How Does Wing Shape Affect Airplane Lift and Pressure Differences?

[member 529879]

I'm confused about how an airplane generates lift. If I'm correct the wing of the plane is bent so air can flow over the top of the wing faster than the bottom of the wing, the faster fluids somehow apply less pressure resulting in a net upward force. Why though does a bent wing allow air to flow faster over the top, and why do fast moving fluids apply a lesser pressure than slow ones?

 

[phinds]

There have been approximately 6,420 threads on this forum discussing it, so I suggest a forum search.

 

[rootone]

The simplest way to describe it is that wing's angle of attack forces air downward, so (Newton's third law) - reaction of the wing is to be forced up.
This also causes 'drag', so the aerofoil is contoured to minimise the element of drag.
The details of it can get get surprisingly complex, but that's the basic idea.
You COULD have a complete flat wing, but the induced drag would make it very inefficient

 

[russ_watters]

Why though does a bent wing allow air to flow faster over the top , and why do fast moving fluids apply a lesser pressure than slow ones?

Both of those are largely related to the Venturi effect (just a one-sided venturi):
http://en.wikipedia.org/wiki/Venturi_effect

 

[russ_watters]

The simplest way to describe it is that wing's angle of attack forces air downward, so (Newton's third law) - reaction of the wing is to be forced up.

While that's basically true it can be enough of a simplification that people lose sight of the fact that the top surface is a bigger contributor to the lift than the bottom surface. And in this case, the OP was asking about what's going on with the top surface anyway.

 

[boneh3ad]

Both of those are largely related to the Venturi effect (just a one-sided venturi):
http://en.wikipedia.org/wiki/Venturi_effect

This is in no way similar to a so-called "one-sided" Venturi effect. The Venturi effect describes a system with finite boundaries. The flow over a wing has effectively infinite boundaries. The two situations are not the same.

 

[boneh3ad]

While that's basically true it can be enough of a simplification that people lose sight of the fact that the top surface is a bigger contributor to the lift than the bottom surface. And in this case, the OP was asking about what's going on with the top surface anyway.

This is at best misleading and at worst untrue. You can't decouple the top and bottom surfaces. Without one, the flow over the other does not develop the same way. Also, the pressure on a wing's surface is meaningless without the pressure on the other side of the wing since the net force only comes from factoring in both. If you only consider the top surface, you will get negative lift, so there's no way the upper surface can be the "bigger contributor".

 

[russ_watters]

This is in no way similar to a so-called "one-sized" Venturi effect. The Venturi effect describes a system with finite boundaries. The flow over a wing has effectively infinite boundaries. The two situations are not the same.

The air provides its own boundary in low-speed flow when it is essentially incompressible. The velocity increase/pressure drop happens because it is being "squeezed", just like in a venturi. This is similar to the principle behind an aerospike engine, where air pressure helps constrain the flow:
http://en.wikipedia.org/wiki/Aerospike_engine

This is at best misleading and at worst untrue. You can't decouple the top and bottom surfaces. Without one, the flow over the other does not develop the same way.

I'm aware they can't be completely decoupled. Nevertheless, it is typical for the top surface to produce more of the lift and the OP was asking about the top surface, whereas the other answer implied it was about the bottom. The reality is that for most people what is going on with the bottom surface is intuitively obvious but what is going on with the top surface is not.

Also, the pressure on a wing's surface is meaningless without the pressure on the other side of the wing since the net force only comes from factoring in both. If you only consider the top surface, you will get negative lift, so there's no way the upper surface can be the "bigger contributor".

That isn't true. Both the top and bottom surface have pressure profiles that are measured/expressed as gauge pressure because the default is atmospheric pressure. As a result, the pressure on the top surface is measured to be negative. For a simplified/idealized example, a flat-bottom airfoil with the bottom parallel to the airflow would essentially just have atmospheric pressure below it and all of the lift generated by the top surface. Here's a sample graph of a pressure profile (not a flat bottom but still showing more of the lift derived from the top surface):

http://www.wfis.uni.lodz.pl/edu/Proposal/image093.gif

 

[boneh3ad]

The air provides its own boundary in low-speed flow when it is essentially incompressible. The velocity increase/pressure drop happens because it is being "squeezed", just like in a venturi. This is similar to the principle behind an aerospike engine, where air pressure helps constrain the flow:
http://en.wikipedia.org/wiki/Aerospike_engine

The fact that it is incompressible makes it less likely to "provide its own boundary," which does not accurately reflect what actually occurs. The velocity increase happens because he design of the trailing edge sets a certain separation point and the conservation laws then require the velocity to increase greatly over the upper surface to maintain physical equilibrium. For an explanation of why the Venturi approach is wrong, see this link:
https://www.grc.nasa.gov/www/k-12/airplane/wrong3.html

Also, that is not how an aerospike engine works. Aerospikes are, first and foremost, compressible flow devices and the Venturi effect is irrelevant to the phenomenon. The constraint by the outer atmosphere here (and importantly, variable constraint) is due to some of the unique features of compressible flows.

I'm aware they can't be completely decoupled. Nevertheless, it is typical for the top surface to produce more of the lift

So which is it? You can believe both things.

The shape of the top certainly contributes to lift. You cannot measure lift an just the top surface, though. The top definitely plays a role, but it does so as a part of the whole shape.

That isn't true. Both the top and bottom surface have pressure profiles that are measured/expressed as gauge pressure because the default is atmospheric pressure. As a result, the pressure on the top surface is measured to be negative. For a simplified/idealized example, a flat-bottom airfoil with the bottom parallel to the airflow would essentially just have atmospheric pressure below it and all of the lift generated by the top surface. Here's a sample graph of a pressure profile (not a flat bottom but still showing more of the lift derived from the top surface):

http://www.wfis.uni.lodz.pl/edu/Proposal/image093.gif

First, I know what gauge pressure is. In fact, I typically would give zero points if a student uses gauge pressure where absolute pressure is required because it's that important to keep straight.

