| The phenomena of flow and heat and mass transfer are common basic problems in many fields of science and engineering,such as energy,environment,aerospace and so on.They are usually described by two kinds of basic mathematical models: Navier-Stokes(NS)equations and nonlinear convection-diffusion equation(NCDE).This kind of problem usually has the characteristics of multi-scale,nonlinearity,strong coupling and complex boundary,and its solution is challenging.With the rapid development of modern science and technology,numerical simulation has become one of the important means to study this kind of problems.Therefore,it is of great scientific significance to develop more efficient and robust numerical methods to solve these two kinds of equations.In the past 30 years,mesoscopic lattice Boltzmann(LB)method based on gas dynamics theory has attracted wide attention because of its simplicity,efficiency and easy handling of complex boundary and multi-field coupling problems.Due to the lack of stability of the classical single-relaxation-time LB model,the study of multi-relaxation-time LB(MRT-LB)model has attracted extensive attention.Although there have been many achievements in the research of MRT-LB model,there are still some problems in the theoretical framework of LB model.Firstly,due to the limitation of discrete velocity direction,the LB model often does not satisfy Galilean invariance,which leads to the instability of numerical methods.To solve this problem,a new method is to use the central moment to construct a LB model to reduce the influence of Galilean invariance loss.However,in the existing studies,the solution of incompressible NSEs is still simulated in the form of weakly compressibility.In addition,in order to eliminate numerical slip caused by boundary schemes,the description of discrete velocity space in existing studies is incomplete,and it is usually a MRT-LB model with orthogonal transformation matrix.In view of the above issues,we have carried out relevant research,which are mainly divided into the following aspects:(1)A multi-relaxation-time lattice Boltzmann model based on discrete Hermite moments is constructed.This thesis puts forward the viewpoint of block orthogonalization.Using the Hermite polynomial orthogonality,the uniform expression of equilibrium distribution function is given for both NS equations and NCDE.The commonly used equilibrium distribution function is a special case of the current form.And taking NCDE as an example,we constructed a discrete Hermite moment MRT-LB(HMRT-LB)model.The stability of the model,the choice of order of equilibrium distribution function and the influence of free parameters are discussed in detail.(2)The central Hermite moment LB model for incompressible NS equations is constructed.Firstly,the HMRT-LB model is constructed for another important class of equations,incompressible NS equations.Then,based on matrix transformation and combining with the idea of central moment,the central Hermite moment MRT-LB(CHMRT-LB)model which can accurately recover the incompressible equation is constructed.Furthermore,the transformation matrix of this model is simplified effectively,and the simplified central Hermite moment MRT-LB(SCHMRT-LB)model is developed based on this.The stability of HMRT-LB,CHMRT-LB and SCHMRT-LB models is compared by numerical examples,and it is found that SCHMRT-LB model has the best stability among these models.(3)The discrete effects of several common boundary schemes are analyzed.Firstly,the equivalent difference scheme is obtained based on the evolution equation of the natural moment LB model,and it is proved that the lower triangular transformation matrix does not change the form of the difference scheme,that is,the HMRT,CHMRT and SCHMRT models constructed above are consistent with the analysis using the natural moment model directly.Then,the discrete effect of different boundary schemes(including anti-bounce-back,bounce-back and non-equilibrium extrapolation boundary schemes)are analyzed,and the relaxation factors and weight coefficients are given to eliminate the numerical slip caused by the boundary schemes.The accuracy of the obtained relation is verified numerically.(4)A simplified central Hermite moment thermal LB(SCHMRT-TLB)model is constructed and applied to the study of natural convection.The SCHMRT-TLB model is constructed by using the above mentioned models.The SCHMRT-LB model is a bidistributed SCHMRT-LB model,that is,the SCHMRT-LB models are used to deal with the NS equations of flow field and the NCDE of temperature field.A numerical example is given to verify the validity of the model,and the natural convection problem with high Rayleigh(Ra)number under different working conditions is studied by using the model.In short,this paper we not only give the general expression of equilibrium distribution function,but also develops several LB models based on discrete Hermite moment and central Hermite moment for general convection-diffusion equation and incompressible NS equations.The relationship to eliminate the numerical slip at the boundary schemes of the model is given.In addition,based on the above work,we apply the SCHMRT-TLB model to investigate the effects of different working conditions on the flow field distribution,temperature distribution and heat transfer efficiency of natural convection at high Ra number.All these work lay a necessary foundation for further study of the numerical application of LB method in the field of mass and heat transfer. |
