# Another question on Bernoulli's principle

Discussion in 'Hydrodynamics and Aerodynamics' started by quequen, Apr 29, 2014.

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### quequenSenior Member

The relationship between pressure and velocity stated by Bernoulli's principle leads me to a misunderstanding. I thought that an increase in flow speed means an automatic reduction in flow pressure. That is not the case, as can be seen with this simple experiment:
Keep a paper sheet vertically from the edge, blow hardly over one face only, parallel to the face: the sheet remains quiescent.
There is a high difference in flow velocity betweeen sheet faces, but that velocity difference doesn't produce any pressure difference.

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### baeckmoHydrodynamics

You have added energy to the "flow side", and there is a mixing process with entrained flow coming in from three sides. This mixing will result in a slight pressure recovery downstream.

If you repeat the experiment with two paper sheets at a distance of about three to five cm apart, blowing between them, the pressure change will be obvious. In this case, the inflow from the ends is not strong enough to disturb the process and the more or less one-dimensional energy exchange according to Bernoulli will be seen.

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### daiquiriEngineering and Design

Although the chosen example of a paper sheet is apparently simple and trivial, it is neither simple nor trivial at all. A complete answer could occupy a separate chapter in an book about aerodynamics. It involves concepts like Bernoulli equation, mass equation, viscosity, jet dynamics, mixing layers and aeroelasticity. It is also probably one of the most prominent examples of a misuse of the Bernoulli's equation in the schools around the world - perhaps second only to the infamous explanation of the generation of lift over an airfoil which relates the flow path lengths over the upper and lower surfaces of the airfoil.
however, it is 2 a.m. here and I really have no chance to write much about it right now, but will try to give you a couple of hints on it tomorrow.
Cheers!

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### philSweetSenior Member

Don't get too upset, getting down to the root of the problem isn't easy. Einstein famously flubbed the whole thing when he designed an airplane that didn't work.

In order to get between two points in a fluid flow, you can travel along the streamlines for a while, then jump across some stream lines to get to the second point. The idea here is that if you jump perpendicular to the stream lines when you jump, the two equations that cover P vs V along and across are partly independent of each other. The streamwise equation depends only on the acceleration along the streamline and the other one depends only on the acceleration normal to the streamline (ie a curved streamline). Furthermore, since each point in the fluid has exactly one pressure and velocity at a given instant, it doesn't matter - it can't matter - which path we take to get from one to the other. So what we are looking for is two equations, one along the stream line which relates the pressure to the velocity (and acceleration along a streamline), and the second which relates the pressure jump across a steam line to velocity (and acceleration across the streamline), such that it doesn't matter what path we take getting from any point to any other point. Bernoulli's equation turns out to be the right thing for the streamline equation.

Another effect you may be confusing with the Bernoulli equation is called the Coanda effect. If you blow down a little away from the paper, it will move towards the jet. This has to do with the jet entraining air differently on the farside and paper side of the jet, and the jet bends a little towards the paper. If you blow right down the edge of a flat paper, there should be very little deflection.

Think about it - if you had a curled piece of paper - curled towards the jet - and you blew down on it - it isn't going to curl more, is it? Of course not, it's going to straighten out. It's all governed by the curvature of the surface (and the curved streamlines that it creates). Flat piece of paper - no movement. Bernoulli's equation just keeps the peace by ensuring that the integral of the acceleration along the streamline from far upstream to a point near the paper causes a pressure that is exactly the same as the integral of the pressure jumps across the curved streamlines from the far away to the point near the paper.

There is a tree native to south Florida called a Sea Grape and it sheds large, circular, perfectly flat leaves. We had the biggest, baddest leaf blower made and you couldn't get those things to budge. They just sat still on the grass or sand while you hit them with 100 mph jets.

<I've changed the bit about the coanda effect.>

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### jehardimanSenior Member

Actually the easiest way to think about it is this.

Bernoulli's equation is about the conservation of energy and where you draw the boundary. Basically Bernoulli's equation says that in the absence of other energy input or removal, then the change in kinetic energy has to be made up is a corresponding change in potential energy (i.e. pressure...note that in actual practice other things are involved like internal energy, etc.). In the case of the horizontal sheet of paper, the paper forms a boundary between the mass flow above and the mass flow below, there the differential potential energy works against the paper. In the vertical case, both sides of the paper have the same potential differential in the horizontal plane between the potential energies, causing action parallel to the surface, not perpendicular to it, i.e. flow goes around the surface with no change in the potential.

Baeckmo's example is a case where that that flow is interrupted, i.e. the flow of energy cannot go around, because of the other piece of paper interrupts (or you could say forms a counter flow) which prevent the horizontal transfer of energy around the first piece of paper. This forms the basis of circulation theory's need for the Kutta Condition which alleviates the need for Navier-Stokes analysis of lift.