Second, plotted there is the pressure coefficient, the definition of which includes a numerator that is equivalent to gauge pressure, so in that, you are correct. That is where it ends. Just because the pressure coefficient is negative does not mean there is somehow negative pressure on the upper surface. The force on a given surface due to pressure is always dependent on the absolute pressure, not gauge pressure, and absolute pressure is always positive. The pressure on the upper surface is always downward. A more negative pressure coefficient just means there is less downward force on the top to counteract the larger pressure on the bottom.

Third, a flat-bottomed airfoil would not necessarily have atmospheric pressure on its underside. This would generally only be true if the leading and train edges were sharp in a way such that the bottom approximated a flat plate, otherwise the curvature of the leading edge would accelerate the flow.

Even if the bottom is atmospheric, the total lift force is still the integrated sum of the absolute pressure on the bottom minus that on the top. The force due to the pressure on the top is still downward. The importance of the upper surface contour is therefore effectively to make the force on the upper surface less downward for a given upward force on the bottom. This is what I mean by being unable to decouple the two.

 

[A.T.]

the top surface is a bigger contributor to the lift than the bottom surface.

If you push a body with 20N from one side and 15N from the opposite side, which side "is a bigger contributor" to the net force on the body?

 

[A.T.]

Both the top and bottom surface have pressure profiles that are measured/expressed as gauge pressure .

That is just a convention. You can express them relative to any reference pressure you want. But the forces on the two surfaces are unambiguous and a function of absolute pressure.

 

[A.T.]

The force due to the pressure on the top is still downward. The importance of the upper surface contour is therefore effectively to make the force on the upper surface less downward for a given upward force on the bottom. This is what I mean by being unable to decouple the two.

Also visualized here:

 

[rcgldr]

Also visualized here ... (youtube video with basic description of lift)

From the video at about 1:10 into the video, pressure is reduced on the top "partially because it's shielded by the wing" as the wing travels forwards through the air. So the air fills in what would otherwise be a void by following the upper surface of a wing, or in the case of a stall, with turbulent flow consisting of vortices or mostly one very large vortice.

A wing doesn't have to be curved. Flat wings and/or symmetrical wings are reasonably efficient on small model gliders, from the dime store type models to the larger ones like this one with nearly symmetrical airfoil.

http://www.4p8.com/eric.brasseur/glider2.html

Aerobatic aircraft, both full scale and models, use symmetrical airfoils.

 

[A.T.]

A wing doesn't have to be curved.

True, and I don't think anyone implied otherwise.

 

[rcgldr]

... Why though does a bent wing allow air to flow faster over the top ...

A wing doesn't have to be curved.

True, and I don't think anyone implied otherwise.

From the original post, "bent wings", which I thought he meant curved (as opposed to dihedral).

 

[stedwards]

You're all wrong. Lift is due to the bound vortex, in superposition with the airstream, induced be viscous drag. :P

Am I making this up? Curved foils, or a planar foil at a positive angle of attack will not induce lift in an inviscid fluid. Invent any shape you wish. Immerse it in an inviscid stream. It will not produce a component of force perpendicular to the stream.

 

[A.T.]

You're all wrong.

I don't think that Newtons Laws are wrong. The momentum transferred by the air molecules hitting the bottom side must be greater than by those hitting the top side. The "why?" of this fact can be explained and discussed (almost endlessly) in various ways. Here are discussion of the point you make:
http://physics.stackexchange.com/questions/46131/does-a-wing-in-a-potential-flow-have-lift

 

[Delta2]

The lift can be explained either by using bernoulli's principle (which principle explains why in a fluid like air, in regions of higher velocity the pressure is lower) or using Newton's laws.

According to wikipedia the explanations are equivalent https://en.wikipedia.org/wiki/Bernoulli's_principle#Misunderstandings_about_the_generation_of_lift.

Bernoulli's principle can be derived from Newton's 2nd law. https://en.wikipedia.org/wiki/Bernoulli's_principle#Derivations_of_Bernoulli_equation

However i prefer bernoulli's principle explanation because it is not so crystal clear how Newtons law aplly in fluid's particles that flow, we are used to apply Newton's laws in point particles or rigid bodies.

As to why the velocity becomes higher in the upper surface of the wing it is because the air has to travel bigger distance in the same time.

 

[A.T.]

the air has to travel bigger distance in the same time.

Why?

 

[Delta2]

Why?

It is basically because of the equation of continuity and because we assume incompressible flow.
In speed's well below the speed of sound the flow of air is incompressible.

 

[russ_watters]

It is basically because of the equation of continuity and because we assume incompressible flow.
In speed's well below the speed of sound the flow of air is incompressible.

While Bernoulli's equation is correctly applied (change in speed of the airstream creates a pressure change), the equal transit time idea is not correct. The different paths result in different times reaching the trailing edge (counterintuitively, air from the top surface reaches the trailing edge first). Continuity applies along a single streamline, not when comparing two different streamlines and in order for the wing to do anything, the air behind it can't look the same as the air in front of it, otherwise nothing happened when the wing passed.

 

[HuskyNamedNala]

You're all wrong. Lift is due to the bound vortex, in superposition with the airstream, induced be viscous drag. :P

Am I making this up? Curved foils, or a planar foil at a positive angle of attack will not induce lift in an inviscid fluid. Invent any shape you wish. Immerse it in an inviscid stream. It will not produce a component of force perpendicular to the stream.

The lift can be explained either by using bernoulli's principle (which principle explains why in a fluid like air, in regions of higher velocity the pressure is lower) or using Newton's laws.

According to wikipedia the explanations are equivalent https://en.wikipedia.org/wiki/Bernoulli's_principle#Misunderstandings_about_the_generation_of_lift.

Bernoulli's principle can be derived from Newton's 2nd law. https://en.wikipedia.org/wiki/Bernoulli's_principle#Derivations_of_Bernoulli_equation

However i prefer bernoulli's principle explanation because it is not so crystal clear how Newtons law aplly in fluid's particles that flow, we are used to apply Newton's laws in point particles or rigid bodies.

