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I was wondering if the polar curve has differences in the takeoff due to the smaller extension of the flaps (this is half that in the landing) that changes the AOA for $ C_ {la} = 0 $. In addition, should I calculate the landing polar on approach?

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Yes, in most cases. How large? That depends.

A classic simple flap does these things to lift:

  • Adds an approximately constant amount of lift at a given AoA in the linear range. For small flaps deflections $\delta_f$ (up to about 15°, typical for takeoff), this amount is proportional to the deflection. After that, flow separates from the flap, and the lift return per deflection diminishes, down to zero in some cases.
    • For real wings of finite span, the added lift will depend on AoA itself, causing slight reduction of $C_{L\alpha}$. However, if we start considering real flaps that occupy only part of the span, situation will get more complicated; in some cases characteristics may improve with small deflections due to better lift distribution.
  • Loss of lift at stall becomes more abrupt, often much more.

This can be illustrated by this example from Hoerner, Fluid-Dynamic Lift, Chapter V:

Lift of a flapped rectangular model wing with A = 6

Fowler flaps will increase the $C_{L\alpha}$ derivative.

However, this by itself does not show much how the polar changes, apart from the distribution of $\alpha$ along it. We need to know how drag changes as well. As it happens, there is more added drag than lift, often dramatically:

L/D for flaps

Source: ibid.

Thus, overall and typically, the polar will shift up and right, more to the right than up. The polar coefficient (how much the curve 'slopes' to the right proportionally to the square of lift) may change either way, but usually not by much.

However, note that flaps are not the only difference between takeoff and landing. For propeller-driven aircraft, propwash will affect the polar significantly, and obviously more so on takeoff than on landing. In this question there is an example of such polar.

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The $C_L(\alpha)$ curve changes with flap/slat extension, so it should logically follow that the polar will also change, since the induced drag depends on the lift.

Furthermore the parasite drag will most likely change as well due to a changing airfoil surface.

So yes, the polar needs to be recalculated for all flight configurations.

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  • $\begingroup$ This should be accepted as the correct answer. $\endgroup$ – JZYL Jul 11 '19 at 14:36
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    $\begingroup$ @jimmy i was actually going to add a few images to illustrate now that the bot bumped this. $\endgroup$ – AEhere supports Monica Jul 11 '19 at 14:47

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