A recent note analyzed the minimum turning radius of an airplane in terms of its airspeed and angle of bank. Unfortunately, some misconceptions concerning the underlying physics were introduced. This note is intended to clarify those areas

Ford, William F.

English

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NASA-T_-11 Z76 _ /'_"_ /j
SIAM
VoL 38, No. 2, pp. 3_9-3t2, lune 1996 0O7
/
CLARIFICATION OF "TURN PERFORMANCE OF AIRCRAFT"*
WILLIAMF.FORDt
Abstract.Arecentnoteanalyzedtheminimumturningradiusofanairplaneintermsofitsai_peedandangleof
bank.Unfortunately,somemisccyaceptionsconcerningtheunderlyingphysicswereinlroduced.Thisnoteis intended
toclarifythoseareas.
Keywords,aircraftperformance,loadfactor,lift
AMSsubjectclassifications.76(325,93A30
Introduction. Wolper's recent note [1] on an airplane's turn performance, although well
motivated and mathematically correct, contains some statements about the aerodynamics that
are either incorrect or misleading. Here are the major difficulties, in the order in wldch they
are encountered.
1. The first paragraphof §1 states that an airplane turns because of its horizontal com-
ponent of lift. This common belief is only half true.
2. The second paragraphdeclares that in analyzing the turn performance of an airplane
we may essentially ignore variation of the lift coefficient CL. This is false.
3. The fourth paragraphspeaks of an airplane in level flight without enough lift to support
its weight and states that the airplane is then "called stalled." This is confusing and incorrect.
4. In §2, the second paragraphstates that if the airspeed in the turn is too low for the angle
of bank, the airplane stalls and the turn radius becomes infinite. This contains a (potentially
dangerous) hidden assumption.
1. Why an airplane turns. To explain this one must recognize that an airplane is an
extended, nonsymmetric solid body, which therefore has five degrees of freedom. It follows
that analysis of the turn must address both oenter-of-mass motion and motion relative to the
center-of-mass.
An airplane does not turn "because" of lift. However, a pilot may choose to employ lift
as one of the two ingredients necessary for the turn. Those ingredients are
1. a horizontal force at rightangles to the flight pathused to change the airplane's direction
of flight and
2. a mechanism for rotating the airplane's longitudinal axis, and thereby the horizontal
force, as the turn progresses.
Note that a horizontal force alone may not produce a turn. For instance, if the pilot rolls
into a moderate bank, but then applies opposite rudder to prevent the fuselage from rotating
about the vertical axis, a simple translation will result (albeit in a new direction). This is called
a slip,
Also note that the pilot is not restricted to using lift as the horizontal force. For instance,
if the pilot keeps the wings level but yaws the airplane with the rudder, the thrust now has
a component lateral to the flight path that will supply the necessary horizontal force, and a
turn will result. (This turn will be less efficient, of course, since at cruise speeds the thrust
produced by a light airplane is 5 to 6 times smaller than its lift.)
In a normal turn that uses a bank, the necessary mechanism for keeping the horizontal
force at right angles to the flight path is the relative wind, as it strikes the horizontal stabilizer
at the tail of the airplane. In effect, the airplane acts like a weather vane.
*ReceivedbytheeditorsMarchI, 1995;acceptedforpublicationMay3, 1995.
tNASALewisResearchCenter,Cleveland,OH44135.
3O9
https://ntrs.nasa.gov/search.jsp?R=19970019927 2020-06-16T01:59:51+00:00Z
310 CLASSROOMNOTES
In the yaw-induced turn of the second example, the pilot keeps the rudder deflected to
produce a moment that counteracts the weathervaning tendency. Interestingly enough, in this
situationthestabilizer-ruddercombinationisactinglikea smallverticalwing,producinga
liftatthetailthatisbasicallyhorizontaland oppositetothedirectionofturn.
Incasualconversation,a phraselike"liftmakes theairplaneturn"isperfectly acceptable,
butintheclassroomorthepages ofa scientificjournal,a higherstandardofaccuracyshould
be our goal.
2. The coefficientoflift.As theoncoming wind sweeps overthecambered,slightlyup-
tiltedwings ofanairplane,anaerodynamicforceisproduced.Thatforcemay beresolvedinto
two components,one paralleltotheoncoming wind ortheflightpath,theotherperpendicular.