As to why the velocity becomes higher in the upper surface of the wing it is because the air has to travel bigger distance in the same time.

"Curved" means cambered.

While Bernoulli's equation is correctly applied (change in speed of the airstream creates a pressure change), the equal transit time idea is not correct. The different paths result in different times reaching the trailing edge (counterintuitively, air from the top surface reaches the trailing edge first). Continuity applies along a single streamline, not when comparing two different streamlines and in order for the wing to do anything, the air behind it can't look the same as the air in front of it, otherwise nothing happened when the wing passed.

1) Inviscid fluids don't exist in reality. What you are describing is potential flow, who's solution to a complicated shape like an airfoil is the superposition of sources and sinks. With that said, you can superimpose a source solution with a vortex solution or simply use a doublet, solve a system of equations (for the source and vortex strengths) such that the flow is tangent to the airfoil. This will give you a pretty good approximation of the lift generated prior to significant separation. Further, you can couple this solution with a code that resolves the boundary layer to get some pretty accurate "rough" estimations of airfoil performance. I think it would be a real interesting experiment to try to generate (or halt) turbulence in a superfluid, since this is as lose as we can get to an inviscid flow.

2) The "equal transit time" theory is crap. I don't know who came up with it or why it is even considered valid. Get it out of your head.

3) I personally don't like the "Bernoulli principle" explanation of lift. For one, it doesn't really answer anything with detail (what about airfoil shape, what about delta wings, how do slotted flaps work, etc). Bernoulli's principle is only valid for inviscid flow and does not take into account irreversible losses due to viscosity, but it is a good explanation for low angles of attack.

Here is a good example: how do delta wings generate lift? The Bernoulli principle would state that the air is accelerated over the top of the wing relative to the bottom of the wing. Well, that is partially true. The air is accelerated but it behaves nothing like a straight wing. On a delta wing the air separates near the leading edge and develops a strong vortex over the upper surface. The vortex maintains a low pressure region which is responsible for phenomenal gains in lift at moderate and high angles of attack. This is why the concord needed to land at almost a 45 degree angle - at low angles of attack and low speeds the delta wing does not maintain a vortex over the upper surface and thus will not generate enough lift.

Here is my favorite example: the dragonfly wing. At low Reynolds numbers (low velocities with a small wing, all else being equal) the dragonfly wing (which looks like a crushed piece of cardboard) has superior performance to a smooth wing that you might see on an airliner or drone! Why is this? Well it turns out that the grooves on the dragonfly wing force the smooth air (we call this laminar flow) over the wing to become chaotic and generate turbulent rotational motion. The rotational motion (small vortices not entirely unlike those on the delta wing) gets "trapped" in the grooves, maintaining low pressure on the upper surface, thus creating lift.

Here is a paper with pretty pictures. The jargon gets pretty technical so don't worry about reading it if you don't have a solid background in fluids
http://www.aere.iastate.edu/~huhui/paper/journal/2008-JA-Corrugation.pdf

If you want to explain lift you have to take into account these and other odd cases.

To tell you the truth, I can't think of a very simple explanation for why an airfoil generates lift. I want to say that the geometry of the airfoil develops a pressure gradient across the wing via acceleration and deceleration of the flow. Depending on your geometry you can have mild accelerations which can be explained roughly by Bernoulli's principle, or you can have significant changes in the flow field such as the development of a vortex structure which also lead to the development of a pressure gradient across the wing. Thin airfoil theory states lift is proportional to (and can be calculated from) circulation, which is basically a measure of the rotation of the fluid around the airfoil.

 

[A.T.]

the equal transit time idea is not correct

As demonstrated here at 0:15:

 

[russ_watters]

2) The "equal transit time" theory is crap. I don't know who came up with it or why it is even considered valid. Get it out of your head.

I don't know who first came-up with the idea, but it sticks around because it makes intuitive sense to some people and there is a void of explanation of the issue. The idea is basically that the air must look the same after the wing passes as before, so air going over the top must meet the air going over the bottom. As another easy consequence, all of the extra speed comes from adding a Y-component of speed to the air. But once you recognize that the air behind the wing can't look like the air in front of it (otherwise the wing and air didn't exchange any momentum), it becomes easier to discard it.

3) I personally don't like the "Bernoulli principle" explanation of lift.

Well, like it or not it is an essential component of lift both in principle and in practice. But it seems like what you actually don't like is that it isn't 100% of the picture. But there's certainly nothing wrong with needing more than one physical principle, simultaneously, to explain a phenomena.

Moreover, when first explaining a concept, you can't just jump right in with the most complicated example, you have to start with the simple/basic/typical examples and build from there. So:

If you want to explain lift you have to take into account these and other odd cases.

No you don't, at least not unless you need for some reason to write a complete history/course on aerodynamics. The OP asked about lift on an airplane and there is no need to bring up the completely separate issue of how insects fly when describing how a basic airplane flies. What you describe is akin to someone asking a simple question about Newtonian gravity and instead answering it by bringing in General Relativity.

To tell you the truth, I can't think of a very simple explanation for why an airfoil generates lift. I want to say that the geometry of the airfoil develops a pressure gradient across the wing via acceleration and deceleration of the flow.

That isn't quite right. Differences in velocity have to be accompanied by accelerations that made them happen, but it is the velocity difference, not the (across the airfloil) acceleration itself that generates the lift.

Otherwise, personally I don't see why such a simple explanation isn't good enough for someone's first introduction to lift. It is highly applicable to most every-day situations where one encounters (non-living-thing) lift.

What I think people do a bad job of, though, is explaining how/why the shape of the airfoil causes the speed change in the air. That was exactly the question the OP asked and no one else actually attempted to answer it. That void is what let's the "equal transit time" fallacy creep in.

