This test case simulates the natural convection problem in a square cavity at \( Ra=10^6 \) (convection is dominant Rayleigh number) and \( Pr=0.71 \) (Prandtl number). The aims of this test case are to:

validate the discretization of the Navier-Stokes equations with regular (or not) grids;
validate the discretization of the Energy equation;
validate the coupling between these two equation thanks to the Boussinesq approximation.

Configurations

Domain is square, its opposite corner coordinates being \((0,0,0)\) and \((0.1,0.1,0.1)\). We solve the Navier-Stokes and the Energy equations on the whole domain.

The following configuration is used:

Velocity boundary conditions:

Boundary	Condition
left	Wall
right	Wall
bottom	Wall
top	Wall
back	Symmetry
front	Symmetry

Temperature boundary conditions:

Boundary	Condition	Value
left	Dirichlet	\( T = T_{min} = 299.5 K \)
right	Dirichlet	\( T = T_{max} = 300.5 K \)
bottom	Neumann	\( \partial_{\mathrm n}T = 0 \)
top	Neumann	\( \partial_{\mathrm n}T = 0 \)
back	Neumann	\( \partial_{\mathrm n}T = 0 \)
front	Neumann	\( \partial_{\mathrm n}T = 0 \)

Density is computed thanks to a linear law:

\( \rho \ = \ \rho_0 \ (1 - \beta_T \ (T - T_0) ) \)

Runtime parameters

The gravity term of the Navier-Stokes equations is activated and the Boussinesq approximation chosen.

Comments

Second order spatial convergence is expected on the wall Nusselt number and mean velocity intensity.

To check the symmetry of the code, a rotation of the configuration is tested and produce same results up to computer precision. This done in the validation script of the code. Spatial convergence is done on the above configuration.

Results

The grid size starts from 32 in every direction to 512.

Temperature field and streamlines of the velocity feld, obtained with a 256x256 grid, are presented figure 1.

Figure 1: Temperature field and streamlines of the velocity field

Second order convergence is observed on left and right Nusselt numbers, as well as on mean velocity magnitude (see Table below). We observe small differences between left and right Nusselt number due to second order boundary condition implementation and second order wall temperature gradient computation. Nevertheless this difference converges with the refinement of the mesh. A reference solution, given in [1], is Nu=8.8252.

Mesh	Left Nusselt	Order	Right Nusselt	Order	Velocity	Order
32	1.049056e+01	n/a	1.049131e+01	n/a	3.792175e-03	n/a
64	9.184279e+00	n/a	9.184525e+00	n/a	3.681106e-03	n/a
128	8.901375e+00	+2.207075	8.901414e+00	+2.206584	3.638767e-03	+1.391398
256	8.842457e+00	+2.263531	8.842462e+00	+2.263758	3.627719e-03	+1.938184
512	8.829294e+00	+2.162209	8.829294e+00	+2.162534	3.624949e-03	+1.995758

References

[1] P. Le Quéré, Accurate solutions to the square differentially heated cavity at high Rayleigh number. Comput. Fluids, 20, (1991), 19–41.