Mattig formula

Crocco's theorem is a fluid dynamics theorem relating the velocity, vorticity, and stagnation pressure (or entropy) of a potential flow. The theorem was first written by Italian scientist Luigi Crocco, a son of Gaetano Crocco.

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: $V\times \omega =v\mathrm{\nabla }{p}_0$
2. Entropy: $T\frac{ds}{dn}=\frac{d{h}_0}{dn}+||V||\omega$
3. Momentum: $\frac{\partial V}{\partial t}+\mathrm{\nabla }(\frac{V^2}{2}+h)=V\times 𝝎+T\mathrm{\nabla }s+𝐟,$

In the above equations, $V$ is the velocity vector, $\omega$ is vorticity, $v$ is specific volume, $p_0$ is stagnation pressure, $T$ is temperature, $s$ is specific entropy, $h$ is specific enthalpy, $𝐟$ is specific body force, and $n$ is the direction normal to the streamlines. All specific quantities (entropy, enthalpy, and body force) are specific per unit mass.

References

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