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".