881 research outputs found

    Hybridisable Discontinuous Galerkin Formulation of Compressible Flows

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    This work presents a review of high-order hybridisable discontinuous Galerkin (HDG) methods in the context of compressible flows. Moreover, an original unified framework for the derivation of Riemann solvers in hybridised formulations is proposed. This framework includes, for the first time in an HDG context, the HLL and HLLEM Riemann solvers as well as the traditional Lax-Friedrichs and Roe solvers. HLL-type Riemann solvers demonstrate their superiority with respect to Roe in supersonic cases due to their positivity preserving properties. In addition, HLLEM specifically outstands in the approximation of boundary layers because of its shear preservation, which confers it an increased accuracy with respect to HLL and Lax-Friedrichs. A comprehensive set of relevant numerical benchmarks of viscous and inviscid compressible flows is presented. The test cases are used to evaluate the competitiveness of the resulting high-order HDG scheme with the aforementioned Riemann solvers and equipped with a shock treatment technique based on artificial viscosity.Comment: 60 pages, 31 figures. arXiv admin note: substantial text overlap with arXiv:1912.0004

    Positivity-preserving and entropy-bounded discontinuous Galerkin method for the chemically reacting, compressible Navier-Stokes equations

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    This article concerns the development of a fully conservative, positivity-preserving, and entropy-bounded discontinuous Galerkin scheme for simulating the multicomponent, chemically reacting, compressible Navier-Stokes equations with complex thermodynamics. In particular, we extend to viscous flows the fully conservative, positivity-preserving, and entropy-bounded discontinuous Galerkin method for the chemically reacting Euler equations that we previously introduced. An important component of the formulation is the positivity-preserving Lax-Friedrichs-type viscous flux function devised by Zhang [J. Comput. Phys., 328 (2017), pp. 301-343], which was adapted to multicomponent flows by Du and Yang [J. Comput. Phys., 469 (2022), pp. 111548] in a manner that treats the inviscid and viscous fluxes as a single flux. Here, we similarly extend the aforementioned flux function to multicomponent flows but separate the inviscid and viscous fluxes. This separation of the fluxes allows for use of other inviscid flux functions, as well as enforcement of entropy boundedness on only the convective contribution to the evolved state, as motivated by physical and mathematical principles. We also discuss in detail how to account for boundary conditions and incorporate previously developed pressure-equilibrium-preserving techniques into the positivity-preserving framework. Comparisons between the Lax-Friedrichs-type viscous flux function and more conventional flux functions are provided, the results of which motivate an adaptive solution procedure that employs the former only when the element-local solution average has negative species concentrations, nonpositive density, or nonpositive pressure. A variety of multicomponent, viscous flows is computed, ranging from a one-dimensional shock tube problem to multidimensional detonation waves and shock/mixing-layer interaction

    Nouvelles constructions de méthodes de volumes/éléments finis pour les écoulements transsoniques/supersoniques compressibles

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    Thèse numérisée par la Direction des bibliothèques de l'Université de Montréal

    Automatic symbolic computation for discontinuous Galerkin finite element methods

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    The implementation of discontinuous Galerkin finite element methods (DGFEMs) represents a very challenging computational task, particularly for systems of coupled nonlinear PDEs, including multiphysics problems, whose parameters may consist of power series or functionals of the solution variables. Thereby, the exploitation of symbolic algebra to express a given DGFEM approximation of a PDE problem within a high level language, whose syntax closely resembles the mathematical definition, is an invaluable tool. Indeed, this then facilitates the automatic assembly of the resulting system of (nonlinear) equations, as well as the computation of Frechet derivative(s) of the DGFEM scheme, needed, for example, within a Newton-type solver. However, even exploiting symbolic algebra, the discretisation of coupled systems of PDEs can still be extremely verbose and hard to debug. Thereby, in this article we develop a further layer of abstraction by designing a class structure for the automatic computation of DGFEM formulations. This work has been implemented within the FEniCS package, based on exploiting the Unified Form Language. Numerical examples are presented which highlight the simplicity of implementation of DGFEMs for the numerical approximation of a range of PDE problems

    Numerical simulation of temperature-dependent flow dynamics in drilling operations

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    Master's thesis in Petroleum engineeringThis thesis is a description of the research and improvements on solving hyperbolic conservation laws by defining a hybrid explicit/implicit numerical scheme. Former researches have been done by (Evje and FlĂĄtten 2005) on the weakly implicit mixture flux method (WIMF) for the isothermal two-phase flow model. The research consists of proposing a semi-implicit numerical scheme for a two-phase flow system by defining a hybrid model incorporating the advection upstream splitting method (AUSMD) to develop an implicit scheme conjugated with an upwind explicit flux. While the WIMF scheme can demonstrate precise resolution of the moving discontinuity, it is bounded by a CFL condition restricting the timestep and grid sizes for numerical simulation (Evje and FlĂĄtten 2005). The aim of the current research is to formulate and systematically code a numerical scheme named X-Force predictor-corrector, as a contribution to previous works of (Evje and FlĂĄtten 2005) and (Evje, FlĂĄtten et al. 2008). The new research has been done on a hybrid scheme consisting of pressure-based and density-based steps to traverse from Isothermal Euler model in the previous works to full Euler model by associating energy equations. The material stated in the thesis is based on unpublished work by Tore FlĂĄtten and Trygve Wangensteen in a collaboration for TechnipFMC with respect to their FlowManager process surveillance software

    Microscopically implicit-macroscopically explicit schemes for the BGK equation

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    In this work a new class of numerical methods for the BGK model of kinetic equations is introduced. The schemes proposed are implicit with respect to the distribution function, while the macroscopic moments are evolved explicitly. In this fashion, the stability condi- tion on the time step coincides with a macroscopic CFL, evaluated using estimated values for the macroscopic velocity and sound speed. Thus the stability restriction does not depend on the relaxation time and it does not depend on the microscopic velocity of ener- getic particles either. With the technique proposed here, the updating of the distribution function requires the solution of a linear system of equations, even though the BGK model is highly non linear. Thus the proposed schemes are particularly effective for high or moderate Mach numbers, where the macroscopic CFL condition is comparable to accuracy requirements. We show results for schemes of order 1 and 2, and the generalization to higher order is sketched
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