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Does this event really cause the other event, according to accepted theory?

This question is in regards to the text, "the flow following the upper surface contributes strongly to the downward-turning action."

To paraphrase, one event - "the flow follows the upper surface" - causes (or more precisely, is a contributing cause of) a second event - "the air turns downward".

Is this in accordance with accepted science? If not, I think that the text should be reworded to convey what was meant.

I think I may understand the important point that the author was trying to convey. I think the author meant to refer to a baseline of 'atmospheric pressure on all subsurfaces'--which applies when there is no flow--and compare that to the conditions of flight, where the deviation from the baseline is greater on the upper surface than on the lower (so the "contribution" relative to the baseline is mostly from the changed conditions on the upper surface, not the lower as intuition would suggest). But that is very different I think than what was said. The fact is that under conditions of lift, the upper flow is following the surface of the wing, more or less, "because of" (if we must allow a causal interpretation) the excess of pressure from the air above over that applied by the wing. And the pressure on the upper surface of the wing under conditions of lift is opposing, not enhancing lift.

Mark.camp (talk) 23:44, 1 December 2014 (UTC)

At some point in the article's history, the idea expressed by this passage was followed by some discussion of airflow on both sides of the wing and the point made that some explanations only involve air being deflected by the bottom of the wing. (sometimes called "bullet theroy" or "skipping stone theory"). This was accompanied by a graph showing the pressure distribution along the top and bottom of a wing where the pressure difference along the top was substantially larger than along the bottom. At some point this was excised, whether by accident or not.

Which is to say that your interpretation of the intent is correct. I can see where what is actually written may be interpreted differently than what was meant. A suggestion for how to reword would be welcome. I'll take a look at earlier versions and see if we edited something out that shouldn't have been. Personally, I don't read the passage as a statement of cause and effect but I can see how someone might interpret it that way. Mr. Swordfish (talk) 00:24, 2 December 2014 (UTC)

Sorry, I didn't say why I thought the current text suggested a causal relationship, so I will provide some examples that clarify why I inferred one. (I address this point of yours before answering your good request for proposed wording, since if I'm off a bit in my reading then no change is needed.)

  The addition of fertilizer to the poor soil common to the region contributes 
  strongly to the higher corn yields.
  (The sentence implies, for me, that "the addition of fertilizer" is a contributing cause of "higher corn yields")

  Having three unusually strong candidates for the lower house contributed strongly to the party's gains in that election.  
  (The sentence implies for me that "having three unusually strong candidates for
  the lower house" was a contributing cause of "the party's gains in that election."

  Adding two extra cylinders to the V6 engine contributed strongly to the increased torque
  in the new engine.  (The sentence implies, in my reading, that "adding two extra 
  cylinders to the V6 engine" was a contributing cause of "the increased torque in the new engine".

In these cases, which are I hope analogous to the text under discussion, it seems that the first event ("addition of fertilizer") is implicitly claimed as being a contributing cause of the second ("increased corn yields").

Does my reading of the original text seem reasonable to you?

Mark.camp (talk) 23:38, 7 December 2014 (UTC)

I am not sure there is a significant problem here:

 The higher yields from fertilizer are predicated on it being poor soil. (this is specifically mentioned in the 
 example sentence)

 The strong election gains are predicated on the election being for the lower house. (this is not exactly 
 mentioned in the example, although the description "that election" might suggest it)

 The increased torque is predicated on the existing cylinder capacities remaining constant. (this is not mentioned 
 at all in the example sentence)

 The contribution of the upper surface flow is predicted on the downward turning of the upper surface itself. (this 
 is not mentioned in the example sentence)

Whether these assumptions are made explicit in the given sentence is neither here nor there since they are, one way or another, trivially evident from the overall discussion. One might write, "the flow following the downward-turning upper surface contributes strongly to the downward-turning action of the flow," but I personally think that this would be unnecessarily pedantic. There comes a point where the sheer amount of word salad begins to confuse. Whatever the present version says in terms of phrasing, the physical mechanism it is explaining is obvious enough, and at the end of the day that is what language is for. Steelpillow 10:54, 8 December 2014 (UTC)

Why I now find "The Statment" to be problematic

The resulting force upwards is equal to the time rate of change of momentum of the air deflected downwards.

— "The Statement"

Since I don't know quite where to insert this into the wall of text above I'm starting fresh with a new section. I'm also starting more or less from the beginning. Please bear with me, and thanks in advance for your patience.

Let's take a look at a simple model, i.e. a plane flying straight and level at a constant speed. In this model, the only things are the plane, the atmosphere, and the ground. After decomposing the total aerodynamic lift into it's components we have four forces on the plane: lift, drag, weight, and thrust. Lift is opposed by weight and thrust is opposed by drag. Since the plane is flying straight and level L+W=0 and D+T=0. With no net force in either the vertical or horizontal direction, there is no acceleration and consequently no momentum change. For the plane, dp/dt = 0.

The ground, of course, is stationary, so for the ground dp/dt is also zero.

The only other thing in the model is the air, and if its momentum is not constant we have a violation of conservation of momentum - how can the air experience a change in momentum without anything else experiencing an equal but opposite change in momentum? So the air (ie the atmosphere as a whole) must have constant momentum, that is dp/dt for the atmosphere as a whole is zero. We don't need to compute integrals of momentum flux and take the limit as the boundary goes to infinity to arrive at this, although making those calculations gives the same result.

However, there are subsets of the air that experience non-zero dp/dt. And if we take just the right subset, it can be shown that for that subset of the atmosphere dp/dt = -L. It is reasonable to interpret the lift physically as the result of the momentum change of that subset of the atmosphere. (not everyone agrees with or likes that interpretation, but it is supported by many reliable sources)

There's nothing special about the vertical direction in the argument above. Looking at the horizontal component of momentum change, thrust is equal in magnitude to drag so the net force in the horizontal direction is zero. For the plane, the horizontal component of dp/dt is zero as well. The ground is just as stationary as it was a couple of paragraphs ago, so we can conclude that for the air the horizontal component of dp/dt is zero.

This does not contradict the fact that the propeller or jet is pushing a fairly large amount of air backwards. Unlike with lift, there seems to be no controversy over whether air is pushed backwards by the engine. ( NASA has a concise treatment of thrust and momentum change at http://www.grc.nasa.gov/WWW/k-12/airplane/thrsteq.html). The physics in the horizontal and vertical directions is basically the same - if you take a close look at the propeller blades they're just small airfoils generating lift. One can say that the "real" reason the propeller drives the plane forward is the imbalance of pressure between the front and back of the propeller blades, but most folks are satisfied with the 'push air backwards' idea, so much so that for many simple explanations thrust isn't even explained - most people just get it without any explanation. In any case, looking at the Talk page of the Thrust article I don't see any long drawn out discussions about how to explain thrust, the article simply says it's a reaction force and this unremarkable statement has collected few remarks on the talk page.

So what does all this have to do with the statement? Well, one point of the exercise is to show in an easy-to-follow manner that when you consider "the air" as the entire atmosphere then the rate of momentum change of "the air" is zero, not -L. So, if a reader gets the idea that "the air" means the whole atmosphere rather than just a subset then including the statement as previously worded will give the wrong impression and mislead the reader. However, if we provide context then some version of the statement would be appropriate.

Another point is to show that dp/dt=0 is not inconsistent with some subset of air being accelerated - either downward as occurs with lift or backwards as occurs with thrust. Some subset of the air is accelerated downward with dp/dt = -L; some subset is being accelerated backwards with dp/dt = -T. I don't know how important it is to specify exactly what subset(s) meets this criteria, but I do think it's important to specify that it is a subset and not the entire atmosphere, that is if we are going to include it at all.

Moving forward, I'll try to collect in one place the cites that relate lift to momentum change of air. then we can put on our wikipedia editor hats and evaluate the material provided by the various reliable sources and perhaps craft a subsection that reflects that material. Mr. Swordfish (talk) 20:08, 8 December 2014 (UTC)

The analogy with thrust is a good one. Since a propeller is a rotating aerofoil, there's little difference in the physics, except for geometry.

I can see why someone familiar with Lanchester's "Principle of No Momentum" might initially see The Statement as a claim that the atmosphere as a whole is given downwards momentum. The trouble with that interpretation is that for it to be true, the atmosphere would have to fall through the ground. That is a nonsensical assumption, which highlights the straw man in the argument that The Statement is false.

It's true that momentum can only be calculated for a specified portion of air. I think you've expressed the point well, that that subset of air (and the momentum that is communicated to the rest of the air) exists whether the location of that subset it is specified or not.

I think we've all agreed from the outset that talking about control volumes at this early stage of the article is more likely to confuse the reader than help them understand how the physical principles expressed in Newton's laws contribute to lift.

I think once some text for the proposed new section is drafted it will be much easier to agree the facts, and decide what changes (if any) should be made to the section on Newton's laws to improve the coherence of the article as a whole.

Burninthruthesky (talk) 10:15, 9 December 2014 (UTC)

I'd second that gathering and correlating sources is far the best way forward. And do you know, I have clean forgotten what "the statement" said, it was so long ago. Perhaps it is better not to remind ourselves but to just stick with the sources and a clean sheet? — Cheers, Steelpillow (Talk) 11:06, 9 December 2014 (UTC)

"The atmosphere as a whole" isn't the only definition of "the air" for which a reader is likely to mistakenly think The Statement is true. Another potential trap is the idea that The Statement is true for "the local airflow" in the neighborhood of the foil and that it is only when you look farther afield that other effects creep in. This is a misconception that's been expressed by several participants in this discussion and may still be held by some of them.

Burninthruthesky seems to have a strange definition of a "straw man". It's obvious to him that The Statement is false for the atmosphere as a whole, so the mentioning of that as an example is a "straw man"? The Statement is also false for "the local airflow", which is perhaps not so obvious. So I disagree with the idea that it's not important to discuss the "location" where this "subset" with dp/dt = -L "exists". I think it's important to know that "the local airflow" isn't that location.

In terms of basic physics, a propeller is different from a wing in important ways, and the analogy between thrust and lift is a poor one for the following reasons:

• Even in a reference frame moving with the airplane, the propeller blades are moving and doing work on the fluid, leaving behind a slipstream with higher total-pressure than in the rest of the field. In even the simplest model of propeller flow, like Rankine's actuator-disc model from 1859, the boundary of the slipstream is a vortex sheet that has no counterpart in the flow around a 2D lifting foil.

• A propeller has a substantial effect on the part of the flow that passes through the disc and becomes the slipstream, and relatively little effect on the rest of the flow. Thus the kind of model used by Clancy and by Chris Waltham, in which only a limited part of the stream is assumed to be affected, applies reasonably well to propeller flow. It doesn't apply so well to the flow around a lifting foil.

• Cross-stream pressure gradients play only a minor role in propeller flows but play a key role in the momentum balance in airfoil flows. This is to be expected, given that thrust is a streamwise force, and lift is a cross-stream force.

So I don't think analyses of thrust are much help in understanding lift.

I also second Mr. Swordfish's initiative to gather the sources.

That said, I have some reservations about including direct quotes in the article's list of sources. I understand that it makes things easier for the reader in a way. But it also has a downside. It isn't generally practical to provide a long enough quote to avoid the "out-of-context" problem, and the result can be an incomplete and possibly misleading view of what the quote really represents. In a quote of reasonable length you often get only the author's conclusion, not the assumptions he made or the analysis he used to reach that conclusion. I'd propose that instead of a direct quote we should consider paraphrasing the findings, including a brief summary of the assumptions and methods of analysis.

Now that we're collecting sources, I propose that we also collect comments by those who take the time to read the sources, e.g. brief summaries of the models and assumptions that were used, and resulting criticisms etc. My preference would be to append comments directly under each source to make things easy to correlate and to facilitate later discussion.

