I Turbomachinery Flow Physics
1 Introduction, Turbomachinery, Applications, Types ............ 3
1.1 Turbine ................................................. 3
1.2 Compressor .............................................. 7
1.3 Application of Turbomachines ............................ 9
1.3.1 Power Generation, Steam Turbines ................. 9
1.3.2 Power Generation, Gas Turbines ................... 9
1.3.3 Aircraft Gas Turbines ........................... 10
1.3.4 Diesel Engine Application ....................... 12
1.4 Classification of Turbomachines ........................ 13
1.4.1 Compressor Types ................................ 13
1.4.2 Turbine Types ................................... 14
1.5 Working Principle of a Turbomachine .................... 14
References .................................................. 14
2 Kinematics of Turbomachinery Fluid Motion ................... 15
2.1 Material and Spatial Description of the Flow Field ..... 15
2.1.1 Material Description ............................ 15
2.1.2 Jacobian Transformation Function and Its
Material Derivative ............................. 17
2.1.3 Spatial Description ............................. 20
2.2 Translation, Deformation, Rotation ..................... 21
2.3 Reynolds Transport Theorem ............................. 25
References .................................................. 27
3 Differential Balances in Turbomachinery ..................... 29
3.1 Mass Flow Balance in Stationary Frame of Reference ..... 29
3.1.1 Incompressibility Condition ..................... 31
3.2 Differential Momentum Balance in Stationary Frame of
Reference .............................................. 32
3.2.1 Relationship between Stress Tensor and
Deformation Tensor .............................. 34
3.2.2 Navier-Stokes Equation of Motion ................ 36
3.2.3 Special Case: Euler Equation of Motion .......... 38
3.3 Some Discussions on Navier-Stokes Equations ............ 41
3.4 Energy Balance in Stationary Frame of Reference ........ 42
3.4.1 Mechanical Energy ............................... 42
3.4.2 Thermal Energy Balance .......................... 45
3.4.3 Total Energy .................................... 48
3.4.4 Entropy Balance ................................. 49
3.5 Differential Balances in Rotating Frame of Reference ... 50
3.5.1 Velocity and Acceleration in Rotating Frame ..... 50
3.5.2 Continuity Equation in Rotating Frame of
Reference ....................................... 52
3.5.3 Equation of Motion in Rotating Frame of
Reference ....................................... 53
3.5.4 Energy Equation in Rotating Frame of
Reference ....................................... 55
References .................................................. 57
4 Integral Balances in Turbomachinery ......................... 59
4.1 Mass Flow Balance ...................................... 59
4.2 Balance of Linear Momentum ............................. 61
4.3 Balance of Moment of Momentum .......................... 66
4.4 Balance of Energy ...................................... 72
4.4.1 Energy Balance Special Case 1: Steady Flow ...... 77
4.4.2 Energy Balance Special Case 2: Steady Flow,
Constant Mass Flow .............................. 78
4.5 Application of Energy Balance to Turbomachinery
Components ............................................. 78
4.5.1 Application: Accelerated, Decelerated Flows ..... 79
4.5.2 Application: Combustion Chamber ................. 80
4.5.3 Application: Turbine, Compressor ................ 81
4.5.3.1 Uncooled Turbine ....................... 81
4.5.3.2 Cooled Turbine ......................... 82
4.5.3.3 Uncooled Compressor .................... 83
4.6 Irreversibility and Total Pressure Losses .............. 84
4.6.1 Application of Second Law to Turbomachinery
Components ...................................... 85
4.7 Flow at High Subsonic and Transonic Mach Numbers ....... 87
4.7.1 Density Changes with Mach Number, Critical
State ........................................... 88
4.7.2 Effect of Cross-Section Change on Mach Number ... 93
4.7.3 Compressible Flow through Channels with
Constant Cross Section ......................... 101
4.7.4 The Normal Shock Wave Relations ................ 109
4.7.5 The Oblique Shock Wave Relations ............... 115
4.7.6 The Detached Shock Wave ........................ 119
4.7.7 Prandtl-Meyer Expansion ........................ 119
References ................................................. 122
5 Theory of Turbomachinery Stages ............................ 123
