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Theory on lift

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Theory on lift

Old 22nd Nov 2016, 23:30
  #361 (permalink)  
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Yes, thank you very much John and as always, Owain!
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Old 28th Jul 2018, 13:17
  #362 (permalink)  
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Just having a bit of a trawl through old threads and came across this one.

Now that I think upon it, we haven't had a good set to on lift (that I can recall) for some time.

We probably have a great many new chums who wouldn't have seen this history .. so, I shall play "moderator troublemaker" and bring it back to the top to see what may transpire.

If there be little interest then it will sink rapidly back into the bog whence it came ..
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Old 28th Jul 2018, 13:35
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Originally Posted by john_tullamarine View Post
Just having a bit of a trawl through old threads and came across this one.

Now that I think upon it, we haven't had a good set to on lift (that I can recall) for some time.

We probably have a great many new chums who wouldn't have seen this history .. so, I shall play "moderator troublemaker" and bring it back to the top to see what may transpire.

If there be little interest then it will sink rapidly back into the bog whence it came ..
Ok John! (you asked for it)

How does the traditional theory of lift as taught on most groundschools and even at GCSE, explaiin how a paper aeroplane flies? Or a symmetric foil section?

Ive also done the bernoulli sums on a standard NACA section and considering 2D flow only, increased velocity due to "further to go"...it explains at maximum about 5% of the lift actually generated in accordance with the standard lift equation and experimental Cl figures.

Still check whether the starting vortex forms in the bath with my kids and their toy aeroplane!
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Old 28th Jul 2018, 13:54
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Add to the list - lift produced by a sail on a yacht versus the "equal tranit time" explaination often given when describing Bernoulli?
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Old 28th Jul 2018, 19:05
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No problem with a little revival.

My take is - one can fly a barn door (or a paper airplane) with no camber and no significant "Bernoulli" lift, simply with enough AoA and power (or gliding force from gravity).

But the process is much more efficient using every possible physical effect one can. Whether one is the Wright Bros. with a pitiful little 12-hp engine, or today's commercial jets trying to save evey penny on fuel while going fast.

Lift is a lot of little pieces, which in the final analysis, together push down (or in the case of Bernoulli, pull down) a mass of air equal to the mass of the aircraft, holding it up against the force of gravity (Newton's 3rd). Actually, probably slightly more than the weight of the aircraft, due to losses to tip vortices and such, but with a net vector "down" that is equal to aircraft weight. Newton, Bernoulli, Coanda, Kutta-Joukowski - they all identified things that contribute something to lift, or rearranged the pieces to look at them from a different angle.

But unless there is room and time to write a thick textbook covering all those pieces, any "popular" explanation of lift has to be abbreviated, and thus only partly correct. It is a bit like "Is light waves or particles?" - the correct answer is "both" or "all of the above."

In theory, one could just put a tube full of people on top of a 12,000m girder, and move it around on wheels. The weight of the craft (mass x G) is supported by the girder, and eventually transferred to the wheels and thus the ground. Lift does the same thing, and someone has even detected the "weight" of an aircraft flying in the flight levels, as a tiny increase in local baro pressure (tiny because the weight is distributed over of billions of square inches, in a wide, spreading "footprint" below the aircraft).
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Old 28th Jul 2018, 22:00
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Interesting. So, for me, I wish we didnt delude people with fudges to explain WHY aircraft fly. Instead, It should be explain the theory of lift is a highly complex topic, with the overall production of lift explained through a number of effects which by themselves do not explain fully the theory of lift, however, assist with the understanding of what is going on in terms of flow fields and on the 2D level.

In ,my mind, there are a few truths that can be stated which at a deeper level, assist understanding of lift. They are:

