Structure function tensor equations with triple decomposition

TL;DR

This paper derives exact budget equations for the coherent and stochastic parts of the second-order structure function tensor, extending AGKE by including phase as a variable, and applies it to analyze phase-dependent turbulence in a flow with periodic forcing.

Contribution

The paper introduces $ ext{φ}$AGKE equations that incorporate phase as an independent variable, enabling detailed phase-resolved analysis of turbulence interactions.

Findings

Phase-dependent turbulent structures are influenced by harmonic wall oscillations.

The $ ext{φ}$AGKE reveal the interplay among mean, coherent, and stochastic fields at different phases.

Flow control effects on turbulence are characterized in phase space.

Abstract

Exact budget equations are derived for the coherent and stochastic contributions to the second-order structure function tensor. They extend the anisotropic generalised Kolmogorov equations (AGKE) by considering the coherent and stochastic parts of the Reynolds stress tensor, and are useful for the statistical description of turbulent flows with periodic or quasi-periodic features, like e.g. the alternate shedding after a bluff body. While the original AGKE describe production, transport, inter-component redistribution and dissipation of the Reynolds stresses in the combined space of scales and positions, the new equations, called $\varphi$AGKE, contain the phase $\varphi$ as an additional independent variable, and describe the interplay among the mean, coherent and stochastic fields at the various phases. The newly derived $\varphi$AGKE are then applied to a case where an exactly periodic external forcing drives the flow: a turbulent plane channel flow modified by harmonic spanwise oscillations of the wall to reduce drag. The phase-by-phase action of the oscillating transversal Stokes layer generated by the forcing on the near-wall turbulent structures is observed, and a detailed description of the scale-space interaction among mean, coherent and stochastic fields is provided thanks to the $\varphi$AGKE.