On the implementation of moment transport equations in OpenFOAM: Boundedness and realizability

In this contribution some crucial numerical aspects concerning the implementation of quadrature-based moment methods (QBMM) into the open-source Computational Fluid Dynamics (CFD) code OpenFOAM are discussed. As well-known QBMM are based on the simple idea of solving a kinetic master equation, not in terms of the underlying number density function (NDF), but in terms of the moments of the NDF itself, via moment transport equations. These numerical aspects are in fact very often overlooked, resulting in implementations that do not satisfy the properties of boundedness and realizability for the moments of the NDF. Boundedness is an important property (i.e., moments of the NDF have to be bounded between some minimal and maximum values), that in turn depends on the initial and boundary conditions. Boundedness can be guaranteed by using a consistent approach with respect to the constraints imposed on the transport variables, such as dispersed phase volume fraction. Realizability is instead related to the existence of an underlying NDF that corresponds to a specific moment set. It is well-known that time and spatial discretization schemes can corrupt a moment set unless they are specifically designed to preserve realizability. One popular choice is to use ad-hoc pseudo high-order schemes to ensure realizable moment set is obtained. In this work, moment transport equations are implemented in the CFD code OpenFOAM using a similar form proposed by Weller (2002) for preserving the boundedness of the volume fraction, together with the numerical schemes of Vikas et al. (2011) for ensuring moment realizability. The effectiveness of such implementation is illustrated on two simple examples, taken from our work on the simulation of fluid-fluid multiphase systems.