There can be no lift without Viscosity. Without viscosity, the flow will not remain adhered to the wing and work can therefore not be done on the flow. The wing slips through the flowfield and does not impact it. As there is no change in flow direction, there is no change in inertia and thus no downwash.
There can be no lift without considering the 3D flowfield.
There can be no lift without a circulatory flow, which by thought, can be tied to the requirement of viscosity in the production of lift. Circulatory flow is not a fudge to explain lift. It actually occurs (dont believe me, move your hand through water at an angle of attack. you can observe the reverse circulation depart the trailing edge as your hand starts to move).
The net effect of the complex flow around a wing is the generation of downwash, the effect of which due to Newtons third law exerts a force upon the wing and thus doing work. Without viscosity, we cannot have downwash as work cannot be done on the flow by the wing.
An ideal wing that is 100% efficient would convert 100% of downwash into lift with no drag, other than that induced by the rearwards deflected lift vector (what we call induced drag).
We dont live in the ideal world. Ignoring compressibility,, the effect of viscosity upon which lift depends is responsible for the generation of non-lift dependent drag.
In the compressible world (Re dependent, big planes, assume M >0.4), the impact of compressibility on the flow field (particularly pressure field) can both positively and negatively impact both lift and drag.
The 3D flowfield significantly impacts the properties of the wing, particularly the impact upon downwash across the span.


Just my musings. Please feel free to rip apart my mental model!
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Old 28th Jul 2018, 22:31
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Originally Posted by pattern_is_full View Post
No problem with a little revival.
True. After all, it's not as if we're going to get posters who haven't bothered to read the thread so far, or have but can't remember what was said ...
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Old 29th Jul 2018, 11:50
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Well that killed that conversation....
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Old 29th Jul 2018, 17:57
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Fortunately President Trump has directed the FAA to stop teaching the fake news theories and instead teach his approved description of lift and thrust.


LIFT DEMONS AND THRUST PIXIES

Title of Paper: The Role of Lift Demons and Thrust Pixies in Heavier Than Air Flight

Publication Date: 2004

Abstract: The role of Lift Demons in aeronautics was first explained in 1994 by Mary Shafer (NASA). Since then, Shafer's work has been explored and revised. This paper summarises advances in Lift Demon technology over the last decade.

Keywords: Lift Demons, Thrust Pixies, lemon fondant

Authors: Sarah Hartwell, DEF Smith, Peter Rieden, Gavin Bull

Question: How did the secret information on Lift Demons make it into the public domain? I've been a practising Aircraft Performance Engineer for the past 26 years and have always tried to explain how aeroplanes fly by using the official public explanations regarding Bernoulli, airfoils and other such rot. Civilians just weren't ready for the truth. In fact, we generally don't speak about the magic directly. Most of our plans and estimates usually end with the phrase "and then a miracle happens".

Answer: The science of Lift Demons was declassified in 1994, throwing this topic wide open for discussion and research.

According to Shafer "Lift is caused by Lift Demons. These little, invisible demons hold on to the leading and trailing edges of the aircraft and lift it into the air by flapping their wings (so, in a reductionist sense, lift is actually caused by feathers). Some of the demons are a little confused and they hold on backwards, causing drag. The reason that planes stall at high alpha is that the leading edge demons get scared and let go when they can't see the ground anymore. Lift demons have good taste and don't like to look at ugly aircraft, so they hold on backwards on ugly planes. That's why gliders have so much lift and so little drag and why F-4s have lots of drag." This, however, did not address what gives lift to the Lift Demons' wings. Again, according to Shafer "Feathers. The multiple filaments on feathers trap the air molecules and they struggle to escape, which causes the action-reaction that we call lift. Bat wings don't have feathers but they're hairy and that works just about as well (air molecules are a little claustrophobic)."

Because Lift demons hold on backwards on ugly planes to avoid looking at the planes, drag can be calculated using CDU (Coefficient of Drag due to Ugliness) used in correlating aircraft ugliness against Lift Demons' unwillingness to hold on forwards and provide lift.

It was originally believed that Lift Demons got their lift from smaller Lift Demons whose lift was, in turn, produced by even smaller Lift Demons leading to the "Infinite Demons Theory" as proposed by the great Greek philosopher-scientist Miasma. However, with the revival of scientific knowledge that eventually ended the Dark Ages, it was realised that this situation unresolvable according to Zeno's paradox. The "Infinite Demons Theory" works in many problems of engineering significance, however, real understanding requires that the ether be introduced into the analysis at some point. The "Ether Concept" explains why planes fly more efficiently at higher altitudes. This is an absolute necessity when studying orbital and interplanetary travel where, it is believed, many of the Lift Demons are unable to breathe.

