Understanding Flight: Pressure Distribution & the Science Behind Airplanes

[arildno]

Before I deal with angles of attack, and how this concept can be seen as related to curvatures/centripetal accelerations, I'll focus on a perhaps trivial feature we can see in a normal streamline picture:

In order to join the curved portion of a streamline directly above the wing profile with the straight, horizontal portions of the same profile (i.e, the shape in infinity), we need to add "small" circular arcs in front of and behind the wing of opposite curvature sign than the sign of the arc in the region directly above the wing.
("small" means either here a short curved segment, or very slight curvature on average.)
A similar argument holds, of course, for streamlines in the lower domain.

But, drawing normal lines from the wing to infinity through these portions clearly indicate that there are regions at the upper airfoil with HIGHER pressure than the free-stream pressure. These are of course the regions in the vicinity of the stagnation pressures at the leading and trailing edges.

That is, when we draw a typical realistic streamline diagram with a smoothly tangential flow at the trailing edge (i.e, consistent with the Kutta condition), we see that this is equivalent with placing the stagnation pressures AT the edges (where they belong).
That is, the Kutta condition could equal well be written in specifying where we want the stagnation pressures to be, and that is essentially how russ' first link writes the condition.

This should be taken as our first indication that the Euler equations (equations of motion governing inviscid flow) are possibly defective compared with say, the full Navier-Stokes equations; that is:
If we have to specify (in the stationary case) where the stagnation pressures shall be, in addition to the normal boundary conditions, how can we be sure that the unique solution of the time-dependent Euler equations (starting from the plane at rest in the ground frame) will converge towards the stationary solution (stationary, that is, as seen from the wing's rest frame) which fulfills the Kutta condition?
As it happens, it doesn't...

 

[PBRMEASAP]

In the steady state case, is the non-uniqueness of the solution a result of the flow region being multiply-connected? I guess I'm still trying to think of it in terms of potential flow, which I know I will have to abandon soon when we get to viscosity.

 

[arildno]

In the steady state case, is the non-uniqueness of the solution a result of the flow region being multiply-connected? I guess I'm still trying to think of it in terms of potential flow, which I know I will have to abandon soon when we get to viscosity.

Correct, although I am not sure if "multiply-connected" is the right topological term (that reveals my topological incompetence, I guess..)
As I've learned it, uniqueness of the solution to the Laplace equation requires that every simple, closed curve contained within the domain is reducible, i.e., basically that they can be "shrunk" to a single point while remaining within the domain.
(If we think of "infinity" as a closed curve, this ought to be the same as saying that the boundary of the domain is a multiply connected set..I think..)
Clearly then, a closed curve about the wing cannot fulfill this demand.
Uniqueness of the solution can then be found by specifying the circulation about the body.
The Kutta condition is really that circulation specification which places the stagnation points at the leading and trailing edges.
Most commonly, the Kutta condition is written in saying that the velocity field at the trailing edge must be finite there.

 

[PBRMEASAP]

As I've learned it, uniqueness of the solution to the Laplace equation requires that every simple, closed curve contained within the domain is reducible, i.e., basically that they can be "shrunk" to a single point while remaining within the domain.

Yeah, that's the way I remember it. Since there is a nonzero circulation around the wing, there must be some point(s) inside the wing where Laplace's equation isn't satisfied. So the domain can't be simply-connected.

 

[arildno]

Effective curvatures&angles of attack:

While it is important to know that an inviscid fluid cannot generate a lift, it should not be ignored that an inviscid fluid is perfectly able to maintain/sustain a lift.
This can essentially be regarded as a consequence of Kelvin's theorem, which states that the circulation on a specific material curve in a barotropic, inviscid fluid with conservative body forces remains constant through time.

So, I am going to focus on the lift-sustaining, stationary phase in this reply as well, before going over to analyze the lift-generating phase.

Andrew Mason, among others, has pointed out the importance of the angle of attack in lift calculations.

Let us focus on the airfoil with a straight-line underside, and a curved upper side, and give the wing a slight positive angle of attack, i.e, so that the vector normal at the leading edge has for example the vector representation $$\cos\alpha\vec{i}+\sin\alpha\vec{j},\alpha>0$$
$$\alpha$$ is then the "actual" angle of attack.
If our wing had been a flat plate, the lift can be shown to increase with the plate's actual angle of attack; a curved, real airfoil of non-zero thickness can be assigned an "effective" angle of attack, which is that angle a flat plate would need to gain the same lift as the curved airfoil does.
Hence, we can see that the "effective" angle of attack depends on two important features of a real wing:
1. The wing's "actual" angle of attack
2. The wing's geometry.
Information of the effective of attack of a particular wing is contained in knowing the mean-camber line of the wing (in slender wing theory).

It should be emphasized that the flat-plate approximation is just about the only practical procedure through which we may calculate accurate lifts (apart, that is, from a big Laplace solver); however, I find the concepts of curvatures and centripetal accelerations to be more illustrative of the physics involved, when we want to develop an intuitive image of what happens in flight.

Hence, I will develop the concept of "effective curvatures" which replaces "effective angles of attack".

Now, in order to gain a measure of the turning of the flow in the upper&lower fluid domains (which is related to effective curvatures), let us note the following for our airfoil with a positive angle of attack:
1. The "outward, leaving" tangent to the underside at the trailing edge has the representation $$-\cos\alpha\vec{i}-\sin\alpha\vec{j}$$
The fluid in the lower domain can then be said to have rotated from a strictly horizontal flow, through an angle $$\alpha$$ downwards.
Note that this places the center of curvature relevant for a streamline in the lower fluid domain beneath the streamline; i.e, we must expect that the pressure at the actual underside of the foil has increased, relative to the free-stream pressure.

