Date of Award


Document Type


Degree Name

Master of Science in Engineering (MSE)


Mechanical and Aerospace Engineering

Committee Chair

Sarma Rani

Committee Member

Kader Frendi

Committee Member

S. S. Ravindran


Turbulence., Navier-Stokes equations--Numerical solutions., Fluid mechanics.


In direct numerical simulations (DNS) of isotropic turbulence, statistical stationarity is achieved by artificially forcing, i.e. adding energy to, the low-wavenumber scales of turbulence. In this work, the effects of two such forcing schemes on the relative positions and velocities of heavy, monodisperse, “point” particle pairs are studied. The first forcing scheme considered is a deterministic method in which the turbulent kinetic energy is maintained constant [34] by resupplying the energy dissipated during a time step to the velocity components in a low-wavenumber band. The second is the stochastic forcing method of [13], where one adds a random acceleration to the fluid momentum equation at the low wavenumbers. The stochastic approach involves three main input parameters that allow us to estimate, a priori, the approximate value of the Taylor micro-scale Reynolds number Reλ that can be obtained in a DNS run. One of these parameters is the correlation time scale Tf of the random acceleration. Among our objectives is to assess the effects of varying Tf on the relative-motion statistics of particle pairs. Direct numerical simulations of isotropic turbulence containing disperse particles were undertaken using both the deterministic and the stochastic forcing schemes for three grids sizes (1283 , 2563 , and 5123 ). At each grid size, DNS runs based on the stochastic forcing were performed for five values of the forcing time scale Tf = TE/4, TE/2, TE, 2TE, and 4TE, where TE is the large-eddy time scale obtained from the corresponding DNS run with deterministic forcing. Thus, six DNS runs (one deterministic and five stochastic) were performed for each grid resolution, with Reλ held nearly constant (varying by less than 5%) among these runs. The nominal values of Reλ were ≈ 80, 131, and 210 for the three grids. In each DNS run, heavy, monodisperse particles were tracked corresponding to twelve Stokes numbers ranging from Stη = 0.05 to 10, where Stη is the Stokes number based on the Kolmogorov time-scale τη. The motivation was to determine how the applied forcing impacted particle-pair relative motion in the Stokes number regime of Stη < 1. We focus our attention on three statistics quantifying the relative positions and velocities of particles: the radial distribution function (RDF), and and the probability density function (PDF) of the component of pair relative velocity along the separation vector (Ur). Using the RDF and the PDF P(Ur), we computed the particle collision kernel for the various DNS cases. The pair statistics are also compared with those from the deterministic DNS study of [15], whose objective was to study the effects of variation in Reλ and Stη on the relative-motion statistics. At all three Reynolds numbers, we find that the forcing method and the time scale Tf have a noticeable effect on the RDFs for 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 considering the effects of Rey, it is seen that the RDFs computed from the DNS with deterministic forcing were more sensitive to Rex variation than those obtained from DNS with stochastic forcing of various time scales. For Stn < 0.4, we notice that the deterministic RDFs decreased significantly as Rex varied from 80 to 131, but then increased as Re, increased to 210. In the study of (15), the RDFs were found to be essentially independent of Rex for Str 3 1. But, a distinguishing feature of their study was that they hold the fluctuating RMS velocity Urms, mean dissipation rate (€), and the resolution parameter kmaxn constant even as Re, is increased (kmax and n are the maximum resolved wavenumber and the Kolmogorov length scale, respectively). In the current study, however, our focus was only on keeping Rex fixed across the various forcing cases. The effects of forcing, Stokes number, and pair separation on the relative velocity PDFs are shown. It is seen that the shapes of the PDFs were little affected by forcing. Using the RDF and P(U.,), we also computed the collision kernels. At the two lower Rex, collision kernels were found to be weakly dependent on Re, for Stn < 1, but showed significant increase with Re, for Stn 31. However, when Rex is increased to 210, the collision kernel is seen to mcrease at all Stokes numbers.



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