Title: Symmetry breaking in turbulent flow statistics -- Rotation and Reflection 

Abstract: 

A question of interest in studying turbulent velocity
statistics is whether or not such statistics are spatially
isotropic. The benchmark work of Kolmogorov (1938) postulated that
there exists a range of scales much smaller than the (anisotropic)
large scales within which turbulence recovers isotropy. This is the
"local isotropy" assumption of Kolmogorov and lead to the derivation
of exact results for third-order velocity statistics.

Since most turbulent flows are apparently anisotropic in the large
scales we would like to quantify the degree to which the rotational
symmetry-breaking penetrates into the small scales and better
understand the local isotropy assumption. To do this we use SO(3)
group decomposition methods to disentangle various anisotropic
contributions which contaminate the isotropic scaling expected from a
Kolmogorov-type argument. We show that, for the type of statistics
usually studied, the small-scales in turbulence are not strictly
isotropic but rather display a hierarchy of scaling exponents with the
leading exponent coming from the isotropic contribution.

There is another symmetry-breaking which can occur in isotropic flows
due to the presence of helicity -- the breaking of
reflection-symmetry. I will show new results which predict the scaling
behavior due to the breaking of reflection-symmetry in the velocity
field; this result yields the counterpart to the Kolmogorov result for
reflection-symmetric flows. Experimental and numerical results from
various sources will be presented where appropriate.