Misapplication of Bernouilli's principle

[Andrew Mason]

I could not help but notice in my daughter's first year university physics text what I think is misconception of Bernouilli's principle.

The illustration of Bernouilli's principle is a truck with a tarpaulin on the box. When the truck speeds down the highway, the tarp lifts. This is supposed to illustrate Bernoulli's principle.

The problem here is that the air is not moving. The truck is. So how can this be an illustration of Bernoulli's principle?

The same misconception frequently occurs with airplane wings. They do not create lift due to Bernoulli's principle. They create lift due to turbulence of the air flow.

AM

 

[FredGarvin]

It's a misapplication to be sure. This page has some nice graphics of the recirculation behind a truck: http://www.mathematik.uni-dortmund.de/~featflow/album/catalog/lkw2_low_2d/data.html#HEADLINE2

The bottom vector plot is a good one. I wonder why they don't talk about flow separation in these physics texts?

The only problem I have is with your statement that the truck is moving, not the air. If you were to measure the total pressure of the air at the truck bed going down the road or in a wind tunnel, would they be any differernt? No. How else could you explain the effectiveness of windtunnel testing?

 

[marcusl]

The same misconception frequently occurs with airplane wings. They do not create lift due to Bernoulli's principle. They create lift due to turbulence of the air flow.

AM

Hmm. Airplane designers go to great lengths to eliminate turbulence over wings. Laminar flow is everything. Turbulence reduces lift and increases drag.

 

[Office_Shredder]

Is there a difference between the air moving and the truck moving? I recall being given similar examples myself... I never saw a problem with it

 

[Andrew Mason]

Is there a difference between the air moving and the truck moving? I recall being given similar examples myself... I never saw a problem with it

The only problem I have is with your statement that the truck is moving, not the air. If you were to measure the total pressure of the air at the truck bed going down the road or in a wind tunnel, would they be any differernt? No. How else could you explain the effectiveness of windtunnel testing?

There is no difference between the air moving or the object moving if all you are interested in is relative motion of air and object. But that has nothing to do with Bernoulli's principle. A windtunnel may or may not model real atmospheric conditions, depending on how it is set up, so I don't think you can say there is necessarily no difference.

Bernoulli's principle relates the speed of a fluid to its energy density. Pressure is potential energy density (Force x distance/Volume). If the total energy of the fluid does not change, and if the kinetic energy of the fluid in the system increases, its potential energy (pressure + gravitational potential) must decrease. That, it seems to me, is the principle.

That principle has no application to a truck (or wing) moving through a fluid where the fluid experiences no change in speed.

Assuming the windtunnel is created by the release of a volume of air under high pressure through a constricted opening, Bernoulli's equation applies to the windtunnel system. Whether the pressure of the high speed air is less than atmospheric pressure (ie. whether the tarp will go up or down) will depend on the applied high pressure and the speed change.

AM

 

[russ_watters]

Hmm. Airplane designers go to great lengths to eliminate turbulence over wings. Laminar flow is everything. Turbulence reduces lift and increases drag.

That isn't strictly true. There are two different kinds of drag: pressure drag and skin friction drag.

-Skin friction drag is significantly lower with laminar airflow than with turbulent, but...
-Pressure drag is generally significantly lower with turbulent airflow than with laminar airflow. And...
-In most circumstances, pressure drag is significantly greater than skin friction drag, hence:

-Golf balls have dimples to make airflow turbulent and vastly reduce drag.
-Airplanes have vortex generators (those little tabs lined up atop the wing) to cause the air to become turbulent.

Turbulent airflow is also (paradoxically) more stable than laminar. You can make a wing that has laminar airflow over its entire surface (the P-51 is one famous example), but even a bird crapping on the wing will utterly destroy the benefit of the laminar airflow.

 

[russ_watters]

There is no difference between the air moving or the object moving if all you are interested in is relative motion of air and object. But that has nothing to do with Bernoulli's principle. A windtunnel may or may not model real atmospheric conditions, depending on how it is set up, so I don't think you can say there is necessarily no difference.

All motion is, by definition, relative. So whenever you face a situation where you can consider one object or another in motion or stationary, it is utterly irrelevant which is which. The rules always work the same.

That said, typically for aerodynamics, you assume the object to be stationary and the air to be moving. But where's the problem?: if you're sitting in the cab of the truck, is the truck moving wrt to you? Clearly, no. Is the air moving wrt you? Clearly yes.

