Complex, 3D mixing of single- and multi-phase flows, in particular by injection of gas and creation of bubble plumes, occurs in a number of situations of interest in energy technology, process and environmental engineering, etc. For all these applications, the basic need is to determine the behaviour of the bubble plume and the currents induced by the ascending gas plume in the surrounding liquid and thereby the consequent mixing in the body of the liquid.
A six-equation, two-fluid model was utilized and transient calculations were performed to study the plume growth, the acceleration of the liquid due to viscous drag, and the approach to steady-state conditions. All calculations were performed using the commercial CFD code CFX4, with appropriate modifications and code extensions to describe the interphase momentum forces and the turbulent exchanges between the phases. Since the k-e is a single-phase model, an extended version was used, with extra source terms introduced to account for the interaction between the bubbles and the liquid. A new model was advanced to relate turbulent bubble dispersion to statistical fluctuations in the liquid velocity field, affecting the drag and lift forces between the phases. The model is able to account for the dispersion of bubbles due to the random influence of the turbulent eddies in the liquid, such as the empirical Turbulent Dispersion Force, and has the advantage that no fitting coefficients need to be introduced.
The interphase forces are not the only source of empiricism: the above-mentioned extra source terms introduced into the k-e model, are patch-ups which introduce ad hoc empirical coefficients which can be tuned to get good comparison with the data. Further, the hypothesis of turbulence isotropy has still to be rigorously proved with clean experimental data. The Reynolds Stress Models (RSMs), which are in principle appropriate for this kind of flow (since equations are solved for each component of the Reynolds stress tensor), are unstable and not robust enough, and it is difficult to achieve convergence even for single-phase flows. Therefore, attention was focused on Large Eddy Simulation (LES) turbulence models.
The main advantage of LES for this class of flows is that it captures directly the interactions of the bubbles with the resolved large-scale structures up to the size of the grid (close to the bubble diameter), whereas the interaction with the subgrid scales can be modelled. In other words, the turbulent dispersion of the bubbles is due only to the largest structures, which are calculated directly with LES. Since this is a new area of study, many open questions will need to be addressed: a universally-accepted, two-phase subgrid model does not exist, and the influence of the grid on the simulation is also not clear, since this determines the scales that are going to be resolved. To pursue this approach, the LES model was implemented into CFX-4. First, a single-phase test case has been calculated to validate the model against the data of GEORGE ET. AL., 1977. Second, a simple case (a 3D box with homogeneous distribution of bubbles) has been run to study the modifications induced by the bubbles on the turbulence of the system and the effect of the filter (mesh size). The results have been obtained with the SMAGORINSKY, 1963 subgrid model and were compared with the experimental data of LANCE & BATAILLE, 1991, finding that the turbulence intensities increase with the mesh size, and the optimum configuration requires a mesh comparable to the bubble diameter; otherwise the liquid velocity fluctuations profile is not captured at all, meaning that the grid is too coarse. The idea recalls the Scale-Similarity Principle of BARDINA ET AL., 1980.
Taking advantage of this experience, two more elaborate situations, closer to reality, were analyzed: the case of a turbulent bubbly shear flow in a plane vertical mixing layer , with calculations compared against the data of ROIG, 1993; and the case of the bubble plume, with calculations compared against the data of ANAGBO & BRIMACOMBE, 1990. A study on the importance of the lift force has been carried out and the results were similar in both cases, with an optimum lift coefficient of 0.25. The results showed good agreement with the experiment, although a more detailed study of bubble-induced turbulence (or pseudoturbulence) is required. The GERMANO ET AL., 1991 dynamic procedure was successfully tested and a new subgrid scale model for the dispersed phase that requires no empirical constants, was introduced.