Natural convection 2D

We solve a two-dimensional natural convection problem without using the Boussinesq or low-Mach number approximations. The problem setup is shown in Figure below.

A large temperature difference is applied to the vertical walls, while the horizontal walls are thermally insulated. The specific heat used is c_v = 717.5 J/kg K, Sutherland’s law in the form given in Figure is used for computing the viscosity, while the conductivity  is computed by assuming that the Prandtl number Pr = c_p / is 0.71. The gravity g is taken to be 0.00295924. These values are chosen so as to yield a Rayleigh number (based on reference values of rho₀ = 0.588415 kg/m³ and theta₀ = 600 K) of 10⁶.

The material is air at standard conditions.

Computations sees a time step of 1 s, and a regular grid made of squares of 0.01 m side. The diagonal incompressible tau of 2007 is adopted.

Details of the simulations can be found in the files fluid_ic_bc.hpp, props.txt, setup.txt and mesh.geo.

The vertical velocity and temperature variations (normalized as v_y /0.05021228 and (theta−600)/600, respectively) along the line y = 0.5 are shown in Figure below.

The time evolution of temperature and velocity fields follows below:

Reference

C.S. Jog “A finite element method for compressible viscous fluid flows” (2011) Int. Jour. Num. Meth. Fluids