Executive Summary
It is now well established that meso-scale structures, whose characteristic size is on the order of a few millimeters, arise in rapid gas-solid flows. These structures significantly affect the overall flow behavior and should therefore be accounted for in CFD simulations.
Unfortunately, the meso-scale, structures cannot be resolved adequately in CFD simulations of risers of practically relevant dimensions. As coarse-grid simulations of such gas-solid flows do not resolve the meso-scale structures, quantities such as the effective inter-phase drag and the net rate of dissipation of pseudo-thermal energy are not properly accounted for in the computations. Therefore, the coarse-grid simulations, which have been published in the literature, should be viewed with some skepticism, as there is little basis to argue that these results are indeed true solutions of the differential equation models one is trying to solve.
In the last four years, we have made significant progress in understanding the origin of the meso-scale structures. It is established that they arise as a result of the inertial instability associated with the relative motion between the gas and particle phases and/or inelastic collisions, both of which are local events occurring on a length-scale comparable to the size of the meso-scale structures. This allowed us to assemble a tentative sub-grid model to account for the effects of the (unresolved) meso-scale structures in coarse-grid simulations. The proposed sub-grid model interrogates the stability of uniform motion on a length scale smaller than the grid size of a coarse-grid simulation and incorporates corrections to quantities such as effective drag, etc. accordingly, In that sense, it is indeed based on the differential equation model, which one is trying to solve. The sub-grid model is internally consistent, in the sense that, as the grid size goes to zero, the sub-grid corrections become smaller and smaller. Thus, the intent of the sub-grid model is not to change the original system of differential equations one is trying to solve, but to help us simulate the macro-scale structures correctly without having, 0 to resolve the meso-scale structures. Some of the elements of the sub-grid model are speculative and remains to be tested.
We then embarked on a program of research aimed at gathering statistics on fluctuations associated with the meso-scale structures in a model two-phase flow problem, so that we can verify the validity of the speculative elements of the sub-grid model. This work is in progress and some of the initial results on fluctuation statistics are described in this report. Results obtained at different levels of solids loading manifest qualitatively similar fluctuation statistics, giving us hope that a validated sub-grid model is indeed within reach,