Forcing in DNS of stationary isotropic turbulence and its effects on the closure of diffusion current in the PDF kinetic equation for high-inertia particle pairs

Abstract

The relative motion of inertial, monodisperse particle pairs in stationary isotropic turbulence is investigated by evolving the Langevin stochastic differential equations governing the pair relative velocities and separations in the limit of Stokes number ≫1. The stochastic force in the Langevin equations is evaluated from the pair relative-velocity-space diffusivity that is equal to 1/τv2 multiplied by the time integral of the Eulerian two-time correlation of fluid velocity differences seen by particle pairs that are nearly stationary (τv is the particle viscous relaxation time). The Eulerian two-time correlation of the fluid velocity-difference field is computed using direct numerical simulation (DNS) of isotropic turbulence seeded with fixed particles. Since numerical forcing of the energy-containing eddies is employed to achieve stationarity in DNS, the impact of the forcing scheme on the two-time correlation is quantified as a function of separation r and the Taylor micro-scale Reynolds number Reλ. As the DNS-computed correlation is needed to evaluate the stochastic force in Langevin simulations (LS), a key objective is to quantify the effects of forcing scheme on the LS predictions of pair relative motion. Deterministic forcing (DF) and stochastic forcing (SF) are employed to achieve stationarity in the DNS runs. In SF, one needs to specify the time scale Tf of the Uhlenbeck–Ornstein processes that constitute the forcing. The Eulerian two-time correlations were computed using both DNS with DF and DNS with SF, the latter for Tf=TE/4,TE/2,TE,2TE, and 4TE, where TE is the large-eddy turnover time obtained from the corresponding DNS with DF. It is seen that the correlations obtained from DNS with DF are higher than those from DNS with SF for all five Tf’s, the differences being substantially more pronounced at larger separations and for higher Reλ. The two-time correlations computed using DNS with DF and DNS with SF were then utilized to perform Langevin simulations for particle Stokes numbers based on Kolmogorov time scale, Stη≥10. For Reλ=80, the pair relative-velocity variances predicted by the Langevin simulations based on DF were not significantly different from those predicted by the Langevin simulations based on SF. However, for Reλ=130 and Stη=10,20, the variances obtained from LS based on DF were significantly higher than the variances from LS based on SF. The overprediction of relative velocity variances by LS-DF may be attributed to the slower decay of Eulerian two-time correlations when deterministic forcing is employed. The effects of forcing scheme on the radial distribution function (RDF) computed from the Langevin simulations are less pronounced compared to its effects on the variances. Nevertheless, for Reλ=130 and separations r=L/2 and L, where L is the integral length scale, the RDF values obtained from LS based on DF were higher than those from LS based on SF.