List of Tables
2.1 Type of possible errors in a computation. . . 18
2.2 Statistical Averageh iproperties (Assuming valid the Ergodic Principle) 23 2.3 Some One-Dimensional Symmetric Filters and their relative Transfer Functions. . . 28
2.4 Homogeneous Isotropic Spatial Filter Properties . . . 29
2.5 ResolvedTurbulent Kinetic Energy Budget . . . 34
3.1 Structures statistics Reτ=180 (Example): Trigger . . . 87
3.2 Subdivision of a Turbulent Plane Channel in three zones based on the distance from the wall. . . 91
3.3 Structures statistics Reτ=180 (Example): Enstrophy comp . . . 92
3.4 Structures statistics Reτ=180 (Example): Enstrophy in Zone 2 . . . 93
3.5 Percentage of Volume,υ,υx(Enstrophy) found in noise and structures versus total quantities; entire computational field . . . 98
3.6 Interesting Structures’ Ratios to compute . . . 105
3.7 Structures statistics Reτ=180 (Example): Volume percentage . . . 106
4.1 Distances from the wall whereE11andE22have been computed . . . . 139
5.1 Turbulent Plane Channel at equilibrium at Reτ=180. Grid Geometry for the Low Res. Channel Grid and the reference Direct Numerical Simulation (DNS). . . 165
5.2 Different Sub Grid Stress Models and accuracies for the Low Res. Channel Grid simulations. . . 165
5.3 Turbulent Plane Channel at equilibrium at Reτ = 180. Comparison between the LESsimulations of Table 5.2 respect three macroscopic quantities. . . 171
5.4 Turbulent Plane Channel at equilibrium at Reτ=180. Grid Geometry for the Medium Res. Channel Grid and the referenceDNS. . . 172
5.5 Turbulent Plane Channel at equilibrium at Reτ = 180. Comparison between theS M Fs of Low Res. Channel Grid and the current Medium Res. Channel Grid . . . 176
5.6 Turbulent Plane Channel at equilibrium at Reτ = 180. Comparison between some raw data of Low Res. Channel Grid and the current Medium Res. Channel Grid. . . 176
xix
5.7 Turbulent Plane Channel at equilibrium at Reτ = 180. Comparison
between theDand theQcriteria for the Medium Res. Channel Grid. . . 189
5.8 Turbulent Plane Channel (TPC) at equilibrium at Reτ = 180. Grid Geometry for the High Res. Channel Grid and the referenceDNS. . . . 192
5.9 Turbulent Plane Channel at equilibrium at Reτ = 180. Comparison between theS M Fs of Medium Res. Channel Grid and the biased sam- pling of High Res. Channel Grid. . . 196
6.1 Bulk Coefficients for the flow behind a BluffBody. . . 219
6.2 BluffBody: Structures’ Macroscopic Factors . . . 248
8.1 Type of possible errors in a computation. . . 283
8.2 Names and types of the external software packages used byMiOmafor theHEAD CVSbranch.. . . 293
8.3 MiOma’s file types version numbers . . . 300
10.1 MiOma TPCat equilibrium at Reτ=180 Grid Geometry . . . 348
10.2 MiOmaParallel Speed-Up Estimation for a Turbulent Plane Channel . . 351
10.3 Turbulent Plane Channel used for Inlet Conditions for the Flat Mounted Cube in a Channel . . . 361
10.4 Comparison of the dimension of the recirculation bubbles with refer- ence Simulations . . . 364
11.1 Provisional Road-Map for the next twoMiOmaPoint Releases . . . 370
11.2 Provisional Road-Map for the successive twoMiOmaPoint Releases . . 371
List of Figures
1.1 Thesis Structure . . . 4
2.1 Leonardo’s “Old Man and Water” drawing on paper. . . 9
2.2 Example of large scale Vortex Shedding . . . 13
2.3 Isotropic Turbulence Energy Spectrum . . . 16
2.4 Example of a slide rule (original idea of William Oughtred 1650 c.a.). . 19
2.5 Map of stations for meteorological measurements as proposed by Richard- son. . . 20
