1.1 Brief Overview of the Next Chapters . . . . 3

Contents

1 Introduction 1

1.1 Brief Overview of the Next Chapters . . . . 3

2 Turbulence: an Overview, and Tools for Its Study 7 2.1 Etymology of a Word . . . . 7

2.2 Turbulence in the scientific research . . . . 9

2.2.1 Hot Wire Anemometry . . . 10

2.2.2 Energy Cascade Concept and Local Isotropy . . . 11

2.3 Navier-Stokes and Turbulence . . . 18

2.3.1 Interesting snippets in the story of Numerical Methods . . . 19

2.3.2 Direct Numerical Resolution of Navier-Stokes Equations . . . . 21

2.3.3 Reynolds Averaged Navier Stokes Equations . . . 22

2.4 Large Eddy Simulation . . . 26

2.4.1 Resolved Turbulent Kinetic Energy Budget . . . 33

2.5 Sub Grid Stress Modeling . . . 35

2.5.1 Fully Dissipative Sub Grid Stress Models . . . 35

2.5.1.1 Smagorinsky Model . . . 36

2.5.1.2 Filtered Structure Function Model . . . 39

2.5.1.3 Dynamic Modeling . . . 41

2.5.1.4 Un-resolved DNS . . . 42

2.5.2 Self-Similar Sub Grid Stress Model . . . 43

2.5.3 Conclusive Remarks on the Sub Grid Stress Models . . . 45

2.6 Coherent Structures . . . 46

2.6.1 A quick overview . . . 46

2.7 High Performance Computing and Parallelism . . . 48

2.8 Free and Open Source Software . . . 54

2.8.1 Digital Rights Management and Future of Free Computing . . . 56

I Study of Coherent Structures with the Legacy LES code 59 3 Coherent Structures 63 3.1 The Coherent Structures Idea . . . 63

3.2 Vortex Dynamics and Coherent Structures . . . 66

3.2.1 Vorticity Definition . . . 67

vii

3.3 Classical Vortex . . . 69

3.3.1 Peak of Vorticity . . . 70

3.3.2 Pressure Minimum . . . 72

3.3.3 Vorticity Peak & Pressure Minimum . . . 73

3.4 Advanced Vortex Modeling . . . 73

3.4.1 The Vortex Core . . . 76

3.5 Advanced Vortex Detection Criteria . . . 77

3.5.1 Gradient of Velocity Tensor: Eigenvalues Approach . . . 77

3.5.2 Gradient of Velocity Tensor: Second Invariant Approach . . . . 79

3.5.3 Jeong & Hussain Approach . . . 80

3.6 Latest Developments for Vortex Definition . . . 82

3.7 Vortex Detection Procedure . . . 83

3.7.1 Histograms . . . 88

3.7.2 Trigger Level Determination . . . 90

3.8 Single Structure Identification and Classification Procedure . . . 93

3.8.1 Structure Core . . . 94

3.8.2 Structure External Layer . . . 96

3.8.2.1 Practical Implementation . . . 99

3.9 Statistical Tools . . . 102

3.9.1 Structures’ Geometry . . . 102

3.9.2 Structure’ Fluid Dynamics Properties . . . 104

3.9.3 Joint Probability Distribution Functions ( JPDF s) . . . 107

3.9.4 Vortical Structures Orientation in Space . . . 108

3.9.4.1 Equivalent Ellipsoid . . . 110

3.9.4.2 Procedure Justification . . . 111

3.9.5 Conditional Sampling . . . 112

3.9.6 Vortical Structures Ensemble Averaging . . . 116

3.9.6.1 Initialization . . . 116

3.9.6.2 Iterative Procedure . . . 118

3.9.7 Ensemble Average and Coherent Structures . . . 120

4 Numerical solution of LES equations 123 4.1 Incompressible Filtered Navier-Stokes Equation Solution Method . . . 123

4.1.1 Predictor-Corrector Procedure . . . 124

4.1.2 Modified Pressure update mechanism & Time Advancement . . 126

4.1.2.1 Stability Limits . . . 128

4.1.2.2 Time Stepping procedure Time Accuracy . . . 129

4.2 Spatial Discretization . . . 132

4.2.1 Aliasing and Arakawa Formulation . . . 133

4.3 Numerical Discretization . . . 134

4.4 Improved Accuracy . . . 135

4.5 Upwinding, Sub Grid Stress Model and Boundary Conditions for the

Modified Pressure . . . 145

Contents ix

4.6 Multi Domain Technique . . . 147

4.6.1 Specifics of the Multi Domain for the VKI’s Environmental & Applied Fluid Dynamics Department LES legacy code . . . 149

