Evaluation of methods for calculating volume fraction in Eulerian–Lagrangian multiphase flow simulations

The present work addresses numerical methods required to compute particle volume fraction or number density. Local volume fraction of the l  th particle, αlαl, is the quantity of foremost importance in calculating the gas-mediated particle–particle interaction effect in multiphase flows. A general multiphase flow with a distribution of Lagrangian particles inside a fluid flow discretized on an Eulerian grid is considered. Particle volume fraction is needed both as a Lagrangian quantity associated with each particle and also as an Eulerian quantity associated with the grid cell for Eulerian–Lagrangian simulations. In Grid-Based (GB) methods the particle volume fraction is first obtained within each grid cell as an Eulerian quantity and then the local particle volume fraction associated with any Lagrangian particle can be obtained from interpolation. The second class of methods presented are Particle-Based (PB) methods, where particle volume fraction will first be obtained at each particle as a Lagrangian quantity, which then can be projected onto the Eulerian grid. Traditionally, the GB methods are used in multiphase flow, but sub-grid resolution can be obtained through use of the PB methods. By evaluating the total error, and its discretization, bias and statistical error components, the performance of the different PB methods is compared against several common GB methods of calculating volume fraction. The standard von Neumann error analysis technique has been adapted for evaluation of rate of convergence of the different methods. The discussion and error analysis presented focus on the volume fraction calculation, but the methods can be extended to obtain field representations of other Lagrangian quantities, such as particle velocity and temperature.