What is the meaning of vorticity and circulation?

Question

In fluid kinematics I can't understand the meaning of these terms : vorticity and circulation.
Can somebody give me a description of these terms so that a lay person can understand them easily?

Answer 1

There are fundamental types of motion (or deformation) for a fluid element: translation, rotation, linear strain and shear strain. Usually all these types of motion occur concurrently which makes the analysis of fluid dynamics somehow difficult.

One can express the rate of translation vector mathematically by the velocity vector $\vec{V}$: $$\vec{V} = u\vec{i} + v\vec{j} + w\vec{k}$$

When it comes to expressing the rate of rotation of a fluid element it becomes quite challenging, Why? because a fluid element translates and deforms as it rotates, imagine an initially rectangular fluid element that starts to rotate while each line of the rectangule having a different angular velocity than the other. You can check White's book for the complete derivation but we can express the rotation vector $\vec{\omega}$ for now as follows:

$$\vec{\omega} = \frac{1}{2} [(\frac{\partial w}{\partial y} - \frac{\partial v}{\partial z})\vec{i} + \frac{1}{2} (\frac{\partial u}{\partial z} - \frac{\partial w}{\partial x})\vec{j} + \frac{1}{2} (\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y})\vec{k}]$$

Putting it simply as half the curl of the velocity vector (just a mathematical manipulation): $$\vec{\omega} = \frac{1}{2}\vec{\nabla}\times \vec{V}$$

Now, let us define a vector which is called Vorticity vector which is twice the angular velocity (we just got rid of the silly $\frac{1}{2}$) and we might call it $\xi $: $$\vec{\xi} = \vec{\nabla}\times \vec{V}$$

Okay, enough with the math. What does it mean?

For an arbitary point in a flow field:

• Any fluid element (particle) that occupy that point having a non-zero vorticity, that point is called rotational.
• Vice versa, Any fluid element (particle) that occupy that point having a zero vorticity, that point is called irrotational which means particle is not rotating.

Flow from A to B is rotational (has voriticity) while flow from A to C is irrotational (has no vorticity).

You can find many examples for rotational flows such as in wake regions behind blunt bodies and flow through turbomachines.

According to this lecture:

• Circulation and vorticity are the two primary measures of rotation in a fluid.

• Circulation, which is a scalar integral quantity, is a macroscopic measure of rotation for a finite area of the fluid.

• Vorticity, however, is a vector field that gives a microscopic measure of the rotation at any point in the fluid.