 

[insightful]

I found this really interesting:

http://en.wikipedia.org/wiki/User:J_Doug_McLean/sandbox#Momentum_balance_in_lifting_flows

Pull quote:

"When a ground plane is present, there is a pattern of higher-than-ambient pressure on the ground below an airplane in flight, as shown on the right[109] For steady, level flight, the integrated pressure force associated with this pattern is equal to the total aerodynamic lift of the airplane and to the airplane's weight."

I assume this is true for any altitude, i.e., not just low-level (ground effect) flight.

 

[HuskyNamedNala]

Yes, pressure ultimately drives lift. Another way to compute the lift on an airfoil is to sum the variation of pressure along the surface and multiply by the chord.

 

[russ_watters]

Mod note: Argumentative posts and responses deleted. If you didn't receive a warning, your posts were fine. Sorry for the lost effort.

 

[russ_watters]

A couple of instructive quotes from my copy of Anderson's "Introduction to Flight":

"No matter how complex the flow field, and no matter how complex the shape of the body, the only way nature has of communicating an aerodynamic force to a solid object or surface is through the pressure and shear stress distributions which exist on the surface. These are the basic funamental sources of all aerodynamic forces." [italics by author]

"...as emphasized in the previous chapters, the funamental source of lift is the pressure distribution over the wing surface...
An alternate explaantion is sometimes given: The wing deflects the airflow downward...From Newton's third law, the equal and opposite reaction produces a lift. However, this explanation really involves the effect of lift, and not the cause."

"A third argument, called the circulation theory of lift,, is sometimes given for the source of lift. However, this turns out to be not so much an explanation of the lift per se, but rather more of a mathematical formulation for the calculation of the lift..."

It also notes that the definition of lift coefficient includes the dynamic pressure term from Bernoulli's equation.

It also discusses the different "types" of pressure, an argument often had on PF:
"When an engineer or scientist uses the word "pressure," it always means static pressure unless otherwise identified...
A second type of pressure is commonly utilized in aerodynamics, namely, total pressure." Later: "dynamic" pressure (I usually say "velocity" pressure).

Unfortunately, it does not discuss the mechanism for the airflow speeding-up along the top surface of the airfoil.

 

[russ_watters]

Ah, found it. It's hidden in the discussion of continuity. The word/name "Venturi" never appears in the book, but continuity is discussed using a thought experiment concept called "stream tubes". Stream tubes are the missing link between the Venturi tube and the flow over the airfoil.

A stream tube is an imaginary collection of streamlines, that form a tube. A Venturi tube is a stream tube, but you can also grab a bundle of streamlines flowing over a wing as a stream tube. The flow through the stream tube follows the same area/velocity ratio as the Venturi. And (from a source more descriptive than Anderson:

These streamlines form a tube that is impermeable since the walls of the tube are made up of streamlines, and there can be no flow normal to a streamline (by definition). This tube is called a streamtube. From mass conservation, we see that for a steady, one-dimensional flow, the mass-flow rate is constant along a streamtube. In a constant density flow, therefore, the cross-sectional area of the streamtube gives information on the local velocity...

As an example, consider steady, constant density flow over a cylinder...

In this region the streamlines come closer together, and the area between them decreases. Since the density is constant, the velocity must increase according to the principle of mass conservation. For constant density flow, wherever the area between streamlines decreases, the velocity increases. This is exactly similar to what happens with constant-density flow through a duct or a pipe --- if the area decreases, the velocity increases so that the volume flow rate is maintained constant (volume flow rate out must equal volume flow rate in by continuity). [emphasis added]

http://www.princeton.edu/~asmits/Bicycle_web/streamline.html

So: the velocity on the top surface increased because the streamlines got closer together. Why did the streamlines get closer together? Because they encountered an object that deflected them away from their previous direction of motion, but the streamlines further above didn't move out of the way enough to maintain the spacing. And this is "exactly similar" to what happens in real tube such as a Venturi tube.

 

[boneh3ad]

So: the velocity on the top surface increased because the streamlines got closer together. Why did the streamlines get closer together? Because they encountered an object that deflected them away from their previous direction of motion, but the streamlines further above didn't move out of the way enough to maintain the spacing. And this is "exactly similar" to what happens in real tube such as a Venturi tube.

Almost. The difference is that with a variable-area duct, you are enforcing the outer boundaries of the streamtube. With a cylinder or a wing or other object with air moving over it, only one side of that streamtube is enforced, so it is not a perfect analogue. What Smits means in the page you cite is that given a set of streamlines over an object, you can note that as they pass over the object they get closer together, so using those streamlines, you can treat them using the same basic rules as a variable-area duct. You can't determine the actual streamlines by drawing an arbitrary box around the flow area and treating it like a streamtube, though, since there is nothing enforcing that upper boundary as a streamline.

 

[rcgldr]

"not 100% Bernoulli". Near the peak of the upper surface, the air changes direction from upwash to downwash, and in general follows the upper convex surface unless or until the flow detaches. Within a streamline, this involves a component of acceleration perpendicular to the flow, and this perpendicular component of acceleration and a corresponding component of pressure gradient don't involve a coexistent change in speed, just a change in direction (the diversion of flow). The reduced pressure will also coexist with acceleration in the direction of flow, but since some of the total acceleration is perpendicular to the flow within a streamline, Bernoulli is violated.

 

[Delta2]

"not 100% Bernoulli". Near the peak of the upper surface, the air changes direction from upwash to downwash, and in general follows the upper convex surface unless or until the flow detaches. Within a streamline, this involves a component of acceleration perpendicular to the flow, and this perpendicular component of acceleration and a corresponding component of pressure gradient don't involve a coexistent change in speed, just a change in direction (the diversion of flow). The reduced pressure will also coexist with acceleration in the direction of flow, but since some of the total acceleration is perpendicular to the flow within a streamline, Bernoulli is violated.

I am not sure what you trying to say here, that Bernoulli is violated in direction perpendicular to the streamline? That might be so , but Bernoulli's principle is holding along the streamline. The perpendicular to the streamline acceleration doesn't affect the term $v^2/2$, it is like it doesn't exist for the Bernoulli along the streamline.

