Influence of the heat transfer model on the estimation of mass transfer

The efficient design and performance of turbopumps in rocket propulsion systems demands a robust numerical tool predicting the phenomenon of cavitation in cryogenic fluids. Building robust models for this complex physics, according to a not-large set of experimental data, is very challenging. In fact, cryogenic fluids are thermo-sensitive, and therefore, thermal effects and strong variations in fluid properties can alter the cavitation properties. This work illustrates how thermal effects can be estimated considering both convective and conductive heat transfer. The Rayleigh-Plesset (RP) equation is coupled with a bubbly flow model to assess the prediction of thermal effects, and used in order to simulate some reference experimental test-cases in literature. Moreover, some tuning parameters, not measured experimentally, such as initial volume vapor phase α0 and initial radius bubbles R0 and the specific coefficient of the heat transfer models are treated like epistemic uncertainties in a probabilistic framework, permitting to obtain numerical error bars for some quantities of interest, and then to perform a robust analysis of the thermal effect

Quantifying uncertainties in a Venturi multiphase configuration

Modeling the complex physical structures of cavitating flows makes numerical simulation far to be predictive, and still a challenging issue. Understanding the role of physical and parametric uncertainties in cavitating flows is of primary importance in order to obtain reliable numerical solutions. In this paper, the impact of various sources of uncertainty on the prediction of cavitating flows is analyzed by coupling a non-intrusive stochastic method with a cavitating CFD solver. The proposed analysis is applied to a Venturi tube, where experimental data concerning vapor formation are available in literature. Numerical solutions with their associated error bars are compared to the experimental curves displaying a large sensitivity to the uncertainties of inlet boundary conditions. Furthermore, this is confirmed by computing the hierarchy of most predominant uncertainties by means of an ANOVA analysis. Finally, a simple algorithm is proposed in order to provide an optimized set of parameters for the cavitation model, thus permitting to obtain a deterministic solution equal to the most probable one when considering physical inlet uncertainties

Construction d'un modèle thermodynamique fiable et robuste pour les mélanges liquide-vapeur

La prédiction numérique des effets de transfert de masse dans les écoulements diphasiques est un outil fondamental dans plusieurs domaines tels que la production d'énergie, les moteurs aérospatiaux,... Un des problèmes est lié au traitement du mélange liquide-vapeur, notamment au niveau thermodynamique. Dans la littérature, de nombreuses équations d'états sont proposées ; néanmoins, on préfère en général utiliser des équations 'convexes', c'est-à-dire, qui présentent une vitesse du son toujours réelle au-dessous de la courbe de saturation dans le mélange liquide-vapeur. L'équation d'état 'Stiffened Gas (SG)' en est un exemple, qui permet un bon compromis entre la facilité d'implémentation et la précision au niveau thermodynamique. Cependant, son utilisation dans la phase gaz ne garantit pas la prise en compte des effets de gaz réel, qui requièrent des lois beaucoup plus complexes généralement non-convexes. Cette étude se concentre sur la formulation d'un algorithme innovant de couplage fort entre un modèle de type SG et une équation d'état complexe quelconque pour la modélisation de la phase gazeuse, basé sur des données expérimentales. En outre, on souhaite considérer les incertitudes inhérentes au modèle, qui concernent certains paramètres thermiques et calorifiques. L'algorithme proposé sera basé sur un cadre bayésien et des techniques de quantification d'incertitudes, permettant la prise en compte d'incertitudes sur les mesures et le modèle. Il réalisera le couplage des deux équations grâce à une calibration de paramètres, cohérente par rapport à la thermodynamique des fluides considérés. La méthode sera appliquée à plusieurs cas d'étude, utilisant différents fluides et différentes lois d'état pour la phase gazeuse. Enfin, un résultat de simulation d'écoulements fluides incluant le nouveau modèle thermodynamique sera montré

Bayesian Inference of Model Error in Imprecise Models

International audienceModern science makes use of computer models to reproduce and predict complex physical systems. Every model involves parameters, which can be measured experimentally (e.g., mass of a solid), or not (e.g., coefficients in the k − ε turbulence model). The latter parameters can be inferred from experimental data, through a procedure called calibration of the computer model. However, some models may not be able to represent reality accurately, due to their limited structure : this is the definition of model error. The "best value" of the parameters of a model is traditionnally defined as the best fit to the data. It depends on the experiment, the quantities of interest considered, and also on the supposed underlying statistical structure of the error. Bayesian methods allow the calibration of the model by taking into account its error. The fit to the data is balanced with the complexity of the model, following Occam's principle. Kennedy and O'Hagan's innovative method [1] to represent model error with a Gaussian process is a reference in this field. Recently, Tuo and Wu [3] proposed a frequentist addition to this method, to deal with the identifiability problem between model error and calibration error. Plumlee [2] applied the method to simple situations and demonstrated the potential of the approach. In this work, we compare Kennedy and O'Hagan's method with its frequentist version, which involves an optimization problem, on several numerical examples with varying degrees of model error. The calibration provides estimates of the model parameters and model predictions, while also inferring model error within observed and not observed parts of the experimental design space. The case of non-linear costly computer models is also considered, and we propose a new algorithm to reduce the numerical complexity of Bayesian calibration techniques

