Multiscale numerical methods for passive advection-diffusion in incompressible turbulent flow fields

TL;DR

This paper introduces a multiscale numerical method for passive advection-diffusion in incompressible turbulent flows, combining Fourier-based velocity decomposition with hierarchical local grids to efficiently approximate macroscopic behavior across multiple scales.

Contribution

It presents a novel seamless multiscale approach that integrates Fourier decomposition and multigrid-like hierarchies to efficiently simulate passive scalar transport in turbulent flows.

Findings

Method captures scale interactions effectively.

Achieves linear computational complexity.

Successfully approximates continuous spectrum flows.

Abstract

We propose a seamless multiscale method which approximates the macroscopic behavior of the passive advection-diffusion equations with steady incompressible velocity fields with multi-spatial scales. The method uses decompositions of the velocity fields in the Fourier space, which are similar to the decomposition in large eddy simulations. It also uses a hierarchy of local domains with different resolutions as in multigrid methods. The effective diffusivity from finer scale is used for the next coarser level computation and this process is repeated up to the coarsest scale of interest. The grids are only in local domains whose sizes decrease depending on the resolution level so that the overall computational complexity increases linearly as the number of different resolution grids increases. The method captures interactions between finer and coarser scales but has to sacrifice some of the interaction between the fine scales. The proposed method is numerically tested with 2D examples including a successful approximation to a continuous spectrum flow.