Turbulence: does energy cascade exist?

TL;DR

This paper investigates the existence of energy cascade in turbulence by developing correlation tests that reveal irreversible energy transfer from large to small scales, supported by experimental and model data analysis.

Contribution

It introduces novel correlation functions to test energy transfer irreversibility and demonstrates the existence of a dynamical cascade in turbulence.

Findings

Confirmed irreversibility of turbulent flow.

Linked irreversibility to finite time singularities.

Established the presence of a cascade in the inertial domain.

Abstract

To answer the question whether a cascade of energy exists or not in turbulence, we propose a set of correlation functions able to test if there is an irreversible transfert of energy, step by step, from large to small structures. These tests are applied to real Eulerian data of a turbulent velocity flow, taken in the wind grid tunnel of Modane, and also to a prototype model equation for wave turbulence. First we demonstrate the irreversible character of the flow by using multi-time correlation function at a given point of space. Moreover the unexpected behavior of the test function leads us to connect irreversibility and finite time singularities (intermittency). Secondly we show that turbulent cascade exists, and is a dynamical process, by using a test function depending on time and frequency. The cascade shows up only in the inertial domain where the kinetic energy is transferred more rapidly (on average) from the wavenumber $k_{1}$ to $k_{2}$ than from $k_{1}$ to $k'_{2}$ larger than $k_{2}$.