# Why do airplanes fly?

PBRMEASAP
arildno said:
Besides, I would like to add, what is pertinent in a flight discussion is the pressure distribution NORMAL to the wing, i.e, the typical vertical pressure distribution.

Since Bernoulli's equation relates quantities along a streamline, rather than across them, I do not find Bernoulli's equation as the most natural starting point for the discussion of the flight phenomenon.

Then what makes an airplane fly?

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The wing pushes air down; Newton's third law pushes the wing up. Bernoulli's law has little or nothing to do with it.

- Warren

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chroot said:
The wing pushes air down; Newton's third law pushes the wing up. Bernoulli's law has little or nothing to do with it.
The angle of attack of the wing is important. But if air being pushed down is the only explanation, why would the top shape of the wing, particularly above the leading edge of the wing, matter?

AM

marlon
Indeed Andrew, Newton's third law is not the only important part here. Bernouilli's law tells us which structure the wings has to be

marlon

ramollari
chroot said:
The wing pushes air down; Newton's third law pushes the wing up. Bernoulli's law has little or nothing to do with it.

- Warren

All phenomena are explained by the Newton's laws. But it is more convenient to think in terms of Bernoulli's law for moving fluids.

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ramollari said:
All phenomena are explained by the Newton's laws. But it is more convenient to think in terms of Bernoulli's law for moving fluids.
I don't see how Bernoulli's law applies. Bernoulli's law is based on energy conservation. Here you have a wing striking air and imparting energy to it. You don't have an closed system in which air pressure is converted to kinetic energy of the flow.

What you have is air being deflected up by the leading edge. The air that would be sitting on top of the wing as the wing moves is, therefore, moving upward above the wing, leaving a partial vacuum above the trailing edge of the wing.

The result of all this, is downward movement of air because the air underneath is pushing up on the wing, the air has to move down. But the mechanism is a little more subtle than the wing just pushing the air down (although that is part of it - angle of attack).

AM

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Andrew Mason said:
What you have is air being deflected up by the leading edge. The air that would be sitting on top of the wing as the wing moves is, therefore, moving upward above the wing, leaving a partial vacuum above the trailing edge of the wing.
But that's just it: the center of lift is much further forward than that. Its only about 1/4 to 1/3 of the way back, near the thickest part of the wing. That's because that's where the speed of the air is highest, and thus (according to Bernoulli's eq) the pressure is lowest.

So, Bernoulli's does explain part of it, is just not the whole story. Also, Newton's 3rd is more an effect than a cause or an explanation. In a high angle of attack situation, its easy to see why air gets directed down, but that doesn't explain how you can get lift at zero geometric aoa.

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You made a few important remarks concerning irrotational flow in the other thread, I'll hope to get back to those later on.

I will focus here on the (in 2-D) TWO integrated expressions we can make of Newton's 2. law related to streamlines, in the stationary case:
1. The quantity which is conserved ALONG the streamline (i.e, what is given in Bernoulli's equation.
2. The integral of Newton's 2. law ACROSS the streamlines (Crocco's theorem)
Since the "stationary" case is only possible in the wing's rest frame, my comments will use this as the frame of reference henceforth (note that in the ground frame, in which the fluid is at rest in infinity, the time-dependent position of the wing will mean that the equivalent velocity field is time-dependent, according to the coordinate transformation given by Galilean relativity.)

But first, a few comments on chroot's post:

chroot gives an absolutely correct description of a flight situation, in that if the net effect on the air from the wing is a downwards deflection of the air, then by Newton's 3.law the air must impart an upwards force on the wing, i.e, lift.
However, I tend to regard this analysis as a GLOBAL analysis, in that it looks at a control volume of air surrounding the wing and calculates the net momentum flux out of that control volume.
This is, of course, both a permissible and intelligent way of viewing the problem, but what I would like to proceed with here, is what I call a LOCAL analysis, i.e, directly relating the air's acceleration in the vicinity of the wing and the forces acting upon it.
That is, Newton's 2.law locally applied on the wing.
I'll post more a bit later.

