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.