Q and R

Q and R are two invariants of the velocity gradient tensor. Following (mostly) the notation used in Davidson, these are defined as:

${\begin{aligned}Q&={\frac {1}{2}}\left(W_{ij}W_{ij}-S_{ij}S_{ij}\right)&&=-{\frac {1}{2}}A_{ij}A_{ji}\\R&={\frac {1}{3}}\left(S_{ij}S_{jk}S_{ki}+{\frac {3}{4}}\omega _{i}\omega _{j}S_{ij}\right)&&={\frac {1}{3}}A_{ij}A_{jk}A_{ki}=\det(A_{ij})\end{aligned}}$

where

${\begin{aligned}A_{ij}&={\frac {\partial u_{i}}{\partial x_{j}}},&&{\text{is the velocity gradient tensor}}\\S_{ij}&={\frac {1}{2}}(A_{ij}+A_{ji}),&&{\text{is the strain rate tensor}}\\W_{ij}&={\frac {1}{2}}(A_{ij}-A_{ji}),&&{\text{is half the rotation tensor}}\end{aligned}}$

and $\omega _{i}$ is the i-th component of vorticity. Note, most authors define R with a negative as $-{\frac {1}{3}}A_{ij}A_{jk}A_{ki}$

By the definitions of viscous dissipation, $\epsilon$, and enstrophy, $\Omega$, and using the relation between $W_{ij}$ and the rotation tensor, we can write

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

which is a very useful formulation for computing Q.

Where Q is large and positive the flow has intense enstrophy, whereas large negative Q is a region of strong strain. The precise meaning of large is not yet fully known.

The value of R specifies regions of vortex stretching (positive R) and compression (negative R). Combined, R and Q will specify the type of vortex stretching or strain.

See Davidson for more information.