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National Advisory Committee for Aeronautics, Technical Notes - Notes on Propeller Design - III - The Aerodynamical Equations of the Propeller Blade Elements

The air flow and the air force created by all elements of
the prOpeller blades lying in a ring located between two concen—
tric circles around the propeller-axis are independent of what
happens in other rings. The aerodynamic equations of such blade
elements are established and their application is discussed.

Since the air forces of welledesigned propellers are uniform—
ly distributed along the blades, the blade elements in each con—
centric ring around the propeller axis have properties practical—
ly-independent of those in other rings. Hence the design or the
analysis of a propeller has to be performed by designing or ana-
lyzing consecutively the elements of a number of such rings. 1
proceed immediately to the discussion of this problem. ,

One such short piece of the blade is generally considered
as a wing moving on a Spiral path. The angle of each blade ele—
ment may always be measured from its direction of zero lift.

There are two_important angles: The angle a of inclination to -
the propeller plane and the angle B of attack relative to the
air. The air force of the blade element has one component in the
direction of the relative motion and another one at right angle
to it. The former is due to the viscosity of the air and will be
considered later. It is small in general and dces not materially
alter the character of the air flow, but it can by no means be
neglected. t may be called drag, and the other force may be call-
ed lift. By dividing the lift by the area of the blade element
and by the dynamical pressure of the relative velocity between
blade element and air, a lift coefficient CL may be defined.
The lift coefficient thus obtained obeys the laws established in
the ordinary wing theory for infinite aspect ratio. In particu—
lat, it is never very different from 2 TI’ B~

The lift of the blade element, the relative velocity between
it and the air, and the absolute velocity of the blade elements
may be divided into components parallel and at right angles to
the propeller axis. The components of the relative velocity may
be Written u and v, and those of the absolute velocity; U and V,
where .v and V are parallel to the axis. It may be assumed that
V at the same time is the velocity of flight.