On the scalar probability density function transport equation for binary mixing in isotropic turbulence at supercritical pressure

Abstract

Effects of Soret cross-diffusion, whereby concentration diffusion occurs in the presence of either temperature or pressure gradients, are investigated in the context of the binary isotropic turbulent mixing problem at supercritical pressure with emphasis on results relevant to probability density function (PDF) modeling. Direct numerical simulations (DNS) are conducted for compressible isotropic turbulent binary mixing of heptane with nitrogen at mean pressure and temperature equal to 45 atm and T=700 K, respectively. The formulation is based on a cubic real gas state equation, and includes generalized forms for heat and mass diffusion derived from nonequilibrium thermodynamics and fluctuation theory. Results from two simulations at 1923 resolution are compared. One case is based on the complete diffusion formulation, whereas in the second simulation only “standard” Fickian and Fourier mass and heat flux terms are considered as a basis for comparisons. The evolutions of the two mixing processes are shown to be nearly the same at early times; however, at long times the pressure gradient based Soret diffusion acts as a forcing function resulting in a statistically stationary scalar fluctuation distribution. In this case the scalar variance achieves a stationary value, in contrast to the exponential decay associated with purely Fickian diffusion. The mechanical to scalar time scale ratio, CY=τu/τY, is increased by a factor of approximately 3 by Soret effects during this latter mixing regime. Soret diffusion is shown to substantially alter the forms of the conditional expected diffusion and the conditional expected dissipation at these long times. A commonly used model based on an assumed proportionality between the conditional diffusion and the scalar fluctuation is tested, and is found to perform poorly in the presence of cross diffusion at long mixing times. Several new conditional expectations resulting from cross diffusion are measured. Implications of the results for extension of PDF models to the high pressure regime are discussed.