Crocco's theorem

Crocco's theorem is an aerodynamic theorem relating the flow velocity, vorticity, and stagnation pressure (or entropy) of a potential flow. Crocco's theorem gives the relation between the thermodynamics and fluid kinematics. The theorem was first enunciated by Alexander Friedmann for the particular case of a perfect gas and published in 1922:

${\frac {D\mathbf {u} }{Dt}}=T\nabla \,s-\nabla \,h$

However, usually this theorem is connected with the name of Italian scientist Luigi Crocco, a son of Gaetano Crocco.

Consider an element of fluid in the flow field subjected to translational and rotational motion: because stagnation pressure loss and entropy generation can be viewed as essentially the same thing, there are three popular forms for writing Crocco's theorem:

1. Stagnation pressure: $\mathbf {u} \times {\boldsymbol {\omega }}=v\nabla p_{0}$
2. Entropy (the following form holds for plane steady flows): $T{\frac {ds}{dn}}={\frac {dh_{0}}{dn}}+u\omega$
3. Momentum: ${\frac {\partial \mathbf {u} }{\partial t}}+\nabla \left({\frac {u^{2}}{2}}+h\right)=\mathbf {u} \times {\boldsymbol {\omega }}+T\nabla s+\mathbf {g} ,$

In the above equations, $\mathbf {u}$ is the flow velocity vector, $\omega$ is the vorticity, $v$ is the specific volume, $p_{0}$ is the stagnation pressure, $T$ is temperature, $s$ is specific entropy, $h$ is specific enthalpy, $\mathbf {g}$ is specific body force, and $n$ is the direction normal to the streamlines. All quantities considered (entropy, enthalpy, and body force) are specific, in the sense of "per unit mass".