Lift as a consequence of streamline arguments

[arildno]

A recently closed thread dealt with the generation of aerodynamic lift, in particular tied to a severe and justified criticism of the "equal transit time"-principle, which often is also called the "Bernoulli effect".


However, from what I read, no one showed how streamline arguments can properly be used, but only derailed them on the mistaken assumption that such arguments necessarily had to rely on the fallacious "equal transit time"-principle.

This is why I reopen this issue.

1) 2-D theory
I will focus on the lift in the 2-D case.
Clearly, reality is always 3-D, but if the wing is sufficiently long, we may expect a region of local 2-D flow about the wing (strip theory).

2) The upper&lower curves constituting the wing are streamlines in the fluid.
This is certainly not necessarily the case; separation of streamlines from the wing will occur at high speeds or too sharp curvatures in the wing's profile.

However, the streamline assumption is an important special case, and my argument will also give a simple description of why separation might occur under high speeds/sharp curvatures.

3) Wing's rest frame analysis
This is rather conventional, and we make the following initial assumptions:
a) Far upstream, the air velocity is $$U\vec[i}$$
b) The induced disturbance of the fluid velocity due to the presence of the wing is restricted to a zone about the wing and downstream indicated by some finite vertical distance measured from the wing.
That is, if you go vertically upwards or downwards some distance, you'll end up in the undisturbed free stream (horizontal) velocity region.
c) Since the buoyancy force of air is negligible compared the weight of the wing, we'll neglect it from now on.

4)Zero angle of attack
For simplicity, we will assume that the chord connecting nose and trailing edge of the wing is practically horizontally aligned.

5) Crocco's theorem.
Given that the upper surface of the wing is CURVED (downwards), we should ask ourselves:
Since the fluid particles following the wing experience centripetal acceleration, what must the VERTICAL pressure distribution be above the wing, in order to produce that acceleration?

(NOTE: We are therefore interested in the pressure distribution ACROSS streamlines, rather than ALONG. Crocco's theorem is the analogue integration of Newton's 2.law across streamlines, whereas Bernoulli's equation is valid along the streamline)
Clearly, since we have downwards curvature on all the streamlines above the wing, up to the region where we end up in the straight-line free stream-region, we must have that a typical measure of the pressure on the upper surface of the wing, $$p_{u}$$ should be LESS than the free stream pressure $$p_{0}$$, that is:
$$p_{u}<{p}_{0}$$

Now, assume that we have a slight positive curvature on the LOWER surface of the wing.
Proceeding vertically downwards to the free-stream, we see that the associated measure of the pressure, $$p_{l}$$ also fulfills the inequality:
$$p_{l}<{p}_{0}$$
(In the case of negative/zero curvature on the downside, the proper measure of pressure on the lower surface would typically be greater or equal to the free-stream pressure).



If now we assume the upside's curvature is stronger than the downside's curvature, it is reasonable that $$p_{u}\leq{p}_{l}$$ as long as:
The typical air velocity on the upside is either of the same order as the air velocity on the downside, or greater.



By this argument, the inequality $$p_{u}\leq{p}_{l}$$ looks quite natural, and we see that the pressure distribution necessary for the centripetal acceleration associated with stream-line movement, will typically generate a LIFT.


No use has been made of the fallacious "equal-transit-time" principle here.

We also see that the streamline behaviour places a demand after a specific pressure distribution, and it is natural to expect that if the required centripetal acceleration is "too large" for the fluid to cope with, SEPARATION will occur.
But since the required centripetal acceleration increases with fluid velocity and curvature, it is natural to assume that stream-line behaviour only will occur if these parameters are sufficiently small.


We might also use streamline arguments to show that the lift will be proportional to the product of the fluid density, air velocity and CIRCULATION about the wing (i.e, in accordance with Kutta-Jakowski's theorem).

 

[alexbib]

why do you say that the equal transit-time principle is fallacious? Is it experimentally false that the air above takes the same time to go across the wing as the air below?

 

[pervect]

why do you say that the equal transit-time principle is fallacious? Is it experimentally false that the air above takes the same time to go across the wing as the air below?

Both experiment and the correct flow equations (the Euler equations) show that the equal transit time explanation is wrong. See for instance

http://www.grc.nasa.gov/WWW/K-12/airplane/wrong1.html

(which, however, disucsses the theory more than the measurements)

 

[Clausius2]

Arildno,

After being read you're analysis I have little to discuss. As I said before I'm not an expert on lift (you seem to have read a bit more about that by the way). Take a look at this:

https://www.physicsforums.com/showthread.php?t=54326

That was my explanation to one guy who questioned about the famous principle. That principle is used when people want to compare upper and lower velocities. This only can be done by continuity. People usually adopt an equal transit time and different path length in each side, so that they apply bernoulli equation and relates both pressures. But we know from Kelvin theorem that just behind the airfoil must be a counterspinning vortex. Therefore, two molecules just at the airfoil front do not reach the rear part just at the same time. Moreover, Euler equations predict a tangential discontinuity (not real) at the rear.

What is actually true is that the responsible of lift are pressure forces over the wing. After all, upper pressure must be less or equal to lower pressure in order to sustain the wing. How is it reached that pressure difference?. The effect of loosing a part of static pressure due to the curvature (and subsequent flow acceleration) is right. But quantifying the lift generated by that method is a bit difficult in my opinion. Behind your argument is Bernoulli equation too.

As you have mentioned it, the responsible of lift is the Circulation:

$$\oint_{wing} \overline{u}\cdot\overline{dl}$$

A net circulation of the flow will generate lift. The problem here is to find a physical meaning for the circulation, which doesn't appear to be an easy thing for me. Your explanation about curvature is closely related to circulation. At first sight the non zero circulation of the flow would mean a global unbalance in velocity along the wing, and therefore a difference of pressures. That global unbalance is caused mainly by the wing shape.

This is my poor collaboration to this subject.

EDIT:

2) The upper&lower curves constituting the wing are streamlines in the fluid.
This is certainly not necessarily the case; separation of streamlines from the wing will occur at high speeds or too sharp curvatures in the wing's profile.

Upper and lower wing lines are always stream lines:

$$u=\frac{\partial \psi}{\partial y}$$

$$v=-\frac{\partial \psi}{\partial x}$$ in a local curvilinear system (x,y). and psi=stream function.

If there is no suction trough the wing surfice, v=0 and

$$\psi=constant$$ along the wing profile. (x local direction).

 

[arildno]

I will continue with a few remarks to fill out my own account, prior to going into an exchange of ideas and replies (which I'll get to in a while):
1)Recapitulation of first post
The difference in curvature in the wing profile plus the observation of upper&lower surfaces as streamlines, ought us to expect that the required centripetal acceleration on the upper side is somewhat larger than that on the underside, since c.a. is proportional to curvature (the proportionality factor being the squared speed).

