# Applications of Bernoulli's principle

Discussion in 'Highly Technical' started by Biftheunderstudy, Jan 13, 2012.

1. ### Biftheunderstudy Senior member

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I'm giving a short presentation on the Bernoulli principle pretty soon and I'd like to solicit some suggestions from the community.

So far I've determined that I'd like to cover:
Aerofoils, astrophysical jets, and a topic that I recently came across...
That one is a combination of Liouville's theorem and flows in phase space (I can't seem to find much out about this aside from coming across it on a video lecture by Leonard Susskin)

If anyone has any interesting examples that would be great, the crazier the math the better.

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3. ### Gibsons Lifer

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The math might not be crazy, but how about the lifting force on a roof from strong (tornado) winds? You could model different roof shapes.

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4. ### CycloWizard Lifer

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Flushing a toilet.

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5. ### Paperdoc Golden Member

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Widely used in measuring systems. Examples:
Pitot tubes for air speed
Differential pressure measurements across a fixed orifice in a pipeline to measure fluid velocity and hence volume flow rate

"Thermocompressors" - special shaped piping devices that allow you to mix two steam sources of different pressures to yield an intermediate-pressure output, but with no moving parts to maintain

The Dyson "Bladeless Fan" and it claimed "multiplier" effect.

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6. ### dkozloski Diamond Member

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The Coanda effect.

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7. ### Hacp Lifer

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Tank Draining.

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8. ### Biftheunderstudy Senior member

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Thanks for the replies everyone.

I really like the dyson air multiplier as an example, I've got lots of images and some vague explanations (enough that I can explain how the Bernoulli effect works in it). Does anyone have some more detailed explanations? A technical document with air speed measurements or specs would be awesome. Really, it would be nice to be able to calculate the air speed and pressure coming out of the aperture on the device.

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9. ### edcarman Member

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Have you seen this?

It doesn't have too much detail in the way of calculations, but the velocity contour plot and streamline pictures might be useful.

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10. ### eLiu Diamond Member

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I would be *extremely* careful with how you present Bernoulli's principle with respect to airfoils and the generation of lift.

In particular, *in general* saying that the air is moving faster over the top than the bottom increases a net upward pressure force is misleading. For example, with a lifting airfoil, clearly the pressure on the upper surface is less than the pressure on the lower surface. So Bernoulli predicts that the air is moving faster over the top than the bottom. But why is that difference in speed happening? If you say the top surface is longer than the bottom surface, then I offer this: an arbitrarily thin, rigid, flat plate generates lift if it is angled against the incoming flow. The top/bottom surfaces are the same length.

But there is a nice example where that pressure difference/speed difference argument works out nicely. Consider a rotating cylinder in a real (viscous) fluid; say the cylinder rotates clockwise and the flow travels left to right. Due to viscous effects, the cylinder will locally slow down the airflow near it's bottom-most point & locally speed up flow near the top-most point.

In my opinion, the more clear & more obvious description of lift generation is just a simple conservation of momentum. The fluid encounters a solid object and it is deflected (imagine that flat plate again). It took some force to deflect the air which we recognize as lift on the body. It holds together for the cylinder example too; if you look at the flow-field, the deflection caused by the cylinder's rotation is clear.

If the flow you're considering is 1) inviscid, 2) steady, 3) irrotational, and 4) incompressible, then yes Bernoulli's principle can be used to precisely predict lift. Without 4), Bernoulli doesn't exist (Bernoulli is the special case of Euler's Equation [along a streamline] in incompressible flow; [steady] Euler's equation [along a streamline] arises from taking the Navier-Stokes equations and removing viscosity). All real fluids are compressible so 4) is not a realistic assumption, but many real fluids are "imcompresible enough" under the right circumstances for this to be useful. Without 3), Bernoulli is only valid along streamlines. Without 1), Bernoulli isn't valid at all. 2) isn't hard to achieve & in most aerospace applications, 1) implies 3). But removing 1) means you're no longer working in the real world since every real fluid has viscosity.

However in many situations, the effects of viscosity can be localized to a thin "boundary layer" around the body (e.g., airfoil). So consider an "extended" body that includes the original body PLUS all the areas where viscosity is important. Now consider this inviscid, steady, irrotational flow about the new extended body. You can use Bernoulli to estimate the lift generated by this extended body and as long as the extended body is not "too far different" from the original body, the prediction will be usable. Alternatively, as in the rotating cylinder example, Bernoulli can come in handy to motivate why a physical phenomenon occurs (e.g., rotating cylinder generates lift) even if it cannot be used to make precise predictions.

In the aerospace world, pitot-static tubes are a much better application of the value of Bernoulli (here, viscous effects are so minor that they can be neglected). Another classic example is that of the Venturi, a concept used in wind tunnel design.

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11. ### Throckmorton Lifer

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Bernoulli's Principle isn't why planes fly. They fly by redirecting airflow downward. That's why pitch is used to adjust the amount of lift the wings produce.The FAA tests ask a pilot how a wing generates lift, and the pilot has to answer "Bernoulli's Principle". Then when he flies the plane, he modulates lift by angling the wings and flaps with no regard for Bernoulli and his principle. Interesting isn't it?

There is one thing I don't understand though. Every explanation says that Bernoulli accounts for a small portion of lift, then they also say that according to the equations it generates the same lift as redirecting airflow. How can both be true? What exactly are the equations based on? Are they set up specially to match observations?

http://www.grc.nasa.gov/WWW/k-12/airplane/bernnew.html

http://www.johndcook.com/blog/2009/01/26/bathtub-drains-and-airplane-wings/

http://amasci.com/wing/airfoil.html

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Last edited: Jan 18, 2012
12. ### eLiu Diamond Member

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You didn't read my post at all did you

Well, that nasa page has the story exactly right. Didn't read the other links. There are "two ways" to explain lift:
1) Newton's 3rd law (as you indicated, the lifting body changes the direction of airflow; it must have exerted a force on the air to do this)
2) Pressure on the upper wing suface is lower than presure on the lower wing surface. Net upward pressure = lift
In reality these things are one and the same. Let me explain in more detail...

