# Fluid mechanics, why does the air flow faster over the wing?

1. Oct 3, 2010

### MaxManus

Why does the air flow faster over the wing? In my fluid mechanics course we used streamlines and a half body to show that the air flow faster over a half circle, but I do not understand why it does so.

2. Oct 3, 2010

### rcgldr

In the case of a half cylinder (flat on bottom, circular on top) "airfoil", it wouldn't generate lift unless there was a large angle of attack. You'd have higher pressure, slower speed upwash at the front, then air would accelerate normal (perpendicular) to the flow if it's mostly a laminar flow and follows the upper surface near the peak to avoid creating a void (a Coanda like effect), or if's turbulent, a vortice would form in order to avoid creating a void, depending on the speed, and rate of curvature of the half cylinder (viscosity is also a factor, but I'm assuming this is air, not helium). In the laminar flow case, towards the rear of the upper surface, pressure would increase and the flow would decrease in speed.

A web site with a casual explanation of how wings work:

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

The half cylinder laminar flow case would be similar to the hill case in section 4.3 of the web page linked to below, and the turbulent flow case would be similar to the roof case further down on that same web page. Note that in the case of the pipe example on that web page, even if the leading edge of the pipe creates a vortice,the pressure is still lower than ambient across the opening, and some carburetors use a tube protuding into a venturi pipe with the opening perpendicular or aft of the relative flow to reduce pressure futher still to draw more fuel into the flow, with an intended turbulent vortice to also atomize the fuel. The old Flit Gun pump sprayer used a similar method, a thin vertical pipe positioned with it's open end within the flow generated by a hand pump with a narrow outlet.

http://user.uni-frankfurt.de/~weltner/Mis6/mis6.html

Last edited: Oct 3, 2010
3. Oct 8, 2010

Thanks.

4. Oct 8, 2010

### DrDu

The flow at the end of the wing has to fulfill the so called Kutta Joukowski condition. I. e., the steam lines from above and below the wing have to be parallel and of equal velocity. That determines the degree of circulation around the wing and the difference of velocity below and above wing.

5. Oct 8, 2010

### K^2

No, it doesn't. Kutta condition merely tells you that the streams meet, but because of the separation layer between them, it doesn't tell you anything about their relative velocities. In other words, Kutta condition doesn't tell you that the stream over the top of cambered wing has to travel faster.

On the other hand, the fact that air over the wing travels faster isn't directly related to the lift. It does tell you that you have positive circulation, which means the air is deflected downwards, which is how you really compute lift, because you can now use momentum conservation (Kutta Joukowski Theorem). But lift itself is caused by pressure at the boundary, and the air speed at the boundary is zero all around the wing. So you cannot use it as an explanation of lift.

6. Oct 18, 2010

### MaxManus

Thanks, but now I'm not sure if I understand. It might be that I misunderstood my fluid mechanic course, but I believe we used the fact that the air flows faster over the wing and the bernoulli equation to show that there will be a lift. I have read the links and see that there are other things that create lift too, but is it wrong to say that that faster flow over the wing creates lift?

7. Oct 18, 2010

### DrDu

I am not a specialist in fluid mechanics.
Which separation layer? Wouldn't a difference of speed after the wing lead to turbulences which finally carry away circulation until the speeds become equal?

I cannot follow your argumentation that the lift cannot be calculated from the air speed as the air speed vanishes at the wing. The vanishing of the velocity is due to friction.
Bernoulli only applies when friction can be neglected. But this is the case already at distances small compared with the dimesnion of the wing.

8. Oct 18, 2010

### Andy Resnick

This is correct. The Kutta-Zhukovsky theorem states that a vortex of circulation G, moving in a uniform fluid of density p with velocity v_inf (the subscript is there for technical reasons), produces a force directed perpendicular to the direction of motion and the axis of the vortex of magnitude p*v_inf*G per unit length.

This theory neglects vortex tip shedding- the free vortices often seen behind airplanes.

