Simulation of fluid–particles flows: heavy particles, flowing regime, and asymptotic-preserving schemes

Abstract

Abstract. We are interested in an Eulerian-Lagrangian model describing particulate flows. The model under study consists of the Euler system and a Vlasov-Fokker-Planck equation coupled through momentum and energy exchanges. This problem contains asymptotic regimes that make the coupling terms stiff, and lead to a limiting model of purely hydrodynamic type. We design a numerical scheme which is able to capture this asymptotic behavior without requiring prohibitive stability conditions. The construction of this Asymptotic Preserving scheme relies on an implicit discretization of the stiff terms which can be treated by efficient inversion methods. This method is a natural coupling of a kinetic solver for the particles with a kinetic scheme for the hydrodynamic Euler equations. Numerical experiments are conducted to study the performance of this scheme in various asymptotic regimes.

Asymptotic‐preserving schemes for kinetic–fluid modeling of disperse two‐phase flows with variable fluid density

“…The mass conservation for the fluid (the first equation in ) is not include in . In the present situation, it turns out to be natural to use a kinetic scheme, as we did in when dealing with compressible flows. A consequence of this is that, in the limit system, the mass conservation equations for the particles and the dense phase are discretized with the same method.…”

“…The Stokes settling time is given by , with a the particle radius and μ the fluid viscosity. In a typical soot, τ ≈ 10 − 8 s. We refer to for details.…”

Asymptotic-preserving schemes for kinetic-fluid modeling of disperse two-phase flows

“…It is also worth mentioning related works like the local existence of smooth solutions for the case without velocity-diffusion [31], several studies of coupling with the Euler system (i.e. viscosity is sensible only at the scale of the particles) [23,32,33] and systems with energy exchanges [34][35][36].…”

“…We define a regularly spaced and symmetric velocity grid, with step Dv. For the transport term v Á r x f in (24) and (30), we apply the upwind type second order shock capturing schemes (see [36]). Discrete differential operators are defined dimension-by-dimension.…”

“…The derivative with respect to velocity which appears in the acceleration term is also solved by the upwind type second order shock capturing schemes (see [36]). …”