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Although the question of how to calculate the radius of a turn of an aircraft has been answered before using only the speed and the bank angle, the reality is that 2 aircraft with the same speed and bank angle will have a different radius of turn if their wing loading is different. Furthermore, two aircraft with the same speed, bank angle, and wing loading will not necessarily have the same radius of turn. The lift coefficient of the wing must be taken into account.

Since the answer posted before is not complete, can somebody provide an equation that takes into consideration the above factors (and any others that should be contemplated)?

TypeIA
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Aldo Polgeo
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    Why do you think wing loading affects turn radius? It doesn't, but if we can understand your thought process better, maybe we can provide a more useful answer that addresses the misconception rather than rehashing the existing Q&A's. – TypeIA Sep 03 '20 at 15:37
  • Please, see my comment for Answer 1. Thank you – Aldo Polgeo Sep 06 '20 at 18:10

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The formula for turn radius is Velocity squared divided by radial G. And since Radial G, (assuming a level turn) is absolutely dependent on bank angle, the turn radius IS absolutely dependent on Velocity and bank angle. No other factors are necessary.

You are incorrect that two aircraft with different wing loading will have different turn radii at the same airspeed and bank angle. Anyone who has flown formation, even if they are not familiar with the physics/mathematics involved, knows this from experience.

Charles Bretana
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  • From the book Fighter Aircraft Performance of WW2 A Comparative Study (Pilawskii, Erik), you can find: "Aerodynamically speaking, a design's wing loading plus its maximum lift coefficient Cl will determine the ability to turn in a small radius in the shortest possible time". Comparison testing of different fighters showed that they did not have the same turning radius when starting from the same speed and using the same bank angle. That is the reason of my question. – Aldo Polgeo Sep 06 '20 at 18:04
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    Wing loading determines the MAXIMUM turn performance, or, in other words, how tight an aircraft can turn before it stalls. . In other words, a an aircraft with high wing loading, as it turns tighter and tighter, will reach stall at at lower bank angle and turn rate than an aircraft with lower wing loading. But as they both increase bank angle and turn rate, flying together at the same airspeed, they will have identical turn rates, bank angles and radial g. – Charles Bretana Sep 07 '20 at 00:03
  • I also found this study that includes a formula (that I cannot add here as an image) but that shows that: From Eq. (11) it is clear that if we want have a low turn radius, that we would expect to need
    1. Large CL (CLmax)
    2. Low altitude (high density)
    3. High load factor n (nmax)
    4. Low wing loading (W/S)
     – Aldo Polgeo Sep 07 '20 at 00:38
  • The formula I mentioned indicates that the Radius of Turn is inversely proportional to air density, gravity, and lift coefficient. It is also directly proportional to wing loading and load factor ( n / sqrt (n2 - 1). The latter is a function of bank angle. This formula appears to make more sense than the equation presented before (in the previous answer) where only speed and bank angle appear as variables. If this latter formula was complete, an airplane could turn even when the density is zero (and we know an airplane needs air to fly). – Aldo Polgeo Sep 07 '20 at 03:35
  • If we imagine an experiment of an airplane flying at high altitude, there will be a point where the air density will be so small that all the lift is needed to keep the airplane flying. In this case, the radius of turn is (almost) infinite: the plane cannot turn or it would lose altitude. If another airplane, very similar but equipped with a larger wing, (thence lower wing loading) was flying, it could generate more lift. So it could bank slightly without losing altitude and therefore it could turn. Here we have a case where the airplane with lower wing loading enjoys a tighter radius of turn. – Aldo Polgeo Sep 07 '20 at 03:45
  • ALL your three above comments apply ONLY to the MAXIMUM POSSIBLE turn performance (minimum possible turn radius), NOT to the instantaneous turn radius. Instantaneous turn radius, (as described above), is dependant on velocity and radial G, and that is all. All these other factors only affect the capability of the aircraft to generate lift, which affects its capacity to generate radial G. In other words, all the stuff you're talking about only affects HOW MUCH RADIAL G an aircraft can generate at a particular airspeed, I.e., the MAXIMUM radial G it can produce. – Charles Bretana Sep 08 '20 at 20:54
  • @Aldo, It sounds like you may have mis-stated your question. I think you are actually asking about the formula for calculating the MINIMUM turn radius an aircraft can generate. If so, then those other factors you mention ARE relevant. But the ACTUAL turn radius at a given airspeed and bank angle (radial G) is NOT, in any way, dependent on them. It's simple. You can calculate the Minimum POSSIBLE turn radius, or you can calculate the ACTUAL radius the aircraft is flying on at any specific instant. For eg., whatever Wing loading, etc., if bank angle is zero, ACTUAL turn radius is infinite! – Charles Bretana Sep 09 '20 at 14:00
  • Charles, first of all, thank you for taking the time to answer my question. It is appreciated. I know you could be doing something else. – Aldo Polgeo Sep 09 '20 at 22:06
