A numerically convergent Lagrangian–Eulerian simulation method for dispersed two-phase flows

In Lagrangian–Eulerian (LE) simulations of two-way coupled particle-laden flows, the dispersed phase is represented either by real particles or by computational particles. In traditional LE (TLE) simulations, each computational particle is assigned a constant statistical weight, which is defined as the expected number of real particles represented by a computational particle. If the spatial distribution of particles becomes highly non-uniform due to particle–fluid or particle–particle interactions, then TLE simulations fail to yield numerically converged solutions due to high statistical error in regions with few particles. In this work, a particle-laden lid-driven cavity flow is solved on progressively refined grids to demonstrate the inability of TLE simulations to yield numerically converged estimates for the mean interphase momentum transfer term. We propose an improved LE simulation (ILE) method that remedies the above limitation of TLE simulations. In the ILE method, the statistical weights are evolved such that the same physical problem is simulated, but the number density of computational particles is maintained near-uniform throughout the simulation, resulting in statistical error that remains nearly constant with grid refinement. The evolution of statistical weights is rigorously justified by deriving the consistency conditions arising from the requirement that the resulting computational ensemble correspond to a statistical description of the same physical problem with real particles. The same particle-laden lid-driven cavity flow is solved on progressively refined grids to demonstrate the ability of ILE simulation to achieve numerically converged estimates for the mean interphase momentum transfer term. The accuracy of the ILE method is quantified using a test problem that admits an analytical solution for the mean interphase momentum transfer term. In order to improve the accuracy of numerical estimates of the mean interphase momentum transfer term, an improved estimator is proposed to replace the conventional estimator. The improved estimator results in more accurate estimates that converge faster than those obtained using the conventional estimator. The ILE simulation method along with the improved estimator is recommended for accurate and numerically convergent LE simulations.

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