Recap of the terminology
Steady:
q = q(r ), i.e. no time dependence
Two-dimensional:
q = u(x, y ) i + v (x, y ) j.
Incompressibility:
∇ · q = 0.
Stream function ψ:
q = (∇ψ) × k.
Stagnation points:
q(r ) = 0.
Vorticity:
ω = ∇ × q.
Irrotational flow:
ω = 0.
With irrotational flow there exists a velocity potential φ such that
q = ∇φ = (∇ψ) × k.
with
∇2 ψ = ∇2 φ = 0.
MA2741 Week 21, Page 1 of 12
The velocity in Cartesian and polars coordinates
In Cartesian’s ψ = ψ(x, y ) and
q=
∂ψ
∂ψ
i−
j
∂y
∂x
and in polars ψ = ψ(r , θ) and
q=
∂ψ
1 ∂ψ
er −
e .
r ∂θ
∂r θ
Curves of the form ψ =const are the streamlines of the flow.
MA2741 Week 21, Page 2 of 12
Streamlines for a simple shear flow
1
ψ = βy 2
2
and q = βy i.
2
1.5
1
y
0.5
0
−0.5
−1
−1.5
−2
−4
−2
0
x
2
4
MA2741 Week 21, Page 3 of 12
Streamlines for a line source at r = 0
ψ = Aθ
and q =
A
e .
r r
4
3
2
1
0
−1
−2
−3
−4
−4
−2
0
2
4
MA2741 Week 21, Page 4 of 12
Streamlines for a line vortex at r = 0
Γ r ψ(r ) = − ln
2π
a
and q =
Γ
2π
1
e .
r θ
4
3
2
1
0
−1
−2
−3
−4
−4
−2
0
2
4
MA2741 Week 21, Page 5 of 12
A source and sink close together
If Ah = µ is constant and h → 0 then we can show that
ψ(r ) = Ah
(θ1 − θ2 )
sin θ
→µ
.
h
r
r =xi +yj
θ2 θ θ 1
−h/2 0 h/2
Sink r 2 Source r 1
x-axis
MA2741 Week 21, Page 6 of 12
Streamlines for a dipole in the i direction
4
3
2
1
0
−1
−2
−3
−4
−4
−2
0
2
4
MA2741 Week 21, Page 7 of 12
Streamlines for a dipole in the −i direction
ψ = −µ sinr θ
This is the same as before with the arrows reversed.
MA2741 Week 21, Page 8 of 12
Streamlines for uniform flow in the i direction
ψ = Ur sin θ
MA2741 Week 21, Page 9 of 12
Streamlines for flow round a cylinder
4
3
2
1
2
ψ = U r − ar sin θ
0
A
B
−1
−2
−3
−4
−4
−2
0
2
4
MA2741 Week 21, Page 10 of 12
Flow around a cylinder – Reynold’s number, Re=1
When we have some viscosity we get a similar pattern.
Red = 1; t = 20
3
2
y
1
0
−1
−2
−3
−2
0
2
4
6
8
10
x
MA2741 Week 21, Page 11 of 12
Flow around a cylinder – Re=10
When the viscous effects increase flow differs from the no viscosity
case close to the cylinder.
Red = 10; t = 20
3
2
y
1
0
−1
−2
−3
−2
0
2
4
6
8
10
x
MA2741 Week 21, Page 12 of 12