As always the Chinese seem to have been there first. "The Genius of China", tells of one Ko Hung. "Someone asked the Master (Ko Hung) about the principles of mounting to dangerous heights and travelling into the vast inane. The Master said "[...] some have made flying cars with wood from the inner part of the jujube tree, using ox leather straps fastened to returning blades so as to set the machine in motion. Others have the idea of making five snakes, six dragons and three oxen to meet the 'hard wind' and ride on it, not stopping until they have risen to a height of 40 li (about 65,000 feet). The the ch'i [emanation of the sky or perhaps wind] is extremely hard, so much so that it can overcome the strength of human beings. [...] Take dragons, for example; when they first rise they go up using the clouds as steps, and after they attained a height of forty li then they rush forward effortlessly gliding." The jujube tree device sounds like the Osprey. Scholars believe that the term "dragon" referred to the Lift Demon and that the Chinese had harnessed their power before the Western world had even discovered the Lift Demons' existence.

Lift Demons operate differently on different aircraft types. They get dizzy holding onto a spinning prop and become disorientated, thus more of them face the wrong direction which increases the drag. Lift Demons have to be magically bound to jet engines and an Airbus propulsion engineer explained "It takes an average of three day's solid chanting and ritual to get one of those 'engine' things to work. If us witches stopped our hard work, there'd be planes dropping out of the sky all over the place. We should never have let you lot find out about being able to fly, it's been nothing but trouble ever since."

Recognition of Lift Demons has also solved one of aviation's oldest puzzles: Gremlins. There appears to be a connection between Gremlins and Lift Demons. Gremlins may be Lift Demons that have, in some way, turned against the pilot. Instead of assisting him in his task of keeping the machine in the air, they do the opposite. The reason may be that, after reading about the Bernoulli/Newton dichotomy, they have become confused. Much has been written about Gremlins, dating back to WWII. Gremlins have been known to run towards the nose of an aircraft, causing it to dive into the sea, showing that they have at least some knowledge of the principles of flight. These days, most planes carry a full complement of In-Flight Gremlins; although they must be carefully managed, their presence generally inhibits the transformation of rogue Lift Demons into Gremlins.

Further study of Lift Demons was carried out by an aero-industry weights engineer who submitted the following scientific paper in 2004 to celebrate the anniversary of Shafer's ground-breaking article:

"There are still people in this company who think we weigh aircraft to find out how much they weigh, not to calculate stresses. Of course we need to know how much the thing weighs. How are we ever going to know how many Thrust Pixies we need to get the thing off the ground if we don't know the weight? Or should that be "Lift Demons"? Pixies have largely fallen into disrepute - something about Bernoulli not being representative in unbounded conditions and cause and effect being transposed in the Newtonian model.

In fact the use of Lift Demons on civil aircraft programmes is generally not that good an idea. The Demon binding contract tends to specify payment in blood or souls. This is readily achievable with aircraft of military function, but frowned upon in civilian circles as they may attempt to acquire payment outside of the terms of their binding contract. Lift Demons are not used on Elf bombers. We don't talk about Lift [Thrust] Pixies too often as it seems to upset the self-loading cargo [passengers]. Pixies require payment in cakes, flowers or nice thoughts. These are readily sourced either from the in-flight catering, or provided cost-free by the passengers. Clearly this would not work well within an operational military environment. Air force cooking is not renowned for the "light and fluffy texture" that Thrust Pixies demand, the availability of flowers might be problematic in desert operations, and nice thoughts may also be hard to find during times of active operations.

There is also a scalability issue. While one rampant Lift Demon would have few problems supporting a fighter aircraft (particularly if there is an immediate prospect of blood), it'll struggle to achieve level controlled flight of a 560 tonne Airbus A380. Use of more than one Lift Demon on the same flight vehicle is contra-indicated (they squabble and eat each other). Communities of Thrust Pixies can be encouraged to work together on the same aircraft by the provision of advanced technologies such as Lemon fondant icing, variegated tulips or in-flight romantic comedies. Ryanair once requested Leprachauns be installed in place of Lift [Thrust] Pixies, but Leprechauns have a mission statement which indicates their desire for monetary gain, and their willingness to search all over the world for it. This makes Lift Leprechauns expensive to keep (gold vs. lemon fondant icing), and makes it difficult to establish a regular route network as the Lift Leprechauns don't like to continuously visit the same locations. By law, aircraft also have to have a full complement of In-Flight Gremlins, but these are generally not a problem unless you feed the Wingtip Vortex Faeries after midnight."

