Is circulation real?

Indeed. But that is where the person using an analogy states from the outset, its limitations..and when the analogy outlives its usefulness, explains to the recipient, where and why.

Thus, an analogy is just a tool, but like all tools, one alone cannot do the job of everything.

 

David,

Sincerest thanks for taking the time to try to explain, I really do appreciate it. I suspect my confusion lies somewhere in the definition of a streamline, I need to read that wiki page slowly. I do not understand how the streamlines in those videos are growing. Surely streamlines are omnipresent (and constant in a steady state airfoil scenario) and shouldn't 'grow' out of the foil like they are shown to do? Daiquiri's V-Vx is exactly what I (previously) thought was the case and what I thought you were trying to explain isn't actually the case.

 

I guess this is the reply to PI Design's doubts expressed in the post #96.

Correct when you say that the flow generally (generally!) does not travel "forward" relative to the airfoil. In fact, as we have seen, the circular flow in the past videos was relative to the undisurbed flow. So, where the circular motion implies that there is a forward component, it should be intended as a forward motion relative to particles of the undisturbed uniform flow. Since the freestream velocity is mostly larger than the x-component of the perturbation circular velocity, the "forward" motion translates into an overall slowing down of the particles on the ventral side of the airfoil.

The vertical motion, however, is visible in the video. Check the motion of the lowest blue dots at the time 0:02 to 0:03.

Also, check what the lowest blue dot does at 0:04. It moves very slightly forward, and than around the leading edge. So, in that particular region, at this particular angle of attack, the perturbation velocity is at least equal or actually higher than the freestream velocity, forcing the particles into an effective forward motion. At a larger AoA this behavior would become even more pronounced.

Cheers

 

But that's the point, isn't it? To generate lift there is no requirement for the air flow on the ventral side to have slowed down relative to the free stream. So the circulation model (or my understanding of it) seems to breakdown at this point.

 

I assume by "ventral side" you mean the lower side of the airfoil.

My guess is you are overlooking the contribution of thickness to the flow around an airfoil.

Let's consider an airfoil with thickness but no camber (for simplicitiy) at zero angle of attack. Since the airfoil is symmetric and at zero angle of attack the flow around the airfoil is symmetric and the circulation is zero. At the stagnation point on the nose the total velocity is zero and the perturbation velocity is the negative of the freestream velocity. Moving rearward along the airfoil's surface the flow accelerates. Around the middle of the airfoil the total velocity will be greater than freestream and the perturbation velocity there will be positive. I'll repeat, the total velocity around mid-chord will be greater than freestream and the perturbation velocity will be positive. Note that the perturbation velocity in this case is due to thickness since there is only circulation.

Now take the same airfoil and change the angle of attack so that it is a small positive amount. The flow around the airfoil will change such that lift is generated and there is a positive circulation. The flow over upper surface witll speed up and the flow over the the lower surface will slow down (except right at the nose due to the stagnation point shifting down). If the amount that the flow over the lower surface slows around mid-chord is less than the previously discussed increase in velocity over freestream due to thickness, the net total velocity around around mid-chord will still be greater than freestream and the perturbation velocity around mid-chord will be positive.

The perturbation (V-Vx) velocity is due to thickness as well as circulation. Overlooking the effects of thickness leads to confusion.

 

The streamlines in the animation are shown as growing out of the airfoil because the person who made the video thought he was generating particle paths instead (per his recent comments on YouTube). It appears that he overlooked in his calculations the fact that the airfoil is moving in the reference frame he used and thus the flow field changes with time relative to that reference frame. Net result was he created streamlines, not particle pathlines.

Here is a simple analogy of how the choice of reference frame affects whether something is constant in time or changes with time. You are riding in a car which is next to a truck. Both the car and the truck are traveling at the same speed. The position of the truck does not change with time relative to you; ie it is steady. I'm standing by the side of the road as the truck goes past. The position of the truck as I see it changes with time; ie it is unsteady.

Neglecting the change from "steady" to "unsteady" due to the change of reference frame is a very easy mistake to make. I've made it and I've seen very experience experts make it.

 

Thanks for that, that makes a lot of sense. Still not sure what that means for circulation in the case of a flat plate, but I think I've had more than my fair share of forum brainpower, so I'll slink away to think about it.
'Ventral' was daiquiri's term. It was new on me, but I looked it up and liked it, so decided to keep it! Learning English of a non-native speaker, eh? Jeesh!

 

For a flat plate at an angle of attack the velocity on the ventral side will be less than freestream except near the nose. Nothing says the velocity on the ventral side has to be greater than freestream, just that depending on thickness and angle of attack it can be greater than freestream.

 

I realise that is the case, but the way I had thought that circulation tries to explain things means that flow on the ventral side could NEVER be greater than the free stream velocity, which is clearly wrong. Which is why I am/was struggling to understand circulation.

At first glance, circulation seems to force the flow on the underside to be affected by an equal and oppsite amount to the flow on the top side, which isn't true.

 

Check again the equation in the post #4. All it tells you is that the circulation is calculated by integrating the velocity field along a closed line contained in the fluid domain.