2. Suppose that with zero angle of attack, the leaving tangent on the upper side has the representation $$]-\cos\beta\vec{i}-\sin\beta\vec{j}$$
By tilting the whole wing with $$\alpha$$ , the leaving tangent makes now the angle $$\alpha+\beta$$ with the negative horizontal.

Hence, the fluid in the upper domain has typically been more strongly turned with the non-zero angle of attack case than in the zero angle of attack case; and centripetal acceleration considerations suggests that the typical pressure drop between the free-stream and the upper foil has increased.
(Or: the upper foil has gained a stronger "effective" curvature)

Combining 1+2, we see that a stronger lift has been produced.
Although it is wrong to assume that all relevant geometric information of the wing is contained in the direction of leaving tangents, the curvature argument serves to make the following insights more intuitive:
1. The upper fluid domain is typically more strongly turned than the lower fluid domain.
2. The formation of the stagnation pressure behind the wing gets a neat illustration:
The fluid from the upper domain comes rushing down with a stronger measure of vertical velocity than the measure of vertical velocity the lower fluid has; i.e, a "collision" occurs where the two half-domains rejoins..
The stagnation pressure is then how the fluid deals with this tendency of the half-domains to collide into each other.
(Note: I do NOT mean that two fluid particles (initially "inseparable) which separated at the leading edge meets up again at the back, as if there existed some physical principle of "equal-transit-time".)
3. If you make the curvature of the underside negative (same as on the upper side), then this ought to boost the lift:
This insight is actually used in flight:
During the acceleration and take-off phases, many planes lowers extensible downwards flaps at the leading (!) and trailing edges.
While increasing the actual DRAG a lot, it is more crucial to gain the benefit of a high pressure zone beneath the wing during acceleration/take-off.
4. We also readily see how negative lifts can be the result of specific wing geometries/orientations.

 

[Andrew Mason]

Coanda effect

So far, no one has mentioned the Coanda effect as an explanation for wing lift. This is the effect (noticed for the first time, apparently, in 1930 by Romanian aircraft engineer Henri Coanda) of a fluid following the shape of a surface (as in water from a tap passing close to the side of a horizontal cylinder, following the cylinder surface instead of going straight down). See:
http://www.aa.washington.edu/faculty/eberhardt/lift.htm

I am having trouble understanding how the Coanda effect pulls air down around the wing. The Anderson/Eberhardt explanation, cited above, is about as good as I have found, but it seems to be an incomplete explanation of the physics involved.

Does anyone know if the Coanda effect works in a vacuum? If you placed a horizontal smooth cylinder in a vacuum chamber and shot a stream of water or air tangential to its surface, does the stream still bend as much?

AM

 

[PBRMEASAP]

3. If you make the curvature of the underside negative (same as on the upper side), then this ought to boost the lift:
This insight is actually used in flight:
During the acceleration and take-off phases, many planes lowers extensible downwards flaps at the leading (!) and trailing edges.
While increasing the actual DRAG a lot, it is more crucial to gain the benefit of a high pressure zone beneath the wing during acceleration/take-off.

This is particularly interesting! I had not noticed this. Apparently I am usually too busy watching my knees knock together to notice what's going on with the wings during takeoff . Also, just to clarify--is the fact that inviscid flow cannot generate lift also a consequence of Kelvin's circulation theorem? That is to say, if initially the flow is irrotational, it stays that way.

So far, no one has mentioned the Coanda effect as an explanation for wing lift. This is the effect (noticed for the first time, apparently, in 1930 by Romanian aircraft engineer Henri Coanda) of a fluid following the shape of a surface (as in water from a tap passing close to the side of a horizontal cylinder, following the cylinder surface instead of going straight down). See:
http://www.aa.washington.edu/faculty/eberhardt/lift.htm

I have not had time to read all this yet, but it looks interesting. Thanks for posting it. I agree that their explanation of the Coanda effect leaves much to be desired, especially since their premise is that the "popular explanation" isn't physical enough. I'll have to think about it some more.

edit: BTW, are you saying that the Coanda effect is produced even when the water does not actually touch the cylinder, but just passes by it closely?

 

[Andrew Mason]

edit: BTW, are you saying that the Coanda effect is produced even when the water does not actually touch the cylinder, but just passes by it closely?

I am not sure. In a vacuum, assuming that the Coanda effect works in a vacuum, I think it has to touch. But in air, it may simply have to contact the layer of air that is trapped next to the surface.

AM

 

[PBRMEASAP]

I read over the Anderson/Eberhardt explanation of the Coanda effect again, and I think there is something missing. They apparently only talk about shear forces due to viscosity. For example:

Because the fluid near the surface has a change in velocity, the fluid flow is bent towards the surface by shear forces.

But I don't think shear forces, either within the water or between the water and the cylinder, are sufficient to create a centripetal force. Or does viscosity also account for cohesive force between fluid elements, as in surface tension? If so, they could have explained that.

One of their complaints about the "popular explanation" is that it uses a shaky argument about velocities to deduce changes in pressure, when in fact the argument should be in the other direction. While I agree with this, it seems to me that they have not explained why there are pressure differences either. They just sort of state it as an obvious fact. For example:

When the air is bent around the top of the wing, it pulls on the air above it accelerating that air downward. Otherwise there would be voids in the air above the wing. Air is pulled from above. This pulling causes the pressure to become lower above the wing

(why? Bernoulli effect?).