Think about it another way: if what you were thinking were true, wind would be a much more serious problem for airplanes.

Bernoulli's principle relates the speed of a fluid to its energy density. Pressure is potential energy density (Force x distance/Volume). If the total energy of the fluid does not change, and if the kinetic energy of the fluid in the system increases, its potential energy (pressure + gravitational potential) must decrease. That, it seems to me, is the principle.

That principle has no application to a truck (or wing) moving through a fluid where the fluid experiences no change in speed.

Kinetic energy is based on velocity. Velocity (motion), is relative, as I said above.

Example:
-Two cars collide head on with each going 20mph.
-Two cars collide head on with one going 10mph and the other going 30mph.

These two scenarios have precisely the same kinetic energy in the collision.

Assuming the windtunnel is created by the release of a volume of air under high pressure through a constricted opening, Bernoulli's equation applies to the windtunnel system. Whether the pressure of the high speed air is less than atmospheric pressure (ie. whether the tarp will go up or down) will depend on the applied high pressure and the speed change.

It depends on the type of wind tunnel, but for a low-speed wind tunnel, the static pressure inside the tunnel is negligibly different from atmospheric pressure.

Though to clarify - most low-speed wind tunnels use fans. Except for some specialized applications, only supersonic wind tunnels use a high-pressure jet and a restriction.

 

[Andrew Mason]

All motion is, by definition, relative. So whenever you face a situation where you can consider one object or another in motion or stationary, it is utterly irrelevant which is which. The rules always work the same.

It is not irrelevant if you are considering the application of the Bernoulli principle to a fluid system. The pressure in the water in the smaller diameter pipe moving from a large diameter pipe and, therefore speeding up, is less than the water in the large diameter pipe. I can't create that pressure drop by keeping the water static and running along side the pipe!

Bernoulli's principle applies only where the fluid experiences a CHANGE in speed - ie an acceleration. The truck moving through the air does not make the air change its speed. This is the fallacy behind the attempt to apply Bernoulli's principle to the air flow over the truck - and indeed to an airplane moving through the air. It has nothing to do with Bernouilli's principle.

Think about it another way: if what you were thinking were true, wind would be a much more serious problem for airplanes. Kinetic energy is based on velocity. Velocity (motion), is relative, as I said above.

Which just proves the point that Bernoulli is irrelevant to the physics of flight.

AM

 

[russ_watters]

It is not irrelevant if you are considering the application of the Bernoulli principle to a fluid system. The pressure in the water in the smaller diameter pipe moving from a large diameter pipe and, therefore speeding up, is less than the water in the large diameter pipe. I can't create that pressure drop by keeping the water static and running along side the pipe!

Huh? No, you're completely missing the point of relative velocity. It is the velocity of the water wrt the pipe that matters, not your velocity wrt the pipe! Heck, if your velocity wrt the pipe and water mattered, you'd screw up every plumbing system in the world every time you started driving in your car! Clearly, an absurdity.

In fact, the example you just gave is exactly the same as what we are talking about here: whether it matters if the medium or the object the medium is moving in/around wrt the other is moving.

Ie, what if the pipe is on a train and the water in the pipe is moving backwards at the same speed that the train is moving forwards. To you, on the ground next to the train, the water is not moving. Will a venturi flow meter on the pipe work? Certainly yes, because the water is moving wrt the pipe. Whether it is moving wrt some other observer is utterly irrelevant.

Bernoulli's principle applies only where the fluid experiences a CHANGE in speed - ie an acceleration. The truck moving through the air does not make the air change its speed. This is the fallacy behind the attempt to apply Bernoulli's principle to the air flow over the truck - and indeed to an airplane moving through the air. It has nothing to do with Bernouilli's principle.

Which just proves the point that Bernoulli is irrelevant to the physics of flight.

AM

No and no. If you are standing next to a road, you can clearly feel that passing cars cause a change in speed of stationary air on the road. What is happening to the air looks different if you are standing next to the road than if you are sitting in the car, but the point is that whether you are sitting in a car at 50mph on the road or sitting in the car in a wind tunnel that is blowing 50mph over the car, the wind and car interact in precisely the same way.

 

[marcusl]

Laminar flow over the majority of the wing surface gives the highest lift to drag ratio. It's difficult to achieve over an entire wing and particularly at supersonic speeds.

