Optimal Moment Sets for Multivariate Direct Quadrature Method of Moments

The direct quadrature method of moments (DQMOM) can be employed to close population balance equations (PBEs) governing a wide class of multivariate number density functions (NDFs). Such equations occur over a vast range of scientific applications, including aerosol science, kinetic theory, multiphase flows, turbulence modeling, and control theory, to name just a few. As the name implies, DQMOM uses quadrature weights and abscissas to approximate the moments of the NDF, and the number of quadrature nodes determines the accuracy of the closure. For nondegenerate univariate cases (i.e., a sufficiently smooth NDF), the N weights and N abscissas are uniquely determined by the first 2N non-negative integer moments of the NDF. Moreover, an efficient product-difference algorithm exists to compute the weights and abscissas from the moments. In contrast, for a d-dimensional NDF, a total of (1 + d)N multivariate moments are required to determine the weights and abscissas, and poor choices for the moment set can lead to nonunique abscissas and even negative weights. In this work, it is demonstrated that optimal moment sets exist for multivariate DQMOM when N ) nd quadrature nodes are employed to represent a d-dimensional NDF with n ) 1-3 and d ) 1-3. Moreover, this choice is independent of the source terms in the PBE governing the time evolution of the NDF. A multivariate Fokker-Planck equation is used to illustrate the numerical properties of the method for d ) 3 with n ) 2 and 3

Numerical Description of Dilute Particle-Laden FLows by a Quadrature-Based Moment Method

The numerical simulation of gas-particle flows is divided into two families of methods. In Euler-Lagrange methods individual particle trajectories are computed, whereas in Euler-Euler methods particles are characterized by statistical descriptors. Lagrangian methods are very precise but their computational cost increases with instationarity and particle volume fraction. In Eulerian methods (also called moment methods) the particle-phase computational cost is comparable to that of the fluid phase but requires strong simplificaions. Existing Eulerian models consider unimodal or close-to-equilibrium particle velocity distributions and then fail when the actual distribution is far from equilibrium. Quadrature-based Eulerian methods introduce a new reconstruction of the velocity distribution, written as a sum of delta functions in phase space constrained to give the right values for selected low-order moments. Two of the quadrature-based Eulerian methods, differing by the reconstruction algorithm, are the focus of this work. Computational results for two academic cases (crossing jets, Taylor-Green flow) are compared to those of a Lagrangian method (considered as the reference solution) and of an existing second-order moment method. With the quadrature-based Eulerian methods, significant qualitative improvement is noticed compared to the second-order moment method in the two test cases

Simulation of Mono- and Bidisperse Gas-Particle Flow in a Riser with a Third-Order Quadrature-Based Moment Method

Gas-particle flows can be described by a kinetic equation for the particle phase coupled with the Navier−Stokes equations for the fluid phase through a momentum exchange term. The direct solution of the kinetic equation is prohibitive for most applications due to the high dimensionality of the space of independent variables. A viable alternative is represented by moment methods, where moments of the velocity distribution function are transported in space and time. In this work, a fully coupled third-order, quadrature-based moment method is applied to the simulation of mono- and bidisperse gas-particle flows in the riser of a circulating fluidized bed. Gaussian quadrature formulas are used to model the unclosed terms in the moment transport equations. A Bhatnagar−Gross−Krook (BGK) collision model is used in the monodisperse case, while the full Boltzmann integral is adopted in the bidisperse case. The predicted values of mean local phase velocities, rms velocities, and particle volume fractions are compared with the Euler−Lagrange simulations and experimental data from the literature. The local values of the time-average Stokes, Mach, and Knudsen numbers predicted by the simulation are reported and analyzed to justify the adoption of high-order moment methods as opposed to models based on hydrodynamic closures