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 coarse-grain concentration and interfacial area in a mixture of two fluids with identical viscosity with no interfacial tension. This results in an efficient computational tool for mixing analysis, able to evaluate mixing dynamics and identify mixing problems such as dead zones (islands), applicable to realistic mixing devices. 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 developed previously. Composition fluctuations smaller than the cell size are represented by cell values of the area tensor which quantifies the amount, shape, and orientation of the interfacial area within each cell. The method is validated by comparison with an explicit interface tracking calculation. We show examples for 2D, time-periodic flows in a lid-driven rectangular cavity. The highly non-uniform time evolution of the spatial distribution of interfacial area can be determined with very low computational effort. Cell-to-cell differences in interfacial area of three orders of magnitude or more are found. It is well known that, globally chaotic flows, the microstructural pattern becomes self-similar, and interfacial area increases exponentially with time. This behavior is also captured well by the extended mapping method. The present calculations are 2D, but the method can readily be applied in 3D problems.