Numerical description of dilute particle-laden flows by a quadrature-based moment method (original) (raw)
Related papers
A quadrature-based moment method for dilute fluid-particle flows
Journal of Computational Physics, 2008
Gas-particle and other dispersed-phase flows can be described by a kinetic equation containing terms for spatial transport, acceleration, and particle processes (such as evaporation or collisions). In principle, the kinetic description is valid from the dilute (non-collisional) to the dense limit. However, its numerical solution in multi-dimensional systems is intractable due to the large number of independent variables. As an alternative, Lagrangian methods ''discretize" the density function into ''parcels" that are simulated using Monte-Carlo methods. While quite accurate, as in any statistical approach, Lagrangian methods require a relatively large number of parcels to control statistical noise, and thus are computationally expensive. A less costly alternative is to solve Eulerian transport equations for selected moments of the kinetic equation. However, it is well known that in the dilute limit Eulerian methods have great difficulty to describe correctly the moments as predicted by a Lagrangian method. Here a two-node quadrature-based Eulerian moment closure is developed and tested for the kinetic equation. It is shown that the method can successfully handle highly non-equilibrium flows (e.g. impinging particle jets, jet crossing, particle rebound off walls, finite Stokes number flows) that heretofore could not be treated accurately with the Eulerian approach.
A quadrature-based third-order moment method for dilute gas-particle flows
Journal of Computational Physics, 2008
Dilute gas-particle flows can be described by a kinetic equation containing terms for spatial transport, gravity, fluid drag, and particle-particle collisions. However, the direct numerical solution of the kinetic equation is intractable for most applications due to the large number of independent variables. A useful alternative is to reformulate the problem in terms of the moments of the velocity distribution function. Closure of the moment equations is challenging for flows away from the equilibrium (Maxwellian) limit. In this work, a quadrature-based third-order moment closure is derived that can be applied to gas-particle flows at any Knudsen number. A key component of quadrature-based closures is the moment-inversion algorithm used to find the weights and abscissas. A robust inversion procedure is proposed for moments up to third order, and tested for three example applications (Riemann shock problem, impinging jets, and vertical channel flow). Extension of the moment-inversion algorithm to fifth (or higher) order is possible, but left to future work. The spatial fluxes in the moment equations are treated using a kinetic description and hence a gradient-diffusion model is not used to close the fluxes. Because the quadrature-based moment method employs the moment transport equations directly instead of a discretized form of the Boltzmann equation, the mass, momentum and energy are conserved for arbitrary Knudsen number (including the Euler limit). While developed here for dilute gas-particle flows, quadrature-based moment methods can, in principle, be applied to any application that can be modeled by a kinetic equation (e.g., thermal and nonisothermal flows currently treated using lattice Boltzmann methods), and examples are given from the literature.
A fully coupled quadrature-based moment method for dilute to moderately dilute fluid–particle flows
Chemical Engineering Science, 2010
Quadrature-based moment methods Computational fluid dynamics Multiphase flow Particle-laden channel flow A third-order quadrature-based moment method for simulating dilute and moderately dilute fluid-particle flows has been implemented with full coupling in a computational fluid dynamics code. The solution algorithm for the particle phase uses a kinetic-based finite-volume technique to solve the velocity moment equations derived from kinetic theory. The procedure to couple the particle-phase volume-fraction and momentum equations with the Eulerian solver for the fluid phase is explained in detail. As an example application, simulations of a particle-laden vertical channel flow at fluid-phase Reynolds number 1379 and particle Stokes numbers 0.061 and 0.61 were carried out. The fluid and particle velocities, particlephase volume fraction and granular temperature were observed to reach a steady state in the case of Stokes number 0.061, while instabilities that led to the formation of structures and initiated the particle segregation process were observed in the case with the higher Stokes number. These results are validated against results from a classical two-fluid model derived from the kinetic theory of granular flows in the small Knudsen number limit, and Euler-Lagrange simulations of the same flow.
