The infrared properties of the energy spectrum in freely decaying isotropic turbulence

TL;DR

This paper investigates the low wavenumber expansion of the energy spectrum in isotropic turbulence, demonstrating that the leading coefficient vanishes under typical conditions and exploring special cases with power-law correlations.

Contribution

It provides a detailed analysis of the conditions under which the low wavenumber expansion coefficients vanish or remain non-zero, including the special case of power-law correlations.

Findings

$E_2(t)=0$ under typical decay conditions

For $f(r,t)= ext{const} imes r^{-3}$, $E_2$ can be non-zero but unphysical

$E_4(t)$ remains constant over time

Abstract

The low wavenumber expansion of the energy spectrum takes the well known form: $ E(k,t) = E_2(t) k^2 + E_4(t) k^4 + ... $, where the coefficients are weighted integrals against the correlation function $C(r,t)$. We show that expressing $E(k,t)$ in terms of the longitudinal correlation function $f(r,t)$ immediately yields $E_2(t)=0$ by cancellation. We verify that the same result is obtained using the correlation function $C(r,t)$, provided only that $f(r,t)$ falls off faster than $r^{-3}$ at large values of $r$. As power-law forms are widely studied for the purpose of establishing bounds, we consider the family of model correlations $f(r,t)=\alpha_n(t)r^{-n}$, for positive integer $n$, at large values of the separation $r$. We find that for the special case $n=3$, the relationship connecting $f(r,t)$ and $C(r,t)$ becomes indeterminate, and (exceptionally) $E_2 \neq 0$, but that this solution is unphysical in that the viscous term in the K\'{a}rm\'{a}n-Howarth equation vanishes. Lastly, we show that $E_4(t)$ is independent of time, without needing to assume the exponential decrease of correlation functions at large distances.