Distributed chaos in turbulent wakes

TL;DR

This paper investigates how spontaneous symmetry breaking influences distributed chaos in turbulent wakes, revealing different spectral behaviors linked to the type of symmetry broken and supported by experimental data.

Contribution

It identifies the dominant correlation integrals and spectral exponents associated with soft and hard translational symmetry breaking in turbulent wakes, connecting theory with experimental observations.

Findings

Soft symmetry breaking leads to spectra with exponent 1/2.

Hard symmetry breaking results in spectra with exponent 1/3.

Bubbles in flows cause a shift to hard symmetry breaking, altering velocity spectra.

Abstract

Soft and hard spontaneous breaking of space translational symmetry (homogeneity) have been studied in turbulent wakes by means of distributed chaos. In the case of the soft translational symmetry breaking the vorticity correlation integral $\int_{V} \langle {\boldsymbol \omega} ({\bf x},t) \cdot {\boldsymbol \omega} ({\bf x} + {\bf r},t) \rangle_{V} d{\bf r}$ dominates the distributed chaos and the chaotic spectra $\exp-(k/k_{\beta})^{\beta }$ have $\beta =1/2$. In the case of the hard translational symmetry breaking, control on the distributed chaos is switched from one type of fundamental symmetry to another (in this case to Lagrangian relabeling symmetry). Due to the Noether's theorem the relabeling symmetry results in the inviscid helicity conservation and helicity correlation integral $I=\int \langle h({\bf x},t)~h({\bf x}+{\bf r}, t)\rangle d{\bf r}$ (Levich-Tsinober invariant) dominates the distributed chaos with $\beta =1/3$. Good agreement with the experimental data has been established for turbulent wakes behind a cylinder, behind grids (for normal and super-fluids) and for bubbling flows. In the last case even small concentration of bubbles leads to a drastic change of the turbulent velocity spectra due to the hard spontaneous symmetry breaking in the bubbles' wakes.