1,721,308 research outputs found
High-order accurate discontinuous finite element solution of the 2D Euler equations
No full textThis paper deals with a high-order accurate discontinuous finite element method for the numerical solution of the Euler equations. The method combines two key ideas which are at the basis of the finite volume and of the finite element method, the physics of wave propagation being accounted for by means of Riemann problems and accuracy being obtained by means of high-order polynomial approximations within elements. We focus our attention on two-dimensional steady-state problems and present higher order accurate (up to fourth-order) discontinuous finite element solutions on unstructured grids of triangles. In particular we show that, in the presence of curved boundaries, a meaningful high-order accurate solution can be obtained only if a corresponding high-order approximation of the geometry is employed. We present numerical solutions of classical test cases computed with linear, quadratic, and cubic elements which illustrate the versatility of the method and the importance of the boundary condition treatment. (C) 1997 Academic Press
An implicit matrix-free Discontinuous Galerkin solver for viscous and turbulent aerodynamic simulations
No full textThis paper presents some recent advancements of the computational efficiency of a Discontinuous Galerkin (DG) solver for the Navier–Stokes (NS) and Reynolds Averaged Navier Stokes (RANS) equations. The implementation and the performance of a Newton–Krylov matrix-free (MF) method is presented and compared with the matrix based (MB) counterpart. Moreover two solution strategies, developed in order to increase the solver efficiency, are discussed and experimented. Numerical results of some test cases proposed within the EU ADIGMA (Adaptive Higher-Order Variational Methods for Aerodynamic Applications in Industry) project demonstrate the capabilities of the method
Numerical evaluation of two discontinuous Galerkin methods for the compressible Navier-Stokes equations
No full textThis paper presents a critical comparison between two recently proposed discontinuous Galerkin methods for the space discretization of the viscous terms of the compressible Navier-Stokes equations. The robustness and accuracy of the two methods has been numerically evaluated by considering simple but well documented classical two-dimensional test cases, including the flow around the NACA0012 airfoil, the flow along a flat plate and the flow through a turbine nozzle. Copyright © 2002 John Wiley and Sons, Ltd
High-order accurate discontinuous Galerkin methods in computational fluid dynamics: from model problems to complex turbulent flows
No full textSchemes with order of accuracy higher than two have become a hot topic in CFD, and the perspective of applying them in industrial aeronautical context is realistic within the next decade. In this course the basics and state of the art of the following methods will be discussed in detail: Discontinuous Galerkin Finite Element methods; Spectral Element and Finite Volume methods; ENO-reconstruction and Residual Based Finite Volume Methods; Residual Distribution schemes. Applications are foreseen in the area of compressible aerodynamics and aeroacoustics. The course is organized in collaboration with the European Union targeted research project ADIGMA, and will bring together a balanced mix of European and American top researchers in the field
High-order discontinuous Galerkin solutions of three-dimensional incompressible RANS equations
No full textThis paper presents the latest developments of the artificial compressibility flux Discontinuous Galerkin (DG) method introduced in [1], extended in [2] to natural convection flows, in [3] to unsteady flows and, more recently, in [4] to turbulent flows. Here we consider the three-dimensional incompressible Reynolds Averaged Navier-Stokes equations (RANS) coupled with the Spalart-Allmaras (SA) turbulence model.The development of efficient high-order RANS solvers is still a difficult task due to the extreme stiffness of the governing equations. For this reason the turbulence model here has been suitably modified, in the source terms and in the diffusion coefficient, in order to prevent unphysical conditions of the turbulent working variable and of one of the closure functions, which sometimes result in numerical instabilities. The reliability, accuracy and robustness of the method were assessed by computing several test cases in simple and real-life configurations: the flow over a sinusoidal bump, the flow field past a sphere in the supercritical regime, the flow field past a delta wing, and the flow around the DLR-F6 wing body transport configuration
A Spalart-Allmaras turbulence model implementation in a discontinuous Galerkin solver for incompressible flows
No full textIn this paper the artificial compressibility flux Discontinuous Galerkin (DG) method for the solution of the incompressible Navier–Stokes equations has been extended to deal with the Reynolds-Averaged Navier–Stokes (RANS) equations coupled with the Spalart–Allmaras (SA) turbulence model. DG implementations of the RANS and SA equations for compressible flows have already been reported in the literature, including the description of limiting or stabilization techniques adopted in order to prevent the turbulent viscosity View the MathML sourceν ̃ from becoming negative. In this paper we introduce an SA model implementation that deals with negative View the MathML sourceν ̃ values by modifying the source and diffusion terms in the SA model equation only when the working variable or one of the model closure functions become negative. This results in an efficient high-order implementation where either stabilization terms or even additional equations are avoided. We remark that the proposed implementation is not DG specific and it is well suited for any numerical discretization of the RANS-SA governing equations. The reliability, robustness and accuracy of the proposed implementation have been assessed by computing several high Reynolds number turbulent test cases: the flow over a flat plate (Re=107Re=107), the flow past a backward-facing step (Re=37400Re=37400) and the flow around a NACA 0012 airfoil at different angles of attack (View the MathML sourceα=0°,10°,15°) and Reynolds numbers (Re=2.88×106,6×106Re=2.88×106,6×106)
DG P-multigrid. Efficient solvers and complex applications
No full textThese lecture notes will focus on some efficiency-related topics relevant to an effective implementation of the p-multigrid technique, presented in the lecture notes “p-Multigrid For Discontinuous Galerkin Methods”, in a high-order DG method.