 

[rcgldr]

I am not sure what you trying to say here, that Bernoulli is violated in direction perpendicular to the streamline?

Yes.

That might be so , but Bernoulli's principle is holding along the streamline. The perpendicular to the streamline acceleration doesn't affect the term v^2/2, it is like it doesn't exist for the Bernoulli along the streamline.

The point here is that lift isn't "100% Bernoulli". The acceleration perpendicular to the flow coexists as part of the reduction in pressure along the upper surface of a wing, but that part of the reduction in pressure is unrelated to the speed (speed^2/2) of the flow, which violates Bernoulli.

 

[Delta2]

If you read the derivation of Bernoulli's principle it should be clear that it holds for inviscid flow along the streamline, regardless if there are components of acceleration perpendicular to the streamline.

In the case of the wing it seems that we apply the principle in direction perpendicular to the streamlines. However all the streamlines start in front of the wing where we assume that the term $v^2/2+gz+p/\rho$ is the same (if we exclude small variations in the term gz). So because Bernoulli's hold along the streamline it will be for a streamline above the wing

$v^2/2+gz_1+p/\rho=v_{above}^2/2+gz_{above}+p_{above}/\rho$

while for a streamline below the wing

$v^2/2+gz_2+p/\rho=v_{below}^2/2+gz_{below}+p_{below}/\rho$

If we neglect the variations in the terms of the form gz, then you can see why bernoulli's holds for the two streamlines.

 

[rcgldr]

If you read the derivation of Bernoulli's principle it should be clear that it holds for inviscid flow along the streamline, regardless if there are components of acceleration perpendicular to the streamline.

My point is this, acceleration perpendicular to the flow (centripetal acceleration) of a streamline coexists with the pressure on the outer side of the streamline being greater than the pressure on the inner side of the streamline. This means the wing experiences more of a decrease in pressure above a wing and more of an increase in pressure below a wing than what Bernoulli would predict based on speeds alone.

 

[Delta2]

My point is this, acceleration perpendicular to the flow (centripetal acceleration) of a streamline coexists with the pressure on the outer side of the streamline being greater than the pressure on the inner side of the streamline. This means the wing experiences more of a decrease in pressure above a wing and more of an increase in pressure below a wing than what Bernoulli would predict based on speeds alone.

Can you tell me where the derivation of Bernoulli's principle goes wrong in the case that perpendicular to the flow acceleration exists? i don't see anything wrong, the perpendicular acceleration produces no work.

The pressure gradient you speak of is in direction perpendicular to the streamline, not along the streamline.

 

[rcgldr]

Can you tell me where the derivation of Bernoulli's principle goes wrong in the case that perpendicular to the flow acceleration exists? i don't see anything wrong, the perpendicular acceleration produces no work. The pressure gradient you speak of is in direction perpendicular to the streamline, not along the streamline.

What I meant by "Bernoulli is violated" by the perpendicular acceleration is the effect it has on the wing, not the effect it has on the energy of the streamlines. The lift force is related to the innermost pressures of the innermost streamlines, and those pressures are affected by perpendicular acceleration of the streamlines, so the lift force is not "100% Bernoulli".

 

[Delta2]

What I meant by "Bernoulli is violated" by the perpendicular acceleration is the effect it has on the wing, not the effect it has on the energy of the streamlines. The lift force is related to the innermost pressures of the innermost streamlines, and those pressures are affected by perpendicular acceleration of the streamlines, so the lift force is not "100% Bernoulli".

You essentially mean that bernoulli is violated between points in different streamlines? If that's true, can you tell me where my reasoning is wrong in post #34.

Well seems there has to be one additional assumption, that the flow is irrotational, for Bernoulli's principle to hold between different streamlines.

 

[rcgldr]

You essentially mean that Bernoulli is violated between points in different streamlines?

No, only that there is a perpendicular pressure gradient (greater on the "outside boundary", lesser on the "inside boundary") within and across a streamline if it's experiencing acceleration perpendicular to the flow of the streamline. It's not related to the overall energy per unit volume of air in the streamline.

For an imaginary situation of attached streamlines with identical flows (except for perpendicular component of acceleration), the pressure exerted onto a convex surface will be less than the pressure exerted onto a flat surface, and the pressure exerted onto a flat surface will be less than the pressure exerted onto a concave surface.

update - The first article linked to in harrylin's post # 40, makes the same comments about curved streamlines and pressure gradients across those curved streamlines.

 

[harrylin]

A couple of instructive quotes from my copy of Anderson's "Introduction to Flight": [..] The wing deflects the airflow downward...From Newton's third law, the equal and opposite reaction produces a lift. However, this explanation really involves the effect of lift, and not the cause." [..]

I agree with those other authors and I wonder how Anderson could make his claim look plausible. How could the upward reaction force be the effect of raising the wing upward?? In fact, the raising upward of a wing as cause will produce a downward reaction force by the air on the wing.

I found the following article useful: Babinsky, Physics Education 2003, 38, p.497
"How do wings work?". Abstract:
The popular explanation of lift is common, quick, sounds logical and gives
the correct answer, yet also introduces misconceptions, uses a nonsensical
physical argument and misleadingly invokes Bernoulli’s equation. A simple
analysis of pressure gradients and the curvature of streamlines is presented
here to give a more correct explanation of lift.
- http://iopscience.iop.org/0031-9120/38/6/001

PS I just found a follow-up by a different author in the same journal:
David Robertson, "Force, momentum, energy and power in flight"
Abstract
Some apparently confusing aspects of Newton’s laws as applied to an
aircraft in normal horizontal flight are neatly resolved by a careful analysis
of force, momentum, energy and power. A number of related phenomena
are explained at the same time, including the lift and induced drag
coefficients, used empirically in the aviation industry.
- http://iopscience.iop.org/0031-9120/49/1/75

 

[boneh3ad]

No, only that there is a perpendicular pressure gradient (greater on the "outside boundary", lesser on the "inside boundary") within and across a streamline if it's experiencing acceleration perpendicular to the flow of the streamline. It's not related to the overall energy per unit volume of air in the streamline.

