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Why exactly is lift coefficient constant with velocity in inviscid flow? (For a given angle of attack, 2D section and fluid, pre-stall etc)

Does it come out in the derivation of lift coefficient or does it just happen to be true?

The usual equation, Cl = Lift Force / (1/2 fluid-density X velocity-squared X area), just enables us to calculate the lift force knowing the velocity because Cl is constant with velocity

Thanks

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We have a natural phenomenon, which is that an airfoil creates lift, and scientists, engineers and others have been studying it for several centuries and came up with different physical-mathematical models describing what is happening, with varying levels of accuracy. One of those models is that you can determine the amount of lift (a force) by multiplying dynamic pressure (proportional to velocity squared), the reference area and a "coefficient of lift", which represents the effects of shape of the wing/airfoil and anything else that's unaccounted for.

Now, in practice, experiments showed that this is an accurate enough model for reasonable speeds, which are low compared to the speed of sound (you only mention "inviscid", but not "incompressible", but I'm assuming it here), and high enough for viscous effects to be limited to a very thin boundary layer, and low enough angles of attack to have no detached flow. Turns out, coefficient of lift is basically constant for a given airfoil at a given angle of attack in the aforementioned conditions and with the definition given above.

"Inviscid flow", which of course does not exist in reality, is another one of those (idealized/simplified) models that people came up with to be able to predict the flow around airfoils and airplanes and all kinds of objects, because they realized that it is accurate enough at least for certain shapes and for certain flying conditions. From the way inviscid flow is defined and how the pressure distribution around an airfoil is determined, it turns out that the coefficient of lift will indeed be exactly constant for a given airfoil and angle of attack.

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  • $\begingroup$ Thanks! Yes, incompressible - sorry I forgot to add that. IIRC wrt Inviscid - thats why the plots in Abbott and Van Doenhoff change slightly with Reynolds especially at higher AoA / separation. My question is more 'Philosophical' 'why' does it 'turn out' that Cl is const (assuming the above)? Is it possible to explain this reality in terms of mechanisms / regimes or other physical things? $\endgroup$ Commented May 20, 2022 at 11:48
  • $\begingroup$ Personally, I feel about that question the same way I feel about "Why do airplanes fly?": Not "because the air goes faster over the top something something Bernoulli" or "the air is deflected downward something something Newton", but simply "Because they do." Newton, Bernoulli etc. are just means to allow us to grasp the phenomenon better. Same with "Why is there gravity?" and other questions like that. If you want to ask "How ..." then the answer is indeed all the different models and explanations that we came up with. $\endgroup$ Commented May 20, 2022 at 13:45
  • $\begingroup$ Thanks. Yes, I see what you mean, our models are just models of reality. I think I am overthinking :) $\endgroup$ Commented May 23, 2022 at 8:48
  • $\begingroup$ @ Raketenolli: Perhaps the answer to my question is that in our equation / model none of the stuff that isnt explicitly referred to in the equation but is lumped into Cl does not change (significantly) with velocity (given invisc, incomp. and attached)? Cl includes foil shape and angle of attack , which dont change with velocity for example? $\endgroup$ Commented May 25, 2022 at 12:01
  • $\begingroup$ Only you know what is an acceptable answer to your question ;-) But if it is inviscid and incompressible it cannot be a detached/stalled flow, and the $c_L$ will actually be perfectly constant for a given shape and angle of attack, not just "not change significantly". $\endgroup$ Commented May 25, 2022 at 13:03
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I guess we should start with saying that over many orders of magnitude of Reynolds number, it would be likely that the coefficient of lift would in fact change. This is why insects, birds, and aircraft use different aerodynamics, depending on their size. However, if we are assuming the presupposition that it doesn't change is true, we can limit this to be for an aircraft in a certain range of viscosities.

Basically, the relative difference in the flow speed, how much faster the upper flow is relative to the lower flow (equivalent to the how much higher the lower pressure is relative to the upper pressure), is independent of the freestream velocity, and is dictated by the properties of the fluid and the geometry of the aerofoil. Even that is not necessary, it turns out that if an aerofoil is thin enough, then even the geometry does not affect the relative velocities or pressures.

So, while in a faster flow the total lift force may be greater, the coefficient of lift is only determined by the angle of attack, in a fluid with viscosity. That is, for a given freestream velocity (the TAS of an aircraft) the relative change in static and dynamic pressure is independent of the initial velocity and is only a function of the angle of attack.

Just to pre-empt any comments, circulation is a difference in flow speed above an aerofoil relative to below and is used to calculate coefficients of lift. It may be more common to talk about this as circulation, but when added to the freestream velocity component from potential flow, the net result is flow going much faster over the upper surface and much slower under the lower surface. And very importantly, when we do experiments in wind tunnels, we measure pressure using manometers so there is a lower pressure above relative to below. These two phenomena are in fact interchangeable, hence the reason we integrate the pressure coefficient along the upper and lower surfaces which is equivalent to the closed contour integral of circulation.

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