2.6 The sphere-teather for meteorological predictions as described by Richard- son . . . 22
2.7 Cut offWave Number ofLESalong the Energy Spectrum . . . 32
2.8 Energy exchange between resolved and unresolved fields. . . 44
2.9 Self Similar Model approach . . . 44
2.10 Shadowgraph of a mixing layer . . . 47
2.11 Moore’s Law . . . 50
2.12 Blue Gene Architecture . . . 53
3.1 Flux tube, the simplex definition, and the most ambiguous, for a “vortex”. 69 3.2 Equipotential lines and current line of an idealized vortex . . . 74
3.3 Velocity profile of an idealized vortex . . . 74
3.4 Velocity profile of the distributed vorticity vortex . . . 75
3.5 Velocity profile of the Oseen vortex . . . 75
3.6 Summary of three-dimensional incompressible flow patterns . . . 79
3.7 Iso-surfaces ofQ=5 for Turbulent Plane Channel . . . 85
3.8 PDF ofQfor Turbulent Plane Channel at equilibrium . . . 86
3.9 Iso-surfaces ofQT L=91 and 200 for a Turbulent Plane Channel . . . . 87
3.10 Percolation Effects for low Trigger Level . . . 88
3.11 Structures statistics Reτ=180 (Example): mass center distribution . . . 89
3.12 Structures statistics Reτ=180 (Example): specific Enstrophy . . . 90
3.13 Subdivision of the Vertical Wise direction in three Zones . . . 91
3.14 Choice of thresholdQT Lbased on the Probability Distribution Function (PDF) of the whole channel. . . 92
3.15 Smallest considered structure and associated volume . . . 95
3.16 Finding interior nodes of a Structure . . . 95
3.17 Vortex Core identification and construction . . . 96
xxi
3.18 Vortex Core growth simplified description . . . 96
3.19 Skin growth simplified description . . . 97
3.20 Identification Algorithm Conflicts log . . . 98
3.21 Successive steps in the structures detection and identification. . . 99
3.22 Detected Mask (blue) and Identified Structures (red) . . . 100
3.23 Structures statistics: volume in Wall Units . . . 103
3.24 Estimation of size of structures from their projections on coordinate axis 103 3.25 Histograms of the structures projections along two Cartesian Axis in Wall Units . . . 104
3.26 Turbulent Plane Channel example of PDF of Stream Wise vorticity modulus . . . 106
3.27 Turbulent Plane Channel example ofPDFof Stream Wise specific En- strophy . . . 106
3.28 Turbulent Plane Channel example ofPDF of Stream Wise convection velocity . . . 107
3.29 Turbulent Plane Channel example ofJPDF for Cartesian projections . . 108
3.30 Turbulent Plane Channel example ofJPDF for specific vorticity . . . . 108
3.31 JPDFof`x−`ycannot be use to determine the tilt angle. . . 109
3.32 Structures’ statistics, example of spatial alignment . . . 110
3.33 Convention for the incidence (α) and tilting (β) angle of the structures. 111 3.34 Structures’ statistics, example ofPDFof incidence and tilt angles (Data from Section 5.3). . . 112
3.35 Equivalent Ellipsoid for a given structure. . . 112
3.36 Volume error between the Equivalent Ellipsoid (data from Figure 5.17) and corresponding structure. . . 113
3.37 Conditional Sampling applied to the Production ofResolvedTurbulent Kinetic Energy Budget . . . 114
3.38 Conditional Sampling applied to the Dissipation ofResolvedTurbulent Kinetic Energy Budget. . . 115
3.39 Ensemble averaging simplified flow-chart . . . 119
3.40 Test structure updating process in Ensemble Averaging . . . 120
4.1 Time Stepping History for a Turbulent Plane Channel at Reτ=180 . . 130
4.2 Time Stepping History for a Turbulent Plane Channel at Reτ=395 . . . 130
4.3 Time Stepping History for a the flow behind a BluffBody . . . 131
