Persistence of reflectional non-symmetry in freely-decaying homogeneous turbulence

We consider freely-decaying incompressible homogeneous helical turbulence whose energy spectrum E(k) and helicity spectrum H(k) are respectively given by E(k)=C k^2 + o(k^2) and H(k)= C_h k^3 +o(k^3) at k -> 0. Here, k is the wavenumber, C is a dynamical invariant, and C_h is a k-independent constant. It is shown that C_h is another dynamical invariant, the O(k^0)-tem of the velocity spectral correlation tensor is time-independent, and the term may be non-reflectional symmetric. A theoretical analysis based on a flow self-similarity and the time-independence suggests the persistence of the reflectional non-symmetry for fully-developed helical turbulence. The decay rates of energy and helicity are obtained by a simple dimensional analysis. We examine the theoretical results by the use of direct numerical simulation of incompressible helical turbulence in a periodic box.