Fluctuation-Dissipation Relations and Statistical Temperatures in a Turbulent von Kármán Flow

Abstract

We experimentally characterize the fluctuations of the non-homogeneous non-isotropic turbulence in an axisymmetric von Kármán flow. We show that these fluctuations satisfy relations, issued from the Euler equation, which are analogous to classical Fluctuation-Dissipation Relations in statistical mechanics. We use these relations to estimate statistical temperatures of turbulence.PACS numbers: 47.27.E-, 05.70.LnFluctuation-Dissipation Relations (FDRs) are one of the corner-stone of statistical mechanics. They offer a direct relation between the fluctuations of a system at equilibrium and its response to a small external perturbation. Classical outcome of FDRs are Einstein or Nyquist relations, or, more generally, measures of the susceptibility, dissipation coefficient or temperature of the system. The hypothesis behind the FDRs restrict their applicability to systems that are close to equilibrium. Theoretical extrapolation of the FDRs to systems far from equilibrium is currently a very active area of research [1]. In this context, experimental tests in several glassy systems have evidenced violation of FDRs [2]. Furthermore, general identities about fluctuations and dissipation, theoretically derived for time-symmetric out of equilibrium systems [3], have been tested in dissipative (non timesymmetric) systems like electrical circuit or turbulent flow [4]. Turbulence is actually a very special example of far from equilibrium system. Due to its intrinsic dissipative nature, an unforced turbulent flow is bound to decay to rest. However, in the presence of a permanent forcing, a steady state regime can be established, in which forcing and dissipation equilibrate on average, allowing the maintenance of non-zero averaged velocities, with large fluctuations covering a wide range of scales. We use measurements performed in a turbulent von Kármán flow to show that there is actually a direct link between these fluctuations and the mean flow properties, in a way analogous to classical FDRs. This approach provides an estimate of effective statistical temperatures of our turbulent flow.Theoretical background and definitions.-Describing turbulence with tools borrowed from statistical mechanics is a long-standing dream, starting with Onsager [5]. Advances in that direction have been recently made for flows with symmetries (2D [6], axisymmetric [7]) using tools developed independently by Robert and Sommeria and Miller [8]. They consider freely evolving flows described by the Euler equation (no forcing and no dissipation). The Euler equation conserves the energy and, for axisymmetric and shear flows, the helicity. In addition, owing to the symmetry, there is conservation of a local scalar quantity along a velocity line (vorticity in 2D, angular momentum for axisymmetry) resulting in a Liouville theorem and additional global conserved quantities as Casimirs of the local scalar quantity. In the MillerRobert-Sommeria theory, the Euler equation develops a mixing process leading to a quasi-stationary state on the coarse-grained scale....

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

“…From the analysis of steady states of a VK flow at very large Reynolds number [6,7], we get B = −4.5 < 0. From the definition of D, we get D ∝ 2ζ B/Re.…”

“…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-equilibrium. This energy is naturally dissipated through molecular viscosity, so that, for well controlled forcing protocols, statistically steady states can be established, that may be seen as the out-of-equilibrium counterpart of the equilibrium states of classical ideal systems [6,7]. Changing the forcing protocol for the VK flow leads to various transitions with associated symmetry breaking.…”

Probing quantum and classical turbulence analogy in von Kármán liquid helium, nitrogen, and water experiments

“…We can also infer that the very nature of the dissipation process does not itself select the large scale flow, justifying a posteriori the statistical physics descriptions of the von Kármán steady states. 18,31 A convenient visualization of the hysteresis cycle relies on the normalized torque difference ∆K p , plotted as a function of θ (see Figure 5); for the Saclay experiments (dashed line), the two branches (b 1 ) and (b 2 ) are clearly visible, the upper (b 1 ) branch, starting from −θ * to θ = 1, and the lower branch (b 2 ), starting from θ = −1 to θ * , drawing a cycle reminiscent of ferromagnetic systems. SHREK data are more scattered, especially compared to the individual torques, but they roughly collapse around the two branches of the classical hysteresis cycle.…”

Is Turbulence a State of Maximum Energy Dissipation?

“…Here, it is the time associated with eigenvalues of the Liouville operator of the processes describing the turbulence action. In the specific example we consider here, namely von Kármán flow, the turbulence is characterized by a symmetry along the rotation axis, which favors stationary states in which angular momentum is mixed along meridional planes [43,44]. In this case, there is a clear connection between the equation obeying the angular momentum and the classical Fokker-Planck equation, Equation (15).…”