5.1 Energy Transfer in Turbomachinery Stages .............. 123
5.2 Energy Transfer in Relative Systems ................... 124
5.3 General Treatment of Turbine and Compressor Stages .... 125
5.4 Dimensionless Stage Parameters ........................ 129
5.5 Relation between Degree of Reaction and Blade
Height ................................................ 131
5.6 Effect of Degree of Reaction on the Stage
Configuration ......................................... 134
5.7 Effect of Stage Load Coefficient on Stage Power ....... 136
5.8 Unified Description of a Turbomachinery Stage ......... 137
5.8.1 Unified Description of Stage with Constant
Mean Diameter .................................. 137
5.8.2 Generalized Dimensionless Stage Parameters ..... 138
5.9 Special Cases ......................................... 140
5.9.1 Case 1, Constant Mean Diameter ................. 141
5.9.2 Case 2, Constant Mean Diameter and Meridional
Velocity Ratio ................................. 141
5.10 Increase of Stage Load Coefficient, Discussion ........ 140
References ................................................. 144
6 Turbine and Compressor Cascade Flow Forces ................. 145
6.1 Blade Force in an Inviscid Flow Field ................. 145
6.2 Blade Forces in a Viscous Flow Field .................. 150
6.3 The Effect of Solidity on Blade Profile Losses ........ 156
6.4 Relationship Between Profile Loss Coefficient and
Drag .................................................. 156
6.5 Optimum Solidity ...................................... 158
6.5.1 Optimum Solidity, by Pfeil ..................... 158
6.5.2 Optimum Solidity, by Zweifel ................... 158
6.6 Generalized Lift-Solidity Coefficient ................. 162
6.6.1 Lift-Solidity Coefficient for Turbine Stator ... 164
6.6.2 Turbine Rotor .................................. 168
References ................................................. 171
II Turbomachinery Losses, Efficiencies, Blades
7 Losses in Turbine and Compressor Cascades .................. 175
7.1 Turbine Profile Loss .................................. 176
7.2 Viscous Flow in Compressor Cascade .................... 178
7.2.1 Calculation of Viscous Flows ................... 178
7.2.2 Boundary Layer Thicknesses ..................... 179
7.2.3 Boundary Layer Integral Equation ............... 181
7.2.4 Application of Boundary Layer Theory to
Compressor Blades .............................. 182
7.2.5 Effect of Reynolds Number ...................... 186
7.2.6 Stage Profile Losses ........................... 186
7.3 Trailing Edge Thickness Losses ........................ 186
7.4 Losses Due to Secondary Flows ......................... 192
7.4.1 Vortex Induced Velocity Field, Law of Bio-
Savart ......................................... 194
7.4.2 Calculation of Tip Clearance Secondary Flow
Losses ......................................... 197
7.4.3 Calculation of Endwall Secondary Flow Losses ... 200
7.5 Flow Losses in Shrouded Blades ........................ 204
7.5.1 Losses Due to Leakage Flow in Shrouds .......... 204
7.6 Exit Loss ............................................. 211
7.7 Trailing Edge Ejection Mixing Losses of Gas Turbine
Blades ................................................ 212
7.7.1 Calculation of Mixing Losses ................... 212
7.7.2 Trailing Edge Ejection Mixing Losses ........... 217
7.7.3 Effect of Ejection Velocity Ratio on Mixing
Loss ........................................... 217
7.7.4 Optimum Mixing Losses .......................... 219
7.8 Stage Total Loss Coefficient .......................... 219
7.9 Diffusers, Configurations, Pressure Recovery,
Losses ................................................ 220
7.9.1 Diffuser Configurations ........................ 221
7.9.2 Diffuser Pressure Recovery ..................... 222
7.9.3 Design of Short Diffusers ...................... 225
7.9.4 Some Guidelines for Designing High Efficiency
Diffusers ...................................... 228
References ................................................. 229
8 Efficiency of Multi-stage Turbomachines .................... 231
8.1 Polytropic Efficiency ................................. 231
8.2 Isentropic Turbine Efficiency, Recovery Factor ........ 234
8.3 Compressor Efficiency, Reheat Factor .................. 237
8.4 Polytropic versus Isentropic Efficiency ............... 239
References ................................................. 240
9 Incidence and Deviation .................................... 241
9.1 Cascade with Low Flow Deflection ...................... 241