PBRMEASAP
chroot said:
The wing pushes air down; Newton's third law pushes the wing up.

I believe that. I would like to know how it happens. Arildno said he would explain.

Bernoulli's law has little or nothing to do with it.

I don't see why the two effects are unrelated.

Andrew Mason said:
I don't see how Bernoulli's law applies. Bernoulli's law is based on energy conservation. Here you have a wing striking air and imparting energy to it. You don't have an closed system in which air pressure is converted to kinetic energy of the flow.

Well, if it turns out that potential flow is a terrible model for airplane flight, then you are right. But in potential flow, energy is not imparted to the infinite fluid around it.

The result of all this, is downward movement of air because the air underneath is pushing up on the wing, the air has to move down. But the mechanism is a little more subtle than the wing just pushing the air down (although that is part of it - angle of attack).

Okay I'm confused. Which air is moving up and which air is moving down? The air pushes up on the wing, causing the air to move down? I'm sure I'm just misunderstanding you.

arildno said:
This is, of course, both a permissible and intelligent way of viewing the problem, but what I would like to proceed with here, is what I call a LOCAL analysis, i.e, directly relating the air's acceleration in the vicinity of the wing and the forces acting upon it.
That is, Newton's 2.law locally applied on the wing.
I'll post more a bit later.

Thanks everyone for your posts. Keep 'em coming :).

scarecrow
I always marveled at how massive an airplane is and yet still get off the ground gracefully.

russ_watters said:
But that's just it: the center of lift is much further forward than that. Its only about 1/4 to 1/3 of the way back, near the thickest part of the wing. That's because that's where the speed of the air is highest, and thus (according to Bernoulli's eq) the pressure is lowest.

That is where Static Pressure is the lowest while dynamic Pressure is highest. Total Pressure Remains the same. I know you know this Tom but I thought I might just add it in for better explination.

Regards,

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I will proceed with a local analysis of the (inviscid) flow over the wing, and its relation to lift.
For the present purposes, I assume that the fluid leaves the trailing edge in a smooth, tangential manner (apart from the formation of a thin wake region, this is what happens in reality, and in inviscid theory is known as the Kutta hypothesis).

Let us glance at the result from chroot's global analysis:
This relates the net downwards deflection of the fluid with the lift force.
But, if the fluid velocity upstream was strictly horizontal, that means that the fluid necessarily have experienced CENTRIPETAL acceleration, i.e, the streamlines must become CURVED when passing about the wing.

Locally speaking, the necessity of the presence of centripetal acceleration is a "trivial" insight, since the wing itself is curved..

But, those forces causing a particle's trajectory to curve, rather than accelerate the particle along a straight line, are the forces ortogonal to the trajectory, rather than the forces tangential to the trajectory.

In the case of the inviscid fluid where we neglect gravity, the force directly related to the curvation of the streamlines is given by the component of the pressure gradient normal to the streamlines, rather than the tangential component of the pressure gradient.

Furthermore, since by global analysis we may conclude that streamlines MUST curve in order for us to have any lift at all, it follows that the component of the pressure gradient most directly relevant for flight is the normal component, rather than the tangential component.
But, Bernoulli's equation essentially relates pressure values as given by the tangential component of the gradient (i.e, through the formation of the dot product between the pressure gradient and the streamline tangent, and then integrating).

From the above, it should seem more natural to fix our attention first upon the insights from Crocco's theorem, rather than upon Bernoulli's equation.

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Now, let's see how the presence of lift is plausible when considering Crocco's theorem, and typical airfoil shapes.
I'll get back to symmetrical wing shapes with an effective angle of attack later.

Now, let our first airfoil consist of a horizontal underside, and a curved form on the upper side, and let gravity be negligible:
We also assume that if we either go infinitely far from the wing horizontally or vertically, we end up in the uniform free-stream with constant pressure.
1. Vertical pressure distribution beneath the wing:
Since the underside is basically horizontal, we may assume that the streamlines underneath are practically straight horizontal lines (as they are in infinity), that is, particles passing beneath the wing don't experience any centripetal acceleration to speak of.
But that means, that the normal component of the pressure gradient on the underside is zero, i.e, a measure of the pressure directly beneath the wing is the free-stream pressure to be found at (vertical) infinity.

