Research On A Mass-conserved Multiphase Lattice Boltzmann Method Based On High-order Difference

The free energy model has been favored by many researchers due to it can solve the problem of interface tracing,restore the Cahn Hilliard equation,satisfy the local mass and momentum conservation,consistent with the thermodynamic theory and so on.However,the numerical instability in simulations would be caused by the original free energy model in the case of large density ratio,and the Galilean invariance is not satisfied when there is a large density gradient near the interface.There has been an ongoing effort to improve the free energy model.Compared with other models,the Z-S-C multiphase LB model,in which the Cahn Hilliard equation is accurately recovered without any additional terms and the Galilean invariance property is kept.In addition,owing to the small change of the average density in this model,it is very stable and efficient,and more importantly,it can be employed to simulate the multiphase flows with large density ratio beyond 1000.Due to the good performance of the Z-S-C model,it has been favored by many researchers.However,due to the presence of the diffusion term and the numerical dissipation caused by the discretization of convection term in Cahn Hilliard equation,the mass of each phase in the Z-S-C model cannot be conserved exactly.On the other hand,the solutions to the gradient of the order parameter are involved in the calculations of chemical potential and velocity in the Z-S-C model,and the accuracies for calculating these gradients are particularly important.The central difference method is often used to calculate the numerical gradient in multiphase models.In general,this method is sufficiently accurate.However,in the diffusion interface method with high density ratio,the results calculated by central difference method may deviate greatly from the theoretical solutions,which may lead to numerical instability of the simulation.Through the above investigations,we know that although great progresses have been made in the numerical study of multiphase flow based on free energy model,the solution to the problems of large density ratio and mass conservation in the model are still an important research subject.In this paper,we aim to improve the original Z-S-C model in terms of the mass conservation and computational accuracy so as to develop a mass-conserved and high accuracy multiphase LB model for simulating multiphase flows,in which the high-order difference is introduced to calculate the gradient of the order parameter to improve its accuracy and a mass correction method is utilized to solve the non-conservative mass problem of the original model.We will investigate the numerical methods for calculating the gradient of the order parameter and reveal the importance of its numerical accuracies,and demonstrate the performances of the improved model by carrying out several testing cases such as the Laplace law,a bubble in a stationary flow,the merging of two bubbles and the bubble rising under buoyancy.As a result,the present model enjoys the advantages:(1)By introducing HDM into the original Z-S-C model,the simulation results from HDM are much better than those from CDM in all the testing cases,the accuracy of the present model has been improved.(2)Whether for simple single static bubble simulation or bubble rising simulation with complex deformation motion characteristics,in all testing cases,through introducing the mass correction method into the original model,the bubble mass is conserved very well,and the problem of non-conservation of mass in the original model has been solved.(3)In the simulations of bubble rising,the Re numbers and the final bubble shapes from the present model are compared with those from the experiments and other models or methods.The results from the present model are in good agreement with the experimental ones,and are better than those from other models and methods mentioned when the Re number is relatively small.(4)In all our simulations,the density ratio of two phases is 1000,the simulations are very stable,and the present model keeps the advantages of the original Z-S-C model,such as large density ratio and stability very well.Due to its high accuracy,mass conservation and large density ratio,the present multiphase model can be expected to be applied reliably on more general complex fluid systems and obtain some good results.