On the effective velocity created by a point vortex in two-dimensional hydrodynamics

The objective of this paper is 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 shear flow considered for the purpose is the 2D temporal mixing layer, which is a time-dependent flow that is statistically homogeneous in the streamwise direction (x) and evolves from a plane vortex sheet in the direction normal to it (y) in a periodi...

We describe two new -- stochastic-geometrical -- methods to obtain reliable velocity field statistics from N-body simulations and from any general density and velocity fluctuation field sampled at a discrete set of locations. These methods, the Voronoi tessellation method and Delaunay tessellation method, are based on the use of the Voronoi and Delaunay tessellations of the point distribution defined by the locations at which the velocity field is sampled. Adjusting themselve...

For turbulence, although the two-point velocity difference u(x+r)-u(x) at each scale r has been studied in detail, the velocity average [u(x+r)+u(x)]/2 has not thus far. Theoretically or experimentally, we find interesting features of the velocity average. It satisfies an exact scale-by-scale energy budget equation. The flatness factor varies with the scale r in a universal manner. These features are not consistent with the existing assumption that the velocity average is ind...

We introduce a stochastic model of two-dimensional Brownian vortices associated with the canonical ensemble. The point vortices evolve through their usual mutual advection but they experience in addition a random velocity and a systematic drift generated by the system as a whole. The statistical equilibrium state of this stochastic model is the Gibbs canonical distribution. We consider a single species system and a system made of two types of vortices with positive and negati...

We introduce two new methods to obtain reliable velocity field statistics from N-body simulations, or indeed from any general density and velocity fluctuation field sampled by discrete points. These methods, the {\it Voronoi tessellation method} and {\it Delaunay tessellation method}, are based on the use of the Voronoi and Delaunay tessellations of the point distribution defined by the locations at which the velocity field is sampled. In the Voronoi method the velocity is su...

The problem of low Reynolds number turbulence in active nematic fluids is theoretically addressed. Using numerical simulations I demonstrate that an incompressible turbulent flow, in two-dimensional active nematics, consists of an ensemble of vortices whose areas are exponentially distributed within a range of scales. Building on this evidence, I construct a mean-field theory of active turbulence by which several measurable quantities, including the spectral densities and the...

The small-scale statistical properties of velocity circulation in classical homogeneous and isotropic turbulent flows are assessed through a modeling framework that brings together the multiplicative cascade and the structural descriptions of turbulence. We find that vortex structures exhibit short-distance repulsive correlations, which is evidenced when they are "tomographically" investigated, by means of planar cuts of the flow, as two-dimensional vortex gases. This phenome...

In arXiv:1004.1407, Flandoli, Gubinelli, and Priola proposed a stochastic variant of the classical point vortex system of Helmholtz and Kirchoff in which multiplicative noise of transport-type is added to the dynamics. An open problem in the years since is to show that in the mean-field scaling regime, in which the circulations are inversely proportional to the number of vortices, the empirical measure of the system converges to a solution of a two-dimensional Euler vorticity...

We explore the velocity fluctuations in a fluid due to a dilute suspension of randomly-distributed vortex rings at moderate Reynolds number, for instance those generated by a large colony of jellyfish. Unlike previous analysis of velocity fluctuations associated with gravitational sedimentation or suspensions of microswimmers, here the vortices have a finite lifetime and are constantly being produced. We find that the net velocity distribution is similar to that of a single v...

We consider freely decaying two-dimensional isotropic turbulence. It is usually assumed that, in such turbulence, the energy spectrum at small wave number, takes the form, where is the two-dimensional version of Loitsyansky's integral. In this paper, we developed a simple model for large scale dynamics of free decay two-dimensional turbulence based on the statistical solution of Navier-Stokes equation. We provide one possible explanation for the large scale dynamics in two-di...