Quadrature-based Moment Methods Research Articles

Recent advances in well-posed Eulerian models for polydisperse multiphase flows

The current state-of-the-art for computational modeling of polydisperse multiphase flows is reviewed and future research directions are discussed. Physics-based computational models at three distinct levels of abstraction: the microscale, mesoscale and macroscale; are discussed and compared. Special emphasis is placed on the relationship between the models at different scales and on how information from the finer scales is used to provide closures at the coarser scales. For disperse multiphase flows, it is argued that the passage from a direct-numerical simulation at the microscale to the kinetic description at the mesoscale is the crucial step for ensuring the validity of the macroscale model. In particular, the passage from the microscale to the mesoscale requires physics-based closures, while the passage from the mesoscale to the macroscale requires mathematical closures. The choices made in the physical and mathematical closures of the spatial fluxes and coupling terms will determine whether the Eulerian model is well-posed. In addition, the use of quadrature-based moment methods for polydisperse particles is presented as an efficient macroscale closure when dealing with a distribution of particle sizes. Examples of monodisperse and polydisperse multiphase flows are provided for cases where the fluid phase is compressible (high speed) and incompressible (low speed).

Performance and accuracy of the basic closure algorithm of quadrature-based moment methods

Quadrature-based moment methods (QBMMs) are numerical methods to close moment equations derived from a population balance equation. There are many variants, most of which are based on the same closure algorithm that can be divided into three subroutines: First, computing the recurrence coefficients of orthogonal polynomials from a moment sequence, second, finding the associated Gaussian quadrature rule by solving an eigenvalue problem, and third, closing the moment equations using that quadrature rule. Since each of these steps may be carried out billions of times in large-scale simulations, knowing the potentials of optimization is the key to making QBMMs as efficient as possible. In this work, we investigate in detail the performance and accuracy of QBMMs with different configurations and over a wide range of input moment sets. Our major findings were the following: Available algorithms to compute the recurrence coefficients are about equally sensitive to the input moment sequence. There are considerable differences in performance, but the contribution to the overall computational costs of the closure algorithm is negligible. The standard method to solve the eigenvalue problem is insensitive to the input moment set in contrast to a less robust but slightly faster alternative that involves solving a Vandermonde system. For relatively simple physical problems, the overall computational costs are mostly determined by the eigenvalue problem, whereas closing the moment equations becomes important in the presence of more complex processes with particle-particle interactions. The complete source code we implemented and used for our numerical investigations is publicly available.

A 3D computational study of the formation, growth and oxidation of soot particles in an optically accessible direct-injection spark-ignition engine using quadrature-based methods of moments

The accurate prediction and assessment of soot emissions in internal combustion engines play a central role in the development of modern, sustainable powertrains. The modeling of soot requires high-fidelity models capturing both the gaseous soot precursors with suitable mechanisms and an accurate description of all physico-chemical processes related to the solid particulate. Semi-empirical models based on acetylene are frequently used but are limited in covering complex fuel compositions.For this reason, we present the coupling of a detailed quadrature-based method of moments (QMOM) soot model to a state-of-the-art flow solver for the simulation of gasoline engines. A close coupling with the underlying gas phase and the additional consideration of polycyclic aromatic hydrocarbons (PAHs) as precursors allow an accurate description of the entire cause-and-effect chain. The fully coupled model is then applied in a 3D-CFD simulation of an optically accessible research engine to investigate the formation, growth and oxidation of soot particles. Experimental high-speed measurements of soot- luminescence and extinction were used for validation purposes. Together with all preceding models along the engine cycle, the newly implemented model is used to identify the root cause of the observed soot formation hotspots. Particular emphasis is placed on the effects of soot oxidation.

