# Three-dimensional Natural Convection in a closed compartment

Introduction

Introduction

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

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.

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

Numercal example

Gometry and node structure

Results: Temperature