Crocco's theorem

{{Short description|Aerodynamic 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:Friedmann A. An essay on hydrodynamics of compressible fluid (Опыт гидромеханики сжимаемой жидкости), Petrograd, 1922, 516 p., [http://books.e-heritage.ru/book/10073889 reprinted] {{webarchive|url=https://web.archive.org/web/20160303235627/http://books.e-heritage.ru/book/10073889 |date=2016-03-03 }} in 1934 under the editorship of Nikolai Kochin (see the first formula on page 198 of the reprint).

:$\backslash frac\{D\backslash mathbf\; u\}\{Dt\}=T\; \backslash nabla\; s-\backslash nabla\; h$

However, usually this theorem is connected with the name of Italian scientist {{ill|Luigi Crocco|it}},Crocco L. [http://dictionnaire.narod.ru/Crocco-1937.djvu Eine neue Stromfunktion für die Erforschung der Bewegung der Gase mit Rotation]. ZAMM, Vol. 17, Issue 1, pp. 1–7, 1937. DOI: 10.1002/zamm.19370170103. Crocco writes the theorem in the form $\backslash scriptstyle\backslash mathrm\{rot\}\backslash ,\backslash mathbf\; u\backslash times\backslash mathbf\; u=T\backslash mathrm\{grad\}\backslash ,S$ for perfect gas (the last formula on page 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: $\backslash mathbf\; u\; \backslash times\; \backslash boldsymbol\; \backslash omega\; =v\; \backslash nabla\; p\_0$ Shapiro, Ascher H. "National Committee for Fluid Mechanics Films Film Notes for 'Vorticity,'" 1969. Encyclopædia Britannica Educational Corporation, Chicago, Illinois. (retrieved from http://web.mit.edu/hml/ncfmf/09VOR.pdf (5/29/11)
2. Entropy (the following form holds for plane steady flows): $T\; \backslash frac\{ds\}\{dn\}\; =\; \backslash frac\{dh\_0\}\{dn\}\; +u\; \backslash omega$ Liepmann, H. W. and Roshko, A. "Elements of Gasdynamics" 2001. Dover Publications, Mineola, NY (eq. (7.33)).
3. Momentum: $\backslash frac\{\backslash partial\; \backslash mathbf\; u\}\{\backslash partial\; t\}\; +\; \backslash nabla\; \backslash left(\backslash frac\{u^2\}\{2\}\; +\; h\; \backslash right)\; =\; \backslash mathbf\; u\; \backslash times\; \backslash boldsymbol\; \backslash omega\; +\; T\; \backslash nabla\; s\; +\; \backslash mathbf\{g\},$

In the above equations, $\backslash mathbf\; u$ is the flow velocity vector, $\backslash omega$ 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, $\backslash mathbf\{g\}$ 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".