Multipoint Statistical Turbulent Dynamics from Hopf Equation Closures

TL;DR

This paper develops a first-principles closure for multipoint turbulence statistics using the velocity increment Hopf equation, providing analytical solutions and promising DNS agreement, advancing theoretical turbulence understanding.

Contribution

It generalizes the closure of the structure function equation to the velocity increment Hopf equation and extends it to N-point equations, enabling analytical and numerical analysis of multipoint turbulence statistics.

Findings

Analytical 3-point structure function transition derived.

Close form Batchelor interpolation matches DNS data.

Method enables further multipoint turbulence predictions.

Abstract

Obtaining accurate multipoint statistics of turbulence is computationally very expensive and therefore these statistics have remained largely unexplored from a theoretical standpoint. In this paper, (i) a first-principles-based closure of the $n$th-order structure function governing equation proposed by Sreenivasan & Yakhot (2021) is generalized to a closure of the velocity increment Hopf equation itself. Then (ii) the closure is further generalized to the $N$-point Hopf equation. Finally, (iii) an example of the method is provided to analytically determine the $3$-point structure function transition between the known $2$-point structure function and the $3$-point fusion rules from the closed $(N=3)$-point velocity increment Hopf equation. The analytical solution takes the form of a Batchelor interpolation and shows promising agreement with preliminary DNS data for the cases examined. Since the $N$-point velocity increment Hopf equation is closed, its solution can be numerically approximated. It is expected that similar methods, applied here to obtain the $2$-point structure function and $3$-point structure function transition, can be used to obtain further analytical predictions of various multipoint quantities to deepen our understanding of turbulence.