Scales of a fluid

TL;DR

This paper investigates the macroscopic scales of viscous fluid flow by applying the classical similarity renormalization group to the Navier-Stokes Hamiltonian, aiming to connect ideal and viscous fluid behaviors and analyze turbulent structures.

Contribution

It introduces a novel approach using the similarity renormalization group to study the Navier-Stokes Hamiltonian and explores the connection between Euler and Navier-Stokes fluids at different scales.

Findings

Identification of the scale where Richardson's 4/3 law applies

Mapping of singular structures between Euler and Navier-Stokes fluids

Analysis of low-order velocity correlators for realistic fluids

Abstract

The flow of a viscous fluid is perturbed by its internal friction which generates heat and leads to a small temperature change. This does not occur for an ideal fluid. We would like to resolve this picture as a function of the dynamical macroscopic scales of both problems. In order to do this we will study the evolution of the Navier-Stokes Hamiltonian with the classical similarity renormalization group in the region of small viscosity. The connection between the Euler and Navier-Stokes fluids will be pursued, but also the viscous structures that arise will be studied in their own right to determine the low-order velocity correlators of realistic fluids such as single-component air and water. The canonical coordinate of the Navier-Stokes Hamiltonian is a vector field that stores the initial position of all the fluid particles. Thus these appear to be natural coordinates for studying arbitrary separations of fluid particles over time. This connection will be pursued and the region where the classic 1926 Richardson 4/3 scaling law holds will be determined. The evolution of the Euler Hamiltonian will also be studied and we will attempt to map its singular structures to those of the small-viscosity Navier-Stokes fluid.