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Note that for the purposes of this question, I am only interested in the scenario described below. By 'steady-state' I mean to exclude from consideration the short-term effects of changing the flap setting during a turn.

In the YouTube video Should I Use Flaps in a Turn?, beginning at 10:59, there is a claim that, at a given airspeed and bank angle, having the flaps lowered will decrease the turn radius. This is followed by a demonstration in a PA-28 where airspeed, bank angle and height are being held as constant as possible (70 kts, 30 degrees and 2,000 feet respectively) through two 360-degree turns: firstly with flaps up and secondly with 25 degrees of flap. It looks pretty convincing if we assume the bank angle and speed were adequately matched in both cases (though there are some variations apparent in those shots showing the panel and forward view during these turns).

The explanation given is this: "The turn radius of an aircraft is directly related to the horizontal component of lift which, as you may remember, is how we get an aircraft to turn. Well, what do flaps do? Yes, they increase our drag but they also increase the amount of lift that our wings produce, so instead of increasing our bank angle (which increases our load factor) we can just lower more flaps. This allows us to tighten those turns up."

The doubt I have over this explanation is that, while (at least in the normal flight regime) lowering the flaps increases the lift at a given airspeed, air density and angle of attack, what's being held constant throughout these tests are airspeed, air density, and the requirement for the vertical component of the lift to be equal to the weight (this last constraint is on account of these tests being conducted at constant altitude, so there is no vertical acceleration).

Given that both tests are performed at the same bank angle, however, this last constraint on the vertical component of lift means that the total lift would be the same in each test, and the same can be said for its horizontal component. It is the latter which acts as the centripetal force for the turn, and the only other variable in determining the radius of the turn is the speed, but that is also the same in both tests.

If this phenomenon is real, what am I missing or getting wrong in this analysis, and how do we calculate or estimate the size of the effect?

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    $\begingroup$ You answered the question yourself in the second-to-last paragraph-- $\endgroup$
    – quiet flyer
    Jun 3 at 13:13
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    $\begingroup$ Well, I thought I had! $\endgroup$
    – sdenham
    Jun 3 at 18:47

1 Answer 1

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No.

... claim that, at a given airspeed and bank angle, having the flaps lowered will decrease the turn radius.

This claim is nonsense. For a coordinated turn, bank angle and speed alone are sufficient to determine radius. This can be derived from basic physics. The equation is:

$$ r = \frac{v^2}{g \times \tan(\theta)} $$

Where $r$ is the turn radius, $v$ is the speed, $\theta$ is the bank angle and $g$ is the acceleration due to gravity.

Note that shape of the airplane is not a variable in that equation. Physics don't care if flaps are extended or not, or if the plane even has flaps, or if it's even a plane at all.

The argument about flaps is based around the fact that they permit lower airspeeds without stalling (at the cost of increased drag). But that's not the claim in your question.

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    $\begingroup$ It's possible that a given aircraft has a different indicated airspeed for the same calibrated airspeed when the flaps are up and down. So this could appear to be the case, I suppose, although in fact the actual explanation would be that the airspeeds are not the same in the two cases. $\endgroup$
    – Chris
    Jun 4 at 2:59
  • $\begingroup$ @Chris-RegenerateResponse Good point! In this PA-28 140 manual it says at 70 mph IAS, CAS is 75 mph flaps up and 72 mph with 40° flaps (at 80 mph IAS they are 83 mph and 81 mph respectively.) This is in the right direction and turn radius goes with the square of the speed, but it does not seem to account for the all difference in the circles, plus only 25° flaps were used. (OTOH, the forward-looking shots from the cabin and the non-circular 'circles' show there was some variation in the flying.) $\endgroup$
    – sdenham
    Jun 4 at 13:42
  • $\begingroup$ @sdenham there may also have been variation in the wind. It's really hard to test something like this in flight! $\endgroup$
    – TypeIA
    Jun 4 at 14:58
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    $\begingroup$ Slightly nitpicky, but in the response it says that physics don't care " it's even a plane at all". In strict interpretation that's true of course, but the formula given assumes in its deduction that we're dealing with something that flies and can point its lift vector by banking (yes, rotary aircraft, and yes you could use the same formula to find by how much you need to "bank" train tracks to run through without lateral acceleration being felt). I guess my point is the assumptions put into the physics models to arrive at a formula shouldn't be dismissed in an offhand remark by accident. $\endgroup$
    – Cpt Reynolds
    Jun 6 at 22:56

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