Strongly coupled fluid-particle flows in vertical channels. II. Turbulence modeling

In Part I, simulations of strongly coupled fluid-particle flow in a vertical channel were performed with the purpose of understanding, in general, the fundamental physics of wall-bounded multiphase turbulence and, in particular, the roles of the spatially correlated and uncorrelated components of the particle velocity.The exact Reynolds-averaged (RA) equations for high-mass-loading suspensions were presented, and the unclosed terms that are retained in the context of fully developed channel flow were evaluated in an Eulerian–Lagrangian (EL) framework. Here, data from the EL simulations are used to validate a multiphase Reynolds-stress model (RSM) that predicts the wall-normal distribution of the two-phase, one-point turbulence statistics up to second order. It is shown that the anisotropy of the Reynolds stresses both near the wall and far away is a crucial component for predicting the distribution of the RA particle-phase volume fraction. Moreover, the decomposition of the phase-average (PA) particle-phase fluctuating energy into the spatially correlated and uncorrelated components is necessary to account for the boundary conditions at the wall. When these factors are properly accounted for in the RSM, the agreement with the EL turbulence statistics is satisfactory at first order (e.g., PA velocities) but less so at second order (e.g., PA turbulent kinetic energy). Finally, an algebraic stress model for the PA particle-phase pressure tensor and the Reynolds stresses is derived from the RSM using the weak-equilibrium assumption

ON THE IMPLEMENTATION OF MOMENT TRANSPORT EQUATIONS IN OPENFOAM TO PRESERVE CONSERVATION, BOUNDEDNESS AND REALIZABILITY

Different industrial scale multiphase systems can be successfully described by considering their polydispersity (e.g. particle/droplet/bubble size and velocity distributions) and phase coupling issues are properly overcome only by considering the evolution in space and time of such distributions, dictated by the so-called Generalized Population Balance Equation (GPBE). A computationally efficient approach for solving the GPBE is represented by the quadrature-based moment methods, where the evolution of the entire particle/droplet/bubble population is recovered by tracking some specific moments of the distribution and the quadrature approximation is used to solve the "closure problem" typical of moment-based methods. In this contribution some crucial computational and numerical details concerning the implementation of these methods into the opensource Computational Fluid Dynamics (CFD) code OpenFOAM are discussed. These aspects are in fact very often overlooked, resulting in implementations that do not satisfy the properties of conservation, realizability and boundedness. These constraints have to be satisfied in a consistent way, with respect to what done with the other conserved transported variables (e.g. volume fraction of the disperse phase) also when higher-order discretization schemes are used. These issues are illustrated on examples taken on our work on the simulation of fluid-fluid multiphase system

Quadrature-based models for multiphase and turbulent reacting flows

The simulation of physical systems requires accurate and robust methods with relatively low cost and it is still the challenge in many applications of engineering processes, specifically in multiphase flow systems. Soot formation, distribution of the aerosols in the atmosphere, reactive precipitation, and combustion modeling are some examples of these processes. Computer simulations of theses systems require a model that can be adapted to that reality. In this study, a quadrature based method of moments (QBMM) is used to address the problems related to the reactive multiphase flow systems. First, the log-normal kernel density function is implemented into the extended quadrature method of moments (Ln-EQMOM). Ln-EQMOM is verified reconstructing the NDF and calculating the moments of a distribution obtained by the linear combination of two log-normal distributions. Later, this numerical procedure is used for problems of aggregation and breakup of fine particles to solve the population balance equation (PBE). The results are compared to the rigorous solutions reported for the cases under consideration \citep{vanni2000}. Finally, the method is verified using two analytically known problems (\textit{e.g.} coalescence and condensation). In comparison to EQMOM with $\Gamma$ kernel density function \citep{yuan2012}, Ln-EQMOM is faster in terms of computations and it preserves the moments more accurately. Then EQMOM with $\beta$ kernel density function is implemented to approximate the solution of the transport equation for the composition probability density function (PDF) of a passive scalar using the Fokker-Planck model to treat the molecular mixing term. The results then compared in a similar condition to those obtained with direct numerical simulation (DNS). The $L_2$ norm of the PDF is reported for two test cases that have been considered. Later the new approach is introduced to address the problems includes the mixing and reaction. Conditional quadrature method of moments (CQMOM) and using the joint composition PDF for the mixture fraction and progress variables, it is possible to address the problems with two consecutive competitive reactions, one reaction and fast reaction, all including the mixing of reactants. direct quadrature method of moments (DQMOM) also expressed for the joint composition PDF. Results obtained with CQMOM and DQMOM are compared with each other. Finally, the CQMOM approach for mixing problems was tested considering two consecutive competitive reactions to verify the implementation and validate the proposed approach. Coupled mixing-PBE approach was then used to investigate polymer aggregation in a multi-inlet vortex reactor (MIVR), typically used to perform flash nanoprecipitation for the production of nanoparticles used in pharmaceutical applications

