Thomas
Turner’s excellent article “High Density Altitude Turn” [“Safety Pilot,” August
2014] is on target. Turner establishes
that the higher the airspeed the greater the turn radius in a level standard
rate turn and that the effect is compounded by high density altitude. He points out the implications of the
phenomena for safer high density altitude flying. This article addresses a related issue: how
to make an emergency turn with the smallest radius consistent with appropriate
safety.
Imagine
yourself flying up a valley with rising terrain. You are only two hundred feet above the
terrain, you are at maximum power, your indicated airspeed is at the maximum
climb angle airspeed – and the terrain is rising faster than you are. You begin to get worried – actually, you are
loaded with adrenalin and shaking like a leaf.
Fortunately,
you have positioned yourself on one side of the valley, so that you have the
maximum room to make a 180 degree turn.
Unfortunately, you have waited too long – there is no longer much room
to make the turn. The terrain is such
that landing is not an attractive option.
Now is
the time to implement that minimum radius level turn procedure you have read so
much about in the POH, and all of those FAA handbooks and other excellent sources
of flying techniques. Uh oh, have you
ever seen a procedure for making a minimum radius level turn in these
materials? If one is there, it is pretty
elusive.
If you
are willing to live with the standard approximation that lift is proportional
to air density, the angle of attack (up to the stalling angle of attack), and airspeed
squared, then there is a simple formula for the radius of a level turn.
The
formula implies that increasing the bank angle reduces the turn radius and that
increasing the angle of attack reduces the turn radius. Consequently, the formula implies that a
minimum radius level turn requires using the steepest feasible bank angle and
the highest feasible angle of attack.
Assuming that sufficient power is available to fly a level turn in this
manner, the formula implies that the following procedure is optimal.
- · Set the angle of attack to just below the stalling angle of attack.
- · Set the bank angle large enough so that level flight requires a g load just below the aircraft’s g limit (4.4 g in most Bonanzas and Debonairs; 3.8 to 4.2 g in Barons and Travel Airs).
- · Set power to maintain level flight.
A 4.4 g
level turn requires a 76.9° bank angle.
The largest feasible angle of attack and g load combination implies an
indicated airspeed just under maneuvering speed (VA), as adjusted
for airplane weight (maneuvering speed is the indicated airspeed at which the
aircraft’s g load reaches its limit at just under the stalling angle of
attack).
Now you
understand why you probably have not seen a procedure for making a
theoretically optimal minimum radius level turn. Trying
to fly a 4.4g steep turn, at substantial airspeed, and at close to the stalling
angle of attack may be optimal in theory but is too likely to be disastrous in
practice. The risk of either
stalling or structural failure is too great.
A safer
procedure is needed that is a better tradeoff between the risk of stalling or
structural failure and the risk of controlled flight into terrain. A safer procedure should not require such
high g forces or flying so close to the stall.
Moreover, why be constrained to a level turn? If there is sufficient power to climb while
holding the desired bank angle and angle of attack, why not use it? If there is sufficient power and speed for a
chandelle (or similar maneuver), why not do it?
If there is insufficient power to hold a level turn at the desired bank
angle and angle of attack, what is the alternative?
A
useful safer procedure must be practical and effective, able to handle a
variety of circumstances, and simple enough to execute without much thinking. One cannot count on thinking well in an
emergency.
The
insight obtained from the minimum radius level turn formula can be brought to
bear to identify a safer procedure that accomplishes all this while increasing
the turn radius about as little as possible.[1] To the extent that your skill level allows,
implement the procedure in a smooth coordinated manner, not as a sequence of
steps.
- · Set full power.
- · Increase the angle of attack until the stall warning horn sounds. Then reduce the angle of attack, slightly, and maintain it.
- · Set the bank angle to just over 60° (65° is the target).
A 65°
bank angle reduces the g load to 2.4, which is manageable, yet increases the
level turn radius by only about 7.5%, relative to a 76.9° bank angle. Staying just below the angle of attack at
which the stall warning horn sounds increases the level turn radius materially,
but substantially reduces the likelihood of stalling. Assuming that the stall warning horn sounds
at about 5-7 knots above the stalling speed in level flight, the procedure’s
reduction in the angle of attack increases the level turn radius about 25%,
relative to an angle of attack just below the stalling angle of attack. Reducing the angle of attack is more costly
than reducing the bank angle. However,
reducing the bank angle further to 60° also is costly, adding about another 5%
to the level turn radius.
If
there is excess power and speed, this procedure produces either a climbing turn,
which helps compensate for the rising terrain, or a (sloppy) chandelle, which
helps compensate for the rising terrain and decreases the horizontal component
of the turn radius. If there is
insufficient power, the procedure approximates the minimum level turn radius
achievable at that power level.
Remember
to apply the procedure judiciously. Do
not take on more than you can handle.
Controlled flight into terrain is preferable to uncontrolled flight into
terrain.
An
appendix with the mathematics, for the geeks, and a spreadsheet that performs
the required calculations for a variety of assumptions is available from the
author at ln2.6931@gmail.com.