I think they would benefit from reading arildno's posts in this thread. I'm not saying their article isn't useful--it is full of neat facts and figures, and I intend to go back and read it some more. But the idea that their explanation could replace the "popular" one is questionable, since it requires an awful lot of explanation and still leaves you high and dry in a couple places.

 

[Andrew Mason]

I read over the Anderson/Eberhardt explanation of the Coanda effect again, and I think there is something missing. They apparently only talk about shear forces due to viscosity. For example: But I don't think shear forces, either within the water or between the water and the cylinder, are sufficient to create a centripetal force. Or does viscosity also account for cohesive force between fluid elements, as in surface tension? If so, they could have explained that.

Right. I can see how water molecules can pull other moving molecules around a surface when they attach to the surface. But I don't see how air molecules can 'pull' on other air molecules like liquid water can.

AM

 

[russ_watters]

Right, that's not something I learned in my aero classes. Viscosity doesn't make water stick to a piece of metal, that's an actual electrical/magnetic attraction (the same attraction responsible for surface tension). That's not the mechanism behind viscosity of air and even if it were, pressure is a much, much bigger effect.

PBRMEASAP, the second quote (and your reaction to it) fits my impression: while a free stream of water is coherent and has no associated static pressure (A_M's statement regarding if it works in a vacuum...), air always has associated pressure. Air is not held to the wing via an attraction to the wing, its held to the wing because its being pushed from above via pressure. Flow separation occurs when that pressure becomes lower than what is necessary to hold the flow to the wing.

 

[russ_watters]

Something else not discussed much: VORTEX GENERATORS and laminar vs turbulent flow. I had a post all typed out, but lost it. So check out the link first...

 

[arildno]

russ:
I think we agree that separation occurs when the fluid is unable to generate the pressure gradient necessary to provide the fluid with the centripetal acceleration the curvature of the surface demands.
Now, if I read you a bit ungenerously, you seem to imply that the too weak pressure gradient is caused by a too low pressure in the inviscid domain outside the surface.
However, the weak gradient might also ensue if the fluid is unable to reduce the pressure at the surface sufficiently..

 

[arildno]

There is an unfortunate tendency of authors loving the Coanda effect to believe, or at least imply, that ONLY viscosity can make streamlines curve (through an adhesion effect).
They then proceed to kill the so-called "equal-transit-time/bernoulli"-explanation (which is easy, since that particular theory is sheer nonsense).

An inviscid fluid is perfectly able to curve its streamlines, but its mechanism for doing so is to create huge pressure gradients.
In fact, an inviscid fluid sees no problem with INFINITE pressure gradients, so an inviscid fluid can generate extremely kinked streamlines!
The adhesive effect of viscosity will, however, help a real fluid to traverse a moderately sharp curve.

EDIT:
Oops!
From what russ says, this cohesion should be thought of more akin to surface tension than viscosity per se. Dumb me..

 

[FredGarvin]

russ:
I think we agree that separation occurs when the fluid is unable to generate the pressure gradient necessary to provide the fluid with the centripetal acceleration the curvature of the surface demands.
Now, if I read you a bit ungenerously, you seem to imply that the too weak pressure gradient is caused by a too low pressure in the inviscid domain outside the surface.
However, the weak gradient might also ensue if the fluid is unable to reduce the pressure at the surface sufficiently..

Perhaps we're looking at it from a slightly different perspective, but I always remember that the separation was from a lack of momentum in the boundary layer to overcome the adverse pressure gradient on the back side of the object. I think we're saying the same thing...

 

[arildno]

Fred Garvin:
We are indeed speaking of the same thing..
However, I prefer to empasize the "adverse" word, in that the pressure further down the surface is too big to allow the streamline defining the boundary layer to follow the actual surface.
The reason I prefer this view is:
Simplify the separated region in letting the "inviscid streamline" pass over a vortex (i.e, constituting the vortex's upper part), with the nether part of the vortex lying at the actual surface.
Thus, there will be a backflow along the surface, which proceeds to whorl up around the vortex center (i.e, turning about 180 degrees around the vortex center).
The "backflow" streamline will lie directly beneath the inviscid streamline once it (i.e, the "backflow" streamline) has turned.
If we sleaze, and say that particles following the backflow streamline experience pure circular motion around the vortex center, then the pressure along the backflow is roughly constant; the pressure gradient formed by that pressure and the pressure in the vortex center providing the particle's centripetal acceleration.

But then it follows that the pressure at the inviscid streamline on the upper side of the vortex must roughly equal the pressure at the surface (since, by continuity of pressure, the pressure in adjacent segments of the inviscid&backflow streamlines must be about equal).
Since the inviscid streamline is a lot less curved than the actual surface, it follows that the pressure at the actual surface is a good deal higher than if the inviscid streamline had been firmly attached to the surface (since the inviscid approximation is good above the inviscid streamline).
This, in my mind, gives a neat illustration of the stalling phenomenon, i.e, the lift collapse experienced in separation.

 

[arildno]

russ and A.M:
Thank you for the new links; I will peruse them with pleasure..