On the realizability and boundedness of the moments in a bubbly gas-liquid pipe flow

The quadrature-based moment methods (QBMM) are widely used to solve the population balance equation (PBE), which provides a useful understanding of the evolution of disperse systems, e.g. bubbly gas-liquid flow. A key element of the methods is the moment-inversion algorithm that fails if the set of moments is not realizable. When the moment transport equations (derived from the PBE) include the convective term, employing standard high-order discretization schemes may cause the non-realizability problem. As a result, several realizable high-order schemes have been developed to address this issue. This work makes use of a bubbly developing pipe flow to investigate the performance of three schemes including 1st- and 2nd-order schemes, which can prevent the non-realizability problem. Eventually, the predictions will be discussed from the point of view of moment boundedness, a property that must be respected to obtain physical predictions

On the Eulerian Large Eddy Simulation of disperse phase flows: an asymptotic preserving scheme for small Stokes number flows

In the present work, the Eulerian Large Eddy Simulation of dilute disperse phase flows is investigated. By highlighting the main advantages and drawbacks of the available approaches in the literature, a choice is made in terms of modelling: a Fokker-Planck-like filtered kinetic equation proposed by Zaichik et al. 2009 and a Kinetic-Based Moment Method (KBMM) based on a Gaussian closure for the NDF proposed by Vie et al. 2014. The resulting Euler-like system of equations is able to reproduce the dynamics of particles for small to moderate Stokes number flows, given a LES model for the gaseous phase, and is representative of the generic difficulties of such models. Indeed, it encounters strong constraints in terms of numerics in the small Stokes number limit, which can lead to a degeneracy of the accuracy of standard numerical methods. These constraints are: 1/as the resulting sound speed is inversely proportional to the Stokes number, it is highly CFL-constraining, and 2/the system tends to an advection-diffusion limit equation on the number density that has to be properly approximated by the designed scheme used for the whole range of Stokes numbers. Then, the present work proposes a numerical scheme that is able to handle both. Relying on the ideas introduced in a different context by Chalons et al. 2013: a Lagrange-Projection, a relaxation formulation and a HLLC scheme with source terms, we extend the approach to a singular flux as well as properly handle the energy equation. The final scheme is proven to be Asymptotic-Preserving on 1D cases comparing to either converged or analytical solutions and can easily be extended to multidimensional configurations, thus setting the path for realistic applications

Application of the Fokker-Planck molecular mixing model to turbulent scalar mixing using moment methods