For example, the closed line could be a rectangle with dimensions 1x1 m.
If the velocity distribution over it's sides was like:

V = 2 m/s on the upper side of the rectangle
V = 1 m/s on the lower side of the rectangle
V = 0 m/s on the remaining two sides​

The circulation would be
Gamma = 2 m/s * 1 m - 1 m/s * 1 m = 1 m^2 / s

The same value of circulation could be obtained if the speed distribution over the rectangle was like:

V = 1 m/s on the upper side of the rectangle
V = 0 m/s everywhere else.​

So, the circulation itself doesn't care or know what is the distribution of velocity along the closed line used for it's calculation. It is just a number obtained by a mathematical integration. I have chosen two random velocity distributions and got the same value of circulation.
In that sense, there is nothing like "the circulation model" which should give you the velocity distribution around an airfoil, and which you should struggle to comprehend.

Hence, the actual distribution of the velocity does not depend on the circulation. The actual velocity field (i.e. the value of the flow velocity and how will the velocity vectors be oriented in space) will depend on other physical conditions which the flow has to satisfy, like the boundary and Kutta conditions, and the Laplace equation - if the flow is incompressible and irrotational. Once the flow field has been calculated which satisfies these conditions, the value of circulation is obtained. Not the other way round.

Cheers

 

LOL

"Ventral" is one of terms used to indicate the pressure (or lower) side of an airfoil which generates a positive lift. By similar animal analogy, the suction (or upper) side is also called "dorsal". A rather informal but often used airfoil nomenclature.

 

Thanks a heap for that guys. I think my mistake was that because you can calculate the circulation from the flow field, I assumed you could caluculate the flow field from the circulation. But you can't. I need to read up.

The brains trust on here is second to none. Thanks again.

 

But it is true. It is strictly true for steady state flow. If you reduce the flow rate through one area, such as beneath the foil, an equal increase must occur in the remaining region- Conservation of mass. If no work is being done, the velocity distribution has to be such that energy is conserved as well.

 

Crumbs. I think my head has circulation goin on!
Surely that implies that if the few on top is being sped up, then the flow underneath is being slowed down?
I'm off to do some reading, before I look any stupider...

 

No no, hold on. That is not generally true. Yes, the flow beneath the foil will be slowed down and the one above it will be sped up, but the speed difference is not simply equal and opposite.

For example, consider the flow over this airfoil:

It shows an airfoil at an angle of attack, and three streamlines (black curved lines):
- one which passes over the dorsal (suction) side of the foil
- one which passes through the stagnation point
- one which passes below the ventral (pressure) side of the foil.
The red and blue areas bounded by two adjacent streamlines are called stream tubes. Since velocity vectors are always tangential to the streamlines, it means all the mass that enters a stream tube on the left will remain contained within it all the way to the far right (where the flow exits the stream tube). In other words, the fluid mass is conserved inside a stream tube.

Now, let's assume that the flow is steady and incompressible (water, for example) and that the points on the extreme left and extreme right sides of the stream tubes are so far away from the airfoil, that the speed there is horizontal and equal to the freestream velocity Vx.
Let's also assume that the red and blue stream tubes have equal width at these extreme left and extreme right points (equal distance between streamlines), so that the same mass of the fluid enters both the red and the blue stream tube.

We know that the foil produces lift, hence the distribution of velocity must be such that the average fluid speed in the upper stream tube has to be higher than the average fluid speed in the lower stream tube. It stems from the Bernoulli's principle, which is valid along any single streamline originated far upstream, and it also stems from the Kutta-Joukowski theorem about relationship between the lift force and the circulation.

But the quantities ΔV1 and ΔV2 in the picture are not necessarily equal and opposite, because the cross-section areas of the stream tubes above and below the airfoil are different and generally not related to each other by a strict geometrical relationship necessary to obtain equal and opposite ΔV. There is also a notable fact that width of stream tubes will vary along the airfoil. In general. for an airfoil which produces a positive lift, the stream tubes will be very thin and densely packed above the airfoil, and much wider below the airfoil.

The only (theoretical) case where ΔV1 = ΔV2 is the thin vortex sheet, which could be an idealization of a sail with no separation (and no mast). In this case the velocity increase right above the vortex sheet equals the velocity decrease of the flow below the sheet:


The difference between a simple vortex sheet and a real airfoil, as DCockey has noted, lies in the thickness.

A numerical example on a real airfoil might be even more helpful. Check this plot of the chord-wise variation of the pressure coefficient around the NACA 0015 airfoil, at 5° angle of attack:


The pressure coefficient is related to the fluid velocity just outside of the boundary layer by the following relationship:

Cp = 1 - (ΔV/Vx)^2​

where ΔV and Vx have the meaning seen before.
The same equation can be used to calculate the increase (or decrease) of fluid velocity ΔV/Vx if the Cp is known:

ΔV/Vx = sq.root(1-Cp) - 1​

Now watch, in the above Cp graph, the two points on the airfoil chord indicated with 1 and 2.

At the point 1:
- on the pressure side of the airfoil, the Cp is zero. Hence, the flow velocity has the same value as the Vx. No increase or decrease.
- on the suction side of the airfoil, the Cp is roughly -1.0. Hence, ΔV = 0.41 Vx (Velocity in this point is around 41% higher than the freestream velocity)
Conclusion: at the point 1, ΔV on the pressure and suction sides are not equal and opposite.

At the point 2:
- on the pressure side of the airfoil, the Cp is again nearly equal to zero. Hence, the flow velocity again has nearly the same value as the Vx. No increase or decrease.
- on the suction side of the airfoil, the Cp is roughly -0.5 Hence, ΔV = 0.22 Vx (Velocity in this point is around 22% higher than the freestream velocity)
Conclusion: at the point 2, ΔV on the pressure and suction sides are again not equal and opposite.

Cheers