Nothing about this (or anything else you have cited in your last few posts) violates Bernoulli's equation. The streamline can be curved and be experiencing a pressure gradient normal to itself and still adhere to the tenets of Bernoulli's equation. It is still a streamline, after all.

In fact, if you already know the flow velocities relative to the wing at the edge of the boundary layer everywhere in the wing, then Bernoulli's equation will very accurately calculate the pressure distribution on the surface and therefore the lift on the wing. Outside the boundary layer, the flow is inviscid, after all. That means the only source of error is the assumption that the wall-normal pressure gradient through the boundary layer is zero, which turns out to be very, very accurate. This is, of course, assuming the boundary layer remains attached. If not, all bets are off.

You essentially mean that bernoulli is violated between points in different streamlines? If that's true, can you tell me where my reasoning is wrong in post #34.

You aren't wrong.

 

[rcgldr]

That means the only source of error is the assumption that the wall-normal pressure gradient through the boundary layer is zero, which turns out to be very, very accurate.

In the article linked to by harrylin (link below) and other articles that I've read, centripetal acceleration of a streamline requires a non-zero wall normal pressure gradient (otherwise centripetal acceleration would not occur).

http://iopscience.iop.org/0031-9120/38/6/001

Then again, how are the flow velocities determined in the first place? Is it possible that the mathematical models to calculate flow velocities are using extremely thin streamlines or somehow take centripetal acceleration into account when calculating effective flow velocities just outside the boundary layer.

Is a streamline that is curved really a streamline? Would the velocity on the "inner" wall be slightly higher than the velocity on the "outer" wall due to the wall normal pressure gradient? Again, using very thin streamlines (like a Calculus approach of using the limit as streamline thickness approaches zero for the streamline just outside the boundary layer), should eliminate or at least minimize this issue.

As a thought exercise, what would the wall normal pressure gradient be for an inviscid flow inside a frictionless hoop with rectangular cross section (as opposed to a circular cross section to make this example simpler)?

 

[boneh3ad]

In the article linked to by harrylin (link below) and other articles that I've read, centripetal acceleration of a streamline requires a non-zero wall normal pressure gradient (otherwise centripetal acceleration would not occur).

Yes, it requires a stream-normal pressure gradient. That applies to the outer inviscid flow where the concept of a streamline makes sense in the first place. The boundary layer invalidates many of the assumptions required to utilize streamlines in this sort of sense, so usually people don't even bother considering them. It turns out, though, that the pressure is very nearly constant through the thickness of a boundary layer, and you can exploit this fact to calculate lift in this example. This approach has been taken for years by airplane designers with great success.

Then again, how are the flow velocities determined in the first place? Is it possible that the mathematical models to calculate flow velocities are using extremely thin streamlines or somehow take centripetal acceleration into account when calculating effective flow velocities just outside the boundary layer.

Streamlines are infinitely thin. A streamline is simply a path that follows the instantaneous velocity vectors through a velocity field, so trying to define a thickness for streamlines is not meaningful.

The actual velocity around an airfoil can be calculated in a number of ways. One "simple" way is what are called panel methods, where you use the potential flow coming from a series of point sources/sinks or vortices to calculate the flow around the airfoil as if viscosity was not important. You use two boundary conditions: impermeability of the airfoil surface (sometimes called the no penetration condition) and by fixing the trailing edge as an imposed stagnation point (the Kutta condition). When you do this, you can calculate the flow over a 2-D airfoil quickly and fairly easily. To correct for the boundary layer, you can use that information to determine the chordwise pressure gradient and calculate the boundary layer, then adjust the shape of the airfoil using the displacement thickness, and re-run the panel method to get a more accurate solution. After several iterations, you get to a solution that converges to a very good approximation of the flow field. This inherently considers streamline curvature.

For more accurate flow field calculation, you can run a more accurate CFD code based directly on the Navier-Stokes equations that has viscosity in it already and varying degrees of modeling. This can be very computationally expensive, but is more accurate. It also inherently handles streamline curvature.

Is a streamline that is curved really a streamline? Would the velocity on the "inner" wall be slightly higher than the velocity on the "outer" wall due to the wall normal pressure gradient? Again, using very thin streamlines (like a Calculus approach of using the limit as streamline thickness approaches zero for the streamline just outside the boundary layer), should eliminate or at least minimize this issue.

Absolutely it is a streamline. Instead of thinking of the flow as a series of streamlines, think of it as a field of vectors, each with a magnitude and direction. A streamline can be drawn by starting at a point and drawing a line an infinitesimal distance away from that point, looking at the velocity vector at the new point, drawing another tiny distance in the direction of that vector, and repeating ad infinitum. In other words, in calculating the flow field, all classical fluid mechanics methods essentially take care of your above-stated concern by default.

As a thought exercise, what would the wall normal pressure gradient be for an inviscid flow inside a frictionless hoop with rectangular cross section (as opposed to a circular cross section to make this example simpler)?

I am not 100% sure what you mean by this. Either way, it is not enough information to come up with an answer.

 

[rcgldr]

Yes, it requires a stream-normal pressure gradient. That applies to the outer inviscid flow where the concept of a streamline makes sense in the first place. The boundary layer invalidates many of the assumptions required to utilize streamlines in this sort of sense, so usually people don't even bother considering them. It turns out, though, that the pressure is very nearly constant through the thickness of a boundary layer, and you can exploit this fact to calculate lift in this example. This approach has been taken for years by airplane designers with great success.

So programs like XFOIL calculate the velocities just outside the boundary layer using an infinitely thin streamline or perhaps not bothering with a streamline model at all? Based on it's documentation, it uses the panel method you mentioned above (or at least it uses panels), and takes into account issues like separation bubbles, and I guess it's somewhere in the mid-range of sophistication in terms of it's calculation of polars for airfoils.