J Doug McLean (talk) 01:57, 10 December 2014 (UTC)

Yes, I know that there is more than one way to misinterpret The Statement in a way which makes it false. What I'm saying is that the language used is not specific enough to imply anything whatsoever about how to calculate the rate it describes. I still believe the control volume analyses which account partly for momentum and partly for pressure do not answer the question posed by The Statement. I see that you now describe it as "misleading" rather than "false". The former is an opinion to which you are entitled and I thank you for moderating your language.

Personally, I have no problem with the suggestion made above to keep quantitative statements out of the Newtonian section and introduce them later if required. I noticed myself that Langeweische doesn't make any quantitative claims.

I look forward to seeing your suggestions for the structure and focus of the new section you propose. Burninthruthesky (talk) 17:00, 10 December 2014 (UTC)

Doug, I think a big reason why you've received so much pushback here on the talk page about your objection to the statement is due to the lack of direct quotes from the sources you cite. Citations and verifiability are what makes the entire wikipedia party happen. While I agree that we need to be careful to not take a single sentence or two out of context, we need more than just a work and a page number to back-up contentious claims.

Misplaced Pages can be a funny place sometimes; I'm sure you are quite capable of looking at a page of equations and concluding in words that "these equations clearly show that the time integral of foo is equal to bar". But unless we have a reliable source that draws the same conclusion putting that statement into the article is synthesis or interpretation which is prohibited. If you publish the same statement in a book or a paper we can use it, but if you just say so here on the talk page we can't. Like I say, Misplaced Pages can be a funny place, but those are the rules of evidence.

So, I'd prefer to see some direct quotes rather than editors' interpretations. Right now, we have an excerpt from your book and the clarifying aside from Waltham about "doing it right" that we can weigh against the imprecise assertions of Smith and the AAPT committee. Another one or two and we're probably there.

Sorry you didn't like the thrust analogy - like any analogy it only goes so far, but I think it serves to illustrate that dp/dt = 0 for the atmosphere as a whole is consistent with non-zero mass flow is some subsets of the atmosphere.

Finally (for now) I'd prefer that the raw material thread not be interrupted by a lot of back-and-forth exchanges amongst the editors. Once that starts happening the thread will become difficult to follow. Instead, I'd suggest a sub-thread for each source below the initial post. I'll start with an example. Mr. Swordfish (talk) 20:45, 10 December 2014 (UTC)

Two examples that may help clarify the discussion

Consider an iceboat oriented so that the wind is perpendicular to its centerline with its sail trimmed in and the brake engaged. The sail generates lift which is parallel to the centerline of the boat and the lift force (L) is opposed by the friction force of the brake (B). We'll assume in this idealized model that the brake is strong enough to hold the boat motionless. L + B = 0, the total force on the boat is zero and therefore it's acceleration and dp/dt are zero.

In this model the boat can't move, the ice doesn't move, and the only other thing is the air. Since momentum is conserved, dp_boat/dt + dp_air/dt = 0 and the net momentum change of the air is zero.

Now, let's release the brake. If we ignore the negligible friction force of the runners (skates), the net force acting on the boat is L. (drag is opposed by the runners that don't move sideways). If F is the total force on the boat we have F = L = ma = dp_boat/dt .

Again, conservation of momentum says dp_boat/dt + dp_air/dt = 0, and in this case dp_air/dt = -L. So in this special case we can say that the lift is equal to the time rate of change of momentum of the air.

When we had the brake engaged, the situation was very similar to a plane in steady level flight, with the brake playing the role of gravity in opposing the lift force. When we release the brake, the situation is similar to the idealized model of a wing in an infinite atmosphere in the absence of gravity. In this second situation, the statement is true even when "the air" is the entire atmosphere. But the usual example when explaining lift is steady level un-accelerated flight in a gravitational field, not a plane flying in the absence of gravity or an iceboat accelerating from a stand still.

To be a bit nit-picky, I should add that in the case of the iceboat, dp_air/dt = -L is only strictly true for a moment - once it begins accelerating the apparent will will move forward and L will not be parallel to the centerline of the boat anymore so L != dp_boat/dt in general. Further, since it is accelerating our usual frame of reference will no longer be an inertial reference frame. And eventually the boat will reach a steady speed at which point dp_air/dt = 0. So I don't think an iceboat accelerating from rest is a good example to place in the article.

Hopefully these two examples (a foil constrained to have no acceleration and a foil that is allowed to accelerate with L being the total net force) will cast a bit of light on how momentum is (or is not) transferred to the air. This thought experiment helped clarify things for me anyway. Mr. Swordfish (talk) 20:52, 15 December 2014 (UTC)

You have changed the statement from the version originally stated (and which I copied above before archiving). This change makes the phrase "the air" more ambiguous and easier to attribute an unintended meaning. Have I missed something or is this effectively a new discussion about a new problematic statement? — Cheers, Steelpillow (Talk) 22:01, 15 December 2014 (UTC)

Apologies. I didn't recall whether the word "downward" appeared in the original, so I went to the talk page archive to find the earliest example. On July 27th, Doug wrote The more specific statement "lift is equal to the time rate of change of momentum of the air" is not correct in general. I recommend deleting this sentence. leaving out the word "downward". I mistakenly took this to be the definitive version of the statement but looking back at the actual draft and subsequent versions of the statement on the Talk page indicates that the word "downward" was in the original. Thank you for restoring it. Mr. Swordfish (talk) 22:15, 15 December 2014 (UTC)

The version I copied also included the word "deflected". This makes a big difference to what one assumes is the air in question. Without it the Statement is ambiguous and requires a suitable preceding remark to give it context. We have lost that context so, if the Statement as now presented is to mean anything at all that can be discussed, it needs that context restoring. — Cheers, Steelpillow (Talk) 11:23, 16 December 2014 (UTC)

Agree that "downward" needs to be part of the statement, otherwise the air deflected in the horizontal direction by the thrust and drag would be included. However, even with "downward" included the statement is still ambiguous. What is meant by "the air"? If it's the entire atmosphere then the statement is false. If parsed narrowly, the statement says that the total momentum change of all the air with a negative vertical deflection is -L and this does not agree with the results of control volume analysis. If "the air" is a carefully defined subset of the atmosphere then the statement is true, but it's not true for arbitrarily chosen subsets and it's not true for most subsets that one would intuitively choose as representative.

Until recently, I was mislead by the statement. Only after reading up a bit on control volume analysis did I realize my intuitive ideas were not supported by rigorous quantitative analysis. I think something like the statement can go into the article if we provide sufficient context, but as it stands it's likely to give the reader the wrong idea. Mr. Swordfish (talk) 16:06, 16 December 2014 (UTC)

Are the sources which convinced you that your ideas were "not supported by rigorous quantitative analysis" listed below? Burninthruthesky (talk) 17:02, 16 December 2014 (UTC)

Yes. Zero net momentum change for the atmosphere as a whole is a fairly standard result. Chapter 8.5 of McLain's book lays it out in words fairly clearly. Google books has a generous excerpt at http://books.google.com/books?id=_DJuEgpmdr8C&q=Manifestations+of+Lift+in+the+Atmosphere+at+Large#v=snippet&q=%22Manifestations%20of%20Lift%20in%20the%20Atmosphere%20at%20Large%22&f=false Here are some quotes:

"...there is no net downward momentum imparted to the atmosphere as a whole and that the lift is reacted by pressure differences on horizontal planes above and below the wing, or on the ground plane, if there is one. We'll also consider how conservation of momentum applies to control volumes that don't encompass the entire atmosphere. ... the lift can show up at the boundaries either as pressure differences on the horizontal surfaces or as fluxes of vertical momentum mainly through the vertical surfaces, or as combinations of the two, depending on the proportions of the control volume."

"Prandtl and Tietjens (1934) showed how in steady level flight the lift is balanced by an overpressure on the ground under the airplane, so that of course there is no need for net momentum transfer."

I've been trying to find the actual passage in Prandtl that supports this, but I haven't yet. In any case, I thought that "the air" in the statement meant the entire atmosphere, or some arbitrary box around the airfoil, but the only interpretation of "the air' that makes the statement true is a tall thin subset of the atmosphere - this is not intuitively obvious. Mr. Swordfish (talk) 21:11, 16 December 2014 (UTC)

Thank you for clarifying. Yes, the net change of momentum of the whole atmosphere is zero, but momentum is necessarily imparted to air within the atmosphere. See my comments on the AAPT paper below. Burninthruthesky (talk) — Preceding undated comment added 10:06, 17 December 2014 (UTC)

Agreed. The net change of momentum of the whole atmosphere is zero. And within the atmosphere there are subsets which experience non-zero momentum change. If one chooses the subset carefully, the rate of momentum change for that subset is equal to -L. But if one chooses a different subset then in general dp/dt != -L. I think the main point of disagreement is over the lack of precision in referring to that particular carefully chosen subset for which dp/dt = -L as simply "the air" or "the air deflected downward". I'm ok with a bit of imprecision in a qualitative introductory section aimed at the lay person as long as it leaves a basically correct impression. Here, I don't think it leaves a basically correct impression. Mr. Swordfish (talk) 16:50, 17 December 2014 (UTC)

It is imprecise to refer to the net momentum of a body of air without defining the body, but we don't do that. What is imprecise about saying that momentum is imparted to air within the atmosphere at a rate equal to lift? Burninthruthesky (talk) 17:48, 17 December 2014 (UTC)

For information, the whole of the disputed section containing what I understand to be "The Statement" can be seen here. Burninthruthesky (talk) 17:24, 16 December 2014 (UTC)

@User:Mr swordfish, I think there is some discussion at cross-purposes here. In saying that "downward" should be included in the Statement, are you implying by omission that "deflected" should not be? That is the word which you removed and which I am more concerned about. To me, the phrase "the air deflected downward" has a very clear meaning which is garbled when any one word is removed. I cannot help but wonder if it is the removal of "deflected" which might have caused the ambiguity which originally confused you. — Cheers, Steelpillow (Talk) 19:30, 16 December 2014 (UTC)

The word "deflected" does not appear in the AAPT version of The Statement or in any other of the sources, as far as I know. Thus adding "deflected" to it is something that has no citable source. And for what it's worth, it's easy to show (though I know of no citable source for this) that adding "deflected" doesn't make The Statement more correct anyway. There is much more negative dp/dt in the region of "the air deflected downward" than just -L. J Doug McLean (talk) 21:15, 16 December 2014 (UTC)

No, I'm not implying anything by omission. And removing whatever word I did was an error on my part for which I apologize (I think I removed "downward' but it doesn't really matter).

I disagree that "the air deflected downward" has a clear meaning. If you take it literally then the statement is false. Mr. Swordfish (talk) 21:24, 16 December 2014 (UTC)

In Clancy's cylinder model (see below) the air which is deflected downwards is described with mathematical precision. If you take "the affected air" to be the air deflected downwards (and what other interpretation is even remotely plausible?) then the Statement is demonstrably true. So I find your assertions to the contrary to be utterly baffling, an absolute failure between us to establish any common use of language. I get the feeling that we agree on the maths but just not on how to describe it. Still, I don't see how we can take this forward between us, so I guess I will have to withdraw from this conversation. — Cheers, Steelpillow (Talk) 23:00, 16 December 2014 (UTC)

I have not read Clancy, but I have the book on order. Unfortunately, I have to get it via inter-library loan so probably won't see it until after the new year. It may well be true that in his model the integral of all the air experiencing downward deflection is equal to -L. But from what I've seen in the excerpt posted here, his cylindrical model is not as accurate as the potential flow model. In the potential flow model, a subset of the air experiences dp/dt = -L, but if you add up all the air being deflected downward the magnitude of dp/dt is larger than |L|.

