Vorticity and circulation are very common terms we encounter while studying aerodynamics and fluid dynamics. However, most of the time we fail to understand the physical significance or intuitive meaning of vorticity and circulation. All the mathematical equations you know become useless when you can’t understand the intuitive meaning. I hope this post will help you understand the concepts of vorticity and circulation in a much better way.
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Both vorticity and circulation are associated with rotation in a fluid. So let’s first understand how rotation phenomena occur in a fluid?
The above picture shows an infinitesimal fluid element in a flow field with velocity gradients in x and y directions. Velocity gradient in x direction means the flow velocity changes gradually as you move towards x-axis, and similar for velocity gradient in y direction. One can clearly see that the velocity at point B is higher than that at point A. Similarly, velocity at point C is greater than that at point A.
One can intuitively figure out that the fluid element will rotate due to the varying velocities at different points. After time ᐃt, the fluid element will rotate as shown below;
Angular velocity is the most fundamental measurement of rotation. In this case, angular velocity of the fluid element is given by;
You can find these derivations in any textbooks of Fluid Dynamics. We can now express the rotational vector as follows:
If you notice this equation properly, the term at the right hand side is actually half of the curl of the velocity field.
The curl of the velocity field is the vector quantity which is equal to twice of the angular velocity. That vector quantity is the vorticity that gives us the measure of local rotation of the fluid element.
What does that mathematical equation mean? Well, if a point in the velocity field has non-zero vorticity then the fluid element at that particular point is rotating(fig 2) . However, if the point has zero vorticity then the fluid particle at that point is not rotating (fig 1). Thus, vorticity is the measure of local rotation.
Circulation is another measure of fluid rotation.
If we consider a closed path c around the fluid flow then circulation is defined as the line integral of the tangential velocity around c.
By applying the Stoke’s theorem, circulation can be related to the vorticity as;
This shows that the circulation actually is the surface integral of the all the vorticities of an area bounded by the curve c. In other words, circulation is the flux of vorticity. And conversely, vorticity at a point can be considered as the circulation per unit area.
Physical significance of vorticity and circulation:
Now that we have discussed about mathematical equations and relations between vorticity and circulation, let’s talk about the physical meaning. Vorticity is a vector quantity and gives the measure of local rotation while circulation is a scalar quantity and it gives the measure of global rotation.
Circulation can be actually thought as the ‘push’ that can be felt while moving along a closed path or boundary. For example, take a water in a bucket or tank and then stir it so that the flow becomes like that of a whirlpool. Now, if you keep any object in that flow, the object will experience the push and that is circulation. If we gather all the circulation in an area and calculate it around a single point, it is called Vorticity. (Circulation / area)
In the same whirlpool you created in the bucket, you can try placing a paper boat and then observe the motion. Sometimes, the boat might not rotate but revolve around the center. In this case, the vorticity at that radial distance is zero. However, we cannot say that the circulation of water inside the bucket is zero. You can try placing the paper boat at the center and find it rotating. Even though there is zero vorticity at other points, there is some finite vorticity at the center and that contributes to the circulation. This gives you idea that circulation is the global measure of rotation.
In conclusion, vorticity and circulation are two primary measures of rotation in fluid flow. Circulation is a macroscopic measure of rotation for a given area of the fluid and is a scalar quantity. However, vorticity being a vector quantity is a microscopic measure of rotation for any point in the flow.
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Also check out this Wikipedia link for more insight on circulation and vorticity https://en.wikipedia.org/wiki/Circulation_(fluid_dynamics)