Mesh Sensitivity & Mesh Independence Study PDF

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CFD Open Series Patch 2.20 Mesh Sensitivity & Mesh Independence Study Edited by : Ideen Sadrehaghighi, Ph.D. Y-Coordinate Sensitivity WRT Max. X-Coordinate Sensitivily WRT Max. Thichness Thickness Y-Coordinate Sensitivity WRT Max. X-Coordinate Sensitivity WRT Max. Camber Camber ANNAPOLIS, MD 1 C...

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Aircraft Wing Shape Opt imizat ion Ideen Sadrehaghighi Numerical sensit ivit y analysis for aerodynamic opt imizat ion: A survey of approaches Jacques Pet er High-fidelit y aerodynamic shape opt imizat ion of wind t urbine blades Trist an Dhert

CFD Open Series Patch 2.20

Mesh Sensitivity & Mesh Independence Study Edited by :

Ideen Sadrehaghighi,

Ph.D.

Y-Coordinate Sensitivity WRT Max. Thichness

Y-Coordinate Sensitivity WRT Max. Camber

X-Coordinate Sensitivily WRT Max. Thickness

X-Coordinate Sensitivity WRT Max. Camber

ANNAPOLIS, MD

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Contents 1

Mesh Sensitivity ........................................................................................................................... 4 1.1 Different Types of Mesh Sensitivity................................................................................................................ 4 1.1.1 Symbolic Differentiation ...................................................................................................... 4 1.1.2 Automatic Differentiation ................................................................................................... 4 1.1.2.1 Symbolic vs. Automatic Differentiation .......................................................................... 4 1.1.3 Finite Differencing ............................................................................................................... 5 1.2 Mesh Sensitivity via Direct Differentiation (DD) ...................................................................................... 5 1.2.1 Surface Modeling Using NURBS........................................................................................... 5 1.3 Adjoint Variable Sensitivity Analysis (AV) .................................................................................................. 7 1.4 Discrete Semi-Analytical Approach in Sensitivity Analysis with Unstructured Free Finite Element Meshes .................................................................................................................................................................. 9

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Case Studies Involving Mesh Sensitivity ........................................................................... 11 2.1 Case Study 1 - 2D Study of Airfoil Grid Sensitivity via Direct Differentiation (DD)................. 11 2.2 Case Study 2 - Using An Adjoint Approach to Eliminate Mesh Sensitivities in Computational Design ................................................................................................................................................................................... 13 2.2.1 Introduction ....................................................................................................................... 13 2.2.2 Mesh Sensitivities via Forward Mode Differentiation ....................................................... 14 2.2.2.1 Adjoint Approach for Eliminating Mesh Sensitivities ................................................... 15 2.2.3 Implementation ................................................................................................................. 16 2.2.4 Consistency of Linearization .............................................................................................. 16 2.2.4.1 Test Case - Slotted Wing–Body Configuration.............................................................. 18 2.2.5 Summary and Conclusions................................................................................................. 19 2.2.6 References ......................................................................................................................... 19 2.3 Case Study 3 - Mesh Sensitivity Through an Algorithm to Detect High - Γ Regions for Unstructured Mesh in 2D ............................................................................................................................................. 22 2.3.1 Algorithm to Compute Γ .................................................................................................... 22 2.3.2 Results ............................................................................................................................... 23 2.3.3 Concluding Remarks .......................................................................................................... 23 2.3.4 References ......................................................................................................................... 23 2.4 Case Study 4 - Grid Resolution Study of a Drag Prediction Workshop Configuration Using the NSU3D Unstructured Mesh Solver.................................................................................................................... 26 2.4.1 Introduction ....................................................................................................................... 26 2.4.2 DLR-F6 Configuration......................................................................................................... 27 2.4.3 NSU3D Solver ..................................................................................................................... 27 2.4.4 Computational Grids.......................................................................................................... 29 2.4.5 Overall Strategy ................................................................................................................. 31 2.4.6 Results ............................................................................................................................... 32 2.4.6.1 Distance Function Sensitivity ........................................................................................ 32 2.4.6.2 Full Navier-Stokes Terms .............................................................................................. 33 2.4.6.3 Grid Convergence Study and Sensitivity to Dissipation Levels ..................................... 36 2.4.7 Conclusions ........................................................................................................................ 41 2.4.8 References ......................................................................................................................... 42

