Application of Millionschikov's Hypothesis to Homogeneous and Isotropic Turbulence Possessing Helicity

Abstract In this manuscript we construct a deductive theory for the stationary homogeneous turbulence, which is invariant under translations and rotations but lacks reflexional symmetry, considering the appropriate forms for the second-and third-order correlation functions. Appropriate invariant forms are considered for some second and third order correlation tensors, formed from the fluctuating components of velocity, acceleration, components of external force and pressures. In the light of the formulation of Millionschikov's hypothesis we would point out further the treatment of first author [4] on the final period decay of isotropic turbulence using the points-merging technique employed here. We include also some enlightments of the process of future work that could be undertaken in this field of research. Mathematics Subject Classification: 76F05, 76D0

An overview of Millionschikov's quasi-normality hypothesis applied to turbulence

In this paper, we examine the zero-fourth cumulant approximation that was applied to fluctuating velocity components of homogeneous and isotropic turbulence by M.D. Millionschikov. Since the publication of the remarkable paper of Millionschikov, many authors have applied this hypothesis to solve the closure problem of turbulence. We discuss here various studies by the other authors on the developments of this hypothesis and their applications to the incompressible velocity temperature, hydrodynamic and magnetohydrodynamic fluctuating pressure fields and the general magnetohydrodynamic turbulence field. Lastly, we discuss broadly the computational difficulties that arise in turbulence problems and their possible refinements. We include also some enlightments of the process of future work that could be undertaken in this field of research