Large-scale hydrodynamic instabilities of periodic helical flows of a given wave number $K$ are investigated using three-dimensional Floquet numerical computations. In the Floquet formalism the unstable field is expanded in modes of different spacial periodicity. This allows us (i) to clearly distinguish large from small scale instabilities and (ii) to study modes of wave number $q$ of arbitrarily large-scale separation $q\ll K$. Different flows are examined including flows that exhibit small-scale turbulence. The growth rate $\sigma$ of the most unstable mode is measured as a function of the scale separation $q/K\ll 1$ and the Reynolds number $\text{Re}$. It is shown that the growth rate follows the scaling $\sigma \propto q$ if an AKA effect [Frisch et al., Physica D: Nonlinear Phenomena 28, 382 (1987)] is present or a negative eddy viscosity scaling $\sigma \propto q^2$ in its absence. This holds both for the $\text{Re}\ll 1$ regime where previously derived asymptotic results are verified but also for $\text{Re}=O\left(1\right)$ that is beyond their range of validity. Furthermore, for values of $\text{Re}$ above a critical value ${\text{Re}}_S^c$ beyond which small-scale instabilities are present, the growth rate becomes independent of $q$ and the energy of the perturbation at large scales decreases with scale separation. The nonlinear behavior of these large-scale instabilities is also examined in the nonlinear regime where the largest scales of the system are found to be the most dominant energetically. These results are interpreted by low-order models.

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• Flow instability