On the decay rate of isotropic turbulence laden with microparticles

This paper is concerned with the two-way coupling effects on the decay rate of isotropic turbulence laden with solid spherical microparticles whose response time, $τp,$ is much smaller than the Kolmogorov time scale $(τp≪τk).$ The particles volumetric concentration, $C0,$ is small enough $(C0∼10−4)$ to neglect particle–particle interactions, and the material density of the particle is much larger than the fluid density $(δ=ρp/ρf≫1).$ We obtain asymptotic analytical solutions for the instantaneous particle velocity and kinetic energy spectrum, in the limit $τp≪τk,$ which indicate that the two-way coupling increases the fluid inertia term in the fluid momentum equation by the factor $(1+C0δ).$ Consequently, the high-wave number components of the spectra of turbulence energy and dissipation develop in time as $∼exp[−2νk2t/(1+C0δ)].$ The net result is a reduction of the decay rate of turbulence energy compared to that of particle-free turbulence (i.e., the one-way coupling case where $C0δ≡0)$⁠. We also perform direct numerical simulation (DNS) of isotropic turbulence laden with microparticles, using the two-fluid (TF or Eulerian–Eulerian) approach developed in an earlier study [Druzhinin and Elghobashi, Phys. Fluids 3 (1998)]. Excellent agreement is achieved between the DNS results and the analytical solution for the particle kinetic energy spectrum. The DNS results show that the two-way coupling reduces the decay rate of turbulence energy compared to that of one-way coupling. In addition, we compare the temporal developments of the turbulence kinetic energy and its dissipation rate, obtained from DNS using TF, with those from DNS using the trajectory (Eulerian–Lagrangian) approach. Satisfactory agreement is achieved between the two approaches, with TF requiring considerably less computational time.