On the development of high order realizable schemes for the Eulerian simulation of disperse phase flows: A convex-state preserving discontinuous galerkin method

In the present work, a high order realizable scheme for the Eulerian simulation of disperse phase flows on unstructured grids is developed and tested. In the Eulerian modeling framework two approaches are studied: the monokinetic (MK) [1] and the Gaussian closures [2, 3]. The former leads to a pressureless gas dynamics system (PGD). It accurately reproduces the physics of such flows at low Stokes number, but is challenging for numerics since the resulting system is weakly hyperbolic. The latter deals with higher Stokes numbers by accounting for particle trajectory crossings (PTC) [4]. Compared to the MK closure, the resulting system of equation is hyperbolic but has a more complex structure; realizability conditions are satisfied at the continuous level, which imply a precise framework for numerical methods. To achieve the goals of accuracy, robustness and realizability, the Discontinuous Galerkin method (DG) is a promising numerical approach [5, 6, 7, 8]. Based on the recent work of Zhang et al. [6], the DG method used is associated to a convex projection strategy, which respects the realizability constraints without affecting the accuracy. The main contribution of this work is to apply one of the latest developments in the field of numerical methods (DG) to physical models, taking into account the free transport and drag terms of the disperse phase flow, which are the building blocks for the Eulerian modeling based on moment methods. DG results are eventually compared qualitatively and quantitatively to the Lagrangian results and to the reference simulations provided by a second order structured MUSCL/HLL finite volume scheme [9, 3]. Through these comparisons, the DG method is shown to be competitive for the description of such flows.