So, perhaps we are just looking at different models? Mr. Swordfish (talk) 16:27, 17 December 2014 (UTC)

Yes, I am sure that we are referencing different models. Some models give dp/dt=0, some 0.5L, some L, some >L. Each model is applicable under different assumptions or conditions, i.e. they are modelling different aspects of the problem, and - crucially to this debate - all are well attested in the literature. Taking a result (or assumption) from one of these aspects of the problem and then complaining that it doesn't match the results derived (or assumed) for a different aspect is at best futile. Asserting that it is therefore "wrong" is nonsensical. If the Statement disturbs you, it is because you and it are addressing different aspects of the problem. — Cheers, Steelpillow (Talk) 17:49, 17 December 2014 (UTC)

I agree. The idea that since these analyses give a numerical value for dp/dt, they all answer exactly the same question is tempting, but wrong.

• Lanchester §112 asks, in effect, how much momentum is transferred per unit time between the foil and the ground, via the atmosphere.
• Prandtl and Tietjens ask how much overpressure is exerted on the ground due to lift.
• The control volume analyses of various other shapes ask how lift can be accounted for by a mixture of pressure and momentum.
• The Statement is only concerned with momentum.

Just to avoid any confusion, I haven't seen any sources saying |dp/dt| > |L|, neither has J Doug McLean. Burninthruthesky (talk) 10:56, 18 December 2014 (UTC)

I think Steelpillow's and Burninthruthesky's line of argument above mischaracterizes the issue.

What we're discussing here is the question of how much integrated dp/dt is imparted to the flow by a lifting foil, and as part of that question we're concerned with how dp/dt is distributed in the field.

Both Clancy's cylinder model and the classical uniform-flow-plus-vortex model address these questions. They are not modeling "different aspects of the problem"; they just model the same flow in different ways. Clancy's model for the velocity field is much more crude than the classical model, and Clancy's model ignores the pressure field, while the classical model models it realistically. Clancy's model assumes dp/dt = -L in the near field (within the cylinder), while finding dp/dt = -L in the classical model requires looking much farther afield (the tall sliver control volume). In this regard, the classical model is realistic, and Clancy's model is not. Comparing the realism of different models in this way is not "futile".

Lanchester's analysis and the other classical analyses don't deal with momentum exclusively, but they do address the question of dp/dt. In that sense they all address the same question, i.e. the value of integrated dp/dt in the flowfield.

True, I have not seen any source saying that control volumes exist for which |dp/dt| > |L|, but it's easy to show that it's true. It's original research and can't be used in the article, but I think it's fair to use as a counterargument against other original research arguments on this page. J Doug McLean (talk) 22:36, 19 December 2014 (UTC)

sources relating momentum transfer and lift

In this section I'm trying to collect source material for a proposed section on momentum transfer and lift. Additions cheerfully accepted, but let's try to keep extensive discussions out of this thread so we can see what raw material we have to work with.

"What supports an airplane aloft? ... Newton has given us the needed principle in his third law: if the air is to produce an upward force on the wing, the wing must produce a downward force on the air. Because under these circumstances air cannot sustain a force, it is deflected, or accelerated, downward.

Newton's second law gives us the means for quantifying the lift force:

F_lift = m∆v/∆t = ∆(mv)/∆t .

The lift force is equal to the time rate of change of the momentum of the air."

Bernoulli and Newton in Fluid Mechanics
Norman F. Smith
The Physics Teacher 10, 451 (1972); doi: 10.1119/1.2352317
http://dx.doi.org/10.1119/1.2352317

"Most of the texts present the Bernoulli formula without derivation, but also with very little explanation. When applied to the lift of an airfoil, the explanation and diagrams are almost always wrong. At least for an introductory course, lift on an airfoil should be explained simply in terms of Newton’s Third Law, with the thrust up being equal to the time rate of change of momentum of the air downwards. See C. Waltham, “Flight without Bernoulli,” Phys. Teach. 36, 457 (Nov. 1998)."

Quibbles, misunderstandings, and egregious mistakes
AAPT Physics Textbook Review Committee
Citation: The Physics Teacher 37, 297 (1999); doi: 10.1119/1.880292
http://dx.doi.org/10.1119/1.880292

"Birds and aircraft fly because they are constantly pushing air downwards:

L=dp/dt (3)

Here L is the lift force and dp/dt is the rate at which downward momentum is imparted to the airflow...

If we were to do this more correctly, we would box in the wing with a control volume of infinite vertical thickness. "

Flight without Bernoulli
C. Waltham
Phys. Teach. 36,457 (Nov. 1998).
http://users.df.uba.ar/sgil/physics_paper_doc/papers_phys/fluids/fly_no_bernoulli.pdf

"Now let’s move on to conservation of momentum: the force exerted on a fluid equals the time rate of change (derivative with respect to time) of its linear momentum. If you exert a force on something, you change its momentum. If you don’t exert a force on something, its momentum stays unchanged or is conserved. This is Newton’s laws, if you choose to call it that. When an airfoil is producing lift, that force does in fact change the vertical component of the airflow’s linear momentum, and the drag force changes the horizontal component of the airflow’s linear momentum. ...Measuring lift by measuring the increase in downward vertical velocity in the flow coming off the trailing edge of the airfoil is conceptually possible. This downward velocity is definitely there and is known as downwash. I have never heard of anyone actually measuring it with sufficient precision to calculate lift, not because it is physically unsound but because it is not a practical experiment."

An Aerodynamicist’s View of Lift, Bernoulli, and Newton
Charles N. Eastlake
THE PHYSICS TEACHER Vol. 40, March 2002
http://www.df.uba.ar/users/sgil/physics_paper_doc/papers_phys/fluids/Bernoulli_Newton_lift.pdf

"There is a widespread notion that an airplane in steady level flight continuously imparts net downward momentum to the atmosphere. ... Thinking in intuitive physical terms, we might also expect the impulse imposed by the airplane on the air (the product Lt) to produce a net vertical momentum in the atmosphere that grows with time. This expectation is not satisfied by the mathematics, however. ... If we expected to see a net downward momentum equal to Lt, the result comes as a surprise: The value of the integral over the semi-infinite space above the ground is zero, which means that the airplane imparts no net downward momentum to the atmosphere in steady level flight over a ground plane, regardless of height above the ground."

Understanding aerodynamics arguing from the real physics sec 8.5
McLean, Doug
Chichester, West Sussex, U.K. : Wiley, 2013.
http://mirlyn.lib.umich.edu/Record/012482734

On basic control-volume analysis of the rate of change of momentum in a moving fluid:

The Dynamics and Thermodynamics of Compressible Fluid Flow Section 1.5
Shapiro, A. H. 1953.
New York: The Ronald Press Company.

McLean: "This analysis shows that for a steady flow the integrated time rate of change of momentum of fluid parcels passing through the interior of a control volume is equal to the integrated (net) flux of momentum through the boundary. This is a basic ingredient in the other analyses cited below."

For the atmosphere as a whole, including a ground plane:

Applied Hydro- and Aeromechanics
Prandtl, L., and O. G. Tietjens. 1934. . New York: Dover Publications. Derivation in connection with figure 150.

McLean: I don't have a copy at hand, so I can't provide a quote, but this is the classic analysis showing that the pressure pattern on the ground constitutes a downward force on the ground, and thus an upward force on the atmosphere, equal to L. The net force on the atmosphere due to the lift, (i.e. the vector sum of the forces exerted by the wing and the ground) is therefore zero, so that the integrated rate of change of vertical momentum for the atmosphere as a whole must be zero.

The Prandtl and Tietjens analysis is for the 3D case. It is easy to show that the same overall conclusion applies in 2D. A citable source for the 2D analysis probably exists, but I don't know of one offhand.

For a circular region centered on the airfoil:

Aerodynamic Theory, vol. 1. Sections B. V. 6 and B. V. 7.
Durand, W. F., ed. 1932.
New York: Dover Publications.

McLean: This is a control-volume analysis of the flow around a 2D lifting body of arbitrary cross-section in an infinite atmosphere, using a circle of large radius as the outer boundary of the volume. It shows that in the far field the flow is independent of the details of the body, and that significant contributions to the pressure and the momentum fluxes at the outer boundary come only from the combination of the uniform flow and the bound vortex. It arrives at a derivation of the Kutta-Joukowski theorem in equation 7.3. Equation 5.6 shows that the flux of vertical momentum across the outer boundary, and thus the time rate of change of vertical momentum in the air in the interior, is equal to only half the lift. Equation 6.6 shows that the integrated vertical pressure force on the outer boundary is upward and equal to half the lift. The net force on the air due to the lift is therefore downward and equal to half the lift, and Newton's second law is satisfied. It is explicitly stated that this result holds regardless of how large the radius of the circle is made.

Reference added by J Doug McLean (talk) 21:08, 16 December 2014 (UTC):

An Introduction to Fluid Dynamics Batchelor, G. K. 1967 Cambridge University Press

Applying the momentum theorem to incompressible inviscid flow around a 2D body of arbitrary cross-section (a general "cylinder") with circulation, and using a control volume bounded by a circle of large radius, Batchelor finds on p. 407:

"It follows from the calculation of the integral that the side-force exerted by the cylinder appears in the fluid far from the body half as a momentum flux and half in the form of a pressure distribution."

For rectangular control volumes:

"For a large rectangular control surface, part of the lift is attributable to pressure and part to momentum, depending on the aspect ratio of the surface. For a square control surface the contributions on the surface due to momentum and pressure are equal; for a tall, long vertical surface the contributions are mainly momentum, while for a streamwise long, flat, horizontal surface the lift is primarily due to pressure. This illustrates that it doesn't make much sense to attribute the lift on an airfoil to either pressure or momentum effect, unless one takes a control surface on the actual airfoil surface, when the lift is indisputably due only to pressure!"

The facts of lift. Section titled "Lift in thin slices: the two dimensional case".
Lissaman, P. B. S. 1996.
AIAA 1996-161.
http://arc.aiaa.org/doi/pdf/10.2514/6.1996-161

For a tall column of air:

"When a loaded aerofoil is dynamically supported by a fluid, we know that its weight is eventually sustained by the surface of the earth, and that the transmission of the stress is effected by the communication of momentum from part to part, and is thereby distributed over a considerable area as a region of increased pressure"

Fig. 62 illustrates the forces on a narrow column of air where W is the weight of the foil acting downwards and the pressure at the base of the column is w. "Consequently the downward momentum imparted per second to the fluid leaving the prism plus the upward momentum received per second from that entering must be equal to W – w."

"When the height at which the aerofoil is sustained is great in comparison with its own dimensions, the area over which the weight is distributed on the earth's surface is obviously also great, and the quantity w becomes negligible. Under ordinary conditions this would usually be the case, so that the weight may be regarded as in no part statically supported."

Aerodynamics. §112 – Aerodynamic support
Lanchester F.W. (1907)
Archibald Constable & Co. Ltd.
https://archive.org/stream/aerodynamicscons00lanc#page/146/mode/2up

"The downwash also varies in the streamwise direction. It reaches its ultimate value little more than a chord length behind the trailing edge; and its mean value at the wing itself can be shown to be one half of this ultimate value." (Page 75)

...

5.15 Lift and downwash

"The lift produced by a wing is imparted to it through the variations in pressure over its surface. This lift force has its reaction in the downward momentum which is imparted to the air as it flows over the wing. Thus the lift of the wing is equal to the rate of transport of downward momentum of the air.

"This downward momentum is measured in terms of the induced downwash described above." (Page 76)

...

"Consider, then, this cylinder of air, as illustrated in Fig. 5.21. The area of cross-section of the cylinder of affected air is 1/4 πb. The rate of mass flow of affected air past the wing is therefore 1/4 πρVb. The rate of transport of downward momentum is therefore 1/4 πρVbw, and this must equal the lift, L." (Page 76)

...

