# How to calculate the radius of a turn as a function of speed, bank, wing-loading, and Cl?

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)?

• 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. Sep 3, 2020 at 15:37
• Please, see my comment for Answer 1. Thank you Sep 6, 2020 at 18:10

## 1 Answer

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.

• 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. Sep 6, 2020 at 18:04
• 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. Sep 7, 2020 at 0:03
• 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) Sep 7, 2020 at 0:38
• 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). Sep 7, 2020 at 3:35
• 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. Sep 7, 2020 at 3:45