"Supersonic laminar flow control has been called the "holy grail" of aerodynamics, because it's the last frontier that can offer significant drag reductions and save airlines, and eventually the flying public, a great amount of money," said Jeffrey Lavell, project manager of the F-16XL Supersonic Laminar Flow Control (SLFC) experiment at NASA Langley Research Center, Hampton, Va.

As an aircraft flies, the friction between the air and the wing creates drag, or resistance, called skin friction drag. Skin friction drag accounts for about half of the total drag on an aircraft. When airflow over the wing becomes turbulent and separates from the wing, skin friction drag increases. Laminar flow, a condition where the airflow over the wings remains smooth and close to the wing, greatly reduces skin friction drag. Smooth, or laminar, flow over a wing can reduce drag and contribute to reduced operating costs by improving fuel consumption and lowering aircraft weight.
from Langley News, October, 1996.

Passive and active Laminar Flow Control (LFC) are used to help. Small amounts of turbulence can be introduced at the leading and trailing edge of wings, especially swept wings, to manage laminar flow separation. In other words, careful introduction of "micro-vortices" using the tabs that Russ refers to (passive LFC) modifies or sweeps away instabilities and can stave off the transition from laminar to turbulent flow or move the separation point towards the trailing edge. Both improve a wing's performance.

Skin drag due to turbulence is significant, as the preceding quote indicates; here's a "brief" that reports a 50% drag reduction after passive LFC is introduced: http://www.nasa.gov/centers/langley/pdf/70865main_FS-2000-06-52-LaRC.pdf"

One Active LFC approach is to reduce turbulence by sucking air into slots or holes at critical areas of the wing. It is often combined with passive LFC for best results. See http://www.nasa.gov/centers/dryden/news/FactSheets/FS-023-DFRC_prt.htm" for an example.

In all cases the goal is to increase the amount or area of laminar flow over the wing surface. Andrew, laminar flow rather than turbulence produces lift.

 

[cesiumfrog]

Example:
-Two cars collide head on with each going 20mph.
-Two cars collide head on with one going 10mph and the other going 30mph.

These two scenarios have precisely the same kinetic energy in the collision.

Odd example or phrasing, since, although those can refer to the same accident in two frames, those obviously do have different total amounts of kinetic energy: eg. from one perspective the wreck keeps moving afterwards.

Bernoulli's equation says energy is conserved, not invariant (indeed the air seems more violent from aboard the truck).

The whole "Misapplication of Bernouilli's principle" thing is quite interesting, because (so far) nobody seems to argue that the experiment will give a result inconsistant with the (mis-)application. This suggests to me that there is subtle reason why the principle is indeed valid in such situations.

In the stationary frame (ie. with respect to the truck station ) we must explain why the acceleration of the truck imparts enough energy to raise the pressure inside, while explaining why the external air changes only in velocity not energy. Airfoil lift is simpler (where the shape intuitively compresses streamlines more on one side) than here (assuming the effect to occur even for carriage covers on long trains along very short track loops, the streamlines appear unpurturbed by the motion).

 

[Andrew Mason]

Huh? No, you're completely missing the point of relative velocity. It is the velocity of the water wrt the pipe that matters, not your velocity wrt the pipe! Heck, if your velocity wrt the pipe and water mattered, you'd screw up every plumbing system in the world every time you started driving in your car! Clearly, an absurdity.

In fact, the example you just gave is exactly the same as what we are talking about here: whether it matters if the medium or the object the medium is moving in/around wrt the other is moving.

My point is that it is not the relative velocity at all. It is a change in velocity of the fluid (ie an acceleration) that causes a reduction in pressure. If the air is stationary and the truck is moving, there is no change in air pressure due to the Bernoulli principle. Any change in pressure is due to the disturbance of air by the truck.

Ie, what if the pipe is on a train and the water in the pipe is moving backwards at the same speed that the train is moving forwards. To you, on the ground next to the train, the water is not moving. Will a venturi flow meter on the pipe work? Certainly yes, because the water is moving wrt the pipe. Whether it is moving wrt some other observer is utterly irrelevant.

If the pipe is moving relative to the water, the speed of the water through the Venturi flow meter will not be constant. The meter constricts flow which causes the water to speed up going through the constriction. The water can't all be moving backward at the same speed as the train is moving forward.

