Research On The High-order Difference Based Pseudo-Potential Multiphase Lattice Boltzmann Method

Multiphase flow is a common phenomenon in nature,the simulation of multiphase flow is a common problem in science and agriculture and industry.Multiphase flow is a complex fluid system,often accompanied by physicochemical changes such as component diffusion,phase transformation,interface generation,before using traditional computational fluid dynamics(CFD)methods such as VOF,Level Set and so on to carry on the numerical simulation,although can describe a few large interface movements,but it is difficult to track a large number of small,discrete interfaces.The Lattice Boltzmann method(LBM)is a new method for modeling and simulating fluid systems developed internationally in the past three decades.It is also a mesoscopic approach with excellence of microscopic molecular dynamic model and macroscopic continuity model,compared with the traditional CFD method,there are obvious advantages,such as a clear physical background,simple algorithm,high computational efficiency,and high degree of parallelism.After nearly 30 years of development and improvement,LBM has become an irreplaceable and new method in computational fluid dynamics,which occupies an important position in the field of multiphase flow,including successful use in complex fluid systems,especially in complex systems that contain problems such as interface dynamics,phase transitions,etc.However,due to the complexity of the multiphase flow problem itself,there are still many challenges in further applying LBM in practice,such as:when the system is balanced,the virtual velocity is large and the density is relatively small.In recent years,people have done a lot of work on how to increase the stability and density ratio of the system simulation,reduce the virtual speed at equilibrium,and improve the thermodynamic consistency and Galileo invariance.Some researchers studied from the point of view of force technology that the discrepancy of macroscopic force in discretization occurred and analyzed from the perspective of restoring the macroeconomic equation,thus put forward many force technologies,of which the precise difference term technology is excellent.There are still many researchers have conducted in-depth studies on the form of non-ideal interaction forces.At present,there are mainly two methods for calculating interaction forces,one is the pseudo-potential model based on Shan-Chen,and the other is thermodynamics and Non-ideal force model based on the perspective of free energy.The free energy non-ideal force model has a solid thermodynamic theoretical basis.Although the two-phase coexistence density curve simulated by this force is in good agreement with the theoretical value of Maxwell's equal area,it still has not obtained ideal results.In the pseudo-potential model,although researchers constantly improve and perfect it,the problems of numerical stability and virtual velocity are still not well solved.In recent years,researchers have proposed a pseudo-potential model of portfolio potential.Because of its high precision,it has been widely circulated and used.However,after further exploration,this paper finds that the model has some problems and there is still a need for in-depth research and improvement.First,we analyze that almost all non-ideal interaction force models(including SC's pseudo-potential model and free energy non-ideal force model)involve gradient calculations,and the calculation of density gradients is particularly important in non-ideal interaction calculations.If the calculation is inaccurate,it will have a significant impact on the results of the multiphase flow simulation.So far,the algorithm for computing gradients in non-ideal forces is the most popular in the traditional center difference algorithm.Therefore,few people have thoroughly studied whether the traditional central difference algorithm is accurate,but numerical differentiation itself is a typical ill-posed problem,a slight inaccuracy in the calculation may cause a large error,and even result in numerical instability.Next,we analyze the influence of the traditional central difference algorithm in the multiphase flow simulation.The typical multiphase flow equilibrium density distribution is approximately a hyperbolic tangent distribution,by analyzing the accuracy of numerical derivative,it is found that the numerical derivative calculated by the center difference method(CDM)has a large deviation from its theoretical solution.Moreover,since there is a deviation between the center difference and the theoretical analysis,will there be a better calculation method than it?We associate high-order differences that are often used in numerical calculations in finite differences.After analysis,the numerical derivatives calculated by the high-order difference method(HDM)are in good agreement with theoretical analysis.Therefore,we introduce HDM to calculate the interaction between particles,which replaces the commonly used CDM,and propose a pseudo-potential multiphase flow lattice Boltzmann method based on high-order difference.This model not only retains the advantages of the pseudo-potential model,but also has the advantages of simple implementation,high efficiency,good parallelism,and high accuracy.In order to verify the performance of the model,multi-phase flow simulation was performed in several aspects,including static and dynamic conditions.After theoretical and numerical tests,the proposed pseudo-potential multiphase flow lattice Boltzmann model based on high-order difference has the following advantages and characteristics:1.In the case of large temperature ranges and large density ratios,the thermodynamic consistency of commonly used state equations can be achieved without any combination of interactions and any additional tunable parameters.2.After analysis in the paper,it is concluded that Shan-Chen's effective density model is consistent with Zhang's potential function in the mathematical theory analysis,and can equally describe the fluid phase transitions of all selected state equations.Therefore,the interaction potential can be selected one of them.3.The predecessors put forward a lot of power technology,among which the precise difference proposed by Kupershtokh is favored by most researchers because of its simplicity,accuracy and irrelevance to the relaxation time.Therefore,the force technology adopted in this paper is precise difference.4.Innovation introduces new numerical methods for calculating interactions.In this paper,through a series of analysis,it is found that the traditional central difference is not the best numerical method for simulating gradients in multiphase flow.Therefore,a numerical calculation method—high-order difference—is introduced.This method is better than the traditional center difference in both the mathematical theory and the hyperbolic tangent function simulation.It can be seen that the use of high-order differentials to simulate multiphase flow should be much better than the traditional central difference.5.The new model is simulated in the two-phase coexistence and real multiphase flow,and the preliminary application of droplet splashing example fully demonstrates that the model is more feasible and practical.It can be found in numerical experiments that the results of high-order differential simulation are very good.6.The new model is stable and accurate.Its maximum density ratio even reaches 10~9,which is enough to meet the requirements of practically all practical applications.Therefore,the research on high-order differential methods applied to multiphase flow has sufficient advantages,and it is expected to obtain some more interesting results.Through the above analysis,not only that the new model has a solid theoretical foundation and good numerical performance,but also reflects the necessity of high-order difference proposed in order to be widely used in the field of multi-phase flow numerical calculation.