Given this rough, first idea, it seems plausible to conclude that the vertical pressure drop on the upside (measured from the region of free-stream velocity) is greater than the pressure drop on the downside.
Hence, it follows that by these rough arguments, a lower (measure of) pressure on the
upside compared to the downside is to be expected, i.e, the conditions for lift is present.
These ideas relies on Crocco's theorem, rather than Bernoulli.
In particular, these ideas are certainly consistent with a failure of the "equal-transit-time"-principle, in that the upper velocity may well be greater than the lower; such a situation would simply sharpen the need for stronger centr. acc. on the upside (rather than acting against these ideas).
However, even in the (unphysical, as we shall see presently) case of strictly equal velocities, the stronger upside curvature necessitates a stronger centripetal accceleration there, and hence, a larger vertical pressue drop on the upside.

Now, let connect these initial ideas to Bernoulli considerations along the respective streamlines.
Since we have been led to expect a lower pressure at the upside, Bernoulli predicts a higher speed on the upside than the downside, which is in perfect accord with our initial ideas that the centripetal acceleration on the upside is probably greater than on the underside.

Note:
This should NOT be considered as an explanation of the generation of lift, but rather as clarifying the various elements present in an interconnected, lift-sustaining whole.

2) Bernoulli's equation and the presence of circulation:
Bernoulli's equation is an eminent strarting point in order to make the relationship between lift and circulation clear on a simple level.
(With a simple level, I mean to emphasize the difference between this approach and the subtle use of solving the problem with the aid of (for simplicity..) potential theory (i.e, a rigourous derivation of Kutta-Jalowski's theorem))

We start with defining the fluid velocity as:
$$\vec{v}=(U+u)\vec{i}+v\vec{j}$$
where (u,v) is the induced velocity field due to the presence of the wing, and (U,0) the free-stream velocity field.

We make the assumptions that the induced velocity field is rather small in comparison to U, and that the wing is very thin.

Now, by Bernoulli, the pressure distribution along streamlines are given by:
$$p=p_{0}+\frac{\rho}{2}U^{2}-\frac{\rho}{2}\vec{v}^{2}\approx{p}_{0}-\rho{u}U$$
according to our assumption of a small disturbance field.

Now, the lift (force per unit length) is given by integrating the vertical pressure component around the wing S:
$$L=-\oint_{S}p\vec{n}\cdot\vec{j}ds\approx\rho{U}\oint_{S}{u}\vec{n}\cdot\vec{j}ds$$
(the constant pressure contribution giving zero).

We now make the pleasing discovery:
$$\vec{n}\cdot\vec{j}=\vec{i}\cdot\vec{s}$$
where $$\vec{s}$$ is the local tangent vector.
Since we also have: $$\vec{s}ds=d\vec{s}$$
we find:
$$L\approx\rho{U}\oint_{S}(u\vec{i})\cdot{d}\vec{s}$$

Now, we make a sleazy use of the idea that the wing is THIN:
We approximate the wing by a simple horizontal line segment L!
Hence, we get:
$$L\approx\rho{U}\oint_{S}(u\vec{i})\cdot{d}\vec{s}\approx\rho{U}\oint_{L}\vec{v}\cdot{d\vec{x}}\approx\rho{U}\Gamma$$
where
$$\Gamma=\oint_{S}\vec{v}\cdot{d\vec{s}}\approx\oint_{L}\vec{v}\cdot{d\vec{x}}$$
is the circulation about the wing.

The curve integral on the horizontal line segment L $$\oint_{L}(u\vec{i})\cdot{d\vec{x}}$$
can clearly be replaced by $$\oint_{L}\vec{v}\cdot{d\vec{x}}$$
since $$d\vec{x}$$ is normal to the vertical part of the velocity field,
and the integral $$\oint_{L}(U\vec{i})\cdot{d\vec{x}}=0$$
since the curve integral traverses the same line twice.

To the order of this thin-wing approximation then, the horizontal velocity is discontinuous in the vertical coordinate at the horizontal line segment representing the wing, and this discontinuity ensures/is consistent with the presence of a non-zero circulation about the wing.

Bernoulli therefore shows us, that in so far that we have a non-zero lift, there has to be a corresponding non-zero circulation about the wing (or, if you like, vice versa).

I have used some rather sleazy maneuvers here to show the connection between circulation and lift, and a few comments should be made in connection with a proper derivation:
a) The horizontal line segment approximation makes a lot more sense in a (semi-) rigourous derivation, since it is actually an approximation of the mean-camber line of the wing rather than the wing itself (if my memory serves me right).
b) Properly formulated as a boundary-value problem, the boundary demand of zero normal velocity on the wing is of crucial importance; this piece of information haven't been explicitly used in my simplistic version.

c) In accordance with the requirement of conservation of circulation on a material curve (i.e, Kelvin's theorem), a proper treatment will find a counter-spinning vortex at the trailing edge, balancing the net circulation around the wing.

A comment on the "equal-transit-time"-principle:
Assuming the validity of the thin-wing approximation, my treatment shows that to the same order of accuracy, we should expect that principle to be VIOLATED, rather than affirmed (by the discontinuity of the horizontal velocity).

3) Angle of attack&effective curvature:
It is of some interest to see how the angle of attack concept ties in with a measure of curvature of the wing profiles.

Let us for this analysis assume a symmetrically shaped wing about the chord connecting nose and trailing edge.
From the previous discussion, we see that if this wing is horizontally aligned, we cannot really expect the presence of lift, since the curvatures are strictly equal. (The required pressure drops must typically be the same).

Let the tangent vector on the upper side at trailing edge in this (horizontal, no-lift) case be:
$$\vec{t}_{u}=\cos\theta\vec{i}-\sin\theta\vec{j}$$
whence it follows that the tangent vector on the lower side at the trailing edge is:
$$\vec{t}_{l}=\cos\theta\vec{i}+\sin\theta\vec{j}$$

We note therefore, (rather roughly) that the rotational displacement of the incoming flow $$U\vec{i}$$ around a given surface is given by the angle $$\theta$$

Let us now see what happens if we tilt the wing with an angle $$\alpha$$ with respect to the horizontal:
Then, the respective tangent vectors at upper and lower sides satisfy:
$$\vec{t}_{u}=\cos(\theta+\alpha)\vec{i}-\sin(\theta+\alpha)\vec{j}$$
$$\vec{t}_{l}=\cos(\theta-\alpha)\vec{i}+\sin(\theta-\alpha)\vec{j}$$
That is, over the same displacement (from "nose" to trailing edge), the fluid passing along the upper side experience a stronger rotation than that part of the fluid passing on the underside.
(It is readily seen that the downwards displacement of fluid flow is consistent with effective curvature ideas).

