Effects of deterministic and stochastic forcing schemes on the relative motion of inertial particles in DNS of isotropic turbulence

Abstract

In direct numerical simulations (DNS) of homogeneous isotropic turbulence, statistical stationarity is achieved by artificially forcing the low-wavenumber scales of turbulence. In this study, the effects of two such forcing schemes on the relative positions and velocities of heavy, monodisperse, “point” particles in isotropic turbulence are studied. The first forcing scheme considered is a deterministic method that maintains the turbulent kinetic energy constant by resupplying the energy dissipated during a time step to the velocity components in a low-wavenumber band. The second is the stochastic forcing scheme of Eswaran and Pope [Comput. Fluids 16, 257 (1988)] that involves adding a forcing acceleration to the fluid momentum equation at the low wavenumbers. The forcing acceleration is based on a combination of six independent Uhlenbeck-Ornstein (UO) random processes. An input parameter to the stochastic scheme is the correlation time scale Tf of the UO processes. Among our objectives is to assess the effects of varying Tf on the relative-motion statistics of inertial particle pairs.Direct numerical simulations of homogeneous, isotropic turbulence containing disperse particles were undertaken using both the deterministic and stochastic forcing schemes for three grids sizes (1283, 2563, and 5123). For each grid size, DNS runs based on the stochastic forcing were performed for five values of the UO correlation time scale Tf = TE/4, TE/2, TE, 2TE, and 4TE, where TE is the large-eddy time scale obtained from the DNS run with deterministic forcing at the same grid size. Thus, six DNS runs (one with deterministic forcing and five with stochastic forcing) were performed for each grid resolution, with Reλ held nearly constant (varying by 2–3%) among these runs. The nominal values of Reλ were ≈ 80, 131, and 210 for the 1283, 2563, and 5123 grids, respectively. In each DNS run, heavy, monodisperse particles were tracked corresponding to twelve Stokes numbers ranging from Stη = 0.05 to 40, where Stη is the Stokes number based on the Kolmogorov time-scale. The motivation was to understand how the applied forcing impacted particle-pair relative motion in the three main Stokes number regimes, namely Stη < 1, Stη ∼ 1, and Stη > 1. We focus our attention on three statistics quantifying the relative positions and velocities of particles: the radial distribution function (RDF), the variance 〈Ur2〉 of the component of pair relative velocity along the separation vector (Ur), and the probability density function (PDF), P(Ur). Using the RDF and the PDF P(Ur), we also computed the particle collision kernel. The pair statistics obtained from the various forcing cases are compared among each other, as well as with those from the DNS study of Ireland et al. that employed only the deterministic forcing [J. Fluid Mech. 796, 617 (2016)].At all three Reynolds numbers, we find that the forcing method and the time scale Tf have a noticeable effect on the RDFs for both Stη < 1 and Stη > 1. For Stη < 1 (at a given Reλ), the differences between the RDFs for the various forcing cases increased with Stokes number, reaching a maximum around Stη = 0.4. However, for Stη ∼ 1 (Stη = 0.7 and 1), the RDFs seem to be relatively unaffected by the forcing schemes. When Stη > 1, we find that the RDFs for the various forcings differed the most at Stη = 2, with the differences decreasing thereafter for higher Stokes numbers. When considering the effects of Reλ, it is seen that the RDFs computed from the DNS with deterministic forcing were more sensitive to Reλ variation than those obtained from DNS with stochastic forcing.For Stη < 1, the variance 〈Ur2〉 showed only a weak sensitivity to the forcing scheme at the two lower Reynolds numbers, but a much clearer dependence is seen for Reλ = 210. When Stη ≥ 1, the forcing has a relatively significant impact on the variances at all Reynolds numbers, and these effects are amplified as Reλ is increased. Using the RDF and the PDF P(Ur), we also computed the collision kernels. At the two lower Reλ, collision kernels were found to be weakly dependent on Reλ for Stη < 1, but showed significant increase with Reλ for Stη ≳ 1. However, when Reλ is increased to 210, the collision kernel is seen to increase at all Stokes numbers.