A unified framework to study compressible turbulence statistics

Compressible turbulence is characterised by fluctuations in thermodynamic variables of temperature, pressure and density, in addition to velocity. To perform any formal analysis on the compressible turbulence statistics, one would need to derive their governing equations. In incompressible turbulence, where only velocity statistics are of interest, many approaches such as the (i) multi-point moment, (ii) probability density function (PDF) and (iii) Hopf-functional approaches can be utilised to derive the governing equations for turbulence statistics. However, in compressible turbulence, the multi-point moment approach becomes quite cumbersome, as there are many variables of interest. Probability density functions provide a unified framework through which any order (multi-point, multi-variable) statistics equations could be obtained by simply taking the moments of the PDF equation. In this talk, we discuss how a multi-point PDF for compressible turbulence (obeying ideal gas law) can be defined and derive its governing equation. The multi-point PDF equation is integro-differential in nature and exhibits a closure problem with n-point statistics dependent on (n+1)- and (n+2)-point statistics.