4
Twodimensional flows
If the flow is in the plane, and there are only two independent variables
x,y
, the flow prob
lem is much simplified. Moreover, we will see that some of the physics is fundamentally
different from three dimensions. One fact we know of already is that no vorticity is created
in two dimensions,
ω
is simply convected with the flow. Of course, real flows are never
truly twodimensional; however if the geometry extends very far in one direction (think
of the wing of an aeroplane), a twodimensional description is a good approximation.
4.1
Flow past a cylinder
Find the flow around a stationary cylinder in a steady stream
U
=
U
ˆ
x
. This is very
closely analogous to the flow around a sphere; the effect of the sphere is modelled by a
(twodimensional) dipole, which is the derivative of a source, whose velocity field behaves
like
u
r
∝
1
/r
(section 1.10). Thus the potential is
φ
= ln
r
, and a dipole in the
μ
direction
has potential
φ
= (
μ
·
∇
) ln
r
=
μ
·
x
r
2
,
The ansatz for the velocity potential is thus
φ
=
Ux
+
μ
·
x
r
2
,
and so
u
=
U
ˆ
x
+
μ
r
2
−
2(
μ
·
x
)
x
r
4
.
Now the boundary condition for
r
=
R
is
u
·
n
=
u
·
x
/r
= 0, and from the velocity field
we find
u
·
n
=
U
x
r
−
μ
·
x
r
3
.
Thus the boundary condition is satisfied if we choose
μ
=
R
2
U
.
Thus the final answer for the potential is
φ
=
U
bracketleftbigg
x
+
R
2
x
r
2
bracketrightbigg
,
(35)
and for the velocity field
u
=
R
2
r
2
[
U
−
2
n
(
U
·
n
)] +
U
.
(36)
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Now we plug this into Bernoulli’s equation to compute the pressure:
u
2
2
+
p
ρ
=
U
2
2
+
p
atm
ρ
.
On the surface
u
2
vextendsingle
vextendsingle
r
=
R
= (2
U
−
2
n
(
U
·
n
))
2
= 4
U
2
−
4(
U
·
n
)
2
= 4
U
2
(1
−
cos
2
θ
) = 4
U
2
sin
2
θ,
so in other words
p
=
p
atm
+
ρU
2
2
(1
−
4 sin
2
θ
)
.
The pressure is once more
symmetric
about
θ
=
π/
2, i.e.
about the midsection of the
cylinder. It follows that the total force on the cylinder is again zero.
4.2
Nonuniqueness of the potential
One particularly important aspect of twodimensional flow is that any solid body placed
in the flow domain will create a domain that is no longer simply connected.
Defn
: A closed curve
C
is reducible
in a domain
D
if it can be shrunk to a point
without ever leaving
D
. If every closed curve is reducible then
D
is simply connected
.
E.g.
(i) if
D
is the interior of a circle, then it is simply connected.
(ii) if
D
is the
exterior of a circle then it is not
simply connected.
D
i
ii
D
Example
: Let
D
be the domain
a<r<b,
0
<θ<
2
π
in cylindrical polars.
D
is not
simply connected.
We now place a point
vortex
(introduced at the beginning of chapter 3) at the centre,
which creates a flow
u
= (0
,
Γ
/
2
πr,
0)
.
The strength of the vortex is measured in terms of its circulation Γ.
Since the flow is
irrotational, there exists a velocity potential s.t.
u
r
=
∂φ
∂r
= 0
, u
θ
=
1
r
∂φ
∂θ
= Γ
/
(2
πr
)
.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '11
 Eggers
 Fluid Dynamics, Cartesian Coordinate System, complex potential, xxxxxxxxxxxxxxxxx

Click to edit the document details