Internal Flow: Concepts and Applications

Cover Page......Page 1
About the Book......Page 2
Title......Page 4
ISBN 0521343933......Page 5
Contents (with page links)......Page 6
Preface......Page 18
Acknowledgements......Page 21
Letters......Page 23
Symbols......Page 26
Subscripts......Page 27
Superscripts and overbar symbols......Page 29
1.1 Introduction......Page 30
1.3 Dynamic and thermodynamic principles......Page 31
1.3.1 The rate of change of quantities following a fluid particle......Page 32
1.3.3 Thermodynamic states and state change processes for a fluid system......Page 33
1.3.4 First and second laws of thermodynamics for a fluid system......Page 35
1.4.1 Equations of state......Page 37
1.4.2 Specific heats......Page 38
1.5 Relation between changes in material and fixed volumes: Reynolds’s Transport Theorem......Page 40
1.6 Conservation laws for a fixed region (control volume)......Page 42
1.7 Description of stress within a fluid......Page 44
1.8.1 Force, torque, and energy exchange in fluid devices......Page 48
1.8.1.2 Torque on a fluid in a control volume......Page 49
1.8.1.4 The steady flow energy equation and the role of stagnation enthalpy......Page 50
1.9.1 Conservation of mass......Page 53
1.9.2 Conservation of momentum......Page 54
1.10 Splitting the energy equation: entropy changes in a fluid......Page 55
1.10.1 Heat transfer and entropy generation sources......Page 56
1.11 Initial and boundary conditions......Page 57
1.11.1 Boundary conditions at solid surfaces......Page 58
1.11.2 Inlet and outlet boundary conditions......Page 59
1.12 The rate of strain tensor and the form of the dissipation function......Page 60
1.13 Relationship between stress and rate of strain......Page 63
1.14 The Navier–Stokes equations......Page 66
1.14.1 Cartesian coordinates......Page 67
1.14.2 Cylindrical coordinates (x, axial; θ, circumferential; r, radial)......Page 68
1.15 Disturbance propagation in a compressible fluid: the speed of sound......Page 69
1.16 Stagnation and static quantities......Page 70
1.16.1 Relation of stagnation and static quantities in terms of Mach number......Page 71
1.17.1 Incompressible flow......Page 72
1.17.3 Dynamic similarity......Page 73
1.17.4 Compressible flow......Page 74
1.17.5 Limiting forms for low Mach number......Page 75
2.2 The assumption of incompressible flow......Page 77
2.2.1 Steady flow......Page 78
2.3 Upstream influence......Page 80
2.3.1 Upstream influence of a circumferentially periodic non-uniformity......Page 81
2.3.2 Upstream influence of a radial non-uniformity in an annulus......Page 83
2.4.1 Normal and streamwise accelerations and pressure gradients......Page 85
2.4.2 Other expressions for streamline curvature......Page 86
2.5 Quasi-one-dimensional steady compressible flow......Page 89
2.5.1 Corrected flow per unit area......Page 90
2.5.2 Differential relations between area and flow variables for steady isentropic one-dimensional flow......Page 92
2.6 Shock waves......Page 94
2.6.1 The entropy rise across a normal shock......Page 95
2.6.2 Shock structure and entropy generation processes......Page 97
2.7 Effect of exit conditions on steady, isentropic, one-dimensional compressible channel flow......Page 100
2.7.1 Flow regimes for a converging nozzle......Page 101
2.7.2 Flow regimes for a converging–diverging nozzle......Page 103
2.8.1 Pressure rise and mixing loss at a sudden expansion......Page 105
2.8.2 Ejector performance......Page 107
2.8.3 Fluid force on turbomachinery blading......Page 109
2.8.4 The Euler turbine equation......Page 112
2.8.5 Thrust force on an inlet......Page 113
2.8.6 Thrust of a cylindrical tube with heating or cooling (idealized ramjet)......Page 115
2.8.7 Oblique shock waves......Page 116
2.9.1 Features of boundary layers in ducts......Page 118
2.9.2 The influence of boundary layers on the flow outside the viscous region......Page 120
2.10.1 Qualitative considerations concerning flow separation from solid surfaces......Page 123
2.10.2 The contrast between flow in and out of a pipe......Page 125