Thus, we should expect that an effective curvature difference is typically proportional to the angle of attack (one might make a bit more detailed arguments here..).

Which of these concepts we would like to use when concerned with lift, ought therefore largely to be a matter of preference; the ideas developed by one technique should be translatable into corresponding ideas by the other.

 

[arildno]

Clausius2:
Just one comment to this:

The problem here is that just behind the airfoil the flow is no longer irrotational. Therefore, the Bernoulli principle is not applicable.

This is untrue! (EDIT: See addendum)
In the stationary, case, we have:
$$\frac{D\vec{v}}{dt}=(\vec{v}\cdot\nabla)\vec{v}=\nabla(\frac{1}{2}\vec{v}^{2})+\vec{c}\times\vec{v},\vec{c}\equiv\nabla\times\vec{v}$$
Note:
I think I've got the right sign on the cross-product, but a possible sign error here is of no consequence further on.

The field formulation of the momentum equation can therefore be written as:
$$\nabla(\frac{1}{2}\vec{v}^{2}+\frac{p}{\rho}+gz)+\vec{c}\times\vec{v}=\vec{0}$$
Let now $$d\vec{s}$$ be tangent to a streamline S.
Hence, $$d\vec{s}$$ is of course parallell to $$\vec{v}$$, and Bernoulli's equation is simply the curve integral along S between two arbitrary points $$\vec{x}_{1},\vec{x}_{0}$$ of the tangential component of the momentum equation:
$$\oint_{S}(\nabla (\frac{1}{2}\vec{v}^{2}+\frac{p}{\rho}+gz)+\vec{c}\times\vec{v})\cdot{d}\vec{s}=\oint_{S}\nabla (\frac{1}{2}\vec{v}^{2}+\frac{p}{\rho}+gz)\cdot{d}\vec{s}=(\frac{1}{2}\vec{v}^{2}+\frac{p}{\rho}+gz)\mid_{\vec{x}_{1}}-(\frac{1}{2}\vec{v}^{2}+\frac{p}{\rho}+gz)\mid_{\vec{x}_{0}}=0$$

Hence, Bernoulli's equation is valid even if $$\vec{c}\neq\vec{0}$$

A vortex is generated behind the airfoil, in part because of the principle you mentioned is not yielded, and because of the Kelvin's Theorem.

Kelvin's theorem is valid for inviscid (not necessarily irrotational) flow, and the production of the counter-balancing vortex can be seen to conform to this.


Alternatively, using potential (that is, irrotational) theory on the lift phenomenon, generates a singularity of circulation strength equal to the circulation about the wing.
The production of singularities means of course, that the potential theory has been pushed to its maximum usefulness, where vortices appear as singularities whose effects cancel each other so that irrotationality is (just barely) preserved.

ADDENDUM:
In Norway, the pressure distribution in the fluid domain for irrotational/potential flow (stationary or not) goes by the name of "Euler's equation for the pressure".
But, as I now recall, this equation is often referred to as "Bernoulli's equation".
I now think it is this "Bernoulli's equation" you were referring to, rather than the equation for the pressure distribution in the stationary case along streamlines.
In that case, you're of course correct in saying that we should expect some trouble on the trailing edge.

Finally, I'd like to say that I think it is conceptually somewhat easier to realize the existence of a non-zero circulation about the wing as a result of a pressure difference (i.e, lift conditions), than the other way around. In that case, it is, in effect Bernoulli's equation which predicts a significant velocity difference between top and bottom, i.e, net circulation.

I know that this flies in the face of the standard interpretation, but I, for one, feel more comfortable with this version.

 

[krab]

I know that this flies in the face of the standard interpretation, but I, for one, feel more comfortable with this version.

So do I. And BTW, I want to thank you for your lucid and constructive analysis. The other thread provided a lot of heat and no light. I must say the Mentors in that case were no help whatever.

 

[Integral]

So do I. And BTW, I want to thank you for your lucid and constructive analysis. The other thread provided a lot of heat and no light. I must say the Mentors in that case were no help whatever.

I have very limited knowledge of this topic. If I don't know about something I tend to stay quite and watch. Clearly the previous thread was not going in a reasonable direction, which is why I finally locked it. This thread on the other hand has been on a much better track. I also appreciate the careful development, unfortunately, I am still out of my league in its evaluation. Thank you all for providing a lucid approach to the problem.

 

[FUNKER]

Also with this argument NASA website had the 5 most common fallacious theories of lift and the equal time theory was the number 1, just check the websiteskies

 

[Clausius2]

Clausius2:

This is untrue! (EDIT: See addendum)

Yes, you're right. The way you derived Bernoulli equation was one of the questions of some exam I did some time ago... . In fact I said it wrong. I appreciate too your effort at clearing it up. It can be considered as your gift to us for Christmas, isn't it? .

I agree with all your arguments, so I will try to summarize:

-Lift force is closely related to circulation of velocity. The circulation is closely related too with the wing shape, in particular with its assymetrical shape or at least with the attack angle if symmetrical (at this point I don't agree with the "centripetal force" you referred to, because it doesn't appear at all in the formulation as you may see). The circulation represents an unbalance of velocities, which ultimately causes a pressure difference as Bernoulli yields. But the principle of equal transit times has no sense.

-That principle is not valid due to the fact of the discontinuity at the trailing edge (in the case of vanishing viscosity) caused by Kelvin's Theorem. The flow remains irrotational except in the weak, where a counter-spinning vortex is formed in order to conservate the zero initial circulation.

What do you think?.

EDIT: as a cautionary note, and adding my ignorance to the pool, I have to say that potential theory is not my best, as it has been demonstrated in this thread. In fact, my proffessor never taught us nothing of potential flow, so the little I know is in my own. On the other hand, our knowledge in compressible flow is higher than the mean. As it is said in Spain, you cannot have all things in this life (that's spanglish ).

 

[4newton]

You need to look at lift on the atomic scale in order to understand the mechanism that results in lift. The shape of the wing is the important part. The molecules of air going over the top of the wing are attracted to the surface of the wing (gravitational force). As the molecules are diverted part of there momentum is transferred to the wing and the velocity of the air molecule is slowed down. The wing is shaped to try and maintain optimum distance between the air and the wing to provide the greatest lift. The attachment shows this function.

If the wing is not curved as also shown in the attachment the air is only able to transfer momentum to the surface for a very short distance. If pressure differential resulted in lift this would provide the most lift the fact that it provides little or no lift is proof that lift has nothing to do with pressure differential.

