Efficient monolithic projection method with staggered time discretization for natural convection problems. (December 2019) - Pan, Xiaomin; Kim, Ki-Ha; Choi, Jung-Il
Highlights: We propose VDMPM-STD by improving FDMPM-1P using staggered time discretization. The VDMPM-STD allows an explicit decoupling of momentum and energy equations. The VDMPM-STD is computationally efficient with the second-order accuracy. The VDMPM-STD provides accurate predictions of 2D and 3D RBC problems. Abstract: For a more efficient algorithm, we introduce staggered time discretization to improve the previous method (Pan et al., 2017), fully decoupled monolithic projection method with one Poisson equation (FDMPM-1P), to solve time-dependent natural convection problems. The momentum and energy equations are discretized using the Crank–Nicolson scheme at the staggered time grids, in which temperature and pressure fields are evaluated at half-integer time levels ( n + 1 2 ), while the velocity fields are evaluated at integer time levels ( n + 1 ). Numerical simulations of two-dimensional (2D) Rayleigh–Bénard convection show that the proposed method is more computationally efficient and stable than FDMPM-1P, while preserving the second-order spatial and temporal accuracy. Further, the proposed method provides accurate predictions of 2D Rayleigh–Bénard convection under different thermal boundary conditions for a Rayleigh number up to 10 10, three-dimensional turbulent Rayleigh–Bénard convection in the range of 1 × 10 5 – 2 × 10 7 in horizontal periodic domain, and three-dimensional turbulent Rayleigh–Bénard convection in the range of 1 × 10 6 – 1 × 10 7 in bounded domain. Finally, we theoretically confirmed that the proposed method is second-order in time and it is more stable than FDMPM-1P for small Ra .