PBRMEASAP:
I had thought to post a bit upon the (strong version of ) D'Alembert's paradox, i.e., how an inviscid fluid will fail to generate a lift (and not only the lack of drag), and why a viscous fluid evades that paradox; I guess I'll leave that till tomorrow..
While Kelvin's theorem does, indeed, predict that we can describe the fluid motion as irrotational, the primary reason for the lack of lift-generation, is that the initial condition makes the velocity potential a CONTINUOUS function of the spatial coordinates.
Note that, the point vortex has an associated DISCONTINUOUS velocity potential (in the angle, when described in polar coordinates); that's effectively why it can maintain a non-zero circulation (and hence, lift).
Since the initial condition of the wing at rest relative to the fluid (or, the fluid everywhere at rest), the velocity potential describing it is continuous, and D'Alembert's paradox will develop.

 

[PBRMEASAP]

Russ:
Thanks for the awesome link. As I was reading it, I was thinking, "hey, this is just like the dimples on golf balls." Come to find out, there is a link at the bottom that talks about that too! When they say that the turbulence/vortex motion adds energy to the flow, helping the flow speed up to overcome the adverse pressure, how does it do that?
----------------
I still get confused reading through all these different explanations though. I have a gut feeling they are all basically saying the same thing, even if they wouldn't readily admit to it. There seem to be some "chicken and egg" paradoxes with the flow separation. In the link on vortex generators, they say that the boudary layer gets pushed away by the adverse pressure on the trailing edge. This is certainly true, but flow would also separate because of good ol' centrifugal force if it weren't for pressure pushing it against the wing. Momentum doesn't really keep the flow moving along the wing, as they seem to imply. So it must really be a delicate balance of adverse pressure free stream pressure, and turbulence that decides whether flow stays attached or doesn't.
------------
arildno:
I look forward to hearing about D'Alembert's paradox. That one has been bothering me for a while.

 

[arildno]

----------------
I still get confused reading through all these different explanations though. I have a gut feeling they are all basically saying the same thing, even if they wouldn't readily admit to it. There seem to be some "chicken and egg" paradoxes with the flow separation. In the link on vortex generators, they say that the boudary layer gets pushed away by the adverse pressure on the trailing edge. This is certainly true, but flow would also separate because of good ol' centrifugal force if it weren't for pressure pushing it against the wing. Momentum doesn't really keep the flow moving along the wing, as they seem to imply. So it must really be a delicate balance of adverse pressure free stream pressure, and turbulence that decides whether flow stays attached or doesn't.
------------

Yeah, separation IS difficult (and I'm certainly no expert on it)
Now, the momentum perspective (which Fred Garvin notes) is somewhat curious, in that if it were more (tangential) momentum, it ought to be more difficult to warp it around a curve.

My own (very private!) resolution is as follows:
Let us consider a fluid element ("FE") "sitting" at a point where separation might occur.
Let us say that there is oppositely directed momenta on either side of "FE" (I.e, some backflow at the backside of "FE")
If now the momentum contained in the fluid approaching "FE" is a lot bigger than the momentum contained in the fluid on the other side of "FE" (that is, in the "backflow side") , then this ought to generate anet pressure force on "FE" so that it is dislodged from its position and rushes downstream (thereby eliminating backflow at that point on the surface).
Consider what would happen, however, if there weren't any net pressure force acting on "FE". Then, it would remain in place, and due to the momentum crushing onto it from both sides, a stagnation pressure would develop WITHIN "FE".

But that stagnation pressure would then force the onrushing fluid to veer off the surface..

 

[arildno]

The lift-generation phase: Failure of inviscid theory and the role of viscosity

Andrew Mason (and in earlier thread, Jeff Reid) has pointed out that if the wing moves forwards, then a region behind&above it becomes evacuated, and that this indubitably occurring process must have some relation to lift.
It turns out that the evacuation (or, rather evacuation rate) is not as such directly responsible for lift; rather, it is the completely different response to such evacuation a viscous fluid displays (compared to the response of an inviscid fluid) which generates the actual lift.


We need therefore to study in detail the inviscid fluid's response to an evacuation rate in order to appreciate the role of viscosity.
Let us work within the ground frame, with both the wing and the air initially at rest.
Also, I will solely concern myself with the development at the trailing edge; let the underside be horizontal, and the upper side of wing curved.

Now, give the wing an acceleration (or, if you like, a jump velocity).
In order to illustrate the evacuation rate, let us draw a following picture:
Draw the "previous" curve the upper side inhabited.
At the bottom, that is the position where the trailing edge was situated, draw a small horizontal segment to the trailing edge's new position, and draw the upper side where it now is.
Thus, we have drawn an evacuated region, which is bounded below by the horizontal segment, and whose sides are the curved outlines of the upper side of the wing.
Since the region is evacuated, the fluid elements adjoining it, will experience a net pressure force from the ambient fluid so that they rush into the evacuated zone.
Now, pay attention to the fluid element directly beneath the HORIZONTAL line segment.
Clearly, this will be accelerated UPWARDS into the evacuated zone, and will, in fact, hug the upper side as it speeds onwards.
That is, a BACKFLOW is created along the upper side, and a stagnation point will develop somewhere on the upper side, when the back&up-flowing fluid collides with downrushing fluid.
This can then be negotiated as follows:
The uprushing fluid bounces through a 180 degrees turn, i.e, twisting its velocity to gain the same direction as the rest of the fluid.