9.1.1 Conformal Transformation ....................... 241
9.1.2 Flow Through an Infinitely Thin Circular Arc
Cascade ........................................ 250
9.1.3 Thickness Correction ........................... 256
9.1.4 Optimum Incidence .............................. 256
9.1.5 Effect of Compressibility ...................... 258
9.2 Deviation for High Flow Deflection .................... 259
9.2.1 Calculation of Exit Flow Angle ................. 261
References ................................................. 263
10 Simple Blade Design ........................................ 265
10.1 Conformal Transformation, Basics ...................... 265
10.1.1 Joukowsky Transformation ....................... 267
10.1.2 Circle-Flat Plate Transformation ............... 267
10.1.3 Circle-Ellipse Transformation .................. 268
10.1.4 Circle-Symmetric Airfoil Transformation ........ 269
10.1.5 Circle-Cambered Airfoil Transformation ......... 271
10.2 Compressor Blade Design ............................... 272
10.2.1 Low Subsonic Compressor Blade Design ........... 273
10.2.2 Compressors Blades for High Subsonic Mach
Number ......................................... 279
10.2.3 Transonic, Supersonic Compressor Blades ........ 280
10.3 Turbine Blade Design .................................. 281
10.3.1 Graphic Design of Camberline ........................ 282
10.3.2 Camberline Coordinates Using Bèzier Curve ...... 283
10.3.3 Alternative Calculation Method ................. 285
10.4 Assessment of Blades Aerodynamic Quality .............. 287
References ................................................. 290
11 Radial Equilibrium ......................................... 291
11.1 Derivation of Equilibrium Equation .................... 292
11.2 Application of Streamline Curvature Method ............ 300
11.2.1 Step-by-step solution procedure ................ 302
11.2 Compressor Examples ................................... 306
11.3 Turbine Example, Compound Lean Design ................. 309
11.3.1 Blade Lean Geometry ............................ 310
11.3.2 Calculation of Compound Lean Angle
Distribution ................................... 311
11.3.3 Example: Three-Stage Turbine Design ............ 313
11.4 Special Cases ......................................... 316
11.4.1 Free Vortex Flow ............................... 316
11.4.2 Forced vortex flow ............................. 317
11.4.3 Flow with constant flow angle .................. 318
References ................................................. 319
III Turbomachinery Dynamic Performance
12 Dynamic Simulation of Turbomachinery Components ............ 323
12.1 Theoretical Background ................................ 324
12.2 Preparation for Numerical Treatment ................... 330
12.3 One-Dimensional Approximation ......................... 331
12.3.1 Time Dependent Equation of Continuity .......... 331
12.3.2 Time Dependent Equation of Motion .............. 333
12.3.3 Time Dependent Equation of Total Energy ........ 334
12.4 Numerical Treatment ................................... 339
References ................................................. 340
13 Generic Modeling of Turbomachinery Components .............. 341
13.1 Generic Component, Modular Configuration .............. 342
13.1.1 Plenum as Coupling Module ...................... 343
13.1.2 Group 1: Modules: Inlet, Exhaust, Pipe ......... 345
13.1.3 Group 2: Recuperators, Combustion Chambers,
Afterburners ................................... 346
13.1.4 Group 3: Adiabatic Compressor and Turbine
Components ..................................... 348
13.1.5 Group 4: Diabatic Turbine and Compressor
Components ..................................... 350
13.1.6 Group 5: Control System, Valves, Shaft,
Sensors ........................................ 352
13.2 System Configuration, Nonlinear Dynamic Simulation .... 352
References ................................................. 356
14 Modeling of Inlet, Exhaust, and Pipe Systems ............... 357
14.1 Unified Modular Treatment ............................. 357
14.2 Physical and Mathematical Modeling of Modules ......... 357
14.3 Example: Dynamic behavior of a Shock Tube ............. 360
14.3.1 Shock Tube Dynamic Behavior .................... 361
References ................................................. 365
15 Modeling of Recuperators, Combustors, Afterburners ......... 367
15.1 Modeling Recuperators ................................. 368
15.1.1 Recuperator Hot Side Transients ................ 369