2. Vertical pressure distribution above the wing:
By assuming the typical negative curvature of the top foil, the pressure must increase upwards from the wing in order for the fluid to traverse the curve as determined by the wing.
Extending that increase up to infinity in the vertical direction, we may conclude that the typical pressure at the upper foil must be LOWER than the free-stream pressure.

But, combining 1+2 indicates the presence of lift..

Now, we may invoke Bernoulli:
Knowing that the pressure is typically lower on the upper side than the lower side, the measure of the velocity at the top of the foil must be greater than the measure of the velocity at the downside, i.e, we have a net CIRCULATION about the wing.
The relation between lift and circulation is known as Kutta-Jakowski's theorem.

Note that the "increase" of velocity at the top foil relative to the underside is consistent with the presence of a stronger centripetal acceleration on the upper side.

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Not to pick on you, warren, but...
chroot said:
The wing pushes air down; Newton's third law pushes the wing up. Bernoulli's law has little or nothing to do with it.
the reason I don't like this explanation is it sometimes leads to this erroneous description of the issue:
Andrew Mason said:
What you have is air being deflected up by the leading edge. The air that would be sitting on top of the wing as the wing moves is, therefore, moving upward above the wing, leaving a partial vacuum above the trailing edge of the wing.
Consider a flat-bottom wing at zero geometric aoa. Using Newton's laws and applying a little logic does imply that air is deflected up, causing a downforce on the front third of the wing and the air is sucked back down by the back 2/3 of the wing (and that's claimed in another thread as well). But that would cause no net lift, only a clockwise moment. But that isn't what happens - in fact, there is a counterclockwise moment and positive lift.

While Newton's laws can be used to calculate the net quantity of lift, they don't describe the airflow over the wing itself.

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PBRMEASAP
arildno said:
In the case of the inviscid fluid where we neglect gravity, the force directly related to the curvation of the streamlines is given by the component of the pressure gradient normal to the streamlines, rather than the tangential component of the pressure gradient.

Furthermore, since by global analysis we may conclude that streamlines MUST curve in order for us to have any lift at all, it follows that the component of the pressure gradient most directly relevant for flight is the normal component, rather than the tangential component.
But, Bernoulli's equation essentially relates pressure values as given by the tangential component of the gradient (i.e, through the formation of the dot product between the pressure gradient and the streamline tangent, and then integrating).

From the above, it should seem more natural to fix our attention first upon the insights from Crocco's theorem, rather than upon Bernoulli's equation.

Great. I am actually following you so far (and thanks for your detailed explanation, by the way). I think I can pinpoint the part of this explantion that confuses me. While I understand that the force on an element of fluid is proportional to the gradient of the pressure there, I do not see why one must distinguish between normal and tangential components of the pressure gradient at the wing surface. The pressure itself is responsible for the force on the wing surface. So even though Bernoulli's law generally relates pressures along streamlines (tangential to the wing), I don't see how that makes it any less valid in this case, since it does predict the pressure at the surface.

Of course, you arrived at this result from a different angle that is very enlightening. It just seems to me that Jukowski's theorem and Bernoulli's theorem are related in a way that makes it hard to say that one completely solves the problem, while the other is less important or practically irrelevant. Could you shed some more light on the distinction?

Also, what is Crocco's theorem? I am not familiar with it.

And please continue with the explanation, if you don't mind. I'm getting a lot out of it.

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You are completely right that one cannot dismiss Bernoulli's equation, i.e, basically the tangential component of Newton's 2.law; but neither must one dismiss that component of Newton's 2.law which is normal to the streamlines.
This is, however, what is ordinarily done when people try to argue from Newton's 2.law, and solely use the tangential integral (Bernoulli's equation).

