Higher-Order Correlations Between Thermodynamic Fluctuations in Compressible Aerodynamic Turbulence

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.