Pressure-Correction Projection FEM for Time-Dependent Natural Convection Problem

Abstract

Pressure-correction projection finite element methods (FEMs) are proposed to solve nonstationary natural convection problems in this paper. The first-order and second-order backward difference formulas are applied for time derivative, the stability analysis and error estimates of the semi-discrete schemes are presented using energy method. Compared with characteristic variational multiscale FEM, pressure-correction projection FEMs are more efficient and unconditionally energy stable. Ample numerical results are presented to demonstrate the effectiveness of the pressure-correction projection FEMs for solving these problems.

A novel characteristic variational multiscale FEM for incompressible natural convection problem with variable density

“…Hence, the temperature contours nearly parallel to the right side plate. Through the results, we can see that our results conform with Wu et al ’s (2017a) very well.…”

“…For natural convection problem, which has a wide range of applications in many research fields (Wu et al , 2017b, 2016), the density difference in the fluid occurring due to temperature gradient is the driving mechanism of fluid motion[1]. In case that the density variation is small, it can be modeled by using a Boussinesq approximation, which treats the density as a constant but with an added buoyancy force, and most literature studies the constant density natural convection based on the Boussinesq approximation (Boland and Layton, 1990; Du et al , 2015; Feng et al , 2011; Huang et al , 2015, 2013, 2012; Liao, 2012, 2010; Si et al , 2014; Szumbarski et al , 2014; Su et al , 2017a, 2017b, 2014a, 2014b; Sun et al , 2011; Davis, 1983; Wang et al , 2018a, 2018b; Wu et al , 2015b, 2017a, 2016; Zhang et al , 2016, 2018]). However, in most geophysical flows and many other situations, fluid motion is usually driven by large temperature differences, which results in a considerable density change and the Boussinesq approximation is no longer valid.…”

Stability and convergence of two‐grid Crank‐Nicolson extrapolation scheme for the time‐dependent natural convection equations

“…Among these numerical schemes, the Crank‐Nicolson extrapolation scheme is almost unconditional stability, while Crank‐Nicolson/Adams‐Bashforth scheme requires some restrictions on the time step and mesh size. Therefore, in this paper, we consider the Crank‐Nicolson extrapolation scheme for the natural convection problem, our work is extension and supplement the previous works and provide some new stability and convergence results for the numerical solutions. At the same time, based on He, He and Li, and He et al, for the 3D Navier‐Stokes equations, we also consider the Crank‐Nicolson extrapolation scheme for 3D time‐dependent natural convection problem.…”

“…Due to the coupling of three variables among the natural convection equations, finding the numerical solutions becomes a difficult task, much attention has been attracted in recent years. For example, we can refer to Du et al, Wu et al, Zhang et al, and Zhang et al for the variational multiscale methods, for the projection method, for the decoupled method, for iteration methods, and the references therein, all above mentioned works are one order schemes and 2D case.…”

Recovery-based error estimator for the natural-convection problem based on penalized finite element method

“…Natural-convection (NC) problem, namely, buoyancy-driven flows which occur frequently in practical problems are atmospheric fronts, katabatic winds in nature, room ventilation, heating general, nuclear reaction systems, fire control in life, dense gas dispersion, natural ventilation, solar collectors, insulation with double pane window and cooling of electronic equipment in industry (Boland and Layton, 1990a;Çıbık and Kaya, 2011;Sankhavara and Shukla, 2006;Wu et al, 2017aWu et al, , 2017bSu et al, 2017;Schmidt-Nielsen, 2011;Wu et al, 2016) [1]. In this article, a posteriori recovery-based error estimator based on penalized FEM is presented to solve the NC problem.…”