We need the full vector equations here (i.e, what happens in "both" directions), otherwise we simplify our "explanation" to the point of misconstruction.
EDIT:
The basic weakness by trying to use only Bernoulli's equation (i.e, the tangential integral relation), is that you don't have any solid arguments for why the velocity should be higher on the upper side (and hence, lower pressure).

By connecting pressure differences to (effective) curvatures (or, rather, centripal accelerations), you DO get a rather powerful argument.
But that requires an analysis of the dynamics normal to the streamlines..
I'll get back tomorrow.

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PBRMEASAP
russ_watters said:
the reason I don't like this explanation is it sometimes leads to this erroneous description of the issue: Consider a flat-bottom wing at zero geometric aoa. Using Newton's laws and applying a little logic does imply that air is deflected up, causing a downforce on the front third of the wing and the air is sucked back down by the back 2/3 of the wing (and that's claimed in another thread as well). But that would cause no net lift...
Yes, I was thinking the same thing.

arildno said:
The basic weakness by trying to use only Bernoulli's equation (i.e, the tangential integral relation), is that you don't have any solid arguments for why the velocity should be higher on the upper side (and hence, lower pressure).

By connecting pressure differences to (effective) curvatures (or, rather, centripal accelerations), you DO get a rather powerful argument.
But that requires an analysis of the dynamics normal to the streamlines..
I'll get back tomorrow.
Okay, sounds good.

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russ_watters said:
Not to pick on you, warren, but... the reason I don't like this explanation is it sometimes leads to this erroneous description of the issue: Consider a flat-bottom wing at zero geometric aoa. Using Newton's laws and applying a little logic does imply that air is deflected up, causing a downforce on the front third of the wing and the air is sucked back down by the back 2/3 of the wing (and that's claimed in another thread as well). But that would cause no net lift, only a clockwise moment. But that isn't what happens - in fact, there is a counterclockwise moment and positive lift.
It is true that the upward deflection of air by the leading edge creates a downward force. But I don't see why the downward force cannot be less than the lift created by the resulting vacuum above the wing. The lift is created by a different mechanism: the pressure differential between the top and bottom surface of the wing. But perhaps I haven't thought it through enough.

Here is how I would calculate the downward force:

$$F_{down} = v_ydm/dt = v_y\rho dV/dt = v_y\rho A_{le}ds/dt = \rho A_{le}v^2sin\theta$$ where $A_le$ is the vertical cross-section area of the leading edge, $v_y$ is the vertical component of the upwardly deflected air, v is the speed of the wing relative to the air and $\theta$ is the upward angle of the deflected air.

The upward lift is the pressure differential x wing area - F_down. So:

$$(P_{bottom} - P_{top}) A_w - \rho A_{le}v^2sin\theta = F_{up}$$ where A_w is the area of the whole wing.

So if:

$$\Delta PA_w > \rho A_{le}v^2sin\theta$$ you should get lift.

I am not sure how to determine the pressure difference between the top and bottom surfaces! I'll have to think about it. But I don't see that the pressure difference is strongly related to the vertical speed of the deflected air. But as I say, I may be missing something.

My sense is that you should get just as much pressure difference if the deflection is at a small upward angle. What is important is that the wing cross section has to encounter a sufficient volume of air as it moves so that enough air is moving upward above the wing as the wing passes under it.

AM

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Andrew Mason said:
It is true that the upward deflection of air by the leading edge creates a downward force.
No, it is not. Most of the lift is generated on the leading third of the wing:

http://www.diam.unige.it/~irro/profilo_e.html
http://www.centennialofflight.gov/essay/Theories_of_Flight/Two_dimensional_coef/TH14G2.htm [Broken]

Caveat: I mentioned earlier a moment: it is clockwise with respect to the geometric center of the airfoil (if the leading edge is to the left), but counterclockwise with respect to the center of lift.
The lift is created by a different mechanism: the pressure differential between the top and bottom surface of the wing.
But even if there is no pressure change on the bottom surface (either positive or negative), a wing can still produce lift.

