As the increasing of computer power and the requirement for accurate solutions for more challenging problems, the high-order methods become good choices for the computational fluid dynamics. In recent years, many high-order numerical methods have been developed for the Euler and Navier-Stokes equations. The gas-kinetic scheme (GKS) based on the BGK model has been developed systematically for the Navier-Stokes solutions, and successfully applied for the continuum flow simulations from nearly incompressible to hypersonic viscous and heat conducting flows. In this thesis, we mainly focus on the development of high-order gas-kinetic scheme for the Euler and Navier-Stokes equations.

In many high-order schemes, with highly complicated initial condition, the Riemann solutions are still...[ Read more ]

As the increasing of computer power and the requirement for accurate solutions for more challenging problems, the high-order methods become good choices for the computational fluid dynamics. In recent years, many high-order numerical methods have been developed for the Euler and Navier-Stokes equations. The gas-kinetic scheme (GKS) based on the BGK model has been developed systematically for the Navier-Stokes solutions, and successfully applied for the continuum flow simulations from nearly incompressible to hypersonic viscous and heat conducting flows. In this thesis, we mainly focus on the development of high-order gas-kinetic scheme for the Euler and Navier-Stokes equations.

In many high-order schemes, with highly complicated initial condition, the Riemann solutions are still implemented because it is the only exact solution we know, even though it gives a first-order dynamics. Compared with the first order dynamics in the Riemann solutions, the whole curve from a discontinuous flow distribution around a cell interface interacts through particle transport and collision in the determination of the flux function in the gas-kinetic scheme. With the implementation of Chapman-Enskog expansion, both inviscid and viscous flows can be calculated in one framework, and no special treatment for the viscous terms is needed.

Because of these advantages, the high-order gas-kinetic scheme was developed for two-dimensional flow based on the WENO reconstruction. In this thesis, we will extend the high-order scheme to three-dimensional computation. Due to the multidimensionality and the spatial-temporal coupling of the scheme, the Runge-Kutta time stepping and Gaussian point integrations are avoided, which will greatly simplify the computation. The high-order scheme is extended for the moving-mesh computation based on the WENO reconstruction as well. The numerical results show that the high-order gas-kinetic scheme is effective to simulate complicated flows.

To achieve the high-order accuracy, a large stencil is needed in the WENO reconstruction, especially for unstructured meshes. The large stencil will lead to difficulties for parallel computing. In this thesis, a compact third-order gas-kinetic scheme is presented for the Euler and Navier-Stokes equations on the unstructured meshes. Based on the third-order gas evolution model, the time-dependent solution of gas distribution at a cell interface can provide both numerical fluxes and point-wise flow variables. Thus, it is possible to develop a compact third-order gas-kinetic scheme by the use of cell averaged values and point values at the cell interface from the neighboring cells. The weighted least square procedure is used to achieve a third-order accuracy. For the flow with strong discontinuity, the shock detection technique is also used. The numerical tests demonstrate that the compact scheme is robust for the flows with strong discontinuities and accurate for the smooth flow solutions.

The formulation of one-stage gas distribution function becomes extremely complicated when we develop even higher order schemes, especially for multidimensional computation. Due to the temporal accuracy in the gas-kinetic flux function, the use of the two-stage Lax-Wendroff time stepping method [44] provides a reliable framework for the further development of higher-order gas-kinetic schemes. With this time stepping method, based on a second-order flux function, a two-stage fourth-order gas-kinetic scheme is developed with a fifth-order WENO reconstruction. The scheme not only reduces the complexity of the flux function, but also improves the accuracy of the scheme. Most importantly, the robustness of the fourth-order GKS is as good as the second-order one. Many numerical tests, including many difficult ones for the Navier-Stokes solvers, have been used to validate the fourth-order method. Following the two-stage time-stepping framework, a fifth-order scheme can be developed as well with the use of the third-order gas-kinetic flux function, in which both first-order and second-order time derivatives are used in the time stepping method.