Quasi-2D Geophysical or engineering flows see sometimes important changes in their structure leading to oscillations between very distinct solutions. These phenomena can be interpreted as phase transitions between different equilibrium states which become meta-stable. The atmosphere in the Northern Hemisphere (Corvellec [1]) displays either 'zonal' or 'blocked' behaviours, having a huge impact on the weather. Similarly in rod bundle flows inside nuclear cores, the transverse mean flow can experience sudden changes of direction which are difficult to predict (Bieder and al. [2]).

In order to understand this phenomenon, we use a theory based on statistical mechanics proposed by [3], [4] and [5]. Equilibrium states of the 2D Euler equations can be computed from a variational problem consisting in maximizing an entropy function (related to enstrophy) while conserving kinetic energy and circulation inside the domain. We obtain the most probable equilibrium states depending on control parameters and geometry here restricted to the representative configuration of a ring-shaped domain.

We have solved numerically this problem and obtained the different caloric curves and phase diagrams. A bifurcation between 1-eddy solution ('zonal') and 2-eddy solution ('blocked') has been identified, confirming the existence of meta-stable states in flows containing a central obstacle.

[1] : M. Corvellec, Transitions de phase en turbulence bidimensionnelle et géophysique, Thèse de doctorat, Université de Lyon, 2012.
[2] : U. Bieder, F. Falk and G. fauchet, LES analysis of the flow in a simplifieded PWR assembly with mixing grid, Progress in Nuclear Energy, Volume 75, August 2014, Pages 15-24, ISSN 0149-1970.
[3] : J. Miller, Statistical mechanics of Euler equations in two dimensions, Phys. Rev. Lett., 65(17):2137-2140, 1990.
[4] : R. Robert and J. Sommeria, Statistical equilibrium states for two-dimensional flows, J. Fluid Mech., 229:291-310.
[5] : A. Naso, P.-H. Chavanis, and B. Dubrulle. Statistical mechanics of two-dimensional Euler ows and minimum enstrophy states. The European Physical Journal B, 77(2):187-212, 2010.