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russ_watters said:
No, it is not. Most of the lift is generated on the leading third of the wing:.
I didn't say that there was downward acceleration. I just said there was a downward force caused by the upward deflection of air. There has to be. That is just Newton's third law. That doesn't mean the front of the wing turns down. I also said that the pressure differential between the top and bottom surfaces overcame that downward force and provided upward acceleration.

AM

PBRMEASAP
arildno said:
The basic weakness by trying to use only Bernoulli's equation (i.e, the tangential integral relation), is that you don't have any solid arguments for why the velocity should be higher on the upper side (and hence, lower pressure).

Just thought I'd toss this out there:

What about continuity? Sufficiently far above the wing the streamlines flatten out again. So if you consider the flow between the upper wing surface (from the front stagnation point to the rear one) and one of those flat streamlines, the average velocity should increase over the thick part of the wing to make up for the decrease in area. I think this is equivalent to the curvature argument--just from a different view. Because in order for the continuity argument to really work, you first have to conclude that the curvature of the streamlines is changing the fastest near the wing surface. Otherwise it would be unclear exactly where the extra "fast moving" fluid is. Just a thought.

Russ,

Those links you gave are great. Could you explain more about the circulation? Specifically, the part about the clockwise vs. counterclockwise, center of lift vs. geometric center.

Thanks

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PBRMEASAP:
I'll just give a first comment to russ' excellent links.
In the first of these, it is made clear that viscosity plays a crucial role in the generation of lift.
I had not yet reached that point in my description, but basically, it is what justifies the first assumption I made, namely that the air leaves the trailing edge in a smooth, tangential manner (the Kutta condition).

However, I thought it most accessible to start with describing how the pressure distribution is in the lift-SUSTAINING situation; the effect of viscosity is so subtle that I think it is difficult to appreciate it before we have a clear picture of how the pressure works.

So, from what I can see, there is no disagreement between the picture given in russ' first link and my own description.

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Andrew Mason said:
I didn't say that there was downward acceleration.
Fair enough, I guess I misunderstood. In that other thread, it is claimed that that model will actually produce no net lift and a clockwise moment due to the molecules bouncing off the leading edge of the wing. You stopped short of that.
PBRMEASAP said:
Russ,

Those links you gave are great. Could you explain more about the circulation? Specifically, the part about the clockwise vs. counterclockwise, center of lift vs. geometric center.
Here's where it starts to get complicated. I'll try to keep it simple, not just for your benefit, but for mine - I had a hard enough time learning it, much less trying to teach it (hence, I'm a mechanical engineer now).

Circulation was mentioned before, but the main reason it comes into play in the first link is that the Kutta-Joukowski theorem is a simplified model which, among other things, ignores viscosity. Check out the description and depictions of flow around a cylinder in the first link (bottom row of pics, 3rd from right).

By inducing circulation (think: curveball in baseball), you not only create downwash behind the cylinder, but you also create upwash in front of it. The whole flow field around the cylinder is rotating.

It can be said that an airfoil is shaped in a way designed to produce such circulation.

PBRMEASAP
Russ:

I'm with you so far. Here is the real-world example I had in mind. When you "slice" a ping-pong/tennis/golf ball, a positive lift is introduced. In this case the circulation is clockwise. Of course, the reason for the circulation around a spinning ball is different from that of the wing--the no-slip condition causes the ball to pull the air around with it. Even though the physical reason for this is viscosity, you can still model it in potential flow by combining a doublet, vortex, and a uniform stream (at least I think that's how it goes). So my question is about the direction of the circulation. This seems to be a positive lift generated from a clockwise circulation. Is this relative to the geometric center or lift center? And what exactly is the center of lift?

