Quantifying the errors of the particle-source-in-cell Euler-Lagrange method

Abstract

The particle-source-in-cell Euler-Lagrange (PSIC-EL) method is widely used to simulate flows laden with particles. Its accuracy, however, is known to deteriorate as the ratio between the particle diameter~($\smash{d_\text{p}}$) and the mesh spacing~($h$) increases, due to the impact of the momentum that is fed back to the flow by the Lagrangian particles. Although the community typically recommends particle diameters to be at least an order of magnitude smaller than the mesh spacing, the errors corresponding to a given $\smash{d_\text{p} / h}$ ratio and/or flow regime have not been systematically studied. In~this paper, we provide an expression to estimate the magnitude of the flow velocity disturbance resulting from the transport of a particle in the PSIC-EL framework, based on the $\smash{d_\text{p} / h}$ ratio and the particle Reynolds number, $\smash{\text{Re}_\text{p}}$. This, in turn, directly relates to the error in the estimation of the undisturbed velocity, and therefore to the error in the prediction of the particle motion. We show that the upper bound of the relative error in the estimation of the undisturbed velocity, for all particle Reynolds numbers, is approximated by $\smash{(6/5)\,d_\text{p} / h}$. Moreover, for all cases where $\smash{d_\text{p} / h \lesssim 1/2}$, the expression we provide accurately estimates the value of the errors across a range of particle Reynolds numbers that are relevant to most gas-solid flow applications ($\smash{\text{Re}_\text{p} < 500}$).