Bernoulli, lift and cause & effect

[haushofer]

TL;DR Summary

Bernoulli, lift and cause & effect: looking for a clear-cut cause-and-effect and analogy.

Dear all,

for a book I'm writing I'm trying to understand the generation of lift by wings on a conceptual level. Some papers I used are given at the end. I won't talk about all the misunderstandings out there concerning the generation of lift. I'm interested, also from a pedagogical level, in the very reason why it's so damn hard to give a clear-cut cause-and-effect for how lift is generated.

Of course, in this explanation Bernoulli's law is invoked. Bernoulli's law relates pressure differences to velocity differences along a single airflow. It's a statement of energy conservation, saying that 'when pressure drops, velocity increases' and vice versa. This can be compared to dropping a stone, and saying that 'when height decreases, speed increases' and vice versa. Being a statement about energy conservation, nothing is said here about cause and effect, but for the dropping stone we know that the cause of both the decreasing height and increasing speed is gravity. Likewise, if you follow a fluid parcel along a flowline, Newton's second law tells us that the particle's velocity changes due to a pressure difference along the flow. So the pressure difference causes the velocity change, and not the other way around; we can at most conclude that if an air parcel's velocity changes, there must be a pressure difference which caused this. But then comes the paper by Doug McLean about lift (Aerodynamic Lift, Part 2: A Comprehensive Physical Explanation). In it, he says the following:

How the lift and the flow details are tied
together in a set of mutual interactions:
As the flow is forced to follow the predominantly downward-
sloping surfaces of the airfoil, a set of mutual interactions
is established between the lift force, the pressure field,
and the velocity field. This is not a linear sequence of one-way
cause-and-effect relationships; the relationships are all reciprocal.
Nor are the relationships ordered in time; in a steady
flow they are all simultaneous.
[...]
At the overall flowfield level, the lift force and the pressure
field support each other in a mutual interaction: The pressure
field exerts the upward lift force on the airfoil, and at the
same time the existence of the pressure field is a result of the
equal-and-opposite downward force exerted by the airfoil
on the air. The relationship is reciprocal, consistent with the
reciprocity between action and reaction inherent in Newton’s
third law.
[...]
To grasp the pressure-velocity interaction intuitively, it
helps to note that a pressure difference can exist only because
the air acted on by the pressure difference is able to “push
back” against the unbalanced pressure force. When a parcel of
air is subjected to different pressures on opposing sides, the
parcel’s neighbors exert a net force on the parcel as illustrated
in Fig. 3. According to Newton’s third law, this force must be
opposed by an equal-and-opposite “pushback” exerted by the
parcel on its neighbors. The “pushback” is provided by the
inertia of the air in the parcel as it is accelerated by the
pressure difference, in accordance with Newton’s second law. This is why
the mass of the air is important, and why lift depends on air density.
So the pressure field that exerts the lift force arises
as part of a mutual interaction with the lift force itself and at the same time
is sustained in a mutual interaction between the pressure
and the vector velocity of the flow. Upward and downward
deflections of the flow and different flow speeds above and
below the airfoil are all essential parts of this interaction. The
pressure differences follow naturally from Newton’s second
and third laws and from the fact that the flow along the surface
is forced to follow the predominantly downward-sloping
contours of the airfoil associated with angle of attack and/or
camber. And of course the fact that the air has mass is crucial
to the interaction.

From: Doug Mclean, "Aerodynamic Lift, Part 2: A Comprehensive Physical Explanation." In ‘The American Association of Physics Teachers’. AAPT, 2018.

To be honest, I don't really get this, but the main lesson here is that the simple "pressure difference causes velocity difference but not the other way around"-thinking is, according to Mclean, wrong. So I was thinking about another analogous case: voltage difference and current. Usually, a voltage difference causes a current, not the other way around. However, we know from e.g. the Hall effect that if we put a conductor in an external magnetic field, a current inside that conductor can deflect, cause a charge building up at one side of the conductor, and cause a voltage difference perpendicular to the original direction of the current. A current of test charges would be driven in the perpendicular direction by this Hall voltage, which is caused by the influence of an external magnetic field on the original current. So my question: is this situation of the conductor and Hall effect with its external magnetic field comparable to a parcel of air ("current") being influenced by the movement of a wing ("the external magnetic field") such that we get a mutual interaction? Can this analogy shed more insight into how a pressure drop accelerates an air parcel around a wing, while on its turn this accelerating air parcel influences the pressure around the wing? And, of course: how would you describe the exact cause of the arising pressure regions around the wing? If I start moving a wing through air, air will collide among others at the front of the wing, such that pressure is build up there and increases. But then?