2.10.3 Flow through a bent tube as an illustration of the principles......Page 127
2.10.4 Flow through a sharp edged orifice......Page 129
3.1 Introduction......Page 133
3.2 Vorticity kinematics......Page 134
3.2.1 Vortex lines and vortex tubes......Page 136
3.2.2 Behavior of vortex lines at a solid surface......Page 139
3.3 Vorticity dynamics......Page 140
3.4 Vorticity changes in an incompressible, uniform density, inviscid flow with conservative body force......Page 141
3.4.1 Examples: Secondary flow in a bend, horseshoe vortices upstream of struts......Page 143
3.4.2 Vorticity changes and angular momentum changes......Page 146
3.5 Vorticity changes in an incompressible, non-uniform density, inviscid flow......Page 148
3.5.1 Examples of vorticity creation due to density non-uniformity......Page 150
3.6 Vorticity changes in a uniform density, viscous flow with conservative body forces......Page 151
3.6.1 Vorticity changes and viscous torques......Page 153
3.6.2 Diffusion and intensification of vorticity in a viscous vortex......Page 154
3.6.3 Changes of vorticity in a fixed volume......Page 156
3.7 Vorticity changes in a compressible inviscid flow......Page 157
3.8.1 Kelvin’s Theorem......Page 159
3.9.1 Uniform density inviscid flow with conservative body forces......Page 161
3.9.2 Incompressible, non-uniform density, inviscid flow with conservative body forces......Page 163
3.10.1 Circulation generation due to shock motion in a non-homogeneous medium......Page 164
3.11 Rate of change of circulation for a flxed contour......Page 166
3.12 Rotational flow descriptions in terms of vorticity and circulation......Page 167
3.12.1 Behavior of vortex tubes when Dt /Dt = 0......Page 168
3.12.2 Evolution of a non-uniform flow through a diffuser or nozzle......Page 169
3.12.3 Trailing vorticity and trailing vortices......Page 171
3.13 Generation of vorticity at solid surfaces......Page 173
3.13.1 Generation of vorticity in a two-dimensional flow......Page 174
3.13.2 Vorticity flux in thin shear layers (boundary layers and free shear layers)......Page 178
3.13.3 Vorticity generation at a plane surface in a three-dimensional flow......Page 180
3.14 Relation between kinematic and thermodynamic properties in an inviscid, non-heat-conducting fluid: Crocco’s Theorem......Page 181
3.14.1.1 Flow downstream of an inlet guide vane (stationary blade row) in a turbomachine......Page 182
3.14.1.3 Flow downstream of a non-uniform strength shock wave......Page 183
3.15 The velocity fleld associated with a vorticity distribution......Page 185
3.15.1 Application of the velocity representation to vortex tubes......Page 187
3.15.3 Surface distributions of vorticity......Page 188
3.15.4 Some specific velocity fields associated with vortex structures......Page 189
3.15.5 Numerical methods based on the distribution of vorticity......Page 192
4.1 Introduction......Page 195
4.1.1 Boundary layer behavior and device performance......Page 196
4.2.1 Plane surfaces......Page 199
4.3.1 Boundary layer integral thicknesses......Page 202
4.3.2 Integral forms of the boundary layer equations......Page 205
4.4.1 Laminar boundary layer behavior in favorable and adverse pressure gradients......Page 206
4.4.2 Laminar boundary layer separation......Page 208
4.5 Laminar–turbulent boundary layer transition......Page 211
4.6.1 The time mean equations for turbulent boundary layers......Page 213
4.6.2 The composite nature of a turbulent boundary layer......Page 216
4.6.3 Introductory discussion of turbulent shear stress......Page 218
4.6.4 Boundary layer thickness and wall shear stress in laminar and turbulent flow......Page 220
4.6.5 Vorticity and velocity fluctuations in turbulent flow......Page 222
4.7 Applications of boundary layer analysis: viscous–inviscid interaction in a diffuser......Page 224
4.7.1 Qualitative description of viscous–inviscid interaction......Page 226
4.7.2 Quantitative description of viscous–inviscid interaction......Page 227
4.7.4 Turbulent boundary layer separation......Page 230