No and no. If you are standing next to a road, you can clearly feel that passing cars cause a change in speed of stationary air on the road. What is happening to the air looks different if you are standing next to the road than if you are sitting in the car, but the point is that whether you are sitting in a car at 50mph on the road or sitting in the car in a wind tunnel that is blowing 50mph over the car, the wind and car interact in precisely the same way.

This is only true if the moving air in the wind tunnel has the same (lateral) pressure as static air. Then I would agree with you.

The fact that the wind tunnel could be equivalent to a truck moving and keeping the air static shows that the air pressure is not reduced because the truck is moving relative to the air. My point is illustrated by the absurdity of the suggestion that it could be otherwise.

AM

 

[Gokul43201]

AM's OP is right in that the raising of the tarp is not a direct outcome of the Bernoulli equation - you can not compare energy densities between points that do not lie along the same streamline while you simultaneously claim to be applying Bernoulli. Moreover, it's also true that it's not the relative velocity between air and truck that's important but the relative velocity between the air above the tarp and the air below it. However, if the air below the tarp can be approximated to a boundary layer that is nearly stationary with respect to the truck, then, for the sake of estimation, the first relative velocity (above) becomes useful.

 

[marcusl]

AM, we part ways at the last sentence below taken from your second post:

Bernoulli's principle relates the speed of a fluid to its energy density. Pressure is potential energy density (Force x distance/Volume). If the total energy of the fluid does not change, and if the kinetic energy of the fluid in the system increases, its potential energy (pressure + gravitational potential) must decrease. That, it seems to me, is the principle.

That principle has no application to a truck (or wing) moving through a fluid where the fluid experiences no change in speed.

But the fluid does change speed. Air moving around a classic asymmetric airfoil (flat on bottom, rounded on top) must traverse a greater distance on top than that on bottom in a given time, even at zero angle of attack, so its speed is higher on top compared to bottom. Total energy is conserved but KE on top is greater than on bottom. By your statement of Bernoulli's principle, this makes the pressure energy density lower on top than on bottom, giving lift. It doesn't matter whether the air moves or the wing moves so long as we are far from any boundaries.

Bernoulli's principle and laminar flow give lift.

What happens with the truck depends on its shape and on boundary effects (since we are close to stationary pavement in this case); if the truck has something of an airfoil shape, then it and therefore the tarp could experience classic lift. The tarp is also likely to undergo chaotic motion due to turbulent flow (this is particularly visible for a plastic tarp covering a load in the bed of a pickup truck, which is not shaped like an airfoil).

This lift is a well understood and serious problem for race cars, because reduced weight on the tires at speed reduces cornering and braking ability. Hence the use of air dams to reduce flow under the car, and spoilers or wings on top to counteract the body lift.

 

[arildno]

There exists NO physical principle that states that to joined particles at the front, the one going over the curved top, the other going beneath, has to meet again at the back.
In fact, this is FALSE.

It is the induced CENTRIPETAL acceleration over the top that can be related to a pressure difference (and hence, lift), rather than trhe tangential acceleration implicit in Bernoullli's principle.

 

[marcusl]

There exists NO physical principle that states that to joined particles at the front, the one going over the curved top, the other going beneath, has to meet again at the back.
In fact, this is FALSE.

It is the induced CENTRIPETAL acceleration over the top that can be related to a pressure difference (and hence, lift), rather than trhe tangential acceleration implicit in Bernoullli's principle.

A vacuum would result if they don't match up, which would be filled in a chaotic way. This does happen at high Reynolds numbers where inertial effects dominate viscous ones; it is turbulent flow. At low numbers (low speeds) air can be assumed incompressible and inviscid. Then laminar flow is described by streamlines that match up at the back.

 

[marcusl]

Expanding on physical principles: Looking again at an ideal fluid, Laplace's equation holds so flow is characterized by streamlines and velocity by equipotential lines. In two dimensions, these form an analytic pair which is why complex variables were often used to analyze low speed fluid flow before the computer age. The match up of particles at the back of a wing then follows from the continuity equation and from the equation for circulation (since air in front of the wing is irrotational, then net circulation around the airfoil must remain zero).

 

[Andrew Mason]

Let's make it really simple. Let us assume that the air over the truck box is not disturbed by the truck's motion through the air (let's say that the air above the level of the truck top is separated from the air below the truck top and all you have is a stationary body of air and truck box moving underneath it). In that situation, there is relative motion between the two volumes of air (the air in the box and the air above the tarp). It is quite obvious that the relative motion does not create a difference in pressure.

