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Multipoint Statistical Turbulent Dynamics from Hopf Equation Closures

Mark Warnecke

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.

Multipoint Statistical Turbulent Dynamics from Hopf Equation Closures

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 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 -point Hopf equation. Finally, (iii) an example of the method is provided to analytically determine the -point structure function transition between the known -point structure function and the -point fusion rules from the closed -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 -point velocity increment Hopf equation is closed, its solution can be numerically approximated. It is expected that similar methods, applied here to obtain the -point structure function and -point structure function transition, can be used to obtain further analytical predictions of various multipoint quantities to deepen our understanding of turbulence.
Paper Structure (8 sections, 83 equations, 1 figure)

This paper contains 8 sections, 83 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Structure Function Hopf Equation Closure and Generalization
  4. $N$-point SY Hopf Closure
  5. 3-point Structure Function Transition
  6. Conclusion
  7. $N$-point velocity difference Hopf equation
  8. Structure Function Equation from the Pressure Closure

Figures (1)

  • Figure 1: The normalized even $3$-point structure function plotted for various $n$ and $p$ using the analytical derivation from the Hopf closure (solid lines) and from about $80,000$ samples of the iso32768 DNS dataset from the Johns Hopkins Turbulence Database (JHTDB), for details about the simulation see JHUTurbulenceYeung2025ExascaleTurbulence.