Particle concentration in homogeneous shear turbulence simulated via Lagrangian and equilibrium Eulerian approaches

Direct numerical simulation is conducted to study particle concentration in particle-laden homogeneous shear turbulence. The equilibrium assumption in which the velocity of particles is approximated in terms of the velocity and acceleration of the fluid phase is assessed for this flow by analyzing the radial distribution function (RDF), the probability density function (PDF) of the number density of particles, the autocorrelation of particle Lagrangian velocity, and the particle turbulent diffusivity for various particle time constants. A good agreement between exact and equilibrium Lagrangian particles is observed for these statistics when nondimensional particle time constant tau(*)(p)< 0.6. In this work, we also introduce a new methodology for measuring the anisotropy of the Lagrangian particle concentration through calculating two-dimensional RDFs. It is shown that particles are the most and least accumulated in the streamwise and cross-stream directions, respectively. Also, a methodology is introduced to extract a spatially smooth number density of particles from discrete Lagrangian particles by employing a locally averaged formulation. In this work, the equilibrium Eulerian equations are reformulated for the case of homogeneous shear turbulence as well and the concentration of particles are simulated in a Eulerian framework by solving the corresponding transport equation. A good agreement between the exact Lagrangian and equilibrium Eulerian results is observed. The governing equations of the fluid phase along with the concentration transport equation of the particle phase are solved by a pseudospectral method.