The velocity difference between the top and the bottom of the wing is the result of the transfer of momentum to the wing. Turbulence in back of the wing is the result of the slowing of the air over the top of the wing. The most turbulence is when the plane is landing and the velocity differential is the greatest.
Many small planes get knocked out of the sky when trying to land behind a large slow moving plane.

In icing conditions it takes only a small amount of frost to decrease the lift on the wing. This is due to a loss of laminar flow over the surface with an effective larger separation of the air and the wing with a small turbulence at the surface of the ice.

The problem a pilot has in icing conditions is that he is able to achieve lift off with the help of ground effect but as he tries to pull up he finds he has no more lift or a lot less than he is use to. Instinct tells him to pull up more. All during this time he will not have a stall warning. Stall warning only tells you when the air over the leading edge of the wing is turbulent. He will continue to increase the angel of attack until the plane has no lift from the wing or he does get turbulence on the leading edge of the wing and stalls.

 

[Clausius2]

Sorry but I cannot agree with your comments.

You need to look at lift on the atomic scale in order to understand the mechanism that results in lift.

That's not necessary at all. Navier-Stokes equation gives us several information without the need of looking at atomic scales.

The molecules of air going over the top of the wing are attracted to the surface of the wing (gravitational force).

That's false. Gravitational force is negligible at this flow regimes. The Froud number is usually so large as to neglect the Buoyancy terms.

The wing is shaped to try and maintain optimum distance between the air and the wing to provide the greatest lift. The attachment shows this function.

It has no sense. As far as I am concerned, none engineer thinks of that when designing a wing. The airfoil shape is more related to Arildno's explanation.

If pressure differential resulted in lift this would provide the most lift the fact that it provides little or no lift is proof that lift has nothing to do with pressure differential.

The unique forces acting on the wing are pressure forces. Therfore, differential pressure is the ultimate responsible of lift. When separation occurs at high Reynolds# or high attack angles, such differential pressure dissapears and the boundary layer becomes disrupted. Boundary layer attachment provides the pressure transmission to the body. Once the separation has overcome, pressure at the cross layer section is not uniform and might be fluctuant.

Turbulence in back of the wing is the result of the slowing of the air over the top of the wing. The most turbulence is when the plane is landing and the velocity differential is the greatest.

Turbulence? Who has talked about turbulence?. Potential theory is capable of explaining lift without turbulence. I don't think the most turbulence is reached when the plane is landing, moreover I think due to the ground effect the layer is proner to be attached. Moreover, turbulence has nothing to do with slowing down any flow. It is a problem of stability. What you actually meant was separation of the boundary layer, which has NOTHING to do with turbulence.

 

[|2eason]

The molecules of air going over the top of the wing are attracted to the surface of the wing (gravitational force). As the molecules are diverted part of there momentum is transferred to the wing and the velocity of the air molecule is slowed down

Congratulations! You alone have uncovered this huge conspiracy. Physicists have tried to keep this a secret for decades now. If the public knew that things could defie gravity by using gravity, we would all wave our hands and be sucked into outer space.

 

[arildno]

Now, I'll get back to various objections&replies given so far, but I would like to commend pervect for the excellent link provided.
(I HIGHLY recommend this link!)

My analysis is fully in accordance with what is presented there; what I've given is essentially the local analysis consistent with the global TURNING of the flow given there.
I'll prepare some more comments in a while.

 

[arildno]

4Newton:
Please remember that the "velocity field" and "pressure" are large-scale structures, compared to molecular dimensions.
From a molecular point of view the "velocity field" should ALWAYS be understood as an a quantity averaged over a humungous number of individual molecules.

Navier-Stokes/Euler, are properly seen from the molecular point of view, as equations predicting the evolution of this averaged quantity, rather than predicting the velocity profile of an individual molecule.

"Pressure" is a large-scale variable closely related to the average collision rates between molecules (i.e, how strongly molecules bump into each other).

"Pressure", and for that matter "viscous effects" are therefore not unconnected with molecular behaviour (as you seem to imply), but are the net aggregate effects of gazillions of molecular interactions.

What it means that whenever the Euler/Navier-Stokes equations are ACCURATE, is that we have a situation WHERE ALL DOMINANT MOLECULAR EFFECTS HAVE BEEN ACCOUNTED FOR, in our modelling of these in large-scale quantities.

Since aerodynamic modelling has been shown to give extremely accurate results (when using, say the Prandtl approximations), this means that there don't exist other dominant molecular effects than those implicitly contained in our approximations.

 

[4newton]

Clausius2

That's not necessary at all. Navier-Stokes equation gives us several information without the need of looking at atomic scales.

There is never a need to look deeper into a problem if you are happy with what you have. As an engineer I am sure you were told that you did not need to understand the physics of a problem to work a solution.

That's false. Gravitational force is negligible at this flow regimes. The Froud number is usually so large as to neglect the Buoyancy terms.

Again you are looking only at the macro understanding of the engineer. You have only two forces in nature that can have an effect on the wing, the gravitational force or a charge force. The air must impart momentum to the wing is some form. We know that the air on the bottom of the wing is not pushing up because the airflow under the wing is not disturbed. We also know that diverted pressure is not able to produce the lift otherwise a wing at a 45-degree angle would produce the most lift.

You may think of forces with different names such as viscosity, boundary layer or what ever but they can only still be the result of mass attraction, gravity.

It has no sense. As far as I am concerned, none engineer thinks of that when designing a wing. The airfoil shape is more related to Arildno's explanation.

As you can see from the posts everyone, except me, thinks the concept of lift is well understood. You can see that this well understood concept is different for everyone. Even the NASA link ends with a doubt. Engineers don’t have a problem. The equations work well enough to let man fly. This however does not lead to advancement of new ideas. If my idea is correct then the material of the surface of the wing should have some effect on lift. If the wing is coated with a thin layer of lead or gold the required area for equal lift may be reduced.

The unique forces acting on the wing are pressure forces. Therefore, differential pressure is the ultimate responsible of lift. When separation occurs at high Reynolds# or high attack angles, such differential pressure disappears and the boundary layer becomes disrupted. Boundary layer attachment provides the pressure transmission to the body. Once the separation has overcome, pressure at the cross layer section is not uniform and might be fluctuant.

Again what is the force of the boundary layer attachment? Remember your choices are limited in nature.

Turbulence? Who has talked about turbulence?. Potential theory is capable of explaining lift without turbulence. I don't think the most turbulence is reached when the plane is landing, moreover I think due to the ground effect the layer is proner to be attached. Moreover, turbulence has nothing to do with slowing down any flow. It is a problem of stability. What you actually meant was separation of the boundary layer, which has NOTHING to do with turbulence.

