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The present work aims at investigating the ability of a Kinetic-Based Moment Method (KBMM) to reproduce the statistics of turbulent particle-laden flows using the Anisotropic Gaussian (AG) closure. This method is the simplest KBMM member that can account for Particle Trajectory Crossing (PTC) properly with a well-posed mathematical structure Vie et al. (2015). In order to validate this model further, we investigate here 3D turbulent flows that are more representative of the mixing processes, which occurs in realistic applications. The chosen configuration is a 3D statistically-stationary Homogeneous Isotropic Turbulence (HIT) loaded with particles in a very dilute regime. The analysis focuses on the description of the first three lowest order moments of the particulate flow: the number density, the Eulerian velocity and the internal energy. A thorough numerical study on a large range of particle inertia allows us to show that the AG closure extends the ability of the Eulerian models to correctly reproduce the particle dynamics up to a Stokes number based on the Eulerian turbulence macro-scale equal to one, but also highlights the necessity of high-order numerical schemes to reach mesh convergence, especially for the number density field.

International audience; 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 hyper-bolic 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 moments 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.

In the framework of fully Eulerian simulation of disperse phase flows, the use of a monokinetic closure for the kinetic based moment method is of high importance since it accurately reproduces the physics of low inertia particles with a minimum number of moments. The free transport part of this model leads to a pressureless gas dynamics system which is weakly hyperbolic and can generate \(\delta \)-shocks. These singularities are difficult to handle numerically, especially without globally degenerating the order or disrespecting the realizability constraints. A comparison between three second order schemes is conducted in the present work. These schemes are: a realizable MUSCL/HLL finite volume scheme, a finite volume kinetic scheme, and a convex state preserving Runge-Kutta discontinuous Galerkin scheme. Even though numerical computations have already been led in 2D and 3D with this model and numerical methods, the present contribution focuses on 1D results for a full understanding of the trade off between robustness and accuracy and of the impact of the limitation procedures on the numerical dissipation. Advantages and drawbacks of each of these schemes are eventually discussed.