Extending Kolmogorov Theory to Polymeric Turbulence

TL;DR

This paper develops a formalism that extends Kolmogorov's turbulence theory to polymeric flows, accounting for polymers' effects on flow dynamics and structure functions, supported by analytical and numerical evidence.

Contribution

It introduces a new framework based on an extended Kármán-Howarth relation to reconcile polymeric turbulence with classical Kolmogorov phenomenology.

Findings

Extended structure functions follow power-law behaviour in elasto-inertial range

Deviations from classical exponents explained by local dissipation averages

Multiplier statistics exhibit scale-invariance across a wide range of scales

Abstract

The addition of polymers fundamentally alters the dynamics of turbulent flows in a way that defies Kolmogorov predictions. However, we now present a formalism that reconciles our understanding of polymeric turbulence with the classical Kolmogorov phenomenology. This is achieved by relying on an appropriate form of the K\'{a}rm\'{a}n-Howarth-Monin-Hill relation, which motivates the definition of extended velocity increments and the associated structure functions, by accounting for the influence of the polymers on the flow. We show, both analytically and numerically, that the ${\rm p}$th-order extended structure functions exhibit a power-law behaviour in the elasto-inertial range of scales, with exponents deviating from the analytically predicted value of ${\rm p}/3$. These deviations are readily accounted for by considering local averages of the total dissipation, rather than global averages, in analogy with the refined similarity hypotheses of Kolmogorov for classical Newtonian turbulence. We also demonstrate the scale-invariance of multiplier statistics of extended velocity increments, whose distributions collapse well for a wide range of scales.