P.S. I'm not sure how much I'm allowed to copy from the paper by McLean, so please let me know or remove my quote if inappropriate due to copyright.

Doug Mclean, Aerodynamic Lift, Part 1: The Science. In ‘The American Association of Physics Teachers’. AAPT, 2018.
Doug Mclean, Aerodynamic Lift, Part 2: A Comprehensive Physical Explanation. In ‘The American Association of Physics Teachers’. AAPT, 2018.
Dwight Neuenschwander, How Airplanes Fly: Lift and Circulation. In ‘Elegant Connections in Physics’, 2015.
John Denker, See How It Flies. A new spin on the perceptions, procedures, and principles of flight. Online te lezen, 2001.
Holger Babinsky, How do wings work? In ‘Physics Education (2003).
M. D. Deshpande en M. Sivapragasam, How Do Wings Generate Lift? 1. Popular myths, what they mean and why they work. In ‘Resonance’, 2017.

 

[vanhees71]

I think Neuenschwander is the best explanation at this level, though I'm a bit worried that it's hard to explain the math of "circulation" and Kelvin's theorem (I'd also mention Helmholtz though, but that might be my German point of view of the history ;-)) and the vector calculus needed to understand it in a popular-science description.

 

[berkeman]

Have you read through the Insights article by our member @boneh3ad (there's a link to it stickied in the General Engineering forum):

https://www.physicsforums.com/insights/airplane-wing-work-primer-lift/

 

[Lnewqban]

The cause is always the energy consumed by the flying wing, which come from fuel (regular airplanes) or from gravity (gliders).
The effect is always drag, and some lifting effect, according to how much and how well the wing disturbs the surrounding air.

 

[haushofer]

I think Neuenschwander is the best explanation at this level, though I'm a bit worried that it's hard to explain the math of "circulation" and Kelvin's theorem (I'd also mention Helmholtz though, but that might be my German point of view of the history ;-)) and the vector calculus needed to understand it in a popular-science description.

I'll check it out, but this circulation theorem usually goes as follows:

-At the end of the wind there will be vortices due to viscosity circulating anti clockwise;
- Because of Kelvin's conservation theorem there must be a clockwise circulation around the wing;
- This means that airflow must be faster above the wing then below;
- Because of Bernoulli this means that there is a net force up, which we call "lift".

But that doesn't mean the faster airflow causes the pressure drop, so this is not the causal explanation I'm looking for. The theorem only tells us how to calculate lift and why you expect there to be lift, but not how it arises as a mutual interaction between pressure and velocity.

 

[haushofer]

Have you read through the Insights article by our member @boneh3ad (there's a link to it stickied in the General Engineering forum):

View attachment 291462
https://www.physicsforums.com/insights/airplane-wing-work-primer-lift/

Yes, I did. It doesn't adress my question; its main focus is the Bernoulli v.s. Newton discussion ;)

 

[vanhees71]

I'll check it out, but this circulation theorem usually goes as follows:

-At the end of the wind there will be vortices due to viscosity circulating anti clockwise;
- Because of Kelvin's conservation theorem there must be a clockwise circulation around the wing;
- This means that airflow must be faster above the wing then below;
- Because of Bernoulli this means that there is a net force up, which we call "lift".

But that doesn't mean the faster airflow causes the pressure drop, so this is not the causal explanation I'm looking for. The theorem only tells us how to calculate lift and why you expect there to be lift, but not how it arises as a mutual interaction between pressure and velocity.

The crucial point is, however, the circulation argument, and the vortex is forming due to to viscosity. That's often not considered, and using only ideal-fluid flow leads indeed to a contradiction. So the logic is: Through viscosity there's a thin boundary layer around the wing and a vortex is formed at the end of the air foil. Far away from the air foil the ideal-fluid flow is a good approximation again, and thus there must be an opposite circulation around the entire air foil. This makes the velocity gradient above an below the foil, and then the usual argument with Bernoulli's theorem leads to the explanation for the lift.

 

[haushofer]

This makes the velocity gradient above an below the foil, and then the usual argument with Bernoulli's theorem leads to the explanation for the lift.