[1] The procedure is relatively benign. However, it includes a bank angle slightly
exceeding 60o and may result in flight that is classified as
aerobatic (FAR 91.303) or that requires parachutes (FAR 91.307). However, FAR 91.3 provides emergency
authority for such maneuvers.
___________________________
The following analysis provides some theory concerning
optimal minimum radius level turns. The theoretically optimum procedure is
dangerous – and this article does not recommend it. However, the theory is useful for providing
insight about why the safer procedure advocated in the article works.
The
relationships for a level turn are that the vertical component of lift equals
weight and the horizontal component of lift equals mass times centripetal
acceleration.
Lift
equation:
mg=L cos(A)
(1)
Centripetal
acceleration equation:
m(V^2)/R=L sin(A)
( 2)
m= The
aircraft’s mass.
g= Gravitational
acceleration.
L= Lift.
A= Bank
angle.
R= Turn radius.
If you
divide Equation ( 2) by Equation ( 1) and rearrange things, you get:
R=(V^2)/(g tan(A))
( 3)
Equation
( 3) shows that a combination of low
airspeed (in the numerator) and high bank angle (in the denominator) minimizes
the turn radius. At any given bank
angle, you slow down by increasing the angle of attack, so airspeed is
minimized by maximizing the angle of attack.
Therefore, the theory implies that an angle of attack just below the
stalling angle of attack is best, for any bank angle.
To gain
more insight, assume that lift is proportional to the product of air density,
angle of attack, and speed squared, up to the stalling angle of attack. Then the lift equation is:
L=C1*r*a*(V^2)
( 4)
Here, C1 is a constant, r is air density, and "a" is angle of attack.
Substitute
Equation ( 4) into Equation ( 1) and solve the resulting
equation for speed squared. Then,
substitute the result for speed squared into Equation ( 3) and rearrange terms to obtain:
R=C/(r*a*sin(A))
( 5)
Here, C is another constant,.
Equation
( 5) validates Turner’s point. The low air density at high density altitudes
is a killer. Halving the air density
doubles the minimum level turn radius.
Equation
( 5) implies that the minimum radius
level turn is achieved, theoretically, with the highest possible angle of
attack combined with the highest possible bank angle. The highest possible bank angle is the bank
angle at which the g load is just below the aircraft’s g limit. The highest possible angle of attack is just
below the stalling angle of attack.
Since maneuvering speed is determined by the indicated airspeed at which
an angle of attack just below the stalling angle of attack results in the
maximum allowable g load, Equation ( 5) shows that an indicated
airspeed of close to maneuvering speed is, theoretically, optimum for a minimum
radius level turn.
The g
load, G, in a level turn is obtained by replacing lift, L, in Equation ( 1) by Gmg and solving for G.
G=1/cos(A)
( 6)
The ratio of the turn radius,RA, at an arbitrary bank angle, A, to the optimum turn radius, RAo, at the optimum bank angle, Ao, is obtained from Equation (
5). The arbitrary bank is presumed to be less
than the optimal bank.
RA/RAo=sin(Ao)/sin(A)
( 7)
The ratio of the turn radius, Ra, at an arbitrary angle of attack, a, to the optimum turn radius, Rao, at the optimum angle of attack, ao, is obtained from Equation (
5). The arbitrary angle of attack is presumed to
be less than the optimum angle of attack.
Ra/Rao=ao/a
( 8)
Table 1
shows the bank angles corresponding to several g loads. Bonanzas typically have a g limit of 4.4,
which implies a bank angle of 76.9 degrees.
|
TABLE 1
|
|||||||||
|
g Load
|
2.0
|
2.4
|
2.9
|
3.9
|
4.4
|
5.8
|
6.6
|
11.5
|
|
|
Bank Angle
|
60
|
65
|
70
|
75
|
76.9
|
80
|
81.3
|
85
|
|
Table 2
shows the percentage increase in turn radius for several bank angles lower than
76.9 degrees. A moderate reduction of
the bank angle is not very costly in terms of increased turn radius.
|
TABLE 2
– (Theoretical Optimum 4.4 g Load at a 76.9° Bank Angle)
|
||||||||
|
Bank Angle
|
45
|
60
|
65
|
70
|
75
|
76.9
|
||
|
% Higher Turn Radius
|
37.7
|
12.5
|
7.5
|
3.6
|
0.8
|
0.0
|
||
Tables
1 and 2 show that using a 65° bank angle with a g load of 2.4 instead of a
76.9° bank angle with a g load of 4.4 g increases the turn radius only about
7.5%.
Reducing
the angle of attack also increases the turn radius. Table 3 shows the percentage increase in turn
radius for several angles of attack lower than an assumed 15° optimum angle of
attack. Reducing the angle of attack is
relatively costly in terms of increased turn radius.
|
TABLE 3
– Optimum 15°
Stalling Angle of Attack
|
||||||||
|
Angle of Attack
|
15
|
14
|
13
|
12
|
11
|
10
|
||
|
% Higher Turn Radius
|
0.0
|
7.1
|
15.4
|
25.0
|
36.4
|
50.0
|
||
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