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Mesh Independence Study ..................................................................................................... 44 3.1 Preliminaries......................................................................................................................................................... 44 3.2 Case Study 1 - Simulation of Heat Transfer with Heat Source ......................................................... 45 3.3 Case Study 2 - Fluid Dynamics (CFD) Mesh Independency Study of a Straight Blade Horizontal Axis Tidal Turbine .................................................................................................................................... 47 3.3.1 Introduction ....................................................................................................................... 47 3.3.2 Methodology ..................................................................................................................... 48 3.3.2.1 Non-Dimensional Forces Acting on Tidal Turbine Blades ............................................. 48 3.3.2.2 Turbine Model .............................................................................................................. 48 3.3.2.3 Standard k-epsilon (k-ε) Turbulence Model ................................................................. 49 3.3.2.4 Shear Stress Transport (SST) Turbulence Model .......................................................... 49 3.3.3 Mesh Independency Study ................................................................................................ 49 3.3.3.1 Turbulence Model Comparison Study .......................................................................... 51 3.3.4 Conclusions ........................................................................................................................ 52 3.3.5 References ......................................................................................................................... 52

List of Tables Table 1.3.1 Pros & Cons of Different Grid Sensitivity Method (NDV = Number of Design Variable) ......................................................................................................................................................................................................... 8 Table 2.2.2 Comparison of sensitivity derivatives for lift and drag coefficients using various approaches ............................................................................................................................................................................... 17 Table 2.2.1 Schemes Used To Obtain Sensitivities .............................................................................................. 17 Table 2.5.1 Characteristics of Meshes used in Grid Refinement Study ....................................................... 31 Table 2.5.2 Computed Lift and Drag Coefficients at M = 0.75, α = 0 on various grids using three different approaches for computing the distance function.................................................................................. 33 Table 2.5.3 Computed Lift and Drag Coefficients at M=0.75, α=00, on various grids using the multidimensional thin-layer discretization and the extended stencil full Navier-Stokes discretization........................................................................................................................................................................... 33 Table 2.5.4 Computed Lift and Drag Coefficients at M=0.75, CL=0.5, on various fine grids ............... 39 Table 3.3.1 Mesh size, CFD simulation time, and estimated CP for SST model at λ = 5 ...................... 50 Table 3.3.2 Mesh size, CFD simulation time, and estimated CP for k-ε model at λ = 5 .......................... 50

List of Figures

Figure 1.2.1 Six Control Point Representation of a Generic Airfoil ................................................................ 6 Figure 1.2.2 B-Spline Approximation of NACA0012 (left) and RAE2822 (right) Airfoils ..................... 6 Figure 1.2.3 Free Form Deformation (FFD) for Volume Grid with Control Points (Courtesy of Kenway et al.) ............................................................................................................................................................................ 7 Figure 2.1.1 Sample Grid and Grid Sensitivity....................................................................................................... 11 Figure 2.2.1 ONERA M6 grid used for evaluating ................................................................................................ 17 Figure 2.2.2 Convergence rates for direct and adjoint modes. ....................................................................... 18 Figure 2.2.3 Surface grid for slotted wing–body .................................................................................................. 18 Figure 2.3.1 Results for irregular but non-curved triangular grids ( Γ = 0 everywhere ) ................... 22 Figure 2.3.2 Results for an irregular triangular grid over a Joukowsky airfoil. Note that the contours are plotted in (c) and (d) with a restricted range, [0 ; 25], to visualize the variation near the airfoil. The maximum value is 902,788 and the average is 10,505. .......................................................... 24 Figure 2.3.3 Results for an irregular triangular grid over a half-cylinder domain with straight boundaries................................................................................................................................................................................ 25 Figure 2.5.1 Multigrid Convergence Rate using 5 grid levels for NSU3D Solution of Viscous Turbulent .................................................................................................................................................................................. 28