Note however, how this is contrasted with the image of the flow given when the Kutta condition holds:
There, the stagnation point was firmly fixed at the trailing edge, but here, the stagnation point might well be situated somewhere on the top side (the strong pressure there should clearly reduce the lift).
We can also, of course, regard the upflowing fluid to generate counter-acting circulation, and hence, lift-reduction

Now, before I proceed further on the inviscid theory, we can see the role of viscosity clearer:
Take a pencil and thicken the actual upper wing somewhat, illustrating a boundary layer which remains firmly attached to the wing.
Whereas the inrush of fluid into the evacuated region through the curved side of the wing's previous position is unaffected by that strip, not so at all with the uprush of fluid through the horizontal segment!
On that fluid piece, there will be a strong resistive force acting upon it from the boundary layer.

Thus, a viscous fluid actually favours downrush into the evacuated region above uprush hugging the airfoil..
I'll proceed further sometime later..

 

[PBRMEASAP]

Now, before I proceed further on the inviscid theory, we can see the role of viscosity clearer:
Take a pencil and thicken the actual upper wing somewhat, illustrating a boundary layer which remains firmly attached to the wing.
Whereas the inrush of fluid into the evacuated region through the curved side of the wing's previous position is unaffected by that strip, not so at all with the uprush of fluid through the horizontal segment!
On that fluid piece, there will be a strong resistive force acting upon it from the boundary layer.

Ah! So it is the boudary layer that causes the stagnation point to occur at the trailing edge. And that means that viscosity, although dissipative, actually aids in generating and sustaining lift. Very cool.

Now, pay attention to the fluid element directly beneath the HORIZONTAL line segment.
Clearly, this will be accelerated UPWARDS into the evacuated zone, and will, in fact, hug the upper side as it speeds onwards

In potential flow theory, the sharp trailing edge is sort of a singular point, right? What happens to the velocity there? Is it infinite, zero, or finite? In order to be continuous it seems like it would need to go to zero, but I don't know whether the velocity must be continuous on the wing surface or not. Now that I think about it, it must not necessarily be zero because that would be a stagnation point.

 

[arildno]

As to potential theory:
For all FINITE times (i.e, in the time-dependent problem), the velocity at the trailing edge is finite.
But this does not mean at all that when time has gone to infinity, (and the stationary picture in the wing's rest frame has developed) that the velocity must be finite there.
In fact, it isn't.
This highlights yet another mathematical reformulation of Kutta's condition : namely, that the velocity at the trailing edge must be finite. Only a single, non-zero circulation value is able to achieve this.

An unnecessary note perhaps:
D'Alembert's paradox pertains to the stationary, steady motion case. That is, potential theory certainly predicts forces to act upon an object in the non-stationary case, and those are in essence, the forces needed to accelerate fluid volumes with mass. For the general, non-stationary case, these inertial forces tends to swamp the effect of the viscous forces, so that potential theory remains very useful in many time-dependent problems (but not in the evolution of flight..)



As for the viscous case:
Just to clarify, I do not mean that right from the start, there will be no upflow at all.
Rather, there will be some upflow**, but that upflow doesn't carry the amount of momentum which would have been present in the inviscid case, and thus, the fluid is enabled to gradually shed it off in the form of vortices (see russ' excellent link on this process).
Gradually, therefore, the stagnation point will be pushed downwards to the trailing edge (i.e, the establishment of kutta's "condition").
Thus we see that it is precisely BECAUSE viscous forces are dissipative that flight occurs: It is essentially the stronger dissipation of backflow than down/in-flow which tilts the balance in favour of lift-generation.
That is, flight is the effect of a necessarily skewed spatial distribution of dissipation.


**: Not of course, AT the actual surface, but (arbitrarily) close to it, inside the inner part of the boundary layer).

Note that, mathematically speaking, it is that boundary condition we have to discard in the inviscid theory (no tangential velocity) which saves us.
Thus, flight generation is really a striking illustration of a singular perturbation theory:
If we try to only use the "outer" solution of Navier-Stokes (i.e, the solution of the Euler equations), our problem collapses into the evolution of D'Alembert's paradox.
However thin, the "inner" solution (i.e, Navier-Stokes in the boundary layer) cannot be neglected if we want a realistic solution of the problem.

Note:
Just a slight correction to what you said:
It is NOT necessary to take into account the effect of the boundary layer in the MAINTENANCE of the stagnation point at the trailing edge.
Once enough momentum has been imparted to the fluid, and the unequal pressure distribution has been developed, we have a totally different inviscid picture than when we started with everything at rest and uniform pressure:
Now, if we regard the situation from the ground frame, the fluid is already rushing down.
If we therefore look at the evacuation picture again, that downrush is just sufficient to prevent any net upflow, i.e, the stagnation point re-establishes itself at the actual trailing edge.
The inviscid fluid is therefore able to maintain flight.
Let us analyze this new inviscid situation further.
Suppose that we have gained a stationary flight situation, and proceeds to tilt the wing, maintaining its march velocity.
We therefore enter a new, non-stationary phase; what circulation level should we expect to occur once stationary conditions becomes re-established?
When regarding the fluid as inviscid, but with circulation, we ought to expect from Kelvin's theorem that the circulation will remain CONSTANT.
This is certainly true for the circulation on closed, material curves in the fluid; it necessarily remains so for the circulation about the wing in so far as it is correct to assume that the wing itself constitues a closed, MATERIAL curve for the fluid.
Let us suppose it is..(there remains a tiny doubt, however: If the initial material curve gets kinked or something, can't it happen that the wing might pierce it somehow? I'm not entirely sure on this..)