Going to a finite wingspan, the problem was solved by the Prandtl-Lanchester lifting line theory, and from there, found an interesting application in Fletter's rotorship.

http://en.wikipedia.org/wiki/Lifting-line_theory

http://en.wikipedia.org/wiki/Rotor_ship

9. Oct 19, 2010

### DrDu

Dear Andy,

I don't doubt the Kutta Joukowski theorem.
The presence of circulation explains different velocities above and below the wing and by Bernoulli leads also to a difference in pressures. So there is no contradiction with the statement that it is the higher velocity above the wing which is responsible for lift as stated by MaxManus.

The amount of circulation is fixed by the Kutta condition. My point is that after the wing the velocity of the fluid elements passing above and below the wing at the line (or surface) where they meet has to be equal as pressure must be a continuous function.

10. Oct 19, 2010

### Andy Resnick

DrDu,

I'm having a hard time parsing your post- the Kutta condition requires the velocity to be *bounded*, not *continuous*. I'm not sure how well I can demonstrate it, but a derivation can be found in Saffman's "Vortex Dynamics". AFAIK, the relationship between a discontinuous velocity and accelerated flow past a wing pertains to shock waves and stability, but it's been a while since I went through the detailed derivation.

11. Oct 19, 2010

### DrDu

Yes, that is true.
I doubt on physical grounds that in sub-sonic stationary systems a shock wave with discontinuous velocities or pressure can pertain.
A more formal argument is the following:
For some special wing shaped bodies the velocity field around the body can be obtained from the fluid flow around a cylinder via a conformal transformation. Now the flow around a cylinder is an analytical function of r and theta for all values of these parameters and a conformal transformation cannot introduce a line of discontinuities.

For the solutions for a cylinder see 3.93 and 3.94 (alas in German only) in

The equality of the velocities behind the wing is explicitly derived in that script from the Kutta condition in node35 of that script.

Last edited by a moderator: Mar 4, 2015
12. Oct 19, 2010

### Cleonis

Faster flow over the wings is a necessary part of the whole, but it shouldn't be thought of as the key factor.

The most direct example of a setup producing lift is a jet engine or rocket engine that directs its exhaust downward. A man with a jetpack strapped on can hover. If the exhaust is directed in horizontal direction, to the left, then you will accelerate to the right; the action-reaction principle. If you direct the exhaust downward you are using the propulsion to hover (or climb).

(For completeness: another category of lift is buoyant lift, as in the case of a hot air balloon. In the following discusssion I refer to 'all forms of lift', meaning 'all forms of non-buoyant lift', but that's such a mouthful.)

All forms of lift have in common that mass is accelerated downward. In the case of a jetpack exhaust is accelerated downward, helicopter blades accelerate air mass downward, and in the case of an aircraft wing the air mass flowing around the wing is deflected downward. Action-reaction.

As you know, the crudest wing of all is just a flat board, at an angle of attack. That is a horribly inefficient wing design, but it will perform as a wing; it will deflect air mass downward. The reason wings are design for good flow characteristics is of course to optimize performance. As you know, with a flat board air mass will have a strong tendency to separate from the top of the surface. Performance is all about getting the air to not separate from the top surface. That is how you maximize lift given the total available wing area.

Incidentially, in the case of buoyancy (balloon, airship) the lift does come fundamentally from a force difference; in a state of positive buoyancy more force is exerted on the underside of the craft than on the upper side.

Jetpacks, helicopters, and aircraft wings generate lift by accelerating air mass downwards; the lift comes from action-reaction.

Last edited: Oct 19, 2010
13. Oct 19, 2010

### Andy Resnick

DrDu,

I'm still having trouble understanding your questions/comments/points. Does this have to do with the origin of lift (generated by a wing), the origin of circulation/vortices, detailed solutions to either, general solutions to either,... ?

14. Oct 19, 2010

### rcgldr

In the real world, depending on air foil, angle of attack, wing loading, ..., the flows aren't quite equal, there's turbulence and vortices at the trailing edge, as differences in velocity and pressure exist at the trailing edge where the flows merge. Wing tip vortices also significantly interfere with the flow on a typical aircraft (high end gliders with 80+ foot wing spans would be an exception).

