# Is circulation real?

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

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### daiquiriEngineering and Design

It looks like you're a Lagrangian thinker, while I'm Eulerian. It means that you feel more comfortable observing the fluid flow from the fluid flow's point of view (and hence letting the boat move), while my mind works better if I see the boat fixed and water and air moving around it.

Nothing really wrong about either methods - as you said it, they are equivalent. The data set obtained from the Eulerian reference frame can be transformed to Lagrangian reference frame with just few mathematical transformations. Another consequence of the aerodynamic relativity.

Here I disagree. It will depend on what you want to observe. Though the two points of view are aerodynamically equivalent, sometimes the Eulerian point of view (observer on board) will give you the required informations with less effort than the Lagrangian (observer in the water, or in air). Other times Lagrangian will give data which are easier to analyse.

It is true in other branches of fluid-dynamics too. Check this example which refers to the hydraulic study of rivers: http://sfbay.wr.usgs.gov/watershed/drifter_studies/eul_lagr.html

By the way, when it comes to boats, the observer standing on the shore (your favorite) - who is a mix of Eulerian and Lagrangian - in realty has more troubles than other two. He sees the boat moving in one direction at some speed, the air moving in another direction at another speed, and the water current doing something completely different. From that point of view, it is really hard to extrapolate any useful aero or hydrodynamic data. But he is in a preferable position to evaluate the course of the boat. So, again, the best point of view will depend on what one wants to observe.

Cheers

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### daiquiriEngineering and Design

You feel the air displacement due to a nearby object, followed by a deviation from your intended course. You could have taken the example of a baloon passing near a building, the result would be pretty much the same.

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### Mikko BrummerSenior Member

So the balloon idea was not too good, as the push from the sailboat would deviate it. But lets assume it has so much inertia it will just go on with the wind speed and direction...

Then, if I would stop and anchor the balloon in front of the sailboat, I would be in the flow field of the sails and feel the streamlines flow on my face (the local flow). But since I'm travelling with the wind, the wind would be subracted from the sailboat flow field and what would I feel?? Am I making any sense here?

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### Mikko BrummerSenior Member

Nice!

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### markdrelaSenior Member

I think you're using incorrect terminology here.

"Eulerian" and "Lagrangian" refer to how the flowfield is parameterized. They do not refer to frames of reference. The guy on the sailboat, or the shore, or in a hot-air balloon all typically use Eulerian descriptions of the flow. They could also all use Lagrangian descriptions, but this is normally not done in practice.

For the record...

Lagrangian parameterization uses a grid fixed to the material of interest. This approach is used in freshman physics where we track the position of each point mass as x(t), and in solid mechanics where the displacement dx,dy,dz(t;i,j,k) of each grid point i,j,k painted on the material is tracked during deformation.

In fluids we don't care much what happens to a particular fluid particle or where it goes. Instead we care about what happens in a particular region of
space as the fluid goes through it, like the vicinity of the sail or keel. So we use a coordinate grid fixed in space, not painted on the material. The fluid then moves through the grid. This is the Eulerian description used in fluid mechanics. The Eulerian grid is fixed in space relative to some (any!) observer. The grid can be fixed to the boat, or earth, or airmass. They are all valid Eulerian descriptions.

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### daiquiriEngineering and Design

In strictly mathematical terms, you are correct.

I was referring to a somewhat relaxed usage of the two terms, where Lagrangian point of view on the flowfield is the one which "rides" on the fluid particle and hence has to use the substantial derivatives to describe the change of state of the particle. The dual situation to this one would then be the Eulerian point of view, which stands still and lets the fluid particles pass by. Partial (non-convective) derivatives are then used to describe the change of the particles passing through a given point in space.

Hence, a guy on the boat is an Eulerian observer relative to his fixed reference (which is the boat). Both the air and the water flow past the boat, which is considered a fixed reference frame.

On the other side would be the said guy in the baloon (ideally assumed having a zero mass) could then be considered a Lagrangian observer. He moves with the airflow, and sees no air movement relative to him. He just feels a change in static pressure and temperature.

This is pretty much how a difference between Eulerian and Lagrangean specifications of the flow field is explained in several books on fluid dynamics. Do you think it is not a legitimate way of intending the two terms?

Cheers

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### DCockeySenior Member

My understanding is Mikko is interested in being an Eularian observer traveling at the same constant velocity as the undisturbed wind. Sufficently far away from the boat he would be traveling at essentially the same velocity as the surrounding air. Near the boat the difference between his velocity and the velocity of the surrounding air would be due to the presence of the boat (assuming he does not disturb the air).

