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SUMMARYIn many areas of computational fluid dynamics, especially numerical convective heat and mass transfer, the 'Hybrid' and 'Power-Law' schemes have been widely used for many years. The popularity of these methods for steady-state computations is based on a combination of algorithmic simplicity, fast convesgence, and plausible looking results. By contrast, classical (second-order central) methods often involve convergence problems and may lead to obviously unphysical solutions exhibiting spurious numerical oscillations. Hybrid, Power-Law, and the exponential-difference scheme on which they are based give reasonably accurate solutions for steady, quasi-one-dimensional flow (when the grid is aligned with the main flow direction). However, they are often also used, out of context, for flows oblique or skew to the grid, in which case, inherent artificial viscosity (or diffusivity) seriously degrades the solution. This is particularly troublesome in the case of recirculating flows, sometimes leading to qualitatively incorrect results-since the effective artificial numerical Reynolds (or Peclet) number may then be orders of magnitude less than the correct physical value. This is demonstrated in the case of thermally driven flow in tall cavities, where experimentally observed recirculation cells are not predicted by the exponential-based schemes. Higherorder methods correctly predict the onset of recirculation cells. In the past, higher-order methods have not been popular because of convergence difficulties and a tendency to generate unphysical overshoots near (what should be) sharp, monotonic transitions. However, recent developments using robust deferredcorrection solution methods and simple flux-limiter techniques have eliminated all of these difficulties. Highly accurate, physically correct solutions can now be obtained at optimum computational efficiency.KEY WORDS: finite difference; finite volume; artificial viscosity; QUICK, hybrid; power-law EXPONENTIAL-BASED SCHEMESExactly 40 years ago, Allen and Southwell' developed what is now called exponential differencing* for convection-diffusion operators. In the intervening period, several researchers have independently 'rediscovered' and/or approximated the basic elements of the Allen-Southwell scheme. The most notable of these convection-diffusion methods are the 'Hybrid' difference scheme of Spalding3 and the 'Power-Law' difference scheme of Patankar.4 These were combined with the SIMPLE pressure-solver4 in the so-called TEACH code5 developed at Imperial College, giving a robust, general-purpose elliptic-equation solver suitable for solving steady-state Navier-Stokes equations and associated heat and mass transfer problems. Over the past decade or more, the Hybrid and Power-Law methods have been used widely. Their popularity seems to have been increasing in recent years: at the most recent Thermal Problems conference6 in

By finite difference calculations the convective flow in a vertical enclosure heated from a side is investigated. The emphasis in the study is for the flow of air in a window cavity. Detailed plots of the stream patterns and isotherms in a multicellular flow are shown and vertically averaged Nusselt numbers as well as typical local values are given. The calculations compare favorably with the experiments of ElSherbiny, Raithby, and Hollands. A working formula for window spacings is suggested.

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We present numerical solutions of natural convection in a shallow enclosure heated from a side. As a result of hydrodynamic instability transverse cells appear in the flow if the Prandtl number is sufficiently small. Both conducting and insulated top and bottom boundaries were considered. For fluids of small Prandtl number the differences in the flow patterns in these two cases are slight, the strength of the circulation in the cells being somewhat weaker when the boundaries are insulated. This is a result of a more stable flow in this case, caused by the kinetic energy being more vigorously expended in the work against the buoyant forces. Insulated boundaries allow the temperature field to adjust more freely in the end regions leading to crowding of the isotherms there and consequently to larger heat transfer than when the boundaries are conducting.