Environmental Engineering Reference
In-Depth Information
Chapter 15
Streamfunction and Complex Potential
15.1 Streamfunction
The
streamfunction
is another mathematical construct that is of high importance for
models using analytical solutions. Together with the potential the streamfunction
enables the visualization of flow patterns that can hardly be produced by other
methods. In 2D the streamfunction
C
is defined by the equations:
q
x
¼
@C
@y
q
y
¼
@C
@x
(15.1)
The derivatives of the streamfunction are the components of the discharge
vector. In contrast to the potential, the negative of the
y
-derivative delivers the
x
-component of discharge, and the x-derivative delivers the
y
-component of dis-
charge. From the defining equations follows that the streamfunction also fulfils the
potential equation:
2
2
@
@x
2
þ
@
C
@y
2
¼
@
C
@x
@C
@x
þ
@
@y
@C
@y
¼
@
@x
q
y
@
@y
q
x
(15.2)
2
h
2
h
@y@x
¼
@
@x
K
@
h
@y
@
@y
K
@
h
@x
¼ K
@
@x@y
@
¼
0
Streamlines are characterized by the property that the tangentials are perpendic-
ular to the contours of the potential or the head. The mathematical proof utilizes the
connection between streamfunction and discharge vector (
15.1
) as well as the
connection between potential and discharge vector (14.3):
q
x
¼
@'
@x
q
y
¼
@'
@y
(15.3)
