Tuesday, March 10, 2015

Emergency Escape: Flying a Minimum-Radius Turn


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).

  •  ·         If a descent results, reduce the bank angle to maintain level flight.

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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