4.8.1 Similarity solutions for incompressible uniform density free shear layers......Page 231
4.8.2 The mixing layer between two streams......Page 234
4.8.3 The effects of compressibility on free shear layer mixing......Page 237
4.8.4 Appropriateness of the similarity solutions......Page 239
4.9 Turbulent entrainment......Page 240
4.10 Jets and wakes in pressure gradients......Page 241
5.1 Introduction......Page 246
5.2.1 Losses in a spatially uniform flow through a screen or porous plate......Page 247
5.2.2 Irreversibility, entropy generation, and lost work......Page 249
5.2.3 Lost work accounting in fluid components and systems......Page 251
5.3 Loss accounting and mixing in spatially non-uniform flows......Page 254
5.4.1 Entropy generation in boundary layers on adiabatic walls......Page 256
5.4.2 The boundary layer dissipation coefficient......Page 259
5.4.3 Estimation of turbomachinery blade profile losses......Page 262
5.5.1 Mixing of two streams with non-uniform stagnation pressure and/or temperature......Page 263
5.5.2 The limiting case of low Mach number...mixing......Page 266
5.5.4 Mixing losses from fluid injection into a stream......Page 268
5.5.5 Irreversibility in mixing......Page 270
5.5.6 A caveat: smoothing out of a flow non-uniformity does not always imply loss......Page 271
5.6.1 Representation of a non-uniform flow by equivalent average quantities......Page 273
5.6.2 Averaging procedures in an incompressible uniform density flow......Page 274
5.6.2.2 Mass average.........Page 275
5.6.2.3 Mixed out average.........Page 276
5.6.3 Effect of velocity distribution on average stagnation pressure (incompressible uniform density flow)......Page 277
5.6.4 Averaging procedures in a compressible flow......Page 279
5.6.4.1 Effects of inlet entropy and/or stagnation temperature non-uniformity......Page 280
5.6.5.1 Definition and application of the entropy flux average (availability average) stagnation pressure......Page 282
5.6.5.2 Some general principles concerning averaging of non-uniform flows......Page 285
5.7.1 Stagnation pressure averages and integral boundary layer parameters......Page 287
5.7.2 Comparison of losses within a device to losses from downstream mixing......Page 290
5.8 Effect of base pressure on mixing losses......Page 291
5.9.1 Two-stream mixing......Page 296
5.9.2 Mixing of a linear shear flow in a diffuser or nozzle......Page 298
5.9.3 Wake mixing......Page 302
5.10 Losses in turbomachinery cascades......Page 303
5.11 Summary concerning loss generation and characterization......Page 306
6.2 The inherent unsteadiness of fluid machinery......Page 308
6.3 The reduced frequency......Page 310
6.3.1 An example of the role of reduced frequency: unsteady flow in a channel......Page 311
6.4.2 The starting transient for incompressible flow exiting a tank......Page 315
6.4.3 Stagnation pressure variations due to the motion of an isolated airfoil......Page 317
6.4.4 Moving blade row (moving row of bound vortices)......Page 319
6.4.5 Unsteady wake structure and energy separation......Page 321
6.5 Shear layer instability......Page 326
6.5.1 Instability of a vortex sheet (Kelvin–Helmholtz instability)......Page 327
6.5.2 General features of parallel shear layer instability......Page 329
6.6 Waves and oscillations in fluid systems: system instabilities......Page 332
6.6.1 Transfer matrices (transmission matrices) for fluid components......Page 334
6.6.1.1 The transfer matrix for a duct......Page 335
6.6.1.4 The transfer matrix for a screen, perforated plate, or throttle......Page 336
6.6.1.5 The transfer matrix for a compressor or pump......Page 338
6.6.2.2 A model for gas turbine engine system instability......Page 339
6.6.2.3 Static and dynamic instability......Page 340
6.6.2.4 Mechanism for dynamic compression system instability......Page 342
6.6.2.5 Instability in distributed (non-lumped parameter) fluid systems......Page 343
6.6.3 Nonlinear oscillations in fluid systems......Page 344
6.6.3.1 Limit cycle oscillations......Page 346