4.4 Staggered Grid variables arrangement . . . 132
4.5 Small and large grid for 4thorder discretization . . . 135
4.6 Energy transfer coefficient ofG1(r),G2(r) andG?(r) . . . 138
4.7 Spectral behavior ofG?(r) for the first derivative. . . 139
4.8 One-dimensional Energy Spectra Comparison in Stream Wise Direc- tion . . . 141
4.9 One-dimensional Energy Spectra Comparison in Span Wise Direction . . 142
4.10 Comparison in semilogarithmic plot of Energy Spectra . . . 143
List of Figures xxiii
4.11 2ndorder discretization: Ensemble vision of the spectra at the different
stations . . . 144
4.12 4thorder discretization: Ensemble vision of the spectra at the different stations . . . 144
4.13 Collocation points for theFilteredRate of Strain Tensor and Sub Grid Scale (SGS) terms in the staggered grid. . . 146
4.14 Boundary Condition for the Modified Pressure . . . 146
4.15 Example of computational field subdivision in 3 domains . . . 148
4.16 Boundary points update at vertical interface between domains . . . 148
4.17 The Staggered Grid arrangement poses additional programming com- plexity . . . 152
5.1 The simple geometry of the Turbulent Plane Channel with the homoge- neous directions clearly indicated. . . 162
5.2 Homogeneous direction handling: grid and spatial autocorrelation co- efficient . . . 163
5.3 Comparison between the fiveLESsimulations classified in Table 5.2 . . 167
5.4 Comparison between the first four LES simulations classified in Ta- ble 5.2 . . . 168
5.5 Comparison between the first four LES simulations classified in Ta- ble 5.2 regarding the anisotropic part of the Residual Stress Tensor(a to c), ˜ε¯k(d) andDk¯ (e). . . 169
5.6 ResolvedTurbulent Kinetic Energy Budget of the first fourLESsimu- lations classified in Table 5.2 compared with theDNSone. . . 170
5.7 Mean and Root Mean Square (RMS) of velocities for Medium Res. Channel Grid with referenceDNS . . . 173
5.8 Some terms of the AnisotropicFilteredReynolds Stress Tensor for the Medium Res. Channel Grid compared with the referenceDNS . . . 174
5.9 Medium Res. Channel Grid: the six terms of theResolvedTurbulent Kinetic Energy Budget with respect to the referenceDNS. . . 175
5.10 Medium Res. Channel Grid:Single Structure DataPDF of Structures’ projections (`) along the Cartesian Axes and their Specific Vorticity components (ω?) . . . 177
5.11 Medium Res. Channel Grid case: Structures’ Distribution across the height of the channel, and their specific convection velocity. . . 179
5.12 Medium Res. Channel Grid case: Single Structure DataJPDFof Struc- tures’ projections (`) along the Cartesian Axes for the Viscous Wall Region and Log Law Region (Lower Half Channel). . . 180
5.13 “. . . alternating spanwise streaks of lower-and-higher-speed fluid . . . ”Kline [2] (VKI’s Environmental & Applied Fluid Dynamics DepartmentLES legacy code (VKILESlegacy code) &Tecplot™) . . . 181
5.14 Low speed longitudinal streaks formation mechanism . . . 182
5.15 Single Structure DataJPDFof Structures’ specific vorticity (ω?) com- ponents for the Viscous Wall Region and Log Law Region foronly the
lower half channel. . . 183
5.16 Medium Res. Channel Grid, Structures’ spatial alignment, different point of view for the Viscous Wall Region and Log Law Region for only the lower half channel.4Structure mean alignment. . . 184
5.17 Medium Res. Channel Grid, Equivalent Ellipsoid (Figure 3.35) Di- mensions, a more refined way to compute the structures sizes respect to Figure 5.10 and/or 5.12; from the Equivalent Ellipsoid the incidence and tilt angle are derivable as well. . . 186
