| This dissertation focuses on the development of numerical moment methods upto an arbitrary order in the gas kinetic theory. The gas kinetic theory describes thestates of gases in scales between the fluid dynamics and the molecular dynamics.Based on the moment method proposed by Grad in1949, this paper brings out afirst systematical investigation on moment equations of all orders, which realizes aconnection between the continuum model and the kinetic model, and thus exploresa brand new way for the modeling and computation in the field of gas kinetic theory.With some special properties of the moment expansion which are discovered inthe derivation of general Grad-type moment equations, a uniform numerical schemeis developed for all moment systems. The scheme adopts a new algorithm for thecalculation of fluxes, such that the computational efciency is essentially enhancedcomparing with traditional methods. The algorithm enables the numerical simula-tion using high-order moment equations, and moreover, it interprets the momentequations as a semi-discrete Boltzmann equation with the velocity variable dis-cretized. Since the Euler equations can be considered as a special case of Grad’s moment methods with the fewest degrees of freedom, all Grad’s systems act as aseries of computable models connecting the gas dynamics and the gas kinetic theory.In this paper, such a method is called the “numerical moment method”.Grad’s moment method shows a very high efciency while approximating theBoltzmann equation. However, it has some fatal defects which greatly restrict thedevelopment of the moment method, including the appearance of subshocks in theprofile of a shock structure with a moderate Mach number, and the loss of well-posedness due to the lack of hyperbolicity when the fluid is far away from the localequilibrium. Aiming at these two points, two improvements on Grad’s momentequations are carried out in diferent points of view:A special moment closure is introduced such that global hyperbolicity can beobtained for moment equations up to an arbitrary order. A theoretical proofof the global hyperbolicity is given for all resulting systems, so that underappropriate conditions, local well-posedness can be achieved. Meanwhile, allcharacteristic speeds of any system in this series can be analytically obtained,and based on their expressions, a wall boundary condition with required num-ber of equations is given. Through the analysis of characteristic waves, it isfound that this set of moment systems holds a very similar hyperbolic struc-ture with Euler equations, and therefore they can be considered as a newbridge connecting the gas dynamics and the gas kinetic theory in place ofGrad’s moment systems. Such an improvement significantly enhances the ro-bustness of the numerical moment method, and thus the moment method iscapable of processing strongly non-equilibrium fluids.With the techniques of Maxwellian iteration and asymptotic analysis, gen-eral moment equations are regularized by adding some second-order difusionterms. The resulting systems are parabolic, and have infinite propagationspeeds. Such a property eliminates the discontinuities inside the profile ofa shock structure, and thus it can be expected that they are applicable tohigh Mach number flows. This set of equations can be considered as exten-sions of the Navier-Stokes-Fourier equations due to their similarity in both the derivation processes and the forms of expressions, and as the number ofmoments increases, they also realize a transition from the gas dynamics to thegas kinetic theory.For both types of models, the numerical computation can be performed under theframework of numerical moment method, for which the author has developed a pro-gram package with friendly interfaces. Through a number of numerical experiments,the moment method shows significant convergence to the Boltzmann equation, andthe advantages of the globally hyperbolic moment equations and the regularizedmoment equations are validated. |
PDF Full Text Request