Turning Aerodynamics

By Dave Esser

Reprinted from Woman Pilot magazine

Student pilots are first taught the basics of straight and level flight. After practice the basics of aircraft control in keeping the shiny side up is mastered with relative ease. The newfound confidence is easily destroyed when the first turn is attempted. An airplane does not turn in the same way as does a car or boat. There is negative learning transfer from a car where the steering wheel remains deflected until the turn is completed. Oddly, in an aircraft the control wheel is returned to neutral after the desired bank is established. Rudders don’t turn the aircraft, they coordinate the turn.  Then there is the need for increased elevator backpressure to keep the flight path level, and if this were not enough, if a constant airspeed is desired the thrust must also be increased.  So in a level, coordinated turn at a constant airspeed the pilot must adjust every flight control, ailerons, rudder, elevator and thrust.  Then add the task of rolling out at a rate that is appropriate to complete the turn on an assigned heading while again adjusting the ailerons, elevator, rudder, and power, the task involves a lot of coordination, planning, and understanding of aerodynamic concepts.


Let’s begin with how an aircraft turns. Newton’s first law says that we must first create an unbalanced accelerating force to deflect the flight path from a straight line. This centripetal force is created to by the horizontal component of lift when banking the aircraft deflects the lift vector from the vertical.  While the ailerons are deflected creating more lift on one wing than the other, the roll occurs that creates the bank. The greater the aileron deflection, the faster the rate of roll. With additional lift comes additional induced drag. This means the wing being raised is also experiencing more drag, resulting in the need for rudder to keep coordinated. Because this drag tends to yaw the nose in the direction opposite  the turn, it is referred to as adverse aileron yaw. 


Since we are talking about coordinating the turn, lets review slips and skids. Part of the turn coordinator is the inclinometer, or “ball”.  If the ball is in the center all is well. If the ball is deflected in the same direction we are turning, that is a slip, and if deflected in the opposite direction, a skid is occurring. In trying to comprehend the slip and skid think of a high bank racetrack.  If a car is circling to the left around a high bank racetrack too slowly, it cannot “stick” to the track but instead it slips to the inside. See Figure 1. To fix this, one would either have to turn faster to the left, or decrease the bank of the track. In the application to an aircraft if in a slipping turn to the left, one would have to either decrease the bank angle or increase the turn rate to the left. This is done by applying left rudder (the same direction that the ball is deflected, hence the saying, “step on the ball”.


If a racecar is going too fast around this track, it will skid to the outside. See Figure 2. To fix this, one would either have to increase the bank of the track, or slow the rate of the left turn. To an aircraft, that would mean to either increase the bank angle, or decrease the rate of turn to the left with right rudder. Again the rudder applied is the same side as the ball is deflected because in a skid to the left, the ball is on the opposite direction of the turn, or to the right.  Either way when slipping or skidding, the “step on the ball” remedy will work.


In reality, what the rudder is doing is coordinating the rate of turn of the flight path to rate of yaw around the aircraft’s vertical axis. In a 360-degree turn not only must the direction of the flight path change, but also the heading orientation of the aircraft must change at the same rate. If this did not happen, after a 180-degree flight path change, the aircraft would be flying tail first backwards.  So the rudder coordinates the longitudinal axis to the flight path, and/or relative wind.  Many know that the relative wind is striking the fuselage at the same angle the ball is deflected. In other words, if the ball is deflected from center, the relative wind is not aligned with the fuselage, but it is striking the aircraft from that side increasing the parasite drag.  This additional drag is why slips can be utilized to increase the angle of descent without increasing airspeed.  Some aircraft have made use of a yaw string, which was little more than a piece of yarn attached to the aircraft skin visible out the front window. Keeping this yaw string aligned with the fuselage means the aircraft is flying with no sideslip.  As simple as this is, the inclinometer (ball) is just as simple. The inclinometer needs no source of external power, it simply responds to gravitational, and inertial forces. 


