Computing Shocked Flows with High-order Accurate Discontinuous Galerkin Methods

Burgess, Nicholas; Mavriplis, Dimitri J.

doi:10.2514/6.2012-2715

Cited by 12 publications

(6 citation statements)

References 20 publications

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“…In the proposed model, which is referred to as artificial viscosity PDE, the above mentioned discontinuity sensor has been used in the source term, and the diffusion term has been biased by the directional mesh size metrics which is particularly useful for anisotropic meshes. Burgess and Mavriplis [75] have compared the above mentioned methods and concluded that although the PDE-based method results in a more dissipated solution, it shows a more robust and consistent convergence behavior in an adjoint-based adaptive algorithm. In this study, a similar comparison was repeated which resulted to the same conclusion.…”

Section: Artificial Viscositymentioning

confidence: 97%

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An Adjoint-Based hp-Adaptive Petrov-Galerkin Method for Turbulent Flows

Ahrabi

Anderson

Newman

2015

22nd AIAA Computational Fluid Dynamics Conference

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In this study, an adjoint-based hp-adaptation algorithm has been developed within a Petrov-Galerkin finite-element method. The developed mesh adaptation algorithm is able to perform non-conformal mesh adaptations. To consistently account for hanging nodes, the constrained approximation method has been utilized. Hierarchical basis functions have been employed to facilitate the implementation of the constrained approximation. The methodology has been demonstrated on numerous cases using the Euler and Reynolds Average Navier-Stokes (RANS) equations, equipped with a modified SpalartAllmaras (SA) turbulence model. Also, a PDE-based artificial viscosity has been added to the governing equations, to stabilize the solution in the vicinity of shock waves. For accurate representation of the geometric surfaces, high-order curved boundary meshes have been generated and the interior meshes have been deformed through the solution of a modified linear elasticity equation. Fully implicit linearization has been used to advance the solution toward a steady-state. Dirichlet boundary conditions have been imposed weakly and the functional outputs have been modified according to the weak boundary conditions in order to provide a smooth adjoint solution near the boundaries.To accelerate the error reduction in presence of singularity points, an enhanced hrefinement, based on solution's smoothness, has been used. Numerical results illustrate consistent accuracy improvement of the functional outputs for both h-and hpadaptation, and also capability enhancement in capturing complex viscous effects such as shock-wave/turbulent boundary layer interaction.

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Section: Artificial Viscositymentioning

confidence: 97%

“…All the studies in Refs. [46,48,75] have used the DG scheme and to our knowledge, this is the first application of the PDE-based artificial viscosity method in the PG scheme. The following formulation has been taken from Ref.…”

Section: Artificial Viscositymentioning

confidence: 99%

An Adjoint-Based hp-Adaptive Petrov-Galerkin Method for Turbulent Flows

Ahrabi

Anderson

Newman

2015

22nd AIAA Computational Fluid Dynamics Conference

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“…Recent studies [4,5,6,7] also show active efforts to develop high-order methods for hypersonic heating computations. Whereas, even for the low-order finite volume methods, current numerical methods cannot always provide good heating predictions, particularly on non-shock-aligned meshes.…”

Section: Introductionmentioning

confidence: 99%

A unified construction of all-speed HLL-type schemes for hypersonic heating computations

Xie¹,

Tian²,

Zhang³

et al. 2021

Preprint

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In this paper, a unified framework to develop all-speed HLL-type schemes for hypersonic heating computations is constructed. Such a unified construction method combines two effective improving techniques: a shock robustness improvement and a low-Mach number fix. It is implemented by properly modifying the approximate solutions of the local Riemann problem in the HLL framework, resulting in two all-speed HLL-type schemes, namely ASHLLC and ASHLLEM solvers. Results from both numerical analysis and experiments demonstrate that the newly proposed schemes not only preserve desirable properties of their original versions, but are also able to provide accurate and robust solutions for complex flows ranging from low-Mach number incompressible to hypersonic compressible regimes. Thus, both the ASHLLC and ASHLLEM schemes can be used as reliable methods for hypersonic heating computations.

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“…Discontinuity capturing for high speed flows is achieved with use of slope limiters 5,6,[11][12][13][15][16][17] or artificial dissipation. [18][19][20][21] A widely used slope limiter for the DG method is the total variation diminishing (TVB) slope limiter of Cockburn and Shu 5,6,11 that is applicable to quadrilateral and triangular meshes. This limiter was recently extended for unstructured three dimensional elements.…”

Section: Introductionmentioning

confidence: 99%

A dissipative Filter for the Discontinuous Galerkin method

Panourgias

Ekaterinaris²

2015

53rd AIAA Aerospace Sciences Meeting

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Complex three dimensional flows in domains with nontrivial geometry require use of mixed type unstructured meshes. The discontinuous Galerkin (DG) method is an unstructured grid method and development of a unified approach for discontinuity capturing with the DG method when higher order expansions are employed is therefore required. A systematic procedure for incorporating a dissipative mechanism to DG discretizations in unstructured meshes is developed. This adaptive non-linear dissipative mechanism enables capturing discontinuities and steep flow gradients without numerical oscillations. A filter designed and extensively used in the development of low dissipative well-balanced high order accurate finite difference schemes is employed. The filtering operator is adapted to the finite element context of DG discretizations. The filter operator is constructed and applied in the canonical computational domain where it can be applied in a direction per direction basis and then extended for all element types in unstructured meshes using collapsed coordinate transformations. The performance of the proposed dissipative mechanism is demonstrated and evaluated for different orders of expansions for one-dimensional and multidimensional problems with exact solutions. It is further applied for a number of standard inviscid flow test problems including strong shocks interactions to demonstrate the potential of the method. The dissipative mechanism yields results comparable to three-dimensional TVB and hierarchical limiting. Furthermore, the proposed approach is suitable for h/p-adaptivity in order to locally enhance resolution for three dimensional flow simulations that include discontinuities and complex flow features.

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Computing Shocked Flows with High-order Accurate Discontinuous Galerkin Methods

Cited by 12 publications

References 20 publications

An Adjoint-Based hp-Adaptive Petrov-Galerkin Method for Turbulent Flows

An Adjoint-Based hp-Adaptive Petrov-Galerkin Method for Turbulent Flows

A unified construction of all-speed HLL-type schemes for hypersonic heating computations

A dissipative Filter for the Discontinuous Galerkin method

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