A Global, Multi-Scale Simulation of Laminar Fluid Mixing: The Extended Mapping Method

We present a global, multi-scale model of fluid mixing in laminar flows, which describes the evolution of the spatial distribution of the microstructure in the mixture, for two fluids of identical viscosity with no interfacial tension. The flow domain is divided into cells, and large-scale variations in composition are tracked by following the cell-average concentrations of one fluid, using the mapping method of Kruijt, Galaktionov, Anderson, Peters, and Meijer (1999). Composition fluctuations smaller than the cell size are represented by cell values of the area tensor (Wetzel and Tucker, 1999a), which quantifies the size, shape, and orientation of the microstructure within each cell. The method is validated by comparison to an explicit interface tracking calculation. We show examples for two-dimensional, time-periodic flows in a lid-driven rectangular cavity. The spatial distribution of interfacial area is found to be highly non-uniform, with cell-to-cell differences of three orders of magnitude or more. If the flow is globally chaotic then the microstructural pattern becomes self similar, and interfacial area increases exponentially with time. The extended mapping method exhibits numerical diffusion, which affects the quantitative results. This can be minimized by using the largest possible mapping steps. The present calculations are two-dimensional, but the method can readily be applied in three-dimensional problems.