15.1.2 Recuperator Cold Side Transients ............... 369
15.1.3 Coupling Condition Hot, Cold Side .............. 370
15.1.4 Recuperator Heat Transfer Coefficient .......... 371
15.2 Modeling Combustion Chambers .......................... 372
15.2.1 Mass Flow Transients ........................... 373
15.2.2 Temperature Transients ......................... 374
15.2.3 Combustion Chamber Heat Transfer ............... 376
15.3 Example: Startup and Shutdown of a Combustion
Chamber ............................................... 378
15.4 Modeling of Afterburners .............................. 381
References ................................................. 382
16 Modeling of Compressor Component, Design, Off-Design ....... 383
16.1 Compressor Losses ..................................... 384
16.1.1 Profile Losses ................................. 385
16.1.2 Diffusion Factor ............................... 387
16.1.3 Generalized Maximum Velocity Ratio for
Cascade, Stage ................................. 391
16.1.4 Compressibility Effect ......................... 393
16.1.5 Shock Losses ................................... 397
16.1.6 Correlations for Boundary Layer Momentum
Thickness ...................................... 406
16.1.7 Influence of Different Parameters on Profile
Losses ......................................... 407
16.1.7.1 Mach Number Effect .................... 407
16.1.7.2 Reynolds Number Effect ................ 408
16.2 Compressor Design and Off-Design Performance .......... 409
16.2.1 Stage-by-Stage and Row-by-Row Compression
Process ........................................ 409
16.2.1.1 Stage-by-Stage Calculation of
Compression Process ................... 409
16.2.1.2 Row-by-Row Adiabatic Compression ...... 411
16.2.1.3 Off-Design Efficiency Calculation ..... 415
16.3 Generation of Steady State Performance Map ............ 418
16.3.1 Inception of Rotating Stall .................... 420
16.3.2 Degeneration of Rotating Stall into Surge ...... 422
16.4 Compressor Modeling Levels ............................ 423
16.4.1 Module Level 1: Using Performance Maps ......... 424
16.4.1.1 Quasi-dynamic Modeling Using
Performance Maps ...................... 426
16.4.1.2 Simulation Example .................... 427
16.4.2 Module Level 2: Row-by-Row Adiabatic
Compression .................................... 429
16.4.2.1 Active Surge Prevention by Adjusting
the Stator Blades ..................... 430
16.4.2.2 Simulation Example: Surge and Its
Prevention ............................ 431
16.4.3 Module Level 3: Row-by-Row Diabatic
Compression .................................... 436
16.4.3.1 Description of Diabatic Compressor
Module ................................ 437
16.4.3.2 Heat Transfer Closure Equations ....... 439
References ................................................. 442
17 Turbine Aerodynamic Design, Performance .................... 445
17.1 Stage-by-Stage and Row-by-Row Design .................. 447
17.1.1 Stage-by-Stage Calculation of Expansion
Process ........................................ 448
17.1.2 Row-by-Row Adiabatic Expansion ................. 449
17.1.3 Off-Design Efficiency Calculation .............. 454
17.1.4 Behavior under Extreme Low Mass Flows .......... 456
17.1.5 Example: Steady Design and Off-Design
Behavior ....................................... 459
17.2 Off-Design Calculation Using Global Turbine
Characteristics ....................................... 461
17.3 Modeling of Turbine Module for Dynamic Performance
Simulation ............................................ 462
17.3.1 Module Level 1: Using Turbine Performance
Characteristics ................................ 463
17.3.2 Module Level 2: Row-by-Row Expansion
Calculation .................................... 464
17.3.3 Module Level 3: Row-by-Row Diabatic
Expansion ...................................... 465
17.3.3.1 Description of Diabatic Turbine
Module, First Method .................. 467
17.3.3.2 Description of Module, Second
Method ................................ 469
17.3.3.3 Heat Transfer Closure Equations ....... 471
References ................................................. 472
18 Gas Turbine Engines, Design and Dynamic Performance ........ 473
18.1 Gas Turbine Steady Design Operation, Process .......... 475
18.1.1 Gas Turbine Process ............................ 477
18.1.2 Improvement of Gas Turbine Thermal
Efficiency ..................................... 483
18.2 Non-Linear Gas Turbine Dynamic Simulation ............. 485
18.2.1 State of Dynamic Simulation, Background ........ 486
