A diffuse interface approach for disperse two-phase flows involving dual-scale kinematics of droplet deformation based on geometrical variables

The purpose of this contribution is to derive a reduced-order two-phase flow model in- cluding interface subscale modeling through geometrical variables based on Stationary Action Principle (SAP) and Second Principle of Thermodynamics in the spirit of [6, 14]. The derivation is conducted in the disperse phase regime for the sake of clarity but the resulting paradigm can be used in a more general framework. One key issue is the definition of the proper potential and kinetic energies in the Lagrangian of the system based on geometrical variables (Interface area density, mean and Gauss curvatures...), which will drive the subscale kinematics and dissipation, and their coupling with large scales of the flow. While [14] relied on bubble pulsation, that is normal deformation of the interface with shape preservation related to pressure changes, we aim here at tackling inclusion deformation at constant volume, thus describing self-sustained oscillations. In order to identify the proper energies, we use Direct Numerical Simulations (DNS) of oscillating droplets using ARCHER code and recently devel- oped library, Mercur(v)e, for mean geometrical variable evaluation and analysis preserving topological invariants. This study is combined with historical analytical studies conducted in the small perturba- tion regime and shows that the proper potential energy is related to the surface difference compared to the spherical minimal surface. A geometrical quasi-invariant is also identified and a natural definition of subscale momentum is proposed. The set of Partial Differential Equations (PDEs) including the conservation equations as well as dissipation source terms are eventually derived leading to an original two-scale diffuse interface model involving geometrical variables

Contribution à l'étude des instabilités de combustion dans les moteurs-fusées cryotechniques : couplage entre modèles à interfaces diffuses et modèles cinétiques pour la simulation de l'atomisation primaire

Gatekeepers to the open space, launchers are subject to intense and competitive enhancements, through experimental and numerical test campaigns. Predictive numerical simulations have become mandatory to increase our understanding of the physics. Adjustable, they provide early-stage optimization processes, in particular of the combustion chamber, to guaranty safety and maximize efficiency. One of the major physical phenomenon involved in the combustion of the fuel and oxidizer is the jet atomization, which pilotes both the droplet distributions and the potential high-frequency instabilities in subcritical conditions. It encompasses a large sprectrum of two-phase flow topologies, from separated phases to disperse phase, with a mixed region where the small scale physics and topology of the flow are very complex. Reduced-order models are good candidates to perform predictive but low CPU demanding simulations on industrial configurations but have only been able so far to capture large scale dynamics and have to be coupled to disperse phase models through adjustable and weakly reliable parameters in order to predict spray formation. Improving the hierarchy of reduced order models in order to better describe both the mixed region and the disperse region requires a series of building blocks at the heart of the present work and give on to complex problems in the mathematical analysis and physical modelling of these systems of PDE as well as their numerical discretization and implementation in CFD codes for industrial uses. Thanks to the extension of the theory on supplementary conservative equations to system of non-conservation laws and the formalism of the multi-fluid thermodynamics accounting for non-ideal effects, we give some new leads to define a strictly convex mixture entropy consistent with the system of equations and the pressure laws, which would allow to recover the entropic symmetrization of two-phase flow models, prove their hyperbolicity and obtain generalized source terms. Furthermore, we have departed from a geometric approach of the interface and proposed a multi-scale rendering of the interface to describe multi-fluid flow with complex interface dynamics. The Stationary Action Principle has returned a single velocity two-phase flow model coupling large and small scales of the flow. We then have developed a splitting strategy based on a Finite Volume discretization and have implemented the new model in the industrial CFD software CEDRE of ONERA to proceed to a numerical verification. Finally, we have constituted and investigated a first building block of a hierarchy of test-cases designed to be amenable to DNS while close enough to industrial configurations in order to assess the simulation results of the new model but also to any up-coming models.Gardiens de l’espace, les lanceurs de fusée font l’objet d’une amélioration continue et concurrentielle, grâce à des campagnes de tests expérimentaux et numériques. Les simulations prédictives sont devenues indispensables pour accroître notre compréhension de la physique. Ajustables, elles se prêtent parfaitement à la conception et l’optimisation, en particuliers de la chambre de combustion, pour garantir la sureté et maximiser l’efficacité. L’atomisation primaire est l’un des phénomènes déterminants de la combustion du combustible et de l’oxydant, pilotant à la fois la distribution de gouttes et les potentielles instabilités hautes-fréquences en conditions sous-critiques. Elle couvre un large spectre de topologies d’écoulement diphasique, depuis ceux de type phases séparées jusqu’à la phase dispersée, en passant par une région mixte caractérisée par la complexité de la physique à petites échelles et de la topologie de l’écoulement. Les modèles d’ordre réduit constituent de bons candidats pour réaliser des simulations numériques prédictives et relativement peu coûteuses en ressource de calcul sur des configurations industrielles. Cependant, jusqu’à présent ils ne décrivent correctement que la dynamique des grandes échelles et doivent donc être couplés à des modèles de phase dispersée nécessitant un réglage minutieux de paramètres pour prédire la formation du spray. Afin de décrire à la fois les régions mixte et dispersée, l’amélioration de la hiérarchie de modèles d’ordre réduit repose sur quelques principes clefs au cœur de la thèse ci-présente et fournit des problèmes interdisciplinaires faisant appel tant à l’analyse mathématique et la modélisation physique de ces systèmes d’EDP qu’à leur discrétisation numérique et leur implémentation dans des codes de CFD à des fins industriels. Grâce d’une part à l’extension de la théorie des équations de conservation supplémentaires à des systèmes impliquant des termes non-conservatifs et d’autre part à un formalisme de thermodynamique multi-fluide tenant compte des effets non-idéaux, nous proposons de nouvelles pistes pour définir une entropie de mélange strictement convexe et consistante avec le système d’équation et les lois de pression, dans le but de permettre la symmétrisation entropique des modèles diphasiques, de prouver leur hyperbolicité et d’obtenir des termes sources généraux. De plus, en rompant avec la vision géométrique de l’interface, nous proposons une description multi-échelle de l’interface pour décrire un mélange multi-fluide comportant une dynamique interfaciale complexe. Le Principe de Moindre Action a permis de dériver un modèle bifluide à une vitesse couplant grandes et petites échelles de l’écoulement. Nous avons ensuite développé une stratégie de séparation d’opérateurs basée sur la discrétisation par Volumes Finis, et nous avons implémenté le nouveau modèle dans le logiciel industriel multiphysique de CFD, CEDRE, de l’ONERA afin d’évaluer numériquement ce dernier. Enfin, nous avons construit et analysé les fondations d’une hiérarchie de cas tests accessibles à la DNS tout en étant au plus proche de configurations industrielles, dans le but d’évaluer les résultats de simulations du nouveau modèle ou de tout autre modèle à venir

