Crocco's theorem

In aerodynamics, Crocco's theorem relates the flow velocity, vorticity, and stagnation pressure (or entropy) of a potential flow. This 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:^[1]

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

However, usually this theorem is connected with the name of Italian scientist Luigi Crocco [it],^[2] 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: ${\displaystyle \mathbf {u} \times {\boldsymbol {\omega }}=v\nabla p_{0}}$ ^[3]
2. Entropy (the following form holds for plane steady flows): ${\displaystyle T{\frac {ds}{dn}}={\frac {dh_{0}}{dn}}+u\omega }$ ^[4]
3. Momentum: ${\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}+\nabla \left({\frac {u^{2}}{2}}+h\right)=\mathbf {u} \times {\boldsymbol {\omega }}+T\nabla s+\mathbf {g} ,}$

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