Introduction

The first case involves a box that is .05m high (y-axis) and .1m wide(x-axis). The temperature of the bottom wall is 401K and the top wall is 399K. All other walls are assumed adiabatic. The small change in temperature assigned to the two walls are not expected to render significant property changes in air. Thus for this particular case, all air properties(density, laminar kinematic viscosity, specific heat, etc.) are assigned constant values.The relevant non-dimensional group for such a problem is the Rayleigh number. It is based on the height of the domain and the temperature difference between the upper and lower walls.

A 40x80 (3200 cell) regular mesh with no stretching factors is used.

HEAT FLUX: Expected vs. Actual-Expected heat flux: q=3.58W/m^2 (According to the Nusselt number correlations obtained from experiments by Dropkin and Somerscales(1965)). Actual heat flux: q=1.65W/m^2

Note: The seemingly large discrepancy between the two heat flux values may be due to the Nusselt number equation used. This particular equation applies to large aspect ratios and as a result does not sufficiently take into account the effect of the side walls. However,this disagreement in heat flux values is considered to be relatively small and, therefore, is an accurate indication of convergence.

Graphical Representations of Selected Results

Temperature Field (K)

Velocity Profile (m/sec)

Velocity Field (m/sec)

Streamline Contour (m^2/sec)

Analysis:

Although this case is relatively straight forward, it provides an opportunity to understand the physics of the flow in a rectangular region heated from below and to learn to analyze those characteristics using the Phoenics 2.1 software: Only two thousand sweeps are needed to get this run to converge. But again, in view of the small size of the domain and the simplicity of the fluid flow this is a reasonable outcome.

The steady flow consists of a symmetric pair of vortices that transfer heat from the lower wall to the upper wall. Air heated at the bottom wall rises along both the side walls, and loses heat to the top wall. This cool air then drops down in the center region.

The maximum velocities are approximately 3 cm/sec. This appears to be a reasonable consequence of a low Rayleigh number such as this one.