So now that a coordinated turn can be entered and exited, the next challenge is to maintain level flight.  It was said before that the lift vector is deflected from the vertical by an amount equal to the bank angle. In a 45-degree bank, half the lift is in the vertical direction, and half is in the horizontal. Using trigonometry one can figure this ratio for all bank angles. When the aircraft was in level flight, the upward lift was in balance with the downward weight. When this vector is deflected from the vertical, in order to keep the upward vertical component still equal to weight, the overall lift vector must be increased by increasing the angle of attack.  See Figure 3.


So as the bank angle increases, the required angle of attack increases. Eventually the critical angle of attack will be reached and the aircraft will stall at a much higher airspeed than usual. Remember the old adage, “An aircraft can be stalled at any attitude, any airspeed.” How can an aircraft stall at a high airspeed? Load factor. The load factor is calculated by dividing the lift the wings are producing by the weight. In straight and level flight, that load factor is one. Some refer to load factor as “Gs” for gravity. When pulling two “Gs” the load factor is two, and the occupants are pushed down in their seats with twice the force of gravity. In reality they are not just pushing down on their seats, their seats are pushing up on them and the inertial force feels like two Gs.


The stall speed increases as the load factor increases. To be more specific, the load factor increases directly with the square root of the load factor. If, for example, the aircraft were experiencing a load factor of 4, or 4 Gs, the stall speed would double, as the square root of 4 is 2.  The stalling angle of attack is the same, but it is reached at twice the airspeed.  See Figure 4.


Note that the relationship between bank angle and load factor is a cosine function, not a linear (straight line) one. If one doubles the bank angle (100% increase) from 15 degrees to 30 degrees, the load factor hardy increases, but from 60 degrees to 75 degrees (the same 15 degree increase but only 25% increase) the load factor almost doubles. At a 90-degree bank, the load factor is infinite. So how do those fighter jets in air shows perform 90 degree coordinated level turns? They don’t. In order to turn at a 90 degree bank, with no component of lift in the vertical direction, the aircraft could not maintain level flight if it were not for a little top rudder to use the fuselage like an airfoil, and some help from the vertical component of thrust.  In this situation, the turn is not coordinated.


In a level coordinated turn, the load factor is determined by the bank angle. Airspeed and weight have no effect.  All aircraft in a coordinated level bank of a given degree have the same load factor. To determine this load factor, take the inverse of (one over) the cosine of the bank angle. If an aircraft is in a coordinated level turn at 60 degrees of bank, the load factor is 2. Since the cosine of a 60 degree angle is 0.5, and 1/0.5 is 2. Since the stall speed increases with the square root of the load factor, the stall speed increases with the square root of the inverse of the cosine of the bank angle. 

VsBank =  Vs


As the bank angle increases in a level coordinated turn, so does the stall speed. No matter how fast an aircraft is flying there will be a bank angle and load factor that results in reaching the critical stalling angle of attack.  Again the old saying, “An aircraft can be stalled at any attitude, any airspeed.” The critical factor is what is the load factor when the aircraft stalls. The faster the airspeed, the higher the maximum possible load factor. The limit load factor of the aircraft is the highest amount of stress that can be applied without permanent deformation of the structure. The amount the aircraft has been designed to withstand depends on its category.  Aircraft in the normal category must be able to withstand 3.8 positive G’s. It should be noted that as aircraft age, metal fatigue, or possible hidden structural flaws could reduce the amount of stress an aircraft can withstand, so this limit should always be considered with caution.


Aeronautical engineers determine the airspeed at which the aircraft will stall at this limit load factor. The airspeed is called Va or design maneuvering speed. Students learn this is the airspeed above which could cause structural damage if full abrupt control deflections are made.  A somewhat more descriptive definition is that Va is the stall speed at the limit load factor.  Since stall speeds decrease as gross weight decreases, so does maneuvering speed. This is contrary to intuition which would make one think that a lighter aircraft would have a higher maneuvering airspeed than a heaver one, which is opposite in actuality.