A computational fluid dynamics—Population balance equation approach for evaporating cough droplets transport

Airborne diseases, including COVID-19, are transmitted by respiratory droplets, which makes the study of the evolution of these droplets important to control the transmission. However, the evolution of the droplets is complex, being a multiphase, polydisperse, multicomponent system undergoing evaporation. To numerically investigate such multiphase flows, there are mainly two approaches. One is the Eulerian–Lagrangian (E–L) approach, which is widely used due to its ability to trace the dispersion and evaporation of individual droplets. However, this approach generally has high costs and difficulty in post-processing. The other one is the Eulerian–Eulerian (E–E) approach, which, though having lower costs, is less adopted because of its failure to treat the features of polydispersity and evaporation. In order to take advantage of the low-costs of E–E approach, the population balance equation (PBE) is combined with the E–E approach to trace the polydisperse evaporating droplets. Two PBE solving methods, sectional method (SM) and quadrature based moment method (QBMM), are used and compared. The codes are developed based on the OpenFOAM library and their abilities to predict size changes of evaporating droplets, evolution of expelled airflow front, and aerosols concentration are assessed by using the experimental and numerical results in literature. Good agreements with the reported results are found, indicating the reliability of the CFD-PBE approaches. The SM and QBMM are finally applied in the transport of cough droplets in a 3D chamber. The suspending trends of small droplets and the falling trends of the large droplets are obtained by both methods. The droplets are found to be able to travel a distance longer than 2 m, which is valuable for the guidelines of social distancing. Additionally, the advantages and disadvantages of SM and QBMM are discussed.

Positivity-Preserving Lax–Wendroff Discontinuous Galerkin Schemes for Quadrature-Based Moment-Closure Approximations of Kinetic Models

The quadrature-based method of moments (QMOM) offers a promising class of approximation techniques for reducing kinetic equations to fluid equations that are valid beyond thermodynamic equilibrium. In this work, we study a particular five-moment variant of QMOM known as HyQMOM and establish that this system is moment-invertible over a convex region in solution space. We then develop a high-order discontinuous Galerkin (DG) scheme for solving the resulting fluid system. The scheme is based on a predictor–corrector approach, where the prediction is a localized space-time DG scheme. The nonlinear algebraic system in this prediction is solved using a Picard iteration. The correction is a straightforward explicit update based on the time-integral of the evolution equation, where the space-time prediction replaces all instances of the exact solution. In the absence of limiters, the high-order scheme does not guarantee that solutions remain in the convex set over which HyQMOM is moment-realizable. To overcome this, we introduce novel limiters that rigorously guarantee that the computed solution does not leave the convex set of realizable solutions, thus guaranteeing the hyperbolicity of the system. We develop positivity-preserving limiters in both the prediction and correction steps and an oscillation limiter that damps unphysical oscillations near shocks. We also develop a novel extension of this scheme to include a BGK collision operator; the proposed method is shown to be asymptotic-preserving in the high-collision limit. The HyQMOM and the HyQMOM-BGK solvers are verified on several test cases, demonstrating high-order accuracy on smooth problems and shock-capturing capability on problems with shocks. The asymptotic-preserving property of the HyQMOM-BGK solver is also numerically verified.

Conditional moment methods for polydisperse cavitating flows

The dynamics of cavitation bubbles are important in many flows, but their small sizes and high number densities often preclude direct numerical simulation. We present a computational model that averages their effect on the flow over larger spatiotemporal scales. The model is based on solving a generalized population balance equation (PBE) for nonlinear bubble dynamics and explicitly represents the evolving probability density of bubble radii and radial velocities. Conditional quadrature-based moment methods (QBMMs) are adapted to solve this PBE. A one-way-coupled bubble dynamics problem demonstrates the efficacy of different QBMMs for the evolving bubble statistics. Results show that enforcing hyperbolicity during moment inversion (CHyQMOM) provides comparable model-form accuracy to the traditional conditional method of moments and decreases computational costs by about ten times for a broad range of test cases. The CHyQMOM-based computational model is implemented in MFC, an open-source multi-phase and high-order-accurate flow solver. We assess the effect of the model and its parameters on a two-way coupled bubble screen flow problem.

