# Is circulation real?

Discussion in 'Hydrodynamics and Aerodynamics' started by Mikko Brummer, Jan 25, 2013.

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### tspeerSenior Member

Ah, where to begin?

Bernoulli's relationship is simply the conservation of energy for flow that is irrotational and has no heat added or drawn from it. Under these conditions, you basically only have potential energy, in the form of pressure, and kinetic energy. If you exchange one for the other without causing any losses, then Bernoulli's relationship holds. This is a good approximation for most of the flowfield for low-speed flows.

It doesn't work in the boundary layer, because the shear stresses there cause the fluid to rotate. It doesn't work when the flow goes through a shock wave. It doesn't work in the core of a trailing vortex. It doesn't work in wakes. There are lots and lots of flows where it doesn't work - anywhere the conservation of energy is more complicated than the simple exchange of pressure for velocity.

Applying conservation of energy based on Bernoulli's relationship does't explain lift. Lift is a force, and force is equal to the change in momentum. So you need to look to conservation of momentum if you want to understand how lift comes about.

Where Bernoulli's relationship is very useful is in understanding how the lift translates into the loading on each part of the lifting surface. And pressure is easy to measure compared to measuring local velocities. It is very convenient that the pressure is essentially constant across the boundary layer, despite the large differences in speed within the boundary layer (a good example of where Bernoulli doesn't apply), because that means we can measure the pressure at the surface and it will be the same as the pressure outside the boundary layer, allowing us to use Bernoulli's relationship to deduce the velocities outside the boundary layer, without having to introduce anything that would disturb the flow. So Bernoulli is very handy for engineering purposes. But it doesn't explain the origin of lift.

To see that lift is not a matter getting energy from the flowfield, consider two wingsails, operating in the same uniform wind, having the same area, with the same loading, and similar cross section shapes and planform shape. However, one is of moderate aspect ratio and the other has twice the span of the first. If the first is operating at its best lift/drag ratio, the second will have a drag that is approximately 5/8 the drag of the first. (For the first wingsail, profile drag will be 50% of the total drag and induced drag will be 50%. For the second wing, the induced drag will be a quarter that of the first, and profile drag will be the same.) Same lift, different energy.

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### philSweetSenior Member

daiquiri, I agree with your example, but it doesn't conflict with what I said. I divided the plane across the flow into a region, and all-the-rest-of-it. You divided it into 1 region, a second region, and all-the-rest-of-it. You still have yet to consider all-the-rest-of-it, and there is enough to work with such that we will agree in the end.

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### 65 NJunior Member

As a comment to tspeer's post #136 quoting my post I would like to clarify my purpose.
I agree. But my post was not about the origin of lift, neither was the post I quoted. My purpose was not to explain lift, neither by Bernoulli’s principle nor by energy considerations, merely to suspect (erroneously, I see) if you can take energy from the field (like by sails and windmills) without changing it so much that Bernoulli’s principle is not valid strictly enough along streamlines. Lift was an unnecessary word in my post. It was included in my poorly formulated sentence as a necessary phenomenon when taking energy from flow, not opposite. I don’t suspect the current theories of lift, and I understand the connection between momentum change and the circulation path integral of velocity, but there are always both conservation of momentum and conservation of energy (including all forms of energy).

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### JoakimSenior Member

Yes the basic is diffusion of matter and energy, but these become diffusion of enthalpy or temperature, velocity and concentrations with some assumptions often used in fluid dynamics. These come from the Navier-Stokes equations for continuum. I don't see how one can get diffusion of pressure from them.

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### 65 NJunior Member

I was writing about pressure and diffusion at molecular level. Navier-Stokes equations describe the flow of continuum. Diffusion in gases, liquids and solids is a process usually towards some equilibrium, independent of flow, generally described by Fick’s laws. I consider the terms “diffusion of temperature” and “diffusion of pressure” as good or bad, one describing the change or equilibration of temperature by the diffusion of heat, the other as the change or equilibration of pressure by the diffusion of the kinetic energy of molecules.

Just to see if the expression “diffusion of pressure” is used in scientific literature, I searched it by Google Scholar, and found 277 papers, many of them in peer reviewed well-ranked journals. The only paper I opened (quite randomly from the first page, http://131.155.54.17/mate/pdfs/8488.pdf, in Physics of Fluids) used it in conjunction with N-S treatment. The last sentence in the paper, “Results are based on application of inertial-sublayer asymptotics to velocity statistics in exact expressions for kinetic energy and pressure diffusion derived from Navier-Stokes equations,” happened to be related to the last two sentences of Joakim's post.

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### ErwanSenior Member

Hi Everybody,

Rainy Sunday was perfect to read the 10 pages of this topic. A bit intimidating to post with all these rocket scientists and engineers writing here.

In the mid-90's a friend of mines was convoying a 60 feet trimaran (dont remenber wether it was "Banque Populaire" or "Fudji" or another one.

He reported me a very interesting observation regarding circulation.

Sailing windward at night, with a very thin rain just like a kind of spray, he has been able to observe in the beam of his hand-light, a vertical vortex, behind and parrallel to the leech of the mainsail. Just like the train gear vortex you can find in Fluid dynamics books.

