Download - Pressure, Drag and Lift for Uniform Flow Over a Cylinder

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Page 1: Pressure, Drag and Lift for Uniform Flow Over a Cylinder

Pressure, Drag and Lift for Uniform Flow Over a Cylinder

22

2

1yx

aux

2

2

1cosr

aur

22

2

1yx

auy

2

2

1sinr

aur

a2 = 1

Page 2: Pressure, Drag and Lift for Uniform Flow Over a Cylinder

Pressure, Drag and Lift for Uniform Flow Over a Cylinder

2

2

1sinr

aur

2

2

1cos1

r

au

rur

2

2

1sinr

au

ru

Along the cylinder, r = a, the velocity components become:

0ru

sin2uu

uθ is maximum at θ = π/2 and 3 π /2; zero at θ = 0 and θ = π

Page 3: Pressure, Drag and Lift for Uniform Flow Over a Cylinder

The pressure distribution can be obtained using Bernoulli’s equation:

220 2

1

2

1cc upup

cylinder along

2

upstream

20 2

1

2

1cc upup

sin2uu

022

2

1ppuu cc

0222 sin4

2

1ppuu c

022 sin41

2

1ppu c

2

02

21

sin41u

ppc

dimensionless pressure coefficient Cp

Page 4: Pressure, Drag and Lift for Uniform Flow Over a Cylinder

The drag on the cylinder may be calculated through integration of the pressure over the cylinder surface:

dapF cx 2

0

cos

daup

2

0

220 cossin41

2

1

The drag on the cylinder acts parallel to the flow.

dapF cy 2

0

sin

daup

2

0

220 sinsin41

2

1

The lift is perpendicular to the flow:

Fx

Fy

Page 5: Pressure, Drag and Lift for Uniform Flow Over a Cylinder

2

1cos 2

2

r

aur r

r

aur ln

21sin 2

2

2

2

1cosr

au

rur

rr

auu

21sin 2

2

Along the cylinder, r = a, the velocity components become:

0ru

auu

2

sin2

Pressure, Drag and Lift for Uniform Flow Over a Rotating Cylinder

Page 6: Pressure, Drag and Lift for Uniform Flow Over a Cylinder

The pressure distribution can be obtained using Bernoulli’s equation:

cylinder along

2

upstream

20 2

1

2

1cc upup

auu

2

sin2

2

20 2

sin22

1

2

1

aupup c

022

2222

4

sin2sin4

2

1pp

aa

uuu c

0222

222

4

sin2sin41

2

1pp

uaauu c

222

22

2

0

4

sin2sin41

21 uaauu

ppc

dimensionless pressure coefficient Cp

Page 7: Pressure, Drag and Lift for Uniform Flow Over a Cylinder

The drag and lift can be obtained by integrating the pressure over the cylinder surface pc :

uFF yx 0

Still no drag for a rotating cylinder

There is lift proportional to density, upstream velocity, and strength of vortex-- Kutta – Jukowski law

Lifting effect for rotating bodies in a free stream is called Magnus effect

Page 8: Pressure, Drag and Lift for Uniform Flow Over a Cylinder

Example of Pressure, Drag and Lift for Uniform Flow Over a Cylinder

3 m

u = 20 m/s

022 sin41

2

1ppu c

220 sin41

2

1 uppc

The drag on the cylinder may be calculated through integration of the pressure over half the cylinder surface, from 0 to π.

That’ll be with the outside pressure, inside pressure p0 should also be considered:

dappF cx 0

0 cos

dapupFx

0

022

0 cossin412

1

0 0

222 cossin2cos2

duadua

Fx

Page 9: Pressure, Drag and Lift for Uniform Flow Over a Cylinder

0 0

222 cossin2cos2

duadua

Fx

322 sin32

sin2

0

uaua

xF

0xF

3 m

u = 20 m/s

Page 10: Pressure, Drag and Lift for Uniform Flow Over a Cylinder

3 m

u = 20 m/s

dappF cy 0

0 sin

The lift on the object may be calculated through integration of the pressure over half the cylinder surface, from 0 to π.

dapupFy

0

022

0 sinsin412

1

0 0

32 sin2sin2

1dduaFy

3cos31

cos2cos21

0

2uaFy

3

112

2

1

3

112

2

12uaFy

6

5

6

52uaFy

Page 11: Pressure, Drag and Lift for Uniform Flow Over a Cylinder

6

5

6

52uaFy

3

52uaFy 2

3

5uaFy

3 m

u = 20 m/s

2400202.13 yFua mPas

kg 2

Page 12: Pressure, Drag and Lift for Uniform Flow Over a Cylinder

Lift for half-cylinder, 3 m high, influenced by wind (air density)

hurricanetropical stormtropical depres-sion

Gale Force2

3

5uaFy

Page 13: Pressure, Drag and Lift for Uniform Flow Over a Cylinder

Lift for half-cylinder, 1 m high, influenced by flow (water density)

2

3

5uaFy