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Lift From Rotating Cylinder Formula
Lift From Rotating Cylinder Formula
When a cylinder is placed transversely to a relative airflow of velocity v∞, the velocity at a point on the surface of the cylinder is 2v∞ sin θ. Since the flow is symmetrical, however, no lift is produced. (See Fig. 1.)
Fig. 1 - Flow Over a Cylinder
If the cylinder with radius r and length L rotates at angular velocity ω (in units of rad/sec) while moving with a relative velocity v∞ through the air, the Kutta- Joukowsky result (theorem) can be used to calculate the lift per unit length of cylinder.a This is also known as the Magnus effect.
Eq. 1, SI Units
FL / L = ρ v∞ Γ
Eq. 2, U.S.
FL / L = ρ v∞ Γ / gc
Eq. 3
Γ = 2 π r2 ω
Eq. 4
v = 2 v∞ sin θ
Eq. 5
v = 2 v∞ sin θ Γ / ( 2 π r )
Where:
r = radius, ft (mm)
L = length, ft (mm)
ω = angular velocity rad/sec
v∞ = relative airflow
of velocity, ft/sec, (m/sec)
ρ = fluid density, lb/ft3, (kg, m3)
gc = Gavitation consant 32.2 lbm/lb-sec2
v = fluid velocity, in/sec, (mm/sec)
Equation 1 and 2 assumes that there is no slip (i.e., that the air drawn around the cylinder by rotation moves at !), and in that ideal case, the maximum coefficient of lift is 4 π. Practical rotating devices, however, seldom achieve a coefficient of lift in excess of 9 or 10, and even then, the power expenditure to keep the cylinder rotating is excessive.
a A similar analysis can be used to explain why a pitched baseball curves. The rotation of the ball produces a force that changes the path of the ball as it travels.
Reference:
- Civil Engineering Reference Manual, Fifteeth Edition
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