Tracking complex singularities of fluids on log-lattices

TL;DR

This paper explores the existence and properties of complex singularities in fluid solutions on log-lattices, reducing computational complexity and providing new insights into their behavior in high Reynolds number flows.

Contribution

It introduces a novel approach to study complex singularities in fluid dynamics using log-lattices, enabling more efficient analysis of their properties.

Findings

Identified dominant complex singularities in 1D and 3D models.

Derived new scalings for the approach to the real axis.

Analyzed the influence of different dissipation types.

Abstract

In 1981, Frisch and Morf [1] postulated the existence of complex singularities in solutions of Navier-Stokes equations. Present progress on this conjecture is hindered by the computational burden involved in simulations of the Euler equations or the Navier-Stokes equations at high Reynolds numbers. We investigate this conjecture in the case of fluid dynamics on log-lattices, where the computational burden is logarithmic concerning ordinary fluid simulations. We analyze properties of potential complex singularities in both 1D and 3D models for lattices of different spacings. Dominant complex singularities are tracked using the singularity strip method to obtain new scalings regarding the approach to the real axis and the influence of normal, hypo and hyper dissipation.