Occurrence of Irrotational Flows

A question that naturally arises is "Where do we find irrotational flows?". A uniform flow is definitely irrotational. But one hardly finds a uniform flow in nature. Further, there is hardly anything to calculate for a uniform flow.

The other region where we can expect an irrotational flow is away from any solid body. Recall the "Thought Experiment" with two parallel plates (What is a Fluid?)when the space in between is filled with a fluid. Here once the top plate starts moving we have seen that a velocity gradient is set up in the flow normal direction. This gives rise to $\partial u / \partial y$ which contributes directly to vorticity or rotation. As such this flow is NOT irrotational. A similar velocity gradient is set up when a fluid flows past a solid body as shown in Fig.4.15. The velocity right on the body surface is zero and it build up gradually we move in a normal direction away from the body. This region is highly rotational and is called the Boundary Layer. But at some distance form the body this velocity gradient flattens out and the velocity becomes constant in the flow normal direction. This is one of the irrotational regions of flow. As indicated in the figure the flow in the wake of the body is also NOT irrotational.

Figure 4.15: Occurrence of irrotational and rotational regions for flow past a body

Figure 4.16: Occurrence of irrotational and rotational regions for flow through a pipe.

At the entrance to a pipe as shown in Fig.4.16 one has a uniform flow. As the flow enters the pipe, velocity components are forced to be zero on the surface of the pipe. A boundary layer develops and starts to grow. At the beginning one sees a inviscid core encircled by a boundary layer. The flow in the inviscid core is irrotational. However, as we move downstream the boundary layers grow and merge to give a fully developed flow when the entire flow is NOT irrotational.

It is also worth noting that the flow is irrotational wherever Bernoulli equation is valid.

We could foresee from this that an inviscid flow is likely to be irrotational. In fact it is broadly true except in case of High Speed flows where shocks could occur. As indicated in Fig. the region behind a shock in a high speed flow has severe gradients of velocity making $\omega$ not negligible.

Figure 4.17: Rotational flow behind a shock wave in a high speed flow.