We present results of direct numerical simulations of travelling waves in dense assemblies of monodisperse spherical particles fluidized by a liquid. The cases we study have been derived from the experimental work of others. In these simulations, the flow of interstitial fluid is solved by the lattice-Boltzmann method (LBM) and the particles move under the influence of gravity, hydrodynamic forces stemming from the LBM, subgrid-scale lubrication forces and hard-sphere collisions. We first show that the propagating inhomogeneous structures seen in the simulations are in agreement with those observed experimentally. We then use the detailed information contained in the simulation results to assess aspects of two-fluid model closures, namely, fluid–particle drag, and the various contributions to the effective stresses. We show that the rates of compaction and dilation of the particle phase in the travelling waves are comparable to the rate at which the microstructure relaxes, and that there is a pronounced effect of the rate of compaction on the average collisional normal stress. Although this effect can be expressed as an effective bulk viscosity term, this approach would require the use of a path-dependent bulk viscosity. We also find that the effective fluid–particle drag coefficient can be described well with the often-used closure motivated by the experiments of Richardson & Zaki (Trans. Inst. Chem. Engng vol. 32, 1954, p. 35). In this respect, the effect of the system size for determining the drag requires specific care. The shear viscosity of the particle phase manifests small, but clearly noticeable dependence on the rate of compaction/dilation of the particle phase. Our observations point to the need for higher-order closures that recognize the slow evolution of the microstructure in these flows and account for the effects of non-equilibrium microstructure on the stresses.
Direct numerical simulations of dense suspensions: wave instabilities in liquid-fluidized beds