Phase Field-based Lattice Boltzmann Method And Its Application For Complex Interface Evolution

The complex dynamic evolution of phase interface widely exists in different natural phenomena and engineering applications,such as structural phase transition,multiphase fluid flow and dendritic growth.Phase field-based lattice Boltzmann method（PFLBM）is a newly developed interface-capturing method,which inherits the complex interface-capturing ability of phase field method and the advantages of lattice Boltzmann method,which is easy to impose complex boundaries and implement efficient parallel comput-ing,and can realize the efficient capture of large-scale complex interfaces.However,the reported models usually suffer some drawbacks,such as the inability to recover the second-order macroscopic equations,inclusion of many nonlocal difference operations,difficulty in maintaining the upper and lower bounds of the order parameter,and gener-ation of non-physical diffusion or interface instability,which limits the application and promotion of this method.Taking Allen-Cahn equation as an example,we aim to propose a series of improved phase field-based lattice Boltzmann models.The thesis includes two parts:the conservative phase field-based modified lattice Boltzmann models and the non-conservative phase-field optimized lattice Boltzmann models.All of these proposed lattice Boltzmann models have been significantly improved in one or more numerical per-formance indicators,such as accuracy,locality,boundedness and stability.For the conservative Allen-Cahn equation,we proposed a second-order modified lattice Boltzmann model with corrected collision step.The modified model includes a spatial difference term of the distribution function to remove the second-error terms in the traditional model and recover the correct macroscopic equation.Based on high-order multiscale expansion and dimensionless analysis,the leading error terms on the third and fourth orders can be extracted.Theoretical analysis and numerical simulation show that the current second-order modified model can maintain excellent accuracy and bounded-ness in a large range of relaxation time(₂)<3.5).In order to remove the leading er-ror terms on the third-order,a high-order source term-based modified lattice Boltzmann model is proposed.By adjusting the relaxation time parameter,the boundedness of the order parameter can strictly be guaranteed in the high-order modified lattice Boltzmann model,which can be reduced to the order(10^-15).The single-relaxation-time lattice Boltzmann model has bad performances in numer-ical stability.To improve it,an alternative modified lattice Boltzmann scheme is developed within the theoretical framework of the multiple relaxation time-based lattice Boltzmann method.The modified multiple-relaxation-time model is proposed by introducing an off-diagonal relaxation matrix and redesigning the coefficients in the equilibrium distribution function.Based on the multiscale analysis,the second-order macroscopic equation can be recovered exactly.Benchmark tests of interface-capturing and two-phase flow simula-tion show that the present model has advantages in numerical accuracy,order parameter boundedness and interface stability.For the nonconservative phase-field equation,the thesis developed a multiple-relaxation-time lattice Boltzmann model with a simple form and high computational effi-ciency.In this thesis,A reaction-diffusion equation with an antisymmetric diffusion co-efficient matrix is derived from the original anisotropic phase-field equation.Combined with the coordinate transformation,the equivalent diffusion matrix for arbitrary preferred growth orientation can be obtained.For such a reaction-diffusion equation,an anisotropic multiple-relaxation-time lattice Boltzmann model is proposed for dendrite growth simu-lation.In order to improve the computational efficiency,a collision-streaming-correction three-step algorithm is proposed,and a parallel computing program with adaptive mesh refinement is developed.The CPU time using adaptive-refinement-mesh can be reduced to 10%compared with the uniform meshes.When the conservative phase field-based lattice Boltzmann model is extended to the axisymmetric two-phase flow,large pseudo velocities will be produced near the phase interface,which reduces the numerical accuracy of the lattice Boltzmann model.In this thesis,it is found that serious spurious currents are produced near the axis for a certain kind of axisymmetric lattice Boltzmann model,i.e.,the radius-weighted multiphase lattice Boltzmann model.Using high-order multi-scale expansion and dimensionless analysis,the mechanism of the singularity is studied.By introducing a simple source term,a high-order modified model is proposed to eliminate the singularity of spurious currents.The proposed model introduces no additional nonlocal terms,which ensures the locality of parallel algorithm.The magnitude of spurious currents in the modified model can be reduced by 2 to 3 orders of magnitude.Using the anisotropic phase field-based lattice Boltzmann model,different factors that affect the evolution behavior of dendrite growth are studied,including the initial un-dercooling temperature,anisotropic interface energy,kinetic anisotropy and anisotropic thermal diffusion coefficients for pure material,as well as the initial concentration and the ratio of solid-liquid diffusion coefficient for alloy.It expands the application of phase field-based lattice Boltzmann method for dendrite growth in arbitrary orientation.