Entropy supplementary conservation law for non-linear systems of PDEs with non-conservative terms: application to the modelling and analysis of complex fluid flows using computer algebra

International audienceIn the present contribution, we investigate first-order nonlinear systems of partial differential equations which are constituted of two parts: a system of conservation laws and non-conservative first order terms. Whereas the theory of first-order systems of conservation laws is well established and the conditions for the existence of supplementary conservation laws, and more specifically of an entropy supplementary conservation law for smooth solutions, well known, there exists so far no general extension to obtain such supplementary conservation laws when non-conservative terms are present. We propose a framework in order to extend the existing theory and show that the presence of non-conservative terms somewhat complexifies the problem since numerous combinations of the conservative and non-conservative terms can lead to a supplementary conservation law. We then identify a restricted framework in order to design and analyze physical models of complex fluid flows by means of computer algebra and thus obtain the entire ensemble of possible combination of conservative and non-conservative terms with the objective of obtaining specifically an entropy supplementary conservation law. The theory as well as developed computer algebra tool are then applied to a Baer-Nunziato two-phase flow model and to a multicomponent plasma fluid model. The first one is a first-order fluid model, with non-conservative terms impacting on the linearly degenerate field and requires a closure since there is no way to derive interfacial quantities from averaging principles and we need guidance in order to close the pressure and velocity of the interface and the thermodynamics of the mixture. The second one involves first order terms for the heavy species coupled to second order terms for the electrons, the non-conservative terms impact the genuinely nonlinear fields and the model can be rigorously derived from kinetic theory. We show how the theory allows to recover the whole spectrum of closures obtained so far in the literature for the two-phase flow system as well as conditions when one aims at extending the thermodynamics and also applies to the plasma case, where we recover the usual entropy supplementary equation, thus assessing the effectiveness and scope of the proposed theory

Modélisation 5 équations pour la simulation diphasique dans les moteurs à injecteur coaxial

International audienceIn this paper, we propose models and methods for the simulation of two-phase flows in Liquid Rocket Engines (LRE) under subcritical conditions. The numerical strategy consists into coupling models dedicated to different topologies. Actually, we propose a five equation diffuse interface model for the treatment of the dense "separated two-phase flow" near the injector and an Eulerian kinetic based model for the "dispersed two-phase flow" in the chamber. We derive a novel formulation of the 5 equation system to build a robust HLLC type scheme. Then we use a fully Eulerian coupling strategy to take into account for primary atomization. We first run classical test cases in order to validate the numerical methods. Then a simulation on a test case representative to one coaxial injector is performed under subcritical conditions

modélisation d'écoulements diphasiques à deux vitesses pour la simulation d'atomisation de jet

International audienceJet atomization is at the core of many industrial applications such as in cryogenic combustion chambers. Since Direct Numerical Simulations (DNS) of these two-phase flows in real engines are still out of reach, reduced-order models must be built to develop predictive numerical tools. However great care must be taken on the choice of these models in order to reach sound mathematical properties and predictive simulations after a validation process.The contribution of this talk is three-fold. First, we present an Euler-Euler modelling strategy. It uses a novel hierarchy of diffuse interface models, with a proper description of various disequilibrium levels of the mixture inspired from [1,2] and a special attention devoted to the choice of these models to respect the entropy inequality. These diffuse interface models are then coupled to an element of the Kinetic-Based Moment Method (KBMM) for the dispersed flow [3].Secondly, to cope with the strong discontinuities encountered in jet atomization, a robust and accurate numerical method using multi-slope MUSCL technique is applied [4].Finally, relying on the previous two points, simulations of a jet atomization in a cryogenic combustion chamber in subcritical conditions have been conducted and results show thatphysical properties are recovered