Quadrature-based moment methods for kinetic plasma simulations

Quadrature-based moment methods (QBMM) are applied to rarefied gas flow and plasma physics problems represented by the Boltzmann equation or the Vlasov-Poisson system of equations. QBMM are a computationally advantageous alternative to both direct Eulerian and Lagrangian particle solvers, able to provide a noise-free solution to the Boltzmann equation and capture non-equilibrium velocity distribution functions (VDF) without excessive computational cost. To provide a closure of the moment equations, the VDF is assumed to be represented by the sum of weighted kernel functions that are placed at given velocity abscissas. Three QBMM approaches, QMOM (Dirac delta kernels), HyQMOM (hyperbolic QMOM), and EQMOM (Gaussian kernels), are applied to canonical one-dimensional rarefied gas flow and plasma physics problems. A new algorithm for EQMOM that is able to treat pathological VDFs and provide control over large abscissas for more than two nodes is developed. QMOM and EQMOM are suitable for problems where the flow does not have shock-like features (for QMOM) or a discontinuity in the VDF (for EQMOM) and where the VDF is far from non-equilibrium and features strong phase-mixing. HyQMOM is generally applicable to all test problems and yields good representations of both the moments and the VDF. HyQMOM also provides significant wall-time improvement (factor of 10–100) compared to other direct Eulerian solvers.

Flamelet LES of a turbulent pulverized solid fuel flame using a detailed phenomenological soot model

In this work, large-eddy simulation (LES) is conducted for a turbulent pulverized coal flame with a detailed phenomenological model to characterize the soot evolution and a flamelet model to describe the detailed gas phase kinetics. In particular, a quadrature-based moment method (QBMM) is adopted to describe the soot statistics with the physicochemical processes of soot particle nucleation, condensation, coagulation, chemical surface growth and oxidization being considered. The simulation results are compared to the experimental data, including the direct photograph images, particle velocities, laser induced fluorescence (LIF) for polycyclic aromatic hydrocarbons (PAHs), laser induced incandescence (LII) for soot, scanning electron microscope (SEM) with the thermophoretic sampling (TS), and the time-resolved LII (TiRe-LII) for spatial soot particle size distribution. The comparisons show that the ignition position and the mean velocity of the coal particles can be accurately predicted with the proposed flamelet/QBMM/LES method, and an overall good agreement is also obtained when comparing against the PAHs-LIF, soot-LII and SEM images. The importance of the contributions of the different physicochemical processes to the soot particle volume growth is quantified by analyzing the corresponding moment source terms. It is found that for the pulverized coal flame studied, the chemical surface growth is the most important contribution to the soot volume growth at different locations along the streamwise direction. Although the nucleation does not make a significant contribution to soot volume growth, it determines the number density of the primary soot particles formed in the main flow region.

Hybrid quadrature moment method for accurate and stable representation of non-Gaussian processes applied to bubble dynamics.

Solving the population balance equation (PBE) for the dynamics of a dispersed phase coupled to a continuous fluid is expensive. Still, one can reduce the cost by representing the evolving particle density function in terms of its moments. In particular, quadrature-based moment methods (QBMMs) invert these moments with a quadrature rule, approximating the required statistics. QBMMs have been shown to accurately model sprays and soot with a relatively compact set of moments. However, significantly non-Gaussian processes such as bubble dynamics lead to numerical instabilities when extending their moment sets accordingly. We solve this problem by training a recurrent neural network (RNN) that adjusts the QBMM quadrature to evaluate unclosed moments with higher accuracy. The proposed method is tested on a simple model of bubbles oscillating in response to a temporally fluctuating pressure field. The approach decreases model-form error by a factor of 10 when compared with traditional QBMMs. It is both numerically stable and computationally efficient since it does not expand the baseline moment set. Additional quadrature points are also assessed, optimally placed and weighted according to an additional RNN. These points further decrease the error at low cost since the moment set is again unchanged. This article is part of the theme issue 'Data-driven prediction in dynamical systems'.

