A Volume-Averaged Hyperbolic System of Governing Equations for Granular Turbulent Flow Modeling with Phase Change

Abstract

A formulation is developed using volume-averaging and the concept of added mass to derive a hyperbolic system of governing equations for modeling turbulent, dense granular flows. The Large Eddy Simulations (LES) framework is employed for the fluid phase, whereas the solid phase equations are based on enlarged Kinetic Theory concepts. To obtain the LES equations, the volume-averaged equations are filtered and the filtered terms not directly computable from the LES solution are generically modeled. Additionally, the pseudo-turbulent kinetic energy (PTKE) is included in the formulation, and it is shown how its contribution is distinct from turbulence and leads to different terms that must be modeled in the conservation equations. Volume-averaging of the continuity, momentum and energy equations result in many integrals that are used to rigorously define the meaning of terms that have only been included heuristically in existing formulations. Simulations with this model are conducted in a configuration representing the interaction of a turbulent supersonic jet with a bed of solid particles. The results are analyzed to demonstrate hyperbolicity. Comparisons of results from a model including PTKE and one excluding show that the inclusion of PTKE has no role in bestowing hyperbolicity to the model, and furthermore does not affect the macroscopic aspects of the crater. Comparisons between results obtained with a hyperbolic model and a model that is hyperbolic everywhere except in regions of particle/fluid interaction show that the macroscopic crater aspects are different, affecting the crater shape and topography.