"If we consider unit span of an infinite wing, however, the air above this unit span forms part of a cylinder of infinite radius, and its mass is therefore infinite. Since the downward momentum imparted to the air in unit time is finite, and since the mass of the air is infinite the induced downwash velocity must be zero." (page 77)

Aerodynamics,
Clancy, L.J.
Pitman Publishing (1973)

"All attempts to fly in heavier-than-air machines must embody some means of forcing the air downwards so as to provide the equal and opposite reaction which is to lift the weight of the machine."

"...if we reject the idea of flapping wings, we must replace it by some other device which will deflect the air downwards."

Mechanics of flight
Kermode, A.C.
Eighth (metric) edition, 1972.
Pitman Publishing

Reference added by J Doug McLean (talk) 21:08, 16 December 2014 (UTC):

Elements of Practical Aerodynamics Jones, B. 1939 John Wiley and Sons, Inc.

In the context of an analysis of induced drag on p. 82 he makes the following statement about the downward momentum imparted to "the mass of air affected by the wing":

"It has been proven mathematically that the downward momentum in unit time is equal to one-half the lift".

Section added by J Doug McLean (talk) 21:08, 16 December 2014 (UTC):

On the popular qualitative flow-deflection explanation based on Newton's laws:

"The main fact of heavier-than-air flight is this: the wing keeps the airplane up by pushing the air down.

Stick and Rudder - An Explanation of the Art of Flying. Langewiesche, W. McGraw-Hill Education

"In momentum-based explanations, it is generally argued that the airfoil produces a flowfield in which some of the air is "deflected" downward and thus has downward momentum imparted to it. To acquire downward momentum, the air must have a downward force exerted on it by the airfoil, and thus, by Newton's third law, the airfoil must have an upward force exerted on it by the air."

Understanding Aerodynamics -- Arguing from the Real Physics sec 7.3.1.7 McLean, D. Chichester, West Sussex, U.K. : Wiley, 2013.

Section added by J Doug McLean

On the Newtonian theory of lift

"The fluid itself is postulated as a collection of individual particles that impact directly on the surface of the body, subsequently giving up their components of momentum normal to the surface, and then traveling downstream tangentially along the body surface. That fluid model was simply a hypothesis on the part of Newton; it did not accurately model the action of a real fluid, as Newton readily acknowledged. However, consistent with that mathematical model, buried deep in the proof of Proposition 34 is the result that the force exerted by the fluid on a segment of a curved surface is proportional to sin^2 (theta), where theta is the angle between the tangent to the surface and the free-stream direction. That result, when applied to a flat surface (e.g. a flat plate) oriented at an angle of attack alpha to the free stream, gives the resultant aerodynamic force on the plate:

R = rho*V^2*sin^2(alpha)

This equation is called Newton's sine squared law...."

A History of Aerodynamics Anderson, J. D. , Jr. Cambridge University Press

(work in progress - to be continued) I'll try to track down the cites provided by Doug McLean and see if I can pull out the relevant direct quotes instead of relying on his summaries. Not that I don't believe his summaries, it's just that we'd be remiss as editors if we just took his word for it. Mr. Swordfish (talk) 21:33, 8 December 2014 (UTC)

Comments on the sources above

Please make any comments below here, so that we can keep the listing of sources clean and uncluttered.

• Comments on Prandtl and Tietjens

"...the integrated rate of change of vertical momentum for the atmosphere as a whole must be zero." Thus for the most obvious assumption a reader is likely to make regarding what is meant by "the air" (i.e. the atmosphere as a whole), The Original Statement is false. - DougMcLean (from earlier)

• Comments on Durand

It is explicitly stated that this result holds regardless of how large the radius of the circle is made. Thus a large circle is another example of a region of "the air" for which a reader might reasonably expect The Original Statement to apply, but for which it is in fact false. - DougMcLean (from earlier)

• Comments on Lissaman

According to Lissaman's results, if "the air" is taken to be the air in a rectangular box surrounding the airfoil, The Original Statement isn't even close to being true unless the box is a tall, slender sliver, and even then it isn't strictly true until the vertical dimension of the box is taken to infinity. Steelpillow quotes the section of my book that describes the result for the infinitely tall, slender sliver, the only control-volume shape for which The Original Statement has been shown to be true, and interprets it as being "in support of The Statement". A balanced recounting of what my book says would also quote the discussion in connection with figure 8.5.4, which deals with other control-volume shapes for which The Original Statement isn't true. - DougMcLean (from earlier)

• Comments on the AAPT paper

The quote from the AAPT textbook committee given above is the entire discussion of lift in that paper. The paper itself is a 25 pager devoted to reviewing seven high school physics textbooks, and that's all the space they had to address lift. As such it is quite cursory and lacks context. They suggest seeing Waltham's paper for more details and context. Waltham in turn cautions that to do it "correctly" requires a control volume of infinite thickness.

The AAPT does not explicitly say dp/dt of the entire atmosphere (or the "local flow" for that matter) is equal to -L. By not defining what they mean by "the air" they leave an imprecise statement that can be interpreted in different ways, one of which is correct and most of which are incorrect. If we repeat it with the same imprecision it is likely that our readers will latch on to one of the incorrect interpretations.

In sum, I would not give it the same weight as other more in-depth treatments of momentum transfer. Mr. Swordfish (talk) 21:09, 10 December 2014 (UTC)

The AAPT describe how much momentum is imparted to air within the atmosphere per unit time, consistent with results obtained by Lanchester, Lissaman and others.

I find it hard to think of a better form of words to describe this rate simply and concisely without going into details of a mathematical proof. Burninthruthesky (talk) 09:20, 16 December 2014 (UTC)

I agree with Mr. Swordfish that the AAPT paper doesn't deserve "the same weight as other more in-depth treatments of momentum transfer", and that "to describe this rate simply and concisely without going into details" would be misleading. J Doug McLean (talk) 21:08, 16 December 2014 (UTC)

• Comments on Lanchester

Lanchester §112 says that it is generally possible to show that dp/dt = -L. He goes on to clarify the exception commonly known as ground effect. Burninthruthesky (talk) 10:13, 11 December 2014 (UTC)

Without the use of calculus, he derives the total directly from Newton's laws.

The pressure w is entirely accounted for on the surface of the earth. Burninthruthesky (talk) 07:20, 13 December 2014 (UTC)

Lanchester's §112 does not say "that it is generally possible to show that dp/dt = -L." His analysis only shows that it is possible if you make one particular assumption for the shape of the control volume.

Lanchester's §112 amounts to a verbal version of a control-volume analysis for a tall-sliver control volume in contact with the ground. Lissaman (1996) published the corresponding analysis for the free-air case. The result is the same whether there is a ground plane or not, i.e. that as the height of the sliver goes to infinity the pressure force at the bottom (and top in the free-air case) vanishes, and thus dp/dt = -L for a control volume that is infinitely tall compared to its width.

Lanchester's result is essentially the same as Lissaman's and doesn't support an unapologetic version of The Statement any more than Lissaman did. The pressure is "entirely accounted for", but it's effect vanishes only because the height of the sliver is taken to infinity and the width is kept finite. J Doug McLean (talk) 21:08, 16 December 2014 (UTC)

Lanchester's §112 does not say "that it is generally possible to show that dp/dt = -L."
— User:J Doug McLean

Yes it does. Providing the right physical conditions exist, it is always possible to make the assumptions which show dp/dt is practically equal to -L. Lanchester suggests that when an aerofoil is in ground effect, w may become significant, in which case it would not be possible to show dp/dt = -L. He characterises the former situation as "ordinary conditions" which would "ususally be the case".

I only mentioned the fact that Lanchester accounts for pressure on the surface of the earth in response to your previous comment suggesting Lanchester supports your argument against The Statement.

It is clear from previous discussion and your obvious expertise in the subject that you need no explanation of the physics. So it is a mystery to me why you repeatedly misrepresent what has been written by others. You jest that "chuckling at the mistakes of others" is "part of the fun of being an aerodynamicist". I have not made a mistake here. We are WP:NOTHERE for an exercise in schadenfreude.

I've no doubt that that Lanchester's §112 is essentially the same as Lissaman's result. Both show that momentum is imparted to air within the atmosphere at a rate equal to lift. I don't see anything misleading about describing this rate as a simple fact in an introductory text, as advocated by the AAPT. You may disagree, but they are experts at educating people. Burninthruthesky (talk) 12:07, 17 December 2014 (UTC)

What I've said repeatedly is that the only published sources that use a realistic flow model and find dp/dt = -L to be true are those using a very-tall-sliver control volume. These sources make no claim that dp/dt = -L is true in any more general sense than that. So what, specifically, have I misrepresented?

You still seem to have concluded that The Statement is somehow true in a more general way. To me, this doesn't seem consistent with the physics or the published evidence. Just because dp/dt = -L applies in one particular control volume doesn't mean it applies to "the atmosphere" in any general sense. Yes, the tall sliver control volume "exists within" the atmosphere, but so do other control volumes for which dp/dt is different from -L. I see no basis for thinking one control volume is the "correct" one and the others are not. In general, the force exerted on the air by the foil is manifested as a combination of momentum changes and pressure differences in the flowfield. The one control volume for which the integrated pressure differences happen to vanish isn't in any fundamental way more "correct" than the others.

I know of no published source that uses a realistic flow model and shows that dp/dt = -L is true in the general sense you're proposing. The AAPT paper presents no analysis, and Chris Waltham's paper doesn't count on this score because he uses a simplified flow model that omits the crucial effect of the pressure field, as discussed in my comments on Clancy.

No, the Statement is not a "simple fact". It is a statement that is misleading unless it is accompanied by a somewhat arcane caveat. And I'm not making these arguments for fun. I'm here to improve the article by avoiding the inclusion of a misleading statement. J Doug McLean (talk) 22:52, 19 December 2014 (UTC)

In answer to your question, I have already clarified why I said, "Lanchester §112 says that it is generally possible to show that dp/dt = -L. He goes on to clarify the exception commonly known as ground effect." My comment is supported by the citation above. Your contradiction takes my words out of context.

The earlier comment I referred to said,

Actually, Lanchester's "deficiencies" section provides no support for The Statement, but instead supports what I've been arguing all along. The "Newtonian medium" (a hail of projectiles that don't interact with each other) is a poor model for flows of real fluids.
— User:J Doug McLean

This is a rebuttal for a proposition that was never made. The point of discussion was whether or not The Statement is true.

Several sources clearly explain how to calculate the result dp/dt = -L. As I have explained repeatedly, all of them account only for momentum. There are other calculations which account partly for pressure differences and for a smaller proportion of momentum. Your repeated assertion that other results for dp/dt are different, yet somehow answer the same question defies common sense. Again it is a rebuttal for a proposition that the cited authors did not make.