Leaving tiny doubts aside then, we should expect that if you tilt a wing (changing its geometry from a dynamical perspective) in a lift-sustaining inviscid fluid, then that tilting wouldn't have any effect whatsoever on the lift which would ensue once stationary conditions re-establishes itself. That is, the lift would be the same as it were initially.
Hence, tilting this wing would typically involve the evolution of a VIOLATION of Kutta's condition, we will NOT be able to gain the actual lift-change you would experience in a real, viscous fluid (where the Kutta condition will re-establish itself for the new geometry).

On further reflection, that tiny doubt can be reduced into something minuscule:
Since we know that the initial velocity distribution can be described by potential theory (essentially, a translatory field plus a point vortex distribution with the singularities hidden away inside the wing), tilting the wing should mathematically induce a redistribution of the point vortices inside the wing so that the boundary conditions remain fulfilled, and the net circulation kept constant during the time-dependent phase.
Since the potential solution is evidently a solution of the Euler equations, whatever doubt remains, is whether or not the Euler equations specify a unique solution or not..

 

[vanesch]

Then what makes an airplane fly?

And now for the stupid answer of the week:

The pilot !

cheers,
Patrick.

 

[arildno]

And now for the stupid answer of the week:

The pilot !

cheers,
Patrick.

I think that is a very good answer!

 

[arildno]

I must say that I find Eberhardt's "explanation" rather worthless.
Air is not actively pulled down by some Coanda hand; once a pressure gradient forms, air is accelerated in the direction of lowest pressure, whether or not that means that a given fluid element's path merely becomes curved or if the path remains straight-lined (with acceleration along that).

Another weakness is their confusion about Newton's laws.

Let us see in some detail how a GLOBAL analysis should be done (in the stationary case):
1. Assume that the wing's rest frame is an INERTIAL frame.
So, if there is a net lift force from the wing, there exists an independent external force acting upon the wing so that the velocity remains constant.
(gravity is a good example, we will assume this in the following)

2. Let us describe the problem in the wing's rest frame.
Let us surround the wing "W" by an annular control volume "V" of fluid, let for example the outer boundary of the annulus be a simple square S.
Then, Newton's 2.law expressed for the fluid momentarily enclosed in V:
$$\vec{P}+\vec{R}+\vec{W}=\int_{S+W}\rho\vec{v}\vec{v}\cdot\vec{n}dS$$
Where:
a) $$\vec{W}=\int_{V}\rho{\vec{g}}dV$$ is the weight of the air in the control volume
b)$$\vec{P}$$ is the surface forces acting upon S from the ambient air
(when neglecting viscous forces, that is the net pressure force)
c) $$\vec{R}$$ the force from the wing onto the fluid contained in "V"
d) $$\int_{S+W}\rho\vec{v}\vec{v}\cdot\vec{n}dS=\int_{S}\rho\vec{v}\vec{v}\cdot\vec{n}dS$$ the net momentum flux through the boundaries of V; since $$\vec{v}\cdot\vec{n}=0$$ on W, it isd only through S there is a momentum flux.

Now, the lift L is, by Newton's 3.law equal to the negative vertical component of $$\vec{R}$$ that is, we have:
$$L=\vec{P}\cdot\vec{j}-M_{air}g-\int_{S}\rho{v}\vec{v}\cdot\vec{n}dS$$
where "v" is the vertical velocity component.
Thus, only if we can disregard the other force terms acting upon our "V", can we state:
$$L\approx-\int_{S}\rho{v}\vec{v}\cdot\vec{n}dS$$
This will usually be the case if we let "V" be big enough.
Hence, my use of the word "GLOBAL".

Note, however, that there is full use of Newton's 2.law here, but FOR THE FLUID!
By invoking Newton's 3.law, we find the force on THE WING.

Eberhardt's miserable use of Newton's 1.law is best left uncommented..

 

[PBRMEASAP]

Note:
Just a slight correction to what you said:
It is NOT necessary to take into account the effect of the boundary layer in the MAINTENANCE of the stagnation point at the trailing edge.
Once enough momentum has been imparted to the fluid, and the unequal pressure distribution has been developed, we have a totally different inviscid picture than when we started with everything at rest and uniform pressure:
Now, if we regard the situation from the ground frame, the fluid is already rushing down.
If we therefore look at the evacuation picture again, that downrush is just sufficient to prevent any net upflow, i.e, the stagnation point re-establishes itself at the actual trailing edge.
The inviscid fluid is therefore able to maintain flight.

Okay, I see that now.

When regarding the fluid as inviscid, but with circulation, we ought to expect from Kelvin's theorem that the circulation will remain CONSTANT.
This is certainly true for the circulation on closed, material curves in the fluid; it necessarily remains so for the circulation about the wing in so far as it is correct to assume that the wing itself constitues a closed, MATERIAL curve for the fluid.

I've actually been wondering about this part myself. At least in the stationary inviscid picture, the fluid is flowing past the wing surface. So even though there's a net circulation, that doesn't necessarily mean the same fluid particles are constantly flowing around the wing. So I'm not really sure how to apply Kelvin's theorem here. Maybe there is a similar result for an "Eulerian" (stationary) curve in the flow?

Let us suppose it is..(there remains a tiny doubt, however: If the initial material curve gets kinked or something, can't it happen that the wing might pierce it somehow? I'm not entirely sure on this..)

Oh, are you taking the material curve to be an ever-expanding curve that encloses the wing's current and initial positions? I guess that does work...hadn't thought of that. Whether or not the curve can become kinked or broken is a good question...I don't see why not.

