Free turbulent shear layer in a point vortex gas as a problem in nonequilibrium statistical mechanics

Abstract

This paper attempts to unravel any relations that may exist between turbulent shear flows and statistical mechanics through a detailed numerical investigation in the simplest case where both can be well defined. The flow considered for the purpose is the two-dimensional (2D) temporal free shear layer with a velocity difference ΔU across it, statistically homogeneous in the streamwise direction (x) and evolving from a plane vortex sheet in the direction normal to it (y) in a periodic-in-x domain L×±∞. Extensive computer simulations of the flow are carried out through appropriate initial-value problems for a "vortex gas" comprising N point vortices of the same strength (γ=LΔU/N) and sign. Such a vortex gas is known to provide weak solutions of the Euler equation. More than ten different initial-condition classes are investigated using simulations involving up to 32000 vortices, with ensemble averages evaluated over up to 103 realizations and integration over 104L/ΔU. The temporal evolution of such a system is found to exhibit three distinct regimes. In Regime I the evolution is strongly influenced by the initial condition, sometimes lasting a significant fraction of L/ΔU. Regime III is a long-time domain-dependent evolution towards a statistically stationary state, via "violent" and "slow" relaxations [ P.-H. Chavanis Physica A 391 3657 (2012)], over flow time scales of order 102 and 104L/ΔU, respectively (for N=400). The final state involves a single structure that stochastically samples the domain, possibly constituting a "relative equilibrium." The vortex distribution within the structure follows a nonisotropic truncated form of the Lundgren-Pointin (L-P) equilibrium distribution (with negatively high temperatures; L-P parameter λ close to -1). The central finding is that, in the intermediate Regime II, the spreading rate of the layer is universal over the wide range of cases considered here. The value (in terms of momentum thickness) is 0.0166±0.0002 times ΔU. Regime II, extensively studied in the turbulent shear flow literature as a self-similar "equilibrium" state, is, however, a part of the rapid nonequilibrium evolution of the vortex-gas system, which we term "explosive" as it lasts less than one L/ΔU. Regime II also exhibits significant values of N-independent two-vortex correlations, indicating that current kinetic theories that neglect correlations or consider them as O(1/N) cannot describe this regime. The evolution of the layer thickness in present simulations in Regimes I and II agree with the experimental observations of spatially evolving (3D Navier-Stokes) shear layers. Further, the vorticity-stream-function relations in Regime III are close to those computed in 2D Navier-Stokes temporal shear layers [ J. Sommeria, C. Staquet and R. Robert J. Fluid Mech. 233 661 (1991)]. These findings suggest the dominance of what may be called the Kelvin-Biot-Savart mechanism in determining the growth of the free shear layer through large-scale momentum and vorticity dispersal.

A Renormalization Group Approach to Spontaneous Stochasticity

“…Perturbing the initial vortex-sheet with random "errors" of decreasing magnitude and simultaneously decreasing the viscosity ν, the crucial "forgetting" of initial error v(0) − v (0) was observed in Figure 1 of [22] for the mean-square separation v(t) − v (t) 2 after a sufficient time-interval. Furthermore, after this initial short transient, the spontaneously stochastic ensemble of solutions entered a selfsimilar growth phase with universal statistical properties independent of regularization and noise level, consistent with earlier observations [44].…”

“…A simple Lagrangian manifestation of spontaneous stochasticity is the universal Richardson t 3 -growth in mean-square width of plumes of smoke or of other aerosols advected by a turbulent flow [5], which many observations suggest to be independent of the details of the source and of the mass diffusivity of the aerosol ([42], section 24.3). More recently, it has been argued [22] that Eulerian spontaneous stochasticity explains how universal statistics are attained in a finite time for turbulent mixing layers during their phase of non-stationary, selfsimilar "equilibrium" spread [43,44]. In addition to many hints and indirect pieces of evidence for spontaneous stochasticity, there are a few controlled, precision studies which provide direct corroboration of the phenomenon.…”

The role of the `monopole' instability in the evolution of 2D turbulent free shear layers

“…Beale & Majda 24 , Marchioro & Pulvirenti 25 ) that prove the convergence of the vortexgas to weak solutions of the Euler equations under appropriate limits (see refs. 12,19 for further discussion). The use of the point-vortex approximation has a long history which includes the early work of Rosenhead 26 and subsequently of Hama and Burke 27 , Acton 28 , Delcourt & Brown 29 , Aref & Siggia 30 and others.…”

“…Extensive simulations (using the vortex-gas model, to be described in detail in Sec. 2) of a 2D temporally evolving free shear layer by Suryanarayanan, Narasimha and Hari Dass 19 (SNH henceforth) have provided new insights. In this work, it was shown that a temporally evolving free shear layer in a gas of point vortices, in a streamwise periodic domain L, exhibits three distinct regimes.…”

Ergodicity and spectral cascades in point vortex flows on the sphere

“…The point vortex model, originally developed by Kirchoff [1] as a limiting form of Euler's equations in two dimensions, continues to provide a conceptual and computational tool for understanding inviscid, nonlinear vortex dynamics in both traditional and superfluid turbulence [2][3][4]. The multibody Hamiltonian describing the dynamics of idealized point vortices serves as a paradigm for developing kinetic theories in systems dominated by long-range interactions [5,6].…”