I'm afraid this is going round in circles again. Hopefully in the New Year there will be some progress on improving the article. Burninthruthesky (talk) 12:26, 20 December 2014 (UTC)

• Comments on Clancy

It is interesting that Clancy sees no contradiction between his Newtonian description and the fact that at the trailing edge, only half the momentum has yet been transferred. — Cheers, Steelpillow (Talk) 09:51, 15 December 2014 (UTC)

As far as I can tell, in the idealised infinite situation he goes on to describe Clancy is arguing that although the downward change of momentum in any finite portion of air is now infinitesimal, the sum of infinitely many such portions remains finite and is in fact still equal to the reaction to the lift. — Cheers, Steelpillow (Talk) 16:11, 11 December 2014 (UTC)

Clancy refers to "the affected air". This compares to the phrase "the air" which has caused so much supposed scope for misunderstanding here. Simply inserting the word "affected" as Clancy does would clear that side issue up. — Cheers, Steelpillow (Talk) 09:36, 12 December 2014 (UTC)

I don't think inserting the word "affected" will solve much of anything. According to the model, the entire atmosphere is affected by the presence of the moving airfoil so the "affected air" is all of it. And we have seen that for the entire atmosphere dp/dt =0. A few months ago we tried inserting "the air deflected by the foil" and that didn't do it either. There is some subset of the air for which dp/dt=-L, but I'm at a loss for how to state this in layman's terms without it being so awkward and convoluted that it distracts from the flow of the simple introductory section. Mr. Swordfish (talk) 13:10, 15 December 2014 (UTC)

It is not true that, as you suggest, "According to the model, the entire atmosphere is affected by the presence of the moving airfoil." Nowhere does Clancy's model address the entire atmosphere. It addresses a certain cylinder of air which he first makes finite and then expands to infinite size. I wonder if you are confusing this infinite cylinder with the whole atmosphere? We can be sure that they are not the same thing because within the interior of Clancy's model infinite cylinder, dp/dt = F while we know that within the even more infinite atmosphere which contains his model, dp/dt = 0. "The affected air" in his model is just the air within the rear part of the cylinder, aka "the air deflected downwards". — Cheers, Steelpillow (Talk) 16:52, 15 December 2014 (UTC)

Apologies for not being clear, but I wasn't referring to Clancy's model - I was referring to solutions to the Navier-Stokes equations, Euler equations, potential flow , etc.

In the very simple case of potential flow, the resulting flow field is the superposition of a steady uniform flow (i.e. what the flow would be like in the absence of the foil) and a vortex flow. While the vortex flow field diminishes as one gets farther away from the foil, at least in theory it is non-zero everywhere. So in the potential flow model the entire atmosphere is affected by the foil. More rigorous models (N-S, CFD, etc) give similar non-zero deviations from uniform flow throughout the entire atmosphere. These mathematical treatments are what I meant by "the model".

It appears that when Clancy says "the air" he means some specific subset of the atmosphere which he has defined beforehand and not the entire atmosphere. Fair enough, but unless we provide such a definition to our readers it is likely that they will interpret "the air" as the entire atmosphere. I think we are in agreement that for a non-accelerated foil dp/dt of the entire atmosphere is zero. If so, we're just arguing over what is meant by "the air". If not, then our disagreement is more fundamental. Which is it? Mr. Swordfish (talk) 19:55, 15 December 2014 (UTC)

Ah, in an item headed "Comments on Clancy", I hope you will forgive my misunderstanding which model you were discussing. The sources certainly bear out that the overall dp/dt is zero, I have no problem with that. Perhaps the best approach to "the air" is simply to write the new section intelligibly and not worry about our past usage. My comment on Clancy's usage was really just a flag to that end, in case it came in useful. — Cheers, Steelpillow (Talk) 20:50, 15 December 2014 (UTC)

Clancy himself admits that his flowfield model is "very crude". It assumes that "the air affected" by the wing in 3D is limited to a stream of circular cross-section with diameter equal to the wingspan (fig 5.21) and that this air is uniformly deflected downward by its interaction with the wing. There is no upward turning ahead of the wing or behind as there is in more-realistic flow models.

His model is unrealistic in another respect, and that is that it completely neglects the pressure field. It is assumed that the only force acting on the affected stream is that exerted on it by the wing, and this force is assumed to show up entirely as a rate of change of momentum of the steam of affected air. Thus dp/dt = -L is not really a result of this analysis; it is more of an a priori assumption.

His model becomes a little less unrealistic in 2D, where the wingspan and the diameter of the affected cylinder have gone to infinity, and the entire flow is thus affected, but it is still unrealistic in assuming the flow deflection is uniform, and in neglecting the pressure field.

Both Durand and Batchelor rigorously show that a uniform flow plus a vortex is a good approximation for the far-field flow in 2D, regardless of the details of the airfoil shape. The classical control-volume analyses I've cited use this model and show that the pressure field exerts significant forces on "the air" except in the case of the infinitely tall sliver.

I would not give Clancy the same weight as I would to the classical analyses I've cited, which use a much more realistic model. And I agree with Mr. Swordfish that adding "affected" doesn't clear up what is meant by "the air". J Doug McLean (talk) 21:08, 16 December 2014 (UTC)

I am quite sure that you would give your own work more weight than you give to those whom you criticise. Yawn. Worse, you cherry-pick from Clancy's full text to support your PoV. It also notes that his simplified model is consistent with a more complex analysis he gives later - and when we turn to that later analysis we find that it embraces the very flow-plus-vortex model you approve of. To selectively quote and then claim that Clancy's simplified model is at odds with the flow-plus-vortex model is somewhat invidious. Turning to the classical analyses you have quoted from, we may note that Clancy also writes that the downwash at the trailing edge is only half of its ultimate value, which it achieves a little over a chord-length further downstream. This is entirely supported by the classical analyses you quote. If I had had a different textbook on my shelf introducing these standard results using the standard introductory model, I am sure you would have laid into them equally. To claim that a standard textbook such as Clancy is at odds with the mainstream is to seriously undermine your own position. — Cheers, Steelpillow (Talk) 11:02, 17 December 2014 (UTC)

All of the quotes attributed to Clancy in the "sources" section above are from chapter 5, and all are based on the simplified flow model he uses there. In criticizing this model, I am not "cherry-picking". I'm sticking to the topic under discussion, i.e. the weight these different sources should be given regarding the question of dp/dt in the flow around a lifting foil.

You say that his results from chapter 5 are "consistent" with his results from the more realistic model in chapter 8. That's simply not true with regard to the question we're discussing here. The issue of dp/dt isn't addressed at all in chapter 8. The only part of the book that deals with dp/dt due to lift is chapter 5. So to criticize the model he uses in chapter 5 is not to "selectively quote" him.

The only thing for which Clancy claims the two models yield equivalent results is in the variation of the downwash from the near field to the far field, which does not address the question we're discussing here. The downwash velocity is a local quantity, while the dp/dt we're discussing is an integrated quantity.

In chapter 8 he uses the horseshoe-vortex model for a 3D wing, for which it is said that both the bound vortex and the trailing vortices contribute to the downwash field. But when he says that the downwash in the near field is only half the value in the far field, he's changed gears and he's referring only to the "3D" part of the downwash, the part associated with the trailing vortex system, and omitting the part of the downwash associated with the bound vorticity. I mention these details only to rebut your claim that "This is entirely supported by the classical analyses you quote." No, the analyses I quote deal with integrated dp/dt in a control volume, not with the variation of the downwash velocity downstream. And all but one of them deal with the 2D case, where the downwash velocity doesn't behave at all as Clancy describes for the 3D case anyway.

I'm not criticizing Clancy's book as a whole. I'm criticizing the applicability of the model he uses in chapter 5, and the statements he derives from it, to the question of dp/dt. And yes, I do find that in neglecting the pressure field he's at odds with the mainstream analyses on this particular issue. If you have any actual specific counterarguments to make in this regard, please let us know what they are. J Doug McLean (talk) 22:52, 19 December 2014 (UTC)

• Comments on Chris Waltham's Flight without Bernoulli

In the section titled "A Simple Model", Waltham uses the same kind ot model Clancy uses in sec 5.15, i.e. the model in which the air is affected only within a stream of limited cross-section. He starts with the assumption that the cross-section is rectangular, but he later says that it could just as well be circular. This model is unrealistic for the same reasons I give in my comments on Clancy. Because the model does not represent the flow realistically, it is wrong on some important details, such as dp/dt in the local flow around the wing. Regarding the question of dp/dt, I would not give Waltham the same weight as I would to the classical sources. J Doug McLean (talk) 21:08, 16 December 2014 (UTC)

• Comments on the popular qualitative flow-deflection explanation based on Newton's laws

I have added new section to the sources list: "On the popular qualitative flow-deflection explanation based on Newton's laws". It lists only two sources so far, but more can be found on NASA websites and elsewhere.

I think this kind of thing is the appropriate level of detail for the "Flow deflection and Newton's laws" section, and that we should not include a quantitative dp/dt statement there. J Doug McLean (talk) 21:08, 16 December 2014 (UTC)

• Comments on the "Newtonian" approach

In the "Sources" section I've added a quote from J. D. Anderson describing Newton's theory of lift, based on modeling the flow as a hail of bullets. The flow models used by Clancy and by Chris Waltham are similar to Newton's model in the sense that they assume that only a limited portion of the flow interacts with the foil, and they take no account of a continuum pressure field. Like Newton's theory, these "fire-hose" models are wrong about important details of the flow, such as dp/dt in the near field. As I argue above, Clancy and Waltham should be given less weight as sources on the topic of dp/dt than the classical sources that use a more realistic model. J Doug McLean (talk) 20:03, 20 December 2014 (UTC)

Some diagrams to help clarify the discussion of 'The Statement'

We've created quite a wall of text arguing about the statement, and don't seem to be getting anywhere. Perhaps some pictures can help clarify things.

Here's a diagram of an airfoil generating lift, with the airflow coming in from the left. Ahead of the airfoil in region A the air is accelerated upwards (upwash), in regions B and C the air is accelerated downward, and in region D the air is accelerated upwards as it returns to horizontal flow.

Let's say we define the regions accordingly, where A is the region in front of the foil where dp/dt>0 (upward acceleration) B and C are the regions above and below the foil where dp/dt<0, and D is the region behind where dp/dt>0.

When I encounter the phrase "the air deflected downwards" I parse that to mean the union of regions B and C. Perhaps there's some other interpretation that can be used, but that's what "the air deflected downwards" means to me. If there are other interpretations floating about I'd be happy to hear them.

What can we say about the net momentum transfer of B ∪ C? I do not know of any source that gives this value dp_B∪C/dt. But we can infer how it relates to L by looking at the following sub-region of B ∪ C:

Region E is a tall thin sliver and if we take the limit of dp_E/dt as the height goes to infinity and the width goes to zero we get dp_E/dt = -L. Since this is a proper subset of B ∪ C and all of B ∪ C has negative dp/dt, E must have a lower magnitude of momentum change than all of B ∪ C. That is, for B ∪ C |dp/dt| > |L|. Earlier, Doug claimed to have calculated it to be -1.6L, which sounds plausible.

Now, I'm not suggesting that the above analysis (along with my crude diagrams) go into the article - it's certainly too arcane for the intro section and it's not ready for "prime time" as a later section. But I hope that my fellow editors can read it and understand why I have a problem with saying "the time rate of change of momentum of the air deflected downwards" is equal to the lift. By my interpretation of 'the air deflected downwards' the statement is incorrect. It took me a while to come around to this view - recall that I actually wrote the statement and included it in my re-draft last summer - but I now see that it's problematic. Comments? Mr. Swordfish (talk) 17:21, 22 December 2014 (UTC)

I am having trouble following your summary:

1. The "width" which goes to zero is not defined - the width of what? Is it of the aerofoil chord or just of the column E, or perhaps the span orthogonal to the screen?
2. Whatever the "width" is, I find it hard to justify the statement that "as ... the width goes to zero we get dp_E/dt = −L" (unless you are also implying that as the width goes to zero, L also tends to zero, which is not very helpful).

If you believe what you wrote, then I can indeed understand why you have a problem with the Statement. But more fundamentally, I think you are still confusing a broad statement of the Newtonian principle with an attempt at detailed analysis. The Statement is expressing the Newtonian principle in a few words, while you are seeking to interpret some of those words ab initio, i.e. without prior acceptance of the principle. That is bound to end in tears. — Cheers, Steelpillow (Talk) 14:51, 23 December 2014 (UTC)

Mr. Swordfish presents a clear, detailed analysis of the problem with The Statement. There is one substantive problem with the analysis as presented, which Steelpillow has picked up on, and that is that it doesn't make sense for the width of column E to go to zero. But this problem doesn't invalidate Mr. Swordfish's conclusion.