Since we know that the initial velocity distribution can be described by potential theory (essentially, a translatory field plus a point vortex distribution with the singularities hidden away inside the wing), tilting the wing should mathematically induce a redistribution of the point vortices inside the wing so that the boundary conditions remain fulfilled, and the net circulation kept constant during the time-dependent phase

That argument makes sense to me. Maybe we don't need Kelvin's theorem.

 

[PBRMEASAP]

I must say that I find Eberhardt's "explanation" rather worthless.
Air is not actively pulled down by some Coanda hand; once a pressure gradient forms, air is accelerated in the direction of lowest pressure, whether or not that means that a given fluid element's path merely becomes curved or if the path remains straight-lined (with acceleration along that).

Another weakness is their confusion about Newton's laws.

Hehe. I was so impressed by the facts and figures they quoted about how much air the wing deflects that I just assumed their application of the momentum principle was correct. But as you've shown, its misleading to attribute all of the momentum flow through the control surface to the force of the wing on the air, if you take too small a control volume. And as for the Coanda effect, they didn't have enough facts and figures to convince me that is a major factor in directing the flow of air.

And now for the stupid answer of the week:

The pilot !

In a recent survey, 9 out of 10 pilots agreed with your answer :-)

 

[arildno]

There is a good topological argument for why a material curve should not usually get broken up:
Consider the positions of a material curve at times "t" and "t+dt".
Since the constituent particles have finite velocities, we should expect that we can map the "t" curve onto the "t+dt" curve through a CONTINUOUS transformation (that is, given "sufficient" closeness of points on the "t" curve, their images will be satisfactorily close on the "t+dt"-curve.)
But, can a continuous transformation effect the radical topological change from "closed" to "not closed" (think of the famous rubber band analogy of topology)?
This seems very unlikely; I am in fact, quite convinced it is untrue.
From what I can see, such a pathology might only occur at points where such a curve gets tangentially kinked, or other such effects which signal a form of breakdown.

As you readily can see, topology is NOT a strong side of mine..

However, from what I can see, it boils down to the following issue:
Given an arbitrary initial velocity distribution, is there always a unique solution to IBV-problem posed by the Euler equations?
I haven't studied uniqueness conditions sufficiently to give a rigourous proof either way, but the lack of uniqueness for the Euler equations would astound me..

I very much suspect that my "doubt" is just yet another lamentable result of my general ignorance..

 

[arildno]

I find it important to do some more Anderson&Eberhardt bashing.
This is a very revealing quote:

So how does a thin wing divert so much air? When the air is bent around the top of the wing, it pulls on the air above it accelerating that air downward. Otherwise there would be voids in the air above the wing. Air is pulled from above. This pulling causes the pressure to become lower above the wing. It is the acceleration of the air above the wing in the downward direction that gives lift. (Why the wing bends the air with enough force to generate lift will be discussed in the next section.)

Clearly, these individuals suffer from a complete misunderstanding of what pressure is.
Since these persons are reputedly employed at Fermi's National Accelerator Laboratory and still suffer from deep misunderstandings, I find it in order to review a few basics on pressure:
Pressure at a "point" is a measure of the typical amount and intensity of molecular collisions at that "point"
The "point" must be understood to be a tiny spatial region which is, however, so large that to speak of averaged quantities within that region (typical examples: velocity, temperature, density, pressure) is useful.
If the region is too small, the merest random drift of molecules into that region would provide wild oscillations in these averages over time; that is, these averaged quantities would essentially lose their usefulness.
As long as our region is big enough to contain gazillions of molecules, statistical arguments leads us to expect that such wild fluctuations in measured quantities die out.
The region is incredibly tiny still, if I remember correctly, the typical linear dimension of such an "element region" for a not-so dilute gas is about $$10^{-7}m$$
(For liquids, like water, I think you can squeeze the linear dimension down at least a couple of orders of magnitude).

Now, the pressure is given as a scalar, and the pressure force onto a surface at our "point" is in the "colliding" direction, i.e, directed along the inwards normal of the surface.
Furthermore, and this is very important:
Since our "point" really contains gazillions of molecules, there should within it be NO PREFERRED DIRECTIONS for the momentum transfers involved in the collisions.
That is, the pressure force at "point" is equally strong in any direction.
Mathematically, this means that the pressure at a point is not a function of the direction of the contact surface normal.


Let us now consider a plate which is originally in contact&rest with a fluid (on one side of the plate, for simplicity). We keep the fluid inviscid, so that the "pure" pressure dynamics comes clearer into focus.
Now, give the plate a jump velocity V directed away from the fluid (it so happens that the argument is easier to visualize in this manner, it is, of course, equally valid when speaking of a finite acceleration and its effect over time).

Now, the pressure force on the plate at a given instant is evidently the accumulated effect of gazillions of molecules striking it at that moment.
The molecules have a random velocity distribution; this also holds for that subset of molecules who happen to have a "colliding" velocity, i.e, those which are actually going to hit the plate.
Let us see what happens in the jump velocity case (with some time gone..):
Can we really say that suddenly there has appeared a tiny strip of complete vacuum between the plate and the fluid?
Not really.
Consider that subset of particles close to the plate which initially had a "colliding" velocity (a lot) bigger than "V". Clearly, these must be regarded to still strike the plate, but instead of say with their original striking velocity $$V_{0}$$ they do so with a new striking velocity $$V_{0}-V$$
Thus, the only molecules which can be said to have been removed from the plate (relative to the case where the plate where at rest) are those whose original collision velocities satisfied the inequality $$0<V_{0}<V$$

Thus, unless V is very large, we cannot really expect a measurable density reduction at the plate.
Since, therefore, in the new position there are still gazillions of molecules who have "followed" the plate, we have in reality established the boundary condition for the macroscopic velocity field, i.e, that at all times, the normal velocity of the fluid equals the normal velocity of the plate.

