ULTRA-SHARP Nonoscillatory Convection Schemes for High-Speed Steady Multidimensional Flow

SUMMARYFor convection-dominated flows, classical second-order methods are notoriouslyoscillatory and often unstable. For this reason, many computational-fluid-dynam-icists have adopted various forms of (inherently stable) first-order upwinding overthe past few decades. Although it is now well known that first-order convectionschemes suffer from serious inaccuracies attributable to artificial viscosity ornumerical diffusion under high-convection conditions, these methods continue toenjoy widespread popularity for numerical heat-transfer calculations, apparentlydue to a perceived lack of viable high-accuracy alternatives. But alternatives areavailable. For example, nonoscillatory methods used in gasdynamics, includingcurrently_opular "TVD" schemes, can be easily adapted to multidimensional incom-pressible 1low and convective transport. This, in itself, would be a major advance fornumerical convective heat transfer, for example. But, as this report shows, second-order TVD schemes form only a small, overly restrictive, subclass of a much moreuniversal, and extremely simple, nonoscillatory flux-limiting strategy which can beapplied to convection schemes of arbitrarily high-order accuracy, while requiringonly a simple tridiagonal ADI line-solver, as used in the majority of general-purposeiterative codes for incompressible flow and numerical heat transfer. The new uni-versal limiter and associated solution procedures form the so-called ULTRA-SHARPalternative for high-resolution nonoscillatory multidimensional steady-state high-speed convective modelling.INTROI) UCTIONFor many years the state of the art in high-speed convective modelling, espe-cially in the field of numerical heat and mass transfer, has been dominated by first-order upwinding, often in the guise of the "Hybrid" scheme of Spalding [1] or, morerecently, the related "Power-Law Differencing Scheme" (PLDS) of Patankar [2].This situation has clearly evolved from an attempt to remedy the infamous problemsof unphysical oscillations and instabilities associated with "classical" central-differ-ence methods under high-convection conditions, using practical grids. The Hybridscheme avoids oscillatory behaviour by switching from second-order central to first-order upwinding for convection (and omitting modelled physical diffusion) whereverthe local component grid Peclet (or Reynolds) number exceeds a value of 2. PLDSand the exponential differencing scheme (EDS) on which it is based [3] involve amore subtle blending strategy, but both are also equivalent to first-order upwindingfor convection (with physical diffusion totally suppressed) for component grid PecletWork funded by Space Act Agreement C-99066-G.