6.6.3.2 Liapunov function description of nonlinear fluid system oscillations......Page 347
6.6.3.3 An energy approach to instability onset......Page 349
6.7 Multi-dimensional unsteady disturbances in a compressible inviscid flow......Page 350
6.8.1.2 Passage of an entropy disturbance through a choked nozzle......Page 353
6.8.2.1 A vorticity disturbance entering a blade row in an incompressible flow......Page 357
6.8.2.2 Vorticity and pressure disturbances entering a blade row in a compressible subsonic flow......Page 359
6.8.4 Irrotational disturbances and upstream influence in a compressible flow......Page 363
6.8.5 Summary concerning small amplitude unsteady disturbances......Page 365
6.9.1 Flow due to an oscillating boundary......Page 366
6.9.2 Oscillating channel flow......Page 367
6.9.3 Unsteady boundary layers......Page 369
6.9.4 Dynamic stall......Page 372
6.9.5 Turbomachinery wake behavior in an unsteady environment......Page 373
7.1.1 Equations of motion in a rotating coordinate system......Page 376
7.1.2 Rotating coordinate systems and Coriolis accelerations......Page 378
7.2 Illustrations of Coriolis and centrifugal forces in a rotating coordinate system......Page 382
7.3 Conserved quantities in a steady rotating flow......Page 384
7.4.1 Non-dimensional parameters: the Rossby and Ekman numbers......Page 386
7.4.2 Inviscid flow at low Rossby number: the Taylor–Proudman Theorem......Page 387
7.4.3 Viscous flow at low Rossby number: Ekman layers......Page 388
7.5 Changes in vorticity and circulation in a rotating flow......Page 392
7.6.1 Inviscid flow......Page 394
7.6.2 Coriolis effects on boundary layer mixing and stability......Page 396
7.7.1 Generation of cross-plane circulation in a rotating passage......Page 398
7.7.2 Fully developed viscous flow in a rotating square duct......Page 402
7.7.3 Comments on viscous flow development in rotating passages......Page 407
7.8.1 Quasi-one-dimensional approximation: irrotational absolute flow......Page 409
7.8.2 Two-dimensional inviscid flow in a rotating diffusing blade passage......Page 411
7.8.3 Effects of rotation on diffuser performance......Page 413
7.9 Features of the relative flow in axial turbomachine passages......Page 414
8.1 Introduction......Page 418
8.2 Incompressible, uniform density, inviscid swirling flows in simple radial equilibrium......Page 419
8.2.1 Examples of simple radial equilibrium flows......Page 420
8.2.2 Rankine vortex flow......Page 422
8.3 Upstream influence in a swirling flow......Page 423
8.4 Effects of circulation and stagnation pressure distributions on upstream influence......Page 426
8.5 Instability in swirling flow......Page 433
8.6.1 Control volume equations for a vortex core......Page 435
8.6.2 Wave propagation in unconfined geometries......Page 437
8.6.3 Wave propagation and flow regimes in confined geometries: swirl stabilization of Kelvin–Helmholtz instability......Page 439
8.7.1 Pressure gradients along a vortex core centerline......Page 440
8.7.3 Applicability of the Rankine vortex model......Page 443
8.8.1 Unconfined geometries (steady vortex cores with specified external pressure variation)......Page 445
8.8.2 Confined geometries (steady vortex cores in ducts with specified area variation)......Page 449
8.8.3 Discontinuous vortex core behavior......Page 451
8.9.1 Swirling flow boundary layers on stationary surfaces and separation in swirling flow......Page 455
8.9.2 Swirling flow boundary layers on rotating surfaces......Page 460
8.9.3 The enclosed rotating disk......Page 462
8.9.4 Internal flow in gas turbine engine rotating disk cavities......Page 463
8.10 Swirling jets......Page 466
8.11 Recirculation in axisymmetric swirling flow and vortex breakdown......Page 469
9.2.1 Qualitative description......Page 475
9.2.2 A simple estimate for streamwise vorticity generation and cross-flow plane velocity components......Page 477
9.3.1 Outflow of swirling fluid from a container......Page 480
9.3.2 Secondary flow in an S-shaped duct......Page 484
