Isotropy, super-isotropy, and the extension of von Karman-Howarth equation: a Lundgren-equation based probability density function approach and its solution to homogeneous isotropic turbulence

We analyze homogeneous isotropic turbulence (HIT) considering Lundgren's (1967), infinite probability density functions (PDF) integro-differential equation hierarchy (IDE), which is a complete description of turbulence statistics. Since there is no mean velocity, the one-point PDF equation vanishes for HIT and the ensuing two-point equation, allows a spherical dimensional reduction. For further dimensional reduction of the higher multi-point equation, we introduce the new concept of super-isotropy. This leads to another significant dimensional reduction and each of the infinite equations then depends only on the spherical radius and the spherical velocity as sample-space variables. The corresponding side conditions of the PDF hierarchy are also derived. To solve the linear system, we formally introduce (i) a product ansatz and (ii) the superposition principle. Using the product ansatz together with the permutation side condition, we derive a new scalar IDE for the PDF that is truly closed, but non-linear. Further side conditions necessitate the superposition of solutions of the latter IDE. With the above approach, we have significantly generalized the Karman-Howarth equation.