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
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
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
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
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
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
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
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
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
Lift for half-cylinder, 3 m high, influenced by wind (air density)
hurricanetropical stormtropical depres-sion
Gale Force2
3
5uaFy