A physically based flux limiter for QUICK calculations of advective scalar transport

Transient, advective transport of a contaminant into a clean domain will exhibit a moving sharp front that separates contaminated and clean regions. Due to 'numerical diffusion' - the combined effects of 'cross-wind diffusion' and 'artificial dispersion' - a numerical solution based on a first-order (upwind) treatment will smear out the sharp front. The use of higher-order schemes, e.g. QUICK (quadratic upwinding) reduces the smearing but can introduce non-physical oscillations in the solution. A common approach to reduce numerical diffusion without oscillations is to use a scheme that blends low-order and high-order approximations of the advective transport. Typically, the blending is based on a parameter that measures the local monotonicity in the predicted scalar field. In this paper, an alternative approach is proposed for use in scalar transport problems where physical bounds C_Low≤ C≤C_High on the scalar are known a priori. For this class of problems, the proposed scheme switches from a QUICK approximation to an upwind approximation whenever the predicted upwind nodal value falls outside of the physical range [C_Low, C_High]. On two-dimensional steady-state and one-dimensional transient test problems predictions obtained with the proposed scheme are essentially indistinguishable from those obtained with monotonic flux-limiter schemes. An analysis of the modified equation explains the observed performance of first- and second-order time-stepping schemes in predicting the advective transport of a step. In application to the transient two-dimensional problem of contaminate transport into a streambed, predictions obtained with the proposed flux-limiter scheme agree with those obtained with a scheme from the literature.