But again: this merely says that, due to a velocity gradient there has to be a pressure gradient. It doesn't explain exactly how this pressure gradient arises. It just says this pressure gradient must be there. It's like finding a body and observe it didn't have a natural cause of death. But that doesn't tell us the cause of death in the first place. Or more physically: If I look at my rock being dropped, consider its energy conservation, and I observe that the rock nearly reaches the ground, then I can conclude that because its height is low, its speed must be high. But that's not a causal explanation. I use Newton's laws for that: gravity is the cause.

So again: what exactly CAUSES the pressure field distribution around the wing? Is McLean right in saying that not only the pressure gradient causes the velocity gradient, but also the other way around? And if so: how do people here understand this mutual interaction? Just consider a plane taking off. The wing starts to move through the air, pushing away the air. What are the precise events after the plane started to move that cause the pressure gradients?

 

[A.T.]

Newton's second law tells us that the particle's velocity changes due to a pressure difference along the flow.

Does it actually say "due to", or just how much velocity changes given a certain pressure difference? The formula of Newton's second law just tells you that force and acceleration are proportional. There is nothing in the formula about causation.

 

[russ_watters]

But again: this merely says that, due to a velocity gradient there has to be a pressure gradient. It doesn't explain exactly how this pressure gradient arises. It just says this pressure gradient must be there...

So again: what exactly CAUSES the pressure field distribution around the wing? Is McLean right in saying that not only the pressure gradient causes the velocity gradient, but also the other way around? And if so: how do people here understand this mutual interaction? Just consider a plane taking off. The wing starts to move through the air, pushing away the air. What are the precise events after the plane started to move that cause the pressure gradients?

There is another thread active where people are discussing derivations of Bernoulli's equation using the work-energy theorem. Here's an example of the common derivation:
https://phys.libretexts.org/Bookshe...ext=To derive Bernoulli's equation, we,−p2)dV.

It feels simple to say that the air is constrained and forced by continuity to accelerate, but continuity is a necessary result, not a mechanism. As you can see from the derivation of Bernoulli's principle/equation, for the fluid to accelerate it must have a force (from the static pressure) applied. So I would agree that it's a mutual interaction causing both the velocity increase and pressure decrease.

But if you really want A Cause, I'd say the shape of the airfoil is what causes lift.

 

[vanhees71]

But again: this merely says that, due to a velocity gradient there has to be a pressure gradient. It doesn't explain exactly how this pressure gradient arises. It just says this pressure gradient must be there. It's like finding a body and observe it didn't have a natural cause of death. But that doesn't tell us the cause of death in the first place. Or more physically: If I look at my rock being dropped, consider its energy conservation, and I observe that the rock nearly reaches the ground, then I can conclude that because its height is low, its speed must be high. But that's not a causal explanation. I use Newton's laws for that: gravity is the cause.

So again: what exactly CAUSES the pressure field distribution around the wing? Is McLean right in saying that not only the pressure gradient causes the velocity gradient, but also the other way around? And if so: how do people here understand this mutual interaction? Just consider a plane taking off. The wing starts to move through the air, pushing away the air. What are the precise events after the plane started to move that cause the pressure gradients?

But Bernoulli's law is just the integral form of energy conservation along a stream line. So when the wing starts to move it creates the vortex at its end due to friction (viscosity), and this necessarily also must generate a circulation around the entire wing such as to have total circulation zero along any closed curve around the entire wing located away from the boundary layer (see Neuenschwandner's paper you quoted above).

 

[haushofer]

Does it actually say "due to", or just how much velocity changes given a certain pressure difference? The formula of Newton's second law just tells you that force and acceleration are proportional. There is nothing in the formula about causation.

Wel, no, not in the mathematical formula. But when I teach Newton's second law, I always state the formula, complementing it with "a force causes an acceleration", not the other way around. Formulas are causally interpreted in physic.

I mean, that's why we speak of "fictitious forces" with e.g. the centifugal force: seeing something deflect doesn't automatically mean an interaction is acting on the object in the form of a force; the causal explanation for the deflection is not the fictitious force. Thats's just a useful bookkeeping device to use Newton's 2nd law in accelerating frames.

 

[haushofer]

But Bernoulli's law is just the integral form of energy conservation along a stream line. So when the wing starts to move it creates the vortex at its end due to friction (viscosity), and this necessarily also must generate a circulation around the entire wing such as to have total circulation zero along any closed curve around the entire wing located away from the boundary layer (see Neuenschwandner's paper you quoted above).

So what are the physical interactions which lead to the pressure gradients according to you? Do you consider it a valid reasoning to say that the circulation theorem implies a velocity gradient, and that this velocity gradient causes the pressure gradient, and hence the lift?

