1 Introduction ................................................. 1
1.1 Flows and Flow Measurements ............................. 1
1.2 Traditional Methods of Flow Measurements ................ 2
1.3 Laser Methods and Laser Doppler Anemometry (LDA) ........ 2
1.3.1 Developments of LDA Fundamentals and
Instrumentations ................................. 4
1.3.2 Developments of LDA Application Methods .......... 5
1.4 Purposeful Flow Measurements and Rational Measurement
Evaluations ............................................. 8
1.5 Purposes of this Book ................................... 9
2 Specifications of Engineering Turbulent Flows ............... 11
2.1 Turbulent Flow Properties .............................. 11
2.1.1 Statistical Views of Flow Turbulences ........... 11
2.1.2 Isotropic and Anisotropic Turbulences ........... 13
2.2 Reynolds Turbulent Stresses ............................ 15
3 LDA Principles and Laser Optics ............................. 19
3.1 Light Wave and Its Propagation ......................... 19
3.2 The Doppler Effect ..................................... 22
3.3 Superposition of Two Plane Light Waves ................. 24
3.4 LDA Principle .......................................... 27
3.5 Fringe Model on the Light Interference ................. 29
3.6 Frequency Shift Method to Resolve the Flow Direction ... 32
3.6.1 Fringe Shift Speed .............................. 34
3.7 Gaussian Beam Properties ............................... 35
3.7.1 Geometrical Specifications of the Gaussian
Beam ............................................ 35
3.7.2 Transmission Performance of the Gaussian Beam ... 38
3.8 Measurement Volume Size ................................ 39
4 LDA Systems ................................................. 41
4.1 Hardware and Optical Components ........................ 41
4.2 Specification of LDA Measurement Volumes ............... 44
5 Basic Data Processing Methods in LDA Measurements ........... 47
5.1 Direct Data Processing for Mean Velocities and
Velocity Fluctuations .................................. 47
5.2 Weighting Facilities of Mean Velocity and
Fluctuations ........................................... 51
6 Linear Transformation of Velocities and Turbulent
Stresses .................................................... 53
6.1 Orthogonal Linear Transformation ....................... 53
6.1.1 Velocity Transformation ......................... 53
6.1.2 Turbulent Stress Transformation ................. 55
6.1.3 Directional Distribution of Turbulent
Stresses ........................................ 55
6.2 Non-orthogonal Transformation .......................... 61
6.2.1 Velocity Transformation ......................... 62
6.2.2 Turbulent Stress Transformation ................. 63
6.3 Graphical Presentation of Turbulent Stresses ........... 65
6.3.1 Ellipse Form of the Turbulence Distribution ..... 65
6.3.2 Expressions of Turbulent Stresses in Mohr's
Stress Circle ................................... 66
7 Tracer Particles and Particle Motion Equations .............. 69
7.1 Effective Forces Exerted on the Particle in the Flow ... 70
7.1.1 Viscous Drag Force .............................. 70
7.1.2 Gravitational and Lift Forces ................... 71
7.1.3 Pressure Force .................................. 72
7.1.4 Force from Added Mass ........................... 73
7.2 Particle Motion Equation ............................... 74
7.3 Particle Motion in the Straight Flow of Constant
Velocity ............................................... 75
7.4 Particle Motion in Nozzle and Diffuser Flows ........... 76
7.4.1 Nozzle Flow ..................................... 78
7.4.2 Diffuser Flow ................................... 80
7.5 Particle Motion in the Oscillation Flow ................ 82
7.5.1 Particle Flows of Small Stokes Numbers .......... 86
7.5.2 Particle Flows of Large Stokes Numbers .......... 88
8 Zero Correlation Method (ZCM) ............................... 89
8.1 Shear Stress Measurements with Non-coincident LDA ...... 89
8.2 Basics of ZCM .......................................... 90
8.3 Extension of ZCM ....................................... 93
8.3.1 Non-orthogonal Velocity Components .............. 93
8.3.2 Three-Dimensional Flow Turbulence ............... 93
8.4 Restriction and Validation of ZCM ...................... 94
9 Dual Measurement Method (DMM) ............................... 97
9.1 Possibility of Resolving the Secondary Flow ............ 97
9.2 DMM in Basic Form ...................................... 99
9.3 DMM with Coordinate Transformation .................... 103
9.4 Extension of DMM ...................................... 105
9.4.1 Direct Component Measurements .................. 106
9.4.2 Method of Using Coordinate Transformation ...... 109
10 Symmetrical Method of 3D-Velocity Measurements ............. 113
11 Non-stationary Turbulent Flows ............................. 117
11.1 Non-stationary Turbulent Flows in the Practice ........ 117
11.2 Time-Resolved Non-stationary Turbulent Flows .......... 119
11.2.1 Method of Linear Least Squares Fitting ......... 119
11.2.2 Linear Trend of the Velocity and the
Calculation Method ............................. 121
11.2.3 Time-Dependent Flow Turbulences ................ 123
11.3 Phase-Resolved Non-stationary Turbulent Flows ......... 126
11.3.1 Method of Linear Least Squares Fitting ......... 127
11.3.2 Linear Trend of the Velocity and the
Calculation Method ............................. 128
