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Distributed chaos in turbulent wakes

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 $\int_{V} \langle {\boldsymbol ω} ({\bf x},t) \cdot {\boldsymbol ω} ({\bf x} + {\bf r},t) \rangle_{V} d{\bf r}$ dominates the distributed chaos and the chaotic spectra $\exp-(k/k_β)^{β}$ have $β=1/2$. 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 $β=1/3$. 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.