Physics:Crocco's theorem

Short description: Aerodynamic 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:

[math]\displaystyle{ \frac{D\mathbf u}{Dt}=T \nabla\,s-\nabla \,h }[/math]

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: [math]\displaystyle{ \mathbf u \times \boldsymbol \omega =v \nabla p_0 }[/math]
2. Entropy (the following form holds for plane steady flows): [math]\displaystyle{ T \frac{ds}{dn} = \frac{dh_0}{dn} +u \omega }[/math]
3. Momentum: [math]\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}, }[/math]

In the above equations, [math]\displaystyle{ \mathbf u }[/math] is the flow velocity vector, [math]\displaystyle{ \omega }[/math] is the vorticity, [math]\displaystyle{ v }[/math] is the specific volume, [math]\displaystyle{ p_0 }[/math] is the stagnation pressure, [math]\displaystyle{ T }[/math] is temperature, [math]\displaystyle{ s }[/math] is specific entropy, [math]\displaystyle{ h }[/math] is specific enthalpy, [math]\displaystyle{ \mathbf{g} }[/math] is specific body force, and [math]\displaystyle{ n }[/math] is the direction normal to the streamlines. All quantities considered (entropy, enthalpy, and body force) are specific, in the sense of "per unit mass".