An extended quadrature method of moments using the beta kernel density function (beta-EQMOM) is used to approximate solutions to the evolution equation for univariate and bivariate composition probability distribution functions (PDFs) of a passive scalar for binary and ternary mixing. The key element of interest is the molecular mixing term, which is described using the Fokker-Planck (FP) molecular mixing model. The direct numerical simulations (DNSs) of Eswaran and Pope [ Direct numerical simulations of the turbulent mixing of a passive scalar, Phys. Fluids 31, 506 (1988)] and the amplitude mapping closure (AMC) of Pope [ Mapping closures for turbulent mixing and reaction, Theor. Comput. Fluid Dyn. 2, 255 (1991)] are taken as reference solutions to establish the accuracy of the FP model in the case of binary mixing. The DNSs of Juneja and Pope [ A DNS study of turbulent mixing of two passive scalars, Phys. Fluids 8, 2161 (1996)] are used to validate the results obtained for ternary mixing. Simulations are performed with both the conditional scalar dissipation rate (CSDR) proposed by Fox [Computational Methods for Turbulent Reacting Flows (Cambridge University Press, 2003)] and the CSDR from AMC, with the scalar dissipation rate provided as input and obtained from the DNS. Using scalar moments up to fourth order, the ability of the FP model to capture the evolution of the shape of the PDF, important in turbulent mixing problems, is demonstrated. Compared to the widely used assumed beta-PDF model [S. S. Girimaji, Assumed beta-pdf model for turbulent mixing: Validation and extension to multiple scalar mixing, Combust. Sci. Technol. 78, 177 (1991)], the beta-EQMOM solution to the FP model more accurately describes the initial mixing process with a relatively small increase in computational cost

Doctor of Philosophy

dissertationThe purpose of this research is the development of mathematical formalisms for the numerical modeling and simulation of multiphase systems with emphasis in polydisperse flows. The framework for these advancements starts with the William-Boltzmann equation which describes the evolution of joint distributions of particle properties: size, velocity, mass, enthalpy, and other scalars. The amount of statistical information that can be obtained from the direct evolution of particle distribution functions is extensive and detailed, but at a computational cost not yet suitable in usable computational fluid dynamics (CFD) codes. Alternatives to the direct evolution of particle distribution functions have been proposed and we are interested in the family of solutions involving the evolution of the statistical moments from the joint distributions. Rather than tracking every single particle characteristic from the joint distribution, transport equations for their joint moments are formulated; these equations share many of the properties of the regular transport equations formulated in the finite volume framework, making them very attractive for their implementation in current Eulerian CFD codes. The information they produce is general enough to provide the characteristic behavior of many multiphase systems to the point of improvement over the current Eulerian methodologies implemented on standard CFD modeling and simulation approaches. Based on the advantages and limitations of the solutions of the ongoing methodologies and the degree of the information provided by them, we propose formalisms to extend their modeling capabilities focusing on the influence of the size distribution in many of the related multiphase phenomena. The first methodology evolves joint moments based on the evolution of primitive variables (size among them) and conditional moments that are approximants of the joint moments at every time step. The second methodology reconstructs completely the marginal size distribution using the concept of parcel and approximate characteristic behavior of the rest of the conditional moments in each parcel. In both approaches, the representation of size distribution plays a fundamental role and accounts for the polydisperse nature of the system. Also, the numerics of the moment transport equations are to be consistent with the theory of general hyperbolic transport equations but the formulation of the discretization schemes are based on the properties of the underlying distribution. A final contribution is presented in the form of an appendix and it analyzes the role of maximum entropy-based methodologies on the formulation of Eulerian moment-based methods. Attempts to derive new transport equations on the framework of maximum entropy methodologies will be considered and reconstruction of distribution strategies will be presented as preliminary results that might impact future research on Eulerian moment-based methods

Development of High Fidelity Soot Aerosol Dynamics Models using Method of Moments with Interpolative Closure

The method of moments with interpolative closure (MOMIC) for soot formation and growth provides a detailed modeling framework maintaining a good balance in generality, accuracy, robustness, and computational efficiency. This study presents several computational issues in the development and implementation of the MOMIC-based soot modeling for direct numerical simulations (DNS). The issues of concern include a wide dynamic range of numbers, choice of normalization, high effective Schmidt number of soot particles, and realizability of the soot particle size distribution function (PSDF). These problems are not unique to DNS, but they are often exacerbated by the high-order numerical schemes used in DNS. Four specific issues are discussed in this article: the treatment of soot diffusion, choice of interpolation scheme for MOMIC, an approach to deal with strongly oxidizing environments, and realizability of the PSDF. General, robust, and stable approaches are sought to address these issues, minimizing the use of ad hoc treatments such as clipping. The solutions proposed and demonstrated here are being applied to generate new physical insight into complex turbulence-chemistry-soot-radiation interactions in turbulent reacting flows using DNS