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### philSweetSenior Member

Baeckmo, I tried, I really did. But I can't see anything. If the sheets are crossways, both eyes are above the top sheet. If the sheets are vertical one eye is on each side.

What am I supposed to see?

Is this effect truly Bernoulli, meaning it would still be there in an ideal fluid flow?

At 3-5 cm separation, the jet is much smaller than the gap. I think that any gap contraction is dominated by the Coanda effect, not Bernoulli.

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### quequenSenior Member

I found a variation of the Baeckmo experiment that works pretty clearly: arrange the 2 sheets as showed on the image, blow into the hole: both sheets collapse to the inside showing low pressure into the hole.

#### Attached Files:

• ###### 2sheets.jpg
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### philSweetSenior Member

Quequen, it is not the jet that is producing the low pressure directly. What is happening is the jet is small compared to the opening. The jet has some mass flow rate. As the jet passes down the slot, it entrains more air and the exiting stream has a higher mass flow than the entering jet. That additional mass flow has to be made up from unentrained room air entering from the top.

So some room air starts from rest and accelerates along its path towards the top of the papers and it curves to align itself with the neighboring flow. You will find that the unentrained room air is causing most of the pressure differential on the paper. The jet probably doesn't even touch the paper, so it has to be the unentrained air.

Baeckmo's way is a simpler and somewhat easier to fathom. Same principle, but the makeup air is induced through the sides, and since it has accelerated, it has a lower pressure compared to the static air on the outside of the sheet.

These are both very round-about ways to explore Bernoulli's equation in terms of cause and effect. In an ideal fluid, there would be no entrainment of air into the jet, and the demonstration wouldn't work. That's a problem because Bernoulli's equation still holds in an ideal fluid. Understanding what's going on here requires an understanding of fluid mixing that goes way beyond the basics of Pressure-Velocity relations.

If you shot stream of pencils down through the slot, what do you think would happen?

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### quequenSenior Member

philsweet, you are right, the sheets collapse only if the jet leaves room to incomming air at the entrance. Off course, if you close de gap between the jet and the sheets at the entrance, there will be no collapse at all. So may be this experiment involves an even more complex phenomena than the first one.

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### quequenSenior Member

Some schollar experiments on Bernoulli's, Coanda's and Venturi's principles found in youtube (just seeing the movies and not paying too much attention to the explanations):
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This one is related to the experiment in post #7:

Venturi machine:

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### ImaginaryNumberImaginary Member

Instructions

Push the pin through the center of the light cardboard square and insert the free end of the pin into the empty hole in the spool. The function of the pin is just to keep the cardboard square centered on the end of the spool. Try to push the cardboard off the by blowing air through the hole in the spool. The harder you try the stronger the Bernoulli force created that holds it in place. With practice you can actually prevent the cardboard from falling out, in an upside down orientation, until you stop creating a Bernoulli effect with your breath.

http://www.physics.umn.edu/outreach/pforce/circus/Bernoulli.html

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### daiquiriEngineering and Design

Attaching a video shot last weekend during a trip to Genoa...

#### Attached Files:

• ###### Bernoulli balloon.zip
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2.8 MB
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### philSweetSenior Member

Thanks a lot Daiquiri, I thought I had a couple of free hours until a pot luck dinner.

The balloon is a puzzler. Getting the thing stable in all five degrees seems improbable.

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### daiquiriEngineering and Design

It was just one of several others in that fountain, behaving in a similar manner. I don't see why it shouldn't be stable. After all, the total energy of the system balloon+water has an absolute minimum in just one point, I guess. That balloon has managed to find his place in the world.
I would say that it has just two degrees of freedom with respect to the water outlet mouth. It can go left-right or towards-from the mouth. The vertical position is the one mutually agreed by Mr. Archimedes and Mr Bernoulli. The rest is governed by Mr. Bernoulli and Mr. D'Alembert through the equilibrium of pressure and inertial forces. Since the balloon has virtually no mass, it reacts promptly to any change in pressures due to external disturbances, immediately getting sucked back into a stable position.
Have a nice potluck dinner! I usually opt for preparing a Russian salad. And it is usually one of the first pots which remains empty during the dinner.
Cheers

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### philSweetSenior Member

Granted, the z position is fixed for all intents, but the body of the balloon can pitch and yaw (but not roll, because it is axisymmetric). So four perturbations have to be considered for stability.

I have a certain reputation with brownies. At pot-lucks , the host just says "everyone sign up to bring a side dish, and Phil will bring the brownies."

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