From: johnson@nrtc.northrop.com (Greg Johnson)
Newsgroups: sci.aeronautics
Subject: angle btwn chord and L+D is monotonic in AOA!
Date: 8 Apr 92 00:28:50 GMT
Reply-To: johnson@nrtc.northrop.com (Greg Johnson)

I've been trying to figure out what's going on aerodynamically when a
helicopter autorotates, and came across an interesting observation about the
lift and drag vectors of airfoils.  (Autorotation is when a helicopter acts
like a maple seed; the engine is not turning the rotor blades.  Rather,
uprushing air is turning the rotor blades, causing them to generate sufficient
lift to sustain a gentle, gliding, unaccelerated descent.)

I found it useful to consider the vector sum of L and D.  (Is there a name for
this vector?  `Total aerodynamic force' or something?  Is there a standard
letter to represent it?)  L+D seems like a `natural' vector to consider; I
would guess it is decomposed into a vector perpendicular to the airstream
(`Lift') and a vector parallel to the airstream (`Drag') primarily for
mathematical convenience.  I'll call L+D `F' for `aerodynamic force'.

Consider the direction of F as a function of angle of attack in a coordinate
system in which the X axis is parallel to the airstream.  (Assume a symmetric
airfoil, and that the airstream is travelling left to right, in a wind tunnel
for example.) At zero AOA, F is on the positive X axis, since lift is zero and
there is some drag.  As angle of attack increases from zero, F rotates
sharply counterclockwise, reaching a maximum of about 85 degrees or so.  This
happens at (ballpark) 6 degrees AOA.  This point is Maximum L/D.  As AOA
increases past 6 degrees up to 18-20 degrees (stall AOA), F rotates rather
slowly clockwise.  (You can determine the angle between the X axis and the F
vector by simply taking arctan( |L| / |D| ) ).

Now, instead of looking at the angle between airstream and F, consider the
following interesting angle:  that between the chord of the airfoil and F.
Think of the airfoil as staying level on the X axis (facing left, say), and the
airstream coming at the airfoil from below and to the left (for positive
AOA's).  Again, at zero AOA the F vector is on the positive X axis.  As we
increase AOA (i.e., as the vector showing wind direction rotates
counter-clockwise), the F vector rotates counter-clockwise.  However, in this
case the F vector continues to travel counter-clockwise as AOA varies from
zero up to stall speed!  Moreover, the F vector goes past 90 degrees, up to a
maximum of about 98 degrees or so.  (The angle between chord and airstream is
just arctan( |L| / |D| ) + AOA, and the observation is that this function
increases monotonically in AOA up to stall speed.  Looking at a standard L/D
curve, if you transform it using arctan it doesn't change much; arctan is
pretty close to a 45-degree line on the range of 0 to 20 degrees.  After L/D
gets over the Max L/D `hump', it starts to decrease, but it doesn't achieve a
slope of -45 degrees until you stall.  So, the AOA + arctan( |L|/|D|) curve has
a positive slope up through stall.)

The fact that the angle between chord and F goes past 90 degrees means that
it is possible to autorotate a helicopter at zero-degree or even positive
collective (blade-pitch) settings.  If the blades are rotating too slowly they
see a high AOA, and F is higher than 90 degrees.  In other words, the
horizontal component of this vector is pointing forward from the perspective of
the airfoil.  Hence, the blades speed up.  (The AOA a blade element sees is
determined by the vector sum of how fast the helicopter is falling and the
velocity of the blade element.)

- Greg Johnson

Disclaimer:  I'm a computer person and not an aerodynamics person.  I play
around with the latter as part of a radio-control flying addiction.