M E 320 Professor John M. Cimbala Lecture 36
Today, we will:
•
Do another example of superposition of irrotational flows – flow over a circular cylinder
•
Start discussing the last approximation of Chapter 10: The Boundary Layer Approx.
b. Example of superposition: Flow over a
circular cylinder
Given: Superpose a uniform stream of
velocity V
∞
and a doublet of strength K at the
origin.
To do: Plot streamlines, and discuss the flow
that results from this superposition.
Solution:
• We simply add up the stream functions for
the two building block flows:
freestream doublet
sin
Vy K r
θ
ψψ ψ
∞
=+=−
.
• But we know that sinyr
θ
= , thus,
sin
sinVr K r
θ
ψθ
∞
=−
.
• For “convenience”, and with hindsight, we
choose to set
ψ
= 0 at r = a.
[It turns out that radius a is a special radius that
becomes the radius of the circle.]
• Set r = a in our equation for the stream
function:
2
sin
0 sin Va K K Va
a
θ
θ
∞∞
=− →=
.
• Then our final expression for
ψ
becomes
2
sin a
Vr
r
ψθ
∞
⎛⎞
=−
⎜⎟
⎝⎠
.
• Plot streamlines: [we plot nondimensionally,
setting x*=x/a and y*=y/a]
• From our equation for
ψ
above, we can
calculate the velocity field from the definition
of
ψ
, i.e.,
1
r
uu
rr
θ
ψ
ψ
θ
∂∂
==−
∂∂
. See text for details. On the cylinder (r = a),
0 2 sin
r
uuV
θ
θ
∞
==−
• We can also define the pressure coefficient,
2
22
1
2
1
p
PP V
CVV
ρ
∞
∞
∞
−
==−
.
• On the cylinder, it turns out that
2
14sin
p
C
β
=−
, where
β
is the angle from the nose.