9.3.3 Streamwise vorticity and secondary flow in a two-dimensional contraction......Page 485
9.3.4 Three-dimensional flow in turbine passages......Page 486
9.4.1 Incompressible uniform density fluid......Page 490
9.4.2 Incompressible non-uniform density fluid......Page 492
9.4.3 Perfect gas with constant specific heats......Page 493
9.5.1 Approximations based on convection of vorticity by a primary flow......Page 494
9.5.2 Flow with large distortion of the stream surfaces......Page 495
9.6 Three-dimensional boundary layers: further remarks on effects of viscosity in secondary flow......Page 498
9.7.1 Absolute vorticity as a measure of secondary circulation......Page 501
9.7.2 Generation of secondary circulation in a rotating reference frame......Page 502
9.7.3 Expressions for, and examples of, secondary circulation in rotating systems......Page 503
9.7.3.1 Secondary flow in a rotating straight pipe......Page 504
9.7.3.2 Secondary flow in a rotating axial to radial bend (impeller)......Page 505
9.8 Secondary flow in rotating machinery......Page 506
9.8.1 Radial migration of high temperature fluid in a turbine rotor......Page 507
9.9.1 Lobed mixers and streamwise vorticity generation......Page 510
9.9.2.1 Effect of the strain rate on mixing......Page 513
9.9.2.2 Mixing enhancement for a two-dimensional laminar vortex......Page 516
9.9.2.3 Mixing enhancement for a distribution of streamwise vorticity......Page 517
9.9.2.4 Estimation of mixing enhancement for a turbulent vortex......Page 518
9.9.3 Additional aspects of mixing enhancement in lobed mixers......Page 520
9.10 Fluid impulse and vorticity generation......Page 523
9.10.1 Creation of a vortex ring by a distribution of impulses......Page 524
9.10.2 Fluid impulse and lift on an airfoil......Page 526
9.10.3 Far field behavior of a jet in cross-flow......Page 528
9.10.3.1 Features of velocity and vorticity flelds for a jet in cross-flow......Page 529
9.10.3.2 An approximate analysis for jet kinematics and vortex pair circulation......Page 530
10.2 Corrected flow per unit area......Page 535
10.3.1 Differential equations for one-dimensional flow......Page 538
10.3.3 Effects of shaft work and body forces......Page 541
10.4.1 Constant area adiabatic flow with friction......Page 546
10.4.2 Constant area frictionless flow with heat addition......Page 547
10.4.3 Results for area change, friction, and heat addition......Page 548
10.5.1 The problem of starting a supersonic flow......Page 551
10.5.2 The use of variable geometry to start the flow......Page 553
10.5.3 Starting of supersonic inlets......Page 554
10.6.1 Turbomachinery blade passages......Page 556
10.6.2 Shock wave patterns in ducts and shock train behavior......Page 557
10.7 Extensions of the one-dimensional concepts – I: Axisymmetric compressible swirling flow......Page 561
10.7.1 Development of equations for compressible swirling flow......Page 562
10.7.2 Application of influence coefficients for axisymmetric compressible swirling flow......Page 566
10.7.2.2 Planar isentropic swirling flow......Page 568
10.7.2.3 Pressure distribution in a swirling flow......Page 571
10.7.2.4 Choking in a swirling flow......Page 572
10.7.3 Behavior of corrected flow per unit area in a compressible swirling flow......Page 573
10.8.1 Introduction to compound flow: two-stream low Mach number (incompressible) flow in a converging nozzle......Page 575
10.8.2 Qualitative considerations for multistream compressible flow......Page 578
10.8.3 Compound-compressible channel flow theory......Page 580
10.8.4 One-dimensional compound waves......Page 583
10.8.5.1 Conceptual solution procedure and non-dimensional parameter specification......Page 585
10.8.5.2 Converging nozzle......Page 586
10.8.5.3 Converging–diverging nozzle......Page 587
10.8.5.5 Experimental results for compound-compressible nozzle flows......Page 591
10.9 Flow angle, Mach number, and pressure changes in isentropic supersonic flow......Page 593
10.9.1 Differential relationships for small angle changes......Page 594
