Laws of turbulence decay from direct numerical simulations

TL;DR

This paper uses extensive direct numerical simulations to analyze the decay laws of homogeneous isotropic turbulence, confirming theoretical decay exponents under specific initial conditions and exploring the behavior of turbulence statistics over time.

Contribution

It demonstrates that classical decay exponents can be observed in simulations with proper initial conditions and discusses the variability of decay exponents in past studies.

Findings

Kolmogorov decay exponent observed under certain initial conditions

Birkhoff-Saffman decay law also observed approximately

Turbulent statistics evolve as predicted by theory, with some invariants not strictly maintained

Abstract

Inspection of available data on the decay exponent for the kinetic energy of homogeneous and isotropic turbulence (HIT) shows that it varies by as much as 100\%. Measurements and simulations often show no correspondence with theoretical arguments, which are themselves varied. This situation is unsatisfactory given that HIT is a building block of turbulence theory and modeling. We take recourse to a large base of direct numerical simulations and study decaying HIT for a variety of initial conditions. We show that the Kolmogorov decay exponent and the Birkhoff-Saffman decay are both readily observed, albeit approximately, for long periods of time if the initial conditions are appropriately arranged. We also present, for both cases, other turbulent statistics such as the velocity derivative skewness, energy spectra and dissipation, and show that the decay and growth laws are approximately as expected theoretically, though the wavenumber spectrum near the origin begins to change relatively quickly, suggesting that the invariants do not strictly exist. We comment briefly on why the decay exponent has varied so widely in past experiments and simulations.