Shock Capturing with PDE-Based Artificial Viscosity for an Adaptive, Higher-Order Discontinuous Galerkin Finite Element Method
The accurate simulation of supersonic and hypersonic flows is well suited to higher-order (p > 1), adaptive computational fluid dynamics (CFD). Since these cases involve flow velocities greater than the speed of sound, an appropriate shock capturing for higher-order, adaptive methods is necessary...
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| Format: | Technical Report |
| Language: | en_US |
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Aerospace Computational Design Laboratory, Dept. of Aeronautics & Astronautics, Massachusetts Institute of Technology
2010
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| Online Access: | http://hdl.handle.net/1721.1/57606 |
| _version_ | 1811097919884886016 |
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| author | Barter, Garrett Ehud |
| author_facet | Barter, Garrett Ehud |
| author_sort | Barter, Garrett Ehud |
| collection | MIT |
| description | The accurate simulation of supersonic and hypersonic flows is well suited to higher-order (p > 1), adaptive computational fluid dynamics (CFD). Since these cases involve flow velocities greater than the speed of sound, an appropriate shock capturing for higher-order, adaptive methods is necessary.
Artificial viscosity can be combined with a higher-order discontinuous Galerkin finite element discretization to resolve a shock layer within a single cell. However, when a non-smooth artificial viscosity model is employed with an otherwise higher-order approximation, element-to-element variations induce oscillations in state gradients and pollute the downstream flow. To alleviate these difficulties, this work proposes a new, higher-order, state-based artificial viscosity with an associated governing partial differential equation (PDE).In the governing PDE, the shock sensor acts as a forcing term, driving the artificial viscosity to a non-zero value where it is necessary. The decay rate of the higher-order solution modes and edge-based jumps are both shown to be reliable shock indicators. This new approach leads to a smooth, higher-order representation of the artificial viscosity that evolves in time with the solution. For applications involving the Navier-Stokes equations, an artificial dissipation operator that preserves total enthalpy is introduced. The combination of higher-order, PDE-based artificial viscosity and enthalpy-preserving dissipation operator is shown to overcome the disadvantages of the non-smooth artificial viscosity.
The PDE-based artificial viscosity can be used in conjunction with an automated grid adaptation framework that minimizes the error of an output functional. Higher-order solutions are shown to reach strict engineering tolerances with fewer degrees of freedom. The benefit in computational efficiency for higher-order solutions is less dramatic in the vicinity of the shock where errors scale with O(h/p). This includes the near-field pressure signals necessary for sonic boom prediction. When applied to heat transfer prediction on unstructured meshes in hypersonic flows, the PDE-based artificial viscosity is less susceptible to errors introduced by poor shock-grid alignment. Surface heating can also drive the output-based grid adaptation framework to arrive at the same heat transfer distribution as a well-designed structured mesh. |
| first_indexed | 2024-09-23T17:07:04Z |
| format | Technical Report |
| id | mit-1721.1/57606 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T17:07:04Z |
| publishDate | 2010 |
| publisher | Aerospace Computational Design Laboratory, Dept. of Aeronautics & Astronautics, Massachusetts Institute of Technology |
| record_format | dspace |
| spelling | mit-1721.1/576062019-04-13T00:05:32Z Shock Capturing with PDE-Based Artificial Viscosity for an Adaptive, Higher-Order Discontinuous Galerkin Finite Element Method Barter, Garrett Ehud The accurate simulation of supersonic and hypersonic flows is well suited to higher-order (p > 1), adaptive computational fluid dynamics (CFD). Since these cases involve flow velocities greater than the speed of sound, an appropriate shock capturing for higher-order, adaptive methods is necessary. Artificial viscosity can be combined with a higher-order discontinuous Galerkin finite element discretization to resolve a shock layer within a single cell. However, when a non-smooth artificial viscosity model is employed with an otherwise higher-order approximation, element-to-element variations induce oscillations in state gradients and pollute the downstream flow. To alleviate these difficulties, this work proposes a new, higher-order, state-based artificial viscosity with an associated governing partial differential equation (PDE).In the governing PDE, the shock sensor acts as a forcing term, driving the artificial viscosity to a non-zero value where it is necessary. The decay rate of the higher-order solution modes and edge-based jumps are both shown to be reliable shock indicators. This new approach leads to a smooth, higher-order representation of the artificial viscosity that evolves in time with the solution. For applications involving the Navier-Stokes equations, an artificial dissipation operator that preserves total enthalpy is introduced. The combination of higher-order, PDE-based artificial viscosity and enthalpy-preserving dissipation operator is shown to overcome the disadvantages of the non-smooth artificial viscosity. The PDE-based artificial viscosity can be used in conjunction with an automated grid adaptation framework that minimizes the error of an output functional. Higher-order solutions are shown to reach strict engineering tolerances with fewer degrees of freedom. The benefit in computational efficiency for higher-order solutions is less dramatic in the vicinity of the shock where errors scale with O(h/p). This includes the near-field pressure signals necessary for sonic boom prediction. When applied to heat transfer prediction on unstructured meshes in hypersonic flows, the PDE-based artificial viscosity is less susceptible to errors introduced by poor shock-grid alignment. Surface heating can also drive the output-based grid adaptation framework to arrive at the same heat transfer distribution as a well-designed structured mesh. This work was made possible by the following sources, • National Defense Science and Engineering Graduate Fellowship • U. S. Air Force Research Laboratory (USAF-3306-03-SC-0001) • The Boeing Company • National Aeronautics and Space Administration (Co-op Agreement NNX07AC70A) 2010-08-27T20:04:52Z 2010-08-27T20:04:52Z 2008-06 Technical Report http://hdl.handle.net/1721.1/57606 en_US ACDL Technical Reports;ACDL TR-08-2 application/pdf Aerospace Computational Design Laboratory, Dept. of Aeronautics & Astronautics, Massachusetts Institute of Technology |
| spellingShingle | Barter, Garrett Ehud Shock Capturing with PDE-Based Artificial Viscosity for an Adaptive, Higher-Order Discontinuous Galerkin Finite Element Method |
| title | Shock Capturing with PDE-Based Artificial Viscosity for an Adaptive, Higher-Order Discontinuous Galerkin Finite Element Method |
| title_full | Shock Capturing with PDE-Based Artificial Viscosity for an Adaptive, Higher-Order Discontinuous Galerkin Finite Element Method |
| title_fullStr | Shock Capturing with PDE-Based Artificial Viscosity for an Adaptive, Higher-Order Discontinuous Galerkin Finite Element Method |
| title_full_unstemmed | Shock Capturing with PDE-Based Artificial Viscosity for an Adaptive, Higher-Order Discontinuous Galerkin Finite Element Method |
| title_short | Shock Capturing with PDE-Based Artificial Viscosity for an Adaptive, Higher-Order Discontinuous Galerkin Finite Element Method |
| title_sort | shock capturing with pde based artificial viscosity for an adaptive higher order discontinuous galerkin finite element method |
| url | http://hdl.handle.net/1721.1/57606 |
| work_keys_str_mv | AT bartergarrettehud shockcapturingwithpdebasedartificialviscosityforanadaptivehigherorderdiscontinuousgalerkinfiniteelementmethod |