His article shows that we've moved on a great deal since 1994. It has been known for some time that Lift Demons are best suited to military uses. In spite of suggestions that Pixies have fallen into disrepute, outside of military aviation it is Lift Demons that are considered passé - they have notoriously short attention spans and the ones used on Harriers have problems discriminating between air and water. Inbreeding in pursuit of the ultimate high performance Lift Demon may have been the cause. In civil aviation, the way ahead still lies with Thrust Pixies, large amounts of lemon fondant icing and in-flight Hugh Grant movies. Many instances of civil planes mysteriously falling out of the sky can be traced to (a) cessation of happy thoughts/lack of lemon fondant for the pixies (resulting in "Pixie Fatigue" or even "catastrophic pixie failure") or to (b) a large amount of happy thoughts/gateau mountain at ground level distracting the pixies from their task. Thrust Pixies dislike Marmite/Vegemite (yeast extract) and since such spreads can cause unhappy thoughts in some passengers, airline caterers are careful to avoid these.

A note on Catastrophic Pixie Failure: Temporary pixie problems resulting in turbulence or sudden loss of altitude causes unhappy thoughts in the passengers (which is why cabin staff and pilots always play down such occurrences). The resulting loss of happy thoughts causes further Pixie failure, worsening the situation. This makes passengers have even more unhappy thoughts and the Pixies become more fatigued; some may even leave the aircraft. Unless counter-acted by large lemon fondant reserves, Pixie Failure reaches catastrophic proportions and the aircraft is doomed.

Modern aircraft designers use Lift Fairies and avoid the whole controversy regarding the used of Lift Demons on civil aircraft programmes. Coming from a less benevolent cast, Lift Fairies are less dependent on nice thoughts or cakes. Fairies tend to less concerned with good and evil and hence make better dual use aircraft. Another major headache today is how to get 707 Lift Fairies at a reasonable price. Many of the older aircraft-specific fairies have become rare, if not unavailable. Aircraft numbering relates to the type and quantity of Lift Demons or Lift Fairies needed. For example a C-130 requires 130 "C class" Lift Demons while an Airbus A380 uses 380 "A" class Thrust Pixies. Aero engineers have a scale of values (proprietary information, not available in the public domain) allowing them to substitute different numbers of Pixies and Demons with different lifting abilities, thus making best use of the available surfaces.

There was short-lived interest in breeding a generic, or at least dual-use, "Thrust Demon" but the blood-loving Lift Demons ate the gateau-loving Thrust Pixies. In vitro techniques failed because their genietic [sic] material is incompatible so there are no hybrid Thrust Demons. DNA sequencing has allowed us to distinguish between many castes of Lift Demon, Thrust Fairy etc. For example due credit should be given to the inelegantly named Fart Fairies without whose bean eating and gas production, no machine could power itself from the earth's surface. The shy Turbine Winder-Rounder Gnomes hide inside those so-called engine nacelles from the public and indulge their serious kerosene drinking problems. Why else would the engines be called Gnome engines? Leading-Edge Leeries give the extra little push that keeps the nose up. The unfortunate tendency for RAF Harriers to crash into the sea has led to speculation that Harriers are equipped with Sirens or that the crashes resulted from an ill-advised experiment in using Water Nymphs (these are better suited to submarine propulsion). In any case, only export versions of the Harrier are likely to use Sirens, while those for the British domestic market use Banshees leased from the Irish.

Thrust Pixies are adequate for civil aircraft, while Lift Demons were good in military applications in years past, but modern Naval Aviation is currently all Angel-powered. The very best ones are, of course, Blue Angels. This is why they report their altitude as "angels twenty" or such. In addition, many Navy pilots claim that black air has no lift, which means they can get all night in to supplement their daytime naps between meals. This shows that Angels and Naval Lift Demons are strictly diurnal. Genetically engineered, military-tolerant Thrust Pixies may be what is needed. Thrust Demons might also have applications in getting Air Force maritime patrol aircraft to stop reaching "prudent limit of endurance" by 2 p.m. (local time) every Friday just after reporting a "possible intruder" submarine in the exercise area, and not be restored to flying status until 9 a.m. Monday, thus leaving the ships to stay out over the weekend looking for the "intruder".

