A Study On The Lattice Boltzmann Method Applied To Thermal Flows

In the recent 30 year,a new numerical method,i.e.,the lattice Boltzmann method(LBM)develops rapidly and has been widely used in complex flows,such as multi-phase flows,porous flows,grain flows,blood flows and so on.LBM achieves great success in the isothermal flows.However,the LBM encounter some difficulties when it is applied to thermal flows.There are still some problems to be solved.This work study several problems which the LBM encounters in thermal flows.Firstly,this work introduce the way of designing the higher order lattice model based on the Hermite polynomial.The equilibrium distribution function is expressed by the eighth order Hermite polynomial and its c++ code are proposed.We also write the code to design the higher order lattice model with Maple.A forth order lattice model of two dimensional,a sixth order lattice model of two dimensional,a eighth order lattice model of two dimensional and a forth order lattice model of three dimensional are testified by the one dimensional shock tube flow.The simulation results and the analytical solutions are consistent very well.Although the Navier-Stokes-Fourier equations(NSF)can be derived from the higher lat-tice models through the Chapman-Enskog expansion,the physical coefficients of the derived NSF,e.g.,the specific heat ratio and the Prandtl number,are fixed.They are not adjustable.This violates physics.To design the lattice model with adjustable coefficients,the Gaussian distri-bution function with the rotational energy is decoupled into two distributions,i.e.,the density distribution function and the energy distribution function.The specific heat ratio can be modified by the additional free degree introduced by the rotational energy and the Prandtl number can be adjusted by the parameter of the Gaussian.The Gaussian distribution is expanded in the Hermite space and the general term formula for the Hermite coefficients of the Gaussian distribution is deduced.We also obtain the forth order Hermite expression of the Gaussian distribution.After these works,the lattice model with ad-justable physical coefficients based on the ES-BGK model is constructed.The thermal Couette flow and the one dimensional shock tube flow are applied to testifying the proposed lattice mod-el.The simulation results agree with the analytical solutions very well.The ES-BGK model has been widely used in the gas kinetics and the accuracy has been discussed in detail.The lat-tice model based on the ES-BGK model has clear physical meaning,and is easy to implement.We only need to modify the code of the equilibrium distribution.It is not necessary to modify the other parts and the structure of the code.We can exploit the existing code to the maximum limit.The boundary condition is one of the key point of the LBM.We compare the accura-cy,stability and the efficiency of the bounce-back scheme(BB),non-equilibrium extrapolation scheme(NEEP),no-equilibrium bounce-back scheme(NEBB)and the kinetics boundary con-dition(KBC).In addition,we propose a boundary condition scheme applied to the higher order lattice,with which the collision-streaming mechanism-one of the most attractive characteristic of the LBM is retained.Flow is often affected by force.In order to add the forcing term to the LBM,we should discretize the forcing term in the velocity space and design the forcing term model.We redefine the fluid velocity and the total energy,and propose a newly forcing term model.From the proposed forcing term model,the NSF can be derived through the Chapman-Enskog expansion.