Numerical simulation of stochastic two-phase flows with a DEM Method coupled to uncertainty quantification scheme

A new scheme for the numerical approximation of a five-equations model taking into account uncertainty quantification (UQ) is presented. In particular, the Discrete Equation Method (DEM) for the discretization of the five-equations model is modified for including a formulation based on the adaptive Semi-intrusive (aSI) scheme, thus yielding a new intrusive scheme (aSDEM) for simulating stochastic two-phase flows. Some reference test-cases are performed in order to demonstrate the convergence properties and the efficiency of the overall scheme. The propagation of initial uncertainties is evaluated in terms of mean and variance of several thermodynamic properties of the two phases

Reliable and robust thermodynamic model for liquid-vapor mixture

Numerical simulation of mass transfer in biphase flows is a fundamental tool in various disciplines. One major issue is related to the thermodynamics of the liquid-vapor mixture. Usually, convex equations of state are used, where a real sound speed can be defined under the saturation curve, such as for exemple the Stiffened Gas (SG) equation. Neverthless, the use of this equation in the gas phase, ban the prediction of real-gas effects, demanding a more complex equation of state, generally non-convex. The aim of this work is to formulate an innovative algorithm for a strong coupling between a SG equation and a whatever more complex equation for the gas phase, using experimental data. The proposed algorithm relies on a bayesian-based method, taking into account model and data uncertainties

About the uncertainty quantification of turbulence and cavitation models in cavitating flows simulations

International audiencePrediction in numerical simulation of turbulent cavitating flows is strongly influenced by the presence of several empirical coefficients.The aim of the present paper is to explore the interaction between the cavitation model and turbulence in terms of uncertainty propagation through an unsteady numerical solver, for assessing the robustness and the accuracy of the physical models at different times. Furthermore, the influence of experimental data in the setting of some turbulence and cavitation model coefficients is investigated by means of a Bayesian approach. Finally, the interest is to provide some innovative insights for improving the understanding of these models for cavitating flows

Numerical and Experimental Investigation of Water and Cryogenic Cavitating Flows

The accuracy of the numerical simulation in the prediction of cavitation in cryogenic fluids is of critical importance for the efficient design and performance of turbopumps in rocket propulsion systems. One of the main challenges remains the efficiency in modeling the physics, handling the multiscale properties and developing robust numerical methodologies. Such flows involve thermodynamic phase transition and cavitation bubbles smaller than the global flow structure. Cryogenic fluids are thermo-sensible, then thermal effects and strong variations in fluid properties can alter the cavity properties. The aim of this work is to address the challenge of efficiently modeling cavitating flows when using water and cryogenic fluids. Because of the complexity of the phenomenon, we focus on improving accuracy of the numerical simulation and on proposing some approaches for a strong coupling between numerics and experiments. We first discuss how to simulate cavitation by means of a mixture model. We specifically address two challenges. The first one is associated with the prediction of thermal effect during the phase transition, requiring the solution of the energy conservation equation. The second challenge is associated to the prediction of the number of bubbles, by considering a transport equations for the bubble density. This study is applied to the numerical simulation of a cavitating flow in a Venturi configuration. We observe an improved estimation of temperature and pressure profiles by using the energy equation and the nucleation model. Secondly, we focus on bubble dynamics. Several forms of Rayleigh-Plesset (RP) equations are solved in order to estimate the temperature and pressure during the collapse of the bubble. We observe that, for high Mach number flows, RP modified with a compressible term can predict the bubble behavior more accurately than the classical form of RP. It is necessary to use a complex equation of state for non-condensable gas (van der Waals) in order to have an accurate estimation of the bubble temperature during the collapse phase. We first apply this approach to the water treatment with cavitation, by proposing a model for the estimation of radicals developed during the collapse of the bubble. Secondly, this equation is modified by adding a term of convective heat transfer at the interface between liquid and bubble and it is coupled with a bubbly flow model in order to assess the prediction of thermal effect. We perform a parametric study by considering several values and models for the convective heat transfer coefficient, hb, and we compare temperature and pressure profiles with respect to the experimental data. We observe the importance of the choice of hb for correctly predicting the temperature drop in the cavitating region and we assess the most efficient models. In addition, we perform an experimental study on nitrogen cavitating flows in order to validate numerical prediction of thermal effect, and in order to assess the fundamental characteristics of the nucleation and the transient growth process of the bubble

Discrete Equation Method (DEM) for the Simulation of Viscous, Compressible, Two-phase Flows

International audienc