A multiphase model for compressible granular–gaseous flows: formulation and initial tests

Abstract

A model for predicting the behaviour of a compressible flow laden with shocks interacting with granular material has been developed and tested. The model consists of two sets of coupled Euler equations, one for the gas phase and the other for the granular phase. Drag, convective, heat transfer and non-conservative terms couple the two sets of governing equations. Intergranular stress acting on the grains is modelled using granular kinetic theory in dilute regimes where particle collisions are dominant and frictional-collisional pressure in dense regions where layers of granular material slide over one another. The two-phase granular-gaseous model, as a result, is valid from dilute to densely packed granular regimes. The solution of these nonlinearly coupled Euler equations is challenging due to the presence of the non-conservative nozzling and work terms. A numerical technique, based on Godunov's method, was designed for solving these equations. This method takes advantage of particle incompressibility to simplify the nozzling terms. It also uses the observation that a Riemann problem is valid in the region where gas can flow between particles and can be used to provide a physically accurate approximation of the non-conservative terms. The model and solution method are verified by comparisons to test problems involving granular shocks and two-phase shock-tube problems, and they are validated against experimental measurements of shock and dense particle-curtain interactions and transmitted oblique granular shocks.

A volume-filtered description of compressible particle-laden flows

“…In low-Mach number flows, it is common to simplify the momentum equation by employing the product rule, i.e., ∇(αp) = p∇α + α∇p, such that the non-conservative nozzeling term p∇α cancels out. As pointed out by Houim and Oran [25], the product rule is not valid when dealing with compressible flows in general. However, assuming the particle phase is incompressible (i.e., the particle density is not a function of fluid pressure), the product rule will hold.…”

“…where the last term on the right-hand side of (13) is typically modeled as a contribution to drag. In the context of compressible flows, p∇α represents a nozzling term that accelerates the gas due to particles restricting the area where fluid can flow [25]. In low-Mach number flows, it is common to simplify the momentum equation by employing the product rule, i.e., ∇(αp) = p∇α + α∇p, such that the non-conservative nozzeling term p∇α cancels out.…”

“…As discussed by Houim and Oran [25], assuming constant particle density the disperse phase continuity equation can be employed to yield…”

“…(25) and the expression above are equivalent, expressing this in terms of volume fraction gradients, as is done in Eq. (25), may be more convenient numerically. It is interesting to note that Houim and Oran [25] arrived at this term via a control volume analysis, in which an 'interfacial pressure' was used at the particle surface and for closure assumed this to be equal to the local gas-phase pressure.…”

МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ ВЗАИМОДЕЙСТВИЯ УДАРНОЙ ВОЛНЫ С ПЛОТНОЙ ЗАСЫПКОЙ ЧАСТИЦ В РАМКАХ ДВУХЖИДКОСТНОГО ПОДХОДА, "Химическая Физика"

“…Во-вторых, подобная постановка соответствует фундаментальной задаче Римана для систем уравнений, описывающих течения двухфазных сред. Формулировка математических моделей и вычислительных алгоритмов, позволяющих рассчитывать широкий спектр режимов течений двухфазных сред -от разреженных до плотных -в рамках подходов механики гетерогенных сред является сегодня очень актуальным и не до конца решенным вопросом [2].…”