Uniform Flow Over a CylinderCombine velocity potential and stream function of uniform flow and doublet

Uniform Flow

xu

yu

Doublet

22 yxx

22 yxy

22 yxxux

2let ua22

2

yxxuaux

22

2

1yx

aux

UniformUniform

a2 = 0.5

Q~

a2 = 0.5

a2 = 0.25

a2 = 0.75

a2 = 3

22 yxyuy

22

2

22

2

1yx

auyyxyuauy

Note that ψ = 0 on the curve x2 + y2 = a2 ;

therefore, x2 + y2 = a2 is a streamline

a2 = 1Stagnation points:

where u = 0

a2 = 1

a2 = 1

22

2

1yx

auy

22

2

1yx

aux

a2 = 1

22

2

1yx

auy

22

2

1yx

aux

Potential flow over a sphere

Uniform Flow Over a Cylinder with Clockwise (negative) Circulation

Combine velocity potential and stream function of uniform flow over cylinderand clockwise vortex

xy

yxaux 1

22

2

tan2

1

2222

2

ln2

1 yxyx

auy

22

22

2

ln2

1

yx

yxauy

a2 = 1

Г = 3π

22

22

2

ln2

1

yx

yxauy

a2 = 1

Г = 3π

22

22

2

ln2

1

yx

yxauy

a2 = 1

Г = 3π

Stagnation points?

2

1costan2

1 2

21

22

2

raur

xy

yxaux

rrauryx

yxauy ln

21sinln

21 2

222

22

2

Location of stagnation points:

2

1cos 2

2

raur

2

2

3

2

2

2

1coscos21cosrau

rura

rau

rur

raur @0

2

1cos 2

2

raur

21sin 2

2

rauru

rarauru

;

21sin0 2

2

au

2

sin2 ua

4sin

Г < 4πua

8660.0sin ]120,60[ oo

U

Г = 4πua

0.1sin o90

U

Г > 4πua

0.1sin