Crocco's set of vertebrae

The vortex law of Crocco (also Crocco's equation ) describes a physical theorem from fluid mechanics established by Luigi Crocco in 1937 . It says that for the steady flow of a frictionless , non- heat-conducting gas , neglecting external forces, the following applies:

${\displaystyle {\vec {v}}\times \operatorname {rot} \,{\vec {v}}=\operatorname {grad} \,h_{0}-T\,\operatorname {grad} \,s}$

Denote herein

• ${\displaystyle {\vec {v}}}$the velocity field of the flow
  • ${\displaystyle \operatorname {rot} \,{\vec {v}}}$the rotation of the velocity field
• ${\displaystyle h_{0}{\dot {=}}h+v^{2}/2}$the enthalpy of rest of the gas
  • ${\displaystyle \operatorname {grad} \,h_{0}}$the gradient of the enthalpy of rest
• T the temperature
• s the entropy ,

where and s are entered as enthalpy and entropy per unit of mass ( specific quantities ). ${\displaystyle h_{0}}$

interpretation

Often one can assume (isoenergetic flow). ${\displaystyle \operatorname {grad} \,h_{0}=0}$

If the flow is also isentropic , then it is also , and Crocco's theorem of vortices follows ${\displaystyle \operatorname {grad} \,s=0}$

${\displaystyle \Rightarrow {\vec {v}}\times \operatorname {rot} \,{\vec {v}}=0}$.

In general it follows from this

${\displaystyle \Rightarrow \operatorname {rot} \,{\vec {v}}=0}$

d. H. the flow is irrotational or irrotational , and there is a potential speed with ${\displaystyle \Phi }$

${\displaystyle {\vec {v}}=\operatorname {grad} \,\Phi }$.

It is then a potential flow .

Crocco's theorem of vortices says that rotation-free flows are isentropic and vice versa, assuming that they are stationary and that friction and external forces are negligible.

literature

• Luigi Crocco: A new current function for exploring the movement of gases with rotation. Z. Angew. Math. Mech., 17 (1), 1937, pp. 1-7. doi : 10.1002 / zamm.19370170103
• Karl Wieghardt: Theoretical fluid mechanics. Universitätsverlag Göttingen, reprint from 1974, ISBN 3-938616-33-4
• Franz Durst: Fundamentals of fluid mechanics. Springer-Verlag Berlin Heidelberg, 2006, ISBN 3-540-31323-0