# Swept Volume Theory

Discussion in 'Hydrodynamics and Aerodynamics' started by Sailor Al, Aug 2, 2022.

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### Sailor AlSenior Member

Those are pretty blunt questions, and I'm not sure this forum is used to such directness. It's late and I have just returned from a fresh afternoon offshore racing, so will prepare a response in the morning.
I'm pretty sure you have have an idea about my response, especially if you read the earlier version of my theory.
Would you like to declare your position on these points?
I'll get back in the morning.

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### BarrySenior Member

The conditions above is that the stream is a wide constant flow / velocity regime prior to being influenced by the airfoil/wing
I had to clean up the prior wording as it was not clear re post 165.

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### Sailor AlSenior Member

@Barry

I've been seriously involved in yacht racing for 20 years and want to understand how sails work. I know that the wind exerts a force on the sail which pushes the boat forward, causes the boat to heel and forces it sideways.

I know that the only way that air, (a gas), can exert a force on a sail (an object) is via pressure, so while I really want to understand how the shape of sails affect the strength and direction of the force, it seems I have to first discover the source of this pressure, and then find out how the shape of the sail influences the pressure.

All the textbooks (Marchaj, Fossati, etc) say a sail works like a wing, so although I don't really care how wings work, it seems I have to understand wings in order to understand sails.

This has led me into the world of aerodynamics and fluid dynamics which appears to contain many theories, many of which, such as Momentum theory, Circulation, Trailing vortex, Lifting line etc. make no mention of pressure at all and generate two apparently independent forces, namely lift and drag. Since my quest is for the source of the pressure, not lift and drag, I was heartened when slogging through Anderson that I read on P 284 (6th Edition):

"The circulation theory of lift is an alternative way of thinking about the generation of lift on an aerodynamic body. Keep in mind that the true physical sources of aerodynamic force on a body are the pressure and shear stress distributions exerted on the surface of the body, as explained in Section 1.5"

Unfortunately the referenced Section 1.5 already assumes the presence of this pressure, so that turned out to be a dead end.

The only theory that appears to address the cause of the pressure is based on Bernoulli's principle, which brings us to the present.

And here's my problem.

Bernoulli's principle describes the behaviour of an incompressible, inviscid fluid, and while I am content to accept that to all intents and purposes air is inviscid, I know for certain that air is not incompressible.

No amount of prevarication about the degree to which it is compressible, or whether is can be considered incompressible below Mach 0.3 or whether water itself is a bit compressible will shake my fundamental knowledge that, at least as far as the air flowing over a sail is concerned, air behaves like an ideal gas, and obeys the Gas laws which define the behaviour of a compressible fluid.

Secondly, even if Bernoulli's Principle were applicable to air (and it obviously doesn't), that would leave two further problems. To generate a pressure drop, Bernoulli's principle requires an increase in the velocity of the fluid.
1. As the current thread will attest, there is no experimental evidence of that increase in velocity. None. Babinsky's video displays no evidence of the velocity increasing. The air over the leeward side of a sail does not flow faster than the apparent wind.

2. There is no theoretical basis which would cause it. McLean, after talking about "diffuse clouds of low pressure" and "a reciprocal or mutual interaction between the pressure field and the velocity field" admits to this section being the one with which he is most dissatisfied and appeals to his audience for assistance to explain it.
If fluid dynamics requires me to accept that Bernoulli's Principle is the only explanation for the lower pressure to leeward and the higher pressure to windward of the sail, then I have to look elsewhere for the answer.

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### BarrySenior Member

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Bernoulli's PRINCIPLE is commonly stated as " an increase in the speed of a fluid occurs simultaneously with a decrease in pressure " If his translated works includes the word FLUID and he was astute enough back then
to include liquids and gasses, that was his principal. I was under the impression that Euler developed the Bernoulli equation. The standard equation equation may be used for incompressible fluids (liquids) in its original form. The assumptions are that liquids are inviscid and incompressible. BUT since then equations have been developed to include the slight changes in density which are a result of compressibility, and are referred to as modified Bernoulli's equation.

