Quantifying Uncertainties from the Grid in CFD Solutions
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Abstract
For most users of CFD, the mesh and the timestep size are the only parts of the solution procedure in which the user has full control. The mesh used must represent the geometry and enable the algebraic analog of the governing PDEs to resolve the relevant flow physics. For realistic engineering problems, the number of grid points or cells that can be used is restricted by either the available computer resource or a need to have a practical turnaround time in generating a solution. With such a constraint, accuracy demands grid points to be placed in regions where they are most needed to resolve the geometry and flow physics (e.g., by r or hrefinement). Unfortunately, this nonuniform distribution could create what are referred to as poorquality cells, which can induce considerable errors in the computed solutions. In addition, it is generally not feasible to do a gridindependent study so that there could be errors from poor quality cells and from inadequate resolution.
This talk begins with a study on gridquality measures that assume gridinduced errors in a CFD solution at a cell is a function of the cell size and shape, the grid distribution around that cell, and the solution computed in the neighborhood of that cell. Several gridquality measures will be presented that account for the vectorial and the tensorial nature of fluid flow, which differ from the second derivatives of pressure or velocity that are commonly used. These measures are evaluated by applying them to flows in an IC engine combustion chamber, an intake manifold, and an exhaust manifold. Next, the basis of the discreteerrortransport equation (DETE) is presented, which recognizes that gridinduced errors at a cell may have nothing to do with the cell or in the neighborhood of that cell because that error may have been generated elsewhere and then transported there. The usefulness of DETE in estimating gridinduced errors is demonstrated by applying it to three PDEs with known exact solutions: the linear advection equation, the linear wave equation, and the inviscid Burger equation with a discontinuity. This is followed by a study on methods for modelling the residual in the DETE. These methods include those that are based on a single grid and those that involve generating CFD solutions on two or more successively refined grids. The usefulness of these models are evaluated by applying them to estimate gridinduced errors in CFD solutions of the following problems: steady flow past a circular cylinder, steady transonic flow about an airfoil, unsteady flow of a translating vortex, and vortex shedding behind a circular cylinder.
Bio
Tom Shih is professor and head of Purdue’s School of Aeronautics and Astronautics. Previously, he was professor and chair of the Department of Aerospace Engineering at Iowa State University (200309). He has also held faculty positions at Michigan State University (19982003), Carnegie Mellon University (198898), and the University of Florida (198388) and was a mechanical engineer at NASA – Lewis (now Glenn) Research Center (198182). He started his undergraduate education at West Virginia University but completed his B.S. degree at the National Cheng Kung University in Taiwan in 1976. He received his M.S.E. and Ph.D. degrees from The University of Michigan at Ann Arbor in 1977 and 1981, respectively. Professor Shih’s research centers on computational fluid dynamics (CFD) – both in developing and improving it as a tool and in using it to study physical problems. He and his students have developed a number of algorithms and codes for studying reacting and nonreacting, compressible and incompressible flows. Algorithms and codes have also been developed for automatic/knowledgebased grid generation and estimating errors in CFD solutions. In using CFD, Shih and his students have studied a wide range of problems in energy, power, and propulsion systems, including piston and Wankel rotary engines, automotive torque converters, control of shockwave/boundarylayer interactions by bleed for supersonic aircraft, aerodynamics of iced airfoils and wings, and internal and film cooling of gas turbine components. In these endeavors, Shih has authored and coauthored more than 200 technical papers in journals and conferences; presented over 160 invited seminars, lectures, and workshops; and served as advisor and coadvisor to 21 PhD and 45 MS students. Professor Shih is a Fellow of ASME and AIAA.
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Birck Nanotechnology Building, Room 1001, Purdue University, West Lafayette, IN
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1. Quantifying Uncertainties from…
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2. CFD has come a long way!
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3. CFD
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4. CFD
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5. Why?
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6. Why?
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7. Sources of Errors in a CFD Sol…
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8. Objective
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9. Outline of Talk
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10. E = f ( F, D, F ), f = algebra…
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11. New GridQuality Measures (Gu …
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12. New GridQuality Measures (Gu …
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13. New GridQuality Measures (Gu …
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14. New GridQuality Measures (Gu …
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15. Construction of E = f (F), f =…
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16. Models
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17. error = f( gridquality measur…
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18. Results
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19. Outline of Talk
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20. Generic Engine  4 valves  pa…
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21. Generic Engine Summary of Simu…
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22. Generic Engine, 50K, CA = 100,…
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23. Generic Engine, 50K, CA = 100,…
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24. Exhaust Manifold
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25. Exhaust Manifold: Hex Mesh Eva…
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26. Exhaust Manifold: Tet Mesh Eva…
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27. Exhaust Manifold: 1LayerPris…
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28. Exhaust Manifold: 2LayerPris…
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29. Intake Manifold
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30. Intake Manifold : Mesh Evaluat…
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31. Outline of Talk
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32. Is error = f(local grid & solu…
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33. Now, we perturb the GRID, maki…
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34. Error in Solution on Perturbed…
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35. When transport equation for er…
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36. Zhang, et al. (2000, 2001) fol…
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37. So, what can we do?
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38. A different approach that disr…
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39. Outline of Talk
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40. Derivation of DETE
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41. Differential vs. FD/FV Operato…
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42. Example Problem
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43. Example Problem
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44. Example Problem
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45. THUS, the errortransport eq s…
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46. Outline of Talk
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47. Test Problem 1: AdvectionDiff…
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48. Modeling the Residual
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49. Predict versus Actual Residual
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50. Predicted versus Actual Error
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51. Test Problem 2: Wave Equation …
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52. Test Problem 3: Inviscid Burge…
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53. Actual Residual for Inviscid B…
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54. Smooth Solutions
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55. Weak Solutions
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56. Outline of Talk
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57. Application to Euler and Navie…
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58. Application to Euler and Navie…
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59. Test Problem
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60. y+ value of first grid points …
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61. Previous Work: SingleBlock Gr…
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62. New Blocking Concept for Singl…
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63. New Blocking Concept for Singl…
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64. SingleBlock Grid
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65. Error in xMomentum
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66. Error in yMomentum
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67. Exact Residual for Continuity …
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68. Outline of Talk
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69. Challenge in DETE: How to mode…
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70. Test Problems
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71. AME Model
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72. AME Model
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73. AME Model
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74. MultipleGrid Approach
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75. Residual Extrapolation Model
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76. Residual Extrapolation Model
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77. Residual Extrapolation Model
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78. Residual Extrapolation Model
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79. Summary
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80. Summary
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81. Acknowledgement
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82. Questions? Comments?
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