The subgrid-scale scalar variance under supercritical pressure conditions

To model the subgrid-scale (SGS) scalar variance under supercritical-pressure conditions, an equation is first derived for it. This equation is considerably more complex than its equivalent for atmospheric-pressure conditions. Using a previously created direct numerical simulation (DNS) database of transitional states obtained for binary-species systems in the context of temporal mixing layers, the activity of terms in this equation is evaluated, and it is found that some of these new terms have magnitude comparable to that of governing terms in the classical equation. Most prominent among these new terms are those expressing the variation of diffusivity with thermodynamic variables and Soret terms having dissipative effects. Since models are not available for these new terms that would enable solving the SGS scalar variance equation, the adopted strategy is to directly model the SGS scalar variance. Two models are investigated for this quantity, both developed in the context of compressible flows. The first one is based on an approximate deconvolution approach and the second one is a gradient-like model which relies on a dynamic procedure using the Leonard term expansion. Both models are successful in reproducing the SGS scalar variance extracted from the filtered DNS database, and moreover, when used in the framework of a probability density function (PDF) approach in conjunction with the β-PDF, they excellently reproduce a filtered quantity which is a function of the scalar. For the dynamic model, the proportionality coefficient spans a small range of values through the layer cross-stream coordinate, boding well for the stability of large eddy simulations using this model

An algebraic-closure-based momentmethod for unsteady Eulerian modeling of non-isothermal particle-laden turbulent flows in very dilute regime and high Stokes number

An algebraic-closure-based moment method (ACBMM) is developed for unsteady Eulerian particle simulations coupled with direct numerical simulations (DNS) of non-isothermal fluid turbulent flows, in very dilute regime and for large Stokes numbers. It is based on a conditional statistical approach which provides a local instantaneous characterization of the dynamic of the dispersed phase accounting for the effect of crossing between particle trajectories which occurs for large Stokes numbers

Algebraic-Closure-Based Moment Method for Unsteady Eulerian Simulations of Non-Isothermal Particle-Laden Turbulent Flows at Moderate Stokes Numbers in Dilute Regime

To model unsteady non-isothermal dilute particle-laden turbulent flows,an algebraic-closure-based moment method (ACBMM) is developed. ACBMM is a Eulerian approach for the dispersed phase conceived to be coupled with direct numerical simulations (DNSs) of the turbulence when an accurate local description of the turbulent mixture is required. It is based on the combination of a conditional probability density function (PDF) approach, which provides local instantaneous Eulerian equations for the low-order moments of the PDF, and appropriate constitutive relations, as algebraic closures, which are necessary to close the set of conservation equations. The computed low-order moments are the mesoscopic particle number density, particle velocity and particle temperature and the unclosed higher order moments are the particle random uncorrelated motion (RUM) stress tensor and the RUM heat flux (RUM-HF) which appear in the particle momentum and enthalpy equations, respectively. The RUM stress tensor is closed by an additional transport equation for the trace of the tensor and a polynomial representation for tensor functions modeling its deviatoric part. The polynomial representation is used in the framework of an assumption of equilibrium of the RUM anisotropy and leads to an explicit algebraic stress model (2EASM). Similarly, the RUM-HF is modeled assuming equilibrium of the scaled heat flux and explicit self-consistent solutions(2EAHFM) are found by analogy with turbulent heat flux models. As 2EAHFM entails the computation of the RUM temperature variance, an additional transport equation is developed for it. By means of an a priori analysis, the algebraic closures developed by the present study are assessed against actual particle Eulerian fields which are extracted from particle Lagrangian simulations coupled with DNS of a temporal non-isothermal particle-laden turbulent planar jet, for various Stokes numbers. Results show that both 2EASM and 2EAHFM are successful in reproducing the unclosed moments up to moderate turbulent-macroscale Stokes numbers allowing the ACBMM to accurately predict the unsteady non-isothermal dispersed phase

A spatial particle correlation-function analysis in non-isothermal dilute particle-laden turbulent flows