4.6.2 Early Critiques to the VKI’s Environmental & Applied Fluid Dynamics Department LES legacy code’s Multi Domain tech- nique . . . 151

4.7 Averaging Procedure for the VKI’s Environmental & Applied Fluid Dy- namics Department LES legacy code . . . 152

4.7.1 Flows with a Dominant Frequency . . . 156

4.8 Comparison between Large Eddy Simulation and reference Direct Nu- merical Simulation or Experiments. . . 157

4.8.1 Statistics and Staggered Grid . . . 160

5 Simulation of coherent structures in attached wall shear layer 161 5.1 Turbulent Plane Channel General Informations . . . 161

5.2 Effects of Sub Grid Stress Model and Discretization Accuracy on Large Eddy Simulation results . . . 164

5.3 Intermediate Medium Res. Channel Grid . . . 171

5.3.1 Eulerian Statistics for the Medium Res. Channel Grid . . . 172

5.3.2 Single Structure Data Statistics for the Medium Res. Channel Grid . . . 176

5.3.3 Considerations for the statistics collected . . . 185

5.3.4 Effect of the Vortex Detection Criterion . . . 188

5.4 Final High Res. Channel Grid . . . 190

5.4.1 Eulerian Statistics for the High Res. Channel Grid . . . 192

5.4.2 Single Structure Data Statistics for the High Res. Channel Grid 192 5.5 0

Order Coherent Structure Ensemble Averaging for Turbulent Plane Channel. . . 199

5.5.1 Considerations for the statistics collected . . . 203

5.6 Conclusions . . . 208

6 Flow Behind a Bluff Body 209 6.1 Points of Interests in Simulating Bluff Bodys . . . 209

6.2 Computational And Numerical Setup . . . 211

6.2.1 Simulation Layout . . . 212

6.2.2 Multi Domain Approach for Bluff Bodys . . . 214

6.2.3 Grid around the Bluff Body . . . 216

6.3 Data Processing and Bulk Coefficients . . . 217

6.3.1 Particular Approach to the Statistics’ Sampling for the wake behind a Bluff Body . . . 219

6.4 Velocity Profiles Comparisons . . . 221

6.4.1 Concluding remarks on the quality of the BB6 Simulation . . . 227

6.5 Flow description and related properties . . . 232

6.6 Alternative approach to the vortical structure identification for the Bluff

Body wake . . . 236

6.6.1 Technique Description . . . 236

6.7 Normal approach to structure identification . . . 241

6.7.1 Detection Algorithm and Trigger Level choice . . . 242

6.7.2 Bluff Body Global statistics . . . 248

6.7.3 Coherent Structures & Bluff Body . . . 249

6.8 Conclusive Remarks on the flow around a Bluff Body . . . 253

7 Conclusions for Part I 255 II Mioma, a complete software Frame Workfor a new LES solver 259 8 MiOma 263 8.1 The lifespan of a Large Eddy Simulation solver. . . 263