Actually, the underlying building blocks of the p-multigrid solver are those of an implicit DG code, named MIGALE, developed over the past years and more recently within the ADIGMA project. The code solves the Euler, Navier-Stokes and the coupled RANS and k-! turbulence model equations. The implicit implementation of the DG method is based on the exact linearization of residuals and on linearly implicit Runge-Kutta schemes for time integration. All the boundary conditions are also implemented implicitly.
Based on the framework of the implicit code, the matrix blocks of the semi-implicit p-multigrid solver are nothing but the matrix blocks of the implicit scheme local to the elements, i.e., the diagonal blocks. On the other hand, the backward Euler smoother employed at the lowest degree of the p-multigrid algorithm is exactly the same used by the implicit solver.
These notes summarize recent results of numerical investigation on how to improve the efficiency of our DG implementation, with special attention to benefits for the p-multigrid approach. For example, the polynomial approximations based on nodal collocation presented in the following should be particularly effective if coupled with the p-multigrid technique. The notes present also new advancements and results of closer investigations on efficiency of the implicit, high-order DG method, which are expected to be important for the p-multigrid and that will serve for assessing its efficiency.
The current capability of the implicit DG approach will be demonstrated by presenting some recent results of high-order turbulent flow computations of fairly complex test cases proposed within the ADIGMA project
Assessment of a high-order discontinuous Galerkin method for incompressible three-dimensional Navier-Stokes equations: Benchmark results for the flow past a sphere up to Re=500
No full textThis paper deals with the implementation of the high-order Discontinuous Galerkin (DG) artificial compressibility flux method [1] into a three-dimensional incompressible Navier-Stokes (INS) solver. The method is fully implicit in time and its distinguishing feature is the formulation of the inviscid interface flux, which is based on the solution of the Riemann problem associated with a local artificial compressibility-like perturbation of the equations.The code has been tested on a wide range of flow regimes considering the flow past a sphere at moderate Reynolds numbers. In order to asses the code reliability and its accuracy in space, up to the sixth order polynomial approximation, and in time, up to the fourth order, both steady (Re=20, 200, 250) and unsteady (Re=300, 500) problems have been approached.With the largest Reynolds number here considered the flow exhibits a complex behavior, even if it still laminar, since undergoes the transition from a regular to an almost chaotic system. For this problem, for which the flow field characteristics are not completely well established and no many data are available in the present literature, detailed results are reported in this paper. © 2013 Elsevier Ltd
High order discontinuous Galerkin methods for the Navier-Stokes equations
No full textThis thesis is devoted to the achievement of numerical methods for the solution of the Navier-Stokes equations, both compressible and incompressible, for fuid dynamics applications. The main objective has been to establish a fair trade-off between the accuracy requirement and the computational cost, since although the higher resolution methodologies can proven to supply advantages as regards the quality of the solution provided, in fact they are often claimed to be excessively costly. This work can be divided into two parts. First we present the implementation of a spectral discontinuous Galerkin methods for the compressible Navier-Stokes set. The co-location strategy in gaussian nodes along with the tensorial computation of the approximating functions has been implemented, for an effective assembly of the discrete operators. This approach has also been joined with a p-multigrid method achieving an overall quite interesting performance enhancement. An extensive numerical validation is provided in order to investigate both the accuracy and the e ciency of the scheme. The second part is dedicated to the solution of the incompressible Navier-Stokes equations. A new semi-implicit algorithm has been devised through a procedure of minimization of the implicitness level necessary to achieve the solvability of the problem. Such an aim required to additively split the system to outline and distinguish the implicitly considered parts from those evaluated explicitly. This scheme clearly takes advantage in term of the memory requirement
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