Three-dimensional Natural Convection in a closed compartment



We consider natural convection of water in a closed shallow vessel, as shown in between Figure 1. The vessel is heated from below, so that the temperature difference between top and bottom is $40 C^{\circ}$. With $H = 0.01 m$ and $\Delta T = 20 C^{\circ}$, the corresponding Rayleigh number is $R_a = 20250$.

Calculated results are shown in Figure 2.(a)-(c) We can see the twelve Benard cells generated in the vessel.

Gorvern equations

$\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u}\cdot \nabla \mathbf{u}) + \frac{1}{\rho_0}\nabla p = \nu \nabla^2 \mathbf{u} + \beta(T- T_0) \mathbf{g}$
$\frac{\partial T}{\partial t} = \kappa \nabla^2 T - \left(\mathbf{u}\cdot\nabla\right)T$
$\nabla\cdot \mathbf{u} = 0$

We shall derive the expression of $F_z$, depending on the temperature $T$. To this end, we consider a fluid at temperature $T_0$ and density $\rho_0$ is a container. The coordinate $z$ is taken in the opposite direction to the gravitational acceleration $\mathbf{g}$. We assume that a lump of the fluid is the container is heated or cooled to the temperature $T$ with its density $\rho$. Then, the buoyancy of magnitude $g( \rho -\rho_0 )[ Nm^{-3}]$ is exerted on the lump of the fluid. If the density difference is small enough, the equation of state of the fluid can be expressed as follows.

$\frac{\rho}{\rho_0} = 1- \beta ( T -T_0 )$

Where $\beta [K^{-1}]$ is the volumetric thermal expansion coefficient. The buoyancy per a unit mass is therefore given by

$F_z = g \left(\frac{\rho_0 - \rho }{\rho_0}\right) = \beta g ( T -T_0 ).$

Numercal example

Gometry and node structure

Figure 1. Geometry and node structure.

Results: Temperature

Figure 2.(a): Temperature along the plane perpendicular to $y$-axis.
Figure 2.(b): Temperature along the plane perpendicular to $x$-axis.
Figure 2.(c): The iso surface of the temperature.