Spiral small-scale structures in compressible turbulent flows

Abstract

We extend the spiral vortex solution of Lundgren 1982 to compressible turbulent flows following a perfect gas law. Lundgren’s model links the dynamical and spectral properties of incompressible flows, providing a k ^−5/3 Kolmogorov spectrum. A similar compressible spatio-temporal transformation is now derived, reducing the dynamics of three-dimensional (3D) vortices stretched by an axisymmetric incompressible strain into a 2D compressible vortex dynamics. It enables to write the 3D spectra of the incompressible and compressible square velocities u ² _s and u ² _d in terms of, respectively, the 2D spectra of the enstrophy and of the square velocity divergence, by use of a temporal integration (Gomez &al. 2001). New numerical results are presented now using 1024² gridpoints; initially, the r.m.s. Mach number is 0.32, with local values up to 0.9, the Reynolds number is 1,400, and x=u ² _s /u ² _d =0.1. A k ^−5/3 inertial behaviour is seen to result from the dynamical evolution for both the compressible and incompressible three-dimensional kinetic energy spectra.