  • Second, since there is a lot of compiled evidence about the difference in turn performance between aircraft (minimum radius of turn without losing altitude) and apparently a general agreement that this performance difference was explained in part by wing loading I wanted to see if somebody had an equation that took every factor into consideration to compute this minimum radius turn. Pilawskii (in his book) mentions that he used a widely employed formula that includes wing loading, Cl, excess power among other factors. – Aldo Polgeo Sep 09 '20 at 22:37
  • The only formula I could get, included as mentioned before wing loading, load factor, air density, gravity and lift coefficient, so I wanted to see if somebody had a more precise version. I was really surprised that the answers I got were that turn radius (minimum turn radius is a type of turn after all) did not depend on wing loading. The formula where turn radius is calculated based only on velocity and bank angle (which affects radial g) and that was presented as a calculation to predict turn radius appeared to me like a simplification for pilots. – Aldo Polgeo Sep 09 '20 at 22:43
  • Why? because at the end what makes an aircraft turn is the interaction of two physical things: moving air and a surface. Remove one and there is no force to that allows your vehicle to turn. Imagine that you have a car moving fast and you place a ramp that tilts the automobile. You have the bank angle and the speed but the vehicle will NOT turn because it does not have wings and therefore does not generate a lift force in the desired direction. If you have an aircraft in a bank at speed and suddenly there is no air, the aircraft does not continue the turn. Surely a formula that does not ... – Aldo Polgeo Sep 09 '20 at 22:53
  • ...include the wing and the air appeared to be incomplete (at least for me) because it predicts that the car will turn and the aircraft will also turn in the absence of air. Also, if the mass of the Earth is reduced to the mass of the moon, the radius of turn should change. As a consequence, I have a problem trying to reconcile other formulas that seem to take into consideration factors that in the real world affect the performance of the turn and the other, simple equation where only two factors enter into play. – Aldo Polgeo Sep 09 '20 at 23:02
  • @Aldo, You are still missing the point. You are talking about an equation to calculate the MINIMUM POSSIBLE turn radius, but complaining about the existing formula for the ACTUAL turn radius. It's the same as measuring the HIGHEST that a person can lift his arm, versus the ACTUAL Height of his/her arm at the moment. Or the HIGHEST winds that an aircraft can take off in, versus the ACTUAL winds when it taxies on the runway. Please read my responses more carefully. – Charles Bretana Sep 10 '20 at 13:14
  • ... and your analogy with the car on a tilted ramp is totally off base. If the car dos not turn it is only because the turning force is opposed by sufficient sideways friction force from the wheels on the surface. If the car was rolling forward, and the wheels were allowed to free-cast the car would indeed turn and the turn radius would EXACTLY be equal to the Velocity divided by the Radial G. In this analogy the radial G would be equal to the sideways component of the force exerted by the tilted surface on the bottom of the wheels, DEPENDANT on the tilt angle of the surface. – Charles Bretana Sep 10 '20 at 13:20
  • And even if there is no air (in your other analogy), you have to understand that the formula for turn radius is R = V^2/Gr. When you see a formula with bank angle in it instead of radial G (Gr), it a simplification for AIRPLANES that assumes that the radial G is due to the sideways component of the lift generated from the tilted wing. The actual formula (R=V^2/Gr) is ALWAYS true, for an aircraft, a satellite, the moon, planets, a car on a track, or even for a kid on a merry-go-round. For an aircraft, the Gr is generated by lift because of the bank angle. – Charles Bretana Sep 10 '20 at 13:25
  • I am actually reading your responses carefully. Curiously, I also have the impression that you are not reading mine with sufficient attention but I try to avoid this type of sentimental language because it is not conducive to arriving at proper conclusions (I do not have evidence that you are not reading my comments carefully and neither do you). Second, I am not "complaining" about the equation of ACTUAL TURN RADIUS. (A) Yes, the equation I talked about is to predict the minimum lever turn radius of an aircraft. I am wondering if this equation is complete since at least one author includes – Aldo Polgeo Sep 10 '20 at 22:18
  • more variables. (B) when I compare the Minimum Turn Radius equation with the Actual Turn radius I noticed two things, first I do not understand the definition of Actual Turn radius, and second, I became intrigued on why using the equation for Minimum Turn Radius it is found that wing loading is a factor but when using the Actual Turn radius it is not. This seems confusing to me, so I went to great lengths to try to explain why it seems odd. In any case, this is my loss and here we are rapidly approaching the point of diminishing returns. – Aldo Polgeo Sep 10 '20 at 22:32
  • Look at https://www.physicsclassroom.com/class/circles/Lesson-1/Mathematics-of-Circular-Motion. Otherwise, I'm done with you. – Charles Bretana Sep 10 '20 at 23:01
  • The conclusion is that turn performance (defined as the ability of an aircraft to turn in the smallest possible radius without losing altitude) is a function of wing loading, Cl, and bank angle among other factors. For special cases, like coordinated, constant speed-bank angle-altitude turns a general physics equation using speed and bank angle as the only variables can be used to calculate the radius of a turn. However, this simplified formula has limited use to compare the max-turn performance of different aircraft (i.e.an aircraft can turn at a speed where another cannot fly and a maximum – Aldo Polgeo Sep 11 '20 at 02:37
  • performance turn bleeds speed, thus the equation is not applicable). As Pilawskii correctly explains, excess power is also an essential factor for a minimum radius turn because power needs to be applied to avoid losing altitude. – Aldo Polgeo Sep 11 '20 at 02:44