Numerical simulation of buoyant mixture flows

Abstract

A ‘mixture model’ for the macroscopic motion of a buoyant suspension is formulated in terms of volume averaged velocities in order to ensure well-posedness of the incompressibility constraint. Conservation laws for mixture volume, momentum and the dispersed phase are complemented by a heuristic closure for the relative motion between the two phases. An efficient hybrid method for numerical simulation of mixture flows in arbitrary two-dimensional and axi-symmetric geometries is presented. The spatial discretisation is based on an h-type finite element method with use of a local, stabilizing upwind finite difference scheme for the advective term. Time-step and incompressibility constraints are decoupled through splitting and pressure-correction methods. The conservation equation for the dispersed phase is treated by a finite volume Roe solver with a slope limiter which ensures second-order accuracy in regions where the volume fraction is varying smoothly. Three separate applications of the code are presented to assess the validity of the various discretisation methods. The classical problem of one-dimensional batch separation is revisited and the exact analytic solution is used to evaluate the performance of the finite volume Roe solver. The results of a previous numerical simulation of spin-up from rest of a mixture are shown to be in good agreement with those produced by the current method. A numerical simulation of gravity settling underneath a curved wall (the Boycott effect) is presented for the intermediate parameter regime where both viscous and inertial effects are important. The first three terms of a Blasius series expansion for the velocity field adjacent to the curved wall are provided for comparison with the numerical results. The numerically obtained velocity profile is observed to adjust slowly to the similarity solution. In addition to verifying the global balances obtained from kinematic considerations, the simulations provide new physical insight about the interior flow-field. Notable features are the possibility of pure-fluid entrainment into the mixture region, and a stratification of the horizontal mixture–pure fluid interface due to an oscillatory vortex motion.