3D Euler about a 2D symmetry plane

Abstract

Initial results from new calculations of interacting anti-parallel Euler vortices are presented with the objective of understanding the origins of singular scaling presented by Kerr (1993) and the lack thereof by Hou and Li (2006). Core profiles designed to reproduce the two results are presented, new more robust analysis is proposed, and new criteria for when calculations should be terminated are introduced and compared with classical resolution studies and spectral convergence tests. Most of the analysis is on a 512 × 128 × 2048 mesh, with new analysis on a just completed 1024 × 256 × 2048 used to confirm trends. One might hypothesize that there is a finite-time singularity with enstrophy growth like Ω ∼ (Tc − t) −γ Ω and vorticity growth like ω ∞ ∼ (Tc − t) −γ . The new analysis would then support γΩ ≈ 1/2 and γ > 1. These represent modifications of the conclusions of Kerr (1993). Issues that might arise at higher resolution are discussed.

Vortex Stretching as a Mechanism for Quantum Kinetic Energy Decay

“…In the first state of spectral development, until the first ring separates off at t = 4.5, high wavenumber spectra are roughly k −3 in all directions for both E I and K ∇ψ . This is similar to spectra in Euler calculations [13]. After t = 4.5, K ∇ψ (k) gradually increases at higher wavenumbers, that is smaller scales, but not E I (k).…”

“…Convergent time-advancement is obtained using a 3rd-order Runge-Kutta, semi-implicit spectral algorithm used for other ideal equations [13] where the nonlinear terms are calculated in physical space, then transformed to Fourier space to calculate the linear terms using complex integrating factors. No-flux domain walls are imposed using cosine transforms and serve as an approximation to those of a superfluid in a container and as a means of generating symmetric initial conditions.…”

“…A new calculation by C. Rorai starting with p = 1 shows cleaner conclusions without changing the overall conclusions about the energetics reported here. This function was applied approximately perpendicular to the trajectory of the vortex lines, and is not perpendicular to the y-axis as in past Euler calculations [13], which assisted in minimizing spots of excess ρ > ρ b .…”

Bounds for Euler from vorticity moments and line divergence

“…The inviscid modification chosen defines 0 with the circulation of the vortices Γ instead of the viscosity ν. This is inspired by an empirical guess used in the analysis of enstrophy growth in Bustamante & Kerr (2008) and gives 0 = Γ = Γ /L 2 , with the choice L = 2L y , the size of the full periodic domain, for the primary calculation analysed.…”

“…The existing analysis tools are unable to simultaneously cover the necessary range of scales in both space and time, while existing methods for mapping vortex tubes onto Eulerian meshes tend to generate ghost images unless ad hoc massaging is applied (Bustamante & Kerr 2008;Hou 2008). This has led to weak and conflicting conclusions that depend upon the numerical method used and the choice of analysis that is applied to the results.…”