The Navier-Stokes Equations are an excellent model for fluid flow on a very large scale. Any fluid is made up of a huge number of molecules. Tracking the behavior of every molecule individually is unbelievably expensive if we're talking about a region of space that's more than a few microns on each side. But if we're looking at an amount of fluid in say the earth's atmosphere, then it turns out that the microscopic behavior of every molecule doesn't really matter. So we can assume that the fluid is a continuum--instead of being made of a whole bunch of tiny particles, it's one continous blob.

Using that assumption, Newton's laws, and some calculus (well, maybe a lot of calculus), you can derive the Navier-Stokes Equations. In 3D, they are a set of coupled partial differential equations describing the conservation of mass, momentum, and energy. These do a great job describing fluid flow; so much so that the enginerring and scientific community accept them basically as law in the macroscopic setting. Unfortunately, solving the Navier-Stokes equations exactly is an unsolved problem except in some very, very special cases. But we can approximate them reasonably well using (big) computers.

The Euler Equations arise from setting viscosity = 0 in the Navier-Stokes Equations. It's a HUGE simplification but Euler Eqns are still very hard to solve. They govern inviscid flow at the most general level. As I mention in my previous post, the Bernoulli equation is a special case of the euler equations. Bernoulli results from considering the euler equations along a single streamline in incompressible (density=constant; reasonable for most liquids and gasses moving at low speed), steady (not time varying) flow. You can only make comparisons across multiple streamlines if the flow is also irrotational (i'm not going to explain this one but suffice it to say, most aerodynamic flows are). When you have all of these conditions, the flow is called "potential" (same thing mathematically as electric potential fields if you're familiar with that). So that should answer your question about where Bernoulli comes from.

At its core, Bernoulli describes how changes in fluid velocity affect changes in fluid pressure.

Anyway, even without big computers, I can apply the Navier Stokes or Euler equations to make interesting qualitative statements about fluid mechanics. For example, imagine an isolated airfoil in "flight". For simplicity, the fluid is inviscid so drag=0. You observed that it feels a lifting force b/c it deflects the air downard. Ok. Now imagine drawing a box surrounding the airfoil*. Say I measure the momentum of the air crossing each face of my box. If I add up (read: integrate) all the momenta, I'll notice that the momentum changed! Why? B/c the airfoil exerted a force on the air. So this change in momentum will be precisely the lift. That was very qualitative. I can make it a little more quantitative by attempting to evaluate all the terms of the Euler Equations. What I will find is that depending on how I draw the box, I could find that:
Lift1 = change in momentum of air; 1) from above
Lift2 = pressure difference of the top/bottom surface of my airfoil; 2) from above
Lift3 = the average of the above two quantities
Note that the value of Lift is fixed, so 1) and 2) produce the same results, Lift1=Lift2. So Lift3=(Lift1+Lift2)/2 is the same thing too. So momentum or pressure, or both are valid ways of explaining lift. It all depends on your frame of reference.

*Ok I'm cutting corners here. Technically the box cannot contain the airfoil. Mathematically you can get around this issue easily but it's hard to explain without a diagram & hard to explain if you aren't familiar with path integrals. So I'll skip it for now...

SO it's not really that Bernoulli accounts for lift. If I knew the exact velocity distribution around an aircraft in *inviscid* *incompressible* *steady* *irrotational* flow, then I could use Bernoulli (which relates velocity to pressure) to compute what the pressure distribution is. From that, I could compute the lift.

In a viscous flow, if I knew the exact velocity distribution around the "extended body" (described in my previous post), I could use Bernoulli to *approximate* the lift. (This turns out to be a good approximation).

However, in both cases, knowing the exact velocity distribution is really hard. So hard that nobody attempts to compute lift by measuring those things physically. But it often is a reasonable way to compute lift on the computer, where you have solved the euler equations (approximately) and have access to the approximate velocity values at any location.

The issue is that people twist Bernoulli. They say silly things like air travels farther over the top of the airfoil than the bottom, so it has to move faster over the top (patently false reasoning for a true statement). Since it's moving faster, the pressure is lower ("true" by Bernoulli). Hence lift is generated. ("True" in quotes b/c you have to be careful where/how you apply Bernoulli in real physical flows, since they aren't incompressible, steady, invscid, and irrotational. With the right assumptions you can get a reasonable approximation.)

The right reasoning is: we observe that lift is generated; one valid explanation is pressure difference. Bernoulli then tells us that hte air is moving faster on the side with lower pressure than the side with higher pressure. If you knew the pressure at every point, you could use Bernoulli to recover the velocity values. But this has absolutely NOTHING to do with the top surface being "longer" or any crazy crap like that.

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13. ### Biftheunderstudy Senior member

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This discussion of lift from aerofoils is exactly why I wanted to talk about them during the presentation, so many people get it wrong, I wanted to do it right. (Given that the talk is finished and I didn't get yelled at, I think I managed in this regard)

There are a lot of interesting facets to Bernoulli's equation, especially when you consider relaxing some of the assumptions and extending it into regimes where it normally doesn't belong...its remarkably good at it, though maybe its incorrect to call it a Bernoulli equation at that point, its clear its analogous though.

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14. ### alkemyst No Lifer

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rear diffusers in automobiles.

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18. ### Anteaus Platinum Member

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If your looking for an everyday application, look at the multi-stage air compressors.

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