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A series of numerical experiments is suggested for the three-dimensionalNavier-Stokes and Euler equations on a periodic domain based on a set of$L^{2m}$-norms of vorticity $\Omega_{m}$ for $m\geq 1$. These are scaled toform the dimensionless sequence $D_{m}=(\varpi_{0}^{-1}\Omega_{m})^{\alpha_{m}}$ where $\varpi_{0}$ is a constantfrequency and $\alpha_{m} = 2m/(4m-3)$. A numerically testable Navier-Stokesregularity criterion comes from comparing the relative magnitudes of $D_{m}$and $D_{m+1}$ while another is furnished by imposing a critical lower bound on$\int_{0}^{t}D_{m}\,d\tau$. The behaviour of the $D_{m}$ is also important inthe Euler case in suggesting a method by which possible singular behaviourmight also be tested.

Dynamics of scaled norms of vorticity for the three-dimensional Navier-Stokes and Euler equations