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The Kutta–Joukowski theorem is applicable for 2D lift calculation as soon as the Kutta condition is verified. When this is the case, there is a circulation $\small \Gamma$ around the airfoil. My question is related to this circulation:

  • What is the physical meaning of the circulation $\small \Gamma$, often represented like this (example 1, example 2, example 3):

    enter image description here
    (Own work)

I'm interested in a simple explanation of the circulation (is air moving around the airfoil? for dummies) and how does this circulation relate to a view of the airflow in a wind tunnel, where there is no apparent air flowing clockwise around the airfoil:

enter image description here
Source: Youtube

The rest of this post is a presentation of the circulation theory which motivates my question, and as I understand it, but is not part of the question.


The Kutta condition is linked to stagnation points, the points where air tubes separate to flow on a given side of the airfoil, and where they join again behind the airfoil. The Kutta condition stipulates the latter point coincides with the airfoil trailing edge:

Wikipedia: A body with a sharp trailing edge which is moving through a fluid will create about itself a circulation of sufficient strength to hold the rear stagnation point at the trailing edge.

enter image description here
(Own work)


According to theory, when the airfoil moves through air, the circulation moves the rear stagnation point to the trailing edge, and then maintains it at this position. When at this position, the circulation is finite and can be used to compute lift with Kutta–Joukowski theorem:

Wikipedia: The theorem relates the lift generated by an airfoil to the speed of the airfoil through the fluid, the density of the fluid and the circulation around the airfoil. [...] The lift per unit span $L^{\prime}$ of the airfoil is given by:

$$L^{\prime} = −\rho_{\infty} V_{\infty} \Gamma$$

where $\small \rho_{\infty}$ and $\small V_{\infty}$ are the fluid density and the fluid velocity far upstream of the airfoil, and and $\small \Gamma$ is the circulation defined as the line integral

$$\Gamma = \oint _{C}V \cdot d \mathbf {s}$$

around a closed contour $C$ enclosing the airfoil and followed in the positive (anti-clockwise) direction.

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    $\begingroup$ That video shows an airfoil shedding vortices, but that has little to do with the main question, that may be formulated as follows: Does a speck of dust caught in the midst of that 'circulation', really revolves round the wing...? or is the Kutta-Joukowski circulation just a useful mathematical construction...? $\endgroup$ – xxavier Jan 6 '18 at 12:40
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Circulation of a fluid around an object by itself will produce no lift. The classic example of this is the spinning cylinder with no other airflow. Viscosity will cause the fluid near a cylinder rotating clockwise to circulate in a clockwise direction around the cylinder. If a left to right horizontal flow is introduced there will be a vector sum of the two flows. This results in the stagnation points near 8 o'clock and 4 o'clock (as opposed to the cylinder with no rotation in the left to right flow having the stagnation points at 9 and 3 o'clock.) The net result of this is the Magnus effect where lift is generated in the 12 o'clock direction.

In your first diagram (typical inviscid flow) there is no circulation. The shape of the airfoil in viscid flow causes the trailing stagnation point to move to the trailing edge (second image -- the Kutta condition). This has the same effect on the airflow as the spinning of the cylinder, in that it creates a clockwise circulation about the airfoil.

The line integral describes, for an arbitrary closed contour around the object, the dot product of the fluid flow velocity vector with the vector path moving around the contour. The simplest contour to analyze is created by following flow streamlines above and below the airfoil and connecting them before and after the airfoil with lines perpendicular to the streamlines.

Since the dot product of perpendicular vectors is 0, the integral along perpendicular portions of the contour are 0. The dot product of parallel vectors is just the multiplication of the scalar values, and since the direction of the contour is reversed between the upper and lower streamline the effect is adding one and subtracting the other. Due to differences in lengths and the different flow speeds (Bernoulli...) along the contour, the integral is non-zero. This number represents the net effective circulation about the airfoil (total flow minus the horizontal flow.)

The interesting thing is that if you extend the contour behind the airfoil far enough to enclose the wake of the airfoil from the beginning of movement, the circulation will be zero as the circulation of total wake is the vector opposite of the circulation about the airfoil.