The perpendicularcomponent isby farthelargerand has been historicallyreferredtoby the
na._e//ft.
The liftL producedby thewings ofan airplaneisgivenby
(1) L = lpV2CL S,
z
where p is the air density, V the airspeed, Cz the coefficient of lift, and S the wing area. In
the usual level turn made with bank angle 0, the lift L has a vertical component L cos 0 equal
to the airplane's weight W and a horizontal component equal to the required centripetal force,
V 2 WV 2
(2) L sin0 = m-- --R gR'
where g is the acceleration of gravity and R the radius of the turn.
If we combine these results one way we find that V 2 = g R tan 0, but if we combine them
another way we get
2W
(3) Cl. _- pV2ScosO,
which shows that, for fixed airspeed in a level turn, CI. is inversely proportional to cos 0. It
is quite clear, therefore, that the coefficient of lift, which at a given airspeed controls the load
on the wing and must be increased to increase bank angle, cannot be dismissed as "essentially
constant" for purposes of analysis.
3. Lift and stall speed, The term _stall speed" is something of a misnomer. It suggests
that an aixplane somehow measures the speed of the oncoming air, and upon finding it below
an acceptable value, promptly refuses to fly. In this view all airplanes parked on the ramp are
stalled.
In fact an airplane wing can be stalled at any akspeed, as the FAA has been emp_'g
for decades without much success. On the other hand, there is a certain limiting airspeed
below which an airplane cannot be maintained in steady level flight. This limiting speed is
often referred to by the FAA as "minimum controllable airspeed," but it has the symbol Vs
and is popularly known as the _stall speedy
The reference, of course, is to the fact that an attempt to maintain altitude, yet fly even
slower, will produce a stall. But the knowledgeable pilot is expected to understand that a
stall -- the sudden breakup of smooth, laminar flow over the wing resulting in asharp reduction
of lift -- is caused not by any change in the speed of the airflow, but by the increase in the
wing's angle of attack beyond a certain critical value.
From a philosophical perspective, decreased airspeed (and anticipated loss of lift) is the
_formar' cause, namely, the motivation for the pilot's decision to increase the angle of attacL
But the direct physical cause of a stall is the increased angle of attack.
CLASSROOMN TES 311
Even without this knowledge, it is apparent that the phrase "level flight without enough
lift to support the weight" is an oxymoron. But steady flight with L < W is not only possible,
it is common and safe, and happens at least twice during every flight. An airplane in steady,
unaccelerated flight, whose lift is less than its weight, is either ascending or descending. (See
the Appendix.)
4. Stalls in the turn. Stalls during a turn occur for the same reason that they do in level
flight: the angle of attack is increased beyond its critical value. They do not occur simply
because the airspeed becomes too low or the bank angle too large.
So what does happen when the airspeed becomes too low or the bank angle too large?
Since the vertical component of lift is now less than the weight, the airplane will begin to
descend while it turns. As long as the pilot does not increase the angle of attack, the airplane
will not stall, nor will it fall out of the sky, nor will its turn radius become infinite. It will
simply descend while it turns.
Prospective commercial pilots practice just such a maneuver while preparing for their
check ride: the maneuver is called a gliding spiral. It is taught primarily as an emergency
procedure, designed to keep the airplane directly over the selected landing area during the
descent, and is typically performed at idle power, low airspeed, and moderately steep bank.
Despite the existence of the gilding spiral in the flight training syllabus, the myth persists
that an airplane will automatically stall ff the airspeed is low and/or the bank is steep. Why?
It is because of the hidden assumption that the altitude must remain constant. I call it hidden
because in so many texts and airplane manuals the typical charts that depict the "increase of
stall speed" with bank angle do not include a warning or legend or note with the crucial proviso
"at constant altitude." My experience as a flight iustructor has convinced me that the failure
to stress this assumption (let alone ignore its existence) is responsible for more accidents in