To reiterate his PRINCIPLE "an increase in the speed of a fluid occurs simultaneously with a decrease in pressure" still holds. (While at higher speeds compressibility and resulting density change may become more significant)

A quick look on the net videos for AIR FLOW through a venturi supports that the increase in speed drops the static pressure in the constriction in the venturi. In fact flow meters for air and other gasses are built around the change of pressure in the venturi based on Bernoulli's Principle.

Bernoulli Equation and the Venturi Effect https://fluidhandlingpro.com/fluid-process-technology/fluid-flow-control-measurement/bernoulli-equation-and-the-venturi-effect/

_________________________________________________________________________________________________________________________________________________________________
The following section covers the two lift producing factors of a wing

The Bernoulli principal applies above the the area above the stagnation point. In order for the air to be accelerated requires a higher pressure source so that the higher pressure air accelerates over the upper control surface. The source of this high pressure source (to accelerate the air) is the stagnation point and the higher pressure gradient volume in front of the stagnation point that accelerates around the leading edge. The only pertinent air pressure is the extremely thin layer right in contact with the wing surface. A contributor showed the pressure distribution over a wing. X axis. So the air above the wing, provides a quasi containment boundary, (mixing of course as it moves down the wing) creating and holding the lower pressure thin layer against the wing. (Coanda)
As this high speed streamline moves toward the trailing edge, it is mixing with the air above and the pressure is increasing closer to ambient but still lower than the bottom wing pressure.

The Babinsky video shows that with an evenly pulsed smoke vertical and parallel array, that the top streamlines exit the trailing edge ahead of the lower smoke streamlines. This is pretty straight forward that the air above the wing
is moving faster than the air below the wing. Bernoulli ( or Coanda which is a Bernoulli manifestation)

So under the stagnation point, on the bottom of the the wing the second contributor to lift is evident. Note that the the first smoke streamline moves down the wing surface BUT the second stream below the first does not penetrate the
first smoke streamline. This indicates that the pressure against the bottom of the wing is higher than the approaching air can penetrate. I will find the equation but explain the mechanism. It takes a force to accelerate a mass of air.
Newton- F = MA As acceleration is a change in velocity, and velocity is a speed and direction, you can create an acceleration by changing the direction. That is what is occurring on the bottom of the wing. The air approaches the wing
horizontally the wing directs the air downward depending on the angle of attack. The larger the change between the inlet angle and the outlet angle for a given air mass, the larger the force produced on the wing.
This is exemplified when a plane drops 30 degrees of flaps coming in for landing. And assume that the plane is flying at say a level flight but slowing down. Two things happen, the dropped flap increases the amount of of the mass of the air that it influences but as well, the outlet angle of the air off the wing increases and there is more lift created.

________________________________________________________________________________________________________________________________________________________________
As with a wing, a sail has two sources of lift.
The first is the same as the bottom of the wing explained above. Ie a mass of air enters the sail and an inlet angle and exits at another producing a force on the windward side. Is there some compression of the air right at the interface
between the air and the sail. Certainly any increase of pressure will cause some compression of a compressible fluid. But this same sail would create lift if it was immersed in water. Ie the change of direction, which is a change in velocity, which is an acceleration, creates a force.

With the lee side of the sail. This is not the same situation as the wing.-------------

I will finish this tomorrow time permitting

F=rho x Q x velocity ( cos theta-1) Theta = 180 - alpha, or the change between the
inlet and outlet angle.

.

Last edited: Nov 6, 2022
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### Sailor AlSenior Member

I guess you ran out of time.

You have egregiously misquoted Bernoulli. By explicitly addressing fluids that are incompressible, he specifically excluded gases. (This is my claim that Bernoulli's Principle does not apply to gases.)
I am not sure what you intended with the youTube video. The experiment has air flowing through a wide glass tube constricted to a narrow tube.
Please show where such a constricting physical boundary exists on a wing, or a sail. (this is my claim that there is no theoretical basis for the existence of accelerated airflow)
If such an experiment were relevant don't you think it would have been referenced by the authorities on the subject?

