Lift From Rotating Cylinder Formula

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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
F_L / L = ρ v_∞Γ

Eq. 2, U.S.
F_L / L = ρ v_∞Γ / g_c

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)
g_c = 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