11.3.3 Phase-Dependent Flow Turbulences ............... 130
12 Turbulent Flow with Spatial Velocity Gradient .............. 133
12.1 Apparent Turbulence Intensity and Related
Quantities ............................................ 135
12.2 Combined Velocity Bias Effect ......................... 140
12.2.1 Mean Velocity .................................. 141
12.2.2 Turbulent Normal Stress ........................ 142
12.3 Method of Resolving the Non-uniform Velocity
Distribution .......................................... 145
13 Flow Measurements Behind the Plane Window: On-axis ......... 147
13.1 Fringe Spacing ........................................ 147
13.2 Shift of the Measurement Volume ....................... 148
13.3 Optical Dispersion and its Negligible Effect .......... 149
14 Flow Measurements Behind the Plane Window: Off-axis ........ 151
14.1 Off-axis Measurements and Velocity Transformation ..... 152
14.2 Fringe Spacing in Measurement Volume and Velocity
Corrections ........................................... 153
14.3 Refraction of Optical Axis and Orientation of the
Measurement Volume .................................... 155
14.4 Two-Dimensional Shift of the Measurement Volume ....... 156
14.5 Astigmatism and its Presence in Transmitting Optics ... 159
14.6 Astigmatism at the Focused Laser Beam Bundle .......... 163
14.6.1 One-time Refraction of a Focused Beam Bundle ... 163
14.6.2 Multiple Refraction of a Focused Beam Bundle ... 169
14.7 Measurement Volume and Its Distortion ................. 170
14.7.1 Single Refraction of Laser Beams ............... 172
14.7.2 Multiple Refractions of Laser Beams ............ 173
14.7.3 Astigmatism at the On-axis LDA Alignment ....... 174
14.8 Signal Qualities and the Lens Dependence .............. 175
14.8.1 Deterioration of Signal Qualities and
Strengths ...................................... 175
14.8.2 Lens Dependence of Signal Qualities and
Strengths ...................................... 176
14.9 Error Sensitivities in Forming the Measurement
Volume ................................................ 179
14.9.1 Beam Separation in the Test Medium ............. 179
14.9.2 Beam Separation After Multiple Refractions ..... 185
14.9.3 Possible Impact on PDA Measurements ............ 185
14.10 Method for Compensation of Astigmatism ............... 186
15 Flow Measurements in Circular Pipes ........................ 191
15.1 Measurements of Axial Velocities ...................... 193
15.2 Measurements of Tangential Velocities ................. 197
15.2.1 Basic Geometrical Relationships ................ 197
15.2.2 Simplifications of Calculations ................ 198
15.2.3 Fringe Spacing and Velocity Corrections ........ 199
15.3 Measurements of Radial Velocities ..................... 200
15.3.1 Accurate Positioning of the Measurement
Volume ......................................... 200
15.3.2 Laser Beam Intersection Angle .................. 205
15.3.3 Fringe Spacing and Velocity Corrections ........ 206
15.3.4 Orientation of the Measurement Volume .......... 207
15.3.5 Determination of Radial Velocities ............. 208
15.3.6 Remarks on the Method .......................... 208
15.4 Optical Aberrations and Measurement Volume
Distortion ............................................ 209
15.4.1 Optical Aberrations in Transmitting and
Receiving Optics ............................... 210
15.4.2 Dislocation of Laser Beam Waists from the
Measurement Volume ............................. 211
16 Fringe Distortion Effects .................................. 219
16.1 Linear Longitudinal Distribution of the Fringe
Spacing ............................................... 220
16.2 Fringe Distortion Number and the Apparent Mean
Velocity .............................................. 221
16.3 Overestimation of the Flow Turbulence ................. 224
17 Velocity Bias Effects ...................................... 227
17.1 Velocity Bias as a Flow Phenomenon .................... 227
17.2 Velocity Bias and the Momentum Flow Rate .............. 229
17.3 Velocity Bias in One-Dimensional Flow Fluctuations .... 231
17.4 Velocity Bias in Two- and Three-Dimensional Flow
Fluctuations .......................................... 235
17.4.1 Velocity Bias in Mean Velocities ............... 236
17.4.2 Velocity Bias in Turbulent Normal Stresses ..... 238
17.4.3 Velocity Bias in Turbulent Shear Stresses ...... 241
18 LDA Application Examples ................................... 243
18.1 High Speed Water Jet Flow in a Pelton Turbine ......... 243
18.2 Measurements of Warp Yarn Speed in a Weaving
Machine ............................................... 247
18.3 Verification of the Shift Frequency in the Laser
Beam .................................................. 249
Appendix A Off-axis LDA Alignment and Measurement Volume
Displacement ............................................... 253
A.1 Laser Beams in the Meridian Plane ..................... 254
A.2 Laser Beams in the Sagittal Plane ..................... 255
A.3 Combination ........................................... 257
Appendix В Laser Beam Orientation Under the Effect of the
Bias Angle δ ............................................... 259
Appendix С Coordinate Transformation of the Reynolds
Stress Matrix .............................................. 263
References .................................................... 267
Index ......................................................... 271
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