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PBRMEASAP said:
Russ:

I'm with you so far. Here is the real-world example I had in mind. When you "slice" a ping-pong/tennis/golf ball, a positive lift is introduced. In this case the circulation is clockwise. Of course, the reason for the circulation around a spinning ball is different from that of the wing--the no-slip condition causes the ball to pull the air around with it. Even though the physical reason for this is viscosity, you can still model it in potential flow by combining a doublet, vortex, and a uniform stream (at least I think that's how it goes).
So far so good - for clarity: airflow is from left to right.
So my question is about the direction of the circulation. This seems to be a positive lift generated from a clockwise circulation. Is this relative to the geometric center or lift center?
That's relative to the airflow, ie, its around the entire object.
And what exactly is the center of lift?
From the cute little animations in that link, if you add up all the arrows showing forces, you get one resultant force from a single point on the wing. That's the center of lift. In a "real" airplane, the center of lift is located slightly behind the center of gravity of the plane in order to produce a counterclockwse moment (torque) that tends to push the nose of the plane down.

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The continuity argument for faster air velocity above than beneath, suffers from the "defect" that one might erroneously conclude that the average velocity in the strip above the wing is somehow a good measure of the fluid velocity AT the wing.
Another is that flow doesn't really become constricted as it does in a tube with solid walls. That's what the argument easily leads us to believe.

Arguing from the actual form of the streamlines (how they curve as determined by the geometry of the wing), as I've done, and show how the pressure distribution must be in order for this to be possible, is IMO, not as easily subject to similar erroneous conclusions.

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PBRMEASAP
arildno said:
Another is that flow doesn't really become constricted as it does in a tube with solid walls. That's what the argument easily leads us to believe.
I didn't know that. From the picture labeled "streamlines" in that first link from Russ, it appears that the streamlines get closer together on the top of the wing. That was what led me to that conclusion.

russ_watters said:
From the cute little animations in that link, if you add up all the arrows showing forces, you get one resultant force from a single point on the wing. That's the center of lift. In a "real" airplane, the center of lift is located slightly behind the center of gravity of the plane in order to produce a counterclockwse moment (torque) that tends to push the nose of the plane down.

Okay, I see what you mean. But the circulation of the velocity is still clockwise, right? I got confused when reading about the shedded vortices (in your link) that conserve angular momentum. It seemed they were going the opposite direction that I would expect.

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From a visualization perspective, the centripetal acceleration view easily makes clear that there is a stronger TURNING of the flow in the upper fluid domain than in the lower.

This empirically correct feature is not easily deducible from, say, the continuity argument.

This yields in my opinion a further reason to prefer the centripetal acceleration argument than the other.

PBRMEASAP
I'll buy that. I also find the centripetal acceleration argument easier to visualize. I was just trying to make the logical connection with the other argument.

And of course, I'm still interested in hearing the rest of your argument, with Crocco's theorem, etc.

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Well, let's take the Crocco's theorem bit:
When you derive Bernoulli's equation for (a not necessarily irrotational) inviscid fluid, you do this by forming the dot product between the equation of motion and the tangent of the streamline and then integrates along the streamline. Right?

Crocco's theorem is exactly the same procedure, but now, you form the dot product between the NORMAL of the streamline, and integrate along the line you then get (the normal line which at all points is normal to the streamline.)
Since the acceleration term along the normal of the streamline at a given point must be the centripetal acceleration, $$-\frac{V^{2}}{\Re}\vec{n}$$, at that point, integrating along the normal in the case of no volume forces yields the following (open) curve integral, symbolically:
$$p_{1}-p_{0}=\oint_{n_{0}}^{n_{1}}\frac{V^{2}}{\Re}dn$$
where I've chosen $$\vec{n}$$ to be the unit normal away from the center of curvature ($$p_{1},p_{0}$$ are the pressure values at the positions $$\vec{x}(n_{1}),\vec{x}(n_{0})$$ respectively.
That is Crocco's theorem.
Needless to say, the integral is practically impossible to evaluate independently; but knowing if we have positive or negative curvatures of the streamlines is sufficient to establish where the pressure is the greater.
From circular motion, we know that the pressure in the direction of the curvature centre must be lower than away from it, in order for the pressure force to provide the required centripetal acceleration.
Crocco's theorem is just a rewriting of this insight.

