The interaction forces in fluid-solid two-phase flows

The interaction force between fluid and solid phases is studied for both steady and unsteady flow situations. As a model, solid particles are assumed to be spherical and are distributed in regular arrays. For steady flows, a numerical method is used to calculate the interaction force for Reynolds number up to about 300. For unsteady flows, an analytical method is used.;To minimize errors due to artificial diffusions, a third-order accurate numerical scheme is developed for the primitive variable form of the Navier-Stokes equations.;The full Navier-Stokes equations are solved numerically for steady flows past regular arrays of cylinders and spheres. The interaction force is compared with both Ergun's (1952) and Richardson-Zaki's (1954) empirical correlations commonly used in multiphase flow modeling. In general, results from these two correlations are different from each other. The difference sometimes can be as large as an order of magnitude. Limited computational resources prevented a full parametric study of 3-D problems, but it appeared that neither equation agrees with 3-D results. For 2-D staggered arrays, however, the results agree with Ergun's correlation, in all likelihood coincidentally.;The interaction force for both fluid and regular arrays in general unsteady motion is obtained analytically for small Reynolds number flows. The force is shown to be composed of the steady drag, the added mass force, the Basset force, and a force directly related to the instantaneous acceleration of the mixture. Due to the presence of the last force component, the interaction force is not symmetric in terms of accelerations of the particle and the fluid. The integration kernel in the Basset force is found to decay exponentially with time, rather than with the inverse square root as is the case of a single sphere. A comparison of present results with other recent studies is also given.