Getting back to the OP, even a flat airfoil with a reasonable angle of attack will end up drawing more air downwards from above than pushing air downwards from below, so the greater difference from ambient pressure above the wing results in a higher maximum speed of the air above.

On a side note, in the case of small balsa models that glide at low speeds, nearly flat air foils aren't that inefficient.

15. Oct 20, 2010

### DrDu

Dear Andy,

well, for me it's also not easy to parse your posts, e.g. expressions like "vortex tip shedding".

In my first post in that thread I wanted to point out the importance of the Kutta condition in understanding the lift of a wing. I think we both can agree on that.

My further comments refered mainly to what K^2 wrote and of which you said "This is correct".
I wondered especially about the "separation layer" and how it may or may not separate sheets of air of different velocity.
You also later wrote:
"AFAIK, the relationship between a discontinuous velocity and accelerated flow past a wing pertains to shock waves and stability, but it's been a while since I went through the detailed derivation." I still cannot believe that there is a discontinuity in the velocity (below velocity of sound) and tried to find some arguments against it. I would be interested in some further explications of your point on that.

16. Oct 20, 2010

### Andy Resnick

DrDu,

That's fair- ok, let's back up a bit- all the way back to the motion of a rigid object through a viscous fluid.

The motion of a sphere/ellipsoid was found a long time ago (say 1845) because the solution is well behaved: in the frame of the ellipsoid, the fluid velocity decays to a constant as 1/r^3, so the disturbance created by the ellipsoid is very localized. Not so with a cylinder- the translation of a cylinder with constant velocity (perpendicular to the cylinder axis) though an infinite fluid creates a disturbance in the fluid that grows without bound.

Oseen linearized the relevant equations and found the drag force is finite- the derivation is in Lamb's book, but we summarized the final results here:

http://www.ncbi.nlm.nih.gov/pubmed/17526573

We'll get to wings shortly- first, let's discuss 'boundary layer separation'. When fluid moves past a body, the no-slip condition ensures that the fluid in contact with the solid moves at the speed of the solid. Thus, there must be a velocity gradient to match the no-slip fluid motion with the far-field fluid motion.

Now, the no-slip condition has a very fuzzy pedigree- it's required to keep the stress finite, but it rules out observed phenomena like wetting. There's lots of material out there trying to reconcile the two, and since it's not that relevant, I'll pass over this point without comment.

Boundary layer flow, which is the region of fluid close to the object, is also a fuzzy concept. There's no clear demarcation between the boundary layer and far-field velocity. A key concept is the Reynolds number- the ratio of inertia to viscosity. Small Reynolds numbers correspond with highly viscous flow (eg Stokes flow, the result above) and high Reynolds numbers correspond to Bernoulli's equation and Euler's equation.

Prandtl was the one who thought of the boundary layer- it's a region of flow where the Reynolds number is important. For fluids with low viscosity (gases), this region of flow is found near the surface of a rigid object. Furthermore, as the Reynolds number increases, the boundary layer thickness decreases and the (local) shear rate is larger.

The usual definitions for boundary layer thickness can be found easily online, but to summarize the result for a flat surface, the thickness 'd' of the boundary layer a distance 'x' from the leading edge is given by d/x =5* Re^0.5, where Re is the local Reynolds number at the distance 'x'. Steven Vogel's "Life in Moving Fluids" covers this topic in an incredibly clear chapter.

Boundary layer separation is a feature of viscous flow past a 'bluff' body, and occurs when the velocity within the boundary layer changes sign from positive (with respect to the body) to negative. The details of where this occurs is complicated, but has to do with the stress gradient. Separation of the boundary layer leads to the phenomenon of 'wake' behind a body moving though a fluid. There is also the phenomenon of 'stall'. When the boundary layer separates, the flow pattern becomes so diffrent from laminar flow that the results from simple inviscid+boundary layer flow have to be abandoned completely.

