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The analysis and boundary optimal control of the nonlinear transport of gas on a network of pipelines is considered. The evolution of the gas distribution on a given pipe is modeled by an isothermal semilinear compressible Euler system in one space dimension. On the network, solutions satisfying (at nodes) the so called Kirchhoff flux continuity conditions are shown to exist in a neighborhood of an equilibrium state. The associated nonlinear optimization problem then aims at steering such dynamics to a given target distribution by means of suitable (network) boundary controls while keeping the distribution within given (state) constraints. The existence of local optimal controls is established and a corresponding Karush-Kuhn-Tucker (KKT) stationarity system with an almost surely non-singular Lagrange multiplier is derived.

This research aims to find the optimal standard k-e turbulence model constants (cµ, c1e, and c2e) for better predicting compressible fluid dynamics in an air jet ejector. The turbulence field in a jet flow plays an important role in influencing the performance of the momentum transfer process at a shear layer in nozzle application for momentum source and mixing process. In this research, some activities have been done before analyzing and optimizing the turbulence model constants, including preliminary turbulence modeling study for compressible flow in the air-jet ejector, verification, and validation with primary experimental data as well as by other secondary data. The preliminary studies in turbulence modeling presented that the turbulence modeling of a 3mm air jet-ejector resulted in a similar trend of the relation between entrainment ratio and motive fluid pressure. The results showed that the sensitive parameters in the standard k-emodel dissipation and diffusion terms, cµ, c1e, and c2e, strongly affected the optimum value of turbulence kinetic energy (k) and dissipation rate (e), compared to the reference model. Better k and e could be obtained by changing the c2e into positively proportional, but the cµ and c1e must be changed with opposite proportionality. It was found that the optimum standard k-e model constants in the case of air-jet ejector with 3 mm nozzle diameter for cµ, c1e, and c2e are 0.05, 1.48, and 1.88, respectively, with the error values for k being -8.88% and e being -17.44%.

A high-pressure field is generated in a circular tube by introducing an unsteady jet from its open end. The head of this jet acts as a piston, driving compression waves ahead of it. The peak value of the induced overpressure is evaluated as a solution of a Riemann problem, wherein the jet head is equivalent to a piston head. The jet head of the driver gas, with a filling pressure of 400 kPa, is equivalent to a piston head moving at 160 m/s. This high-pressure generation scheme through the “piston effect” is useful for industrial applications, including filter cleaning in dust collectors, and as an interesting example of unsteady, compressible fluid dynamics.

Observations at numerous volcanoes reveal that eruptions are often accompanied by continual radio frequency (CRF) emissions. The source of this radiation, however, has remained elusive until now. Through experiments and the analysis of field data, we show that CRF originates from proximal discharges driven by the compressible fluid dynamics associated with individual volcanic explosions. Blasts produce flows that expand supersonically, generating regions of weakened dielectric strength in close proximity to the vent. As erupted materialcharged through fragmentation, friction, or other electrification processtransits through such a region, pyroclasts remove charge from their surfaces in the form of small interparticle spark discharges or corona discharge. Discharge is maintained as long as overpressured conditions at the vent remain. Beyond describing the mechanism underlying CRF, we demonstrate that the magnitude of the overpressure at the vent as well as the structure of the supersonic jet can be inferred in real time by detecting and locating CRF sources.

In this paper, we present an overview of the entropy production in fluid dynamics in a systematic way. First of all, we clarify a rigorous derivation of the incompressible limit for the Navier–Stokes–Fourier system of equations based on the asymptotic analysis, which is a very well known mathematical technique used to derive macroscopic limits of kinetic equations (Chapman–Enskog expansion and Hilbert expansion are popular methodologies). This allows to overcome the theoretical limits of assuming that the material derivative of the density simply vanishes. Moreover, we show that the fundamental Gibbs relation in classical thermodynamics can be applied to non-equilibrium flows for generalizing the entropy and for expressing the second law of thermodynamics in case of both incompressible and compressible flows. This is consistent with the thermodynamics of irreversible processes and it is an essential condition for the design and optimization of fluid flow devices. Summarizing a theoretical framework valid at different regimes (both incompressible and compressible) sheds light on entropy production in fluid mechanics, with broad implications in applied mechanics.

Two methods for solid body representation in flow simulations available in the Pencil Code are the immersed boundary method and overset grids. These methods are quite different in terms of computational cost, flexibility and numerical accuracy. We present here an investigation of the use of the different methods with the purpose of assessing their strengths and weaknesses. At present, the overset grid method in the Pencil Code can only be used for representing cylinders in the flow. For this task it surpasses the immersed boundary method in yielding highly accurate solutions at moderate computational costs. This is partly due to local grid stretching and a body-conformal grid, and partly due to the possibility of working with local time step restrictions on different grids. The immersed boundary method makes up the lack of computational efficiency with flexibility in regards to application to complex geometries, due to a recent extension of the method that allows our implementation of it to represent arbitrarily shaped objects in the flow., Comment: 23 pages, 9 figures, 2 tables. To appear in Geophysical & Astrophysical Fluid Dynamics

