Modélisation physique, mathématique et numérique d’un écoulement de gaz dans des réseaux de tuyaux avec une expansion à faible nombre de Mach

This project aims to develop a model for low Mach flow in pipelines and an industrial code implementing it. The model can describe low mach regimes while avoiding blunt approximations, improving over legacy approaches like Boussinesq; as a result, our program is more accurate. In order to build our model and program, we investigate gas flow at low velocities in a network of pipes. We consider a one-dimensional system of equations obtained by averaging the Navier-Stokes equations for a compressible fluid over the pipe section. In contrast to the classical Boussinesq approximation, we employ the Low Mach Expansion to describe asymptotically compressible effects, aiming for a more accurate solution capable of characterizing flows with significant temperature variations. We first apply the model we obtained thus far to a well-known configuration of pipes called the "thermosyphon.". This setup consists of a loop of two horizontal adiabatic pipes and two vertical pipes with prescribed wall temperatures, resulting in a temperature gradient that drives the flow. The application of the model to this configuration gives us an exact but semi-implicit solution under laminar and steady-state conditions. This solution serves as a benchmark against which we validate our numerical results. By comparing our computed values with the quasi-exact solution, we demonstrate the accuracy and reliability of our approach. To implement the low Mach averaged model, we use a numerical method based on the characteristics method and the projection technique. We incorporate in our algorithm the treatment of periodic conditions and Dirac distributions as derivatives of the discontinuous gravity term at the corners. To generalize the model to more complex configurations of pipes, we propose laws that govern the junctions between multiple pipes. We study the "three-rung ladder," a closed configuration consisting of horizontal adiabatic pipes and vertical ones with imposed wall temperatures to induce a temperature-driven flow. %From this setup, more complex pipeline configurations can be derived. To tackle the challenges junctions pose in this context, we implemented an algorithm in the program capable of ensuring proper transmission conditions. Whenever feasible, we provide quasi-exact solutions under laminar and steady-state conditions to validate our numerical results further. Overall, this study investigates further low Mach number gas flows, employing advanced numerical techniques and validating our findings against established benchmarks.Cette étude porte sur les écoulements de gaz à faible vitesse dans des réseaux de tuyaux, en se concentrant sur un régime caractérisé par un faible nombre de Mach. Nous utilisons un modèle unidimensionnel obtenu en moyennant les équations de Navier-Stokes pour un fluide compressible sur la section d'une conduite. Notre approche utilise le développement asymptotique à faible nombre de Mach pour décrire les effets compressibles de manière plus précise, contrairement à l'approximation classique de Boussinesq qui en est un cas limite lorsque l'élévation de température est assez faible. Nous utilisons un schéma numérique fondé sur la méthode des caractéristiques et la méthode de projection pour traiter ce modèle à faible Mach. Nous présentons des résultats numériques pour une configuration appelée "thermosiphon". Cette configuration consiste en une boucle fermée constituée de deux tuyaux horizontaux adiabatiques et de deux tuyaux verticaux avec des températures de paroi prescrites, qui entraînent l'écoulement. L'algorithme développé permet de prendre en compte des distributions de Dirac, qui peuvent apparaître en terme source dans le modèle pour représenter les coins de la géométrie. La méthode proposée est également adaptée au modèle pour des conditions aux limites de type périodique en plus des conditions de type Dirichlet. Nous incorporons dans notre algorithme le traitement des distributions de Dirac en tant que dérivées du terme de gravité discontinu aux coins et des conditions périodiques. Afin d'établir une référence pour le problème du thermosiphon, nous fournissons une solution exacte mais semi-implicite d'un écoulement laminaire en régime permanent. Cette solution sert de référence pour valider la méthode proposée. Nous proposons également des lois qui régissent les jonctions entre plusieurs conduites et présentons des résultats numériques pour des configurations de conduites plus complexes. Nous nous intéressons en particulier à l'échelle à trois barreaux, une configuration fermée composée de tuyaux horizontaux adiabatiques et de tuyaux verticaux dont les parois sont soumises à des températures imposées afin d'induire un gradient de température qui alimente l'écoulement. qui corresponds à une extension du thermosiphon. À partir de celle-ci, plusieurs configurations générales peuvent être dérivées. Nous développons un algorithme pour garantir des conditions de transmission adéquates aux jonctions, en fournissant autant que possible des solutions stationnaires semi-analytiques pour valider nos résultats numériques. Cette étude contribue à une meilleure compréhension des principes qui régissent les écoulements de gaz à faible nombre de Mach en utilisant des techniques numériques avancées et en les comparant à des références établies

Mathematical and numerical modelling of a gas flow for the transportation of liquefied natural gas