I see you don’t fly an airplane. When landing you extend flaps to create more lift at slow speed. This extends the surface of the wing allowing more lift. More lift imparted to the wing must result in a decrease of the momentum of the air. The momentum of the air is a function of the velocity of the air over the wing. The more momentum the air gives up the slower the air flows over the wing and the greater the velocity difference when the air from the top of the wing meets the air from the bottom.

When ice is on the wing there is a small turbulence at the surface of the ice, this brakes the boundary layer.
The ground effect I am talking about is the extra lift a wing gets from air between the ground and the wing.

 

[4newton]

|2eason

Congratulations! You alone have uncovered this huge conspiracy. Physicists have tried to keep this a secret for decades now. If the public knew that things could defie gravity by using gravity, we would all wave our hands and be sucked into outer space.

thank you for your humor. I of course find it funny for different reasons.

 

[4newton]

arildno

"Pressure", and for that matter "viscous effects" are therefore not unconnected with molecular behavior (as you seem to imply), but are the net aggregate effects of gazillions of molecular interactions.

I am not implying that any of fluid dynamics is incorrect. I am saying that you ignore the molecular mechanism and the basic force of nature in your consideration and as a result you arrive at an incorrect reason for the function. This requires you to develop a complicated reason that has nothing to do with the result. The theories that you are using already understand the mass attraction as the force involved. You do not need a run on theory.

 

[arildno]

4Newton:
Please read what I've actually read, not what you think I have said.

In particular, note that I have taken much care in denying that I have come with an explanation; what I have clarified is how the large-scale quantities in the lift-sustaining situation are related to each other.
That's quite a different thing, IMO.

 

[arildno]

Besides, you have not convinced me at all that you understand what pressure is.

 

[arildno]

" We know that the air on the bottom of the wing is not pushing up because the airflow under the wing is not disturbed."

It is quite clear from such comments that you are in complete misunderstanding of what "air flow" and "velocity field" and "pressure" is.
These are AVERAGED QUANTITIES, look up what that means.
To give you a few hints&advices:
1)
Consider two molecules "1", 2" with internal direction (edited from "distance) vector $$\vec{r}$$
and velocities (for simplicity we'll let all molecules mass be the same):
$$\vec{v}_{1}=\vec{U}+\vec{u}_{1}$$
$$\vec{v}_{2}=\vec{U}+\vec{u}_{2}$$
where the average velocity of sufficiently many molecules in a (rather tiny) region these inhabit will be $$\vec{U}$$.

Now, consider the quantity:
$$(\vec{v}_{1}-\vec{v}_{2})\cdot\vec{r}$$
a)Do you think that has something to do with the value of macroscopic pressure for that region?
b)How often does $$\vec{U}$$ appear in the expression in a)?

2) Consider a (humongous) group of molecules with well-defined average velocity $$\vec{U}$$ (yet contained in a macroscopically tiny region) approaching (really, really close!) the nose of the plane.
Will $$\vec{U}$$ for that group be the free-stream velocity?
Or perhaps something else? (Hint: Have you heard about the term stagnation point?)

3) Your idea of gravitational attraction is laughable.
4) You would do well to get rid of your unwarranted contempt for individuals involved in engineering&classical physics.
As you yourself has laid bare, your level of understanding of physics is on par with that of an ignorant, arrogant schoolboy.


From now on, you're on my IGNORE list, and I would appreciate that you post your idiotic comments elsewhere.

 

[pervect]

I'd like to thank arildno for his kind comments about the links I posted. The Glenn research website has been a favorite of mine for aerodynamic theory for a long time.

To avoid making this thread a carbon copy of the one that was locked, I will respond to 4Newton's personal theories in another thread, in the hope that this thread can stay "on course" and not get hijacked by 4Newton as the last thread did.

 

[arildno]

I plan to make quite a few remarks on some of the solid comments here (particularly from Clausius), but first of all, I would like to thank pervect for a very constructive attitude.

However, there is one very sleazy maneuver in what I've done, which I haven't really argued for:

When I decomposed the velocity field as (U+u,v), I blithely said that (u,v) is always much smaller in magnitude than U.
This is not strictly correct, close to the stagnation point(s), $$u\approx-U$$
that is not small at all compared to U!

Behind my "argument" was therefore an implicit assumption that the contributions from these (tiny) regions doesn't influence the total lift that much; rather than providing some suspect "argument" for this, I'll say that by the procedure I undertook, we were able to glean the correct relation between lift and circulation (Kutta-Jakowski's theorem).

As this comment shows, sleaze works best when noone's looking too closely upon it..

To make a couple of comments about proper procedure here:
1) In a proper, rigourous treatment , it is shown possible that you can split the problem into two subproblems:
a) The lifting problem
b) The drag problem.

The drag problem is concerned with a symmetrically shaped representation
of the wing horizontally aligned (i.e, a no-lift producing shape).

The parts of the total velocity field close to the stagnation point (the existence of which really invalidates my approach) appears within this sub-problem, rather than under a), where the asymmetries responsible for the lift,
is given by the concept of the mean-camber line.

 

[Clausius2]

Arildno, why the hell don't you have the Private Message contact enabled?. Due to the fact I cannot send you a PM, I should tell you do not be afraid if I don't give inmmediate answers in this thread. Thisthread and your explanation is worth of undertanding in detail.

But now I'm a bit under snow with some work...brrrr...wait a minute (or more)!.

 

[rcgldr]

I'll throw in my two bits here. I fly radio control gliders and have read a lot about the airfoil design used. The goal of an airfoil is to produce a certain amount of lift and drag (usually low drag) for a specific air speed range. The requirements of a slow speed aircraft like the human powered aircrafts require near tear drop shaped airfoils that are curved downwards, these are realtivly thick air foils. High speed aircraft require thin and nearly symmetrical airfoils. Hand launch gliders need a bit more chord than usual because of the Reynolds effect, they need the chord distance to compensate for the slow speed. The current trend is to use very light models (9 ounces), with 58 inch wing spans (limit of the HLG rules), and fairly thin airfoils. Larger models use more cambered airfoils, with thickness and amount of camber depending on desired speed range. The new trend is to used airfoils where the trailing edge is adjusted (ailerons and flaps moved together to do this). This gives better speed range, just as is done with commercial jets (which can also vary the chord length).

Remember that the orignal thread was a way to explain lift to a man's daughter, and I think the math part of this has gotten a bit overboard.

