Three-dimensional vorticity dynamics in miscible Hele-Shaw displacements
Seminar Room 1, Newton Institute
We perform three-dimensional DNS simulations of the transient, variable viscosity Navier-Stokes equations in the Boussinesq approximation, coupled to a convection-diffusion equation for a concentration field, to simulate miscible viscous fingers in Hele-Shaw cells. The three-dimensional problem allows for new instabilities and patterns that cannot be captured by traditional gap-averaged modeling. For constant density displacements, the simulations reveal the mechanism by which the initial spanwise vorticity of the base flow, when perturbed, gives rise to the cross-gap vorticity that drives the fingering instability in the classical Darcy sense. Cross-sections at constant streamwise locations reveal the existence of a streamwise vorticity quadrupole that induces fluid transport from the walls of the cell to its center, thereby leading to a new hydrodynamic instability, termed 'inner splitting' that had not been previously reported. If gravity is included, the nature of the two-dimensional base flow and its subsequent instability changes dramatically. The interaction between Saffman-Taylor and Rayleigh-Taylor instabilities can lead to additional splitting events, and it can significantly enhance the mixing rates of the two fluids, thereby altering the overall displacement efficiency.