5.18 Medium Res. Channel Grid, Example of identified structures. . . 187
5.19 Choice of thresholdQT Lbased on thePDF of the whole channel. . . 189
5.20 Medium Res. Channel Grid withDcriterion: Single Structure Data PDF of Structures’ Specific Vorticity (ω?) and Enstrophy (υ?) compo- nents . . . 191
5.21 Mean andRMSof velocities for High Res. Channel Grid with reference DNS . . . 193
5.22 Some terms of the AnisotropicFilteredReynolds Stress Tensor for the High Res. Channel Grid compared with the referenceDNS . . . 194
5.23 High Res. Channel Grid: the six terms of theResolvedTurbulent Ki- netic Energy Budget with respect to the referenceDNS. . . 195
5.24 High Res. Channel Grid case: Structures’ Distribution across the height of the channel, and their volume. . . 197
5.25 High Res. Channel Grid case: Structures’ convective velocities along Stream Wise and Vertical Wise directions. . . 197
5.26 High Res. Channel Grid: Single Structure DataPDF of Structures’ projections (`) along the Cartesian Axes and their Specific Vorticity components (ω?) . . . 198
5.27 High Res. Channel Grid case, limited to the Log Law Region: Single Structure DataJPDFof Structures’ projections (`) and specific vorticity components (ω?) along the Cartesian Axes (Lower Half Channel). . . . 200
5.28 Bottom half channel, counter-clockwise rotation . . . 201
5.29 Top half channel, clockwise rotation . . . 202
5.30 The effective number of structures available for averaging is reduced by multiple factors . . . 203
5.31 Bottom half channel, clockwise rotation . . . 204
5.32 Top half channel, counter-clockwise rotation . . . 204
5.33 Coherent Structures’ Prototype, Ensemble Flow -I- . . . 205
5.34 Coherent Structures’ Prototype, Ensemble Flow -II- . . . 206
6.1 Interaction between braid and rolls for flow at ReH=22,000, simula- tion at 4Hin Span Wise direction used for Coherent Structure investi- gation (BB4case). . . 210
List of Figures xxv
6.2 Flow Geometry and Boundary Condition for the flom behind the Bluff Body . . . 212 6.3 Multi Domain set-up for the flow behind a BluffBody . . . 215 6.4 Computational grid (only odd nodes presented for clarity sake,BB6). . 216 6.5 Close-up vision of the BluffBody grid and instantaneous Vertical Wise
Velocity for the simulation introduced in Section 6.5 (BB4). . . 217 6.6 Modified Pressure Lift Coefficient (clp) duringBB6Statistics’ Gathering. 219 6.7 Modified Pressure Lift and Drag Coefficients during BB4 Statistics’
Gathering. . . 220 6.8 Modified Pressure Coefficient (cp) Distribution around the BluffBody. . 220 6.9 Stream-lines of Time-Averaged flow field within the Computational
Field (BB6) . . . 222 6.10 Stream-lines of Time-Averaged flow field on the BluffBody (BB6). . . . 222 6.11 Wake Centerline Stream Wise and Span Wise mean and fluctuations’
RMSforBB6(solid Vertical Wise, dashed Stream Wise in (b)) . . . 224 6.12 Ensemble Vision of the comparison in the wake of the mean andRMS
of theStream WiseVelocity between theLES(BB6) and the reference experiments. . . 225 6.13 Ensemble Vision of the comparison in the wake of the mean andRMS
of theVertical WiseVelocity between theLES(BB6) and the reference experiments. . . 226 6.14 BB6Stream Wise Velocity and Fluctuations’RMSin three stations (from
top to bottom) −18,18,48 on top of the Bluff Body: comparison with the two sets of reference experiments (2DLaser Doppler Velocimetry (LDV)◦, 1DLDV)[3]. . . 228 6.15 BB6Stream Wise Velocity and Fluctuations’RMSin three stations (from