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Figure 2.5.2 (a): Illustration of medium resolution (3.0 million point) mesh about DLR-F6 wingbody configuration and (b) closeup showing resolution in wing root trailing edge fuselage junction ....................................................................................................................................................................................................... 29 Figure 2.5.3 Comparison between surface resolution for 72 million point grid (a) and 65 million point grid (b) in wing root fuselage junction region............................................................................................... 30 Figure 2.5.4 Comparison between surface resolution for 72 million point grid (a) and 65 million point grid (b) in trailing edge region in the vicinity of 60% span region illustrating spanwise stretching for the 72 million point grid ........................................................................................................................ 31 Figure 2.5.5 Comparison of computed lift (a) and drag (b) coefficients versus the number of grid points to the -2/3 power for transonic (M = 0.75) and subsonic (M = 0.3) cases at 0o incidence ....... 34 Figure 2.5.6 Comparison Of Computed Pressure Drag (A) And Friction Drag (B) Coefficients Versus The Number Of Grid Points To The -2/3 Power For Transonic (M=0.75) And Subsonic (M=0.3) Cases At 0o Incidence.......................................................................................................................................... 35 Figure 2.5.7 Comparison of computed lift (a) and drag (b) coefficients versus the number of grid points to the -2/3 power for transonic (M=0.75) and subsonic (M=0.3) cases at 0o incidence including results computed on the 65 million point grid. .................................................................................... 37 Figure 2.5.8 (a): Variation of computed pressure and viscous drag coefficients on different grids including 65 million point grid for Mach=0.75, 0o incidence case. (b): Comparison of lift coefficient versus incidence computed on workshop grids and 65 million point grid versus experimental values. ......................................................................................................................................................................................... 38 Figure 2.5.9 Comparison of computed drag polar and moment coefficients .......................................... 40 Figure 2.5.10 Comparison of computed drag polar using workshop grids and 65 M point grid versus experimental values............................................................................................................................................... 41 Figure 3.1.1 Effects of Mesh Density on Solution Domain ................................................................................ 44 Figure 3.1.2 Mesh Independence ................................................................................................................................ 45 Figure 3.1.3 Temperature field of heat transfer with heat source; Boundaries are set to fixed temperature ............................................................................................................................................................................. 45 Figure 3.2.1 Center Temperature vs. Number of Grid Cells ............................................................................. 46 Figure 3.3.1 Hydrodynamic Forces Acting on The Airfoil ................................................................................ 48 Figure 3.3.2 (a) 3D model of the straight blade HATT; (b) Non-linear twist distribution .................. 49 Figure 3.3.3 The Power Coefficients of All The Investigated Meshes in Mesh Independency Study ....................................................................................................................................................................................................... 51 Figure 3.3.4 Torque coefficient versus Tip speed ratio for k-ε and SST model medium meshes .... 51

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Mesh Sensitivity

After reviewing relevant literature, it is apparent that one aspect of aerodynamic sensitivity analysis, namely mesh sensitivity, has not been investigated extensively. The mesh sensitivity algorithms in most of these studies are based on structural design models. Such models, although sufficient for preliminary or conceptional design, are not acceptable for detailed design analysis. Careless mesh sensitivity evaluations, would introduce gradient errors within the sensitivity module, therefore, infecting the overall optimization process. Development of an efficient and reliable mesh sensitivity module with special emphasis on aerodynamic applications appear essential. Although mesh sensitivity, in our opinion, could be grouped as one of mesh quality criteria, it is still debatable a, so we leave it as such. But more importantly what is the difference between mesh sensitivity and mesh independence? Very little. Although some argue that grid sensitivity is a real (measured quantity) while mesh independence is merely a mist and cannot be truly achieved. So it dependence who is your audience. Here we homage first mesh sensitivity then mesh independence study.

1.1

Different Types of Mesh Sensitivity

Several methods concerning the derivation of mesh sensitivity equations are currently available. Among the most frequently mentioned are : • • • • • •

Direct (Analytical) Differentiation (DD), Adjoint Variable (AV), Symbolic Differentiation (SD), Automatic Differentiation (AD), ( e.g. Odyssée or ADIFOR) Finite Difference (FD), (Brute Force) Discrete Semi-Analytical Approach

Each technique has its own unique characteristics. For example, the Direct Differentiation, used here, has the advantage of being exact, due to direct differentiation of governing equations with respect to design parameters, but limited in scope. By far, the most used sensitivity analysis, is Adjoint Variables techniques, especially for aerodynamic optimization. Due to apparent popularity, we consider these in more details. 1.1.1 Symbolic Differentiation Manipulates mathematical expressions in the code. If you ever used Matlab or Mathematica, then you probably used it. For every math expression they know the derivative and use various rules (product rule, chain rule) to calculate the resulting derivative. Then they simplify the end expression to obtain the resulting expression. 1.1.2 Automatic Differentiation Manipulates blocks of computer each element of a program (when you define any operation in code, you need to register a gradient for this operation). It also uses chain rule to break complex expressions into simpler ones. 1.1.2.1 Symbolic vs. Automatic Differentiation You might think that Automatic differentiation is the same as Symbolic differentiation (in one place they operate on math expression, in another on computer programs). And yes, they are sometimes very similar. But for control flow statements (`if, while, loops) the results can be very different: symbolic differentiation leads to inefficient code (unless carefully done) and faces the difficulty of converting a computer program into a single expression programs1. 1

Stackoverflow blog.

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1.1.3 Finite Differencing This method is easy implement, but cost insensitive. There are a series of issues with this approach. Accuracy is the main drawback of the method especially for non-linear problems such as those of aerodynamic nature. Cost is also somethin...