Turbulence on Hybrid Lattices: regular grids meet self similar cascade models

Abstract

Turbulent cascades in high Reynolds number regimes span many decades in scale, which can be challenging for numerical simulations. Cascade models that use logarithmic discretization can help, but these have limited capabilities to describe spatial structures or real space details. Hybrid Lattices (HLs) are proposed in order to provide a novel framework to investigate turbulent cascades in Fourier space, combining regular grids and cascade models based on logarithmic discretisations. The approach consists of a regular spectral region, capable of resolving large scale coherent structures, that is extended in a recurrent, asymptotically self similar fashion, spanning many decades in scale. The idea comes from shell models and their extensions, namely the logarithmic lattices, the nested polyhedra models or the spiral chain models. The logarithmic spacing factor of those models, such as the golden ratio, typically prevents interactions between a regular grid and a self similar model. HLs are constructed by expanding a regular region in Fourier space through recurrence relations (e.g. the Fibonacci recurrence relation), generating an asymptotically logarithmic lattice where Cartesian components remain integer multiples of the regular Fourier space grid spacing. This way, the resulting grid preserves triadic interactions linking regular and recurrent regions, allowing energy transfer across these different regions. However, the sharp reduction in triads connecting the two regions creates a bottleneck effect, and in order to address this issue, a region of ghost points at the boundary had to be introduced, efficiently allowing the flux to leak into the recurrent part. The HL framework is then tested for the one dimensional Burgers equation, and for two dimensional incompressible Navier Stokes turbulence.