The only dynamical feature we have gained, is a (significant) pressure DROP at the plate, which clearly follows from the argument above.
(Since the total kinetic energy of a striking molecule ought to be the same as a non-striking one, it follows that the striking molecules have a correspondingly less "tangential" velocity to start with, i.e, the actual amount of momentum transfers in local collisions remains non-directional)
Alternatively, we may say that we will get that pressure drop which is sufficient to accelerate the fluid so that the boundary condition of equal normal velocities is fulfilled..


Thus, there is absolutely no mystery involved in why a fluid tends to remain in contact with a surface, which Eberhardt&Anderson seems to think.
In particular, we don't need to pose the existence of some ghostly hand reaching up from the surface to grab air molecules.

An inviscid fluid is equally capable to fill out voids as a viscous fluid is; the pertinent feature is how either fluid goes about doing just that..
As we have seen, a viscous fluid prefers downrush about the wing, the inviscid fluid is not so picky.

 

[Thomas2]

The one and only reason why airplanes fly is the fact the the orientation and shape of the wing in combination with the velocity relative to the air leads to more molecules hitting the wing from below than from above.
Consider a simple plane surface: if it is resting relatively to the air, the molecules hit both sides of the surface with the same rate and speed according to their density and thermal speed, i.e. there is no resultant net force; now consider the same surface moving such that it is orientated at a certain angle to its velocity vector. If you consider the air as a strictly inviscid medium (i.e. the molecules interact only with the surface but not with each other), then the surface facing into the relative airstream experiences a higher rate and speed of molecules and the other surface a lower (simply by the virtue of the velocity of the surface adding to or subtracting from the average thermal speed of the air molecules). This results in a corresponding net force proportional to cos(alpha) (where alpha is the angle between the normal of the surface and the airstream), which can then be decomposed into the horizontal component i.e. the drag (~ cos^2(alpha)) and the vertical component i.e. the lift (~sin(alpha)*cos(alpha)).

Everything else like the airflow pattern around the wing etc. is only a secondary consequence of this due to the actual viscosity of the air, i.e. hydrodynamics may explain what effect an object moving through air has on the latter, but it does not actually give the causal reason why an airplane flies.

 

[Andrew Mason]

I find it important to do some more Anderson&Eberhardt bashing.

If you look at the Coanda effect as a 'bending' of the streamline, I think I can understand what they are saying. If the wing bends the streamline toward it from below, by Newton's third law, the force will be up. If it bends it away from above, again the force will be opposite or up. Isn't the bending of the streamline a key here?

AM

 

[arildno]

If you look at the Coanda effect as a 'bending' of the streamline, I think I can understand what they are saying. If the wing bends the streamline toward it from below, by Newton's third law, the force will be up. If it bends it away from above, again the force will be opposite or up. Isn't the bending of the streamline a key here?

AM

Bending of streamlines occurs naturally in solely pressure-driven fluids as well.
What they are really saying, is that viscous normal forces are comparable to pressure forces at high Reynolds numbers.
Nothing of what they present suggests that this is the case.

 

[arildno]

The one and only reason why airplanes fly is the fact the the orientation and shape of the wing in combination with the velocity relative to the air leads to more molecules hitting the wing from below than from above.
Consider a simple plane surface: if it is resting relatively to the air, the molecules hit both sides of the surface with the same rate and speed according to their density and thermal speed, i.e. there is no resultant net force; now consider the same surface moving such that it is orientated at a certain angle to its velocity vector. If you consider the air as a strictly inviscid medium (i.e. the molecules interact only with the surface but not with each other), then the surface facing into the relative airstream experiences a higher rate and speed of molecules and the other surface a lower (simply by the virtue of the velocity of the surface adding to or subtracting from the average thermal speed of the air molecules). This results in a corresponding net force proportional to cos(alpha) (where alpha is the angle between the normal of the surface and the airstream), which can then be decomposed into the horizontal component i.e. the drag (~ cos^2(alpha)) and the vertical component i.e. the lift (~sin(alpha)*cos(alpha)).

Everything else like the airflow pattern around the wing etc. is only a secondary consequence of this due to the actual viscosity of the air, i.e. hydrodynamics may explain what effect an object moving through air has on the latter, but it does not actually give the causal reason why an airplane flies.

You evidently understand nothing of the physics of inviscid fluids, and I may add, nothing else in physics either (that is why your papers are consistently rejected).
A flat plate initially at rest (no circulation to begin with) in an inviscid fluid, and which then started to move, would develop D'Alembert's paradox once conditions in the plate's rest frame could be called stationary.

 

[Thomas2]

A flat plate initially at rest (no circulation to begin with) in an inviscid fluid, and which then started to move, would develop D'Alembert's paradox once conditions in the plate's rest frame could be called stationary.

Would it? Why do satellites then fall down after some time due to drag in the upper atmosphere? At a height of 500 km the density of air (mainly atomic oxygen) is about 10^8 cm^-3 which, assuming a collision cross section of 10^-16 cm^2, amounts to a free flight distance of 10^8 cm = 1000 km between collisions of two atoms. Surely more than enough to assume an inviscid gas.

 

[strid]

isnt it so that airplanes can fly upside down?
then this stuff with the wing pushing air down doesn't work really...