shown in Table 2.1, to unsymmetric components of the resulting matrix of the type,
−
uN
i
∂N
j
∂x
−
u∂φ/∂x
and
d
x
d
y
−
vN
i
∂N
j
∂y
(2.133)
which must be added to the symmetric, diffusion components given in (2.125). When this
has been done, equilibrium equations like (2.123) or transient equations like (2.131) are
regained.
Mathematically, equation (2.132) is a differential equation which is not self-adjoint
(Berg, 1962), due to the presence of the first-order spatial derivatives. From a finite element
