Q and R

Q and R are two invariants of the velocity gradient tensor which are defined as such:

• $Q={\frac {1}{2}}(R_{ij}R_{ij}-S_{ij}S_{ij})=-{\frac {1}{2}}A_{ij}A_{ji}$
• $R={\frac {1}{3}}(S_{ij}S_{jk}S_{ki}+{\frac {3}{4}}\omega _{i}\omega _{j}S_{ij})=-{\frac {1}{3}}A_{ij}A_{jk}A_{ki}$

where ${\begin{aligned}A_{ij}&={\frac {\partial u_{i}}{\partial x_{j}}}\\R_{ij}&={\frac {1}{2}}(A_{ij}-A_{ji})\\S_{ij}&={\frac {1}{2}}(A_{ij}+A_{ji})\\\end{aligned}}$

and $\omega _{i}$ is the i-th component of vorticity.

By the definitions of viscous dissipation and enstrophy, $Q$ can be written as

$Q={\frac {1}{2}}(\Omega -{\frac {\epsilon }{2\mu }})$

Where Q is large and positive the flow has intense enstrophy, whereas large negative Q is a region of strong strain.

See Davidson for more information.