A Gauss/anti-Gauss quadrature method of moments applied to population balance equations with turbulence-induced nonlinear phase-space diffusion

Many particulate systems occurring in nature and technology are adequately described by a number density function (NDF). The numerical solution of the corresponding population balance equation (PBE) is typically accompanied by high computational costs. Quadrature-based moment methods are an approach to reduce the computational complexity by solving only for a set of moments associated with the NDF employing Gaussian quadrature rules to close the moment equations derived from the PBE. The evolution of a population of inertial particles dispersed in a turbulent fluid is governed by a PBE with a phase-space diffusion term as a result of random microscale fluctuations. Considerations on the microscopic behavior concerning the momentum exchange between fluid and dispersed particles suggest that this diffusion term is nonlinear and nonsmooth. The resulting integral terms in the derived moment equations entail large approximation errors when using Gaussian quadrature rules for closure. In this work, we propose a modification of the quadrature method of moments (QMOM), namely the Gauss/anti-Gauss-QMOM (GaG-QMOM), making use of anti-Gaussian quadrature formulae to reduce the large errors due to this particular form of diffusion. This new method is investigated in a series of simple one-dimensional test cases with analytical reference solutions. Moreover, we propose a realizability-preserving variation of the strong-stability preserving Runge-Kutta (RKSSP) schemes that is suited to problems involving phase-space diffusion. Besides the observation that the modified second-order RKSSP scheme can serve as a realizability-preserving alternative to the standard RKSSP scheme of the same order, the numerical results reveal that, compared to the standard QMOM, the GaG-QMOM can reduce the large errors by one to two orders of magnitude.

Turbulent flame-wall interaction of premixed flames using Quadrature-based Moment Methods (QbMM) and tabulated chemistry: An a priori analysis

Presumed probability density function (PDF) and transported PDF methods are commonly applied to model the turbulence chemistry interaction in turbulent reacting flows. However, little focus has been given to the turbulence chemistry interaction PDF closure for flame-wall interaction. In this study, a quasi-DNS of a turbulent premixed, stoichiometric methane-air flame ignited in a fully developed turbulent channel flow undergoing side-wall quenching is investigated. The objective of this study is twofold. First, the joint PDF of the progress variable and enthalpy that needs to be accounted for in turbulence chemistry interaction closure models is analyzed in the quasi-DNS configuration, both in the core flow and the near-wall region. Secondly, a transported PDF closure model, based on a Conditional Quadrature Method of Moments approach, and a presumed PDF approach are examined in an a priori analysis using the quasi-DNS as a reference both in the context of Reynolds-Averaged Navier Stokes (RANS) and Large-Eddy Simulations (LESs). The analysis of the joint PDF demonstrates the high complexity of the reactive scalar distribution in the near-wall region. Here, a high correlation between the progress variable and enthalpy is found, where the flame propagation and quenching are present simultaneously. The transported PDF approach presented in this work, based on the Conditional Quadrature Method of Moments, accounts for the moments of the joint PDF of progress variable and enthalpy coupled to a Quenching-Flamelet Generated Manifold. In the a priori analysis both turbulence chemistry interaction PDF closure models show a high accuracy in the core flow. In the near-wall region, however, only the Conditional Quadrature Method of Moments approach is suitable to predict the flame structures.

A quadrature-based moment method for the evolution of the joint size-velocity number density function of a particle population

A quadrature-based moment method for the approximate solution of the generalized population balance equation (GPBE) governing the evolution of the joint size-velocity number density function (NDF) of a particle population is formulated and tested. The proposed method relies on the third-order hyperbolic conditional quadrature method of moments developed for velocity distribution transport. This approach is combined with the conditional quadrature method of moments to incorporate the dependency of the NDF on particle size, leading to an efficient, stable, quadrature method that uses an analytical solution to determine the size-conditioned velocity moments. The incorporation of source terms accounting for aggregation, breakup, and collisions, as well as acceleration terms such as gravity and drag, is performed using a realizable ODE solver. The approach is then demonstrated by considering zero-dimensional cases to verify the correct integration of the source terms. A set of one-dimensional cases involving droplet evaporation and coalescence is used to validate the velocity-dependent source terms. A two-dimensional case of crossing jets of particles with different sizes is used to demonstrate the proposed method in the case of a polydisperse flow with inertial particles. All work has been implemented in the open-source framework OpenQBMM, based on OpenFOAM®.

