Crocco's Theorem

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 Gaetano Crocco.

Because stagnation pressure loss and entropy generation can be viewed as essentially the same thing, there are two popular forms for writing Crocco's theorem:

1. Stagnation pressure: $V \times \omega = \frac{1}{\rho} \nabla p_0$ ^[1]
2. Entropy: $T \frac{ds}{dn} = \frac{dh_0}{dn} + ||V||\omega$ ^[2]

In the above equations, V is velocity, ω is vorticity, ρ is density, p₀ is stagnation pressure, T is temperature, s is entropy, and n is the direction normal to the streamlines.

References

1. ^ Shapiro, Ascher H. "National Committee for Fluid Mechanics Films Film Notes for 'Vorticity,'" 1969. Encyclopaedia Britannica Educational Corporation, Chicago, Illinois. (retrieved from http://web.mit.edu/hml/ncfmf/09VOR.pdf (5/29/11)
2. ^ Liepmann, H. W. and Roshko, A. "Elements of Gasdynamics" 2001. Dover Publications, Mineola, NY.