18.3 Engine Components, Modular Concept, Module
Identification ........................................ 487
18.4 Levels of Gas Turbine Engine Simulations, Cross
Coupling .............................................. 493
18.5 Non-Linear Dynamic Simulation Case Studies ............ 494
18.5.1 Case Study 1: Compressed Air Energy Storage
Gas Turbine .................................... 495
18.5.1.1 Simulation of Emergency Shutdown ...... 497
18.5.2 Case Study 2: Power Generation Gas Turbine
Engine ......................................... 499
18.5.3 Case Study 3: Simulation of a Multi-Spool Gas
Turbine ........................................ 504
18.6 A Byproduct of Dynamic Simulation: Detailed
Calculation ........................................... 507
18.7 Summary Part III, Further Development ................. 510
References ................................................. 511
IV Turbomachinery CFD-Essentials
19 Basic Physics of Laminar-Turbulent Transition .............. 515
19.1 Transition Basics: Stability of Laminar Flow .......... 515
19.2 Laminar-Turbulent Transition, Fundamentals ............ 515
19.3 Physics of an Intermittent Flow ....................... 518
19.3.1 Intermittent Behavior of Statistically Steady
Flows .......................................... 519
19.3.2 Turbulent/Non-turbulent Decisions .............. 520
19.3.3 Intermittency Modeling for Flat Plate
Boundary Layer ................................. 524
19.4 Physics of Unsteady Boundary Layer Transition ......... 525
19.4.1 Experimental Simulation of the Unsteady
Boundary Layer ................................. 527
19.4.2 Ensemble Averaging High Frequency Data ......... 530
19.4.3 Intermittency Modeling for Periodic Unsteady
Flow ........................................... 533
19.5 Implementation of Intermittency into Navier Stokes
Equations ............................................. 536
19.5.1 Reynolds-Averaged Navier-Stokes Equations
(RANS) ......................................... 536
19.5.2 Conditioning RANS for Intermittency
Implementation ................................. 540
References ................................................. 542
20 Turbulent Flow and Modeling in Turbomachinery .............. 545
20.1 Fundamentals of Turbulent Flows ....................... 545
20.1.1 Type of Turbulence ............................. 547
20.1.2 Correlations, Length and Time Scales ........... 548
20.1.3 Spectral Representation of Turbulent Flows ..... 555
20.1.4 Spectral Tensor, Energy Spectral Function ...... 558
20.2 Averaging Fundamental Equations of Turbulent Flow ..... 560
20.2.1 Averaging Conservation Equations ............... 561
20.2.1.1 Averaging the Continuity Equation ..... 561
20.2.1.2 Averaging the Navier-Stokes
Equation .............................. 561
20.2.1.3 Averaging the Mechanical Energy
Equation .............................. 562
20.2.1.4 Averaging the Thermal Energy
Equation .............................. 563
20.2.1.5 Averaging the Total Enthalpy
Equation .............................. 565
20.2.1.6 Quantities Resulting from Averaging
to Be Modeled ......................... 568
20.2.2 Equation of Turbulence Kinetic Energy .......... 570
20.2.3 Equation of Dissipation of Kinetic Energy ...... 576
20.3 Turbulence Modeling ................................... 577
20.3.1 Algebraic Model: Prandtl Mixing Length
Hypothesis ..................................... 578
20.3.2 Algebraic Model: Cebeci-Smith Model ............ 584
20.3.3 Baldwin-Lomax Algebraic Model .................. 585
20.3.4 One- Equation Model by Prandtl ................. 586
20.3.5 Two-Equation Models ............................ 587
20.3.5.1 Two-Equation k-ε Model ............... 587
20.3.5.2 Two-Equation k-ω-Model ............... 589
20.3.5.3 Two-Equation k-ω-SST-Model ........... 590
20.4 Grid Turbulence ....................................... 592
20.5 Examples of Two-Equation Models ....................... 594
20.5.1 Internal Flow, Sudden Expansion ................ 594
20.5.2 Internal Flow, Turbine Cascade ................. 595
20.5.3 External Flow, Lift-Drag Polar Diagram ......... 596
20.5.4 Case Study: Flow Simulation in a Rotating
Turbine ........................................ 597
20.5.5 Results, Discussion ............................ 605
20.5.6 Rotating Turbine RANS, URANS-Shortcomings,
Discussion ..................................... 614
References ................................................. 615
21 Introduction into Boundary Layer Theory .................... 619