Higher order hyperbolic quadrature method of moments for solving kinetic equations

This work attempts to solve two fundamental problems that can be encountered when solving kinetic equations using quadrature based moment methods (QBMM). These two problems are considered crucial problems since they are observed even in the simplest particulate systems. On the one hand, we have that particle trajectory crossing (PTC) events can lead to non-physical accumulation of particles in the solution when the system of transport equations is weakly hyperbolic (i.e. delta-shock formation). On the other hand, the fact that particle segregation events can create moment sets close to the frontier of moment space. These, in turn, can produce non-physical high-velocity fluxes that will significantly reduce the computational efficiency. Given that both problems are fundamental, we limit the framework to one-dimensional free transport of particles. For the first problem we create a novel hyperbolic quadrature moment closure. Although we lack formal analytic proof for a quadrature order (N) higher than three, the numerical examples presented clearly show a well-behaved solution, absent of delta-shocks up to an eight-order quadrature. Our solution to the second problem consists of two minor, though effective, modifications to the general numerical algorithm frequently used for solving kinetic equations using QBMM. The performance of the new algorithm is remarkably stable and efficient, shown by a numerical example.

On Spray Modeling with Quadrature‐Based Moment Methods

AbstractSpray modeling requires a proper representation of the dispersed phase and involves models for a variety of physical phenomena. On the population scale, a properly formulated population balance equation provides a complete statistical description of the particle system. However, in most cases the high dimensionality entails prohibitive computational costs. Quadrature‐based moment methods (QBMM) limit the description of particle populations to a set of only a few moments and thus considerably reduce computational complexity. In order to apply QBMM to sprays, models for many involved physical processes must be formulated consistently in the QBMM‐context. We study secondary droplet breakup with QBMM using a simple well‐established spray breakup model and simplified setups.

Effect of the conditional scalar dissipation rate in the conditional moment closure

In the context of modeling turbulent scalar mixing using probability density function (PDF) methods, the treatment of molecular mixing is of paramount importance. The conditional moment closure (CMC) offers a high-fidelity description for molecular mixing in nonpremixed flows. Recent work has demonstrated that first-order CMC can be implemented numerically using the moments of the conditioning variable and first-order joint moments of the scalar of interest. When solving the CMC using, for example, quadrature-based moment methods (QBMM), a functional form must be chosen for the conditional scalar dissipation rate (CSDR) of the conditioning variable. In prior work, the CSDR was chosen to produce a β-PDF for the conditioning variable (mixture fraction) at steady state. This choice has the advantage that the system of moment equations used in QBMM-CMC can be written in closed form. In this work, an alternative choice for the CSDR is investigated, namely, the amplitude mapping closure (AMC). With the AMC, the moment equations can be closed using the quadrature method of moments incorporated into a realizable ordinary differential equation solver. Results are compared with the β-CSDR closure for binary, passive scalar mixing in homogeneous single- and disperse-phase turbulent flows. It is also demonstrated that the moment formulation of CMC provides a straightforward method for modeling the effect of differential diffusion in the context of CMC.

Evaluation of Quadrature-based Moment Methods in turbulent premixed combustion

Transported probability density function (PDF) methods are widely used to model turbulent flames characterized by strong turbulence-chemistry interactions. Numerical methods directly resolving the PDF are commonly used, such as the Lagrangian particle or the stochastic fields (SF) approach. However, especially for premixed combustion configurations, characterized by high reaction rates and thin reaction zones, a fine PDF resolution is required, both in physical and in composition space, leading to high numerical costs. An alternative approach to solve a PDF is the method of moments, which has shown to be numerically efficient in a wide range of applications. In this work, two Quadrature-based Moment closures are evaluated in the context of turbulent premixed combustion. The Quadrature-based Moment Methods (QMOM) and the recently developed Extended QMOM (EQMOM) are used in combination with a tabulated chemistry approach to approximate the composition PDF. Both closures are first applied to an established benchmark case for PDF methods, a plug-flow reactor with imperfect mixing, and compared to reference results obtained from Lagrangian particle and SF approaches. Second, a set of turbulent premixed methane-air flames are simulated, varying the Karlovitz number and the turbulent length scale. The turbulent flame speeds obtained are compared with SF reference solutions. Further, spatial resolution requirements for simulating these premixed flames using QMOM are investigated and compared with the requirements of SF. The results demonstrate that both QMOM and EQMOM approaches are well suited to reproduce the turbulent flame properties. Additionally, it is shown that moment methods require lower spatial resolution compared to SF method.