LAUREA MAGISTRALEL'ottimizzazione delle condizioni di trasporto di sostanze gassose è un problema molto comune, con svariate limitazioni da tenere in considerazione e cercare di arginare, come per esempio le perdite di carico. Un punto chiave molto importante è il fatto che la velocità all'interno dei canali di trasporto è abbastanza spesso talmente piccola da poter fortunatamente avere la possibilità di introdurre delle ipotesi molto forti nella modellizzazione di tali flussi gassosi come per esempio l'ipotesi di numero di Mach basso. Lo scopo ultimo del progetto su cui ho lavorato è quello di sviluppare un modello e quindi un metodo numerico robusto ed efficiente per trattare il problema precedente. In questo lavoro ho riportato solamente una prima parte del progetto (dato che è stato pensato per essere finito nei tempi di un intero Dottorato), che consiste nel trattare il caso più semplice di un flusso di gas ideale in una dimensione soggetto all'ipotesi di basso Mach suddetta. Schematicamente la struttura di questa presentazione è la seguente: un'introduzione più approfondita al problema, la formulazione di un modello analitico e di un modello numerico, la presentazione dei risultati ottenuti.The optimization of the conditions of transportation of gases is a very common problem with a lot of constraints to take in account and to try to limit like e.g. pressure losses. A key point is the fact that the velocity inside the transportation pipes is quite often small and this leads fortunately to the possibility to make very strong approximations in modeling such gas flows like e.g. the low Mach assumption. The final aim of the project I worked on is to develop a model and then a robust and efficient numerical method for modelling the before mentioned problem. In this work I reported only the first part of the project (since it’s meant to be completed during a Phd), which consists in treating the simpler case of 1-D flow of ideal gases under the low Mach assumptions above mentioned. The outline is the following: a deeper introduction to the problem, the formulation of the model both analytically and numerically and the show of the results obtained

Modélisation physique, mathématique et numérique d’un écoulement de gaz dans des réseaux de tuyaux avec une expansion à faible nombre de Mach

Cette étude porte sur les écoulements de gaz à faible vitesse dans des réseaux de tuyaux, en se concentrant sur un régime caractérisé par un faible nombre de Mach. Nous utilisons un modèle unidimensionnel obtenu en moyennant les équations de Navier-Stokes pour un fluide compressible sur la section d'une conduite. Notre approche utilise le développement asymptotique à faible nombre de Mach pour décrire les effets compressibles de manière plus précise, contrairement à l'approximation classique de Boussinesq qui en est un cas limite lorsque l'élévation de température est assez faible. Nous utilisons un schéma numérique fondé sur la méthode des caractéristiques et la méthode de projection pour traiter ce modèle à faible Mach. Nous présentons des résultats numériques pour une configuration appelée "thermosiphon". Cette configuration consiste en une boucle fermée constituée de deux tuyaux horizontaux adiabatiques et de deux tuyaux verticaux avec des températures de paroi prescrites, qui entraînent l'écoulement. L'algorithme développé permet de prendre en compte des distributions de Dirac, qui peuvent apparaître en terme source dans le modèle pour représenter les coins de la géométrie. La méthode proposée est également adaptée au modèle pour des conditions aux limites de type périodique en plus des conditions de type Dirichlet. Nous incorporons dans notre algorithme le traitement des distributions de Dirac en tant que dérivées du terme de gravité discontinu aux coins et des conditions périodiques. Afin d'établir une référence pour le problème du thermosiphon, nous fournissons une solution exacte mais semi-implicite d'un écoulement laminaire en régime permanent. Cette solution sert de référence pour valider la méthode proposée. Nous proposons également des lois qui régissent les jonctions entre plusieurs conduites et présentons des résultats numériques pour des configurations de conduites plus complexes. Nous nous intéressons en particulier à l'échelle à trois barreaux, une configuration fermée composée de tuyaux horizontaux adiabatiques et de tuyaux verticaux dont les parois sont soumises à des températures imposées afin d'induire un gradient de température qui alimente l'écoulement. qui corresponds à une extension du thermosiphon. À partir de celle-ci, plusieurs configurations générales peuvent être dérivées. Nous développons un algorithme pour garantir des conditions de transmission adéquates aux jonctions, en fournissant autant que possible des solutions stationnaires semi-analytiques pour valider nos résultats numériques. Cette étude contribue à une meilleure compréhension des principes qui régissent les écoulements de gaz à faible nombre de Mach en utilisant des techniques numériques avancées et en les comparant à des références établies.This project aims to develop a model for low Mach flow in pipelines and an industrial code implementing it. The model can describe low mach regimes while avoiding blunt approximations, improving over legacy approaches like Boussinesq; as a result, our program is more accurate. In order to build our model and program, we investigate gas flow at low velocities in a network of pipes. We consider a one-dimensional system of equations obtained by averaging the Navier-Stokes equations for a compressible fluid over the pipe section. In contrast to the classical Boussinesq approximation, we employ the Low Mach Expansion to describe asymptotically compressible effects, aiming for a more accurate solution capable of characterizing flows with significant temperature variations. We first apply the model we obtained thus far to a well-known configuration of pipes called the "thermosyphon.". This setup consists of a loop of two horizontal adiabatic pipes and two vertical pipes with prescribed wall temperatures, resulting in a temperature gradient that drives the flow. The application of the model to this configuration gives us an exact but semi-implicit solution under laminar and steady-state conditions. This solution serves as a benchmark against which we validate our numerical results. By comparing our computed values with the quasi-exact solution, we demonstrate the accuracy and reliability of our approach. To implement the low Mach averaged model, we use a numerical method based on the characteristics method and the projection technique. We incorporate in our algorithm the treatment of periodic conditions and Dirac distributions as derivatives of the discontinuous gravity term at the corners. To generalize the model to more complex configurations of pipes, we propose laws that govern the junctions between multiple pipes. We study the "three-rung ladder," a closed configuration consisting of horizontal adiabatic pipes and vertical ones with imposed wall temperatures to induce a temperature-driven flow. %From this setup, more complex pipeline configurations can be derived. To tackle the challenges junctions pose in this context, we implemented an algorithm in the program capable of ensuring proper transmission conditions. Whenever feasible, we provide quasi-exact solutions under laminar and steady-state conditions to validate our numerical results further. Overall, this study investigates further low Mach number gas flows, employing advanced numerical techniques and validating our findings against established benchmarks