More comments here from this peanut gallery. Except for very slow speed aircraft, turbulent air flow is desired (since laminar air flow will turn turbulent anyway in most cases). Some wings use turbulators to break up the air flow over a wing in a controlled fashion, rather than have the laminar air flow break up over an undesirable portion of the wing.

An explanation for lift should be independent of the frame of reference. When considering the air as a frame of reference, the relative velocities of the air above and below the wing are opposite those if the wing is used as a frame of refenced. Using the relative velocities may be OK for mathematical approximations to lift, but it doesn't explain it. There are stagnant (relative to the wing) areas of air at the front and trailing edges of a wing, both of these move at the same speed, but the pressures are different because work is being done on the air.

The principles for lift are the same if using a flat board, a symmetrical airfoil, or a non-symmetrical airfoil.

Getting back to a wing, when a wing is producing lift, there's an effective angle of attack. The wing mostly accelerates air downwards (lift) and a bit forwards (drag). The main reason you get downwards acceleration of air from above a wing is that the wing passing through air creates a downwards and forwards moving "void", and the air is accelerated towards this low pressure area. It also creates a high pressure area underneath as it simply deflect air downwards (assuming an angle of attack here), but most of the lift is from above a wing, even in the case of flat board. Because air has momentum, this acceleration isn't infinite, and the low pressure area remains, causing air to acclerate towards it. Because of the effective angle of attack, this void is moving downwards as the wing passes through a volume of air, so the air is accelerated downwards. Since the wing also moves forwards, it's tapered to reduce the drag of accelerating air forwards.

The net result is pressure differential on the wing and acceleration of air downwards and forwards. I don't believe in a cause and effect here, both are required for lift. The wing reacts to the pressure differential by creating lift. However, this doesn't come for free. Work has to be done on the air, in order for this pressure differential to exists and be sustained, so the Newton aspect of force equals mass times acceleration, in regards to the acceleration of air downwards and forwards is also required.

Regarding circulation - air accelerates away from higher pressure areas to lower pressure areas. In the case of a wing, some air will circulate around the wing because of the pressure differences. This circulation is affected by the speed, effective angle of attack, and the size/shape a wing. One interesting wing design is a delta wing, which relies on the vortices that flow over the leading angled edge to allow these wings to generate lift at high angles of attack, like 20 to 22 degrees, much more than a conventional wing.

To get back to the answer for that man's daughter, a wing produces lift because it deflects the air downwards, just like a rudder in water, or sticking your hand out in the air stream while traveling in a car. It turns out that for most wings, that most of this deflection is due to the air being pulled downwards from above the wing, rather than being deflected from below the wing. Underneath wing, the air can accelerate away in all directions but upwards through the wing itself, so there's some lift generated. The air above is being drawn towards a downwards and forwards moving void created by a wing, and for most wings, there's more lift generated from above the wing rather than below.

 

[rcgldr]

Note that a box kite is a good example of a totally flat airfoil.

 

[Clausius2]

As an engineer I am sure you were told that you did not need to understand the physics of a problem to work a solution.

This gives the demonstration that you are the worst engineer. Engineers MUST understand the physics of a problem to work a solution. I DO understand the physics of the problem, at the same time I NEGLECT other effects which don't contribute significantly to the final solution.

Do you know who was Ockham? Maybe you should read his principle.

Again you are looking only at the macro understanding of the engineer. You have only two forces in nature that can have an effect on the wing, the gravitational force or a charge force. .

You may think of forces with different names such as viscosity, boundary layer or what ever but they can only still be the result of mass attraction, gravity.

FALSE. Viscous interaction and lift is not a question of gravity. In fact I think they are closer atomically to electromagnetic forces. Wrong again.

If my idea is correct then the material of the surface of the wing should have some effect on lift. If the wing is coated with a thin layer of lead or gold the required area for equal lift may be reduced.

Sure the material has something to do. Do you know what is the molecular rugosity?.

I see you don’t fly an airplane. .

That's TRUE. The unique true thing you have said here.

 

[arildno]

Jeff Reid:
Thank you for a very valuable contribution; from what I can see, all your insights are both important and correct.
However, I hope to show later on, that the issues you've raised are all consistent with my own approach, and, indeed, readily envisaged by it.
I plan therefore to show that our approaches are not mutually inconsistent, but rather as complimentary strategies.

Just one thing, though:
It is evidently true, as you imply, that the upper half of the fluid is more strongly TURNED than the lower half, and in this sense, contributes more to the actual lift.

Of course, I assume you agree with me that the actual pressure force from the upper half onto the air foil always works DOWNWARDS, never upwards.
But, reinterpreted in the terms of relative turning of the flow, it does make sense to say that the dominant contribution to the lift comes from the upper half.

But do you agree with me that envisaging, as I've done, the lift in terms of effective curvature differences that this insight is readily established?

Saying that the lift is connected to the fact that the upper side of the foil is more curved than the downside leads, in my mind at least, readily to the insight that the turning of the upper half of the fluid is stronger than the turning of the lower half of the fluid.
As an additional note on this feature, the excellent link provided by pervect discusses in some detail this stronger turning of the upper half of the fluid
(see for example, the discussion on the "skipping stone" theory)


Clausius2:
You pointed out early on a mistake on my part, in that the surface lines of the air foils are ALWAYS streamlines (contrary to what I said).
This mistake of mine was unfortunate, I can only defend myself that I at this time focused solely upon the inviscid region of the fluid; it is certainly correct to say that the wing surface cannot always be regarded as/approximated as streamlines for the inviscid region.
As long as the boundary layer approximations are accurate, the surface can always be regarded (to leading order accuracy) as a streamline for the inviscid region; when separation occurs, this is no longer true (nor is the pressure distribution in the viscous layer as neat and simple as when the boundary layer approximations can be used).

I will use the remainder of this post to focus on the separation phenomenon; in particular, I would like to post some thought on how this is related to STALLING, i.e., the dramatic lift reduction commonly associated with separation.

Now, separation will typically occur at points on the surface where the wall shear is zero, i.e, close to the separation point, there will be some backflow, so that a SEPARATION BUBBLE/vortex forms between the foil surface, and the streamline roughly denoting the limit betweeen the viscous and inviscid layers of the fluid.

There, is however, a subtlety connected to this which is of crucial importance for our purposes in getting some grip on the stalling phenomenon:

At first, we can say that the centers of curvature for the motion (trajectories)of the fluid in the viscous layer is no longer coincident with with the centres of curvature as determined by the foil surface (that is the case when the Prandtl equations are valid).

Rather, the centres of curvature in the viscous layer has now a complicated spatial distribution, in the form of vortex centres situated in the viscous layer itself.