In his analysis of the tall sliver control volume, Lissaman took the limit as the height goes to infinity, but not as the width goes to zero. With an actual foil in the picture, a proper control-volume analysis of the lift requires that the control volume be wide enough to bracket the projected chord of the foil. I'm sorry if the word "sliver" confused the issue here. I used it to refer to the control volume's proportions, meaning to convey that the control-volume width is small compared to the height, not that it is small compared to the chord. Anyway, even though the width doesn't go to zero in Lissaman's analysis, the pressure difference between the top and bottom vanishes as the height goes to infinity, and the integrated vertical pressure force vanishes, with the result that dp/dt = -L, independent of the width, as long as the width is kept finite.

For E to be a proper subset of B ∪ C, and to not include part of A ∪ B, the width of E must be restricted compared to what Lissaman assumed, with the vertical boundaries pushed up against the leading and trailing edges of the foil. But Lissaman's analysis still applies, and dp/dt = -L for E.

Steelpillow's point 3 is mistaken. He says "E has a smaller magnitude which means that it is closer to zero and is therefore 'greater than' −L." No, E is the subset with dp/dt =-L, so its dp/dt cannot be "'greater than' −L.". Mr. Swordfish's analysis is correct.

Ah, yes, thank you for pointing that out. Too much Christmas spirit in my glass, I fear. I have now deleted that item. — Cheers, Steelpillow (Talk) 22:59, 23 December 2014 (UTC)

Steelpillow is also mistaken when he says "But more fundamentally, I think you are still confusing a broad statement of the Newtonian principle with an attempt at detailed analysis." No, dp/dt = -L for "the air" can be established as a valid "statement of the Newtonian principle" only after a detailed analysis has shown that -L is indeed the resultant force acting on "the air". And analysis has only shown this to be the case for the tall sliver control volume.

Mr. Swordfish's analysis above clearly shows that dp/dt = -L is false for "the air deflected downward". However, we have no citable source for this argument. On the other hand, even if we thought dp/dt = -L was true for "the air deflected downward", we have no citable source that supports including the word "deflected". To stay within what our sources support, our only choice is The Original Statement from the AAPT paper, without the word "deflected".

I agree that The Original Statement in unapologetic form is unacceptably vague. If we include it in the intro section, we must, at a minimum, also specify that the only definition of "the air" for which it's been shown to be true is "a region that is very tall compared to its width", citing Lissaman. As I said long ago, I think the best option is not to make any quantitative statement in the intro section and to discuss the quantitative momentum balance in a new later section.

I'd like to see something like Mr. Swordfish's ABCDE discussion in this new section, but I know of no citable source for it. From the sources I know of, I think the best we can do is to describe the results for the circular and rectangular (tall, square, and flat) control volumes in free air, and the atmosphere with a ground plane. Here are two possible diagrams I've made for the purpose, and I've started drafting candidate text to go with them.

If we mention the "fire-hose" models at all (Clancy, Waltham), it would only be to point out their deficiencies relative to the more rigorous analyses. I'm proposing that it be a new subsection titled "Analyses of the integrated momentum balance in lifting flows", under "Mathematical theories of lift", just after "Circulation and the Kutta-Joukowski theorem". J Doug McLean (talk) 22:36, 23 December 2014 (UTC)

This is just the same old same old. The article is not going to be amended to support a view unsubstantiated by reliable sources. — Cheers, Steelpillow (Talk) 22:59, 23 December 2014 (UTC)

There is one substantive problem with the analysis as presented ... it doesn't make sense for the width of column E to go to zero. Thanks. I have struck that language.

I look forward to seeing the draft section. Agree that anything we put into the article must have a citable source. Mr. Swordfish (talk) 00:46, 25 December 2014 (UTC)

Mr. Swordfish, I suggest you think about how dt should be handled in these calculations. You may find Lanchester §3 helpful.

When making content decisions, we should be guided more by RS than by OR. The novel, unpublished analysis above will not help anyone understand the relevant physics. If you haven't already, I recommend reading Lanchester §112 for a straightforward explanation. Burninthruthesky (talk) 14:58, 27 December 2014 (UTC)

At this point I would like to ask both Burninthruthesky and Steelpillow to elaborate on what they think "the air deflected downward" means. I think we are in agreement that it doesn't mean the entire atmosphere since dp/dt of the entire atmosphere is zero. So, what does "the air" mean? I've given my interpretation and Doug has given several possible reasonable interpretations (of which only one makes the statement true). What's yours? And is it likely that our readers will have the same interpretation?

Agree that we should value RS over OR discussion on the talk page, and I'll keep trying to chase down a copy of Lanchester and read sec 112. Mr. Swordfish (talk) 21:28, 27 December 2014 (UTC)

Since the Statement is just the application of Newton's laws to lift in words, in this context it means, "the air deflected downwards in reaction to the lift force." Not any other air deflected downwards because of some vortex or some distant pressure distribution or some clever analysis or whatnot. It is simply affirming that if we apply F=dp/dt to L, then there is a mass of air whose dp/dt is in reaction to L. Our clever analysis can then identify the location of this mass for us, and different analytical models will identify different locations (e.g. Clancy's firehose). Crucially, it is not saying that this is the only approach, nor even the best approach, just that it is an approach. Other models based on other approaches, say on pressure, will not even yield dp/dt=L because they have already accounted for much or all of L some other way. The sources can then indicate the due weight that each deserves. This is how pretty much every textbook treats it, and I am not aware of widespread misinterpretation among engineering students. I am confident that our readers will be no different. One can really only misinterpret it once one has gained a good deal of detailed knowledge, well beyond the introductory stage at which it is appropriate. — Cheers, Steelpillow (Talk) 22:10, 27 December 2014 (UTC)

Apologies for taking so long to reply.

I don't have any fundamental disagreement with your position. I do think Doug has a point (made elsewhere) that defining "the air" as the region that makes the statement true is somewhat circular, but the fact remains that there is such a region and I'm comfortable eliding over some of the details in the introductory section. To that end, I've prepared a draft in my user space that may help move us towards consensus. I'll introduce that in a new thread. Mr. Swordfish (talk) 19:56, 7 January 2015 (UTC)

Suggested revisions

I have posted a proposed revised version of the article in my sandbox User:J_Doug_McLean/sandbox. Changes from the current version are limited to two places:

1) Under "Flow deflection and Newton's laws" I have removed the quantitative statement about momentum and done a bit of rewording of what remains to be sure that both the second and third laws still get their due.

2) At the end of "Mathematical theories of lift" I have added two new subsections: "Analyses of the integrated momentum balance in lifting flows" and "Newtonian theories of lift".

Everything I've included has a citable source and is presented, I think, from a neutral point of view.

I think removal of The Statement is best for the following reasons:

1) In this part of the article and at this level ("Simplified physical explanations..."), the flow deflection explanation is better off without it. This explanation is usually presented in its qualitative form anyway.

2) Without the qualification that it's only been found to be true for the tall sliver, The Statement is open to misinterpretation. Steelpillow and Burninthruthesky still seem to think that it isn't, but I think their confidence that the typical reader will know how to interpret it correctly is unjustified. I think it's just too easy for an uninitiated reader to assume that "the air deflected downward" could refer to the atmosphere as a whole or at least to some sufficiently large subset of it. The atmosphere as a whole may be a "nonsensical" assumption as Burninthruthesky called it, but I wouldn't expect the typical reader to know that unless we tell him. And I certainly don't expect the typical reader to realize that it's true only for a particular shape of region.

3) A general, unapologetic "dp/dt = -L" isn't consistent with what the mainstream literature says about dp/dt.

The two new sections are pretty self-explanatory. I included a paragraph on Newton's bullet model because I think it's interesting in its own right, it had an impact on early assessments of the practicality of heavier-than-air flight, and it helps put the "fire hose" models in perspective, which I also included under "Newtonian theories".

Looking forward to comments and suggestions.

J Doug McLean (talk) 23:53, 30 December 2014 (UTC)

First of all I would like to thank @J Doug McLean: for all the hard work that has gone into this. The only real issue I have with any of it is the well-worn one we are all familiar with.

Given all the quotations we so carefully collected above, quite how anyone can maintain that the mainstream sources do not support the momentum statement - and right at the introductory stage at that - is beyond me. Mainstream sources do include it, and even mandate it there. Ours is not to reason why or to sanitise it out of the article. It is there in the article, it is reliably cited, and there it must stay. Doug McLean makes some other minor textual changes to this section which are generally good.

The bulk of the new section on "Analyses of the integrated momentum balance in lifting flows" is good, though there is a certain residual defensiveness in the last few sentences which can probably simply be omitted. Also, I would shorten the heading to just "Integrated momentum balance in lifting flows".

The next section - the critique of the momentum model - is useful as far as it goes. However I think it needs a balancing critique of the other simple model we introduce, viz. the pressure model. These critiques could better introduce the whole section on "Mathematical theories of lift" rather than conclude it. Alternatively, they could be confined to their respective introductory subsections on "Limitations of ..." — Cheers, Steelpillow (Talk) 10:52, 31 December 2014 (UTC)

I agree with Steelpillow there is no justification for removing the existing momentum statement. I'm also grateful for progress on improving the article.

One minor point - I haven't been able to find Shapiro to reference the momentum equation, but I have found this page which says (in abbreviated terms):

F = dM/dt = F_B + F_S

I understand that in steady flow, F_B = 0, so F = F_S.

In the proposed new section, I read the last two sentences of the first paragraph as F_S = F_B. Is this the intended meaning? Burninthruthesky (talk) 18:06, 1 January 2015 (UTC)

I have now incorporated what I believe to be the bulk of the proposal into the article, save for two parts: I retained the Statement along with its citations, and I have not copied across the subsection on Newtonian theories of lift because I am not yet sure where to put it. — Cheers, Steelpillow (Talk) 14:01, 4 January 2015 (UTC)

Momentum theorem

I still don't have an answer to my question above. A Google search for "momentum theorem for a control volume" brings back mainly references to Reynolds transport theorem. Is this what is being described? Burninthruthesky (talk) 14:58, 4 January 2015 (UTC) Here is an actual quote from Shapiro:

Eq. 1.13 is usually called the momentum theorem and states that the net force acting instantaneously on the fluid within the control volume is equal to the time rate of change of momentum within the control volume plus the excess of outgoing momentum flux over incoming momentum flux.

— Shapiro

Also, Shapiro's Eq 1.15 is identical to Eq 3.42 in the page I linked above. Burninthruthesky (talk) 17:05, 4 January 2015 (UTC); edited 17:17, 4 January 2015 (UTC) I've left "disputed" tags on these two sentences. I am disappointed they were added to the article despite my unanswered question. Please will someone with more specialist knowledge than myself correct them? Burninthruthesky (talk) 17:36, 4 January 2015 (UTC)

My apologies. I took the citation of Shapiro Section 1.5 at its word, as I was not able to check it and you had referred to it as a "minor point". I do think we need to be clear whether each case is allowing a dynamic flow where the net rate of momentum change within the control volume is non-zero, or restricted to a steady-state flow where the net rate of momentum change within the control volume must be zero. — Cheers, Steelpillow (Talk) 22:17, 4 January 2015 (UTC)

Thank you, and I'm sorry I didn't make myself clear. By "minor", I meant the content probably needed correcting before release, as opposed to not being inherently useful.

I think a link to Reynolds transport theorem may be useful, as I undertsand this momentum theorem directly follows from it.

In this case, I would be a little uncomfortable with adding material to the encyclopedia which I only learned myself yesterday, in the course of verifying another's work. Burninthruthesky (talk) 07:44, 5 January 2015 (UTC)

First to Burninthruthesky's question: Is F_S = F_B the intended meaning of the last two sentences of the first paragraph of the new section? No. That interpretation isn't consistent with what the variables represent.