The mistake, it seems to me, is in failing to recognize that fluid at faster speed has less pressure than fluid at slower speed only where the faster moving fluid has made the transition from the slower moving fluid (and where the energy density of the fluid is constant).

AM

 

[cesiumfrog]

It is quite obvious that the relative motion does not create a difference in pressure.

Not so obvious to me. (I must take a barometer driving sometime.) I have just looked at this problem from two different angles, and obtained almost contradictory results.

Bernoulli's equation simply states energy (per unit volume) constantly (along the flow streamlines) remains equal to the pressure plus (weighted by half the density) the square of the net velocity. So first I compared the energy of a gas in two reference frames.

In the stationary frame energy is proportional to the mean square of velocity (while the mean velocity is zero). Changing frames (<(v+u)^2>=<v^2>+2u0+u^2) the energy (per vol.) is increased by \frac 1 2 \rho u^2 and, therefore, separately applying Bernoulli's eq. in both frames gives that the pressure is equal in both frames. In particular (ignoring the truck's own displacement) this implies that if the tarp is initially sealed airtight then it shouldn't start bulging as the truck speeds up.

For a second approach (following explanations of airfoil lift), since a tarp covered container isn't airtight, some streamlines will go through the container interior. It is now valid to compare the flow in front of the truck with that inside (or trivially, on top of) the container. By simple application of Bernoulli's principle (the relative velocities are very different on the two sides of the tarp) a loosely sealed tarp should indeed bulge when the truck is moving.

 

[Andrew Mason]

Not so obvious to me. (I must take a barometer driving sometime.) I have just looked at this problem from two different angles, and obtained almost contradictory results.

Bernoulli's equation simply states energy (per unit volume) constantly (along the flow streamlines) remains equal to the pressure plus (weighted by half the density) the square of the net velocity. So first I compared the energy of a gas in two reference frames.

In the stationary frame energy is proportional to the mean square of velocity (while the mean velocity is zero). Changing frames (<(v+u)^2>=<v^2>+2u0+u^2) the energy (per vol.) is increased by \frac 1 2 \rho u^2 and, therefore, separately applying Bernoulli's eq. in both frames gives that the pressure is equal in both frames. In particular (ignoring the truck's own displacement) this implies that if the tarp is initially sealed airtight then it shouldn't start bulging as the truck speeds up.

For a second approach (following explanations of airfoil lift), since a tarp covered container isn't airtight, some streamlines will go through the container interior. It is now valid to compare the flow in front of the truck with that inside (or trivially, on top of) the container. By simple application of Bernoulli's principle (the relative velocities are very different on the two sides of the tarp) a loosely sealed tarp should indeed bulge when the truck is moving.

Why would air flow from the inside the truck through the tarp? That can only happen IF there is a pressure difference (and assuming the tarp leaks).

If the air above the tarp is not disturbed by the truck's motion, and if the static pressures are the same to begin with, the truck's motion can't change the pressure of either the air in the truck box or the air above it.

AM

 

[cesiumfrog]

Why would air flow from the inside the truck through the tarp? That can only happen IF there is a pressure difference (and assuming the tarp leaks).

I stand by my explanation.

Moreover we predict different experimental outcomes. What if you do occasion to see a tarp bulge? Would you insisted it is only acting as a spinnaker? I think my explanation partially explains those as well. Can you also explain (another textbook "mis-application"?) why a sheet of paper lifts when you blow over one side of it?

I have never seen a tarp used in a manner that would not allow air flow to leak, and having felt cold drafts under my door on windy days, I don't think flow through a truck's container would require much more pressure difference than the flow outside the truck does.

If the air above the tarp is not disturbed by the truck's motion, and if the static pressures are the same to begin with, the truck's motion can't change the pressure of either the air in the truck box or the air above it.

Indeed, this is in the third paragraph you quoted.

 

[pervect]

A vacuum would result if they don't match up, which would be filled in a chaotic way. This does happen at high Reynolds numbers where inertial effects dominate viscous ones; it is turbulent flow. At low numbers (low speeds) air can be assumed incompressible and inviscid. Then laminar flow is described by streamlines that match up at the back.

I'm pretty sure this is wrong. See for instance

http://www.grc.nasa.gov/WWW/K-12/airplane/wrong1.html

The problem is that it neglects "spin vortices" that appear at the edge of wings even at low speeds.

