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Numerical computations of internal flows over bluff bodies
... On définit le facteur de compressibilité % 0 = E c /E. Porter et al. [28] étudient numériquement une turbulence supersonique décroissante et identifient trois étapes dans la relaxation de l'énergie cinétique de l'écoulement: on observe d'abord la formation de discontinuités de pression capturées par le solveur via un opérateur Piecewise Parabolic Method (PPM; voir [29], [28] pour référence) sur lequel nous reviendrons plus en détail dans le chapitre 3. Les fluctuations des variables primitives associées au système (Eq.(2.35)+Eq.(2.36)+Eq.(2.37)) se développent à partir de là condition initiale imposée via un spectre de Pouquet [11]. Puis un régime supersonique prend place à travers lequel les discontinuités de pression donnent lieu à des feuilles de cisaillement dont on a vu précédement qu'elles évoluaient vers une topologie de rouleaux tourbillonnâmes via l'instabilité de cisaillement. ...
... Dans un cadre plus général, l'utilisation d'une différence centrée comme schéma de discrétisation spatiale de la dérivée des flux souffre de ne pas prendre en compte la nature hyperbolique d'un problème eulérien, c'est-à-dire le sens de propagation de l'information depuis sa source d'émission que constitue le point de calcul. D'autres schémas numériques explicites existent (schéma de Lax-Wendroff, schéma de Mac-Cormack,..., voir [29], [82] pour une revue) mais aucun n'offre la précision des techniques centrées compactes. Pour introduire les propriétés propagatives associées au problème convectif dans le schéma numérique, des solveurs purement hyperboliques ont été développés. ...
... La nature propagative de la perturbation n'est pas prise en compte dans la construction du schéma numérique. En particulier, l'introduction d'un degré de physique supplémentaire dans les schémas de discrétisation spatiale a été rendue possible par le développement de techniques de capture de choc, pour les écoulements supersoniques [29]. Une revue succinte de ces techniques est donnée maintenant. ...
There is a close analogy between the dynamics of primary atomization process in a two phase shear layer and the processes initiating the mixing transition in a monophase layer presenting the same shear: the primary instability and and the elongation of dense fluid "fingers" in the rapid fllow are analogous if the inlet veocity and the initial density ratio are conserved between both cases. Therefore, the primary atomization coincides with the turbulent mixing if the Reynolds number and the Weber number are sufficiently high. The study investigates these conditions using numerical simulation. Because of stiff velocity and density ratios, we use hyperbolic Roe solvers with WENO for the convective parts of Navier-Stoks equations. At small Reynolds number, for density ratio greater than one, the vorticity field resulting from the hydrodynamic instability becomes asymetric. This yeilds the time scale of first 'atomization' been governed by square root of density ratio, as a signature of baroclinic effects. The study also illustrates that at larger Reynolds numbers, the MILES approach can be used for developing ther eulerian modeling of primary atomization, dedicated to applied computations of two phase flows, as LOx H2 injection in a rocket engine.
... The spatial derivatives in Eq. (5) were discretized by means of the second order finite difference approximations, while the time discretization was carried out explicitly by means of the Adam-Bashforth scheme (Hirsch, 2007). The domain of integration is an annulus of external radius R e and internal radius R i . ...
... The numerical grid was made up of 40 grid points which were clustered near d = R i where the temperature profile shows a more complicated trend. The explicit time discretization of Eq. (5), thorough the Adam-Bashworth scheme, makes the time step Δ t constrained by the stability condition k Δ t / (ρC p Δ d 2 ) b 1/4 (Hirsch, 2007), where Δd is the distance between two grid-points. ...
The limits of microwave (MW) heating in situ application for the remediation of hydrocarbon-polluted soils have been investigated by means of a dedicated simulating model. A computer code was expressly developed and applied to simulate the physical phenomena induced by a MW treatment. MW process was modeled by means of the mono-dimensional transient equations of energy, taking into account the interaction between the electromagnetic field and soil and conductivity phenomena. The model was validated by comparison with experimental results from lab-scale activities. Simulations investigated the influence of soil texture and moisture and operating conditions on electric field penetration into the soil and inducted soil temperature profiles.
Main results indicate that all investigated parameters significantly influenced the effectiveness of the heating process. In generally, the sandy soil appeared to be more penetrable by MWs respect to the clayey one and higher electric field and soil temperature values were found for the sandy soil with the lower water content. Results also revealed that, for the successful remediation of a clayey soil, a maximum distance of 76 cm should be considered when applying an incident electric field of 1000 V m-1 for 15 days, whereas a higher distance of 101 cm can be taken in consideration in the case of sandy soil. A further slight distance increase can be achievable in the case of lowest-moisture soil (5%). Overall, the general performance of the treatment can be greatly improved by increasing the incident electric field to a value 1500 V m-1. In this case, a 15-day heating of a sandy soil led to the maximum distance calculated of 145 cm. Obtained results are of scientific and practical interest and represent the basis for further technical, energy and economic studies, useful to better define the limits of the in situ MW treatment real applicability. The proposed model, giving good prediction of the experimental data for electric field and soil temperature variation, represents a powerful and suitable tool to predict the impact of operating conditions on the effectiveness of the MW techniques.
