A zero-mode mechanism for spontaneous symmetry breaking in a turbulent von Kármán flow

Abstract

We suggest that the dynamical spontaneous symmetry breaking reported in a turbulent swirling flow at Re = 40 000 by Cortet et al (2010 Phys. Rev. Lett. 105 214501) can be described through a continuous one parameter family transformation (amounting to a phase shift) of steady states. We investigate a possible mechanism of emergence of such spontaneous symmetry breaking in a toy model of out-of-equilibrium systems. We show that the stationary states are solutions of a linear differential equation. For a specific value of the Reynolds number, they are subject to a spontaneous symmetry breaking through a zeromode mechanism. The associated susceptibility diverges at the transition, in a way similar to what is observed in the experimental turbulent flow. Overall, the susceptibility of the toy model reproduces the features of the experimental results, meaning that the zero-mode mechanism is a good candidate to explain the experimental symmetry breaking. New J. Phys. 16 (2014) 013055 B Saint-Michel et al problem and the emergence of a new set of stable solutions individually breaking the symmetry.Nevertheless, the set of solutions itself respects the broken symmetry to respect Curie's symmetry principle. Spontaneous symmetry breaking is also present in out-of-equilibrium systems, such as forced-dissipative flows. In the case where the dissipation is large, and the fluctuations very small, spontaneous symmetry breaking is well described through classical bifurcation theory [1-3] starting from linear or nonlinear perturbations of the so-called 'basic state', the stationary laminar solution of the Navier-Stokes (NS) equation at low Reynolds number [4].When the fluctuations are much higher, and the symmetry breaking occurs over a turbulent background, however, tools are often missing to model the transition. This is the case for example when the symmetry breaking occurs for the mean state of a turbulent flow. This flow is stationary by construction, but differs from an usual basic state in the sense that it is solution of the ensemble time-averaged NS equation, differing from the plain NS equation via a Reynolds stress tensor. This Reynolds stress represents the influence of all the degrees of freedom of the flow onto its average, and can, in general, only be computed via full solution of the NS equation. Therefore, the problem of instability of a mean turbulent flow cannot currently be tackled analytically or is too demanding numerically, unless a prescription (parameterization) of the Reynolds stress is provided. In the case of the plane Couette turbulent flow, for example, this was attempted by Tuckerman et al [5] via the K − closure model.In the present paper, we explore a new way to tackle the problem using tools inspired from statistical physics applied to a well-controlled laboratory model of spontaneous symmetry breaking, such as the von Kármán (VK) flow. In this system, the flow is inertially forced by two counter-rotating impellers with blades, providing the necessary energy injection to set the system out-of-eq...

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