B How Do You Prove This Vector Calculus Identity from Turbulence?

[FluidStu]

The following identity is found in a book on Turbulence:

Can someone provide a proof of this identity? It isn't listed in the list of vector calculus identities on Wiki.

Thanks

 

[RUber]

Check here:
http://www-pord.ucsd.edu/~rsalmon/chap4.pdf
Equations 1.2 to 1.4

 

[Charles Link]

Apparently $\omega$ is defined as $\omega=\nabla \times u$. This one is actually a variation of a well-known E&M vector identity (i.e. a vector calculus identity used quite often in E&M coursework): $\nabla (a \cdot b)=a \cdot \nabla b +b \cdot \nabla a + \ a \times \nabla \times b + \ b \times \nabla \times a$ $\$ where $a=b=u$. (This one is found on the cover of J.D. Jackson's E&M textbook as well as other E&M textbooks.)