It's also my understanding that the lowest pressure and highest velocity over the top of a wing typically occur at or near the point of greatest curvature?

 

[boneh3ad]

What do you mean by "the streamline model"? XFOIL, along with any other panel code (or really any other CFD code) calculates the velocity field directly. If then want to view the streamlines later, you calculate them from the velocity field that you already have. You will find that the regions of higher flow velocity occur in regions where the streamlines are closest together. This may or may not be the same location as the sharpest streamline curvature depending on the situation. Off the top of my head, I would guess the sharpest curvature for an airfoil is near the leading edge stagnation point, though I haven't done any math to check that.

 

[rcgldr]

The boundary layer invalidates many of the assumptions required to utilize streamlines in this sort of sense, so usually people don't even bother considering them. ... panel methods ... This inherently considers streamline curvature. ... For more accurate flow field calculation, you can run a more accurate CFD code based directly on the Navier-Stokes equations that has viscosity in it already and varying degrees of modeling. This can be very computationally expensive, but is more accurate. It also inherently handles streamline curvature.

What do you mean by "the streamline model"?

Badly worded on my part, I meant taking streamline curvature into account, which you already answered in your previous post.

Part of whether or not Bernoulli is violated depends on the frame of reference. If using the air as a frame of reference, and noting what occurs as a wing passes through a volume of air, a wing performs work on the air, resulting in a non-zero velocity of the air (mostly downwards, somewhat forwards), at the moment the affected air's pressure returns to ambient. If using the wing as a frame of reference, in the case of an idealized wing, the oncoming flow is diverted with no change in speed. For a real wing, some energy is lost in the process. Newtons laws hold from any inertial frame of reference.

For a basic description of lift, my issue with Bernoulli is that it doesn't deal with the relationship between neighboring streamlines, such as the effects related to streamline curvature. The Newton explanation is macroscopic, only looking at the overall effects of diverted flow: lift is related to the net average downwards acceleration of affected air, drag is related to the net average forward acceleration of air, so it might serve as an explanation, but it's not very useful as a basis for a mathematical model of an airfoil. Neither Bernoulli or Newton explain why air flow would tend to remain attached to the upper surface of a flat or convex surface (with a reasonable angle of attack).

 

[boneh3ad]

Part of whether or not Bernoulli is violated depends on the frame of reference. If using the air as a frame of reference, and noting what occurs as a wing passes through a volume of air, a wing performs work on the air, resulting in a non-zero velocity of the air (mostly downwards, somewhat forwards), at the moment the affected air's pressure returns to ambient. If using the wing as a frame of reference, in the case of an idealized wing, the oncoming flow is diverted with no change in speed. For a real wing, some energy is lost in the process.

That flow diversion happens regardless of which frame of reference you use to view the problem. The thing is, in either case, the losses to lead to this effect occur due to viscosity and are confined to the boundary layer, so Bernoulli's equation still holds out in the free stream. After all, Bernoulli's equation can certainly be used in the presence of an external body force so long as that force is irrotational (conservative), and without considering viscosity, all of the forces acting here are conservative. You wouldn't ever use Bernoulli's equation in the boundary layer anyway.

Bernoulli's equation can be derived directly from the equations of motion using only the assumptions of irrotationality (which implies inviscid), incompressibility, and steady flow. You can generalize it to unsteady flows, although the conditions of validity change (cf. Karamcheti).

For a basic description of lift, my issue with Bernoulli is that it doesn't deal with the relationship between neighboring streamlines, such as the effects related to streamline curvature.

I still do not understand what yo mean here. There is no issue of the relationship between neighboring streamlines. Streamlines do not affect the flow; they are a concept that can be calculated to help visualize and understand the flow field after the flow field has been calculated (or measured). No one draws the streamlines and then uses that to deduce the flow field. You can sometimes substitute the streamfunction into the governing equations and calculate the flow field that way, but you are still calculating the whole field and can place a streamline at any arbitrary location and back the whole-field velocity out of the solution. Streamline curvature only means that there is something diverting the flow in some fashion, which we already know just by defining the problem. There is nothing more to it than that.

The Newton explanation is macroscopic, only looking at the overall effects of diverted flow: lift is related to the net average downwards acceleration of affected air, drag is related to the net average forward acceleration of air, so it might serve as an explanation, but it's not very useful as a basis for a mathematical model of an airfoil.

True, it is essentially useless for trying to calculate the lift on some arbitrary airfoil shape. There is no method for this invoking Bernoulli's equation that is not harder to implement than at least one alternative method that could be used in the same situation. That doesn't change the fact that the particular view of the phenomenon is physically correct, though.

Neither Bernoulli or Newton explain why air flow would tend to remain attached to the upper surface of a flat or convex surface (with a reasonable angle of attack).

It would tend to remain attached because nature abhors a vacuum. If the air flow (or water flow in the case of a hydrofoil) did not remain somehow attached to the surface, there would be a vacumm bubble in its place, a massive pressure gradient pushing the fluid back toward the surface, and therefore the fluid would tend to be attracted back toward the surface anyway. In the case of a separated airfoil, there is still air by the surface, only it is slower and circulating as a result of the pressure gradient along the surface. That, of course, causes stall in many cases.

Of course, that doesn't answer the question that many people really have. Why does the fluid move faster over the top of the airfoil (or equivalently, why does a net circulation develop around an airfoil)? It all comes back to viscosity. If you put an airfoil shape into an ideal, inviscid fluid, then you would have zero lift, as the leading stagnation point and trailing stagnation point would set up in locations that allow the air before and after the shape to remain undisturbed. Generally, this means the trailing edge stagnation point ends up somewhere above the airfoil for a positive angle of attack. You would also measure no drag, a phenomenon known as D'Alembert's paradox. Of course we know this isn't what happens in a real fluid, and the difference is viscosity and boundary-layer separation.