Your next section is bewildering. You start with "The source of this high pressure source (to accelerate the air) is the stagnation point" and then blather on about " higher pressure gradient volume", "quasi containment boundary" and come to a conclusion "... the pressure is increasing closer to ambient but still lower than the bottom wing pressure."
If the source of the high pressure is the stagnation point, why do you need an argument to conclude the proposition? Your argument goes: Stagnation Point -> pressure -> acceleration -> pressure. It's circular!
The Snark I am hunting is the source of the pressure difference.

Yes, the air is flowing faster over the top of the wing than the bottom. That is not the issue. That has never been the issue. The issue is whether the air over the top of the wing is flowing faster than the free stream. There is no evidence in the Babinsky video that it does. (This is my claim that there is no experimental evidence of accelerated airflow)
Why not? Fossati says it is, Marchaj says it is. This is the reason I have had to delve into the murky waters of fluid dynamics in the first place.
If a sail is not "like a wing" then you are simply confirming my conclusion that the pursuit of fluid dynamics is inimical to the pursuit understanding how sails work.

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### BarrySenior Member

Actually, I have not quoted anyone had you cared to read my wording
Bernoulli's PRINCIPLE is commonly stated as " an increase in the speed of a fluid occurs simultaneously with a decrease in pressure "

So your argument that " Bernoulli by explicitly addressing fluids that are incompressible, he specifically excluded gasses" and therefore that a stream of air surrounded by a non moving stream of air, DOES NOT EXHIBIT LOWER PRESSURE
THAN THE SURROUNDING AIR.
This is what you are saying.
I included the venturi tube as this drop in pressure in the faster flowing smaller diameter is at lower pressure than the wider diameter slower moving air thinking that you would recognize that this set of parameters exits when water is
exchanged for the air.

So back to my earlier post and I am not sure why you chose to ignore it. It was not blunt, but direct.

And as an aside wrt venturis: It was during our first year engineering Fluid Mechanics course lab that we did these velocity and subsequent pressure measurements with air and water, using pitot tubes (for dynamic pressure) in all three sections to confirm flow rates, static port outlets in all three diameters, attached to inclined plane manometers. Pretty easy stuff.
__________________________________________________________________________________________________________________________________________________________________

The area in front of a stagnation point has a pressure gradient along its streamline as the air impacts on the stagnation point. The smoke streamline clearly shows the streamline impacting the stagnation point. This higher pressure area
will not just sit and hold a captive designated amount of molecules. It is at higher pressure and surrounded by lower pressure volumes. So it provides an increase in speed as it expands.
You can see the streamline next to the wing surface re establish itself and curve around the leading edge and down the top of the wing.
Constrained (until pressure diffusion and mixing occurs.)
To illustrate the concept of high pressure air moving into a lower pressure adjacent area, simply blow up a balloon to a high pressure, and put a pin to it. As the balloon explodes, the air within the balloon moves out at an accelerated speed until the pressure is equalized.
___________________________________________________________________________________________________________________________________________________________________
QUOTE (Mine) With the lee side of the sail. This is not the same situation as the wing.

My premise is that with a wing, the lift is a combination of Newtonian forces on the bottom and the "resulting lower pressure above the wing caused by the increase of speed of the air" The acceleration of the air driven off the high pressure stagnation zone. The video coupled with the aforementioned pressure distribution earlier in the thread pretty much shows this.

My premise is that with a wing that the windward side of the sail is governed by the same Newtonian forces, simplified, the sail changes the direction of a mass flow rate between the inlet angle and outlet angle. (By the way, this is the basic
method when designing turbine blades. On the back side, I would put forth that the lower pressure than ambient is caused by wake effects which lowers the pressure
_________________________________________________________________________________________________________________________________________________________________

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### Sailor AlSenior Member

If it is true "that a stream of air surrounded by a non moving stream of air, DOES does exhibit lower pressure....", then it's not due to Bernoulli's Principle, but either way, you are still faced with the two other issues that I have raised:
Where is the evidence of the increased air velocity?
What is the theoretical origin of such an increase in velocity?

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### gonzoSenior Member

Pressure is force/area. The force on a sail is from pressure differential. You are mixing pressure and force as if they are equivalent.

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### gonzoSenior Member

Where is the proof? If you agree water is "a bit compressible", then it is compressible. The difference between air and water is one of degree and not of behavior.