8.2 The MiOma (R)Evolution . . . 266

8.2.1 Transition from Fortran to C . . . 268

8.2.2 Free and Open Source Software . . . 268

8.3 Professional Code Development Approach . . . 269

8.3.1 Concurrent Versions System (CVS) . . . 269

8.3.1.1 M.i.O.m.a. Chronology . . . 271

8.3.2 Documenting the code . . . 274

8.3.3 Embedding the Documentation inside the Code . . . 276

8.3.3.1 doxygen’s Return On Investment . . . 277

8.3.3.2 doxygen’s Generated Documentation . . . 278

8.3.3.3 M.i.O.m.a. and doxygen’s perspective . . . 281

8.3.4 gcc, Makefiles and “Paranoid” Flags . . . 282

8.3.4.1 C Compiler . . . 282

8.3.4.2 gcc Compile Flags and Optimization Levels . . . 283

8.3.4.3 M.i.O.m.a. is makefile Driven . . . 284

8.3.5 Warning and Error Handling At Run Time . . . 289

8.3.6 Agile Programming & Code Reuse . . . 290

8.3.7 Robust & Fault Tolerant Programming . . . 291

8.4 M.i.O.m.a. Software Dependencies . . . 292

8.4.1 Software Libraries . . . 295

8.4.1.1 Portable Extensible Toolkit for Scientific computations(PETSc) and Functions Hierarchy . . . 295

8.4.1.2 Data Format : Unidata Network Common Data Format297 8.4.1.3 Miscellaneous Libraries . . . 300

8.4.2 MAXIMA . . . 302

8.4.3 Development and Debugging Software . . . 307

Contents xi

8.4.4 Plotting and Flow Visualization Tools for M.i.O.m.a. . . 309

8.4.5 Post-Production, Optimization & Collaborative Software . . . . 313

8.4.6 M.i.O.m.a. Dedicated Forums and Bug-Tracking Servers . . . . 315

8.4.6.1 Online Tutorials . . . 317

8.5 Conclusions . . . 319

9 MiOma Approach to the Numerical solution of LES equations 321 9.1 Continuity with the Past . . . 321

9.2 M.i.O.m.a. and Homogeneous Directions . . . 323

9.3 Load Balancing and Domain Auto-Partitioning . . . 325

9.4 Incompressibility Constrain Handling in the MiOma Solver . . . 328

9.4.1 Modified Pressure Boundary Conditions in the MiOma Optic . 331 9.4.2 Incompressibility Constrain Linear Solver Solution Strategy . . 332

9.5 Viscous and Convective Terms Treatments in MiOma . . . 333

9.5.1 Use of High Resolution Central Scheme for the convective terms 334 9.6 On the fly generation of Inlet Profiles . . . 337

9.7 Miscellanea and Concluding Remarks . . . 341

10 First Simulations produced in the MiOma Environment 347 10.1 Testing the core of the MiOma Frame Work, the LES solver. . . 347

10.2 MiOma Simulation of a Turbulent Plane Channel at Re

= 180 . . . 348

10.2.1 Early test of Parallel Efficiency . . . 351

10.3 Flat Mounted Cube in a Channel Simulation . . . 353

10.3.1 MiOma Flat Mounted Cube in a Channel Multi Domain Dis- cretization . . . 356

10.3.2 Inlet Conditions . . . 359

10.3.3 Flat Mounted Cube in a Channel Results . . . 361

10.4 Concluding Remarks . . . 368

11 Future Perspectives and Conclusions for Part II 369 11.1 Conclusions . . . 369

III Back Matter 375 12 Conclusions to the Thesis 377 Bibliography 381 A Sub Grid and Resolved Mean Flow Kinetic energy Budgets 393 A.1 SGS kinetic Energy Budget . . . 393

A.2 Kinetic Energy of the Mean Resolved Flow . . . 395

B Resolved Turbulent Kinetic Energy Budget 397 B.1 Derivation . . . 397

C MAXIMA Usage 401

C.1 Implicit Filtering Filter Size : ∆

or 2∆

? . . . 401

D Estimation of Algorithmic errors 407

D.1 Algorithmic Error for the Recursive Statistics using error graphs. . . 407 E High Res. Channel Grid Supplementary data 411 E.1 Velocities fluctuations Budgets . . . 411

F M.i.O.m.a. Documentation 417

F.1 Man Pages generation methods . . . 417 F.2 General Documentation using doxygen . . . 425 F.3 doxygen’s Data-Structure Documentation . . . 432 G Data Format, debugging tools and Concurrent Versions System (CVS)

Evaluation 437

G.1 Overview . . . 437

H MAXIMA In Details 461

H.1 Overview . . . 461

I OpenDX Usage 515

I.1 Overview . . . 515 J MiOma Function Optimization Approach 525 J.1 MiOma Solver Divergence Computing Function . . . 525

Index 533