Evaluation of the equilibrium Eulerian approach for the evolution of particle concentration in isotropic turbulence

In the current work, the accuracy of the equilibrium Eulerian approach in evolving the particulate concentration field is evaluated by comparing it against the Lagrangian approach, for varying particle response time and terminal velocity. In particular, we compare the statistics of preferential accumulation and gravitational settling of particles in a cubic box of isotropic turbulence. Twelve simulations corresponding to four values of nondimensional particle response time, τp=0.05, 0.1, 0.2, 0.4, and three values of nondimensional terminal velocity, |Vs|=0.5,2,4 are considered. The equilibrium Eulerian approach obviates the need to solve additional governing equations for the particle velocity field. It, however, involves evolution of the particle concentration field using the equilibrium Eulerian velocity field. A spectral diffusion term is included in the particle concentration equation to provide an essentially non-oscillatory behavior to the solution. There is good agreement between the equilibrium Eulerian and Lagrangian statistics for small particles. With increasing particle size, the equilibrium Eulerian approach tends to somewhat overestimate particle preferential concentration in regions of excess strain-rate over rotation-rate compared to the Lagrangian approach. Over the entire range of parameters considered, the equilibrium approach provides a good approximation to the actual mean and rms fluctuating settling velocities of the particle.