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There is no molecule in the air which actually revolves around the airfoil in the way you would normally think of it. Circulation is a mathematical concept used to explain the motion of air from a frame of reference bound to the wing. It is useful in understanding relative motion above and below the wing.

A similar situation might be a person walking toward the back of a train. The person can walk at 2mph and the train runs at 80mph, so is the person going forward or backward? The answer depends on your frame of reference: backward if you are on the train, forward if you standing by the tracks. Don't even ask about the direction from space.)

The simplest way to think of it is that airfow above the wing is moving faster than that below the wing, which gives the wing its lift. The cause is immaterial. For illustration assume a mach .8 aircraft has mach .88 airflow above its wing and mach .72 below. All molecules move to the trailing edge. If you want to compare these two flows, it is useful to subtract out the aircraft forward speed of .8 leaving mach +0.08 above the wing and -0.08 below, which defines the circulation. The negative speed (forward) below the wing only exists mathematically.

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  • $\begingroup$ Interesting analogy with the train, but I am still not clear on this circulation concept. Because from the relative position of an observer on the wing, the air molecules are still moving from the leading edge to the trailing edge. Can you help me better understand what this means? $\endgroup$ – Michael Hall Jul 29 at 21:17
  • $\begingroup$ @MichaelHall The simplest way to think of it is that airfow above the wing is moving faster than that below the wing, which gives the wing its lift. The cause is immaterial. For illustration assume a mach .8 aircraft has mach .88 airflow above its wing and mach .72 below. All molecules move to the trailing edge as you suppose. If you want to compare these two flows, it is useful to subtract out the aircraft forward speed of .8 leaving mach +0.08 above the wing and -0.08 below, which defines the circulation. The negative speed (forward) below the wing only exists mathematically. $\endgroup$ – Pilothead Jul 30 at 19:59
  • $\begingroup$ Got it, thanks! $\endgroup$ – Michael Hall Jul 30 at 22:43
  • $\begingroup$ Just because there's net circulation doesn't mean the fluid is rotating around the airfoil. That's not the definition is circulation. $\endgroup$ – Jimmy Jul 31 at 0:26
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I very much like @Gerry's answer. It illustrates the principle of lift through potential theory very well.

I would like to add that circulation doesn't mean that fluid particles are rotating around the airfoil. In fact, even a simple rotating cylinder in an inviscid/irrotational flow would have well-defined streamlines flowing from upstream to downstream.

Rather, the integral definition of circulation in the OP is defined on a closed contour around the velocity vector field, not on a trajectory of any fluid particle. Intuitively, then, circulation illustrates how much a uniform flow has turned.

From Wolfram

Streamlines of rotating cylinder

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  • $\begingroup$ My understanding so far: circulation is a mathematical circulation of a field of vectors around a contour (nothing physical). We can use a holomorphic transform of this to obtain this. Moving the rear stagnation point to the trailing point starts the downwash and creates a transient opposite actual / physical vortex behind, which then disappears (more). But that's no so clear, that's why no answer is yet selected. $\endgroup$ – mins Jul 31 at 9:12
  • $\begingroup$ @mins Do you have a reference on who said these two are holomorphic? The first one is a non-lifting cylinder. As explained by Gerry, the circulation around the cylinder is zero. The second one has a non-zero circulation if you enclose the airfoil with any closed curve. The stagnation points are not the cause for circulation. If you explain what is not clear, maybe we can adjust our answers. $\endgroup$ – Jimmy Jul 31 at 15:09
  • $\begingroup$ I've found several mentions of conformal mapping (e.g. at Nasa), which according this (end of second page) is a holomorphic function. $\endgroup$ – mins Jul 31 at 21:56
  • $\begingroup$ @mins Yes, you can transform the circulation of an airfoil to that of a rotating cylinder. But the first image in your previous comment was a non-rotating cylinder. So there is no mapping between the two. Kutta condition dictates where the stagnation point should be on the airfoil, but itself does not give rise to lift. Circulation does. $\endgroup$ – Jimmy Jul 31 at 22:28
  • $\begingroup$ Ok, I got this image from a page (unfortunately in French), where the cylinder wasn't rotating, but a holomorphic transform was yet mentioned. $\endgroup$ – mins Aug 1 at 6:32

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