A pseudopotential multiphase lattice Boltzmann model based on high-order difference

Abstract

The hyperbolic tangent function is usually used as a reliable approximation of the equilibrium density distributions of a system with phase transitions. However, analyzing the accuracies of the numerical derivatives, we find that its numerical derivatives computed by central difference method (CDM) may deviate significantly from its analytical solutions, while those computed by high-order difference method (HDM) can agree very well. Therefore, we introduce HDM to evaluate the interparticle interactions instead of popular CDM, and propose a pseudopotential multiphase lattice Boltzmann model based on high-order difference method. The present model not only retains the advantages of the pseudopotential model, such as easy implementation, high efficiency, full parallelism and so on, but also achieves higher accuracies. To verify the performances of this model, several multiphase flow simulations are conducted, including both stationary and dynamic situations. Firstly, full thermodynamic consistencies for the popular equations of state have been achieved in large temperature range and at large density ratio, without any combining interaction and any additional adjustable parameter of interaction. Secondly, with high-order difference, either the interparticle interaction proposed by Shan-Chen or by Zhang-Chen can equally depict the phase transitions of the fluids with all selected equations of the state. These numerical * Corresponding authors.agreements based on HDM are consistent to the theoretical analysis that the two models are mathematically identical. Thirdly, the present model is stable and accurate at a wider temperature range. Lastly, our newly proposed model can be easily and reliably applied to various practical simulations and expected to obtain some more interesting results.

Chemical-potential multiphase lattice Boltzmann method with superlarge density ratios

“…Eventually, the sheet breaks up and form secondary droplets. These observations are consistent to the previous reports [30,33]. The researchers also found that the spread radius r obeys the power law…”

“…This results in a rather steep phase interface, which is similar to a hyperbolic tangent function. In these regions, CDM causes some considerable errors, and even makes mathematically equivalent formulas behave like different algorithms [30]. Therefore, it is necessary to introduce more accurate method to compute derivative and gradient in multiphase simulations, especially in the transition region.…”

Multiphase curved boundary condition in lattice Boltzmann method

“…Therefore, the boundary conditions, which must reflect the nonideal effect of the wetting boundary in multiphase simulations, should involve the consideration of the nonideal force; otherwise, the calculating errors are intolerable. Furthermore, in single-component two-phase systems, for example water and vapor, the liquid-gas density ratio can reach around 100 ∼ 1000 times, and the density profile across the phase transition region is nonlinear and approximate to a hyperbolic tangent curve [46], so the interpolation-based boundary conditions produce large calculating errors at three-phase contact regions. Even at the liquid-solid and gas-solid interfaces, the wetting boundaries would remarkably change the adjacent fluid density and lead to sizable interpolating errors.…”

Spurious currents suppression by accurate difference schemes in multiphase lattice Boltzmann method

“…10 Although the angle deviation of the force can be reduced by high-order isotropic schemes and the spurious currents are reduced to a minimum level, it introduces more neighbor points, which makes the boundary realization more complicated and incurs additional computational costs [16,24]. Moreover, the accuracy of the discrete gradient mainly determines the errors in the numerical calculation of multiphase flow simulations, and the large numerical errors may make the simulation results inaccurate [70]. For example, it can be clearly seen from Fig.…”