Unified Gas-Kinetic Scheme

The main aim of this thesis is to investigate and develop a unified scheme which is able to simulate the flows from continuum regime to rarefied regime. Academican Hsue-shen Tsien divided the whole flow regime into continuum flow, slip flow, transitional flow and free molecular flow according to their degrees of rarefaction. The continuum flow follows the continuum medium assumption. The study of the continuum flow often bases on the fluid dynamics for continuum medium. The Euler equations and Navier-Stokes equations are the governing equations which are often used in continuum flows. While, in the slip flow, transitional flow and free molecular flow regimes, the continuum hypothesis can not hold due to the rarefaction. The study of such flows often bases on the gas-kinetic theory or the molecular dynamics. In the scope of numerical simulations, the finite difference method and finite volume method for solving the Navier-Stokes equations are widely used in continuum regime. While, for the rarefied regimes, the direct simulation Monte Carlo method is often used, which uses one model molecule representing a large amount of real molecules. By calculating the transport and collision of one model molecule, this method can obtain the flow behavior. Since both the Navier-Stokes methods and direct simulation Monte Carlo methods have their limitation of flow regimes, a unified method which can simulate the whole flow regime is useful. The methods that can be classified into a unified method have some common characteristics such as the using of Boltzmann model equation, the discrete velocity space and the evolution of both microscopic distribution function and macroscopic physical variables. The Boltzmann model equation used in the unified scheme is investigated in this thesis. The discretization of velocity space and the numerical scheme are also investigated using both theoretical and numerical methods. By introducing the model of energy relaxation between translational and rotational freedoms, a new unified gas-kinetic scheme is developed which expands the application of unified gas-kinetic scheme from monatomic gases to diatomic gases which are the main components of earth’s atmosphere. Finally a three dimensional code of unified gas-kinetic scheme is developed for aerodynamic and aeroheating computations in distributed parallel computing environment. The presented work has practical significance for the aerodynamic and aeroheating predictions for equipment such as aerospace plane and micro-electromechanical systems, especially for the cases which would experience multiple flow regimes according to the time sequence or have multi-scales in a single flow field.The content of this research is as follows:Investigates the methods based on gas-kinetic theory including the deterministic method represented by the discrete velocity method and the statistical method such as the direct simulation Monte Carlo method. The main features of these methods are investigated along with their advantages and disadvantages. The differences of the operator splitting method used in earlier gas-kinetic method and the unsplitting method used in the unified gas-kinetic scheme are analyzed in a theoretical way along with the reason why the unsplitting method leads to a unified scheme.Since the unified gas-kinetic scheme uses the Bhatnagar-Gross-Krook-like model equation as its governing equation. The presented work investigates a series of model equations that can be used in the unified gas-kinetic scheme. A shock structure case and homogenous non-equilibrium relaxation case were carried out using difference model equations for comparison study. The degrees of approximation of difference model equations to the Boltzmann equation are also investigated theoretically.For the evolution of the discrete distribution function, the unified gas-kinetic scheme has a finite volume framework whose flux terms are constructed using the local analytical solution of the governing equation. By using theoretical and numerical methods, the presented work investigates the construction of the local analytical solution.By introducing the model of relaxation of inertial energy, a new unified gas-kinetic scheme is developed which expands the application of unified gas-kinetic scheme from monatomic gases to diatomic gases which are the main components of earth’s atmosphere. The relationships between the new governing equation and the macroscopic Navier-Stokes equation are derived. The treatment of inertial freedom and the detailed scheme are also discussed. Then, the effectiveness of new unified gas-kinetic scheme is tested numerically using test cases.Finally a three dimensional code of unified gas-kinetic scheme for aerodynamic and aeroheating computations is developed in distributed parallel computing environment. The MPICH is used for information messaging. The code is validated using three dimensional testing cases.