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PBRMEASAP
Aha! That is very neat. Does the V^2/R come from the Vx(curl V) term in Euler's equation?

russ_watters said:
Circulation was mentioned before, but the main reason it comes into play in the first link is that the Kutta-Joukowski theorem is a simplified model which, among other things, ignores viscosity. Check out the description and depictions of flow around a cylinder in the first link (bottom row of pics, 3rd from right).

By inducing circulation (think: curveball in baseball), you not only create downwash behind the cylinder, but you also create upwash in front of it. The whole flow field around the cylinder is rotating.

It can be said that an airfoil is shaped in a way designed to produce such circulation.
That is EXACTLY one of the main points my texts make about lift generation. You can accuarately model a lift producing body with inviscid flow and circulation:
"Since viscous effects are of minor importance in the generation of lift, it should be possible to calculate the lift force on an airfoil by integrating the pressure distribution obtained by the equations governing inviscid flow past the airfoil. That is, the potential flow theory discussed should provide a method to determine the lift."

"The predicted flow field past an airfoil with no lift (i.e. a symmetrical airfoil with zero angle of attack) appears to be quite accurate (except for the absence of thin boundary layer regions). However, the calculated flow past the same airfoil at a non-zero angle of attack (but small enough to avoid BL separation) is NOT PROPER AT THE TRAILING EDGE. In addition, the calculated lift for the non-zero angle of attack is zero-in conflict with the known fact that such airfoils produce lift."
Almost done...I promise.
"The unrealistic flow situation can be corrected by adding an appropriate clock-wise swirling flow around the airfoil (flow moving left to right). The results are twofold: (1) The unrealistic behavior at the trailing edge is eliminated and (2) the average velocity on the upper surface of the airfoil is increased while that on the lower surface is decreased. From the Bernoulli equation concepts, the average pressure on the on the upper surface is decreased and that on the lower is increased. The net effect is to change the the original zero lift condition to that of a lift-producing airfoil...The amount of circulation needed to have the flow leave the trailing edge smoothly is a function of the airfoil geometry and can be calculated using potential flow (inviscid) theory."

PBRMEASAP
arildno:

I see now that the v^2/R comes from the full (v . grad)v term. I had to go look up what curvature is, but now I think I see how you got it.

Take the dot product of Euler's equation with the unit normal:

$$\mathbf{v} \cdot \nabla \mathbf{v} = - \frac{\nabla p}{\rho}$$

$$\mathbf{n} \cdot (\mathbf{v} \cdot \nabla \mathbf{v}) = - \mathbf{n} \cdot \frac{\nabla p}{\rho}$$

Then use the definition of the (radius of) curvature to get

$$\frac{1}{\Re} = \mathbf{n} \cdot \frac{d \mathbf{\tau}}{ds} = \mathbf{n} \cdot \{ \frac{\mathbf{v}}{v} \cdot \nabla (\frac{\mathbf{v}}{v}) \} = \frac{\mathbf{n} \cdot (\mathbf{v} \cdot \nabla) \mathbf{v}}{v^2} \ \ - \ \ \frac{(\mathbf{n} \cdot \mathbf{v})(\mathbf{v} \cdot \nabla v)}{v^3}$$

Where d(tau)/ds is the derivative of the unit tangent along the streamline. The second term on the far right drops out because (v . n) = 0. Then we can substitute into the Euler equation to get

$$\frac{v^2}{\Re} = - \mathbf{n} \cdot \frac{\nabla p}{\rho}$$

Of course, like you said, it must be true because you know that whatever it is making the streamlines curve has to be a centripetal force, i.e V^2/R.

Fred:

When you add the swirl velocity that makes the Kutta condition hold, does it change the shape of the wing surface? I mean in the potential flow model, not physically, of course.

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So far no one has really mentioned surface smoothness as an essential part of wing lift. If wings get a tiny film of ice they stall. Particularly critical is the leading edge. Why does that happen?

AM