A Gaussian moment method for polydisperse multiphase flow modelling

The accurate prediction of multiphase flow when particles are differentiated by a set of “internal” variables, such as size or temperature, can pose modelling and numerical challenges. Although Lagrangian particle methods can provide predictive simulations of a wide spectrum of complex multiphase problems, they can become prohibitively expensive as the number of particles becomes large. Alternatively, Eulerian approaches have the potential to improve the computational efficiency of multiphase flows, but classical methods produce modelling artifacts or do not properly treat the local statistical dependence between the particle velocities and internal variables, or between the internal variables themselves. In this paper an extension is proposed of the classical Gaussian ten-moment model from gaskinetic theory to a model for the treatment of a dilute particle phase with an arbitrary number of internal variables based on an entropy-maximization argument. Unlike previous formulations, this new model provides a set of first-order robustly-hyperbolic balance laws that include a direct treatment for the local statistical variance of each variable, as well as the covariance between the internal variables or the internal variables and particle velocity. A study of the wave speeds of the general hyperbolic system is presented. To demonstrate an example application, the model is then specialized for polydisperse flows that are subject to viscous fluid drag as well as gravitational and buoyancy forces. The complete eigenstructure of this fifteen-moment polydisperse Gaussian model (PGM) is presented and the PGM is shown to maintain a physically realizable distribution function for all admissible initial conditions. Finally, several illustrative low-dimensional problems are studied to demonstrate the predictive capabilities of the new model.