Environmental Engineering Reference
In-Depth Information
11.2 The Euler Equations and the Bernoulli Theorem
There are situations in which the friction can be neglected. For frictionless
ideal
fluids
, i.e. fluids with zero viscosity, the
Euler
6
equations
are written in modern
notation as:
r
@
@t
v
þ r
v
r
ð
Þ
v
r
f
þrp ¼
0
(11.5)
which were published by Euler in 1750.
Utilizing potential theory for fluid mechanics is the classical approach (“classi-
cal hydrodynamics” according to Prandtl and Tietjens
1957
), developed already by
the Bernoulli's
7
and Euler. The works of Euler in Berlin and St. Petersburg not only
mark the completion of classical fluid mechanics (Szab
´
1987
), but also the origin
of an approach by which natural phenomena in the laboratory or in the field are
described by differential equations and their solutions.
The general Navier-Stokes equations have few analytical solutions (for an
example see Sidebar 11.1). Thus, they usually have to be solved by special software
packages utilizing numerical methods, such as finite differences, finite elements, or
finite volumes. The use of numerical methods by applying the
pdepe
command was
already described in Part I of the topic. However,
pdepe
can be used for 1D
problems only. For higher dimensional problems the MATLAB
partial differen-
tial toolbox has to be applied, which is not described here. In the following chapters,
it is the aim to examine the use of MATLAB
®
for potential flow.
Potential flow is an umbrella term for a technique to obtain analytical solutions.
Analytical solutions are explicit formulae for the unknown variables, sometimes
also referred to as
closed form solutions
(Narasimhan
1998
). If a flow field is
irrotational, i.e. if the condition
8
®
6
Leonard Euler (1707-1783), Swiss mathematician.
7
Johann Bernoulli (1667-1748), Daniel Bernoulli (1700-1782), Swiss mathematicians.
8
'
(
x
1
,
y
1
,
z
1
)
T
' denotes the cross product for vectors, which for vectors r
1
¼
and r
2
¼
(
x
2
,
y
2
,
z
2
)
T
is defined by:
T
r
1
r
2
¼ x
2
y
3
x
3
y
2
; x
3
y
1
x
1
y
3
; x
1
y
2
x
2
y
1
ð
Þ
(
v
x
,v
y
,v
z
)
T
as second, one obtains:
If the nabla-operator is used as first vector and v
¼
T
@y
v
z
@
@
@z
v
y
;
@
@z
v
x
@
@x
v
z
;
@
@x
v
y
@
r
v
¼
@y
v
x
For the special case of flow in the 2D (
x
,
y
)-plane follows the equation:
T
0
;
0
;
@
@x
v
y
@
r
v
¼
@y
v
x