This experience is probably easy to repeat as long as you have thin rain and can sail at night.

Cheers Everybody

EK

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### markdrelaSenior Member

That expression refers to diffusion CAUSED BY pressure fluctuations in a turbulent flow. It does not refer to diffusion of pressure itself. There is no such thing.

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### daiquiriEngineering and Design

I don't have my books on gas dynamics here with me, but I don't think the pressure is even defined at molecular level. It is a macroscopic averaged force per unit area caused by a change of momentum of fluid molecules due to mutually repulsive intermolecular interactions. At a molecular level, where the number of interactions per unit area can vary considerably in both time and space at a very fast rate, it is not called pressure. I just don't remember what is it called instead...

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### 65 NJunior Member

Now this seems to be a question about terminology. For me, the treatment of gases at molecular level (basically same what daquiri writes above) means that the gas is considered consisting of separate molecules which move and collide (more or less) randomly among other molecules, but you can e.g. simulate possible paths of molecules, beyond fluid treatment . So you are neither treating the gas as continuum nor studying a single molecule. Likewise, studying solids at atomic level is not a study of the structure and prorerties of a single atom or the bulk (continuum) material. - In molecular level treatment of gas (defined as above), pressure, temperature, and density are still statistical/averaged variables, connected neither to a single molecule, nor to nanoscale space/time elements.

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### 65 NJunior Member

In the abstract and text the author uses expressions like…diffusion of pressure and kinetic energy…, …lateral diffusion of kinetic energy and pressure...,…diffusion of kinetic energy and pressure, the terms on the left-hand side of Eq…. Within the treatment he gets into “well-known Poisson equation for pressure”, (equivalent to Fick’s second law of diffusion), where pressure is in the Laplace operator term, as the quantity which diffuses, not in the “cause” term (field, potential etc depending on the phenomenon, constants etc). Nowhere is mentioned diffusion caused by pressure and kinetic energy. He uses the obtained pressure gradient terms to treat the redistribution of turbulence production, but not using diffusion treatment… but that is beyond my scope.
For me, it looks like he uses and accepts (like the reviewer(s) and the editor) the expression "diffusion of pressure" (itself), in the same sense as I tried to use it.

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### markdrelaSenior Member

That's simply bad terminology. Or maybe they are using lame analogies.

Diffusion refers to movement of a quantity in response to a gradient of that quantity or a closely-related quantity. e.g.
Heat moves in response to a temperature gradient (if conductivity is finite).
Momentum moves in response to a velocity gradient (if viscosity is finite).

Pressure does not obey any such relation, so it does not diffuse in that sense.

PS
By "X moves" I mean "there is a flux of X through space"

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### 65 NJunior Member

quote: “Diffusion refers to movement of a quantity in response to a gradient of that quantity or a closely-related quantity. e.g. …”

In the article, the energy production from stochastic fluctuations in a turbulent flow is calculated according to statistical fluid dynamics. Different terms present turbulence production, kinetic energy (of molecules) production, dissipation and a viscous term. The space-dependent kinetic energy term causes a kinetic energy and pressure gradient, driving the diffusion of kinetic energy and pressure. This happens through collisional movements of molecules, without net mass flow.

quote: "By "X moves" I mean "there is a flux of X through space" “

There is a flux of kinetic energy (of molecules) and the associated pressure through space. This is described by the usual diffusion treatment.

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### Mikko BrummerSenior Member

Thanks for the hint, David! I ordered the book from Amazon, perfect reading for me... I have always wondered how it is that someone can come even to think of such models as the circulation theory... the book sheds light on this all, the people involved, and shows how it's never an achievement of a single person, but a group of like-minded people drawing from others' achievements. A must-read for anyone interested in the history of aeronautics. Great reading, thank you!

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### Mikko BrummerSenior Member

The book inspired me to look further into the topic of this thread. It would appear that revealing circulation is merely a frame of reference thing: I ran a simple 2D-simulation on an airfoil moving at 5 m/s in still air - looking at the airfoil in the reference frame of an outsider standing on the earth, while the wing would be flying over him. Now, there is no moving fluid, no uniform flow mixing with the circulatory flow, but simply the motion of the wing in still air. What you see is the velocity field created by the moving wing, and it certainly looks like the circulatory motion created by a "bound vortex".

We are so used to looking at the flow field from the pilot's frame of reference, that of someone moving with the wing - which is equivalent to that of a windtunnel experiment, with the wing staying still and the air moving - that the actual flow field around a moving wing in the air (or a keel or rudder in the water) does look quite suprising. The fluid really appears to circulate around the nose of the airfoil, turning back on it, almost piercing through the upper surface if not the wing would be moving forward and just in time out of its way.

The first animation shows velocity vectors, and the second markers moving with fluid.

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### Mikko BrummerSenior Member

In the pilot's or the windtunnel's reference frame, we get the familiar streamlines-look of the flow. There, the circulation created by the wing is mixed with the uniform flow (free stream), masked by it so circulation will not show. In the animation, I have a uniform 5 m/s flow from the right and markers moving with the flow (all animations are slow motion, in reality they only last a second or so).

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