Strictly axisymmetric turbulence, i.e. turbulence governed by the Navier-Stokes equations modified such that the flow is invariant in the azimuthal direction, is a system intermediate between two- and three-dimensional turbulence. Recent numerical simulations showed in particular that this system allows for an inverse energy cascade, responsible for the generation of large scale coherent structures, and a direct helicity cascade towards small scales [1, 2]. As predicted from theorical works using statistical mechanics tools [3, 4], different behaviors were obtained for swirling and for non-swirling flows. Let us recall here that in the latter the toroidal (azimuthal) movements are negligible with respect to poloidal (non-azimuthal) ones, while in the former the toroidal velocity is of the same order as its poloidal components. We investigate here the transition from the non-swirling to the swirling regime.

 
The system is investigated by direct numerical simulation. For this, we have adapted a code based on a fully spectral method [5] and allowing to integrate the Navier-Stokes equations in a cylindrical domain. The spectral forcing used to maintain a stationary level of turbulence allows to control separately the energy injection in the toroidal and in the poloidal directions. We investigate the transition from non-swirling to swirling regimes by measuring a swirl indicator γ, defined as the ratio between the toroidal and poloidal energy components, as a function of the ratio of the toroidal/poloidal forcing coefficients. The pure non-swirling case then corresponds to a swirl indicator γ = 0.

Our results show the relationship between this swirl indicator γ and the toroidal/poloidal forcing ratio. The numerical data exhibits a bifurcation behavior between the non-swirling (2D2C, 2 dimensions-2 components) and the swirling (2D3C, 2 dimensions-3 components) regimes. The transition occurs for a forcing ratio close to 1. We then propose a model based on the dynamical equations of the toroidal and poloidal energy components in order to understand this sudden change of flow regime. This model shows an excellent agreement with the numerical results.

References
[1] B. Qu, W. J. T. Bos, and A. Naso. Direct numerical simulation of axisymmetric turbulence. Phys. Rev. Fluids, 2:094608, 2017.
[2] B. Qu, A. Naso, and W. J. T. Bos. Cascades of energy and helicity in axisymmetric turbulence. Phys. Rev. Fluids, 3:014607, 2018.
[3] N. Leprovost, B. Dubrulle, and P.-H. Chavanis. Dynamics and thermodynamics of axisymmetric flows: Theory. Phys. Rev. E, 73(4):046308, 2006.
[4] A. Naso, R. Monchaux, P.H. Chavanis, and B. Dubrulle. Statistical mechanics of Beltrami flows in axisymmetric geometry: Theory reexamined. Phys. Rev. E, 81:066318, 2010.
[5] S. Li, D. Montgomery, and W. B. Jones. Inverse cascades of angular momentum. Journal of plasma physics, 56(03):615-639, 1996.