A fully coupled fluid-particle flow solver using quadrature-based moment method with high-order realizable schemes on unstructured grids

Kinetic Equations containing terms for spatial transport, gravity, fluid drag and particle-particle collisions can be used to model dilute gas-particle flows. However, the enormity of independent variables makes direct numerical simulation of these equations almost impossible for practical problems. A viable alternative is to reformulate the problem in terms of moments of the velocity distribution function. A quadrature method of moments (QMOM) was derived by Desjardins et al. [1] for approximating solutions to the kinetic equation for arbitrary Knudsen number. Fox [2, 13] derived a third-order QMOMfor dilute particle flows, including the effect of the fluid drag on the particles. Passalacqua et al. [4] and Garg et al. [3] coupled an incompressible finite-volume solver for the fluid-phase and a third order QMOM solver for particle-phase on Cartesian grids. In the current work a compressible finite-volume fluid solver is coupled with a particle-phase solver based on third-order QMOM on unstructured grids. The fluid and particle-phase are fully coupled by accounting for the volume displacement effects induced by the presence of the particles and the momentum exchange between the phases. The success of QMOM is based on the moment inversion algorithm that allows quadrature weights and abscissas to be computed from the moments of the distribution function. The moment-inversion algorithm does not work if the moments are non-realizable, which might lead to negative weights. Desjardins et al. [1] showed that realizability is guaranteed only with the 1st-order finite-volume scheme that has excessive numerical diffusion. The authors [5, 6] have derived high-order finite-volume schemes that guarantee realizability for QMOM. These high-order realizable schemes are used in this work for the particle-phase solver. Results are presented for a dilute gas-particle flow in a lid-driven cavity with both Stokes and Knudsen numbers equal to 1. For this choice of Knudsen and Stokes numbers, particle trajectory crossing occurs which is captured by QMOM particle-phase solver

A Gaussian moment method and its augmentation via LSTM recurrent neural networks for the statistics of cavitating bubble populations

Phase-averaged dilute bubbly flow models require high-order statistical moments of the bubble population. The method of classes, which directly evolve bins of bubbles in the probability space, are accurate but computationally expensive. Moment-based methods based upon a Gaussian closure present an opportunity to accelerate this approach, particularly when the bubble size distributions are broad (polydisperse). For linear bubble dynamics a Gaussian closure is exact, but for bubbles undergoing large and nonlinear oscillations, it results in a large error from misrepresented higher-order moments. Long short-term memory recurrent neural networks, trained on Monte Carlo truth data, are proposed to improve these model predictions. The networks are used to correct the low-order moment evolution equations and improve prediction of higher-order moments based upon the low-order ones. Results show that the networks can reduce model errors to less than 1% of their unaugmented values

Four-field Hamiltonian fluid closures of the one-dimensional Vlasov-Poisson equation

We consider a reduced dynamics for the first four fluid moments of the one-dimensional Vlasov-Poisson equation, namely, the fluid density, fluid velocity, pressure and heat flux. This dynamics depends on an equation of state to close the system. This equation of state (closure) connects the fifth order moment—related to the kurtosis in velocity of the Vlasov distribution—with the first four moments. By solving the Jacobi identity, we derive an equation of state which ensures that the resulting reduced fluid model is Hamiltonian. We show that this Hamiltonian closure allows symmetric homogeneous equilibria of the reduced fluid model to be stable

On the efficiency and robustness of the core routine of the quadrature method of moments (QMOM)

Three methods are reviewed for computing optimal weights and abscissas which can be used in the Quadrature Method of Moments (QMOM): the Product-Difference Algorithm (PDA), the Long Quotient-Modified Difference Algorithm (LQMDA, variants are also called Wheeler algorithm or Chebyshev algorithm), and the Golub--Welsch Algorithm (GWA). The PDA is traditionally used in applications. It is discussed that the PDA fails in certain situations whereas the LQMDA and the GWA are successful. Numerical studies reveal that the LQMDA is also more efficient than the PDA

Simulation of particle flow in an inertial particle separator with an Eulerian velocity re-associated two-node quadrature-based method of moments

This paper presents research into practical simulations of particle flow in inertial particle separators (IPS) for helicopters and tilt rotor aircraft. The flow field of the carrier gas is predicted by means of the two-equation k-ϵ turbulence model. An Eulerian methodology is used to trace the particle trajectories of foreign particles such as droplets, ice and sand. To predict the characteristics of particle wall bouncing in dilute particle flow, the velocity re-associated two-node quadrature-based method of moments (VR-QMOM) is used. The particle distribution in the IPS is predicted for various particle sizes and these are compared with results from a Lagrangian particle tracking method. The particle-wall interactions and the separation efficiencies are studied for solid particles bouncing off perfectly elastic walls and an IPS shell coated with the M246 alloy which changes the coefficients of restitution. The simulated separation efficiencies predicted by the Eulerian method are compared with the simulation using the Lagrangian method over a range of particle sizes. The VR-QMOM method is seen to reproduce the particle bouncing and trajectory crossing behavior and to agree well with the Lagrangian method for predicted separation efficiencies. The new VR-QMOM method is shown to be an accurate and convenient alternative to established Lagrangian approaches

Beyond pressureless gas dynamics: Quadrature-based velocity moment models

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 $\delta$-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.Comment: Submitted to Communication in Mathematical Science

Development of high-order realizable finite-volume schemes for quadrature-based moment method

Kinetic equations containing terms for spatial transport, gravity, fluid drag and particle-particle collisions can be used to model dilute gas-particle flows. However, the enormity of independent variables makes direct numerical simulation of these equations almost impossible for practical problems. A viable alternative is to reformulate the problem in terms of moments of velocity distribution. Recently, a quadrature-based moment method was derived by Fox for approximating solutions to kinetic equation for arbitrary Knudsen number. Fox also described 1st- and 2nd-order finite-volume schemes for solving the equations. The success of the new method is based on a moment-inversion algorithm that is used to calculate non-negative weights and abscissas from moments. The moment-inversion algorithm does not work if the moments are non-realizable, meaning they do not correspond to a distribution function. Not all the finite-volume schemes lead to realizable moments. Desjardins et al. showed that realizability is guaranteed with the 1 st-order finite-volume scheme, but at the expense of excess numerical diffusion. In the present work, the nonrealizability of the standard 2 nd-order finite-volume scheme is demonstrated and a generalized idea for the development of high-order realizable finite-volume schemes for quadrature-based moment methods is presented. This marks a significant improvement in the accuracy of solutions using the quadrature-based moment method as the use of 1st-order scheme to guarantee realizability is no longer a limitation