F_B is the body force acting throughout the volume of the fluid, which for an electrically neutral fluid is just gravity. In aerodynamics we usually neglect both gravity and the background hydrostatic pressure gradient that goes with it, under the assumption that they cancel each other. Thus F_B = 0 results from neglecting gravity. Whether the flow is steady or unsteady has nothing to do with it.

F_S is the sum of the surface forces (pressure and viscous shear stress) acting on the control volume boundaries. In the airfoil analyses I've cited it includes both the -L' imposed by the foil and the integrated pressure force on the outer boundary. The volume integral I refer to in the sentences in question is the integral of the material rate of change (material derivative) of momentum in the interior, which is not at all the same thing as F_B. Note that the two integrals on the RHS of equation 3.42 of this page represent the volume- and surface-integral parts of dM/dt. The fact that dM/dt has two parts has nothing to do with the decomposition of F into F_B and F_S, and the first term on the RHS is not equal to the first term on the LHS.

So yes, F_B = 0, so that F = F_S, but it's because we neglect gravity, and not because the flow is steady. And no, F_S = F_B does not follow, and is not the intended meaning of the last two sentences of the first paragraph of the new section. These two sentences follow from the fact that the first integral on the RHS of Shapiro's Eq 1.15 is zero for steady flow, and that the entire RHS represents the same quantity as the RHS of the first equation (unnumbered) of section 1.5, i.e. the instantaneous time rate of change of the momentum of the material system that occupies the control volume at time t, commonly referred to as the material derivative. The two sentences don't "need correcting". They are consistent with Shapiro, and the "disputed" tags should be removed.

The Reynolds transport theorem as described in the linked article is similar to Shapiro's Eq 1.15, but it uses d/dt to represent a more general kind of time derivative, the total time derivative for some quantity contained within the volume as it evolves in time, including the effects of movement of the boundaries of the volume in the general case. The result is formally the same as Shapiro's only for the special case in which the boundaries of the volume move with the flow as described under "Form for a material element". I think this article is unlikely to help anyone understand Shapiro because it provides less supporting detail than Shapiro does.

Steelpillow wrote:

"I do think we need to be clear whether each case is allowing a dynamic flow where the net rate of momentum change within the control volume is non-zero, or restricted to a steady-state flow where the net rate of momentum change within the control volume must be zero."

Are you saying that The Statement dp/dt = -L can be true only in unsteady flow? That would make no sense. Actually, all of the sources we list that bear on the dp/dt issue analyze the flow in the frame of the foil and assume the flow is steady in that frame. But the rate of change of momentum within a control volume that's stationary in that frame can still be non-zero in Shapiro's sense of the material derivative. And although Chris Waltham and the AAPT don't say so explicitly, dp/dt in their versions of The Statement has to be referring to the material derivative. Otherwise, if it referred to the conventional partial derivative with respect to time at a fixed location, it would be zero for steady flow, as you say, and The Statement would be false for any control volume. And even I'm saying that there's one control volume for which The Statement is true. J Doug McLean (talk) 06:40, 7 January 2015 (UTC)

I am certainly not saying that, as you say that would make no sense. Equations applicable to dynamic momentum distribution will presumably differ from the simpler equations required for steady state. Some of the remarks posted or referenced appeared to encompass the dynamic situation, and that concerned me. For example your new material contains this: "The momentum theorem states that the integrated force exerted at the boundaries of the control volume (a surface integral), is equal to the integrated time rate of change (material derivative) of the momentum of fluid parcels passing through the interior of the control volume (a volume integral). For a steady flow, the volume integral can be replaced by the net surface integral of the flux of momentum through the boundary." Grammatically, the addition of the caveat "For a steady flow" to the second sentence would suggest that the first sentence encompasses unsteady, aka dynamic, flow. Since the momentum theorem is not yet properly defined and cited, I cannot judge whether this is so. Some other materials left me similarly concerned. I have no knowledge of such dynamic situations, nor any references on my bookshelf, or I would be able to comment more sensibly. — Cheers, Steelpillow (Talk) 12:14, 7 January 2015 (UTC)

I see my assumption that each term on the LHS of (3.42) as discussed equates to the same term on the RHS doesn't necessarily follow from what is written, but that's how I read it. Anyway, this equation shows the total force is equal to a volume integral plus a surface integral and it requires some calculation to follow your argument. In shorthand (with the integral contents omitted), Shapiro's Eq 1.15 is ∑F = ∂/∂t ∫_c.v. + ∮_c.s.

As you say, In steady flow the first RHS term disappears, so
∑F = ∮_c.s.

The first equation of 1.5 is a statement of Newton II in the x-direction. In general Newton II is
∑F = d/dt (mV)

So
d/dt (mV) = ∮_c.s.

I'm starting to think this wording may not be incorrect, so I have changed the tags. However, Shapiro 1.5 doesn't directly state any of the three equations above.

The wording in question is:

The momentum theorem states that the integrated force exerted at the boundaries of the control volume (a surface integral), is equal to the integrated time rate of change (material derivative) of the momentum of fluid parcels passing through the interior of the control volume (a volume integral). For a steady flow, the volume integral can be replaced by the net surface integral of the flux of momentum through the boundary.

The idea that either of the relationships derived above for steady flow conditions are "The momentum theorem" is not consistent with any of the sources I have seen.

I don't see where Shapiro says that d/dt (mV) is the "material derivative" or that this term involves a volume integral.

Your assertion that in the discussed (3.42), "the first term on the RHS is not equal to the first term on the LHS" implies that neither term on the RHS is equal to the corresponding term on the LHS. Yet the quotation above seems to imply the opposite: i.e. the second term on the RHS (a surface integral) is equal to the surface forces, "the integrated force exerted at the boundaries of the control volume" and "the net surface integral of the flux of momentum through the boundary".

Anyway, this discussion has already taken up far more of my time than I wish. I am not going to embark on an undergraduate course in Fluid Mechanics just so I can continue to protect this article from dubious and misleading information, as I have done for many months now. Burninthruthesky (talk) 12:09, 7 January 2015 (UTC); last edited 06:54, 9 January 2015 (UTC)

Also, this page says,

Physically, the linear momentum equation states that the sum of all forces applied on the control volume is equal to the sum of the rate of change of momentum inside the control volume and the net flux of momentum through the control surface.

Burninthruthesky (talk) 15:46, 7 January 2015 (UTC)

From a reader's perspective, all he needs to know in this context is that in steady flow, momentum changes to air passing through the control volume can be accounted for at its boundaries. I think a simpler form of words would be more understandable. Burninthruthesky (talk) 07:43, 8 January 2015 (UTC)

Looking at other sources, as far as I can tell the momentum theorem states that the integrated rate of change of momentum within the control volume equals the sum of the integrated forces acting on the internal volume plus the integrated forces acting on the surface. (e.g. George Emanuel; Analytical Fluid Dynamics, Second Edition, pp447-448 ) That is rather different from the definition currently given in the article. One can add that for a steady state flow where the force acting on the interior is zero, the integrated rate of change of momentum within the control volume equals the integrated forces acting on the surface, like this:

The momentum theorem states that the integrated rate of change of momentum within the control volume equals the sum of the integrated forces acting on the internal volume plus the integrated forces acting on the surface. For a steady state flow where the force acting on the interior is zero, the integrated rate of change of momentum within the control volume simply equals the integrated forces acting at the boundary.

I would be happy for this to replace the "wording in question" quoted above. Any objections? — Cheers, Steelpillow (Talk) 16:35, 9 January 2015 (UTC)

Thanks for suggesting a new wording and for the further reference. All the sources I've seen have presented the momentum theorem in the same form . I see this equates a force to the sum of volume integral and surface integral components of momentum, but I'm not sure how to express it accurately in words. Burninthruthesky (talk) 07:01, 10 January 2015 (UTC); edited 07:06, 10 January 2015 (UTC)

If the suggested wording can be reliably cited, I've no objection to its addition to the article. Burninthruthesky (talk) 11:55, 10 January 2015 (UTC)

Specific wording does not need to be cited verbatim (or we would fall foul of plagiarism), however their meaning needs to be clear and that meaning needs to be cited properly. This is what I have tried to do. — Cheers, Steelpillow (Talk) 12:35, 10 January 2015 (UTC)

Without plagiarising, we must somehow present the facts which are directly supported by the sources. Does Emanuel say the Momentum Theorem is:

• the splitting of momentum into surface and volume components of force (14.1)
• the splitting of momentum into surface and volume components of momentum (14.2)
• the splitting a force into surface and volume components of momentum (14.3)
• something else derived later?

I'm pretty certain some of those bullets are not the correct answer, but I'm not sure which is. Burninthruthesky (talk) 13:11, 10 January 2015 (UTC)

Emanuel gives the equation of integrals. I paraphrased that equation in words. The LHS is the momentum-change integral, the RHS is the sum of the two force integrals, effectively your 14.1. One could write the equation, which is not plagiarism but equally is not especially helpful unless one explains it in words as well. It would be better included in an article on fluid dynamics in general. — Cheers, Steelpillow (Talk) 13:31, 10 January 2015 (UTC)

I was attempting to describe Emanuel's 14.1, not make my own. I agree the suggested text above is also a verbal description of that equation. My question is whether that equation is in fact "The Momentum Theorem", since Emanuel describes it as "Newton's second law". Burninthruthesky (talk) 14:49, 10 January 2015 (UTC); edited 15:39, 10 January 2015 (UTC)

Ah, silly me. My apologies. I'll have to find time to digest Emanuel a bit more carefully. We also have 14.9 to consider. — Cheers, Steelpillow (Talk) 16:56, 10 January 2015 (UTC)

A lot of issues to deal with here. To begin: Burninthruthesky wrote:

"I don't see where Shapiro says that d/dt (mV) is the "material derivative" or that this term involves a volume integral."

Good point. He doesn't explicitly use the term "material derivative", though that is the widely accepted term for the kind of time derivative he refers to as d/dt(mV). He also doesn't explicitly say the term involves a volume integral, but he does define it as "the time rate of change of the total x-momentum of the system", which for non-uniform flow can be quantified only in terms of a volume integral. Shapiro's notation is confusing in this regard, and Emanuel would be a better source to cite. The RHS of his eq 14.2 is the same quantity as Shapiro's d/dt(mV). It is clearly a material derivative (for which the capitalized D/Dt is the common notation), and it is clearly a volume integral. So I think my wording is correct and well supported by Emanuel, just not well supported by Shapiro. Thanks for pointing this out.

Burninthruthesky wrote:

"The idea that either of the relationships derived above for steady flow conditions are "The momentum theorem" is not consistent with any of the sources I have seen."

I don't understand your basis for saying this. Your relationship
∑F = ∮_c.s. is identical to Shapiro's eq 1.15, for the case of steady flow, where the first term on the RHS is zero. And this is the final equation in a section headed "Working Form of Momentum Theorem".

Burninthruthesky wrote:

"Your assertion that in the discussed (3.42), "the first term on the RHS is not equal to the first term on the LHS" implies that neither term on the RHS is equal to the corresponding term on the LHS. Yet the quotation above seems to imply the opposite: i.e. the second term on the RHS (a surface integral) is equal to the surface forces. "

You're right about what is implied, but that doesn't mean it's contradictory. The two second terms are equal only in the special case in which the flow is steady, for which the first term on the RHS is zero, and the body force (the first term on the LHS) is neglected, as it usually is in aerodynamics. This is the special case used in all of the airfoil analyses.

Burninthruthesky wrote:

"From a reader's perspective, all he needs to know in this context is that in steady flow, momentum changes to air passing through the control volume can be accounted for at its boundaries. I think a simpler form of words would be more understandable."

