The influence of particle inertia on the two-way coupling and modification of isotropic turbulence by microparticles

The objective of this paper is to examine the modulation of isotropic decaying turbulence by particles whose diameter is smaller than the Kolmogorov length scale, and their response time, τp, is smaller than the Kolmogorov time scale, τk (hence microparticles), and the influence of increasing the particle inertia on the two-way coupling. The particle volume fraction is considered small enough, so that particle–particle interactions are neglected. On the other hand, the particle material density ρp≫ρf, the fluid density, and the mass loading of the particles is large enough to modify the carrier flow. The particle Reynolds number is smaller than unity, and the gravitational settling of the particles is neglected. We obtain an asymptotic analytical solution describing the spectrum of the instantaneous two-way coupling source term, Ψp(k,t), in the equation for the fluid turbulence kinetic energy (TKE) spectrum, E(k,t), as a series in powers of the ratio (τp/τk). Recent results of Druzhinin and Elghobashi [Phys. Fluids 11, 602 (1999)] for particles whose τp≪τk show that to the zeroth order in (τp/τk), Ψp(k,t) is proportional to the fluid spectral dissipation function, ε(k,t). In the present paper, the asymptotic solution is extended up to the first order in (τp/τk) and is applicable for particles with small but finite inertia. We also perform direct numerical simulation (DNS) of particle-laden isotropic turbulence using the Eulerian–Lagrangian approach. The results obtained for particles whose τp⩽0.4τk show that both the TKE and its dissipation rate, ε(t), as well as the spectral transfer of the fluid kinetic energy, are increased by the two-way coupling as compared to the particle-free case, and the increase is more pronounced for smaller τp. The asymptotic solution for the two-way coupling source term spectrum, Ψp(k,t), is found in good qualitative and quantitative agreement with the numerical results. Both the asymptotic solution and the DNS results for the instantaneous source term spectrum, Ψp(k,t), show that as the particle response time is increased, the magnitude of the maximum of Ψp(k,t) is reduced and its location is shifted toward higher wave numbers, as compared to the limiting case τp≪τk. The DNS results also show that for particles with sufficiently high inertia (whose τp⩾0.5τk), a negative peak of Ψp(k,t) is created at low wave numbers, whereas the fluid spectral energy transfer is reduced, as compared to the one-way coupling case. The development of the negative peak of Ψp(k,t) is accompanied by a well-pronounced preferential accumulation of particles. The net two-way coupling effect is the reduction of the TKE by particles with sufficiently high inertia (whose τp=0.8τk in our DNS), as compared to the particle-free flow. In this case, our results are in qualitative agreement with the DNS results of Boivin et al. [J. Fluid Mech. 375, 235 (1998)], who considered particles whose τp⩾1.26τk. Therefore, our results show that there occurs a qualitative transition in the two-way coupling effect of particles on isotropic turbulence as the particle response time is increased from τp≪τk, in the limit of microparticles, to τp≃τk, for particles with finite inertia. In the case of microparticles (whose τp≪τk), the instantaneous spectrum of the two-way coupling source term, Ψp(k,t), is positive at all wave numbers so that the particles add the energy to the fluid motion and increase the turbulence kinetic energy, as compared to the one-way coupling case. On the other hand, in the case of particles with higher inertia (whose τp≃τk), the positive contribution of the source term, Ψp(k,t), is reduced at high wave numbers whereas a negative peak of Ψp(k,t) is created at low wave numbers. In this case, the net two-way coupling effect is the reduction of the TKE by the particles, as compared to the one-way coupling case.