Two-point Turbulence Closures in Physical Space

TL;DR

This work develops a predictive two-point turbulence closure formulated entirely in physical space, starting from incompressible HIT to enable extensions to more complex flows where spectral methods struggle. Building on EDQNM ideas, it employs quasi-normality, Markovian modification, and eddy relaxation to close the hierarchy of two-point moments while preserving linear terms exactly and maintaining nonlocal pressure effects through a physical-space formulation. Verification against DNS and spectral EDQNM demonstrates accurate reproduction of HIT statistics, inertial-range scaling, and stationary forcing behavior, validating the approach for predictive use. The framework offers a pathway to handle anisotropy and inhomogeneity with explicit two-point statistics, albeit with significant computational cost, which can be mitigated through FMM, SO(3) decompositions, and precomputation strategies, broadening applicability to compressible and discontinuous flows where spectral methods falter.

Abstract

This work presents a predictive two-point statistical closure framework for turbulence formulated in physical space. A closure model for ensemble-averaged, incompressible homogeneous isotropic turbulence (HIT) is developed as a starting point to demonstrate the viability of the approach in more general flows. The evolution equation for the longitudinal correlation function is derived in a discrete form, circumventing the need for a Fourier transformation. The formulation preserves the near-exact representation of the linear terms, a defining feature of rapid distortion theory. The closure of the nonlinear higher-order moments follows the phenomenological principles of the Eddy-Damped Quasi-Normal Markovian (EDQNM) model of Orszag (1970). Several key differences emerge from the physical-space treatment, including the need to evaluate a matrix exponential in the evolution equation and the appearance of triple integrals arising from the non-local nature of the pressure-Poisson equation. This framework naturally incorporates non-local length-scale information into the evolution of turbulence statistics. Verification of the physical-space two-point closure is performed by comparison with direct numerical simulations of statistically stationary forced HIT and with classical EDQNM predictions for decaying HIT. Finally, extensions to inhomogeneous and anisotropic turbulence are discussed, emphasizing advantages in applications where spectral methods are ill-conditioned, such as compressible flows with discontinuities.

Figures (10)

• Figure 1: Change of the longitudinal and lateral functions for various times using the linearized HIT equations with local differentiation matrices. The Batchelor spectrum was used as the initial condition. Blue lines represent the physical space model and black the spectral model.
• Figure 2: Normalized derivatives of the longitudinal function showing decay at large r-values and sharp features for realistic spectra with a Kolmogorov inertial range.
• Figure 3: Decaying longitudinal and lateral functions with initial Batchelor spectrum. Blue lines represent the physical space model and black the spectral model. Results are shown for 2.5 eddy turn-over times.
• Figure 4: Energy spectrum $E(\kappa)$ predictions from the spectral and physical closures for various time intervals starting from the Batchelor spectrum. Blue lines represent the physical space model and black lines the spectral model. Physical space model results are truncated at higher wavenumbers due to non-physical artifacts arising from the transformation.
• Figure 5: Normalized turbulent kinetic energy and integral length scale evolution for decaying HIT with initial Batchelor and Saffamn spectrum. Time scaling laws (red lines) are plotted at $t=3.5$, when the inertial range is sufficiently established. Blue lines represent the physical space model and black the spectral model.