Study On Highly Efficient And Accurate Discrete Unified Gas Kinetic Schemes

Computational fluid dynamics is an effective tool to study the critical problems in the fields of energy,materials,chemical engineering and bio-engineering,such as shale gas exploitation,fuel cell,microfluidic chip and etc.The mesoscopic method based on the Boltzmann equation has received more and more attention due to its clear physical background,wide-scale applicability,and considerable computational efficiency.As a deterministic mesoscopic method with second-order accuracy,the discrete unified gas kinetic scheme(DUGKS)method combines the advantages of both lattice Boltzmann equation(LBE)and unified gas kinetic scheme(UGKS),including the simplified flux evaluation scheme,flexible mesh adaption and the asymptotic preserving properties.From the continuous flow to the rarefied flow,the DUGKS method shows good numerical performance and application potential.With the increasing requirement of accuracy and real-time computation for numerical methods,it is of great importance to develop high-precision and high-efficiency DUGKS methods for engineering applications.However,the improvements of the DUGKS method in the aspects of accuracy and computational efficiency are restricted,due to the complicated nonlinear collision term in the Boltzmann equation and the huge number of discrete particle velocities for the rarefied flow.At the same time,in the widely considered low-speed incompressible flow,the compressible effect existed in the original DUGKS may bring some undesirable errors.In order to solve the above problems,we conduct the following researches:To eliminate the compressible effect,a more accurate incompressible DUGKS model is developed in this thesis.Meanwhile,a general external force term is added in DUGKS to promote the development of multi-field coupling models such as thermal,multiphase and chemical reactions.And the non-equilibrium extrapolation scheme is also introduced,which can extend the ability of DUGKS to treat different boundary conditions including the velocity and pressure boundary conditions.In this thesis,we introduce the history of high-order mesoscopic methods and point out the main reason for limiting the mesoscopic method to achieve higher-order accuracy.An efficient third-order DUGKS is also presented for simulating continuum and rarefied flows.By employing two-stage time-stepping scheme and the high-order DUGKS flux reconstruction strategy,third-order of accuracy in both time and space can be achieved in the present method.It is also analytically proven that the second-order DUGKS is a special case of the present method.Typical benchmark tests are carried out for comprehensive evaluation of the present method.It is observed in the tests that the present method is advantageous over the original DUGKS in accuracy and capturing delicate flow structures.Moreover,the virtual memory consumed is only about 1.3 times over the second-order method.At the same time,due to the applicability of large time steps,the total computation time of the third-order method is even better than the second-order DUGKS method under certain conditions.In the simulation of rarefied fluid flow,numerous discrete particle velocities are commonly used by mesoscopic methods to ensure the accuracy of the calculation,which will cause the significant increase in the computational complexity,computational time-consuming,and virtual memory usage.It restricts the mesoscopic methods for the further application seriously.To solve this problem,a Reduced Order Method(ROM)is employed to optimize the discrete particle velocity space and the ROM-DUGKS method is developed.The specific process can be summarized as follows: the optimal subset of discrete velocity space is selected by Gappy method;after the original discrete particle velocity is approximated by the discrete empirical interpolation method;then the distribution function is evolved by the DUGKS method;finally,the approximate solution to the problem can be obtained.This method can reduce the number of discrete particle velocity with preserving the expected numerical accuracy.And the purposes of improving the computational efficiency and reducing the computational resource consumption can be achieved.It is worth noting that this optimization method is also applicable to other discrete velocity mesoscopic methods.The lid-driven flow in the three-dimensional cavity,which contains many complex flow phenomena,is a very important numerical case.In this thesis,DUGKS method is employed to study the three-dimensional lid-driven cavity flow in continuous and rarefied fluid areas.Compared with the reference data,the accuracy of DUGKS in studying three-dimensional problems is proved.The three-dimensional topology of the flow is studied,and the differences of the flow structure in the cubic cavity and the deep cavity are pointed out.In this thesis,the vortex topology in the deep cavity is analyzed at different Reynolds numbers.It can be found that the splitting phenomenon of the secondary vortex occurs between Re = 500 and 600.Besides,the three-dimensional lid-drive flow in the micro-cavity is also investigated.Compared with the DSMC method,DUGKS has better numerical accuracy and stability.The three-dimensional flow structure in the micro-cavity at Kn = 0.1 is given.In summary,the detailed research of the DUGKS method with high efficiency and high precision is conducted.An incompressible DUGKS is established to improve the accuracy for the simulation of incompressible flow.We also develop the DUGKS method with three-order accuracy in both time and space.And the reduced-order strategy is employed to improve the computational efficiency of DUGKS in simulating rarefied flows.In addition,the DUGKS is also used to study the three-dimensional lid-driven cavity flow in continuous and rarefied fluid areas,and the three-dimensional topology of the flow is analyzed.The related work in this thesis is also valuable for the study of other mesoscopic methods in terms of improving computational efficiency and numerical accuracy.