Two-point Statistical Turbulence Closure in Physical-space

This work presents a two-point statistical closure for incompressible homogeneous isotropic turbulence (HIT) using a formulation that does not require transformation to wave-space. The formulation mimics the ideas from the spectral Eddy-Damped Quasi-Normal Markovian (EDQNM) closure, where instead of relying on the Fourier transform to convert the Navier-Stokes PDE into an ODE, a discrete approach is taken and the evolution equation of the longitudinal function, the analogue of HIT energy spectrum in physical space, is derived. This method preserves the near-exactness of the linear terms, a key feature of RDT. Several key differences arise from the physical treatment, such as the need to solve a matrix exponential in the evolution equation and appearance of 3D integrals due to non-locality of the pressure-Poisson equation. This new method allows one to incorporate non-local length-scale information naturally into the evolution equations for turbulence statistics. This framework will eventually be extended to inhomogeneous flows and will be beneficial in applications where spectral-methods are mathematically ill-conditioned such as compressible flows with discontinuities.