A numerical study on the drag law of a gas bubble using dynamic body force method

Abstract

For a gas bubble rising in quiescent liquid, most previous drag force models correlate the drag coefficient Cd to dimensionless parameters such as the Reynolds number Re, the Eötvös number Eo, and the Morton number Mo. However, it is still an open question if the current models can be applied to non-buoyancy-driven bubbles in a wide range of parameters. In the present study, we investigate the effects of Reynolds number Re and Weber number We on the drag force coefficient Cd of bubbly flows. To obtain a converged drag force, a dynamic body force model is developed, which constraints the bubble at a fixed position. A large number of numerical simulations are carried out, spanning in the (We, Re) parametric space. The correlations of Cd to We and Re are studied, respectively. We find that for low We, as the bubble keeps spherical, its drag coefficients can be well predicted by the previous spherical bubble models. For medium–high We, when Re is low, the viscous force contributes most, and therefore, the bubble deformation or We plays an insignificant role on the drag. In contrast, when Re is high, the pressure force or shape drag becomes dominant, which is strongly dependent of We. Therefore, we can identify three different regions, spherical, We-dependent, and We-independent, in the parametric space. Finally, a new empirical drag force model for a non-buoyancy-driven bubble is proposed based on the Cd-Re and Cd-We correlations obtained by our numerical simulations. The current model is superior to the previous drag models, with predicting errors within 20%, by compared to the direct numerical simulation and experimental data. More profoundly, the current model can apply to a wider range of parameters, that is, 1≤Re≤1000 and 0≤We≤20, particularly for high We, when the bubble deformation cannot be neglected.