Three-dimensional Navier-Stokes simulations of buoyant, vertical miscible Hele-Shaw displacements

Gravitationally and viscously unstable miscible displacements in vertical Hele-Shaw cells are investigated via three-dimensional Navier-Stokes simulations. The velocity of the two-dimensional base-flow displacement fronts generally increases with the unfavourable viscosity contrast and the destabilizing density difference. Displacement fronts moving faster than the maximum velocity of the Poiseuille flow far downstream exhibit a single stagnation point in a moving reference frame, consistent with earlier observations for corresponding capillary tube flows. Gravitationally stable fronts, on the other hand, can move more slowly than the Poiseuille flow, resulting in more complex streamline patterns and the formation of a spike at the tip of the front, in line with earlier findings. A two-dimensional pinch-off governed by dispersion is observed some distance behind the displacement front. Three-dimensional simulations of viscously and gravitationally unstable vertical displacements show a strong vorticity quadrupole along the length of the finger, similar to recent observations for neutrally buoyant flows. This quadrupole results in an inner splitting instability of vertically propagating fingers. Even though the quadrupole's strength increases for larger destabilizing density differences, the inner splitting is delayed due to the presence of a secondary, outer quadrupole which counteracts the inner one. For large unstable density differences, the formation of a secondary, downward-propagating front is observed, which is also characterized by inner and outer vorticity quadrupoles. This front develops an anchor-like shape as a result of the flow induced by these quadrupoles. Increased spanwise wavelengths of the initial perturbation are seen to result in the formation of the well-known tip-splitting instability. For suitable initial conditions, the inner and tip-splitting instabilities can be seen to develop side by side, affecting different regions of the flow field.