In dilute gas-solid turbulent flows, as that encountered, for example, in pulverized coal combustion processes, the correct prediction of the non-isothermal/reactive particle-laden turbulent mixture relies on the accuracy of the modeling of the local and unsteady particle behavior, which affects the hydro-thermodynamic coupling and the heat transfer and transport in and between the phases and at wall. In very dilute mixtures composed of highly inertial solid particles, such a local and unsteady behavior is the result of the particle interactions with very distant and independent turbulent eddies, namely with different dynamic and thermal turbulent scales. Such interactions strongly modify the local particle velocity and temperature distributions, changing the local evolution of the properties of the dispersed phase. Their knowledge is thus crucial when modeling unsteady particle-laden turbulent flows. In this work, the focus is on the particle temperature distribution. Its characterization is provided by means of an analysis of the two-particle correlation functions in the frame of the direct numerical simulation of non-isothermal homogeneous isotropic, statistically stationary, turbulent flows

Étude théorique et numérique de la modélisation instationnaire des écoulements turbulents anisothermes gaz-particules par une approche Euler-Euler

Le contexte général de cette thèse s'inscrit dans le cadre de la modélisation eulérienne instationnaire des écoulements turbulents anisothermes gaz - particules. La modélisation de ces écoulements est cruciale pour de nombreuses applications industrielles et pour la prédiction de certains phénomènes naturels. Par exemple, la combustion diphasique dans les moteurs automobiles et aéronautiques est précédée par l'injection et la dispersion de carburant liquide dans la chambre de combustion. Les phénomènes mis en jeu exigent alors une prédiction locale tenant compte du caractère instationnaire de l'écoulement turbulent et de la présence de géométries complexes. De plus, de nombreuses études expérimentales et numériques récentes ont mis en évidence le rôle prépondérant de l'inertie des particules sur les mécanismes de dispersion et de concentration préférentielle en écoulement turbulent. Ceci rend donc indispensable la prise en compte de ces mécanismes dans la modélisation diphasique. Au cours de ce travail de thèse, une approche eulérienne locale et instantanée a été développée pour prédire les écoulements gaz-particules anisothermes et turbulents. Elle est basée sur l'approche statistique du Formalisme Eulérien Mésoscopique (MEF) introduite par Février et al. (JFM, 2005). Cette approche a été ici étendue aux variables thermiques pour la prise en compte du caractère anisotherme de l'écoulement. Cette approche a été ensuite utilisée dans le cadre de la méthode des moments (Kaufmann et al., JCP, 2008), et un système d'équations locales et instantanées pour la phase dispersée a été proposé. La modélisation au premier ordre exige la fermeture des moments de second ordre apparaissant dans les équations de la quantité de mouvement et de l'énergie. La proposition de telles relations constitutives fait l'objet d'une partie de la thèse. Afin de fournir une méthode capable de prédire le comportement local, instantané et anisotherme de la phase dispersée dans des configurations `a une échelle réaliste, les équations pour la phase dispersée ont été filtrées et une modélisation aux grandes échelles (LES) est effectuée. Cette modélisation étends, par la prise en compte des variables thermiques, le travail de Moreau et al. (FtaC, 2010) sur l'approche LES Euler-Euler en conditions isothermes. L'approche complète est enfin appliquée aux résultats de simulation numérique d'un jet plan turbulent gazeux froid, chargé en particules, dans une turbulence homogène isotrope chaude monophasique. ABSTRACT : The aim of this thesis is to provide an Eulerian modeling for the dispersed phase interacting with unsteady non-isothermal turbulent flows. The modeling of these flows is crucial for several industrial applications and for predictions of natural events. Examples are the combustion chambers of areo engines where the combustion is preceded by the injection and dispersion of liquid fuel. The prediction of such phenomena involves a local modeling of the mixture for taking into account the unsteady behavior of the turbulent flow and the presence of complex geometries. Moreover, many experimental and numerical studies have recently highlighted the significant role of the particle inertia on the mechanisms of dispersion and preferential concentration. Accounting for such mechanisms is therefore essential for modeling the particle-laden turbulent flows. In this thesis, a local and instantaneous Eulerian approach able to describe and to predict the local behavior of inertial particles interacting with non-isothermal turbulent flows has been developed. It is based on the statistical approach known as Mesoscopic Eulerian formalism (MEF) introduced by Février et al. (JFM, 2005). The statistical approach has been extended to the thermal quantities in order to account for the non-isothermal conditions into the modeling. This formalism is then used in the framework of the moment approach (Kaufmann et al., JCP, 2008) and a system of local and instantaneous equations for the non-isothermal dispersed phase has been suggested. The first order modeling requires to close second-order moments appearing in momentum and energy equations. The proposal of such constitutive relations makes the object of a part of this study. In order to provide an Eulerian approach usable in real configurations at industrial scale, the equations of the dispersed phase are filtered and the approach developed in the framework of the Large-Eddy Simulations. From the work of Moreau et al. (FTaC, 2010), the Eulerian-Eulerian LES approach is then extended to non-isothermal conditions. The whole modeling is then a priori tested against numerical simulations of a cold planar turbulent particle-laden jet crossing a homogeneous isotropic decaying hot turbulence

