Eulerian and Lagrangian scaling properties of randomly advected vortex tubes

Abstract

We investigate the Eulerian and Lagrangian spectral scaling properties of vortex tubes, and the consistency of these properties with Tennekes’ (1975) statistical advection analysis and universal equilibrium arguments. We consider three different vortex tubes with power-law wavenumber spectra: a Burgers vortex tube, an inviscid Lundgren single spiral vortex sheet, and a vortex tube solution of the Euler equation. While the Burgers vortex is a steady solution of the Navier–Stokes equation, the other two are unsteady solutions of, respectively, the Navier–Stokes and the Euler equations. In our numerical experiments we study the vortex tubes by subjecting each of them to external ‘large-scale’ sinusoidal advection of characteristic frequency f and length scale ρ.Not only do we find that the Eulerian frequency spectrum ϕE(ω) can be derived from the wavenumber spectrum E(k) using the simple Tennekes advection relation ω ∼ k for all finite advection frequencies f when the vortex is steady, but also when the vortex is unsteady, and in the Lundgren case even when f = 0 owing to the self-advection of the Lundgren vortex by its own differential rotation.An analytical calculation using the method of stationary phases for f = 0 shows that for large enough Reynolds numbers the combination of strain with differential rotation implies that ϕL(ω) ∼ ω−2+Const for large values of ω. We verify numerically that ϕL(ω) does not change when f ≠ 0. With the Burgers vortex tube we are in a position to investigate the spectral broadening of the Eulerian frequency spectrum with respect to the Lagrangian frequency spectrum. A spectral broadening does exist but is different from the spectral broadening predicted by Tennekes (1975).