There are no Anti-Gravity Demons so a different approach has to be taken in this field of research. Current research into inertia-free propulsion has shown great promise through the use of properly modified felines. Butter is spread evenly over the felines back. When the creature is tossed lightly into the air, the third law of universal fate dictates that it must land butter side down. However this does not occur due to the intervention of the feline landing axiom (feet first). The above conflicting forces result in a stable hover. The subject felines have demonstrated the ability to control their own velocity at will. The only loose ends delaying the full commercialisation of this process is the matter of persuading the felines to (a) work in teams; (b) not lick off the butter and (c) follow a flight plan. As there seems to be a deficiency in feline herding instincts, any suggestions would be appreciated by the researchers involved.
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Old 29th Jul 2018, 18:09
  #370 (permalink)  
 
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…..explaiin how a paper aeroplane flies? Or a symmetric foil section?
Circulation and vortices.
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Old 30th Jul 2018, 03:04
  #371 (permalink)  
 
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Originally Posted by VinRouge View Post


Ok John! (you asked for it)

How does the traditional theory of lift as taught on most groundschools and even at GCSE, explaiin how a paper aeroplane flies? Or a symmetric foil section?


The tired old "symmetrical airfoil" argument is only appealing to those who believe that the stagnation point (point where airflow divides between under the wing and over the wing) on a symmetrical airfoil operating at a non-zero angle of attack is located on the airfoil's axis of symmetry. It's not. There are numerous wind tunnel photographs of symmetrical airfoils with smoke generator streamlines switch demonstrate the fallacy of this.

Originally Posted by VinRouge View Post
Ive also done the bernoulli sums on a standard NACA section and considering 2D flow only, increased velocity due to "further to go"...it explains at maximum about 5% of the lift actually generated in accordance with the standard lift equation and experimental Cl figures.
*sigh* the "equal transit time" strawman. That would only prove something if the equal transit time was some fundamental element of bernoulii's principle. In fact it is not. In fact, the air flowing over the "top" of the airfoil arrives at the trailing edge of the airfoil well before the air flowing beneath. This can be seen in wind tunnel photos and footage of pulsed smoke steamers flowing over air foils.
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Old 31st Jul 2018, 18:07
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Originally Posted by A Squared View Post
The tired old "symmetrical airfoil" argument is only appealing to those who believe that the stagnation point (point where airflow divides between under the wing and over the wing) on a symmetrical airfoil operating at a non-zero angle of attack is located on the airfoil's axis of symmetry. It's not. There are numerous wind tunnel photographs of symmetrical airfoils with smoke generator streamlines switch demonstrate the fallacy of this.


*sigh* the "equal transit time" strawman. That would only prove something if the equal transit time was some fundamental element of bernoulii's principle. In fact it is not. In fact, the air flowing over the "top" of the airfoil arrives at the trailing edge of the airfoil well before the air flowing beneath. This can be seen in wind tunnel photos and footage of pulsed smoke steamers flowing over air foils.
Sigh. You completely missed my tongue in cheek jibe in that post. My original point if you read my later post, is that we shouldn't be teaching incorrect science as fact to those at GCSE and even A Level Physics.

Having 30+ hours in labs in sub, trans and supersonic tunnels, I can testify to the above and stagnation point/ Kutta condition.

Unless your repose was as equally tongue in cheek of course.
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Old 31st Jul 2018, 18:36
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Originally Posted by Lyman View Post
Put propellers in a tube, and spin them, will there be thrust? Yes.
While this statement is technically true, in reality no jet engine puts "propellers in a tube." Unlike propellers which accelerate air, those "blades inside the tube" of a jet engine compress air. The compressed air is then accelerated in the nozzle of the engine. Similar principles, yet very different.

Last edited by KenV; 31st Jul 2018 at 18:53.
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Old 1st Aug 2018, 00:49
  #374 (permalink)  
 
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in reality no jet engine puts "propellers in a tube."
It does if you consider the loads imposed by the various components. Here the compressor is providing some 19,000 lbs of thrust. You can even see the principal on which the SR-71 derives the majority of its thrust from the inlet, where the diffuser here is providing some 2,000 lbs thrust. You'll note that aft of the burner cans very little thrust is developed, the vast majority of the loads being drag, 46,678 lbs drag against 2,419 lbs thrust. The compressor is providing 33% of the total thrust load, a prop in a tube.

