Discrete unified gas kinetic scheme with a force term for incompressible fluid flows

The discrete unified gas kinetic scheme (DUGKS) is a finite-volume scheme with discretization of particle velocity space, which 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. However, similar to standard LBE, the DUGKS can also be considered as a compressible scheme, and the compressible effect may bring some undesirable errors when it is used to investigate incompressible fluid flows. To eliminate the compressible effect, in this work a new DUGKS with a force term is developed through modifying the equilibrium distribution function. And simultaneously, the non-equilibrium extrapolation (NEE) scheme is also introduced to treat the velocity and pressure boundary conditions. To illustrate the capacity of the present DUGKS, we first performed some numerical simulations of two-dimensional steady and unsteady flows, and conducted a comparison between the present DUGKS and the original one. The results indicate that the present DUGKS can reduce the compressible effect efficiently, and the NEE scheme is also consistent with the second-order accuracy of DUGKS. We then extended the present DUGKS to study the three-dimensional lid-driven flows (LDF) in cubic and deep cavities, and found that the present results are in good agreement with available benchmark results, which indicates the present DUGKS is also accurate and efficient in the study of three-dimensional problems. At last, the structures of vortex in the cubic and deep cavities are also considered, and the symmetric affiliated vortices aside the secondary vortex at R e ź 600 can be observed in the deep LDF.