10.9.2 Relationship for finite angle changes: Prandtl–Meyer flows......Page 596
10.10 Flow field invariance to stagnation temperature distribution: the Munk and Prim substitution principle......Page 598
10.10.1 Two-dimensional flow......Page 599
10.10.2 Three-dimensional flow......Page 601
10.10.3 Flow from a reservoir with non-uniform stagnation temperature......Page 602
11.1 Introduction: sources of heat addition......Page 604
11.2 Heat addition and vorticity generation......Page 606
11.3 Stagnation pressure decrease due to heat addition......Page 608
11.4.1 The H–K diagram......Page 611
11.4.2 Flow processes in ramjet and scramjet systems......Page 615
11.5 An illustration of the effect of condensation on compressible flow behavior......Page 619
11.6 Swirling flow with heat addition......Page 621
11.6.1 Results for vortex core behavior with heat addition......Page 625
11.7.1 Equations for flow with heat addition and mixing......Page 628
11.7.2 Two-stream mixing as a model problem – I: Constant area, low Mach number, uniform inlet stagnation pressure......Page 630
11.7.3 Two-stream mixing as a model problem – II: Non-uniform inlet stagnation pressures......Page 633
11.7.4 Effects of inlet Mach number level......Page 634
11.8.1 Lobed mixer nozzles......Page 636
11.8.2 Jets......Page 638
11.8.3 Ejectors......Page 639
11.8.4 Mixing of streams with non-uniform densities......Page 642
11.8.5 Comments on the approximations......Page 643
12.1 Introduction......Page 644
12.2 An illustrative example of flow modeling: two-dimensional steady non-uniform flow through a screen......Page 645
12.2.1 Velocity and pressure field upstream of the screen......Page 646
12.2.3 Matching conditions across the screen......Page 649
12.2.4 Overall features of the solution......Page 651
12.2.5 Nonlinear effects......Page 654
12.2.6 Disturbance length scales and the assumption of inviscid flow......Page 655
12.3.1 Flow through a uniform inclined screen......Page 657
12.3.2 Pressure drop and velocity field with partial duct blockage......Page 658
12.3.3 Enhancing flow uniformity in diffusing passages......Page 660
12.4 Upstream influence and component interaction......Page 663
12.5 Non-axisymmetric (asymmetric) flow in axial compressors......Page 666
12.5.1 Flow upstream of the compressor......Page 667
12.5.2 Flow downstream of the compressor......Page 668
12.5.3 Matching conditions across the compressor......Page 669
12.5.4 Behavior of the axial velocity and upstream static pressure......Page 670
12.5.5 Generation of non-uniform flow by circumferentially varying tip clearance......Page 673
12.6.1 Turbine engine effects on inlet performance......Page 674
12.6.2 Strut-vane row interaction: upstream infiuence with two different length scales......Page 676
12.7 Unsteady compressor response to asymmetric flow......Page 677
12.7.1 Self-excited propagating disturbances in axial compressors and compressor instability......Page 680
12.7.2 A deeper look at the effects of circumferentially varying tip clearance......Page 682
12.7.3 Axial compressor response to circumferentially propagating distortions......Page 683
12.8 Nonlinear descriptions of compressor behavior in asymmetric flow......Page 684
12.9 Non-axisymmetric flow in annular diffusers and compressor–component coupling......Page 687
12.9.1 Quasi-two-dimensional description of non-axisymmetric flow in an annular diffuser......Page 690
12.9.2 Features of the diffuser inlet static pressure field......Page 692
12.9.3 Compressor–component coupling......Page 695
12.10 Effects of flow non-uniformity on diffuser performance......Page 697
12.11 Introduction to non-axisymmetric swirling flows......Page 702
12.11.1 A simple approach for long length scale non-uniformity......Page 704
12.11.3 Flow angle disturbances......Page 706
12.11.5 Overall features of non-axisymmetric swirling flow......Page 707
12.11.6 A secondary flow approach to non-axisymmetric swirling flow......Page 711
References......Page 712
Index (with page links)......Page 729