Infinite Lundgren Hierarchy of Turbulence: Isotropy, Super-isotropy and a Finite Dimensional Eigenvalue Problem

We analyze Lundgren’s infinite multipoint hierarchy of probability density functions (PDFs) in the context of homogeneous isotropic turbulence (HIT). To account for the physical properties of HIT, we express the PDF in terms of the scalar invariants under the SO(3) group of rotations, yielding new coordinates for a homogeneous isotropic PDF and leading to a dimensional reduction of the problem. Furthermore, we transform Lundgren’s hierarchy to a description in spatial increment variables in order to obtain a description consistent with the approach chosen by von Kármán and Howarth for the description of two-point velocity moments. For further dimensional reduction compared to the homogeneous isotropic case, a small scale approximation is introduced, which greatly simplifies the hierarchy. For the simplified hierarchy, a solution approach of superposed eigenfunctions is introduced. Together with the side conditions of the hierarchy, this leads to a formally closed eigenvalue problem.