Eckhaus-like instability of large scale coherent structures in a fully turbulent von K\'arm\'an flow

The notion of instability of a turbulent flow is introduced in the case of a von K\'arm\'an flow thanks to the monitoring of the spatio-temporal spectrum of the velocity fluctuations, combined with projection onto suitable Beltrami modes. It is shown that the large scale coherent fluctuations of the flow obeys a sequence of Eckhaus instabilities when the Reynolds number $\mathrm{Re}$ is varied from $10^2$ to $10^6$. This sequence results in modulations of increasing azimuthal wavenumber. The basic state is the laminar or time-averaged flow at an arbitrary $\mathrm{Re}$, which is axi-symmetric, i.e. with a $0$ azimuthal wavenumber. Increasing $\mathrm{Re}$ leads to non-axisymmetric modulations with increasing azimuthal wavenumber from $1$ to $3$. These modulations are found to rotate in the azimuthal direction. However no clear rotation frequency can be established until $\mathrm{Re}\approx 4\times 10^3$. Above, they become periodic with an increasing frequency. We finally show that these modulations are connected with the coherent structures of the mixing shear layer. The implication of these findings for the turbulence parametrization is discussed. Especially, they may explain why simple eddy viscosity models are able to capture complex turbulent flow dynamics.