  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.
