Inequalities and realizability constraints between thermodynamic fluctuations in compressible aerodynamic turbulence

The dilute-gas equation-of-state imposes exact nonlinear relations between variances and correlations of the fluctuations of thermodynamic variables. Correlation coefficients in particular satisfy a nonlinear compatibility relation. Applying the positivity of variances and the covariance inequalities on these exact nonlinear relations yields inequalities that constitute realizability constraints for the thermodynamic turbulence structure, valid for any compressible turbulent dilute-gas flow. These relations are illustrated using direct numerical simulation data for compressible turbulent plane channel and boundary-layer flows. At the limit of moderately compressible turbulence, the exact nonlinear identities and inequalities can be linearized yielding approximate relations that are practical in approximately mapping the thermodynamic turbulence structure in terms of two independent structure parameters. The nonlinear terms that determine the range of validity of the linear moderately compressible regime are assessed in terms of fluctuation amplitudes and order-3 correlation coefficients and correlated with the external flow Mach number.