Civil Engineering Reference
In-Depth Information
hence
2
2
ψ ψ
x z
φ
and
ψ
are known respectively as potential and stream functions. If
φ
is given a particular constant value
then an equation of the form h
=
a constant can be derived (the equation of an equipotential line); if
ψ
is given a particular constant value then the equation derived is that of a stream or flow line.
Direct integration of these expressions in order to obtain a solution is possible for straightforward cases.
However, in general such integration cannot be easily carried out and a solution obtained by a graphical
method in which a flow net is drawn has been used by engineers for many decades. Nowadays, however,
much use is made of computer software to find the solution using numerical techniques such as the finite
difference and finite element methods. Nevertheless, the method for drawing a flow net by hand is given
in Section
2.10.3
for readers interested in learning the techniques involved.
∂
∂
+
∂
∂
0
=
2
2
2.10 Flow nets
The flow of water through a soil can be represented graphically by a flow net, a form of curvilinear net
made up of a set of flow lines intersected by a set of equipotential lines.
Flow lines
The paths which water particles follow in the course of seepage are known as flow lines. Water flows from
points of high to points of low head, and makes smooth curves when changing direction. Hence we can
draw, by hand or by computer, a series of smooth curves representing the paths followed by moving water
particles.
Equipotential lines
As the water moves along the flow line it experiences a continuous loss of head. If we can obtain the
head causing flow at points along a flow line, then by joining up points of equal potential we obtain a
second set of lines known as equipotential lines.
2.10.1 Hydraulic gradient
The potential drop between two adjacent equipotentials divided by the distance between them is known
as the hydraulic gradient. It attains a maximum along a path normal to the equipotentials and in isotropic
soil the flow follows the paths of the steepest gradients, so that flow lines cross equipotential lines at
right angles.
Figure
2.7
shows a typical flow net representing seepage through a soil beneath a dam. The flow is
assumed to be two dimensional, a condition that covers a large number of seepage problems encountered
in practice.
From Darcy's law q
=
Aki, so if we consider unit width of soil and if
Δ
q
=
the unit flow through a flow
channel (the space between adjacent flow lines), then:
∆
q b l k i bki
= × × × =
where b
=
distance between the two flow lines.
In Fig.
2.7
the figure ABCD is bounded by the same flow lines as figure A
1
B
1
C
1
D
1
and by the same
equipotentials as figure A
2
B
2
C
2
D
2
. For any figure in the net
Δ
q
=
bki
=
bk
Δ
h/l, where
Δ
h
=
head loss between the two equipotentials
