# 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{align} 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{align}}

where {\begin{align} 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{align}}

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.