Lie Symmetries of Characteristic Function Hierarchy in Compressible Turbulence

Compressible turbulence is characterized by fluctuations in velocity as well as thermodynamic quantities. A unified framework to study flow and thermodynamics statistics in compressible turbulence is facilitated by a probability density function (PDF) or a characteristic function (CF) approach. In compressible flow regime, characteristic function offers advantages over its Fourier transform pair, the PDF. While the governing PDF equations are non-local in nature, CF equations do not contain any integral terms, thus simplifying the symmetry analysis. We compute the point-symmetries of the multi-point CF hierarchy by generalizing the symmetry groups obtained from single-, two- and three-point CF equations. As the CF equations are linear in nature, 'superposition principle' leads to two symmetry groups in addition to the ones seen in Euler equations. The validity of each of the symmetries at different fluid and flow parameters is discussed. We also present the group invariant solutions of various key statistics of compressible turbulence.