RESUMO
We introduce a model of interacting singularities of Navier-Stokes equations, named pinçons. They follow non-equilibrium dynamics, obtained by the condition that the velocity field around these singularities obeys locally Navier-Stokes equations. This model can be seen as a generalization of the vorton model of Novikov that was derived for the Euler equations. When immersed in a regular field, the pinçons are further transported and sheared by the regular field, while applying a stress onto the regular field that becomes dominant at a scale that is smaller than the Kolmogorov length. We apply this model to compute the motion of a pair of pinçons. A pinçon dipole is intrinsically repelling and the pinçons generically run away from each other in the early stage of their interaction. At a late time, the dissipation takes over, and the dipole dies over a viscous time scale. In the presence of a stochastic forcing, the dipole tends to orientate itself so that its components are perpendicular to their separation, and it can then follow during a transient time a near out-of-equilibrium state, with forcing balancing dissipation. In the general case where the pinçons have arbitrary intensity and orientation, we observe three generic dynamics in the early stage: one collapse with infinite dissipation, and two expansion modes, the dipolar anti-aligned runaway and an anisotropic aligned runaway. The collapse of a pair of pinçons follows several characteristics of the reconnection between two vortex rings, including the scaling of the distance between the two components, following Leray scaling tc-t.
RESUMO
This paper investigates the universality of the Eulerian velocity structure functions using velocity fields obtained from the stereoscopic particle image velocimetry (SPIV) technique in experiments and direct numerical simulations (DNS) of the Navier-Stokes equations. It shows that the numerical and experimental velocity structure functions up to order 9 follow a log-universality (Castaing et al. Phys. D Nonlinear Phenom. 1993); this leads to a collapse on a universal curve, when units including a logarithmic dependence on the Reynolds number are used. This paper then investigates the meaning and consequences of such log-universality, and shows that it is connected with the properties of a "multifractal free energy", based on an analogy between multifractal and thermodynamics. It shows that in such a framework, the existence of a fluctuating dissipation scale is associated with a phase transition describing the relaminarisation of rough velocity fields with different Hölder exponents. Such a phase transition has been already observed using the Lagrangian velocity structure functions, but was so far believed to be out of reach for the Eulerian data.