As the aircraft banks, the radius of turn decreases, and the rate of turn increases. The higher the bank angle, the greater the horizontal component of lift or centripetal accelerating force.  The greater this horizontal acceleration force, the smaller the radius of curvature.  Using the formula;

  r  =   vKTAS2 / (11.26 tan f )

the radius of a turn in feet can be determined by taking the square of the velocity in knots divided by the sum of the constant 11.26 multiplied by the tangent of the bank angle. Note that the radius of turn increases with the square of the velocity. If for example an aircraft doubles its airspeed, the radius of turn quadruples.


Another measure of turn is rate of turn, used in instrument flight.  The rate is the expressed in degrees per second with 3 degrees per second (two minute 360 degree turn) defined as standard rate.  As the bank angle increases, the rate of turn also increases.  Using the formula;  

ROT     =   (1091 tan f) / vKTS

The rate of turn in degrees per second is found by multiplying the constant 1091 times the tangent of the bank angle and dividing by the true airspeed in knots.  A rule of thumb for light training aircraft is standard rate bank angle is 10% of the airspeed plus 7. With this rule of thumb, if flying at 100 knots the desired bank angle would be 10 +7 or 17 degrees.


One might intuitively assume incorrectly that the rate of turn increases with a constant bank angle if the airspeed increases.  If imagining a car going around a race track, it would seem logical that the faster the car travels around the track, the quicker the 360 degrees can be completed, and therefore the faster the rate.  In an aircraft however, with a constant bank angle, if the airspeed increases, the radius of turn increases with the square of this speed. The circumference distance that must be traveled also was squared so the track just got a lot bigger. Even though we are traveling faster, the distance to be traveled is increasing faster than our speed.  Therefore, the time it takes to complete the 360-degree turn increased and our rate of turn decreased.


Lets end this with the abolition of one other misconception. This concerns the existence of centrifugal force.  Imagine a driver is making a turn with her seatbelt fastened while her date, sitting next to her is not wearing his.  As she rounds a corner, her date appears to be repelled to the far side of the seat. Amused, and possibly happy with this apparent force she calls it “centrifugal” force.  The reason for this apparent force is that our frame of reference was being accelerated. If we were in an inertial (not accelerating) frame of reference, such as in a hot air balloon above, we would have seen that the driver was being accelerated by a centripetal force, created by the friction of the tires, toward the center of the turn. Her date was under the influence of Newton’s first law and continued on a straight line until acted upon by the outside force of the right hand door.  See Figure 3. Some people attempt to argue the existence of centrifugal force with the demonstration of a bucket of water swung overhead. They claim the reason the water stayed in the bucket was proof that centrifugal force is pushing the water to the outside.  What is actually happening is that the bucket is being accelerated toward the center of the turn (the shoulder of the one swinging the bucket) at a rate faster than the acceleration of gravity would cause the water to fall out of the bucket.  So Virginia, there is no such thing as centrifugal force.  

Publisher; Note: The original article will soon be on womanpilot.com



David Esser
Professor and Associate Chair
Embry-Riddle Aeronautical University
600 S Clyde Morris Blvd
Daytona Beach, FL 32114-3966
United States
Dr. Esser was a member of the Embry-Riddle Aeronautical University Flight Department from 1981 to 1995, and has been a member of the Aeronautical Science Department since that time. His undergraduate degrees are from Embry-Riddle in Aviation Management, Computer Science, and Aeronautical Science. In 1989, he was awarded the M.S. in Aeronautical Science from Embry-Riddle, graduating at the top of his class.
Professor Esser completed the Ph.D. Degree in Organization and Management Leadership from Capella University, again graduating at the top of his class. His dissertation pertained to airline Advanced Qualification Training and Line Check Safety Audits to validate Threat and Error Mitigation techniques. It also involved Flight Operations Quality Assurance and Aviation Safety Action Program data collection. He holds FAA Airline Transport Pilot certificate with a Type Rating in Airbus A319/320: Advised Graduate Research Projects and Thesis in Topics of CRM, AQP, Flight Data Monitoring, and FOQA.


About the Author

Chicago, Illinois

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