# Higher-Order Correlations Between Thermodynamic Fluctuations in Compressible Aerodynamic Turbulence

## TL;DR

This paper explores how thermodynamic variables like pressure, density, and temperature fluctuate in turbulent flows, focusing on higher-order correlations and nonlinear effects.

## Contribution

The study derives exact equations for higher-order correlations and validates them using direct numerical simulations of compressible turbulence.

## Key findings

- Nonlinear terms from density-temperature fluctuations significantly affect third-order correlations.
- Exact equations linking third-order moments are validated using DNS data for compressible turbulent flows.
- Linearized relations help identify independent moment 4-tuples for further analysis.

## Abstract

This paper studies the exact and approximate relations between fluctuations in thermodynamic variables (pressure, density and temperature) that are imposed by the dilute-gas (Z=1) equation-of-state (EoS), which is a satisfactory approximation of air thermodynamics for a wide range of pressures and temperatures. It focuses on triple- and higher-order correlations, extending previous studies that concentrated on second-order moments, with emphasis on the mathematical relations, which are generally valid independently of the particular flow configuration. Exact equations are developed both involving only single-variable moments and relating the correlations between variables. These contain nonlinear terms generated by the density-temperature fluctuation product in the fluctuating EoS. The importance of the nonlinear terms in the 6 exact equations between the 10 third-order moments is assessed using DNS (direct numerical simulation) data for compressible turbulent plane channel (TPC) flows and analyzed using general statistical inequalities involving third-order and fourth-order moments. The corresponding linearized system between third-order moments is studied to determine approximate relations and 4-tuples of linearly independent moments. These mathematical tools are then used to analyze TPC flow DNS data on the triple correlations between the thermodynamic variables.