We are concerned with the suitability of the main models of compressible fluid dynamics for the Lighthill problem for shock diffraction by a convex corned wedge, by studying the regularity of solutions of the problem, which can be formulated as a free boundary problem. In this paper, we prove that there is no regular solution that is subsonic up to the wedge corner for potential flow. This indicates that, if the solution is subsonic at the wedge corner, at least a characteristic discontinuity (vortex sheet or entropy wave) is expected to be generated, which is consistent with the experimental and computational results. Therefore, the potential flow equation is not suitable for the Lighthill problem so that the compressible Euler system must be considered. In order to achieve the non-existence result, a weak maximum principle for the solution is established, and several other mathematical techniques are developed. The methods and techniques developed here are also useful to the other problems with similar difficulties., 20 pages, 4 figures, To appear in: SIAM Journal of Mathematical Analysis, 2020

This paper concerns the traveling wave formation in macroscopic traffic flow models. The dynamics involved in this problem is described following a close analogy to compressible fluid dynamics. It is well known that vehicle clusters appear along a highway when the homogenous steady state taken as a reference is linearly unstable. The cluster properties are determined in an approximate way in terms of the parameters proper to each model and are compared between them.

We address the conservation of angular momentum for cell-centered discretization of compressible fluid dynamics on general grids. We concentrate on the Lagrangian step which is also sufficient for Eulerian discretization using Lagrange+Remap. Starting from the conservative equation of the angular momentum, we show that a standard Riemann solver (a nodal one in our case) can easily be extended to update the new variable. This new variable allows to reconstruct all solid displacements in a cell, and is analogous to a partial Discontinuous Galerkin (DG) discretization. We detail the coupling with a second- order Muscl extension. All numerical tests show the important enhancement of accuracy for rotation problems, and the reduction of mesh imprint for implosion problems. The generalization to axi-symmetric case is detailed.

A finite difference scheme is presented for the inviscid terms of the equations of compressible fluid dynamics with general non-equilibrium chemistry and internal energy.

In this work, we study hyperbolic systems of balance laws. The first part is devoted to compressible fluid dynamics, and particularly to the lifespan of smooth or piecewise smooth solutions. After presenting the state of art, we show an extension to more general gases of a theorem by Grassin.We also study shock waves solutions: first, we extend T. T. Li's approach to estimate the time of existence in the isentropic spherical case; second, we develop Whitham's ideas to obtain an approximated equation satisfied by the discontinuity surface. In the second part, we set up a new model for a roundabout. This leads us to study a multi-class extension of the macroscopic Lighthill-Whitham-Richards' model. We study the traffic on an infinite road, with some points of junction. We distinguish vehicles according to their origin and destination and add some boundary conditions at the junctions. We obtain existence and uniqueness of a weak entropy solution for the Riemann problem. As a complement, we provide numerical simulations that exhibit solutions with a long time of existence. Finally, the Cauchy problem is tackled by the front tracking method. In the last part, we are interested in scalar hyperbolic balance laws. The first question addressed is the control of the total variation and the stability of entropy solutions with respect to flow and source. With this result, we can study equations with non-local flow, which do not fit into the framework of classical theorems. We show here that these kinds of equations are well posed and we show the Gâteaux-differentiability with respect to initial conditions, which is important to characterize maxima or minima of a given cost functional., On étudie dans ce travail des systèmes de lois de conservation hyperboliques. La première partie étudie le temps d'existence des solutions régulières et régulières par morceaux de la dynamique des fluides compressibles. Après avoir présenté l'état de l'art en matière de solutions régulières, on montre une extension d'un théorème de Grassin à des gaz de Van der Waals. On étudie ensuite les solutions ondes de chocs : on poursuit l'approche de T. T. Li pour estimer leur temps d'existence dans le cas isentropique à symétrie sphérique, et l'approche de Whitham afin d'obtenir une équation approchée vérifiée par la surface de discontinuité. Dans une deuxième partie, motivée par la modélisation d'un rond-point en trafic routier, on étudie une extension multi-classe du modèle macroscopique de Lighthill-Whitham-Richards sur une route infinie avec des jonctions. On différencie les véhicules selon leur origine et leur destination et on introduit des conditions aux bords adaptées au niveau des jonctions. On obtient existence et unicité d'une solution au problème de Riemann pour ce modèle. Des simulations numériques attestent que les solutions obtenues existent en temps long. On aborde enfin le problème de Cauchy par la méthode de front tracking. La dernière partie concerne les lois de conservation scalaires. La première question abordée est le contrôle de la variation totale de la solution et la stabilité des solutions faibles entropiques par rapport au flux et à la source. Ce résultat nous permet d'étudier des équations avec flux non-local. Une fois établi leur caractère bien posé, on montre la Gâteaux-différentiabilité du semi-groupe obtenu par rapport aux conditions initiales.

