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 = 2RU_slip/v for 0.03 φ 0.5 and 5 Re 2000. Here R, Uslip and v 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 CD(Reeff)-relation of the suspension and the CD(Re)-relation of an individual bubble, where the effective Reynolds number Reeff = 2RU_slip/veff is based on the effective viscosity veff which depends on the properties of the suspension. For Re 100, we find v_eff v/(1-0.6φ^1/3), which is in qualitative agreement with previous proposed correlations for CD in bubble suspensions. For Re 100, on the other hand, we find veff RUslipφ, which is explained by considering the turbulent kinetic energy levels in the liquid phase. Based on these findings, a correlation is constructed for CD(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).