Miscible rectilinear displacements with gravity override

Rectilinear homogeneous miscible displacements with gravity override are analysed by means of direct numerical simulations on the basis of the vorticity–streamfunction formulation of the governing equations. The vorticity-based point of view offers the advantage of clearly attributing the dominant flow characteristics to the effects of viscosity contrast, density difference, impermeable boundary conditions, or interactions among the above. Basic considerations regarding the vorticity field show that in an integral sense the coupling between viscosity and gravity vorticity is predominantly one way in nature, in that the gravity vorticity can amplify the viscous vorticity, but not vice versa. In particular, the vorticity point of view provides an explanation for the formation of the gravity tongue in terms of a focusing mechanism, which results from the combined action of the unfavourable viscosity gradient and the potential flow field generated by the interaction of the gravitational vorticity with the horizontal boundaries. This potential velocity field locally enhances the uniform global displacement velocity near the upper boundary, and thereby amplifies the viscous fingering instability along this section of the interface. In some parameter ranges, the gravity tongue exhibits interesting interactions with the viscous fingers next to it, such as pinching and partial merging. The influence of the Péclet number, the viscosity and density contrasts, and the aspect ratio on the dynamic evolution of the displacement is investigated quantitatively. 1. Introduction The exploration of porous media flows involving fluids of different viscosities and/or densities has covered several decades. Early and relatively simple theoretical models for gravity-dominated flows were developed by Dietz (1953), cf. also the recent work by Yortsos (1991), and subsequently extended by Sheldon & Fayers (1962) as well as Fayers & Muggeridge (1990). These models are reasonably successful in predicting the rate of propagation of the so-called gravity tongue that often develops when density effects dominate. At the other end of the spectrum, displacements governed by viscosity effects have been investigated extensively as well, experimentally, theoretically, and by means of computations, dating back to the pioneering work of Hill (1952). In particular, the linear instability of the front responsible for the formation of viscous fingers has been analysed, along with their nonlinear growth