... Therefore, trajectory-derived LCS ridges/patches may be similar to ridges/patches of the c field as Pe → ∞ if c 0 is uniform, or if c 0 is such that relevant regions are appropriately sampled. However, this may be impractical because numerically advecting high Péclet number flows is prone to accuracy/stability problems [154]. We show in Fig. 6 situations in which we use for c 0 uniform and random 16 distribution. ...
The notion of a Lagrangian Coherent Structure (LCS) is by now well established as a way to capture transient coherent transport dynamics in unsteady and aperiodic fluid flows that are known over finite time. We show that the concept of an LCS can be generalized to capture coherence in other quantities of interest that are transported by, but not fully locked to, the fluid. Such quantities include those with dynamic, biological, chemical, or thermodynamic relevance, such as temperature, pollutant concentration, vorticity, kinetic energy, plankton density, and so on. We provide a conceptual framework for identifying the Generalized Lagrangian Coherent Structures (GLCSs) associated with such evolving quantities. We show how LCSs can be seen as a special case within this framework, and provide an overarching discussion of various methods for identifying LCSs. The utility of this more general viewpoint is highlighted through a variety of examples. We also show that although LCSs approximate GLCSs in certain limiting situations under restrictive assumptions on how the velocity field affects the additional quantities of interest, LCSs are not in general sufficient to describe their coherent transport.
... As regards the numerical approach, the spatial derivatives in Eq. (6) were discretized by means of the second order finite difference approximations while the time discretization has been carried out explicitly by means of the Adam-Bashforth scheme [45]. The domain of integration is an annulus of external radius R e and internal radius R i . ...
This work aims to obtain essential data for the in situ application of microwave (MW) heating for hydrocarbon-polluted soil remediation. For this purpose, lab-scale experiments were performed and a dedicated computer code was developed and applied to simulate the phenomena induced by a MW treatment. MW process was modelled by means of the mono-dimensional transient equations of energy taking into account the interaction between the electromagnetic field and soil and conductivity phenomena. The model was validated by comparison with results from lab-scale experiments.
... The most general description of a fluid flow is obtained from the conservation laws, more commonly known as the Navier-Stokes equations. They represent mass conservation, conservation of momentum and conservation of energy [23,32]. The flow conditions observed during our filling tests are laminar. ...
... The requirement for increasing performance and accuracy of these devices is very strong. Among the wide variety of measurement principles, the vortex shedding flow meter is well regarded to be reliable, robust and flexible (Alsalihi et al. [1]). The application range of the vortex shedding device reaches from liquids through saturated steam to pure gases. ...
The flow rate measurement of liquid, steam, and gas is one of the most important areas of application for today’s field instrumentation. Vortex meters are used in numerous branches of industry to measure the volumetric flow by exploiting the unsteady vortex flow behind a blunt body. The classical Kármán vortex street behind a cylinder shows a decrease in Strouhal number with decreasing Reynolds number. Considering the flow behind a vortex shedding device in a pipe the Strouhal-Reynolds number dependence shows a different behaviour for turbulent flows: a decrease in Reynolds number leads to an increase in Strouhal number. This phenomenon was found in the experimental investigations as well as in the numerical results and has been confirmed theoretically by a stability analysis.
... We use a higher order Gaussian integration formula for triangular elements (see e. g. [Hir88]) for all computations that require the approximation of functions on T that are not constant on all elements T j ∈ T , e. g. for the evaluation of initial values (see (3.15) on page 64) or the computation of the L 1 -norm in (4.1). ...