This is also a good point. I originally included the wording about the volume integral of the material derivative because that's the form of the momentum theorem that relates to the "time rate of change of momentum of the air" in The Statement in the intro section, and I think I had the editing community in mind more than the general reader. I agree that the simpler form is better for the article, and I've made the change in my sandbox User:J_Doug_McLean/sandbox.

Steelpillow proposed replacing "the wording in question" with the following:

"The momentum theorem states that the integrated rate of change of momentum within the control volume equals the sum of the integrated forces acting on the internal volume plus the integrated forces acting on the surface. For a steady state flow where the force acting on the interior is zero, the integrated rate of change of momentum within the control volume simply equals the integrated forces acting at the boundary."

There are two reasons this isn't satisfactory:

1) As I explained before, setting the body-force term ("the sum of the integrated forces acting on the internal volume") to zero comes from neglecting gravity and has nothing to do with whether the flow is steady or unsteady.

2) This version doesn't mention the surface-integral form for the momentum term, which is the form used in the airfoil analyses that follow.

Burninthruthesky asks which of the equations is actually the "momentum theorem". I think a reasonable reading of the sources is that the "momentum theorem" can be expressed in several forms and that it can be any of the equations that relate the integrated force to the integrated rate of change of momentum and/or momentum flux. In Emanuel that would be eq 14.1, 14.3, 14.4, or 14.9. It is not eq 14.2, 14.5, 14.6, 14.7, or 14.8 because they deal with forces or momentum changes separately, not with the second-law relationship between them.

In light of the above discussion I have changed the words in question to the simpler form suggested by Burninthruthesky, and cited both Shapiro and Emanuel. See in my sandbox User:J_Doug_McLean/sandbox. J Doug McLean (talk) 00:19, 11 January 2015 (UTC)

I take on board much that has been said, and apologise again for looking at the wrong equation. But I am still puzzled by a couple of points in Doug McLean's latest proposal. It says, "For a steady flow, the momentum theorem states that the integrated force exerted at the boundaries of the control volume is equal to the integrated flux of momentum through the boundary." But in explaining his equation 14.3 Emanuel remarks, "the left side represents the sum of all the applied forces that act on the fluid inside the control volume." and later in explaining 14.9, "the force is therefore provided by a volumetric integral plus an integral over a stationary surface." i.e. Emanuel's generic analysis of fluid flows does not support two aspects of the new version:

• Emanuel describes the force as acting within the control volume, McLean as acting at its boundary.
• Emanuel does not observe that in air, gravity may be ignored and the momentum volume integral is zero.

I am not saying that Doug's analysis is necessarily wrong, only that Emanuel does not appear to support it fully. Does Shapiro? — Cheers, Steelpillow (Talk) 13:28, 11 January 2015 (UTC)

The Statement and Newtonian theories of lift

Regarding the proposed deletion of The Statement, I see nothing in Misplaced Pages policy that says that something "must" remain in an article just because it's already there and has a citable source. Nor do I see anything that says that removal of something that has a citable source requires a direct and citable refutation. No, it looks to me like we as editors are free to make changes based on weighing the sources available to us, even if those sources don't explicitly refer to each other. So it appears to me that Steelpillow's contention that The Statement "must stay" is unfounded. If I'm wrong, show me the specific policy wording.

When Steelpillow insists that "mainstream sources" support The Statement, he's being unduly selective. Yes, some of the sources support The Statement in unapologetic form, but all of the sources taken together, on balance and weighted according to the quality of their analyses, clearly show that The Statement is true only with qualifications. If we present The Statement without the qualifications, we're presenting a biased and misleading impression of what sources on this topic actually say.

The only published analyses we have that support The Statement in unapologetic form are based on the "firehose" model for the flow (Chris Waltham and Clancy). In my posts of 17and 19 December I presented detailed arguments as to why these sources should be accorded less weight than those that use the classical model based on uniform flow plus a vortex. I invited specific counterarguments and so far there have been none.

Likewise, Mr. Swordfish invited Burninthruthesky and Steelpillow to elaborate on what they think "the air deflected downward" means. Only Steelpillow responded, and his answer amounted to "It means whatever it has to mean to make The Statement true, depending on what flow model you prefer". That's an unsatisfactory answer in general, but especially when one of the models we're considering (the "firehose") assigns an unrealistic spatial distribution to dp/dt, as Clancy himself admits.

This is not a question of a choice between a "momentum model" and a "pressure model", as a matter of style or personal taste. No, a proper application of the momentum theorem requires that all of the forces exerted on the air be taken into account, including the pressure force. The "firehose" analyses ignore the pressure force, while the classical analyses properly include it. Thus in terms of the basic physics, these two types of analysis are not of equivalent quality. Steelpillow suggests that the new section on Newtonian models "needs a balancing critique of the other simple model we introduce, viz. the pressure model." No, there is no "balancing" of that kind to be done. The classical model is not just a "pressure model". It accounts for both momentum and pressure, and assigns them their realistic locations in the field. It has no faults that rise to the same level as the faults of the "firehose" model. If you disagree with this assessment, please state your specific counterarguments.

The classical analyses (Durand, Batchelor, and Lissaman) clearly deserve more weight than the "firehose" analyses. And they clearly show that The Statement in unapologetic form is misleading. Thus it should either be deleted or properly qualified. This is not "a view unsubstantiated by reliable sources". It is a conclusion supported by an appropriately weighted and balanced consideration of all of the sources. Nor does it constitute synthesis, though I would also argue that the (WP:SYNTH) policy doesn't apply to a decision to omit something.

I agree with shortening the heading to "Integrated momentum balance in lifting flows". Regarding the overall organization of "Mathematical theories of lift", I think it's best just as it is, starting with the basic principles, followed by the predictive theories (the ones that actually predict lift starting with the airfoil shape). Those that merely relate one thing to another but don't make actual predictions, such as the Kutta-Joukowski theorem, the momentum-balance analyses, and the Newtonian theories, are not theories at the same level as the predictive theories and should not precede them. This is an ordering that befits an encyclopedia article as opposed to a textbook.

The "Newtonian theories of lift" belongs where I proposed putting it, at the end of Mathematical theories....". None of it should be moved to the introductory section.

J Doug McLean (talk) 06:40, 7 January 2015 (UTC)

Doubters of Misplaced Pages's policies and guidelines may find it useful to read WP:REMOVAL and, on the matter of neutrality, WP:NPOV. — Cheers, Steelpillow (Talk) 12:14, 7 January 2015 (UTC)

Re-reading the proposed section on Newtonian theories of lift, it is really two unconnected parts. The first is a summary of early theories and would go better as the start of a "Historical development" of theories of lift, the second is a critique of the firehose model and while that might be a useful topic to work in somewhere, frankly I find the treatment presented to be unbalanced. Neither part is fit to be moved into the present article as it stands, nor do I intend to engage with the author on their improvement. — Cheers, Steelpillow (Talk) 12:47, 7 January 2015 (UTC)

Your feedback that you don't see the connection between the two parts of "Newtonian theories...." is useful. Thank you. I have revised the second part to make the connection clearer. See User:J_Doug_McLean/sandbox. Beyond that, I have already explained why I struck the balance I did in my criticism of the firehose model. If you have specific suggestions for improving the balance, let's hear them. Simply calling the treatment "unbalanced" isn't helpful.

You are the only one who has expressed opposition to including "Newtonian theories...." in the article, so at this point yours would seem to be a minority view. Does anyone else oppose my adding this new subsection? J Doug McLean (talk) 00:26, 11 January 2015 (UTC)

Yes. Burninthruthesky (talk) 08:08, 12 January 2015 (UTC)

Proposed re-draft of "Flow deflection and Newton's laws"

Doug McLean recently proposed a re-draft of this section, however it was not met with much support. (personally, I'd support it were there consensus to go in that direction) I've created another draft that will hopefully move us in the direction towards consensus: https://en.wikipedia.org/User:Mr_swordfish/Lift#Flow_deflection_and_Newton.27s_laws

The main ideas are:

• I think we all agree that there is a region of air for which dp/dt = -L. The draft uses a instead of the; by using the indefinite article I hope to avoid the confusion that may result in referring to "the air" without specifying what is meant by "the air".
• I've added a cite which I believe is sufficient to support the language that is in the draft:

"...if the air is to produce an upward force on the wing, the wing must produce a downward force on the air. Because under these circumstances air cannot sustain a force, it is deflected, or accelerated, downward. Newton's second law gives us the means for quantifying the lift force: F_lift = m∆v/∆t = ∆(mv)/∆t. The lift force is equal to the time rate of change of momentum of the air." Norman F. Smith "Bernoulli and Newton in Fluid Mechanics" The Physics Teacher 10, 451 (1972); doi: 10.1119/1.2352317 http://dx.doi.org/10.1119/1.2352317

• I've moved some of the material around so that the first paragraph deals with the third law and the second paragraph deals with the second law. That seems like a logical organization.
• Some other minor tweaks such as referring to the 'air flow above the wing' rather than the 'air that follows the upper surface'

I invite the other editors to comment. I hope that I have crafted language that is technically correct, explains the momentum transfer idea as recommended by the AAPT, and is understandable by the lay reader. Suggestions for improvement cheerfully accepted. Mr. Swordfish (talk) 20:44, 7 January 2015 (UTC)

I did edit the article section to include some of Doug McLean's minor suggestions and, from that perspective, I am happy with it as it is. I do not think that Mr. Swordfish's version changes the substance in any way. No version can please everybody, so I have no strong opinion as to whether we stick with what we've got or go for Mr. Swordfish. Either way, the new citation is useful. — Cheers, Steelpillow (Talk) 21:36, 7 January 2015 (UTC)

I appreciate the work Mr. Swordfish has done here to try to bring us together. But although this proposal is less bad than the current version, it doesn't fix the problem. True, "a volume of air" isn't as likely to be misinterpreted as referring to the atmosphere as a whole, but it still begs the question: What volume of air does it refer to?

Couching The Statement in terms of "a volume of air" instead of "the air" or "the air deflected downward" changes the substance a bit, but not much. And it still gives prominence to the quote from Cliff Swartz, which makes The Statement in its most unapologetic form, a statement that I've said all along and Mr. Swordfish has recently argued is problematic.

We've been through this over and over, and I thought that recently Mr. Swordfish had come to agree with me on this: The only assumption for which authoritative published analyses have shown The Statement dp/dt = -L to be true, i.e. the tall sliver control volume, is so specific that making The Statement without spelling out the assumption is misleading. The proposed new wording doesn't get us past this problem.

If we're going to make the quantitative statement dp/dt = -L, we must spell out what body of air it's true for. The wording wouldn't have to be overly technical or mathematical, but it would have to be specific. Here's my idea of the minimum that would be required to replace Mr. Swordfish's second paragraph and to be technically correct:

In accordance with Newton's second law, some of the air surrounding the airfoil is accelerated downward. The rate at which downward momentum is imparted to the air can be calculated, but it depends on what portion of the air is included in the accounting. Depending on the shape of the region considered, pressure differences acting on the outer boundary of the region offset some of the downward force exerted on the air by the airfoil, reducing the rate at which momentum is imparted, as described below under "Momentum balance in lifting flows". For a column of air that extends to large distances above and below the airfoil and is relatively narrow, the pressure differences are not a factor, and the rate at which downward momentum is imparted is equal to the lift.

This gets the momentum-transfer message across in a more accurate way, and it isn't that much longer than the current version or the proposed draft. It cites a different set of sources (Lissaman, Durand, Batchelor) from the proposed draft (Swartz, Clancy, Smith), but I've already made detailed arguments as to why that reflects an appropriate weighting. I would still prefer to keep this section qualitative, an option that both Mr. Swordfish and Burninthruthesky have said they could live with. But if we are to make a quantitative Statement, I'd urge you to consider a version with a technically correct qualification, as suggested above.

J Doug McLean (talk) 01:14, 11 January 2015 (UTC)

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