Resolvent analysis of turbulent pipe flow laden with low-inertia particles*

We extend the resolvent framework to turbulent flows laden with low-inertia particles. The particle velocities are modelled using the equilibrium Eulerian model, which is assumed to be valid for Stokes numbers up to 1. We analyse a vertical turbulent pipe flow with Reynolds number equal to 5300 based on diameter and bulk velocity, Froude numbers F r = −4, −0.4, 0.4, 4 and Stokes numbers St+ = 0−1. A direct numerical simulation (DNS) for a pipe with a length of 7.5 diameters is performed with the particles released uniformly at the inlet. The resolvent formulation can reproduce the physical phenomena observed in inertial particle flows, such as localized high concentration due to the vortical centrifuge effect, turbophoresis and gravitational effects. It also reveals that upward flow increases particle concentration in the log layer while downward flow increases concentration near the centre of the pipe: both features have been observed in previous Lagrangian simulations as well as experiments. The main effect of Stokes number is the amplification of the gain for resolvent modes with smaller streamwise wavelengths, resulting in an increase of the local scale clustering of particles and turbophoresis.