Extension of resolvent framework to turbulent flows laden with low-inertia particles

We extend the resolvent framework, originally formulated for single-phase turbulent flows, to two-phase flows with low-inertia particles. The particle velocities are modelled using the equilibrium Eulerian model. We analyse the turbulent flow in a vertical pipe with Reynolds number of $5300$ (based on diameter and bulk velocity), for Stokes numbers St⁺=0-1 and different Froude numbers, as well as the special case where gravity is omitted (1/Fr_z=0). The resolvent formulation can predict (even with a crude forcing model) the physical phenomena observed in Lagrangian simulations of particulate flows, such as particle clustering and gravitational effects. As for single-phase flows, the operator is low rank over a significant range of wavenumbers and frequencies around the critical layer. When gravity is present, there are two critical layers, one for the velocity field and one for the concentration field. The model also correctly predicts the interaction of near-wall vortices with particle clusters. Overall, the resolvent operator provides a useful framework to explain and interpret many features observed in Lagrangian simulations. Further work will consider the extraction of the operator directly from the Lagrangian particle positions. Success in this endeavour will make the resolvent framework applicable to two-phase flows with high St⁺ number particles.