Preferential concentration of particles in isotropic turbulence: a comparison of the Lagrangian and the equilibrium Eulerian approaches

Abstract

In this brief note, we present a study of the preferential concentration of particles in isotropic turbulence using the fast Eulerian method [Int. J. Multiph. Flow 27 (2001) 1199]. The advantage of the fast Eulerian method is that it obviates the need to solve additional governing equations for the particulate phase. For small particle response times, Ferry and Balachandar [Int. J. Multiph. Flow 27 (2001) 1199] show that a unique equilibrium Eulerian particle velocity field, v( x, t), independent of initial conditions can be obtained. The resultant expansion in τ p is an explicit function of the fluid velocity field, u( x, t), and its spatial and temporal derivatives. In the current work, a truncated expansion of the particle velocity field, accurate to O( τ p), is used to calculate statistics of particle dispersion and preferential accumulation in isotropic turbulence. These statistics are compared with those from the Lagrangian evolution of particles. There is good agreement for small τ p, where the theory is best applicable. Preferential concentration properties of dense particles as well as bubbles are studied. It is demonstrated that instantaneous velocities of exact Lagrangian particles agree well with those evaluated by interpolating the Eulerian particle velocity field to the particle positions. In addition, particles evolved using the Eulerian velocity field exhibit similar long-time spatial distribution as the exact particles, indicating that there is no accumulation of errors in time. Specifically, the concentration of dense particles in high strain-rate regions and of bubbles in high vorticity regions is accurately captured by the Eulerian method.