Curved shocks and vorticity generation

Question

In fluid dynamics, it is known that curved shocks (which are common, e.g., in bowshocks) generate vorticity. I am looking for a quantitative derivation.

Answer 1

What you are looking for is called Crocco's theorem. For a steady inviscid flow it is possible to write the momentum equation as

$$\mathbf{u} \times \mathbf{\omega} = \nabla h_0 - T \nabla s \,.$$

This equation shows that changes in the total enthalpy or entropy can result in rotational flow. Typically it is assumed that the total enthalpy across a shock is constant, so $\nabla h_0 = 0$ everywhere, in front of and behind the shock.

This is not true when considering the entropy. In front of the shock the entropy is uniform in space, and inside the shock the entropy jumps up, but behind the shock the entropy is no longer uniform when the shock is curved. This is because a curved shock causes a different jump in entropy at different locations, since the shock strength varies. For example, a streamline going through a stronger portion of the shock has a larger jump in entropy than a streamline going through a weaker portion of the shock. Therefore, for a curved shock the entropy is not uniform behind the shock (the entropy gradient $\nabla s \ne 0$).

This observation demonstrates why the flow behind a curved shock is rotational. Since the entropy is not uniform behind a curved shock, Crocco's theorem states that $\mathbf{u} \times \mathbf{\omega} \ne 0$. In practice this means that the vorticity is not zero behind a curved shock ($\mathbf{\omega} \ne 0$).

For more discussion I suggest reading pages 254 to 256 of John Anderson's "Modern Compressible Flow", which discusses this topic in more detail.