This is all fine, but what does it have to do with an airfoil? The conformal mapping of a cylinder to an airfoil results in a solution that (excluding stall) has negligible boundary layer separation. However, the *inviscid* flow pattern- a limiting flow pattern- around a bluff body has, AFAIK, remained resistant to solution by either theory, calculation, or experiment.

So now what? Now we introduce the idea of vorticity and circulation. Consider the flow of an inviscid incompressible fluid past a body started from rest by conservative forces. The motion is not unique, unless the flow is continuous. If the flow is continuous, the Helmholtz-Kelvin theorem states that the velocity is the gradient of a single-valued potential. If discontinuous flow is allowed, by for example allowing the creation of a vortex sheet from separated flow, the unsteady flow no longer has a unique solution.

This has the best images I could find right now:

http://www.biketechreview.com/aerodynamics/misc/466-skinsuits-and-boundary-layers?start=2

Vortices have 'circulation' associated with them, and so again, by the Helmholtz-Kelvin theorem (circulation is conserved), there is circulation associated with the wing to compensate for the generation of a vortex sheet- the vortices detach from the wing, forming a "von Karmen vortex street":

http://www.simerics.com/animation/karman_vortex_street_experiment.gif

The Kutta condition comes into play simply because an airfoil cross-section is a conformal map of a circle.

There's a lot here; let me pause and ask you what makes sense/what does not make sense

17. Oct 20, 2010

### rcgldr

A lot of this seems to be going beyond what the OP was asking.

For a simple answer, air is sort of like a rope, pulling it has more effect than pushing it. If an object is pushing the air (advancing), a lot of the affected air will tend to flow around the object rather than getting pushed forwards by the object. However if an object is pulling the air (retreating), the air is forced to follow the object in order to fill in what would otherwise be a void.

In the case of a wing, the upper surface is pulling air downwards, and the lower surface is pushing air downwards. The flow that is "going around" the wing occurs just ahead of the wing and around the wing tips. Because the pressure above is lower and the pressure below is higher, the air flow is diverted upwards ahead of the wing. The air "going around" the wing decreases the overall lift, but the end result is that for a typical wing, there's more pressure reduction above a wing than pressure increase below the wing, and this corresponds to a greater maximum speed of flow above the wing.

Last edited: Oct 20, 2010
18. Oct 20, 2010

### DrDu

I think I have a good understanding on how a boundary layer forms. In old times I even wrote a paper on multiple scales techniques (although in QM, not fluid mechanics).

I also saved a copy of Prandtls collected works from the library which is here on my shelf.

I am confused. I thought the *inviscid* flow pattern to be known, at least for a cylinder?
What is a "bluff" body? I can't find a really fitting translation into German.

I always thought that after a wing ideally there is no von Karman street, only the vortex formed when starting? But maybe that is an (over-)idealization.
In a von Karman sheet, the direction of the vortices alternates. So I would expect that at least on the mean the velocity above and below the street is equal.

Well, thank you for your pacience. I am enjoying very much this discussion.

19. Oct 20, 2010

### Andy Resnick

Me, too!

I only have a moment to respond, I can do more later:

1) 'Bluff' bodies are the opposite of slender bodies, 'blunt' may be a more descriptive term:

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

2) I am unaware of a complete and general solution for Re -> 00 flow past a solid body. I haven't pored over the literature lately, tho.

3) We haven't considered airfoils of finite length yet- it turns out the dominant mechanism of vortex shedding from a finite airfoil is from the ends, not along the length.

http://pilotsweb.com/principle/art/v_sheet.jpg

More later...

20. Oct 20, 2010

### DrDu

ad 1) ok, so the contrary of streamlined
ad 2) naively, I thought that inviscible is the same as an ideal fluid. However, the limit Re-> infinity (or nu->0) is a singular limit and so it will not coincide with the nu=0 case.
However, I have no good idea how this will lead to turbulence.
ad 3) that I know. In 3d, the vortex lines are closed.

But you still avoided my primary question. Is the velocity behind a streamlined object continuous or not?