 

[cjl]

The physical interaction that causes the pressure gradient, ultimately, is the viscosity at the sharp trailing edge enforcing the location of the rear stagnation point. All the rest of it kind of falls out of the necessity of the rear stagnation point staying colocated with the sharp trailing edge, including circulation, velocity differences, and pressure gradients.

 

[A.T.]

Wel, no, not in the mathematical formula. But when I teach Newton's second law, I always state the formula, complementing it with "a force causes an acceleration", not the other way around. Formulas are causally interpreted in physic.

Maybe such causal interpretations are the root of the problem. They create the expectation, that everything in mechanics should explainable in the form a linear cause-effect chain. But this type of reasoning already breaks down for simple mechanical feedback loops, like the gear box below.

 

[haushofer]

Maybe such causal interpretations are the root of the problem. They create the expectation, that everything in mechanics should explainable in the form a linear cause-effect chain. But this type of reasoning already breaks down for simple mechanical feedback loops, like the gear box below.

That's a great analogy! Yes, it would make sense that this is the root of the problem, considering the fact that nowhere such a causal interpretation has yet been given (afaik). Thanks for your post!

 

[haushofer]

The physical interaction that causes the pressure gradient, ultimately, is the viscosity at the sharp trailing edge enforcing the location of the rear stagnation point. All the rest of it kind of falls out of the necessity of the rear stagnation point staying colocated with the sharp trailing edge, including circulation, velocity differences, and pressure gradients.

I'm not sure this is satisfying. Why not just say the ultimate cause is the wing itself? That's also true, but not very enlightening.

 

[cjl]

I agree that I'm not sure it's satisfying, and yes, ultimately the reason for the trailing edge stagnation point being where it is is the shape of the wing. Unfortunately though, airflow around wings is complicated, and the more satisfying and more enlightening answers that people tend to give have the unfortunate property of also not being correct.

 

[boneh3ad]

Answers aren't always satisfying. Ultimately, fluid flows in a continuum are governed by the Navier-Stokes equations, which are elliptic partial differential equations. What happens at one point in the flow field has the potential to effect every other location in the flow field, so finding a direct causal relationship in a situation like this is not only difficult, but also not very enlightening. McLean is correct.

Ultimately, I agree with @cjl here (as is often the case). The action of viscosity enforcing the rear stagnation point at a "sharp" trailing edge is a constraint on the flow field that ultimately gives rise to the velocity and pressure distributions. Without that fixed point, the fluid will still interact with the wing but no lift will be generated since no net circulation will develop.

This is intimately related to the fact that, in an inviscid flow, you predict zero drag and must artificially impose a circulation or else you will also predict zero lift. The circulation you impose in that case is tailored to "artificially" fix the trailing stagnation point at the trailing edge (i.e. enforce the Kutta condition) so that the Kutta-Joukowski theorem can be invoked.

 

[A.T.]

... the more satisfying and more enlightening answers that people tend to give have the unfortunate property of also not being correct.

And also not general. For example, any mention of the sharp trailing edge in the explanation already constrains the applicability of the explanation, and might give people the wrong idea that you cannot generate lift without it. While in fact any shape that isn't symmetric to the flow is likely to produce some lift.

 

[rcgldr]

For cause and effect, the cause is a wing with an effective angle of attack moving at some velocity through the air, and the effect is the affected air is diverted downwards (assuming level flight).

A relatively simple explanation is that a wing diverts the relative air flow "downward". The air is essentially deflected downwards by the bottom surface of a wing. The air will mostly follow the upper surface of a wing to fill in what would otherwise be a void as the wing sweeps out a volume of air as it travels through the air, as long as the speed and rate of curvature of the flow are not excessive.

The wing exerts a downwards force onto the air, coexistent with the air exerting an upwards force on the wing.

The diverted flow is a curved flow, and the curvature of flow coexists with a pressure gradient perpendicular to the flow, lower on the inside of the curve, greater on the outside of the curve. The pressure on the upper surface of a wing (inside of a curve) is less than the pressure on the lower surface of the wing (outside of a curve)

The air will also accelerate in the direction of flow from higher pressure zones to lower pressure zone, and there is a coexistent Bernoulli relationship between lower pressure and higher velocity in the direction of flow.

 

[haushofer]

What you call "coexistence", is this a result of the underlying differential equations being nonlinear? I.e. that the nonlinear character of the equations prevents us from a simple cause-and-effect explanation?