Couplage d'une hierarchie de modèles à interface diffuse avec des méthodes de moments basées sur l'équation semi-cinétique pour la simulation de l'atomisation de spray pour les moteurs de fusée cryogéniques

International audienceJet atomizations play a crucial role in many applications such as in cryogenic combustion chambers, thus must be thoroughly studied to understand their impact on high-frequency instabilities. Since direct numerical simulations of these two-phase flows in a real configuration of an engine are still out of reach, predictive numerical tools must be developed using reduced-order models. However great care must be taken on the choices of these models in order to both have sound mathematics properties and lead to predictive simulations. The contribution of this work is three-fold. First, we present an original fully Eulerian modelling strategy. It relies on the coupling of a hierarchy of diffuse interface models with a Eulerian kinetic-based moment method (KBMM). Special attention will be given to the description of various disequilibrium levels for the diffuse interface model, which describes the separated and mixed zones. A member of the KBMM hierarchy will accurately describe the polydisperse evaporating spray generated through atomization. Second, to cope with the strong discontinuities encountered in jet atomization, a robust and accurate numerical method using multi-slope MUSCL technique will be applied. The extension of the proposed strategy to the various levels of the diffuse interface models will be discussed. Third, relying on the previous two points, large eddy simulations of a jet atomization in a cryogenic combustion chamber in subcritical conditions are presented using various levels of modelling

Modélisation mathématique d'écoulements multi-fluides par symétrisation entropique

International audienceJet atomizations play a crucial role in many applications such as in cryogenic combustion chambers, thus must be thoroughly studied to understand its impact on high frequencies instabilities. Since direct numerical simulations of these two-phase flows in a real configuration of an engine are still out of reach, predictive numerical tools must be developed using reduced-order models. However great care must be taken on the choices of these models in order to both have sound mathematics properties and lead to predictive simulations after a validation process. The contribution of this work is three-fold. First, we present an original fully Eulerian modelling strategy. It relies on the coupling of a hierarchy of diffuse interface models with a Eulerian kinetic-based moment method (KBMM). Special attention will be given to the description of various disequilibrium level for the diffuse interface model, which describes the separated and mixed zone. A member of the KBMM hierarchy will accurately describe the polydisperse evaporating spray generated through atomization. Second, to cope with the strong discontinuities encountered in jet atomization, a robust and accurate numerical method using multi-slope MUSCL technique will be applied. The extension of the proposed strategy to the various levels of the diffuse interface models will be discussed. Third, relying on the previous two points, large eddy simulations of a jet atomization in a cryogenic combustion chamber in subcritical conditions are presented using various levels of modelling. Numerical results of jet atomization on the test bench MASCOTTE (ONERA). should eventually be obtained

Validation strategy of reduced-order two-fluid flow models based on a hierarchy of direct numerical simulations

International audienceIn industrial applications, the use of reduced-order models to conduct numerical simulations on realistic configurations as a predictive tool strengthen the need of assessing them. In the context of cryogenic atomization, we propose to build a validation strategy of large scale two fluid models with subscale modelling based on a hierarchy of direct numerical simulation test cases to qualitatively and quantitatively assess these models. In the present work, we propose a validation of these reduced-order model relying on DNS on an hierarchy of specific test cases. We propose in this work to investigate an air-assisted water atomization using a planar injector. This test case offers an atomization regime, which makes it worthy to eventually validate our reduced-order models on a cryogenic coaxial injection

Initiation of a validation strategy of reduced-order two-fluid flow models using direct numerical simulations in the context of jet atomization

International audienceIn industrial applications, developing predictive tools relying on numerical simulations using reduced-order models nourish the need of building a validation strategy. In the context of cryogenic atomization, we propose to build a hierarchy of direct numerical simulation test cases to assess qualitatively and quantitatively diffuse interface models. The present work proposes an initiation of the validation strategy with an air-assisted water atomization using a coaxial injector

Derivation of a two-phase flow model with two-scale kinematics and surface tension by means of variational calculus

International audienceThe present paper proposes a definition of a two-phase interface that relies on a probability density function. This definition enables to introduce a scale separation in the definition this interface and to define fields that characterize the geometry of the interface. Relying on these fields, we propose a two-phase flow model that is able to account for small and large scale separation of the interface description by means of supplementary convected geometric variables. The model accounts for two-scale kinematics and two-scale surface tension. At large scale, the flow and the full geometry of the interface may be retrieved thanks to the bulk variables and the volume fraction, while at small scale the interface dynamics is accurately recovered through the interfacial area density fluctuation and the mean curvature