The proposed research is on development and simulation of a multiscale model for heterogeneous particulate flows, in which the motion of a very large number of small solid particles is coupled with a Newtonian incompressible fluid. The problem derives from particle transport, sedimentation, fluidization and separation processes that are common in many applications. Currently, there are two major approaches to simulate such flows: one of them is based on averaged equations, the other one - on direct numerical simulations. The first approach requires homogeneity in the flow which is often violated in applications, while the second one is currently limited to a relatively small number (compared to applications) of particles. However, practical applications involve millions and billions of particles whose small size leads the problem of interest to span two length scales: fine scale - the scale of particles, and coarse scale - the scale of the application. In developing the proposed approach the variational multiscale framework is employed, in which the problem is split into two problems corresponding to the coarse and fine scales. It is assumed that in each coarse grid the particle distribution can be of three different regimes depending on their concentrations: dilute, moderate and dense. The main goal of the proposed study is to design a multiscale method that can handle heterogeneous flows in all three regimes in a unified way. The key idea of the proposed approach is, after separating the two scales, to use properly chosen subgrid models adaptively to represent micro-scale flow and particle dynamics on a coarse grid. In the dilute limit, a modified viscosity is used to represent averaged effects. In the dense limit, the discrete network approximation method to approximate the subgrid effects is utilized. Network models allow avoiding detailed fine-scale simulations in the dense regimes and provide substantial CPU savings. In the intermediate regime, fine-scale features within a target coarse-grid block are resolved. The issues to be focused on are (1) coarse coupling, (2) boundary conditions for local problems, (3) approximation of subgrid effects, (4) effective solution strategies.
The proposed project provides tools to simulate heterogeneous particulate flows of small rigid particles in an incompressible fluid which have been of great interest for the last decade and are commonly encountered in many applications and fundamental fluid mechanics. The complexity of this problem due to the presence of two different length scales is enormous and only very recently the development of numerical techniques for direct numerical simulations and computing power allowed for some attempts to numerically resolving flows containing a relatively large number of particles. However, existing direct numerical simulations methods have been developed for flows where the maximal number of particles to be handled does not exceed a couple dozens of thousands while for practical applications much larger amount is of primary importance. This dictates a need in developing a state-of-the-art predictive multiscale model for heterogeneous particulate flows which allows one to obtain a detailed and accurate representation of the flow, and which is the main focus of the proposed research.
The principal goal of the conducted under DMS-1016531 research program is to develop novel efficient tools for the numerical simulation of the multiscale heterogeneous particulate flows. Under the investigation are the composite media consisting of a large number of particles dispersed in a host medium where the mechanical properties of the constituents are vastly different. There are certain challenges in traditional numerical treatment of these media, because of which the issue of developing of the aforementioned tools has received increased attention in recent years. Conducted research activities feature the interplay of modeling, analysis and simulation techniques for particulate flows, and include: - analytical studies of mechanisms that govern interactions between medium components and make the corresponding problems challenging; - developing efficient schemes for numerical approximation for certain heterogeneous flows with fine features; - devising strategies to enforce relevant physical and mathematical constraints in numerical approximations; - simulations of certain particulate flows using the devised methods; - derivation of benchmark problem solutions that allows for a quantitative comparing the results obtained with different methods/codes; and - advancing existing methods of the discrete network approximation, among others. Developed methods and tools are expected to substantially advance the field of multiscale modeling and simulation of particulate composites, which is of great interest in science and engineering. Research projects have provided great opportunities for the students of University of Houston (UH) Mathematics Department to participate in conducted studies. In particular, six UH Mathematics Department students ranging from undergraduate to PhD level including minority ones have received valuable training in both applied and computational mathematics beyond the classroom and also gained hands-on experience how to carry out an independent research. Results and ideas of the conducted studies have been disseminated through presentations at seminars and talks at workshops both in the US and abroad in addition to the published papers.