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### Doug HalseySenior Member

So nothing can convince you that, if you extended curves like this all the way to zero Mach number, they wouldn't be close enough to horizontal?

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### gonzoSenior Member

In that case, since the authorities on the subject have not conducted an experiment, it would mean by your standard that it is not relevant. In fact, your whole argument is irrelevant since the authorities have not seen fit to mention it.

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### baeckmoHydrodynamics

It seems that the discussion is missing a few facts about Bernoulli’s theorem. All matter is elastic at temperatures above 0 K, and the theorem is a tool to follow the conversion of all the different energy forms found in a specific continuum, as it undergoes changes of state.

As energy is indestructible and can not be created, the complete version of the theorem must include expressions for all the four “classical” energy forms that interact in the process:

(W is energy; m is mass; g is earth acceleration; p is pressure; ρ is density; c is velocity; u is specific internal energy)

· Geodetic energy (related to the vertical position re a common datum level): W1=mgh.
· External (pressure) energy (related to the local static pressure): W2=mp/ρ.
· Kinetic energy (function of velocity): W3=mc^2/2.
· Internal energy (function of temperature): W4=mu.

Additional energy forms (chemical, electromagnetical aso) must be included where they take part of the energy transformation (fi flame propagation or plasma behaviour).

Expressed as energy per kg, the total energy content is then written in the more complete form:

W/m=(gh+p/ρ+u+c^2/2); [Nm/kg]

In the “Bernoulli light” version that has been discussed, two factors (gh and u) have been omitted, which leads to unclarity. The geodetic (“head”) term is important if the study involves significant variation in elevation (pumping liquid between two levels or compensating for height in atmospheric pressure aso). In the study of free gas flow past an object, the geodetic energy is considered constant and thus left out.

In this context pressure is always linked to the density of the substance (“all matter is elastic”), which is why we refer to mass flow (not volume flow). Whether the fluid’s elasticiy has a significant influence on the energy transformation results or not, depends on the context. For example, in steady flow, liquids are considered incompressible. But in transient flows the pressure wave velocities depend on the elasticity, which then is crucial.

For gases, the influence of elasticity is more complex, since a change of state will influence both density and internal energy. Again, excluding the elasticity depends on the context; for low gas speeds (~0,25 x speed of pressure waves), the elasticity has negligible influence compared to pressure and velocity, and is often left out. Gas behaviour follows the thermodynamic laws and must be considered for higher velocities or pressure ratios above ~1,8.

Now, there is rational reason to keep the W4 term both for liquids and low speed gases, since the unavoidable fluid friction and turbulence will manifest as an increase in temperature, ie internal energy. Often, you will see a “loss” term, either pressure or velocity loss instead, but strictly speaking there is no loss, but an increase of internal (low temperature) energy, which is only a loss if it can not be recovered somehow.

So, instead of the truncated Bernoulli teorem with only the pressure and kinetic energy factors, the internal energy should be included for stringency, even for low speed, incompressible cases.

The consequence of the basic Bernoulli energy equation is that wherever a change in static pressure can be observed, there must be a corresponding, and compensating change in the other energy forms. In order to create a force on a surface in a free flow, the flow, or part thereof, has to be deflected. To deflect the path of a moving fluid there has to be a pressure difference across the path, and if there is a pressure change, there will be a velocity change, unless there is a change in elevation or chemical energy et c.

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### Doug HalseySenior Member

Here are two pages from a basic aero textbook (Theoretical Aerodynamics by L.M. Milne-Thomson) explaining the issue you have. What, exactly,do you find so hard to accept?

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### Sailor AlSenior Member

Nope, As I said, the only way a gas can exert a force on an object is by pressure.
Again, Nope. As far as sailboats and aeroplanes are concerned, water is incompressible and air is compressible. Not a matter of degree: absolutely.
I'm sure they have!
Yes I believe it is irrelevant - it was @Barry who thought it was relevant.

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### Sailor AlSenior Member

Are you proposing an amendment to Bernoulli's Principle? It's been around for a long time, I think you might have trouble getting it accepted!

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