The aim of this thesis is the investigation of liquid-vapor flows. Vapor together with liquid phases occurin many applications like cavitation problems, cooling and boiling processes or the breakup of liquid jets. Since e.g. turbine blades and ship propellers can bedestroyed by cavitation, this research is of high industrial interest. Both phases are transported by the flow and undergo phase transitions. The governing mathematical equations are expressed by the basic conservation laws for mass, momentum (and energy) together with suitable equations of state for all phases. Additionally, for the mass transfer across the phase boundaries special treatment is necessary. A very important issue of this thesis concerns the simultaneous treatment of phase transition and compressible flow. Up to now the dynamics of pure phase transition (Stefan problems, Cahn-Hilliard problem, Landau-Ginzburg equation, etc.) as well as the dynamics of compressible (viscous) flow have been studied very extensively but separately. The most basic experiment on liquid vapor flows considers the dynamics of a single vapor bubble in a container filled with liquid. If the outer pressure of the liquid is decreased to vapor pressure, then the liquid vapor interface starts to move: We have a dynamic phase boundary with mass transfer. In more complex settings this occurs during the process of cavitation. Lord Rayleigh discovered that pressure waves emitted during the process of cavitation near rigid walls may damage the walls. Mathematical models for liquid vapor phase transformations can be divided into two classes: diffuse interface models and sharp interface (SI) models. The first class takes into account the internal structure of a phase boundary and resolves it as a steep but continuous transition. In the second class phase boundaries are discontinuous transitions of the thermodynamical variables. In this thesis we are looking at the SI model and are dealing with the Euler equations together with a non monotone equation of state (e.g. van-der-Waals). At present reliable and efficient numerical methods for one--phase flow in multiple space dimensions are available. In contrast, compressible two--phase flow with phase transition is still a major challenge for numerics. As basic one--dimensional initial value problem for SI models we will analyze the Riemann or shock tube problem. The knowledge of the unique exact solution of this problem is of highest importance for both: analysis and numerics. The solution of this Riemann problem is the basis for the new numerical method to solve the multidimensional version of the SI model. Finally, we test the new algorithm on several problems with known exact solution and report on a number of numerical experiments. In dieser Arbeit untersuchen wir dynamische Phasenübergänge in kompressiblen Medien. Wir betrachten die isothermalen Eulergleichungen der Gasdynamik zusammen mit einer nichtmonotonen Zustandsgleichung und einen so genannten Sharp-Interface Ansatz für die Modellierung der Phasenübergänge. Im Gegensatz zum Diffuse-Interface Ansatz werden Phasenübergänge hier als unstetige Wellen modelliert. Dies führt auf ein hyperbolisch-elliptisches System von Gleichungen auf das die klassische Theorie für hyperbolische Gleichungen nicht anwendbar ist. Wir untersuchen das zugehörige Riemann-Problem analytisch und entwickeln ein neues numerisches Verfahren auf unstrukturierten Gittern zur Approximation von Lösungen. Das Verfahren kann leicht auf höhere Raumdimensionen erweitert werden. Wir führen numerische Tests in 1D und 2D durch.
A comprehensive analysis of the mechanisms and parameters which control the self-ignition and the stabilization of Hydrogen-Air combustion in supersonic shear layers is presented. This analysis is based on extensive numerical results obtained for unpremixed and premixed basic flow configurations, namely mixing layers and co-flowing jets, boundary layers and oblique shock waves.
Coarse-grid ocean models parameterise the effects of mesoscale eddies by means of a mixing operator which diffuses tracers along surfaces of constant potential density, called isopycnals. We establish the consistency conditions, the spatio-temporal accuracy error and the computational stability conditions for the small-slope isopycnal mixing operator.
A numerical study is conducted of the interaction between the leading oblique detonation wave stabilized by a wedge ramp of finite length and the expansion waves generated by the sudden deflection of the wedge surface. The two-dimensional Euler equations for a reactive gas mixture are solved using a cell-centered finite volume technique on unstructured meshes. Mesh refinement and coarsening techniques are used to concentrate the computational volumes in regions of interest of the flowfield. A detailed chemical kinetic mechanism is employed to describe the combustion process in the mixtures of hydrogen and air considered. Computational results show that a Chapman-Jouguet oblique detonation wave is obtained for intermediate wedge angles within the range of stable detonations. For wedge angles near to the maximum angle that supports stable oblique detonations, decoupling of the oblique detonation wave is observed.
The paper presents a computational system based upon formal principles to run spatial models for environmental processes. The simulator is named SimuMap because it is typically used to simulate spatial processes over a mapped representation of terrain. A model is formally represented in SimuMap as a set of coupled sub-models. The paper considers the situation where spatial processes operate at different time levels, but are still integrated. An example of such a situation commonly occurs in watershed hydrology where overland flow and stream channel flow have very different flow rates but are highly related as they are subject to the same terrain runoff processes. SimuMap is able to run a network of sub-models that express different time–space derivatives for water flow processes. Sub-models may be coded generically with a map algebra programming language that uses a surface data model. To address the problem of differing time levels in simulation, the paper: (i) reviews general approaches for numerical solvers, (ii) considers the constraints that need to be enforced to use more adaptive time steps in discrete time specified simulations, and (iii) scaling transfer rates in equations that use different time bases for time–space derivatives. A multistep scheme is proposed for SimuMap. This is presented along with a description of its visual programming interface, its modelling formalisms and future plans.
A numerical method for direct simulations of boiling flows is presented. The method is similar to the front tracking/finite difference technique of Juric and Tryggvason [Int. J. Multiphase Flow 24 (1998) 387], where one set of conservation equations is used to represent the mass transfer, heat transfer, and fluid flow in the liquid and the vapor, but improves on their numerical technique by elimination of their iterative algorithm. The justification of the mathematical formulation is presented and the numerical method and the code is validated by comparison of the results with the exact solutions of a few analytical problems. A grid refinement test for film boiling on a horizontal surface shows the convergence of results.
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