Turbulence modulation in particle-laden channel flow: the particle inertial effects

The particle inertial effects on turbulence modulation in particle-laden channel flow are investigated through four-way coupled point-particle direct numerical simulations, with the mass loading fixed at $0.6$ and friction Stokes number $St^+$ varying from $3$ to $300$. A full transition pathway is realized in sequence from a drag-enhanced to a drag-reduced flow regime, before asymptotically approaching the single-phase state as $St^+$ increases continuously up to 300. For the first time, a set of transport equations for the particle phase is derived analytically to interpret the inter-phase coupling, in the context of the point-based statistical description of particle-laden turbulence. By virtue of this, two dominant mechanisms are substantially identified and quantified: a positive, particle-induced extra transport, which decreases monotonically with $St^+$, and a negative, particle-induced extra dissipation, which depends non-monotonically on $St^+$. The coupling of these two mechanisms leads to a direct contribution of particle phase to the shear stress balance, turbulent kinetic energy, and Reynolds stress budgets. As a consequence, with the increase of particle inertia, the self-sustaining cycle of near-wall turbulence transitions from being augmented to being suppressed and, eventually, recovers to the single-phase situation. This gives rise to an indirect effect, manifested by the non-monotonic modification of Reynolds shear stress and turbulent production rate. Taken together, comprehensive interplays between particle-modified turbulent transport, particle-induced extra transport and dissipation are analyzed and summarized, providing a holistic physical picture composed of consistent interpretations of turbulence modulation.