top to bottom) 88,148,178 in thenear wake: comparison with the 2DLDV set of experiments [3]. . . 229 6.16 BB6 Vertical Wise Velocity and Fluctuations’ RMS in three stations
(from top to bottom) −18,18,48 on top of the BluffBody: comparison with the two sets of reference experiments (2DLDV◦, 1DLDV)[3]. 230 6.17 BB6 Vertical Wise Velocity and Fluctuations’ RMS in three stations
(from top to bottom) 88,148,178 in thenear wake: comparison with the 2DLDVset of experiments [3]. . . 231 6.18 BB6Examples ofFilteredReynolds Stress Tensor ( ¯σxz) in two stations. 232 6.19 Multi Domain set-up for the flow behind a BluffBody for the Coherent
Structures investigation . . . 233 6.20 BB4Flow Visualization for the BluffBody: shedding sequence seen
from the top . . . 234 6.21 BB4Flow Visualization for the BluffBody: shedding sequence seen
from the side . . . 235 6.22 Structure identification: Span Wise rolls -I- . . . 238 6.23 BluffBody Structure identification: reference pattern . . . 239
6.24 BluffBody Structure identification: Span Wise rolls’ tracking . . . 240
6.25 BluffBody Structure identification: Span Wise rolls’ Velocities. . . 241
6.26 BB4Structures identification: histograms -I- . . . 243
6.27 BB4Structures identification: histograms -II- . . . 244
6.28 BB4Structures identification: histograms -III- . . . 245
6.29 BB4Structures identification: histograms -IV- . . . 246
6.30 Examples of Identified Structures in the wake of the BluffBody. . . 250
6.31 BluffBody Structures’ Probability Distribution Functions of projection along the Cartesian Axes and specific vorticity. . . 251
6.32 BluffBody Structures’ Probability Distribution Functions of Convec- tion Velocities and barycenter distributions along Stream Wise and Ver- tical Wise directions. . . 252
8.1 VKILESlegacy code code schematic evolution through time and de- velopers . . . 265
8.2 statcvs-xmlgenerated plots for theHEAD v1.0betabranch ofMiOma.272 8.3 Progression By module ofHEAD v1.0betabranch ofMiOma. . . 273
8.4 Cervisiaexamining the history of a file and comparing its presence on different branches. . . 273
8.5 Cervisia, with supporting software likeMeldallows for a rapid and productive peruse of theCVSrepository. . . 273
8.6 Graphical User Interface (GUI) for the MiOma Pre-Processor, safely contained in theCVSrepository. . . 274
8.7 Example of Dependency Graphs generable viadoxygen. . . 280
8.8 doxygencan provide excellent support for documenting Data-Structures.281 8.9 MiOmaSimplified Build Directory Hierarchy . . . 286
8.10 Each domain needs to posses some nodes from other domains via five ghost zones. In this figure the ghost zones for one side of one domain are illustrated. . . 296
8.11 Arrangement in memory of the three domains of the figure on the left; filled zone mean memory local to the processor, trailing ones need to be retrieved via parallel communication. . . 296
8.12 TheMiOma Solver usesGNU Scientific Library(gsl) as an in- termediate format between the data written on disk as Unidata Network Common Data Format (NetCDF) and used in memory as Portable Ex- tensible Toolkit for Scientific computations (PETSc). . . 302
8.13 MiOma utilizes MAXIMA to compute the discretized formulas for the convective terms . . . 306
8.14 Example ofOpenDXUsage for plotting Vortical Structures for aTPCat Reτ=180 . . . 311
8.15 OpenDXcan be programmed to offer user interfaces like(a)and(b) via Visual Programs made ofVisual Moduleslike(c), that allow for an interactive analysis of the data, as illustrated in(d). . . 312
List of Figures xxvii
