Quantifying Uncertainties from the Grid in CFD Solutions

By Tom I-P. Shih

Aeronautics and Astronautics, Purdue University, West Lafayette, IN

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Abstract

For most users of CFD, the mesh and the time-step 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 turn-around 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 h-refinement). Unfortunately, this non-uniform distribution could create what are referred to as poor-quality cells, which can induce considerable errors in the computed solutions. In addition, it is generally not feasible to do a grid-independent study so that there could be errors from poor quality cells and from inadequate resolution.

This talk begins with a study on grid-quality measures that assume grid-induced 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 grid-quality 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 discrete-error-transport equation (DETE) is presented, which recognizes that grid-induced 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 grid-induced 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 grid-induced 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 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 (2003-09). He has also held faculty positions at Michigan State University (1998-2003), Carnegie Mellon University (1988-98), and the University of Florida (1983-88) and was a mechanical engineer at NASA – Lewis (now Glenn) Research Center (1981-82). 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 non-reacting, compressible and incompressible flows. Algorithms and codes have also been developed for automatic/knowledge-based 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 shock-wave/boundary-layer 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 co-authored more than 200 technical papers in journals and conferences; presented over 160 invited seminars, lectures, and workshops; and served as advisor and co-advisor to 21 PhD and 45 MS students. Professor Shih is a Fellow of ASME and AIAA.

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Cite this work

Researchers should cite this work as follows:

  • Tom I-P. Shih (2012), "Quantifying Uncertainties from the Grid in CFD Solutions," http://nanohub.org/resources/12548.

