Phase space analysis of the heat transfer in a shearless particle-laden turbulent flow

We perform a statistical analysis of single-particle probability density function (pdf) of inertial heavy particles transported by a homogeneous and isotropic, statistically steady turbulent velocity in a thermally inhomogeneous flow which produces a time evolving thermal shearless mixing layer. Numerical results of direct numerical simulations (DNSs) within the Eulerian-Lagrangian point particle approach are used to compute the unclosed terms in the pdf transport equation. We observed that particle temperature derivative d(\Theta_p)/dt conditional on particle position X_p, temperature \Theta_p, and velocity V_p scaled with t^{-1} for any particle Stokes number. This statistical moment is visualized in a reduced order 3-dimensional space, including particle temperature, its position, and velocity in the direction that the temperature field has an inhomogeneity and a thermal mixing layer develops in a self-similar manner. Through the two- and three-dimensional visual inspection of the pdf evolution in the phase space, a self-similar behavior has been observed independent of particle inertia and thermal inertia. We present the two-dimensional map of pdf at the center of thermal mixing at different times and show how the evolution in phase space develops self-similarly for different particle Stokes numbers. Furthermore, a three-dimensional visualization in the whole phase space of the pdf at different times is presented. We also show that the whole dynamics of the flow regime in the physical space can be expressed in a quasi self-similar form and how the delay introduced by the particle thermal inertia allows for self-similarity only in the asymptotic limit of long time.