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arildno

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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 [tex]U\vec[i}[/tex]

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, [tex]p_{u}[/tex] should be LESS than the free stream pressure [tex]p_{0}[/tex], that is:

[tex]p_{u}<{p}_{0}[/tex]

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, [tex]p_{l}[/tex] also fulfills the inequality:

[tex]p_{l}<{p}_{0}[/tex]

(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 [tex]p_{u}\leq{p}_{l}[/tex] 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 [tex]p_{u}\leq{p}_{l}[/tex] 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).

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 [tex]U\vec[i}[/tex]

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, [tex]p_{u}[/tex] should be LESS than the free stream pressure [tex]p_{0}[/tex], that is:

[tex]p_{u}<{p}_{0}[/tex]

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, [tex]p_{l}[/tex] also fulfills the inequality:

[tex]p_{l}<{p}_{0}[/tex]

(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 [tex]p_{u}\leq{p}_{l}[/tex] 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 [tex]p_{u}\leq{p}_{l}[/tex] 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).

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