Unconditionally stable Gauge–Uzawa finite element schemes for incompressible natural convection problems with variable density

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

“…It shows that the C-VMS method deal with high Rayleigh number very well. And from the numerical results, we can see that our results conform with Wu et al ’s (2017b) 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]).…”

“…In this case, the natural convection equations are as follows: where Ω is a bounded domain in ℝ d ( d = 2, 3) with a sufficiently smooth boundary ∂Ω, ρ is the density which is positive, u = ( u 1 , …, u d ) T is the fluid velocity, p the pressure, T the temperature, μ=1Re the dynamic viscosity coefficient, k=μPr the thermal conductivity, g the gravitational force and T 1 the fixed time, Ra, Re and Pr are the Rayleigh number, the Reynolds number and the Prandtl number, respectively. The system (1.1) we consider in this paper is natural convection problem of an incompressible viscous Newtonian fluid with variable density (Szewc et al , 2011; Wu et al , 2017b). And for the sake of simplicity, we will consider the homogeneous boundary condition for velocity, i.e.…”

“…However, very few papers have been dedicated to natural convection problem with variable density. To the best of our knowledge, Wu et al (2017b) constructed unconditionally stable Gauge-Uzawa finite element schemes for natural convection problem with variable density. Szewc et al (2011) presented a new variant of the smoothed particle hydrodynamics simulations and Wang et al (2018c) constructed a novel pressure-correction projection finite element method for natural convection problem with variable density.…”

A Finite Element Algorithm for Nematic Liquid Crystal Flow Based on the Gauge-Uzawa Method

“…Then, Pyo has designed second-order time discrete schemes of the Gauge-Uzawa method for the Navier-Stokes equations [27] and the Boussinesq equations [28], based on a second order backward difference formula. Furthermore, for the incompressible viscous flows with variable density [29], the viscoelastic Oldroyd flows [30], the conduction-convection equations [31], the incompressible magnetohydrodynamics equations [36] and the incompressible natural convection problem with variable density [33], the efficiency and validity of the Gauge-Uzawa method are shown.…”