A numerical source of small-scale number-density fluctuations in Eulerian–Lagrangian simulations of multiphase flows

Eulerian–Lagrangian simulations of multiphase flow are known to suffer from two errors that can introduce small-scale fluctuations in the number-density of the dispersed phase. These errors can be reduced by increasing the number of particles in the simulation. Here, we present results to demonstrate that a third error exists that can also generate small-scale number-density fluctuations. In contrast to the two known errors, this error cannot be lowered by increasing the number of particles. Analysis shows that this error is caused by spatial variation at the subgrid scale in the interpolation error of the fluid velocity to the particle location. If the particle velocity divergence is zero, the particle concentration error remains at the subgrid scale. However, if particles preferentially accumulate either due to their inertia or due to divergence of the underlying fluid-velocity field, this error manifests as number-density fluctuations on the grid scale. The only mechanism of reducing these errors is through higher-order accurate interpolation. By studying two model problems, estimates for the errors are derived. These estimates are shown to be quite accurate for simulations of shock and expansion waves interacting with particles.