Let us fix our attention to a single vortex (in the stationary case):
Locally, therefore, the particle trajectories has the approximate form of concentric "circles" about the vortex centre (that is, after all, the archetypal vortex representation), and let us furthermore assume that the local, associated velocity field is dominantly dependent on the radial variable (measured from the vortex centre).

It then follows, from Navier-Stokes, that only the pressure gradient can provide the centripetal acceleration associated with the vortex motion (viscous forces affects the tangential accelerations).
That is, the pressure must increase radially outward from the vortex centre.

But let us now envisage the separation phenomenon as follows:
At the onset of separation, points of low pressure separate from the foil surface, being displaced by points of (relatively speaking) high pressure, so that the conditions for vortex formation are established (for vortices grazing the foil surface, a rough value for the pressure at the airfoil must provide for the centripetal acceleration about the vortex centre).

But, it was the congregation of such low pressure points on the actual upper airfoil that was instrumental in providing the lift!

Envisaging the separation process in this manner, as a complicated mess where points of low pressure is kicked off the actual airfoil, being replaced by points of higher pressure, the STALLING phenomenon begins looking rather natural.
(I neglect the deviation in the lift due to viscous effects here)

(Note that this process completely disrupts the pressure gradients present in the non-separation case; the neat pressure distribution (in the viscous layer) in the Prandtl approximation has been totally invalidated).


Now, I would like to go a few steps further, in making a somewhat speculative estimate of the pressure distribution on the actual airfoil in the case of separation.
(I must emphasize that this IS rather speculative, and will probably contain some severe distortions of reality).

Let us graphically represent a portion of the upper foil as as a circular arc, and let the inviscid streamline separate fromthe foil, such that we have a roughly "triangular" viscous region between the streamline and the foil.
Now, place a vortex centre somewhere in that region, and draw concentric circles about it such that the outermost circle grazes both the airfoil and the streamline.
Regarding the velocity field in the direct vicinity of the the vortex as typically radial, it follows that the measure of pressure at the contact point of the outermost circle on the airfoil is typically of the same order as the pressure at the contact point on the inviscid streamline (they must basically provide the same centripetal acceleration about the vortex centre).
(We have neglected, amongst other stuff, the impertinent fact that the velocity at the contact point on the airfoil is zero, but non-zero at the streamline. We must therefore think of the velocity on the circe as an average velocity, averaged over the angles).

If we momentarily generalize this insight, a typical measure of the pressure on the air foil is representable by a typical measure of pressure on the SEPARATED inviscid streamline!

But THAT measure of pressure (i.e, on the inviscid streamline) can be found by relating the curvature of the separated streamline with the free-stream pressure by the aid of Crocco's theorem, as I did in the first post.

Since, typically, the curvature of the separated inviscid streamline is significantly less than that of the actual airfoil, this line of thinking provides an effective illustration of the truly dramatic lift reduction associated by stalling...

 

[arildno]

Now, the argument I presented above is decidedly sneaky, and it is my aim here to provide a minimal justification of it in showing that it might be consistent with actual observed features.

That a proposed mechanism is consistent with observed features, is a long shot from saying that the mechanism is TRUE; consistency is merely a minimum requirement that ought to be met.

1) Local analysis: Presence of pressure drag
Let us simplify the issue in assuming that the inviscid region in the upper half separates from the airfoil in a straight-line manner (i.e, no net turning of the upper half of the inviscid fluid has occurred).
Applying Crocco's theorem in the inviscid domain shows therefore that the pressure on the streamline separating the inviscid and viscous domains is of order free-stream pressure.

By my sneaky argument then, the pressure on the backside of the airfoil is of order free-stream pressure (i.e, a complete collapse of lift), whereas in front of the airfoil, the pressure is of order stagnation pressure, i.e, a pressure drag might be expected.
(I'll devote a later post on the topic on how we, in the unseparated, wholly inviscid, case, might envisage the production of the stagnation pressure at the back of the air
foil (i.e, basically the mechanism behind D'Alembert's paradox))

Note that if we were to use Bernoulli's equation along the dividing streamline, we would expect the actual velocity on that streamline to be of order free-stream velocity.
However, note that a constant pressure in an inviscid fluid may support a velocity field
$$U(y)\vec{i}$$ that is, where the velocity vary ACROSS the streamlines.

The fact that the streamline constitutes the boundary of the VISCOUS region,
should tell us that Bernoulli probably fails along this, but Crocco, which is evaluated across the strictly inviscid region might still be applicable.

That is, in this case of separation, we expect that the presence of the airfoil has produced a velocity field $$U(y)\vec{i}$$ in the upper fluid domain, with U(y) increasing towards the free-stream velocity when y goes to infinity.

I'll get back to a more global analysis later on..

 

[Clausius2]

Clausius2:
You pointed out early on a mistake on my part, in that the surface lines of the air foils are ALWAYS streamlines (contrary to what I said).
This mistake of mine was unfortunate, I can only defend myself that I at this time focused solely upon the inviscid region of the fluid; it is certainly correct to say that the wing surface cannot always be regarded as/approximated as streamlines for the inviscid region.
As long as the boundary layer approximations are accurate, the surface can always be regarded (to leading order accuracy) as a streamline for the inviscid region; when separation occurs, this is no longer true (nor is the pressure distribution in the viscous layer as neat and simple as when the boundary layer approximations can be used).

That seems to me more accurate. I (and Blasius too) agree totally with that.

I will use the remainder of this post to focus on the separation phenomenon; in particular, I would like to post some thought on how this is related to STALLING, i.e., the dramatic lift reduction commonly associated with separation.
...(and those all next ideas)
...

An EXCELLENT interpretation. It seems to me a bit rudimentary, but they are right just at the bottom. Good job, Arildno. Well, I have to go now to eat something. Maybe we will discuss about that some day...

 

[arildno]

Thank you for the kind words, Clausius (and Happy New Year, by the way!)

You are, of course, perfectly correct: My analysis IS rudimentary.
It has not been my aim to imply that we don't need careful experimentation and close study of the Navier-Stokes equations in order to fully understand these issues.

However, when faced by exceedingly complicated phenomena (as the lift&stalling situations), there is some value in trying to find out:
How much can we understand by relatively simple and primitive means; in particular, how should we try to develop our INTUITION of these phenomena?
That is, I have attempted to give some rough guide-lines which may be of some educational value, if not of accurate computational value.
Hopefully, it might also serve as a rough, memorial aid for persons not directly studying these phenomena, but want to have an essentially correct mental picture of the processes.

I'll get back sometime later.

 

[rcgldr]

Saying that the lift is connected to the fact that the upper side of the foil is more curved than the downside leads, in my mind at least, readily to the insight that the turning of the upper half of the fluid is stronger than the turning of the lower half of the fluid.