According to potential flow theory, the trailing edge of the airfoil is a singularity point, so in order for the flow field with zero lift predicted by potential flow to occur, there would need to be an infinite velocity around the trailing edge. Clearly that can't happen in real life. Instead, that sharp trailing edge (or any sufficiently abrupt truncation to the end of the airfoil) causes the boundary layer to separate at that point. Essentially, with viscosity and a sufficiently abrupt trailing edge, we are choosing our own rear stagnation point. This is the reason why modeling lift using a panel method, which uses potential flow, requires the use of the artificial Kutta condition. Now with this "artificial" trailing edge stagnation point combined with the requirements for conservation of mass and momentum ensure that the flow velocity over the upper surface is faster in order to enforce that rear stagnation location. This is true whether it is viscosity enforcing the trailing edge stagnation point or the Kutta condition.

Once you have that flow field, you can use Bernoulli's equation if you wish to calculate lift. Calculating drag is much, much more difficult and, in general, can't even be done accurately by modern computer codes.

 

[rcgldr]

Part of whether or not Bernoulli is violated depends on the frame of reference. If using the air as a frame of reference ...

That flow diversion happens regardless of which frame of reference you use to view the problem.

True, but using an ideal wing, and the wing as a frame of reference, the flow can be diverted with no change in speed, and no change in energy (in the ideal case), while using the air (free stream) as a frame of reference, energy is added to the air violating Bernoulli (or at least classic Bernoulli's equation), similar to the case of a propeller (since both a wing and a propeller involve an increase in pressure as air flows across the region swept out by a wing or propeller) as mentioned in this NASA article:

We can apply Bernoulli's equation to the air in front of the propeller and to the air behind the propeller. But we cannot apply Bernoulli's equation across the propeller disk because the work performed by the engine (propeller) violates an assumption used to derive the equation.

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

Maybe it's an issue of terminology depending on the author, for example in some articles, the term Bernoulli only refers to the relationship between a pressure gradient and speed along a streamline, while no special name is given for the relationship between a pressure gradient across a streamline and the corresponding centripetal acceleration. In some aritcles, if energy is changed within a streamline, it's no longer considered to be a streamline, there can be a streamline before and after the region of change in energy, but not across the region of change in energy.

 

[boneh3ad]

True, but using an ideal wing, and the wing as a frame of reference, the flow can be diverted with no change in speed, and no change in energy (in the ideal case), while using the air (free stream) as a frame of reference, energy is added to the air violating Bernoulli (or at least classic Bernoulli's equation), similar to the case of a propeller as mentioned in this NASA article:

The fundamental misunderstanding you have here is that in the situation where the observer rests with the stationary air and watches the wing pass through the medium, the problem is not steady state and therefore Bernoulli's equation in the classical sense does not apply. However, since you there exists an unsteady form Bernoulli equation (see below), you could apply that to the situation just fine.

Regardless, literally no one treats a wing this way, as it is not very informative and is substantially more difficult to calculate anything useful owing to the fact that it is no longer a steady problem. By simply switching to the frame of the wing, you remove all the time dependence and make the problem much more tractable, make it much simpler to visualize for a person, and don't lose any of the relevant physics, i.e. the value of lift remains the same.

So no, Bernoulli's equation as it is classically written does not apply in the stationary observer's frame of reference. My question to you is why does that matter? It applies in the frame of reference of any typical fluid dynamic calculation, in the frame of reference of a wind tunnel, and in the frame or reference of someone sitting on a plane.

Maybe it's an issue of terminology depending on the author, for example in some articles, the term Bernoulli only refers to the relationship between a pressure gradient and speed along a streamline, while no special name is given for the relationship between a pressure gradient across a streamline and the corresponding centripetal acceleration. In some aritcles, if energy is changed within a streamline, it's no longer considered to be a streamline, there can be a streamline before and after the region of change in energy, but not across the region of change in energy.

It is not a terminology issue. In all cases, Bernoulli's equation is
\dfrac{v^2}{2} + \dfrac{p}{\rho} + gz = C
where $C$ is some constant and $g$ is the acceleration due to gravity (or any irrotational body force). In the case where the flow is irrotational, then $C$ is the same everywhere. If the flow is not irrotational, then $C$ differs from one streamline to the next, but the equation can still be applied across a single streamline. Sometimes you will see it generalized to what is sometimes called the unsteady Bernoulli equation, which is
\dfrac{\partial \Phi}{\partial t} + \dfrac{v^2}{2} + \dfrac{p}{\rho} + gz = f(t),
where $\Phi$ is the velocity potential and $f(t)$ changes with time. Either way, the only requirements here are that the flow is irrotational and incompressible (and steady in the first case). Bernoulli's equation is never applicable in a boundary layer, however.

The pressure gradient normal to a streamline does not matter. A streamline can curve and twist in whichever way the flow dictates and it is still a streamline and Bernoulli's equation still holds along it provided the other required conditions are met.

One issue I haven't seen addressed is the pressure gradient across a curved boundary layer, where the inside of that boundary layer has the slowest (or zero) velocity, yet also has the lowest pressure (perhaps restrict this to laminar flow boundary layers).

I've said repeatedly that in the wall-normal direction, it is effectively zero. The pressure at the wall is almost identically equal to the pressure at the edge. See https://www.amazon.com/dp/3540662707/?tag=pfamazon01-20.

 

[rcgldr]

There have been approximately 6,420 threads on this forum discussing it, so I suggest a forum search.

I think phinds is correct, we just should have posted a link to a prior thread.

while using the air (free stream ... ) as a frame of reference ...

Regardless, literally no one treats a wing this way ...

Not mathematically, but there are popular articles that include a description of how a wing works from the perspective of a ground based observer (assuming no wind) or from the air's frame of reference as well as from the wings frame of reference. A couple of articles with a "macroscopic" of how wings work that should answer the questions in the original post.

http://home.comcast.net/~clipper-108/lift.htm

http://www.avweb.com/news/airman/183261-1.html

pressure gradient across a curved boundary layer ... (this was already removed from my previous post, somehow we cross posted here)

Your post #47 also covers part of the original posters question.