The hierarchy of multi-point probability density functions and characteristic functions in compressible turbulence

Abstract

The hierarchies of equations for a general multi-point probability density function (PDF) and its characteristic function (CF) are derived for compressible turbulent flows, obeying the ideal gas law. The closure problem of turbulence is clearly exhibited in each of the approaches, with n-point statistics being dependent on the (n + 1)-point statistics and, for some cases, even the (n + 2)-point statistics. When dynamic viscosity and heat conductivity are dependent on temperature as a power-law, the CF hierarchy could contain fractional derivatives if the exponent is a non-integer. The additional conditions satisfied by all the PDFs and CFs in both the hierarchies are also prescribed. The PDF and CF equations derived in this paper, with the unclosed terms explicitly written in terms of higher order PDF/CF, act as a starting point in constructing symmetry-based invariant solutions of compressible turbulence, analogous to the works of Wacławczyk et al. [“Statistical symmetries of the Lundgren–Monin–Novikov hierarchy,” Phys. Rev. E 90, 013022 (2014)] and Oberlack and Rosteck [“New statistical symmetries of the multi-point equations and its importance for turbulent scaling laws,” Discrete Contin. Dyn. Syst. 3, 451–471 (2010)] for incompressible turbulence.

On the Lie symmetries of characteristic function hierarchy in compressible turbulence

“…In compressible turbulence, as it is critical to examine density, pressure and temperature statistics in addition to velocity, we utilise the following definition for the multi-point CF [16]:…”

“…As a result, the moment hierarchy in compressible turbulence is more involved than its incompressible counterpart. Probability density function (PDF) approach has been utilised to examine statistics of compressible non-reacting and reacting flows [5,4,7,16]. The PDF approach circumvents the issue of multiple hierarchies and gives rise to a single hierarchy of multi-point PDF equations [16].…”

“…Probability density function (PDF) approach has been utilised to examine statistics of compressible non-reacting and reacting flows [5,4,7,16]. The PDF approach circumvents the issue of multiple hierarchies and gives rise to a single hierarchy of multi-point PDF equations [16]. In a similar manner, characteristic function (CF), the Fourier transform of the PDF, gives rise to an alternate hierarchy of infinite multi-point CF equations [16].…”

“…The PDF approach circumvents the issue of multiple hierarchies and gives rise to a single hierarchy of multi-point PDF equations [16]. In a similar manner, characteristic function (CF), the Fourier transform of the PDF, gives rise to an alternate hierarchy of infinite multi-point CF equations [16]. Characteristic functional can also be utilized for the examination of statistics, and the resulting equation is a functional differential equation [7].…”

Edward E. O'Brien contributions to reactive-flow turbulence

“…External specification of a timescale is not necessary and the agreement of predictions with available experimental data for a mixing layer is satisfactory. The hierarchy of multipoint pdf transport equations in compressible turbulence has been investigated, among others, by Fox 68 and more recently by Praturi et al 69…”