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Time

Location

Birck Nanotechnology Building, Room 1001, Purdue University, West Lafayette, IN

Tags

Quantifying Uncertainties from the Grid in CFD Solutions
  • Quantifying Uncertainties from the Grid in CFD Solutions 1. Quantifying Uncertainties from… 0
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  • CFD has come a long way! 2. CFD has come a long way! 110.33333333333333
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  • CFD 3. CFD 156.93333333333334
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  • CFD 4. CFD 175.26666666666668
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  • Why? 5. Why? 216.16666666666666
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  • Why? 6. Why? 332.76666666666665
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  • Sources of Errors in a CFD Solution 7. Sources of Errors in a CFD Sol… 561.4666666666667
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  • Objective 8. Objective 652.8
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  • Outline of Talk 9. Outline of Talk 695.66666666666663
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  • E = f ( F, D, F ), f = algebraic 10. E = f ( F, D, F ), f = algebra… 713.43333333333328
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  • New Grid-Quality Measures (Gu & Shih, 2001) 11. New Grid-Quality Measures (Gu … 836.5
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  • New Grid-Quality Measures (Gu & Shih, 2001) 12. New Grid-Quality Measures (Gu … 891.43333333333328
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  • New Grid-Quality Measures (Gu & Shih, 2001) 13. New Grid-Quality Measures (Gu … 921.9666666666667
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  • New Grid-Quality Measures (Gu & Shih, 2001) 14. New Grid-Quality Measures (Gu … 950.86666666666667
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  • Construction of E = f (F), f = algebraic (data mining) 15. Construction of E = f (F), f =… 973.43333333333328
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  • Models 16. Models 983.23333333333335
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  • error = f( grid-quality measures ) f = algebraic  local 17. error = f( grid-quality measur… 1089.3
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  • Results 18. Results 1129.6333333333334
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  • Outline of Talk 19. Outline of Talk 1144.7
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  • Generic Engine - 4 valves - pan cake combustion chamber 20. Generic Engine - 4 valves - pa… 1158.2666666666667
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  • Generic Engine Summary of Simulations 21. Generic Engine Summary of Simu… 1184.7
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  • Generic Engine, 50K, CA = 100, Y-Plane 22. Generic Engine, 50K, CA = 100,… 1207.6666666666667
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  • Generic Engine, 50K, CA = 100, Y-Plane 23. Generic Engine, 50K, CA = 100,… 1288.5333333333333
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  • Exhaust Manifold 24. Exhaust Manifold 1293.7666666666667
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  • Exhaust Manifold: Hex Mesh Evaluation 25. Exhaust Manifold: Hex Mesh Eva… 1307.8666666666666
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  • Exhaust Manifold: Tet Mesh Evaluation 26. Exhaust Manifold: Tet Mesh Eva… 1350.6333333333334
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  • Exhaust Manifold: 1-Layer-Prism Mesh Evaluation 27. Exhaust Manifold: 1-Layer-Pris… 1352.7
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  • Exhaust Manifold: 2-Layer-Prism Mesh Evaluation 28. Exhaust Manifold: 2-Layer-Pris… 1353.6666666666667
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  • Intake Manifold 29. Intake Manifold 1354.5666666666666
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  • Intake Manifold : Mesh Evaluation 30. Intake Manifold : Mesh Evaluat… 1372.4333333333334
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  • Outline of Talk 31. Outline of Talk 1375.4
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  • Is error = f(local grid & solution) ??? 32. Is error = f(local grid & solu… 1377.7333333333334
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  • Now, we perturb the GRID, making it coarser & poor quality at one location. 33. Now, we perturb the GRID, maki… 1401.6333333333334
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  • Error in Solution on Perturbed Grid 34. Error in Solution on Perturbed… 1415.5333333333333
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  • When transport equation for error is needed 35. When transport equation for er… 1443.5333333333333
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  • Zhang, et al. (2000, 2001) followed the finite-element approach in deriving the error-transport equation 36. Zhang, et al. (2000, 2001) fol… 1482.4666666666667
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  • So, what can we do? 37. So, what can we do? 1542.5666666666666
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  • A different approach that disregards the original PDE in deriving a discrete error transport equation (DETE) 38. A different approach that disr… 1624.0666666666666
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  • Outline of Talk 39. Outline of Talk 1658.5
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  • Derivation of DETE 40. Derivation of DETE 1668.4
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  • Differential vs. FD/FV Operators 41. Differential vs. FD/FV Operato… 1683.9
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  • Example Problem 42. Example Problem 1752.0333333333333
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  • Example Problem 43. Example Problem 1811.9666666666667
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  • Example Problem 44. Example Problem 1861.0333333333333
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  • THUS, the error-transport eq should be defined on the coarse mesh 45. THUS, the error-transport eq s… 1880.3666666666666
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  • Outline of Talk 46. Outline of Talk 1951.6
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  • Test Problem 1: Advection-Diffusion Equation (linear & steady) 47. Test Problem 1: Advection-Diff… 1954.7333333333334
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  • Modeling the Residual 48. Modeling the Residual 1967.8
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  • Predict versus Actual Residual 49. Predict versus Actual Residual 1994.8666666666666
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  • Predicted versus Actual Error 50. Predicted versus Actual Error 2026.6666666666667
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  • Test Problem 2: Wave Equation (linear & unsteady) 51. Test Problem 2: Wave Equation … 2066.1
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  • Test Problem 3: Inviscid Burger Equation (quasi-linear & unsteady) 52. Test Problem 3: Inviscid Burge… 2111.9333333333334
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  • Actual Residual for Inviscid Burger Equation 53. Actual Residual for Inviscid B… 2143.2666666666669
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  • Smooth Solutions 54. Smooth Solutions 2183.3666666666668
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  • Weak Solutions 55. Weak Solutions 2220.8
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  • Outline of Talk 56. Outline of Talk 2223.7333333333331
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  • Application to Euler and Navier-Stokes Eqs 57. Application to Euler and Navie… 2225.8333333333335
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  • Application to Euler and Navier-Stokes Eqs 58. Application to Euler and Navie… 2267.1666666666665
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  • Test Problem 59. Test Problem 2329.6
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  • y+ value of first grid points along iced airfoil 60. y+ value of first grid points … 2336.3666666666668
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  • Previous Work: Single-Block Grids 61. Previous Work: Single-Block Gr… 2354.1
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  • New Blocking Concept for Single-Block Grid 62. New Blocking Concept for Singl… 2406.2333333333331
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  • New Blocking Concept for Single-Block Grid 63. New Blocking Concept for Singl… 2432.8666666666668
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  • Single-Block Grid 64. Single-Block Grid 2436.2666666666669
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  • Error in x-Momentum 65. Error in x-Momentum 2446.8666666666668
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  • Error in y-Momentum 66. Error in y-Momentum 2467.3
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  • Exact Residual for Continuity Eq. 67. Exact Residual for Continuity … 2470.5666666666666
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  • Outline of Talk 68. Outline of Talk 2489.1333333333332
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  • Challenge in DETE: How to model the Residual? 69. Challenge in DETE: How to mode… 2491.5333333333333
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  • Test Problems 70. Test Problems 2551.0666666666666
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  • AME Model 71. AME Model 2572.1333333333332
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  • AME Model 72. AME Model 2616.4333333333334
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  • AME Model 73. AME Model 2619.5666666666666
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  • Multiple-Grid Approach 74. Multiple-Grid Approach 2622.8333333333335
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  • Residual Extrapolation Model 75. Residual Extrapolation Model 2650.2333333333331
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  • Residual Extrapolation Model 76. Residual Extrapolation Model 2667.6666666666665
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  • Residual Extrapolation Model 77. Residual Extrapolation Model 2679.2333333333331
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  • Residual Extrapolation Model 78. Residual Extrapolation Model 2681.4333333333334
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  • Summary 79. Summary 2731.4
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  • Summary 80. Summary 2785.3666666666668
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  • Acknowledgement 81. Acknowledgement 2815.0333333333333
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  • Questions? Comments? 82. Questions? Comments? 2821.6333333333332
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