general aviation than any other cause.
It is true that in Wolper's case the condition "level" is stated, not assumed. Nevertheless, a
sentence that begins "If the airspeed is too low for the angle of bank then the airplane stalls..."
misleadingly implies a causal relation between airspeed and stall that does not exist if the pilot
simply abandons the attempt to maintain altitude.
5. The maximum performmace turn. A maximum performance turn may be defined as
one having minimum turn radius. Since both lift and centripetal acceleration are proportional
to V 2, that quantity can be eliminated from (1) and (2) and the resnlting equation solved for
R, yielding
W
(4) R--
½pgSCL sinO"
Observe that the minimum turn radius is achieved by using maximum coefficient of lift and
maximum bank angle, independent of airspeed.
Assuming from now on that CL has its maximum value (CL)max, we can replace ( CL )max
by its expression in terms of weight and minimum controllable akspeed Vs, obtaining
V 2
R0 Ro = "s.
(5) R==(O)- sinO' g
Of course R0 can be achieved only with a 90 ° bank and consequent altitude loss which, for a
single course reversal, can be estimated as
(6) Ah = = 2g = R0.
312 CLASSROOM NOTES
Evidently, to minimize altitude loss, V should be as large as possible without risking undue
stress on the wings. Every airplane has a particular airspeed, known as the "maneuvering
speed" or "turbulent air penetration airspeed" (denoted by Va ), whose usage guarantees that the
wings will not be accidentally overstressed. For a "Normal Category" airplane Va = _/3_Vs,
and so ifthis airspeed is used, Ah _ 1.3R0.
If a level turn is required then (3) comes into play, and from the form it takes when
CL = (CL)m_,
cosO=-_,
we see that V must be as large as possible to achieve maximum bank angle. Again, Va should
be used for safety, with the result
(7) 0 = COS -1 : VS_2 Ro _-- 1.03R0.
\_1 _-__75°, R_I = _/I - (Vs/V,,)"
There are other ways of defining a maximum performance turn that include climbing dur-
ing the turn; many of these can produce course reversals requiring less than 2R0 of horizontal
airspace.
6. Appendix: Lift and weight. Of the four major forces acting on the airplane (lift,
drag, thrust, and weight) two have, by definition, directions that are fixed relative to the flight
path: lift is always perpendicular and drag is always parallel. Except in rare instances, l the
third force, thrust, also lies along the flight path. The weight, however, is independent of flight
path and is always directed toward the earth.
Now consider an airplane whose flight path is inclined at an angle _ to the horizontal
and is therefore climbing or descending. The weight of the airplane can be resolved into two
components, one of them being a force W cos # perpendicular to the flight path. If the airplane
is in steady state motion, i.e., in equilibrium, then there must be another force perpendicular
to the flight path having exactly the same magnitude. We conclude that L = W cos/_, and
therefore L < W in a steady climb or descent. 2
The limiting case, a vertical climb and descent with zero lift, may be observed at any
airshow in the maneuver known as the "hammerheadY 3 To perform it, the airplane enters
a totally vertical climb, slows down while rising because of its weight, and then, just as its
airspeed reaches zero, gracefully arcs over to a totally vertical descent. Although the climb
and descent portions are not steady, they are straight, and so the single perpendicular force,
lift, must therefore vanish.
Once these facts are fully appreciated by the student a number of misconceptions disap-
pear, and the quantity known as lift can be regarded simply as a force
1. conveniently perpendicular to the tlight path,
2. used in part to counteract gravity and in part for maneuvering purposes,
3. whose production requires an angle of attack that may become close to a critical value,
and thus must be treated with care.
REFERENCE
[1] J. S. Wot.pez, Turn performance of aircraft, SIAM Rev., 36 (1994), pp. 470--473.
t One example is the yaw-induced turn described above.
2It is my experience that, despite all their training, 99 out of 100 pilots are unaware that lift is actually smaller
than weight during a steady climb.
3Actually it is called a "hammezhead stall, _ but this tOO is a misnomer since no stall occurs.


Clarification of 'Turn Performance of Aircraft'

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