Modeling gas flow in a looped thermosyphon with a 1 D low-Mach number expansion

International audienceThis article provides numerical results for a laminar gas flow at small velocities in the "looped thermosyphon", or "natural circulation loop" : a closed configuration composed of two horizontal adiabatic pipes and two vertical pipes with different fixed wall temperature. To this extent, following Paolucci, [39, 40] we construct a low-Mach number model capable of taking into account the periodicity and the discontinuities intrinsic to this configuration. This compressible model is richer than the Boussinesq model since it describes the pressure variation and is adapted to the description of flows driven by large temperature gradients. We settle averaged equations through the pipes of small radius compared to the length, this gives a one dimensional system of equations of mass, momentum and energy with two pressures, a dynamical one and a thermodynamical one only function of time. We construct a quasi-exact solution in a laminar and steady-state regime. We approach the low-Mach averaged 1D Model with a coupled numerical method based on the characteristics method considering the presence of the periodic conditions and the discontinuous gravity term with Dirac distributions as derivatives at the corners. The numerical results are confronted and validated by the aforementioned reference solution to determine theiraccuracy

Hyperbolic reduced model for Vlasov-Poisson equation with Fokker-Planck collision

This paper proposes a reduced model to simulate the one-dimensional Vlasov-Poisson equation with the non-linear Fokker-Planck operator. The model provides the space-time dynamics of a few macroscopic quantities constructed following the Reduced Order Method (ROM) in the velocity variable: the compression is thus applied to the semi-discretization of the Vlasov equation. To gain efficiency, a Discrete Empirical Interpolation Method (DEIM) is applied to the compressed non-linear Fokker-Planck operator. The size of the resulting reduced model is chosen empirically according to the Knudsen number. Furthermore, we propose a correction to the reduced collision operator that ensures the reduced moments to satisfy an Euler-type system. Numerical simulations of the reduced model show that the model can capture the plasma dynamics in different collisional regimes and initial conditions at a low cost

Hyperbolic reduced model for Vlasov-Poisson equation with Fokker-Planck collision

This paper proposes a reduced model to simulate the one-dimensional Vlasov-Poisson equation with the non-linear Fokker-Planck operator. The model provides the space-time dynamics of a few macroscopic quantities constructed following the Reduced Order Method (ROM) in the velocity variable: the compression is thus applied to the semi-discretization of the Vlasov equation. To gain efficiency, a Discrete Empirical Interpolation Method (DEIM) is applied to the compressed non-linear Fokker-Planck operator. The size of the resulting reduced model is chosen empirically according to the Knudsen number. Furthermore, we propose a correction to the reduced collision operator that ensures the reduced moments to satisfy an Euler-type system. Numerical simulations of the reduced model show that the model can capture the plasma dynamics in different collisional regimes and initial conditions at a low cost

Hyperbolic reduced model for Vlasov-Poisson equation with Fokker-Planck collision

