We extend the resolvent framework 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, Froude numbers Frz = −4, −0.4, 0.4, 4
and 1/Frz = 0 (gravity omitted). The governing equations are written in input–output
form and a singular value decomposition is performed on the resolvent operator. As
for single-phase flows, the operator is low rank around the critical layer, and the true
response can be approximated using one singular vector. Even with a crude forcing model,
the formulation can predict physical phenomena observed in Lagrangian simulations,
such as particle clustering and gravitational effects. Increasing the Stokes number shifts
the predicted concentration spectra to lower wavelengths; this shift also appears in the
direct numerical simulation spectra and is due to particle clustering. When gravity is
present, there are two critical layers, one for the concentration field, and one for the
velocity field. For upward flow, the peak of concentration fluctuations shifts closer to
the wall, in agreement with the literature. We explain this with the aid of the different
locations of the two critical layers. Finally, the model 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. The
application of the resolvent framework to higher St+ flows in combination with Lagrangian
simulations is also discussed.