Designing a consistent implementation of the discrete unified gas-kinetic scheme for the simulation of three-dimensional compressible natural convection

Discrete unified gas-kinetic scheme (DUGKS) has been developed as a robust and accurate approach for thermal compressible flow simulations; however, designing an efficient and accurate lattice velocity model to take full advantage of DUGKS remains a challenge. In this study, we apply DUGKS to simulate three-dimensional compressible natural convection in an enclosure with a large temperature difference, without making the Boussinesq approximation. The Chapman–Enskog analysis indicates that the fourth-order moments of equilibrium is needed for the heat flux evaluation in the energy equation, implying that the fourth-order Hermite expansion of equilibrium and thus at least an eighth-order Gauss–Hermite quadrature are needed for accurate simulation of the Navier–Stokes–Fourier system. For this purpose, a highly efficient lattice velocity model, D3Q77A9, is derived, which provides a Gauss–Hermite quadrature of ninth-order accuracy in three dimensions. The accuracy of this D3Q77A9 model is demonstrated by simulating compressible natural convection flows in both two-dimensional and three-dimensional cavities. An error analysis is performed to emphasize the importance of combining a quadrature with an adequate degree of precision and a proper order of Hermite expansion of the equilibrium distribution.