 

[marcusl]

Yeah, looks like I was wrong about that and I was also wrong about the circulation being zero around the airfoil. It's only zero around any path that doesn't enclose the wing. Apologies to arildno and AM on these.

 

[Clausius2]

Wow! A pretty long thread!. I wish I could have time for reading everything you said...but I don't.

I actually don't understand where is the tarp put in the truck, how is it put?. I don't see it. Is it wrapping the whole box?. Only the rear part?.

Anyways, here I have heard statements about the famous Bernoulli equation and trucks and pipes moving. It does NOT MATTER who moves and who stands, IF AND IF you write CORRECTLY the Bernoulli equation. If it is the truck which is moving, the flow is essentially unsteady from a laboratory reference frame. A correction term accounting for the unsteadiness of the flow is then needed. For simplicity assume irrotational flow. Then you would have to add a term like \partial \phi/\partial t in the equation (and this term is not artificial at all, it comes directly from the N-S equations). The fluid is not as rest as you may think.

You should not forget that pressure (as well as force and accelerations) are INVARIANTS under change of reference frame (at least at ordinary speeds), so that no matter what is moving with respect to what your pressure distribution must remain the same.

I have to say, and I hope that I won't offend you saying this, that you are the one that has the misconception. And I have to say also that maybe the book is wrong also (I cannot judge that because I didn't understand the business of the tarp), in part because nowadays there are so many awful books of general fluid mechanics in the market that even the trash guy has written one.

 

[marcusl]

Ok, I'm motivated to try to learn some fluid dynamics. What is a good book to get? (This will be spare-time reading, I can't plow through Lamb or similar tome...)

 

[Clausius2]

Ok, I'm motivated to try to learn some fluid dynamics. What is a good book to get? (This will be spare-time reading, I can't plow through Lamb or similar tome...)

What about Lamb? . It's probably the classic one, but the toughest one too. Batchelor is also very good. If you want something more readable (readable=less number of concepts per line), I advice you the one of Kundu or the one of Spurk.

Nevertheless, the best book I have ever found in fluid mechanics is the one of Van Dyke, "Perturbation Methods in Fluid Mechanics", but is only for someone initiated in asymptotic analysis because it is not a book of general fluid dynamics. It is amazing how this book is written, it is a masterpiece, and you can smell the history of this science throughout its pages. Really tough to read though.

 

[Cyrus]

And I have to say also that maybe the book is wrong also (I cannot judge that because I didn't understand the business of the tarp), in part because nowadays there are so many awful books of general fluid mechanics in the market that even the trash guy has written one.

AHAHAHAHAHAH!

 

[vijay123]

but i thought that the two points of air particles moving up and down a plane wing meet at the same point is an idealisation. this doesn't happen most of the time due to things like air resistance and so on, but if one has to caculate the upward force in the plane's wing, it becomes too tedious to calculate the turbulent flow. hence, we use bournoulli's principle.

but planes can still fly if there wings are upside down, making the net force downwards according to bournioullis principle which is indeed a violation.

 

[Andrew Mason]

but i thought that the two points of air particles moving up and down a plane wing meet at the same point is an idealisation. this doesn't happen most of the time due to things like air resistance and so on, but if one has to caculate the upward force in the plane's wing, it becomes too tedious to calculate the turbulent flow. hence, we use bournoulli's principle.

but planes can still fly if there wings are upside down, making the net force downwards according to bournioullis principle which is indeed a violation.

Which shows that Bernoulli's principle has nothing to do with flight.

AM

 

[cesiumfrog]

but planes can still fly if there wings are upside down, making the net force downwards according to bournioullis principle which is indeed a violation.

After reading the NASA site, my understanding is that Berrnoulli's principle (energy-conservation) correcly determines the amount of lift from the velocity of air above and below the wing (noting the flow is uniform ahead of the wing).

However, in "textbook" applications of Bernoulli's principle we use mass-continuity to first calculate the velocity of the fluid. For a wing (since the flow isn't as tightly constrained) mass-continuity is insufficient to determine the airflow velocity, and we need an additional constraint.

It seems that the mistake is assuming "the flows matches up so particles initially adjacent return so" is a suitable constraint, when it is infact incorrect. Instead the additional constraint should be momentum-conservation (culminating in the Euler equations).