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Kutta-Joukowski Lift
Theorem for a Cylinder
Lift per unit length of a cylinder acts
perpendicular to the velocity (V) and is
given by:
L = ρVG (Lbs/Ft)
Where:
ρ = Fluid Density (slugs/Cu Ft)
G = Vortex Strength (Sq Ft/sec) (G=2.Π.b.Vr)
V = Flow Velocity (Ft/sec)
Vr = rotational speed (Ft/sec) (Vr=2.Π.b.s)
b = radius of cylinder
s = revolutions/sec
pi = 3.14159
Two early aerodynamicists determined the
magnitude of the lift force, Kutta in
Germany and Joukowski in Russia. The lift
equation for a rotating cylinder bears their
names. The equation states that the lift
L per unit length along the
cylinder is directly proportional to the
velocity V of the flow, the
density ρ of the flow, and the
strength of the vortex G that
is established by the rotation.
L = ρ * V * G
The equation gives lift-per-unit length
because the flow is two-dimensional.
(Obviously, the longer the cylinder, the
greater the lift) Determining the vortex
strength G takes a little more
math. The vortex strength equals the
rotational speed Vr
times the circumference of the cylinder. If
b is the radius of the
cylinder,
G = 2.0 * b* pi * Vr
Where pi =3.14159. The rotational
speed Vr is equal
to the circumference of the cylinder times
the spin s of the cylinder.
Vr = 2.0
* b* pi * s
WARNING: Be particularly aware of
the simplifying assumptions that have gone
into this analysis. This type of flow field
is called an ideal flow field. It is
produced by superimposing the flow field
from an ideal vortex centered in the
cylinder with a uniform free stream flow.
There is no viscosity in this model (no
boundary layer on the cylinder) even though
this is the real origin of the circulating
flow! In reality, the flow around a rotating
cylinder is very complex, depending on the
ratio of rotational speed, free stream
speed, angle of incidence of the free stream
flow, viscosity of the fluid, and size of
the cylinder, the flow off the rear of the
cylinder can separate and become unsteady.
But, the simplified model does give
the first order effects; it gives an initial
good prediction of the force on the cylinder
when compared to experiments.
In Conclusion: All that is
necessary to create lift is to turn a flow
of air. We are familiar with the lift
generated by an airplane wing or a curving
baseball. But a simple rotating cylinder
will also create lift. In fact, because the
flow field associated with a rotating
cylinder is two-dimensional, it is much
easier to understand the basic physics of
this problem than the more complex
three-dimensional aspects of a curveball.
However, the details of how a rotating
cylinder creates lift are still pretty
complex. Next to any surface, the molecules
of the fluid will stick to the surface. This
thin layer of molecules will entrain or pull
the surrounding flow in the direction that
the surface moves. If we put a cylinder that
is rotating about the longitudinal axis (a
line perpendicular to the circular cross
section) into a fluid, it would eventually
create a spinning, vortex-like flow around
the cylinder. If we then set the fluid in
motion, the uniform velocity flow field can
be added to the vortex flow. |
