Why You Should Not Use 'Hybrid', 'Power-Law' or Related Exponential Schemes for Convective Modelling—There Are Much Better Alternatives

In 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 convergence, 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. Higher-order 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 deferred-correction 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