8.16 KCacheGring analyzes a profile produced fromValgrindwith skin
Callgrind.. . . 314
8.17 TheMiOma Frame Work aims to create a Knowledge Base using an Electronic Bulletin Board onLAMP. . . 316
8.18 MiOma Bug-Tracking Example . . . 318
9.1 PETScMemory Model . . . 323
9.2 A surface Mounted Cube . . . 324
9.3 The Backward Facing Step geometry . . . 325
9.4 Load Balancing performed usingParMETIS. . . 327
9.5 OpenDXused to visualize the iso-surface of Vertical Wise velocity . . . . 327
9.6 ParMETISpartitioned graph for a Flat Mounted Cube in a Channel . . . 327
9.7 MiOma alternative approach to the Modified Pressure derivatives . . . . 329
9.8 Three-points stencil forP(and dP) derivatives . . . 330
9.9 Poisson Matrix View for Backward Facing Step . . . 331
9.10 Case of the piece wise reconstruction of the variableu. . . 336
9.11 Region of strong shear Resolved using Kurganov and Tadmor [4] High Resolution Central Schemes . . . 337
9.12 Master-Slave Parallel Communication Types . . . 339
9.13 Master-Slave Simulations in case of Flat Mounted Cube in a Channel . . 340
9.14 Slave Simulation taking advantage of Span Wise periodicity . . . 340
9.15 TheMiOmaSolver prepares theNetCDFdata for Flow Visualization in OpenDX. . . 342
9.16 OpenDXartifacts in the rendering of the domains interfaces . . . 343
9.17 MiOma handling of a special case of wall Boundary Condition, the moving wall. . . 345
10.1 MiOma Turbulent Plane Channel Span Wise-Vertical Wise plane cut Iso-Contours . . . 349
10.2 MiOma Turbulent Plane Channel Alternative Instantaneous Plane Cut views . . . 350
10.3 MiOma Turbulent Plane ChannelLES-DNScomparison . . . 352
10.4 MiOma Turbulent Plane Channel Turbulent Viscosities. . . 353
10.5 MiOma Parallel Speed-Up Estimation for a Turbulent Plane Channel . . 354
10.6 Typical flow Topology to be expected for a Flat Mounted Cube in a Channel . . . 354
10.7 Geometry and Dimensions for the case of Flat Mounted Cube in a Channel Simulation. . . 355
10.8 Ensemble Vision of the Experimental Data Set for the Flat Mounted Cube in a Channel . . . 356
10.9 Geometrical Characteristics of the Flat Mounted Cube in a Channel Simulation . . . 358
10.10Time History of the Kinetic Energy for the considered Flat Mounted Cube in a Channel simulation . . . 359 10.11Turbulent Plane Channel for Flat Mounted Cube in a Channel Inlet
Conditions Profiles . . . 360 10.12Iso-surface ofQ=5 with Iso-Contours of Span Wise at distance 0.01H
from the wall. . . 362 10.13Comparison between instantaneous and Averaged On-Plane Vectors for
the Flat Mounted Cube in a Channel . . . 363 10.14Comparison between instantaneous and Averaged On-Plane Iso-Contours
for the Flat Mounted Cube in a Channel . . . 365 10.15Stream Wise Velocity Profiles for the Flat Mounted Cube in a Channel
grouped in four groups. . . 366 10.16Stream Wise RMS Velocity Profiles for the Flat Mounted Cube in a
Channel grouped in four groups . . . 367 10.17Reference Flat Mounted Cube in a Channel Lengths . . . 368 D.1 Error graph of a real function of two variables. . . 408 E.1 Balance of the variance of the velocity fluctuations along the Stream Wise
direction compared with theDNSones. . . 413 E.2 Balance of the variance of the velocity fluctuations along the Span Wise
direction compared with theDNSones. . . 414 E.3 Balance of the variance of the velocity fluctuations along the Verti-
cal Wise direction compared with theDNSones. . . 415 I.1 MiOmaReference Visual ProgramGraphical User Interfaces. . . 516 I.2 Example of theOpenDXEnvironment on a User Desktop. . . 524
