Development of a robust Eulerian–Lagrangian model for the simulation of an industrial solid–fluid system

The discrete elements method (DEM) coupled with computational fluid dynamics (CFD) is widely employed in the numerical simulation of a multiphase flow including solid particles. The stable conditions have been examined in the DEM–CFD simulations, where a semi-implicit algorithm is often emplyed in computation of a solid-fluid flow. The stable DEM conditions can easily be derived based on the oscillation period, and the Courant condition provides stable conditions for CFD. Accordingly, the time step should be given to satisfy them simultaneously in the DEM–CFD method. However, when the drag force acting on the solid particles becomes some extent high, the DEM–CFD simulation cannot be stably performed even if the above conditions are satisfied. In this case, the time step should be decided through trial and error. Although several studies regarding the stability of the DEM–CFD method have been conducted, the theoretical time step for the drag force term has not been derived to date. Additionally, no model has been proposed to solve the instability problem regarding the drag force term in the DEM–CFD method. From this background, this paper aims to theoretically derive a stable condition for the drag force term and to develop a new implicit algorithm for the DEM–CFD method. To illustrate the accuracy and stability of the implicit algorithm, verification tests are performed in a fixed bed system. Subsequently, the DEM–CFD simulations are performed in a fluidized bed to show the capability of the implicit algorithm. Finally, the compatibility of the implicit algorithm with the wall boundary model, composed of the signed distance function and the immersed boundary method, is examined through the validation tests.