A multi-Gaussian quadrature method of moments for simulating high Stokes number turbulent two-phase flows (original) (raw)
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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 ...
A high order moment method for the simulation of polydisperse two-phase flows
Comptes Rendus Mécanique, 2013
In this work, we are interested in the modeling of spray polydispersion in size as well as size-velocity correlations, which may greatly influence the evaporation and the dynamics of the disperse phase. Vié et al. 2011 proposed a new model called Coupled Size-Velocity Moment method (CSVM), which handles the polydispersion using the NDF reconstruction proposed in Kah at al. 2011 for size distribution, a velocity reconstruction and adapted numerical methods for moments evolution. Here, the CSVM is evaluated in a Frozen Homogeneous Isotropic Turbulence. Results shows the ability of the method to capture the statistical information of a turbulent flow with a minimal number of moments. As soon as a numerical scheme for unstructured grid is provided, the CSVM would be an interesting approach for complex simulations in industrial codes. Résumé Une méthode de moments d'ordreélevé pour la simulation d'écoulements diphasiques polydisperses. Dans ce travail, nous nous intéressonsà la modélisation de la polydispersion en taille des sprays et la prise en compte des corrélations taille-vitesse, qui peuvent grandement influencer l'évaporation et la dynamique de la phase dispersée. Vié et al. 2011 ont proposé un nouveau modèle appelé Coupled Size-Velocity Moment method (CSVM), qui prend en compte la polydispersion avec la stratégie de reconstruction de la NDF proposée dans Kah et al. 2011 pour la distribution en taille, une reconstruction de la vitesse, et des méthodes numériques adaptées pour l'évolution des moments. Ici, le CSVM estévalué sur une Turbulence Homogène Isotrope Figée. Les résultats montrent la capacité de la méthodeà capturer toutes les caractéristiques importantes de ce type d'écoulement avec un nombre minimal de moments. Dès qu'un schéma pour maillages non-structurés sera proposé, le CSVM deviendra une approche intéressante pour des simulations complexes avec des codes industriels.
Journal of Fluid Mechanics, 2005
The velocity distribution of dilute suspensions of heavy particles in gas-solid turbulent flows is investigated. A statistical approach -the mesoscopic Eulerian formalism (MEF) -is developed in which an average conditioned on a realization of the turbulent carrier flow is introduced and enables a decomposition of the instantaneous particle velocity into two contributions. The first is a contribution from an underlying continuous turbulent velocity field shared by all the particles -the mesoscopic Eulerian particle velocity field (MEPVF) -that accounts for all particle-particle and fluid-particle two-point correlations. The second contribution corresponds to a distribution -the quasi-Brownian velocity distribution (QBVD) -that represents a random velocity component satisfying the molecular chaos assumption that is not spatially correlated and identified with each particle of the system. The MEF is used to investigate properties of statistically stationary particle-laden isotropic turbulence. The carrier flow is computed using direct numerical simulation (DNS) or large-eddy simulation (LES) with discrete particle tracking employed for the dispersed phase. Particle material densities are much larger than that of the fluid and the force of the fluid on the particle is assumed to reduce to the drag contribution. Computations are performed in the dilute regime for which the influences of inter-particle collisions and fluid-turbulence modulation are neglected. The simulations show that increases in particle inertia increase the contribution of the quasi-Brownian component to the particle velocity. The particle velocity field is correlated at larger length scales than the fluid, with the integral length scales of the MEPVF also increasing with particle inertia. Consistent with the previous work of Abrahamson (1975), the MEF shows that in the limiting case of large inertia, particle motion becomes stochastically equivalent to a Brownian motion with a random spatial distribution of positions and velocities. For the current system of statistically stationary isotropic turbulence, both the DNS and LES show that the fraction of the kinetic energy residing in the mesoscopic field decreases with particle inertia as the square root of the ratio of the total particulate-phase kinetic energy to that of the fluid. † Present address: Michelin, Place des Carmes-Déchaux,
A numerically convergent Lagrangian-Eulerian simulation method for dispersed two-phase flows
2009
In Lagrangian-Eulerian (LE) simulations of two-way coupled particle-laden flows, the dispersed phase is represented either by real particles or by computational particles. In traditional LE (TLE) simulations, each computational particle is assigned a constant statistical weight, which is defined as the expected number of real particles represented by a computational particle. If the spatial distribution of particles becomes highly non-uniform due to particle-fluid or particle-particle interactions, then TLE simulations fail to yield numerically converged solutions due to high statistical error in regions with few particles. In this work, a particle-laden lid-driven cavity flow is solved on progressively refined grids to demonstrate the inability of TLE simulations to yield numerically converged estimates for the mean interphase momentum transfer term. We propose an improved LE simulation (ILE) method that remedies the above limitation of TLE simulations. In the ILE method, the statistical weights are evolved such that the same physical problem is simulated, but the number density of computational particles is maintained nearuniform throughout the simulation, resulting in statistical error that remains nearly constant with grid refinement. The evolution of statistical weights is rigorously justified by deriving the consistency conditions arising from the requirement that the resulting computational ensemble correspond to a statistical description of the same physical problem with real particles. The same particle-laden lid-driven cavity flow is solved on progressively refined grids to demonstrate the ability of ILE simulation to achieve numerically converged estimates for the mean interphase momentum transfer term. The accuracy of the ILE method is quantified using a test problem that admits an analytical solution for the mean interphase momentum transfer term. In order to improve the accuracy of numerical estimates of the mean interphase momentum transfer term, an improved estimator is proposed to replace the conventional estimator. The improved estimator results in more accurate estimates that converge faster than those obtained using the conventional estimator. The ILE simulation method along with the improved estimator is recommended for accurate and numerically convergent LE simulations. (S. Subramaniam). 1 By particle we mean any dispersed-phase element, including solid particles, droplets and bubbles. 2 We use the term 'estimate' in the statistical sense, just as the sample mean
Beyond pressureless gas dynamics : Quadrature-based velocity moment models
2010
Following the seminal work of F. Bouchut on zero pressure gas dynamics which has been extensively used for gas particle-flows, the present contribution investigates quadrature-based velocity moments models for kinetic equations in the framework of the infinite Knudsen number limit, that is, for dilute clouds of small particles where the collision or coalescence probability asymptotically approaches zero. Such models define a hierarchy based on the number of moments and associated quadrature nodes, the first level of which leads to pressureless gas dynamics. We focus in particular on the four moment model where the flux closure is provided by a two-node quadrature in the velocity phase space and provide the right framework for studying both smooth and singular solutions. The link with both the kinetic underlying equation as well as with zero pressure gas dynamics is provided and we define the notion of measure solutions as well as the mathematical structure of the resulting system of four PDEs. We exhibit a family of entropies and entropy fluxes and define the notion of entropic solution. We study the Riemann problem and provide a series of entropic solutions in particular cases. This leads to a rigorous link with the possibility of the system of macroscopic PDEs to allow particle trajectory crossing (PTC) in the framework of smooth solutions. Generalized delta\deltadelta-choc solutions resulting from Riemann problem are also investigated. Finally, using a kinetic scheme proposed in the literature without mathematical background in several areas, we validate such a numerical approach in the framework of both smooth and singular solutions.
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.