I'll think of other simple systems with this property as an analogy.

 

[rcgldr]

What you call "coexistence"

I use the term coexistent to avoid a cause and effect relationship. Once the cause of downwards diversion of the relative air flow has been established (a wing at an effective angle of attack moving through the air), then pressure gradients and acceleration of air (either perpendicular to or in the direction of flow) can be considered to be coexistent aspects of that diversion of the relative flow.

 

[haushofer]

Ok. I think I have a better understanding. Still, it would be nice to see a timelapse/simulation of how pressure-areas and streamlines evolve around a wing which starts to mobe.

 

[cjl]

To find that, the term you'll want to google or search for is "starting vortex", but maybe these videos will give you a good start:

 

[haushofer]

Excellent, exactly what I was looking for, thanks!

 

[cmb]

Summary:: Bernoulli, lift and cause & effect: looking for a clear-cut cause-and-effect and analogy.

A wing directs air downwards.

By thrusting vertical downward momentum to the air, so the aircraft is thrusted upwards.

The low pressure effects, surface velocities and pressures, etc., all the rest is an integral part of all of that.

Without air thrust downwards, there is no lift.

You can generate lift with a completely flat structure. Aerofoil/wing profiling helps a wing's efficiency, but any flat angled surface, tilted up at the leading edge, will generate lift.

 

[cjl]

Air can only interact with the wing through pressure and viscosity. The direction of any viscous forces will be nearly straight back, so the only way to impart a large lift force is through pressure on the surface of the wing. A wing must generate a pressure differential to fly.

By generating a pressure differential, of course, this means that the pressure below the wing is higher than ambient and the pressure above the wing is lower than ambient, so the downward curvature of surrounding air is just an integral part of generating this pressure gradient, and without a pressure differential, there is no lift.

This is of course phrased this way just to show why your post is misleading - the reality is that both the pressure differential and the downwash are integral parts of lift and inseparably tied to each other. You can't claim downwash is primary and pressure gradients (and the associated flow accelerations) are just results of it - they're all part of an inseparable, intertied system.

As for your last point? Sure. A completely flat surface angled up at the leading edge will still have lower pressure and higher velocity along its upper surface though, and a higher pressure lower velocity region on the lower surface, so this isn't actually as much of a "gotcha" for the downwash explanation as you might think.

 

[cmb]

... the downward curvature of surrounding air is just an integral part of generating this pressure gradient, and without a pressure differential, there is no lift.

This is of course phrased this way just to show why your post is misleading

So, how is that different to my comment "The low pressure effects, surface velocities and pressures, etc., all the rest is an integral part of all of that."?

We even used the same adjective, 'integral'.

Fact is, my point being, one way or another one HAS to deflect a bulk of air downwards.

Think of it another way. Let's say you had a 'wing' design consisting of a long rectangular duct ["of negligible mass and thickness"] whose axis/long edges were aligned to the direction of the oncoming air flow, such that oncoming air could enter at the front and exit at the rear. Now imagine that the 'upper' profile of a wing is fixed to the bottom of the inside of this duct and the remaining 'lower' profile part to the upper inner surface of the duct.

If these were bolted to the sides of an aircraft, now the oncoming air could flow over the 'upper' profile and under the 'lower' profile.

But ... it would generate zero lift because the rectangular duct would cause the exiting air to flow out without a vertical momentum, even though the air is flowing over the same wing profiles

So I propose it is not the wing profiles that cause the pressure differential and deflection of air, but rather the deflection of air causes the pressure differentials over the wing.

I've heard it argued often that the air over the top has to flow faster than the bottom because it is a longer distance. I see no logic in that. If I take the long route around a city ring road and my friend takes a route through the middle of the city, there is no fundamental axiomatic reason we'd end up getting to the same point on the far side of the city at the same time. Same with air molecules parting at the leading edge, they really don't 'have to' flow over the top faster, I can't see that argument.

But if the air on the underside is being deflected downwards by simply batting it that direction, and also the air over the top is pulled downwards by the Coanda effect, all of it makes simple sense.

 

[cjl]

So, how is that different to my comment "The low pressure effects, surface velocities and pressures, etc., all the rest is an integral part of all of that."?

We even used the same adjective, 'integral'.

Fact is, my point being, one way or another one HAS to deflect a bulk of air downwards.

And the fact is, my point being, one way or another, one has to have a lower pressure region above the wing and a higher pressure one below it.

