A NUMERICAL AND EXPERIMENTAL INVESTIGATION OF TAYLOR FLOW
INSTABILITIES IN NARROW GAPS AND THEIR RELATIONSHIP TO
TURBULENT FLOW IN BEARINGS
A Dissertation
Presented to
The Graduate Faculty of The University of Akron
In Partial Fulfillment
of the Requirements for the Degree
Doctor of Philosophy
Dingfeng Deng
August, 2007
A NUMERICAL AND EXPERIMENTAL INVESTIGATION OF TAYLOR FLOW
INSTABILITIES IN NARROW GAPS AND THEIR RELATIONSHIP TO
TURBULENT FLOW IN BEARINGS
Dingfeng Deng
Dissertation
Approved: Accepted:
______Advisor Department Chair Dr. M. J. Braun Dr. C. Batur
______Committee Member Dean of the College Dr. J. Drummond Dr. G. K. Haritos
______Committee Member Dean of the Graduate School Dr. S. I. Hariharan Dr. G. R. Newkome
______Committee Member Date R. C. Hendricks
______Committee Member Dr. A. Povitsky
______Committee Member Dr. G. Young ii
ABSTRACT
The relationship between the onset of Taylor instability and appearance of what is
commonly known as “turbulence” in narrow gaps between two cylinders is investigated.
A question open to debate is whether the flow formations observed during Taylor
instability regimes are, or are related to the actual “turbulence” as it is presently modeled
in micro-scale clearance flows.
This question is approached by considering the viscous fluid flow in narrow gaps
between two cylinders with various eccentricity ratios. The computational engine is
provided by CFD-ACE+, a commercial multi-physics software. The flow patterns,
velocity profiles and torques on the outer cylinder are determined when the speed of the
inner cylinder, clearance and eccentricity ratio are changed on a parametric basis.
Calculations show that during the Taylor vortex regime velocity profiles in the radial
direction are sinusoidal with pressure variations in the axial direction even for the case of
the “long journal bearing” (L/D>2). For the concentric case ( ε = 0), both velocity and pressure profiles are axisymmetric and time-independent during the Taylor vortex regime. During the wavy vortex regime the radial velocities maintain their sinusoidal profiles, while pressure varies in both axial and circumferential directions. Both velocity and pressure profiles are non-axisymmetric and time-dependent. An order of magnitude analysis of the Navier-Stokes equation terms shows that the inertia, viscous, pressure and
iii the Reynolds stress terms are equally significant during the transition regime (Taylor to wavy vortex regimes).
Based on these findings, a new model for predicting the flow behavior in long and short journal bearing films in the transition regime is proposed. The new model indicates that the velocity profiles are sinusoidal and depend on the local Reynolds number (or
Taylor number) and the position in the axial direction. Unlike the modified turbulent viscosity of the most accepted models (Constantinescu, Ng-Pan, Hirs and Gross et al.), the viscosity used in the new model is kept at its laminar value. A comparison is made between the results of this model and the four most accepted turbulence models mentioned above.
Experimental torque measurements and flow visualization are performed for three kinds of oils with different viscosities. It is shown that in general there is a good agreement between the numerical and experimental torques except those in turbulent regime. The different flow patterns in the turbulent regime accounts for the difference between the numerical and experimental torques. Comparison between numerical and experimental flow patterns is also made and it shows that they match well in the Couette,
Taylor and Wavy regimes. There is a discrepancy between numerical and experimental flow patterns in Pre-wavy regime. The reason is that the vorticity is strong enough to destroy the organized Taylor vortices but it is not strong enough to hold the particles in the center areas of the vortices during experiments in this regime.
In general there is a good agreement between the numerical and experimental results including torque measurements and flow patterns. The new model for predicting the flow behavior in journal bearing films in the transition regime is justified. iv
ACKNOWLEDGEMENTS
I would like to express my appreciation and gratitude to my advisor, Dr. M.J. Braun, for his continuous and invaluable advisement. I am glad that I met Dr. Braun during my first semester of graduate school. Since then he has been my professional advisor, life mentor and personal friend. It is obvious that I have achieved my goals under the five years’ guidance of Dr. Braun.