A multi-Gaussian quadrature method of moments for gas-particle flows
2010
The purpose of the present contribution is to introduce a new high-order moment formalism for particle/droplet trajectory crossing (PTC) in the framework of large-eddy simulation (LES) of gas-particle flows. Thus far, the ability to treat PTC has been examined by several investigators for direct numerical simulations (DNS) using quadraturebased moment methods based on a sum of Dirac delta functions (Yuan & Fox (2010), Kah et al. (2010)). However, for LES, such methods require too many moments in order to capture both the effect of subgrid-scale turbulence on the disperse phase as well as PTC due to large-scale eddies in a Eulerian mesoscopic framework. The challenge is thus twofold: first, to propose a new generation of quadrature with less singular behavior as well as associated proper mathematical properties and related algorithms, and second to limit the number of moments used for applicability in multi-dimensional configurations without losing accuracy in the representation of spatial fluxes.
A multi-Gaussian quadrature method of moments for gas-particle flows in a LES framework
Proceedings of the Summer Program, 2010
The purpose of the present contribution is to introduce a new high-order moment formalism for particle/droplet trajectory crossing (PTC) in the framework of large-eddy simulation (LES) of gas-particle flows. Thus far, the ability to treat PTC has been examined by several investigators for direct numerical simulations (DNS) using quadraturebased moment methods based on a sum of Dirac delta functions (Yuan & Fox (2010), Kah et al.(2010)). However, for LES, such methods require too many moments in order to capture both the ...
Industrial & Engineering Chemistry Research, 2012
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.
An Euler–Lagrange strategy for simulating particle-laden flows
Journal of Computational Physics, 2013
In this work, a strategy capable of simulating polydisperse flows in complex geometries is employed where the fluid transport equations are solved in an Eulerian framework and the dispersed phase is represented as Lagrangian particles. Volume filtered equations for the carrier phase are derived in detail for variable density flows, and all unclosed terms are discussed. Special care is given to the interphase coupling terms that arise, in order to ensure that they are implemented consistently and that they converge under mesh refinement. This provides the flexibility of using cell sizes that are smaller than the particle diameter if necessary. Particle collisions are handled using a soft-sphere model that has been modified for parallel efficiency. Simulations are carried out for a number of laboratory-scale configurations, showing excellent agreement with experiments.
The Journal of Computational Multiphase Flows, 2009
A detailed study into the turbulent behaviour of dilute particulate flow under the influence of two carrier phases namely gas and liquid has been carried out behind a sudden expansion geometry. The major endeavour of the study is to ascertain the response of the particles within the carrier (gas or liquid) phase. The main aim prompting the current study is the density difference between the carrier and the dispersed phases. While the ratio is quite high in terms of the dispersed phase for the gas-particle flows, the ratio is far more less in terms of the liquid-particle flows. Numerical simulations were carried out for both these classes of flows using an Eulerian two-fluid model with RNG based k- emodel as the turbulent closure. An additional kinetic energy equation to better represent the combined fluid-particle behaviour is also employed in the current set of simulations. In the first part of this two part series, experimental results of Fessler and Eaton (1995) for Gas-Particle ...
Quadrature Method of Moments for the Simulation of Turbulent Reacting Flows
APS Division of Fluid Dynamics Meeting Abstracts, 2003
Computational schemes for turbulent reacting flow systems typically solve the species transport equations using a grid-based Eulerian technique. Such schemes inherently do not contain information about the sub-grid scalar PDF required for the computation of the non-linear reaction source terms and sub-grid scalar dissipation. Though a transport equation for the scalar PDF can be formulated, the high-dimensional equation has to be solved using a computationally expensive particle-based Lagrangian scheme. ...
Canadian Journal of Chemical Engineering, 2020
Polydisperse multiphase flows can be simulated by coupling a population balance model to the multi-fluid model. For the first time, this simulation is carried out using the dual-quadrature method of generalized moments (DuQMoGeM) and its direct version to solve the population balance model. The main disadvantage of these methods is the high computational cost of the embedded cubature. Herein, this challenge was addressed by parallelization on graphics processing units, which resulted in a significant acceleration of the simulations, with speedups that can be larger than 1000. Numerical simulations considering simultaneous particles breakage and aggregation were conducted with direct DuQMoGeM-FC and DQMoM-FC, whose results were different due to the existence of quadrature error in the latter.