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Old 1st Aug 2018, 06:37
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Originally Posted by megan View Post
the SR-71 derives the majority of its thrust from the inlet, where the diffuser here is providing some 2,000 lbs thrust.
I've always disliked statements like these (true though they may be) as it's no different to saying that the forward force in a conventional-drive car is generated by the rear axle. Airflows and pressures are a bit less visible than shafts and gears, but the principle is the same.

I wouldn't be surprised if someone comes up with an inlet design that actually produces a net backward force on the engine mounts....
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Old 1st Aug 2018, 07:30
  #376 (permalink)  
 
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Big Pistons Forever,
Love it!!
Rather overshadows the "Acoustic Lift Theory" --- usually applied to the Brittain Norman Islander ---- when the noise stops, the aeroplane quits flying.

Now back to serious explanations --- in its simplest terms, it is the basic physics, laws of motion, action and reaction, that creates lift.
All the shapes and sizes of aerofoils are just refinements to maximise "lift" for a given power.
The papers of Lawrence Hargrave from Sydney, AU, ( the Wright Flyer was fundamentally a Hargrave Box Kite with an engine --- something acknowledged by the Wright Bros themselves, it's all in the Smithsonian) detail the whirling rig he built, starting with flat plates, moved on from there. with tabulated results for "lift" from various shapes.

As for the "friendly little molecules" that separate at the leading edge, and are miraculously re-united at the training edge, having had many adventure in-between, how many students have been taught that over the years.

As an aside, I have always been fascinated by the history of the development of the Navier-Stokes equations for analysis of fluid flow, to me that brain ranks with Newton and not far behind Einstein.

Tootle pip!!
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Old 1st Aug 2018, 10:04
  #377 (permalink)  
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+1 for post #369 .. should be compulsory reading for all engineering undergraduates ...

As for the "friendly little molecules" that separate at the leading edge, and are miraculously re-united at the training edge, having had many adventure in-between, how many students have been taught that over the years.

I'm presuming that Leadie is being a tad tongue-in-cheek.

However, one hopes that any newchums reading this thread are comfortable with the notion that the truth of the matter includes some extra words -

that don't separate at the leading edge

and are miraculously not re-united at the training edge

how many students, incorrectly, have been taught that over the years.

As to being re-united at the training edge that just doesn't happen and the fact that it doesn't happen is fundamental to lift production.

While I'm on the soap-box, the old nonsense about parcels of air separating up at the front and then joining up at the back ... is, likewise, nonsense. Just doesn't happen .. the bits going over the top win the race handsomely.
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Old 1st Aug 2018, 11:01
  #378 (permalink)  
 
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Isn't it wonderful that, more than a century after the Wright Brothers and half as much again since Cayley, we still can't agree why aircraft fly ?
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Old 1st Aug 2018, 12:59
  #379 (permalink)  
 
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Originally Posted by megan View Post
Ken: in reality no jet engine puts "propellers in a tube."
It does if you consider the loads imposed by the various components..... The compressor is providing 33% of the total thrust load, a prop in a tube.
You missed the entire point of my post. It has nothing to do with "the loads imposed" or even "thrust generated". Yes, the intake system can provide a considerable amount of thrust. But it does so NOT because the blades in the intake system are accelerating the air, but because the blades are compressing the air and then the constriction of the casing acts as a nozzle to accelerate the air. Propeller blades directly accelerate air. The blades in the jet engine are compressors, not accelerators. Propeller blades and jet engine blades behave similarly from a pilot perspective (they provide thrust), but employ fundamentally different principles.

You noted that the SR-71 inlet/diffuser provides a lot of thrust. That is NOT because the inlet/diffuser has a bunch of little propellers in the tube accelerating the air. A fundamentally different principle is being employed.

Last edited by KenV; 1st Aug 2018 at 13:20.
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Old 1st Aug 2018, 13:17
  #380 (permalink)  
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Isn't it wonderful that, more than a century after the Wright Brothers and half as much again since Cayley, we still can't agree why aircraft fly ?

The techo folk have been in agreement for most of that period. I might refer the remainder to readings on such folk as Lanchester (no slouch, an FRS, no less), Kutta. Zhukovsky, Prandtl, and others all of whom were involved in getting circulation theory sorted out. What amazes me is that the pilot theory examination stuff more or less stagnated a long time ago ....

... whilst still adhering to the lift pixies and thrust demons view of life, death and the universe ... which, quite obviously, summarises everything in a most elegant manner ...
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