Environmental Engineering Reference
In-Depth Information
Chapter 14
Potential and Flow Visualization
14.1 Definition and First Examples
Potential and streamfunction are mathematical functions that cannot be observed
directly in the real world, but which turn out to be extremely powerful concerning
the calculation and visualization of 2D flow fields. There are applications for all
types of fluids, for free flow of gases and liquids, as well as for porous media flow.
Electro- and magnetodynamics are other scientific fields where potential theory is
applied extensively.
The notation potential refers to a function
',
from which a flow field is derived
by the gradient of
'. '
is a velocity potential if:
v
¼r'
(14.1)
For steady incompressible fluids (see Chap. 2), for which the continuity equation
r
0 is valid, follows the
potential equation
or
Laplace
1
equation
2
:
¼
v
2
2
in 2D:
@
'ðx; yÞ
@x
2
þ
@
'ðx; yÞ
@y
2
2
r
' ¼
0
;
¼
0
2
2
2
in 3D:
@
'ðx; y; zÞ
@x
2
þ
@
'ðx; y; zÞ
@y
2
þ
@
'ðx; y; zÞ
@z
2
¼
0
(14.2)
The short form, using the
-operator, is valid for 2D and 3D cases. In fluid
∇
has the physical unit of [m
3
/s]. The name potential is
connected with the property that at each location of the model region the flux or
velocity vector can be derived from the gradient of the potential:
dynamics the potential
'
1
Pierre-Simon Laplace (1749-1827), French mathematician and astronomer.
2
The formulation
D
' ¼
0 can be found frequently, which makes sense, as the Laplace operator
is formally defined as
D :
¼rr
.
D
