Speaker
Description
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 formulation consistent with the famous approach chosen by von Kármán and Howarth for the description of two-point velocity moments which may be derived once the PDF is known. The new reduced PDF equation thus represents a higher-level equation for the moment hierarchy. Introducing the new coordinates describing homogeneous isotropic PDFs allows us to dimensionally reduce the hierarchy, finally describing multi-point statistics in HIT with a minimal set of coordinates. For further dimensional reduction compared to the homogeneous isotropic case, invariance of the PDF under one more group of rotations per additional point 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.
