Understanding Airplane Lift: Exploring the Mechanics of Flight

[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.

 

[boneh3ad]

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.

And yet none of them actually refutes that Bernoulli's equation is valid for calculating lift. Treating the airfoil from the stationary observer's frame of reference is a curiosity and can illustrate a couple of interesting things, but in terms of actually modeling or calculating lift, it is very inconvenient and rarely used.

 

[rcgldr]

I think this thread went off on a tangent into the details (like Newton versus Bernoulli, when in fact both apply), when it seems the original post was asking for a more generic description of how wings work, like why a flow follows a convex surface which was explained in post #47 (I call this void theory, air accelerates into what would otherwise be a void ... ). Getting back to the original post:

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?

Assuming bent wing means curved wing, this isn't required. A flat wing can produce lift, and for small balsa type models, it's good enough. Curved wings are more efficient (less drag for the same amount of lift).

As for faster flow over the top of a wing, this only applies from the wings perspective (or anything moving at the same speed as the wing, like the pilot in an aircraft). It's because a wing draws the air downwards from above (see post #47 for why this happens) reducing it's pressure and pushes it downwards from below increasing it's pressure. Air accelerates as it moves from a higher pressure zone to a lower pressure zone, and decelerates when it moves (due to momentum) from a lower pressure zone to a higher pressure zone. So higher speeds coexist with the lower pressure zones above a wing, and lower speeds coexist with higher pressure zones below a wing.

For an observer on the ground (with no wind), again you have the wing drawing air downwards from above and pushing air downwards from below, but the fastest moving air occurs just behind the trailing edge of the wing, mostly downwards (related to lift) and somewhat forwards (related to drag).

For an example of an unusually shaped wing, here's a video of a prototype reentry vehicle called a M2-F2:

 

[stedwards]

"...and why do fast moving fluids apply a lesser pressure than slow ones?"

I should have found this answered in wikipedia bernoulli as the conservation of energy
equation, E = T+V pertaining to an element of fluid, rather than the oblique reference
to Newton.