First, a numerical study is made of the finite-wall effect in the benchmark-configuration buoyant convection in a square cavity at large Rayleigh number. A general formulation, with one vertical sidewall of finite thickness and thermal conductivity, is presented. In the first place, the finite-wall effect for the case of non-pulsating boundary temperature condition is delineated. The energy balance consideration, together with the preceding empirical correlations, leads to a simple formula to predict the temperature at the interior surface of the finite-thickness wall. The analytical predictions are shown to be consistent with the results of full-dress Navier-Stokes numerical solutions. Next, the finite-wall effect for the case of pulsating boundary temperature condition is explored. The numerical results illustrate that the amplitude of oscillating Nusselt number, A(Nu), at the central plane peaks at a particular pulsation frequency. This has been interpreted to be a manifestation of resonance. The finite-wall effect on the shift of resonance frequency is discussed. The temperature oscillation at the interior surface of the solid wall is examined, and the convection-modified model is introduced to describe the alteration in the temperature contrast across the fluid portion. The estimation of the resonance frequency, based on the internal gravity oscillations, is shown to be in accord with the Navier-Stokes numerical solutions.
Second, a numerical investigation is made of three-dimensional natural convection of a Boussinesq-fluid in a vertically-mounted cylindrical container. The boundary conditions are such that the wall temperature $\theta_\Sigma$ is inhomogeneous in the horizontal azimuthal direction but $\theta_\Sigma$ increases in the vertical direction. Interest is confined to flows with globally-stable stratifications and with substantial azimuthal variations in thermal boundary conditions. Comprehensive numerical solutions to the Navier-Stokes equations are obtained. A variety of specific thermal boundary conditions are considered for detailed examination. Flow characteristics are described in broad ranges of principal nondimensional parameters, i. e., the vertical and horizontal Rayleigh numbers, the container aspect ratio and the Prandtl number. Three-dimensional flow patterns are constructed. For large Rayleigh numbers, the azimuthal inhomogeneity of boundary conditions is absorbed in the boundary layers. In the interior core, flow is determined mostly by the azimuthally-averaged temperature boundary condition. Exemplifications are made for two cases: (1) when $\theta_\Sigma$ is vertically uniform, and (2) when $\theta_\Sigma$ is a linear function of height. For both cases, the interior core is stably-stratified, and on the planes of constant height, horizontal motions are present. Vertical and horizontal profiles of major flow variables are plotted. The explicit effect of increasing the vertical gradient of $\theta_\Sigma$ on the global flow structure is delineated. Also, the effect of the frequency (n) of circumferential variation of sidewall temperature is investigated. The frequency n is increased from 1 to 10. For the different value of n, flow and temperature fields are compared with each other. Also, the heat transfer characteristics on the cylindrical surface are presented for various n. As n increases, the flow fields become weaker and convective gain in heat transfer is reduces.
Finally, a numerical study is made of natural convection of a Boussinesq fluid in a rectangular cavity to investigate the effects of the spatially non-uniform boundary temperature. The flow configuration is similar to the benchmark configuration which has thermally-insulated top- and bottom-endwalls and finite temperature difference between two vertical sidewalls. The notable point in the present configuration is that the temperature at one vertical sidewall varies linearly or sinusoidally in space. In the first stage, the case of linear boundary temperature is considered. Here, the slope of wall temperature is altered for a same average value. The maximum temperature deviation(ε), the cavity aspect ratio($A_r$) and the Rayleigh number(Ra) are changed. The flow and temperature fields and the average heat transfer rate are substantially altered for the different slope of boundary temperature. Depending on the slope, the flow field becomes more stable or unstable and a stability criteria is found for Ar, Ra and ε. The simple model equations for the vertical temperature distribution in the interior core give excellent agreements with the numerical results for all Ar, Ra and ε. There is a slope for which the heat transfer rate is minimized for various values of Ra and Ar. Next, the effects of the spatially-periodic boundary temperature are examined. Here, the temperature at the hot sidewall varies sinusoidally in vertical direction about the average value. For a large Rayleigh number, the amplitude, frequency and phase difference of the spatially-periodic boundary temperature, and the cavity aspect ratio are altered to examine the effects of those parameters on the fluid flow and the heat transfer. The flow and temperature fields, and the average heat transfer rate are substantially changed for the problem parameters. A separated secondary flow is found dependent on the phase difference. The variation of the average Nusselt number was explained with an observation of the vertical distribution of the local Nusselt number. For high frequency, the flow and heat transport become less sensitive to the changes of other flow conditions.
