Closures for discrete suspension flow models: new insight from particle-resolved simulations and spatial data filtering

Resolved simulation of disperse multiphase flows is often used as fundamental tool to derive closures for homogenized models like two-fluid models, or so-called discrete particle models [1]. However, it is often not possible to produce high quality numerical grids (especially in the case of moving bodies) due to the complex topology of the fluid domain. Therefore, immersed-boundary methods are often used to model the presence of a dispersed solid phase within a simply connected fluid domain. Another issue of particle-resolved simulations is the large amount of produced data and their processing, especially in the case of transient phenomena.
In our contribution, we first describe a method to impose general boundary conditions (Dirichlet, Neumann or Robin) at immersed surfaces by means of appropriate source terms in the governing equations. This method can be seen as a generalization of our Hybrid Fictitious Domain Immersed Boundary (HFD-IB) method [2] that uses second order interpolation to impose a Dirichlet boundary condition at the particle surface. As the HFD-IB, our generalized method is coupled with the CFD-DEM library CFDEM® [3] to allow simulation of moving bodies. Unlike the HFD-IB that was based on a polynomial reconstruction of the boundary layer, the new method consists in expanding Eulerian fields in Taylor series (up to second order) along the direction normal to the immersed surfaces. These terms are evaluated using field values interpolated at multiple fluid points along the normal direction, as well as by considering the desired boundary condition. Subsequently, the Taylor series expansion is used to evaluate the field values at a surface cell (i.e., a cell intersected by the immersed surface), and this value is then imposed using a direct-forcing approach. In this work, we demonstrate the accuracy and convergence of the proposed method and we show its application to particle-resolved direct numerical simulation of momentum, heat and mass transfer in dense particle beds (Figure 1).
Second, we show how the open-source parallel data processing library CPPPO [4] can be used in conjunction with the new immersed boundary method. CPPPO is a filtering tool capable of performing filtering, sampling and binning operations on-the-fly, i.e. while the solver is running, that features advanced filtering algorithms to reduce the number of parallel communications and increase flexibility (for example, computing covariances, or customizing filtering kernels). Specifically, we show how CPPPO can be used to track the particle-based drag coefficient and Nusselt number in gas-solid suspensions. We show that our results are in agreement with existing literature and that more insight can be obtained using the advanced capabilities of CPPPO: for example, we highlight that closures developed for discrete particle models (i.e., closures for particle-based quantities) are significantly different from those used in two-fluid models (in which ensemble averaged quantities are considered).