Showing papers by "Thierry Gallay published in 2011"

Interaction of Vortices in Weakly Viscous Planar Flows

Abstract: We consider the inviscid limit for the two-dimensional incompressible Navier–Stokes equation in the particular case where the initial flow is a finite collection of point vortices. We suppose that the initial positions and the circulations of the vortices do not depend on the viscosity parameter ν, and we choose a time T > 0 such that the Helmholtz–Kirchhoff point vortex system is well-posed on the interval [0, T]. Under these assumptions, we prove that the solution of the Navier–Stokes equation converges, as ν → 0, to a superposition of Lamb–Oseen vortices whose centers evolve according to a viscous regularization of the point vortex system. Convergence holds uniformly in time, in a strong topology which allows us to give an accurate description of the asymptotic profile of each individual vortex. In particular, we compute to leading order the deformations of the vortices due to mutual interactions. This makes it possible to estimate the self-interactions, which play an important role in the convergence proof.

Three-Dimensional Stability of Burgers Vortices

Abstract: Burgers vortices are explicit stationary solutions of the Navier-Stokes equations which are often used to describe the vortex tubes observed in numerical simulations of three-dimensional turbulence. In this model, the velocity field is a two-dimensional perturbation of a linear straining flow with axial symmetry. The only free parameter is the Reynolds number Re = Γ/ν, where Γ is the total circulation of the vortex and ν is the kinematic viscosity. The purpose of this paper is to show that Burgers vortices are asymptotically stable with respect to small three-dimensional perturbations, for all values of the Reynolds number. This general result subsumes earlier studies by various authors, which were either restricted to small Reynolds numbers or to two-dimensional perturbations. Our proof relies on the fact that the linearized operator at Burgers vortex has a simple and very specific dependence upon the axial variable. This allows to reduce the full linearized equations to a vectorial two-dimensional problem, which can be treated using an extension of the techniques developed in earlier works. Although Burgers vortices are found to be stable for all Reynolds numbers, the proof indicates that perturbations may undergo an important transient amplification if Re is large, a phenomenon that was indeed observed in numerical simulations.

Interaction of Vortices in Weakly Viscous

Abstract: We consider the inviscid limit for the two-dimensional incompressible Navier‐Stokes equation in the particular case where the initial flow is a finite collection of point vortices. We suppose that the initial positions and the circulations of the vortices do not depend on the viscosity parameter ν, and we choose a time T > 0 such that the Helmholtz‐Kirchhoff point vortex system is well-posed on the interval [0, T ]. Under these assumptions, we prove that the solution of the Navier‐ Stokes equation converges, as ν → 0, to a superposition of Lamb‐Oseen vortices whose centers evolve according to a viscous regularization of the point vortex system. Convergence holds uniformly in time, in a strong topology which allows us to give an accurate description of the asymptotic profile of each individual vortex. In particular, we compute to leading order the deformations of the vortices due to mutual interactions. This makes it possible to estimate the self-interactions, which play an important role in the convergence proof.