Application of quadrature-based moment methods to the conditional moment closure

Two solution algorithms are developed for the conditional moment closure (CMC) using quadrature-based moment methods (QBMM). Their primary purpose is to eliminate the necessity of the additional grid for the conditioning variable (e.g., mixture fraction). As in almost every probability-density-function (PDF)-based description, the main technical hurdle is the solution of the molecular-mixing term. Here the focus is on solving this term coupled with chemistry, as opposed to spatial transport. In the first algorithm (referred to as SA-CMC), this is done by semi-analytically solving an equation for the deviation variable of the conditional scalar mean in terms of Jacobi polynomials. The mixture-fraction space is represented by the Gauss–Lobatto quadrature rule and a β-PDF. In the second algorithm (referred to as QBMM-CMC), a closed set of differential equations is written for the joint moments of a scalar and the mixture fraction, thereby eliminating the need to assume a form for the mixture-fraction PDF. Both solution algorithms are tested for multi-step H2 combustion, and the expected results are achieved with at most six quadrature nodes. Since in the proposed algorithms the scalar mean is ensured to be constant during molecular mixing, it is further concluded that the proposed algorithms are accurate, computationally cost-effective, and straightforward alternatives for traditional CMC solution methods.

Solution of the first-order conditional moment closure for multiphase reacting flows using quadrature-based moment methods

The quadrature-based semi-analytical solution for the conditional moment closure (SA-CMC) given in (A. D. Ilgun, A. Passalacqua, and R. O. Fox, “A quadrature-based conditional moment closure for mixing-sensitive reactions,” Chem. Eng. Sci., 226, 2020) eliminates the additional conditioning-space discretization in CMC applications by assuming that the mixture-fraction PDF is well represented by a β-PDF. A Gaussian quadrature provides the mixture-fraction abscissae, and the conditional scalar mean is expressed in terms of Jacobi polynomials. Here, by preserving the computational efficiency of SA-CMC, a novel quadrature-based moment method (QBMM-CMC) is developed for CMC applications, which does not assume the form of the mixture-fraction PDF. Remarkably, by solving the closed forms for the micromixing terms from CMC, exact expressions result for the mixture-fraction moments of any order. Thus, QBMM-CMC covers cases where the mixture-fraction PDF cannot be well represented by a β-PDF and can be applied to disperse multiphase flows with mass transfer (e.g., droplet evaporation). For single-phase and multiphase pure-mixing problems, the QBMM-CMC mixture-fraction moments are observed to deviate from the β-PDF. For single-phase mixing with and without dispersed-phase mass transfer, QBMM-CMC predictions for mixing-sensitive competitive-consecutive and parallel reactions are investigated parametrically.

Numerical Methods for the Solution of Population Balance Equations Coupled with Computational Fluid Dynamics.

This review article discusses the solution of population balance equations, for the simulation of disperse multiphase systems, tightly coupled with computational fluid dynamics. Although several methods are discussed, the focus is on quadrature-based moment methods (QBMMs) with particular attention to the quadrature method of moments, the conditional quadrature method of moments, and the direct quadrature method of moments. The relationship between the population balance equation, in its generalized form, and the Euler-Euler multiphase flow models, notably the two-fluid model, is thoroughly discussed. Then the closure problem and the use of Gaussian quadratures to overcome it are analyzed. The review concludes with the presentation of numerical issues and guidelines for users of these modeling approaches.

A novel finite-volume TVD scheme to overcome non-realizability problem in quadrature-based moment methods

A new finite-volume total variation diminishing (TVD) scheme is proposed for the solution of moment transport equations in quadrature-based moment methods (QBMM). The proposed scheme is capable of preserving important properties of the moments, such as realizability and boundedness. The idea behind the approach is to limit the flux of all the moments at each cell face with the same limiter value. The proposed numerical technique is eventually compared with other realizable schemes developed for the moment transport equations, showing that the method is able to keep the moments realizable and bounded at the same time.