Two-Level Variational Multiscale Method Based On The Decoupling Approach For The Natural Convection Problem

Natural convection problem is a kind of important problems in Computational Fluid Dynamics and typical coupled nonlinear equations which describe incompressible viscous ?uid movement and widely used in the ?eld of science and engineering.However, because of their coupling and nonlinearity, especially, in the case of high Rayleigh number, it’s di?cult to make mathematic analysis and numerical computation, hence, studying and constructing an algorithm with simplicity and high e?ciency is very important.Direct Numerical Simulation natural convection problem with high Rayleigh number usually requires very ?ne mesh size in order to capture the small scales information. This needs computer to increase the storage space to realize the numerical simulation, which directly led to a sharp increase in the computational scale and CPU-time. The VMS method can be used to handle the large Rayleigh number problem. The main idea of VMS method regards the standard ?nite element space as the resolved scales space, and its appropriate projection space as the large scales space, then simulate the in?uence of the unresolved small scales to the resolved small scales through some formulations based on the resolved small scales. The natural convection problem based on the Navier-Stokes equations coupled with a temperature ?eld, so the equations have strong coupling, in addition to the nonlinear term,using Direct Numerical Simulation method to solve the natural convection problem, it takes a large of CPU-time. Therefore, it is very necessary to use decoupling technique and two-level method to improve the computational e?ciency.In this paper, we formulated two-level VMS method based on the decoupling approach for the natural convection problem based on the work of the predecessor.The key idea of the method combines decoupling technology and two-level method with VMS method. Based on the work of the predecessor using VMS method to handle the natural convection problem with large Rayleigh number, we use the decoupling technology to decouple coupling variables, nonlinear term of equation is linearized by using two-level method on ?ne grid, this signi?cantly reduces the computational complexity and scale. The work of this paper not only perfected theoretical analysis of the LG-VMS method for the natural convection problem,but also further used the decoupling technique and two-level method to improve and enhance the computing e?ciency, and gave theoretical analysis of stability and convergence of the algorithm, the results of numerical experiments also verify the theoretical analysis, and show that the algorithm is able to handle the natural convection problem with high Rayleigh number e?ciently and reliably.