Computational fluid dynamics (CFD) is the direct modeling of physical laws in a discrete space and time. The basic physical laws include the mass, momentum and energy conservations, reliable transport process, and consistent local domain of dependence and influence of the evolution solution. In this study, the high-order compact gas-kinetic scheme (GKS) is developed on structured and unstructured meshes based on the direct modeling framework for flow evolution with nonlinearly limited flux transport at a cell interface. The high-order compact schemes have overwhelming advantages in efficiency and accuracy in scale-resolved simulations compared with second-order scheme. The compact scheme in this study means that the update of the variables in a mesh cell only depends on the cell and its...[ Read more ]

Computational fluid dynamics (CFD) is the direct modeling of physical laws in a discrete space and time. The basic physical laws include the mass, momentum and energy conservations, reliable transport process, and consistent local domain of dependence and influence of the evolution solution. In this study, the high-order compact gas-kinetic scheme (GKS) is developed on structured and unstructured meshes based on the direct modeling framework for flow evolution with nonlinearly limited flux transport at a cell interface. The high-order compact schemes have overwhelming advantages in efficiency and accuracy in scale-resolved simulations compared with second-order scheme. The compact scheme in this study means that the update of the variables in a mesh cell only depends on the cell and its closest neighbors. In the construction of high-order compact scheme, subcell flow distributions or the equivalent degrees of freedom (DOFs) beyond the cell-averaged flow variables must be evolved and updated, such as the gradients of the flow variables inside each control volume. Under such a requirement, the direct use of the Riemann solver as the evolution model is not adequate in its dynamics in the construction of high-order compact scheme. The high-order dynamic process has to be modeled on the scales of cell size and time step.

The direct modeling on flow evolution under generalized initial conditions will be developed in this study. In the discrete space, with the cell size resolution, it is impossible to identify the subcell discontinuity except using the shock fitting. As a shock-capturing scheme, the appropriate approach is to assume a continuous subcell flow distribution and contribute all possible discontinuity to the cell interface. Therefore, in discrete space, the real solutions of flow variables on both sides of the interface may become discontinuous. In each cell, the evolved flow variables on interface can be used to update the cell-averaged gradients by Gauss’s law. At the same time, the time-accurate flux function at a cell interface can become a discontinuous function of time, such as a shock wave moving across a cell interface within a time step. Same as the spatial nonlinear reconstruction in the conventional CFD methods, such as total variation diminishing (TVD) and weighted essentially non-oscillatory (WENO) methods, the temporal nonlinear limiter on the time-accurate flux function has to be designed and properly used as well. The direct modeling unifies the nonlinear limiting methodology in both space for the data reconstruction and time for the time-dependent flux function. Under the direct modeling framework, the high-order compact GKS will be constructed on structured and unstructured meshes from 1-D to 3-D cases. The general linear high-order compact reconstruction strategies on structured and unstructured meshes are presented, respectively. To deal with discontinuities, the simplified WENO method with enhancing adaptivity is proposed for the nonlinear reconstruction and possesses high efficiency and accuracy on unstructured mesh.

On the 1-D uniform mesh, the sixth- and eighth-order compact GKS with spectral-like resolution are obtained. The 1-D high-order compact GKS can be extended to the multidimensional case. The spectral-like resolution benefits from the update of additional DOF, i.e., cell-averaged slope. On unstructured mesh, the high-order compact GKS has been developed on 2-D triangular mesh and 3-D tetrahedral mesh, where the schemes can achieve sixth order of accuracy in 2-D and fourth order of accuracy in 3-D from the compact stencils. As a result, the drawback of finite volume method for requiring large stencils is overcome. A variety of numerical tests are presented to validate the high-order compact GKS. The current scheme presents state-of-art numerical solutions under a wide range of flow conditions, i.e., strong shock discontinuity, boundary layer structure, and aeroacoustic wave propagation, using the same nonlinear reconstruction and parameters settings in the code.

The high-order compact GKS is also extended to solve the shallow water equations (SWE) coupled with realistic hydraulic model on triangular mesh. The successful application to SWE and its extended model demonstrates the potential of the high-order compact GKS to be applied to other convection-dominated flow environment.