This paper proposes a reduced model to simulate the one-dimensional Vlasov-Poisson equation with the non-linear Fokker-Planck operator. The model provides the space-time dynamics of a few macroscopic quantities constructed following the Reduced Order Method (ROM) in the velocity variable: the compression is thus applied to the semi-discretization of the Vlasov equation. To gain efficiency, a Discrete Empirical Interpolation Method (DEIM) is applied to the compressed non-linear Fokker-Planck operator. The size of the resulting reduced model is chosen empirically according to the Knudsen number. Furthermore, we propose a correction to the reduced collision operator that ensures the reduced moments to satisfy an Euler-type system. Numerical simulations of the reduced model show that the model can capture the plasma dynamics in different collisional regimes and initial conditions at a low cost

Antenatal magnesium sulphate neuroprotection in the preterm infant

Very preterm infants have high rates of neurological impairments and disabilities. These rates have not diminished as the survival rates have improved. Basic science research suggests that magnesium sulphate before birth can be neuroprotective for the preterm fetus. Some, but not all, observational studies in humans also suggest a protective effect of antenatal magnesium sulphate on cerebral palsy. Four randomised controlled trials of antenatal magnesium sulphate have reported long-term neurological effects in surviving infants, but only one of these was designed specifically to evaluate the long-term effects of treatment. These studies found that, overall, antenatal magnesium sulphate therapy had no significant effect on paediatric mortality or neurological outcomes in the first few years of life, including cerebral palsy, but it was found to lower the rate of motor problems at 2 years of age in one study. The role for antenatal magnesium sulphate therapy as a neuroprotective agent for the preterm fetus is not yet established.Marret S, Doyle LW, Crowther CA and Middleton P

Antenatal magnesium sulfate exposure and acute cardiorespiratory events in preterm infants

ObjectiveAntenatal magnesium (anteMg) is used for various obstetric indications including fetal neuroprotection. Infants exposed to anteMg may be at risk for respiratory depression and delivery room (DR) resuscitation. The study objective was to compare the risk of acute cardiorespiratory events among preterm infants who were and were not exposed to anteMg.Study designThis was a retrospective analysis of prospective data collected in the Eunice Kennedy Shriver National Institute of Child Health and Human Development Neonatal Research Network's Generic Database from April 1, 2011, through March 31, 2012. The primary outcome was DR intubation or respiratory support at birth or on day 1 of life. Secondary outcomes were invasive mechanical ventilation, hypotension treatment, neonatal morbidities, and mortality. Logistic regression analysis evaluated the risk of primary outcome after adjustment for covariates.ResultsWe evaluated 1544 infants <29 weeks' gestational age (1091 in anteMg group and 453 in nonexposed group). Mothers in the anteMg group were more likely to have higher education, pregnancy-induced hypertension, and antenatal corticosteroids, while their infants were younger in gestation and weighed less (P < .05). The primary outcome (odds ratio [OR], 1.2; 95% confidence interval [CI], 0.88-1.65) was similar between groups. Hypotension treatment (OR, 0.70; 95% CI, 0.51-0.97) and invasive mechanical ventilation (OR, 0.54; 95% CI, 0.41-0.72) were significantly less in the anteMg group.ConclusionAmong preterm infants age <29 weeks' gestation, anteMg exposure was not associated with an increase in cardiorespiratory events in the early newborn period. The safety of anteMg as measured by the need for DR intubation or respiratory support on day 1 of life was comparable between groups

Effects of antenatal magnesium sulfate treatment for neonatal neuro-protection on cerebral oxygen kinetics

BACKGROUND: The underlying neuro-protective mechanisms of antenatal magnesium sulfate (MgSO₄) in infants born preterm remain poorly understood. Early neonatal brain injury may be preceded by low cerebral blood flow (CBF) and elevated cerebral fractional tissue oxygen extraction (cFTOE). This study investigated the effect of antenatal MgSO₄ on cerebral oxygen delivery, consumption, and cFTOE in preterm infants. METHODS: CBF and tissue oxygenation index were measured, and oxygen delivery, consumption, and cFTOE calculated within 24 h of birth and at 48 and 72 h of life in 36 infants ≤ 30 wk gestation exposed to MgSO₄ and 29 unexposed infants. RESULTS: Total internal carotid blood flow and cerebral oxygen delivery did not differ between the groups at the three study time-points. Cerebral oxygen consumption and cFTOE were lower in infants exposed to antenatal MgSO₄ (P = 0.012) compared to unexposed infants within 24 h of delivery. This difference was not evident by 48 h of age. Fewer infants in the MgSO₄ group developed P/IVH by 72 h of age (P = 0.03). CONCLUSION: Infants exposed to MgSO4 had similar systemic and cerebral hemodynamics but lower cFTOE compared to nonexposed. These findings suggest reduced cerebral metabolism maybe a component of the neuro-protective actions of antenatal MgSO₄.Michael J. Stark, Nicolette A. Hodyl and Chad C. Anderse