21.1 Boundary Layer Approximations ......................... 620
21.2 Exact Solutions of Laminar Boundary Layer Equations ... 623
21.2.1 Zero Pressure Gradient Boundary Layer .......... 624
21.2.2 Non-zero Pressure Gradient Boundary Layer ...... 626
21.2.3 Polhausen Approximate Solution ................. 629
21.3 Boundary Layer Theory, Integral Method ................ 631
21.3.1 Boundary Layer Thicknesses ..................... 631
21.3.2 Boundary Layer Integral Equation ............... 633
21.4 Turbulent Boundary Layers ............................. 637
21.4.1 Universal Wall Functions ....................... 640
21.4.2 Velocity Defect Function ....................... 643
21.5 Boundary Layer, Differential Treatment ................ 648
21.5.1 Solution of Boundary Layer Equations ........... 653
21.6 Measurement of Boundary Flow, Basic Techniques ........ 653
21.6.1 Experimental Techniques ........................ 653
21.6.1.1 HWA Operation Modes, Calibration ...... 654
21.6.1.2 HWA Averaging, Sampling Data .......... 655
21.7 Examples: Calculations, Experiments ................... 657
21.7.1 Steady State Velocity Calculations ............. 657
21.7.1.1 Experimental Verification ............. 659
21.7.1.2 Heat Transfer Calculation,
Experiment ............................ 660
21.7.2 Periodic Unsteady Inlet Flow Condition ......... 661
21.7.2.1 Experimental Verification ............. 664
21.7.2.2 Heat Transfer Calculation,
Experiment ............................ 666
21.7.3 Application of K-ca Model to Boundary Layer .... 667
21.8 Parameters Affecting Boundary Layer ................... 667
21.8.1 Parameter Variations, General Remarks .......... 668
21.8.2 Effect of Periodic Unsteady Flow ............... 672
References ................................................. 680
A Vector and Tensor Analysis in Turbomachinery ............... 685
A.1 Tensors in Three-Dimensional Euclidean Space .......... 685
A.1.1 Index Notation ................................. 686
A.2 Vector Operations: Scalar, Vector and Tensor
Products .............................................. 687
A.2.1 Scalar Product ................................. 687
A.2.2 Vector or Cross Product ........................ 688
A.2.3 Tensor Product ................................. 688
A.3 Contraction of Tensors ................................ 689
A.4 Differential Operators in Fluid Mechanics ............. 690
A.4.1 Substantial Derivatives ........................ 690
A.4.2 Differential Operator  ........................ 691
A.5 Operator Applied to Different Functions ............. 693
A.5.1 Scalar Product of and V ..................... 694
A.5.2 Vector Product × V ........................... 695
A.5.3 Tensor Product of and V ...................... 695
A.5.4 Scalar Product of and a Second Order
Tensor ......................................... 696
A.5.5 Eigenvalue and Eigenvector of a Second Order
Tensor ......................................... 700
References ................................................. 702
В Tensors in Orthogonal Curvilinear Coordinate Systems ....... 703
B.l Change of Coordinate System ........................... 703
B.2 Co- and Contravariant Base Vectors, Metric
Coefficients .......................................... 703
B.3 Physical Components of a Vector ....................... 706
B.4 Derivatives of the Base Vectors, Christoffel
Symbols ............................................... 707
B.5 Spatial Derivatives in Curvilinear Coordinate
System ................................................ 708
B.5.1 Application of V to Zeroth Order Tensor
Functions ...................................... 708
B.5.2 Application of V to First and Second Order
Tensor Functions ............................... 709
B.6 Application Example 1: Inviscid Incompressible Flow
Motion ................................................ 710
B.6.1 Equation of Motion in Curvilinear Coordinate
Systems ........................................ 711
B.6.2 Special Case: Cylindrical Coordinate System .... 712
B.6.3 Base Vectors, Metric Coefficients .............. 712
B.6.4 Christoffel Symbols ............................ 713
B.6.5 Introduction of Physical Components ............ 714
B.7 Application Example 2: Viscous Flow Motion ............ 715
B.7.1 Equation of Motion in Curvilinear Coordinate
Systems ........................................ 715
B.7.2 Special Case: Cylindrical Coordinate System .... 716
References ................................................. 716
Index ......................................................... 717
|
