On Simulations of High-Density Ratio Flows Using Color-Gradient Multiphase Lattice Boltzmann Models

Abstract

Originally, the color-gradient model proposed by Rothman and Keller (RÀK) was unable to simulate immiscible two-phase°ows with di®erent densities. Later, a revised version of the RÀK model was proposed by Grunau et al. [D. Grunau, S. Chen and K. Eggert, Phys. Fluids A: Fluid Dyn. 5, 2557 (1993).] and claimed it was able to simulate two-phase°ows with high-density contrast. Some studies investigate high-density contrast two-phase°ows using this revised RÀK model but they are mainly focused on the stationary spherical droplet and bubble cases. Through theoretical analysis of the model, we found that in the recovered NavierÀStokes (NÀS) equations which are derived from the RÀK model, there are unwanted extra terms. These terms disappear for simulations of two-phase°ows with identical densities, so the correct NÀS equations are fully recovered. Hence, the RÀK model is able to give accurate results for°ows with identical densities. However, the unwanted terms may a®ect the accuracy of simulations signi¯cantly when the densities of the two°uids are di®erent. For the simulations of spherical bubbles and droplets immersed in another°uid (where the densities of the two°uids are di®erent), the extra terms may not be important and hence, in terms of surface tension, accurate results can be obtained. However, generally speaking, the unwanted term may be signi¯cant in many°ows and the RÀK model is unable to obtain the correct results due to the e®ect of the extra terms. Through numerical simulations of parallel two-phase°ows in a channel, we con¯rm that the RÀK model is not appropriate for general two-phase°ows with di®erent densities. A scheme to eliminate the unwanted terms is also proposed and the scheme works well for cases of density ratios less than 10.

Generalized three-dimensional lattice Boltzmann color-gradient method for immiscible two-phase pore-scale imbibition and drainage in porous media

“…This article does, however, not provide any theoretical validation beyond a simple validation of the Laplace law. Other articles show that this type of CGM may not be able to solve flows adequately with nonunit density ratio in complex test cases and require special adjustment to improve their accuracy [39,49].…”

“…The CGM can therefore be considered a mature tool for both industry and academia for the characterization of porous media. However, all the CGM methods described above still suffer from some deficiencies, as they may lack Galilean invariance to a certain extent and therefore may be inaccurate for simulating flow with density ratio in many practical situations [37,39,41,49,62]. Consequently, they can only simulate multicomponent flow through porous media with a more limited ratio of dynamic viscosity.…”

Three-dimensional lattice Boltzmann method benchmarks between color-gradient and pseudo-potential immiscible multi-component models

“…Approach to increase the Galilean invariance of the CGM has also been proposed. [28][29][30] The MRT has also been introduced to improve the stability of the CGM. [30][31][32] There are also approaches to model the static and dynamic contact angles accurately at a solid boundary.…”

Entropic Lattice Boltzmann Method for Multiphase Flows

“…In spite of a number of refinements (see review, e.g., [9,10] and references therein), existing LB formulations for the multiphase flows are still unable to address complex dynamical effects, such as droplet collisions, in a quantitative fashion. Moreover, many of the attempts at improving the performance of multiphase models were concentrated around simplifying existing approaches by sacrificing the physics of the model [11,12] or resorting to improvements on Shan-Chen type models, where one cannot introduce temperature that is consistent with thermodynamics [7,[13][14][15].…”