On the Direct Numerical Simulation of moderate-Stokes-number turbulent particulate flows using Algebraic-Closure-Based and Kinetic-Based Moment Methods

In turbulent particulate flows, the occurrence of particle trajectory crossings (PTC) is the main constraint on classical monokinetic Eulerian methods. To handle such PTC, accounting for high-order moments of the particle velocity distribution is mandatory. In the simplest case, second-order moments are needed. To retrieve these moments, two solutions are proposed in the literature: the Algebraic-Closure-Based Moment Method (ACBMM) and the Kinetic-Based Moment Method (KBMM). The ACBMM provides constitutive relations for the random-uncorrelated-motion (RUM) particle kinetic stress tensor as algebraic closures based on physical arguments (Simonin et al. 2002; Kaufmann et al. 2008; Masi 2010; Masi & Simonin 2012). These closures rely on the internal energy, namely the RUM particle kinetic energy, which is obtained using an additional transport equation. Alternatively, it is possible to directly solve for the second-order moment by providing a closure for the third-order correlation. The KBMM proposes a kinetic description, that is, the number density function (NDF) is reconstructed based on the resolved moments and on a presumed shape. In the present work, an isotropic Gaussian and the anisotropic Gaussian closure of Vié et al. (2012) are used. The goal of the present study is to provide a first comparison between ACBMM and KBMM, using the same robust numerical methods, in order to highlight differences and common points. The test case is a 2D turbulent flow with a mean shear

Kinetic study and modelling of char combustion in TGA in isothermal conditions

The purpose of this work is the kinetic study of biomass char combustion in isothermal conditions in TGA. This char was obtained from fast pyrolysis of beech bark pellet in a fluidized bed reactor at 850 °C and atmospheric pressure. Kinetic study of isothermal char combustion was performed for temperatures up to 400 °C, oxygen partial pressures ranging from 5065 to 21,273 Pa and a char particles size of 25 μm. Mass transfer effects around and within the crucible were thoroughly characterized by naphthalene vaporization. Oxygen diffusion was found to have no effect on char combustion for temperatures below 400 °C. A novel method including the transfer function of the TGA which describes the variation of oxygen partial pressure just after switching the gas from inert to reactive in the TGA was taken into consideration in the kinetic modelling. Two kinetic models (the Grain Model and the Random Pore Model) were used to determine kinetic parameters. The Grain Model was found to be in very good agreement with experimental data. Values of activation energy and reaction order with respect to oxygen are respectively equal to 124 kJ/mol and 0.74. Besides, the maximum combustion rate commonly observed in the literature during char combustion was found to be the result of the non-uniform oxygen partial pressure in the TGA at the initial stage of the char combustion

Multi-species turbulent mixing under supercritical-pressure conditions: modelling, direct numerical simulation and analysis revealing species spinodal decomposition

