A lattice Boltzmann study on the drag force in bubble swarms

Lattice Boltzmann and immersed boundary methods are used to conduct direct numerical simulations of suspensions of massless, spherical gas bubbles driven by buoyancy in a three-dimensional periodic domain. The drag coefficient C D is computed as a function of the gas volume fraction φ and the Reynolds number Re = 2 RU slip /ν for 0.03 φ 0.5 and 5 Re 2000. Here R , U slip and ν denote the bubble radius, the slip velocity between the liquid and the gas phases and the kinematic viscosity of the liquid phase, respectively. The results are rationalized by assuming a similarity between the C D ( Re eff )-relation of the suspension and the C D ( Re )-relation of an individual bubble, where the effective Reynolds number Re eff = 2 RU slip /ν eff is based on the effective viscosity ν eff which depends on the properties of the suspension. For Re ≲ 100, we find ν eff ≈ ν/(1−0.6φ 1/3 ), which is in qualitative agreement with previous proposed correlations for C D in bubble suspensions. For Re ≳ 100, on the other hand, we find ν eff ≈ RU slip φ, which is explained by considering the turbulent kinetic energy levels in the liquid phase. Based on these findings, a correlation is constructed for C D ( Re , φ). A modification of the drag correlation is proposed to account for effects of bubble deformation, by the inclusion of a correction factor based on the theory of Moore ( J. Fluid Mech ., vol. 23, 1995, p. 749).