Q and R

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

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})


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}

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.