Ok, but the upper side doesn't need to be more curved. Even with a flat board, most of the downwash is produced above the wing. Also with thin airfoils, the curve on the top and bottom are the same, yet these can be fairly efficient.

In the case of a bus traveling down a highway, most of the pressure differentical is due to a low pressure region at the rear of the bus. In the case of a wing, it's a similar thing. When producing lift, a wing has an effective angle of attack (effective angle of attack is zero when no lift is produced). Using the simple flat board wing as an example, at a slight angle of attack, the bottom half is the "front", and the top half is the "rear". Similar to the bus, most of the low pressure region is generated at the "rear".

I like to call this "void" theory. A solid passes through a gas, leaving a moving void (low pressure region) directly "behind" it. The net direction of acceleration of air towards this void determines the amount of lift and drag.

A wing passes through the air with a slight effective angle of attack. Frontal area is small because of the slight angle of attack, so most of this "void" is introduced to the air as a downards moving "void", resulting in lift, and a small amount forwards, resulting in drag.

The reason for the curves in wings it to improve lift to drag ratio for a range of air speeds. Flat bottom wings aren't the most efficient, but they are easier to build than fully cambered wings. How lift is produced is the same for most airfoils. One exception being high drag lifting bodies with flat tops and angled bottoms, these look like fat flat bottom wings turned upside down and backwards. A lot of the force is drag, and most of the lift is deflection off the bottom surface.

 

[Clausius2]

Jeff,

I'm not an expert on aeroplanes, but I think Arildno analysis is emphatized on non symmetric wings with zero angle of attack. He wants to discover the lift phenomena without involving angles of attack, and without having to consider a stream deflection at the rear. He starts with only a nonsymmetric airfoil to discover that such assymmetry can be performed in a way to produce lift.

The example of deflection is usual in turbomachinery. I would like to discuss about the lift phenomena at blades. I would like to know if such pressure differential is the responsible of lift forces on the blades instead of the proper stream deflection as Euler equation of Turbomachinery states. But I have to take lunch just know...

 

[arildno]

Just a first comment:
As I showed early on (post 5) , a symmetric body given a non-zero angle of attack, will induce an EFFECTIVE stronger curvature on the upper side than the lower, by determining that curvature as a measure of how the fluid has typically been turned as passing by the fluid.

I'll get back to this later on, but here is the relevant passage:

3) Angle of attack&effective curvature:
It is of some interest to see how the angle of attack concept ties in with a measure of curvature of the wing profiles.

Let us for this analysis assume a symmetrically shaped wing about the chord connecting nose and trailing edge.
From the previous discussion, we see that if this wing is horizontally aligned, we cannot really expect the presence of lift, since the curvatures are strictly equal. (The required pressure drops must typically be the same).

Let the tangent vector on the upper side at trailing edge in this (horizontal, no-lift) case be:
$$\vec{t}_{u}=\cos\theta\vec{i}-\sin\theta\vec{j}$$
whence it follows that the tangent vector on the lower side at the trailing edge is:
$$\vec{t}_{l}=\cos\theta\vec{i}+\sin\theta\vec{j}$$

We note therefore, (rather roughly) that the rotational displacement of the incoming flow $$U\vec{i}$$ around a given surface is given by the angle $$\theta$$

Let us now see what happens if we tilt the wing with an angle $$\alpha$$ with respect to the horizontal:
Then, the respective tangent vectors at upper and lower sides satisfy:
$$\vec{t}_{u}=\cos(\theta+\alpha)\vec{i}-\sin(\theta+\alpha)\vec{j}$$
$$\vec{t}_{l}=\cos(\theta-\alpha)\vec{i}+\sin(\theta-\alpha)\vec{j}$$
That is, over the same displacement (from "nose" to trailing edge), the fluid passing along the upper side experience a stronger rotation than that part of the fluid passing on the underside.
(It is readily seen that the downwards displacement of fluid flow is consistent with effective curvature ideas).

Thus, we should expect that an effective curvature difference is typically proportional to the angle of attack (one might make a bit more detailed arguments here..).

Which of these concepts we would like to use when concerned with lift, ought therefore largely to be a matter of preference; the ideas developed by one technique should be translatable into corresponding ideas by the other.

Note that therefore the EFFECTIVE curvature concept is an IDEALIZED concept dependent upon foil geometry and actual angle of attack,
BUT THAT IS ALSO TRUE OF THE EFFECTIVE ANGLE OF ATTACK CONCEPT!
The one is no less true or more false than the other, they are complementary concepts.

Note that in my example, I have used a symmetric body with typically negative curvature on the upside and positive curvature on the downside .
The tilted flat plate gets the representation of having effective negative curvature on BOTH sides.

The basic geometric difference in the effective angle approach and effective curvature approach, is that the effective angle approach is a FLAT-PLATE approach (with the same lift dynamics as the wing), whereas effective curvature considers lift in terms of a body of non-zero width with the same lift dynamics as the actual body (I'll describe that ideal body in some detail later; I call it a "suction ovoid")

 

[arildno]

Now, I'll get back to tying together most of my ideas of "local" behaviour with a more "global" analysis, however I must emphasize the following fact about the centripetal acceleration/curvature concepts I've used here:

Most of the time, I have assumed that the lower foil surface has a POSITIVE curvature (typical with all (lift-inducing) symmetric, and many assymetric foils), which by Crocco's theorem indicates that the typical presssure value on the bottom foil is LOWER than the free-stream pressure.
That is, when comparing with the upper foil's negative curvature (which also implies
lower pressure than free-stream), the absolutely crucial lift-sustaining feature is that the upper foil is more strongly curved than the bottom surface.

This has been the dominant picture I've used, and should therefore be kept in mind when reading it.

I will here expand my initial comments on the situation when the bottom profile also should be assigned a NEGATIVE curvature.
Note that by applying Crocco's theorem DOWNWARDS TO THE UNDISTURBED FREE-STREAM, this situation implies that the
PRESSURE AT THE BOTTOM SURFACE IS GREATER THAN THE FREE-STREAM PRESSURE!

That is, we may expect a HEIGHTENED LIFT EFFICIENCY in this case, since the difference between upper and lower pressures has INCREASED!
(The pressure at the top foil is, due to negative curvature, still LESS than free-stream pressure)

This feature is utilized in aircraft during acceleration&take-off phases, in that A RETRACTABLE FLAP is lowered down at the trailing edge, contributing significantly to the build-up of lift (at the expense of heightened drag).
When the plane is up&flying, enough lift is produced by the high velocity, and the flap is retracted in order to reduce DRAG.