Crocco's theorem

Remarks

The steady Euler equation is $(u \cdot \nabla) u = -\nabla p / \rho$. The identity $(u \cdot \nabla) u = \nabla(\abs{u}^2/2) - u \times \omega$ and the thermodynamic relation $\nabla p / \rho = \nabla h - T \nabla s$ turn it into the formula.

Example: behind a curved shock wave in a uniform supersonic stream, $h_0$ is still uniform but the entropy jump differs from one streamline to the next. So $\nabla s \ne 0$, and the theorem forces $\omega \ne 0$: a curved shock creates vorticity.

Conversely, if $h_0$ and $s$ are uniform, then $u \times \omega = 0$, so the flow is either irrotational or has its vorticity parallel to the velocity.

Going deeper

For a student

The theorem is the compressible counterpart of the relation $\nabla H = u \times \omega$ behind Bernoulli's principle. In a plane flow the vorticity is perpendicular to the velocity, so uniform $h_0$ and $s$ force $\omega = 0$ wherever $u \ne 0$. This justifies a velocity potential for such flows, and it shows where this fails, for instance downstream of curved shocks.