A model is developed for describing mixing of several species under high-pressure conditions. The model includes the Peng–Robinson equation of state, a full massdiffusion matrix, a full thermal-diffusion-factor matrix necessary to incorporate the Soret and Dufour effects and both thermal conductivity and viscosity computed for the species mixture using mixing rules. Direct numerical simulations (DNSs) are conducted in a temporal mixing layer configuration. The initial mean flow is perturbed using an analytical perturbation which is consistent with the definition of vorticity and is divergence free. Simulations are performed for a set of five species relevant to hydrocarbon combustion and an ensemble of realizations is created to explore the effect of the initial Reynolds number and of the initial pressure. Each simulation reaches a transitional state having turbulent characteristics and most of the data analysis is performed on that state. A mathematical reformulation of the flux terms in the conservation equations allows the definition of effective species-specific Schmidt numbers (Sc) and of an effective Prandtl number (Pr) based on effective speciesspecific diffusivities and an effective thermal conductivity, respectively. Because these effective species-specific diffusivities and the effective thermal conductivity are not directly computable from the DNS solution, we develop models for both of these quantities that prove very accurate when compared with the DNS database. For two of the five species, values of the effective species-specific diffusivities are negative at some locations indicating that these species experience spinodal decomposition; we determine the necessary and sufficient condition for spinodal decomposition to occur. We also show that flows displaying spinodal decomposition have enhanced vortical characteristics and trace this aspect to the specific features of high-density-gradient magnitude regions formed in the flows. The largest values of the effective speciesspecific Sc numbers can be well in excess of those known for gases but almost two orders of magnitude smaller than those of liquids at atmospheric pressure. The effective thermal conductivity also exhibits negative values at some locations and the effective Pr displays values that can be as high as those of a liquid refrigerant. Examination of the equivalence ratio indicates that the stoichiometric region is thin and coincides with regions where the mixture effective species-specific Lewis number values are well in excess of unity. Very lean and very rich regions coexist in the vicinity of the stoichiometric region. Analysis of the dissipation indicates that it is dominated by mass diffusion, with viscous dissipation being the smallest among the three dissipation modes. The sum of the heat and species (i.e. scalar) dissipation is functionally modelled using the effective species-specific diffusivities and the effective thermal conductivity. Computations of the modelled sum employing the modelled effective species-specific diffusivities and the modelled effective thermal conductivity shows that it accurately replicates the exact equivalent dissipation

Particle-resolved numerical simulations of the gas–solid heat transfer in arrays of random motionless particles

Particle-resolved direct numerical simulations of non-isothermal gas–solid flows have been performed and analyzed from microscopic to macroscopic scales. The numerical configuration consists in an assembly of random motionless spherical particles exchanging heat with the surrounding moving fluid throughout the solid surface. Numerical simulations have been carried out using a Lagrangian VOF approach based on fictitious domain framework and penalty methods. The entire numerical approach (numerical solution and post-processing) has first been validated on a single particle through academic test cases of heat transfer by pure diffusion and by forced convection for which analytical solution or empirical correlations are available from the literature. Then, it has been used for simulating gas–solid heat exchanges in dense regimes, fully resolving fluid velocity and temperature evolving within random arrays of fixed particles. Three Reynolds numbers and four solid volume fractions, for unity Prandtl number, have been investigated. Two Nusselt numbers based, respectively, on the fluid temperature and on the bulk (cup-mixing) temperature have been computed and analyzed. Numerical results revealed differences between the two Nusselt numbers for a selected operating point. This outcome shows the inadequacy of the Nusselt number based on the bulk temperature to accurately reproduce the heat transfer rate when an Eulerian–Eulerian approach is used. Finally, a connection between the ratio of the two Nusselt numbers and the fluctuating fluid velocity–temperature correlation in the mean flow direction is pointed out. Based on such a Nusselt number ratio, a model is proposed for it

Internal gravity waves feedback on a parallel mean flow: modelling of a boundary layer above a sinusoidal topography

We consider a boundary layer flow horizontally homogeneous in the presence of a vertical stratification and of a sinusoidal topography. We present a simple model describing the interaction between the mean flow and the packet of internal waves emitted at the bottom, assuming that it obeys to the laws of the refraction. We focus on the configurations where no critical layers develop and where the waves propagate upward. We show that an equilibrium state exists when the bottom boundary conditions are stationary. With numerical simulations and considering the analytical expression of equilibra, we show that the presence of the waves amplifies the mean flow evolutions