In this work, we study hyperbolic systems of balance laws. The first part is devoted to compressible fluid dynamics, and particularly to the lifespan of smooth or piecewise smooth solutions. After presenting the state of art, we show an extension to more general gases of a theorem by Grassin. We also study shock waves solutions: first, we extend T. T. Li's approach to estimate the time of existence in the isentropic spherical case; second, we develop Whitham's ideas to obtain an approximated equation satisfied by the discontinuity surface. In the second part, we set up a new model for a roundabout. This leads us to study a multi-class extension of the macroscopic Lighthill-Whitham-Richards' model. We study the traffic on an infinite road, with some points of junction. We distinguish vehicles according to their origin and destination and add some boundary conditions at the junctions. We obtain existence and uniqueness of a weak entropy solution for the Riemann problem. As a complement, we provide numerical simulations that exhibit solutions with a long time of existence. Finally, the Cauchy problem is tackled by the front tracking method. In the last part, we are interested in scalar hyperbolic balance laws. The first question addressed is the control of the total variation and the stability of entropy solutions with respect to flow and source. With this result, we can study equations with non-local flow, which do not fit into the framework of classical theorems. We show here that these kinds of equations are well posed and we show the Gâteaux-differentiability with respect to initial conditions, which is important to characterize maxima or minima of a given cost functional., On étudie dans ce travail des systèmes de lois de conservation hyperboliques. La première partie étudie le temps d'existence des solutions régulières et régulières par morceaux de la dynamique des fluides compressibles. Après avoir présenté l'état de l'art en matière de solutions régulières, on montre une extension d'un théorème de Grassin à des gaz de Van der Waals. On étudie ensuite les solutions ondes de chocs : on poursuit l'approche de T. T. Li pour estimer leur temps d'existence dans le cas isentropique à symétrie sphérique, et l'approche de Whitham afin d'obtenir une équation approchée vérifiée par la surface de discontinuité. Dans une deuxième partie, motivée par la modélisation d'un rond-point en trafic routier, on étudie une extension multi-classe du modèle macroscopique de Lighthill-Whitham-Richards sur une route infinie avec des jonctions. On différencie les véhicules selon leur origine et leur destination et on introduit des conditions aux bord adaptées au niveau des jonctions. On obtient existence et unicité d'une solution au problème de Riemann pour ce modèle. Des simulations numériques attestent que les solutions obtenues existent en temps long. On aborde enfin le problème de Cauchy par la méthode de front tracking. La dernière partie concerne les lois de conservation scalaires. La première question abordée est le contrôle de la variation totale de la solution et la stabilité des solutions faibles entropiques par rapport au flux et à la source. Ce résultat nous permet d'étudier des équations avec flux non-local. Une fois établi leur caractère bien posé, on montre la Gâteaux-différentiabilité du semi-groupe obtenu par rapport aux conditions initiales.

The objective is to improve the stability and accuracy of finite volume spatial discretization on unstructured grids. The interest lies in the growing use of finite volumes for large eddy simulation (LES) that requires accurate discretization methods. Another goal is the design of algorithms capable of reconstructing polynomials of higher degree on unstructured grids using only small and compact stencils. The study starts with a general analysis of the reconstruction of polynomials of degree k on unstructured grids, completed by numerical measurements of the convergence rate of the reconstruction error for polynomials of degree 2 and 3. The study presents algorithms for the reconstruction of polynomials on small stencils. Numerical experiments confirm the order of the approximation of these reconstruction methods for quadratic polynomials in 2 dimensions. A theoretical stability analysis exhibits general principles for the design of stable reconstruction methods. A theoretical accuracy analysis, based on the modified equation approach, highlights the errors induced by unstructured grids. The theoretical investigations are completed and confirmed by numerical experiments. The study of slope limiters on unstructured grids formulates algorithms based on a geometric approach.Large eddy simulations of a subsonic flow over a cavity and of a supersonic jet allow the validation and comparison of several discretization features implemented in the code Cedre of Onera.The results of the theoretical stability analysis make it possible to obtain better results for the jet computation on tetrahedral grids., L'objectif est d'améliorer la stabilité et la précision de la discrétisation spatiale de type volumes finis sur des maillages non structurés. La thèse fournit une analyse générale de la reconstruction des polynômes de degré k en maillage non structuré et présente plusieurs algorithmes permettant de reconstruire des polynômes sur de petits voisinages compacts. Une étude théorique de la stabilité établit des principes pour concevoir des méthodes de reconstruction stables. Une étude théorique de la précision caractérise les erreurs induites par le maillage non structuré à l'aide de l'approche de l'équation modifiée. L'étude formule également des algorithmes de limitation en maillage non structuré basés sur une approche géométrique. Toutes les études théoriques sont complétées par des expériences numériques. Les calculs LES d'un écoulement subsonique au-dessus d'une cavité et d'un jet supersonique permettent de valider et comparer plusieurs options de discrétisation spatiale.

The asymptotic behavior as λ → ∞ is analysed for solutions u(λ) of quasilinear symmetric hyperbolic systems of the form A0ut + Ajuxi + λCjuxj = Fu(0, x, λ) = u0(x, λ), where the Ak and F depend on t, x, u, and uλ, but the Cj are constant. The results are applied to the equations of slightly compressible fluid dynamics to obtain a generalized form of the acoustics equations that describe the first-order correction to incompressible flow. The special case of a barotropic fluid was analyzed previously by Klainerman and Majda.