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### DCockeySenior Member

In aerodynamics for a body (boat, aircraft, wing, etc) moving at constant velocity relative to the approaching fluid (air, water, etc) it is very common to use an Eularian reference frame with the coordinate system attached to the body. Variation with time is eliminated which can be a considerable simplification as long as the boundary conditions do not change with time relative to the body.

Also very common is to split the flow around into two components. One component is the undisturbed flow which is what the flow would be without the body present. For a body moving at constant velocity relative to the fluid the undisturbed flow and/or a uniform, constant wind it is a simple uniform flow. The other component is the difference of the total flow less the undisturbed flow which which equals the flow disturbance caused by the body. This component is sometimes referred to as the perturbation flow or perturbation velocity field. It is what an observer traveling with the undisturbted flow would see.

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### DCockeySenior Member

The flow past a sail at 1/3 rig height or past a wing a 1/3 semi-span is not 2 Dimensional. In general the velocity will have a component in the span direction, and streamtubes will change width in the span direction. In some situations these deviations from 2D flow may be small enough to be neglected.

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### markdrelaSenior Member

Yes. Like I said, "Eulerian" and "Lagrangian" apply to different types of flowfield parametrizations. They are rather meaningless adjectives when applied to observers.

My guess is that the books you referred to should have said "body frame" rather than "Eulerian", and "airmass frame" rather than "Lagrangian".

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### daiquiriEngineering and Design

Thanks, I'll take note of this.

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### markdrelaSenior Member

FWIW, the attached figure compares the alternative Lagrangian and Eulerian parametrizations. Both sketches have the same flowfield and also the same frame of reference, so the velocities in the two descriptions are numerically equal at the same x,y,z,t event point.

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• ###### LEcomp.pdf
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### tspeerSenior Member

Circulation is basically a bookkeeping tool, used to distinguish the contribution to the flowfield from lift vs those of thickness and camber. But it's not a property of the air in the same sense as the mass or linear momentum.

It is definitely real in so far as when you subtract the freestream velocity from the local velocities, the result is a circulating flowfield. This is true whether you are talking about making actual measurements, looking at CFD results, or using a boundary element method like a panel code.

I don't think diffusion is relevant to explaining circulation because diffusion is ignored in the physical approximations in which circulation plays its most explicit role. Diffusion is what causes the boundary layer.

The Kutta condition is really a substitute for neglecting the boundary layer. The boundary layer determines the conditions at the trailing edge, which in turn determine the value of the circulation.

I think that if you want to understand where circulation comes from, you could just as easily point to conservation of mass. Conservation of momentum says that the wake leaving the trailing edge has to be deflected to windward because the lift force comes from the change in direction of the flow. Conservation of mass says the fluid that has been deflected sideways has to come from somewhere and it has to be replaced by other fluid.

If it was a pure sink, you'd be getting fluid flowing in from all directions, but it's the whole wake that is is being displaced. So there is really only one option in 2D flow, and that is for the displaced fluid to push the fluid on the pressure side out and around the leading edge, and for the fluid on the suction side to be drawn into area vacated by the deflected fluid in the wake. The result is circulation.

If you consider 3D flow, the same mechanism is responsible for the formation of the trailing vortices. They are needed to adjust to the displacement of the wake, just like the vortices that form at the edges of a canoe paddle drawn through the water. Just like the way the vortices wrap around the tip of the canoe paddle, forming a U, it should be no surprise that the trailing vortices and the circulation around the lifting surface also form a U, whose strength is proportional to the lift being generated.

I don't see diffusion playing much of a role in any of this. Diffusion is very weak where there's little shear and the flow is essentially irrotational. The inertia and pressure forces acting on each blob of the flowfield are much more powerful.

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### Mikko BrummerSenior Member

I didn't mean 2-dimensional, there's no way to escape the induced angle of the finite span. Rather I was referring to a level where the spanwise circulation distribution is at the maximum (does this happen at the mean aero-chord MAC?), where the spanwise flow is at the minimum and the flow as horisontal as possible on both sides of the sail... for a triangular sail this appears to be at about 1/3rd up from the foot, like the second level in the Europe illustration attached. But this is quite inconsequential to my mind experiment.

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• ###### EuropeStreamSurfs.jpg
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### Mikko BrummerSenior Member

Perturbation velocity does ring a bell from a long distant fluid mechanics course. Would this perturbation velocity equal to circulation in that point? Does it make sense/is it by any means correct to talk about circulation in a point? How does ciculation relate to vorticity? As I recall, circulation is sometimes defined as the sum of vorticity in a closed area.

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