Steady-state two-dimensional results obtained from numerical solutions to the transient Navier-Stokes equations are given for laminar convective motion of a gas in an enclosed vertical slot with large horizontal temperature differences. We present results for air using the ideal-gas law and Sutherland-law transport properties, although the results are also valid for hydrogen. Wide ranges of aspect-ratio, Rayleigh-number and temperature-difference parameters are examined. The results are compared in detail with the exact solution in the conduction and fully developed merged boundary-layer limits for arbitrary temperature difference, and to the well-established Boussinesq limit for small temperature difference. It is found that the static pressure, and temperature and velocity distributions are very sensitive to property variations, even though the average heat flux is not. In addition we observe a net vertical heat flux to be the same as that obtained from the Boussinesq equations. We concentrate on the boundary-layer regime, but we present a rather complete picture of different flow regimes in Rayleigh-number, aspect-ratio and temperature-difference parameter space. We observe that, with increasing temperature difference, lower critical Rayleigh numbers for stationary and oscillatory instabilities are obtained. In addition we observe that in some cases the physical nature of the instability changes with increasing temperature difference.

We investigate numerically the transition from laminar to chaotic flow of a Boussinesq fluid with Pr = 0.71 in two-dimensional closed, differentially heated, vertical cavities having aspect ratios near unity. The cavities have rigid conducting sidewalls, and rigid insulating top and bottom walls. The physical nature of the resulting flow is a function of the aspect ratio and Rayleigh number.It is shown that an oscillatory approach to steady-state, oscillatory instabilities, quasi-periodic flow, and chaotic flow exist for the flow regimes investigated. We find that for aspect ratios of approximately three or larger the the first transition from steady-state is due to instability of the sidewall boundary layers, while for small aspect ratios, but larger than ½, it is due to internal waves near the departing corners. For both instabilities we obtain the critical Rayleigh number as a function of aspect ratio and write expressions relating the fundamental frequencies of the oscillatory flow to the Rayleigh number and aspect ratio. When Ra is increased significantly above the first critical value, the flow becomes complex since both types of instabilities can be present. With a further increase in Rayleigh number the flow becomes chaotic and eventually turbulent. The above results are illustrated for different Rayleigh numbers and aspect ratios using time histories, spectral analysis, and streamlines at different values of time.

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Perfect gas exact solutions to the steady Navier–Stokes equations are given for laminar convective motion in open and closed vertical slots with large temperature differences using Sutherland law transport properties. The solutions are valid a few slot widths away from the ends in the asymptotic region where the opposite hot and cold wall boundary layers are fully merged. It is found that the static pressure (in the closed slot) and temperature and velocity distributions (in all cases) are very sensitive to property variations, even though the heat flux may not be. We observe the net horizontal and vertical heat fluxes to be the same as those obtained from the Boussinesq equations. Comparisons with constant property solutions and the well-known Boussinesq limiting solution for small temperature differences are given for examples using air.

We consider a low-aspect-ratio two-dimensional rectangular cavity, differentially heated with an arbitrarily large horizontal temperature difference. Steady-state results obtained from numerical solutions of the transient Navier-Stokes equations are given for air using the ideal gas law and Sutherland law transport properties. We clarify the different flow regimes possible by using numerical results for aspect ratios 0.025 < A < 1 and for Rayleigh numbers (based on height) 10 2 < Ra < 10 9 . We present Nusselt numbers, and temperature and velocity distributions, and compare them with existing data. At high Ra in the Boussinesq limit we show the existence of weak secondary and tertiary flows in the core of the cavity. In addition we present novel solutions in the non-Boussinesq regime. We find that the classical parallel flow solution, accurate in the core of the cavity in the Boussinesq limit, does not exist when variable properties are introduced. For higher Rayleigh numbers, we generalize the well-known analytical boundary layer solution of Gill. The non-Boussinesq solutions show greatly reduced static pressure levels and lower critical Rayleigh numbers. Problem DefinitionConsider a two-dimensional rectangular enclosure of width L and height H filled with a gas. The gas is initially quiescent at a uniform temperature T 0 and pressure p 0 . The walls of the vessel are initially at the same temperature T 0 . At times larger