Splitting method for the combined formulation of the fluid-particle problem

In this paper, a splitting method for the direct numerical simulation of solid-liquid mixtures is presented, where a symmetric pressure equation is newly proposed. Through numerical experiment, it is found that the newly proposed splitting method works well with a matrix-free formulation for some benchmark problems, avoiding an erroneous pressure field which appears when using the conventional pressure equation of a splitting method. When deriving a typical pressure equation of a splitting method, the motion of a solid particle has to be approximated by the `intermediate velocity' instead of treating it as unknowns since it is necessary as a boundary condition. Therefore, the motion of a solid particle is treated in such an explicit way that a particle moves by the known form drag (pressure drag) that is calculated from the pressure equation in the previous step. From the numerical experiment, it was shown that this method gives an erroneous pressure field even for a very small time step size as the particle velocity increases. In this paper, coupling the unknowns of particle velocities in the pressure equation is proposed, where the resulting matrix is reduced to the symmetric one by applying the projector of the combined formulation. It has been tested over some benchmark problems and gives reasonable pressure fields.