Think of it another way. Let's say you had a 'wing' design consisting of a long rectangular duct ["of negligible mass and thickness"] whose axis/long edges were aligned to the direction of the oncoming air flow, such that oncoming air could enter at the front and exit at the rear. Now imagine that the 'upper' profile of a wing is fixed to the bottom of the inside of this duct and the remaining 'lower' profile part to the upper inner surface of the duct.

That has very little relevance to how wings actually work.

If these were bolted to the sides of an aircraft, now the oncoming air could flow over the 'upper' profile and under the 'lower' profile.

But ... it would generate zero lift because the rectangular duct would cause the exiting air to flow out without a vertical momentum, even though the air is flowing over the same wing profiles

Not necessarily. Depending on the profiles you use, the flow behavior will be very different. If the two surfaces are far apart, you probably will still obtain lift, as the flow off the lower surface will still be angled downwards at the rear. If the two surfaces are closer together though, you'll end up with more of a venturi, and because of that, you'll end up with a low pressure region against the upper surface of the duct, which is very different from the high pressure region you get from exactly the same profile being exposed to freestream flow without the opposing wall.

So I propose it is not the wing profiles that cause the pressure differential and deflection of air, but rather the deflection of air causes the pressure differentials over the wing.

Both the deflection of air and the pressure differentials are tied together, and neither can be said to "cause" the other. If anything, both are effects of viscosity causing an enforced rear stagnation point, and the circulation that is caused by pushing this rear stagnation point downwards.

I've heard it argued often that the air over the top has to flow faster than the bottom because it is a longer distance. I see no logic in that. If I take the long route around a city ring road and my friend takes a route through the middle of the city, there is no fundamental axiomatic reason we'd end up getting to the same point on the far side of the city at the same time. Same with air molecules parting at the leading edge, they really don't 'have to' flow over the top faster, I can't see that argument.

Of course that argument is nonsense. As you correctly guess, there's no reason they need to reach the rear at the same time, though counterintuitively, the ones that go the longer distance over the top actually end up reaching the rear first, well ahead of the ones that go underneath.

But if the air on the underside is being deflected downwards by simply batting it that direction, and also the air over the top is pulled downwards by the Coanda effect, all of it makes simple sense.

It might make sense, but it's wrong. Coanda has nothing to do with the air following the upper surface - coanda applies to a jet of fluid, not just bulk freestream flow. Fluid follows the upper surface because the surrounding fluid has nonzero pressure, so it tries to fill in all available space. The air on the underside really doesn't act much like the imaginary string of molecules bouncing off a flat underside either. The overall flowfield of the wing does deflect air downwards, but the deflection, the pressure gradients, and everything else about the flow is interconnected.

@boneh3ad wrote an excellent insight on this if you want to read more:
https://www.physicsforums.com/insights/airplane-wing-work-primer-lift/

 

[cmb]

Consider the following, then, please.

Take a sheet of A4 paper and grip the shorter edge between your fingers. The other end of the paper flops downwards.

Now, draw the paper through the air, and the lower edge rises up!

You only need the pressure from the air underneath it being accelerated down to create lift.

Of course, with an aerofoil design you get both that effect AND the flow of air over the top that stays in laminar flow over the top, due to Coanda (resistance to flow separation from the surface), thus follows the profile of the wing (downwards), making for an efficient wing.

Of course, in the stall this is where the flow separates and this exposes the integral nature of the pressure on the top to the bottom, if the pressure on the top becomes larger than under it (stalled air, high pressure) then of course there can never be lift.

But you only 'need' the extra pressure from underneath. No 'Bernoulli' low pressure on top is necessary, though it is an inevitability and modern wing designs take advantage of that, just not 'essential'. I would say low pressure on the top of the wing is effect not cause, but it's not something that can be teased apart, the two happen together.

To prove that applies to the paper example, here are two alternative variations.

First, glue the top edge of the paper to an A4 sheet of card and then grip them both so the card is held horizontal and the paper flops down. Now draw the arrangement through the air. What happens? The paper STILL lifts, yet air cannot flow over the top because there is a piece of cardboard stopping it.

Second, put lots of holes through the paper with a pencil. Now, the pressure on the bottom can neutralise any low pressure on top. What happens? Still the paper lifts, not so much but it lifts.

In fact you can do it with a string bag and still get the same effect.

Or even more extreme, a rope hanging from a helicopter. The rope still lifts as it is pulled through the air.