Stabilized Finite Element Method for Compressible–Incompressible Diphasic Flows

Billaud, M.; Gallice, Gérard; Nkonga, Boniface

doi:10.1007/978-3-642-11795-4_17

Cited by 2 publications

(2 citation statements)

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“…An important contribution of this pioneering work was the inclusion of the largest eigenvalue of the hyperbolic system into the matrix of algorithmic parameters. More recently, Polner (2005), Billaud et al (2010), and Sevilla et al (2013) applied the SUPG stabilization into their compressible flow formulations. Some later stabilizations were formulated based on the Galerkin Least Squares method.…”

Section: Hff 263/4mentioning

confidence: 99%

Variational multi-scale finite element approximation of the compressible Navier-Stokes equations

Bayona-Roa

Baiges

Codina

2016

International Journal of Numerical Methods for Heat &Amp; Fluid Flow

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Purpose - The purpose of this paper is to apply the variational multi-scale framework to the finite element approximation of the compressible Navier-Stokes equations written in conservation form. Even though this formulation is relatively well known, some particular features that have been applied with great success in other flow problems are incorporated. \ud \ud Design/methodology/approach - The orthogonal subgrid scales, the non-linear tracking of these subscales, and their time evolution are applied. Moreover, a systematic way to design the matrix of algorithmic parameters from the perspective of a Fourier analysis is given, and the adjoint of the non-linear operator including the volumetric part of the convective term is defined. Because the subgrid stabilization method works in the streamline direction, an anisotropic shock capturing method that keeps the diffusion unaltered in the direction of the streamlines, but modifies the crosswind diffusion is implemented. The artificial shock capturing diffusivity is calculated by using the orthogonal projection onto the finite element space of the gradient of the solution, instead of the common residual definition. Temporal derivatives are integrated in an explicit fashion. \ud \ud Findings - Subsonic and supersonic numerical experiments show that including the orthogonal, dynamic, and the non-linear subscales improve the accuracy of the compressible formulation. The non-linearity introduced by the anisotropic shock capturing method has less effect in the convergence behavior to the steady state. \ud \ud Originality/value - A complete investigation of the stabilized formulation of the compressible problem is addressed.Peer ReviewedPostprint (author's final draft

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Section: Hff 263/4mentioning

confidence: 99%

Variational multi-scale finite element approximation of the compressible Navier-Stokes equations

Bayona-Roa

Baiges

Codina

2016

International Journal of Numerical Methods for Heat &Amp; Fluid Flow

View full text Add to dashboard Cite

show abstract

“…An important contribution of this pioneering work was the inclusion of the largest eigenvalue of the hyperbolic system into the matrix of algorithmic parameters. More recently, in references [63][64][65], the SUPG stabilization has been applied into compressible flow formulations. Some later stabilizations were formulated based on the Galerkin Least Squares (GLS) method.…”

Section: Introductionmentioning

confidence: 99%

Adaptive mesh simulations of compressible flows using stabilized formulations

Bayona-Roa¹

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This thesis investigates numerical methods that approximate the solution of compressible flow equations. The first part of the thesis is committed to studying the Variational Multi-Scale (VMS) finite element approximation of several compressible flow equations. In particular, the one-dimensional Burgers equation in the Fourier space, and the compressible Navier-Stokes equations written in both conservative and primitive variables are considered. The approximations made for the VMS formulation are extensively researched; the design of the matrix of stabilization parameters, the definition of the space where the subscales live, the inclusion of the temporal derivatives of the subscales, and the non-linear tracking of the subscales are formulated. Also, the addition of local artificial diffusion in the form of shock capturing techniques is included. The accuracy of the formulations is studied for several regimes of the compressible flow, from aeroacoustic flows at low Mach numbers to supersonic shocks. The second part of the thesis is devoted to make the solution of the smallest fluctuating scales of the compressible flow affordable. To this end, a novel algorithm for $h-$refinement of computational physics meshes in a distributed parallel setting, together with the solution of some refinement test cases in supercomputers are presented. The definition of an explicit a-posteriori error estimator that can be used in the adaptive mesh refinement simulations of compressible flows is also developed; the proposed methodology employs the variational subscales as a local error estimate that drives the mesh refinement. The numerical methods proposed in this thesis are capable to describe the high-frequency fluctuations of compressible flows, especially, the ones corresponding to complex aeroacoustic applications. Precisely, the direct simulation of the fricative [s] sound inside a realistic geometry of the human vocal tract is achieved at the end of the thesis. Esta tesis investiga métodos numéricos que aproximan la solución de las ecuaciones de flujo compresible. La primera parte de la tesis está dedicada al estudio de la aproximación numérica del flujo compresible por medio del método multiescala variacional (VMS) en elementos finitos. En particular, se consideran la ecuación de Burgers unidimensional descrita en el espacio de Fourier y las ecuaciones de Navier-Stokes de flujo compresible escritas en variables conservativas y primitivas. Las aproximaciones hechas para plantear la formulación VMS son ampliamente investigadas; el diseño de la matriz de parámetros de estabilización, la definición del espacio donde viven las subescalas, la inclusión de las derivadas temporales de las subescalas y el seguimiento no lineal de las subescalas son particularidades de la formulación que se analizan para cada una de las ecuaciones consideradas. Además, se incluye la adición de difusión artificial local en forma de técnicas de captura de choque. La precisión de las formulaciones se estudia para varios regímenes del flujo compresible, desde flujos aeroacústicos a bajos números de Mach hasta choques supersónicos. La segunda parte de la tesis está dedicada a hacer asequible la solución de las escalas fluctuantes más pequeñas del flujo compresible. Con este fin, se presenta un algoritmo novedoso para el refinamiento $h$ de las mallas de física computacional usadas en computación distribuida en paralelo. Además, se demuestra la solución en superordenadores de algunos casos de prueba del refinamiento de mallas. También se desarrolla la definición de un estimador de error explícito a posteriori que se puede usar en las simulaciones adaptativas de refinamiento de malla de flujos compresibles; la metodología propuesta emplea las subescalas variacionales como una estimación de error local que induce el refinamiento de la malla. Los métodos numéricos propuestos en esta tesis son capaces de describir las fluctuaciones de alta frecuencia de los flujos compresibles, especialmente los correspondientes a aplicaciones aeroacústicas complejas. Precisamente, la simulación directa del sonido consonántico fricativo [s] dentro de una geometría realista del tracto vocal humano se demuestra al final de la tesis.

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Stabilized Finite Element Method for Compressible–Incompressible Diphasic Flows

Cited by 2 publications

References 5 publications

Variational multi-scale finite element approximation of the compressible Navier-Stokes equations

Variational multi-scale finite element approximation of the compressible Navier-Stokes equations

Adaptive mesh simulations of compressible flows using stabilized formulations

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