High Order Lattice Boltzmann Method For Non-equilibrium Flows

The non-equilibrium gas flows have been widely used in practical engineerings,such as near-space vehicle,micro-electro-mechanical system and shale gas.In the kinetic theory,the Boltzmann equation models the dynamics of a dilute monatomic gas in the whole regime from continuum(hydrodynamic)to free-molecular flow.In the past 30 years,lattice Boltzmann method(LBM)has been widely used because of its intuitive physics,easy implementation and high computational efficiency.Theoretically,it is shown that high-order LBM can approximate the solution of Boltzmann equation.However,the order of the most existing LBM is not enough,and therefore the ability of high-order LBM to capture non-equilibrium effect still needs to be systematically discussed.The complex Boltzmann collision term is the most difficult point and compromises are often made between accuracy and cost.One efficient way is to utilize a simplified collision model,which retains much of the essential physics in the Boltzmann collision term but with a simpler form and better solvability,such as Ellipsoid–Statistical and Shakhov models.However,the relaxation of higher-order moments is not considered or given reasonably,which plays important role in describing the non-equilibrium flows.Therefore,it is necessary to develop higher-order multiple-relaxation-time model.Accurate and effective calculation of the Boltzmann model equation is another important research aspect.Based on the above background,this thesis aims to develop high-order LBM and discusses its ability to describe non-equilibrium flow in detail.A new spectral multiple relaxation times(SMRT)collision model is proposed,which satisfies the physical principles of Galilean invariance and rotational symmetry.The research contents are summarized as follows:The high order LBM up to 39 th order is developed and it is validated that the highorder LBM can accurately and efficiently capture the rarefaction effect;Based on the equivalence between the discrete velocity distribution function and the moment,the highorder moment is essenial to describe the non-equilibrium flow state.With the increase of the non-equilibrium level,more higher-order moments are needed.In the continuum regime,the streaming and collision term are coupled in LBM,so that the time step and grid size are not limited by the average collision time and average free path of gas molecules.Meanwhile,the streaming-collision algorithm avoids interpolation operation.Hence,the LBM has the advantages of low dissipation and high efficiency.Based on the physical principles of Galilean invariance and rotational symmetry,a general phenomenological SMRT collision model of any order is proposed by using the Hermite expansion in the relative velocity coordinate system and the irreducible representation of the rotation group SO(3).The minimum moment element with independent relaxation time is strictly given,and then the discrete relaxation time spectrum is used to describe the collision process in place of the collision integral.The relaxation time of Maxwell molecular model is used to verify the effectiveness of the model in describing non-equilibrium flow.Therefore,the discrete relaxation time spectrum can be regarded as physical parameters to describe the non-equilibrium flow process.It can be expected that for any kind of gas,as long as the discrete relaxation time spectrum is determined,the SMRT model can accurately predict the non-equilibrium flow behavior of the gas.Taking the relaxation times of Maxwell molecular model as an example,the spontaneous Rayleigh Brillouin scattering and normal shock wave structure are used to assess the accuracy of SMRT collision model in describing the non-equilibrium gas flow.The 8th order SMRT model in the spontaneous Rayleigh Brillouin scattering is validated when the Knudsen number reaches 0.6,and the 5th order SMRT model in the normal shock structure can guarantee the accuracy when the Mach number is 5.In addition,it is found that the non-convergence of the Hermite expansion of the velocity distribution function is an important reason for the numerical instability of the simulation of the compressible flow.In summary,this thesis assesses the ability of the high-order LBM to simulate nonequilibrium flows and develops the SMRT satisfying the Galilean invariance and rotation symmetry.Theoretical and numerical analysis show that the SMRT can accurately characterize the relaxation behavior of higher-order moments and can accurately and effectively simulate non-equilibrium flow and continuous flow.