Natural Convection in a Square Cavity

This validation example demonstrates Flow Simulation capabilities to simulate natural convection. Natural convection has attracted a great deal of attention from researchers because of its presence both in nature and engineering applications. In nature, convection cells formed from air raising above sunlight-warmed land or water are a major feature of all weather systems. In engineering applications, convection is commonly visualized in the formation of microstructures during the cooling of molten metals, and fluid flows around shrouded heat-dissipation fins, and solar ponds. A very common industrial application of natural convection is free air cooling without the aid of fans.

Here we will consider a 2D square cavity with a steady-state natural convection, for which a highly-accurate numerical solution has been proposed in Ref. 7 and used as a benchmark for about 40 computer codes in Ref. 8, besides it well agrees with the semi-empirical formula proposed in Ref. 10 for rectangular cavities. This cavity's configuration and imposed boundary conditions, as well as the used coordinate system, are presented in Figure 1. Here, the left and right vertical walls are held at the constant temperatures of T1 = 305 K and T2 = 295 K, accordingly, whereas the upper and bottom walls are adiabatic. The cavity is filled with air.

Figure 1. An enclosed 2D square cavity with natural convection.

The square cavity's side dimension, L, is varied within the range of 0.0111...0.111 m in order to vary the cavity's Rayleigh number within the range of 103…106. Rayleigh number describes the characteristics of the natural convection inside the cavity and is defined as follows:



where is the volume expansion coefficient of air,

g is the gravitational acceleration,

Cp is the air's specific heat at constant pressure,

ΔT = T1 – T2 = 10 K is the temperature difference between the walls,

k is the thermal conductivity of air,

μ is the dynamic viscosity of air.

The cavity's model is shown in Figure 2.

Figure 2. The model created for calculating the 2D natural convection flow in the 2D square cavity using Flow Simulation.

Due to gravity and different temperatures of the cavity's vertical walls, a steady-state natural convection flow (vortex) with a vertical temperature stratification forms inside the cavity. The Ra = 105 flow's prediction performed with Flow Simulation is shown in Figure 3.

Figure 3. The parameters distributions predicted by Flow Simulation in the square cavity at Ra = 105: (a) temperature, (b) X-velocity, (c) Y-velocity, (d) velocity vectors, and (e) streamlines.

A quantitative comparison of the Flow Simulation predictions performed at result resolution level 8 with Ref. 7, Ref. 8 (computational benchmark) and Ref. 10 (semi-empirical formula) for different Ra values is presented in Figure 4 - Figure 6. The Nusselt number averaged over the cavity's hot vertical wall (evidently, the same value must be obtained over the cavity's cold vertical wall) , where qw,av is the heat flux from the wall to the fluid, averaged over the wall, is considered in Figure 4.

Here, the dash line presents the Ref. 10 semi-empirical formula:

,

where D is the distance between the vertical walls and L is the cavity height (D = L in the case under consideration). One can see that the Flow Simulation predictions practically coincide with the benchmark at Ra ≤ 105 and are close to the semi-empirical data.

Figure 4. The average sidewall Nusselt number vs. the Rayleigh number.

The dimensionless velocities of the natural convection flow in the X and Y directions, and (which are maximum along the cavity's mid-planes, i.e., along the vertical mid-plane and along the horizontal mid-plane) are considered in Figure 5. The dimensionless coordinates, and , of these maximums' locations (i.e., for and for ) are presented in Figure 6. One can see that the Flow Simulation predictions of the natural convection flow's local parameters are fairly close to the benchmark data at Ra ≤ 105.

Figure 5. Dimensionless maximum velocities vs. Rayleigh number.

Figure 6. Dimensionless coordinates of the maximum velocities' locations.