I would like to thank my committee members, Dr. Drummond, Dr. Hariharan, Mr.
Hendrick, Dr. Povitsky and Dr. Young, for their comments and guidance during the
course of my dissertation.
I would also like to thank Dale Ertley for his work on building the installation for my
experiments and Bob Shardy for his work on installing and trouble-shooting the CFD
software package.
Additionally I received unlimited support and courage from my friends and family. I
would not have accomplished this without them.
v
TABLE OF CONTENTS
Page
LIST OF TABLES...... xii
LIST OF FIGURES...... xiv
NOMENCLATURE...... xxiii
CHAPTER
I. THEORETICAL INTRODUCTION AND LITERATURE REVIEW...... 1
1.1 Introduction...... 1
1.2 Theories of Fluid Instability in Concentric Cylinders and Literature Review...... 4
1.2.1 Rayleigh’s Stability Criterion...... 4
1.2.2 Linear Theory of the Instability...... 8
1.2.3 Literature Review on Linear Theory...... 12
1.2.4 Weakly Nonlinear Theory...... 15
1.2.5 Literature Review on Weakly Nonlinear Theory...... 18
1.2.6 Theory of Wavy Taylor Vortices and Literature Review...... 20
1.3 Theories of Fluid Instability in Eccentric Cylinders and Literature Review...... 25
1.3.1 DiPrima’s Local Theory and Literature Review...... 25
1.3.2 DiPrima and Stuart’s Non-local Theory and Literature Review...... 31
1.4 Theories of Turbulence in Bearings and Literature Review...... 33
vi 1.4.1 Constantinescu’s Theory...... 33
1.4.2 Ng-Pan Theory...... 37
1.4.3 Bulk Flow Theory of Hirs...... 39
1.4.4 Literature Review on Theories of Turbulence in Bearings...... 42
II. SCOPE OF WORK...... 47
2.1 Numerical Simulations and Calculations of Long and Short Bearings...... 47
2.2 The Nature of “Instability” & “Turbulence” in Small Gap Journal Bearings...... 47
2.3 New Models for Transition Flow of Thin Films in Long and Short Bearings...... 48
2.4 Experimental Verification of the Numerical Results...... 49
III. NUMERICAL ALGORITHM...... 50
3.1 General Introduction to CFD-ACE+...... 50
3.2 Numerical Methodology Adopted by CFD-ACE+...... 50
3.2.1 Discretization...... 51
3.2.2 Velocity-Pressure Coupling...... 55
3.2.3 Boundary Conditions...... 57
3.2.4 Solution Methods...... 58
3.3 Flow Module...... 61
IV. NUMERICAL RESULTS (LONG BEARING)...... 62
4.1 Introduction...... 62
4.2 Geometry and Boundary Condition Applications...... 63
4.2.1 Geometry...... 63
4.2.2 Boundary Condition Applications ...... 67 vii 4.3 Convergence Criteria and Numerical Accuracy...... 67
4.3.1 Gridding and Grid Convergence...... 67
4.3.2 Time Step and Time Step Convergence...... 69
4.4 Multiplicity and Transition of Taylor-Couette Flow...... 71
4.4.1 Introduction...... 71
4.4.2 The Influence of T b on the Number of Taylor Cells...... 73
4.4.3 The Influence of T b on the Number of Waves...... 82
4.4.4 Summary on Multiplicity and Transition...... 88
4.5 Effect of Clearance on Taylor Vortices Induced Instability...... 88
4.5.1 Concentric Cylinder Case with Clearance of 0.01 in...... 89
4.5.2 Concentric Cylinder Case with Clearance of 0.13 in...... 104
4.5.3 Effect of Clearance on Taylor Vortices Induced Instability...... 126
4.6 Effect of Eccentricity Ratio on Taylor Vortices Induced Instability...... 129
4.6.1 Effect of Eccentricity Ratio on the Torque and Magnitude of Critical Taylor Number...... 129
4.6.2 Effect of Eccentricity Ratio on the Friction Factor...... 131
4.6.3 Effect of Eccentricity Ratio on Maximum Vortex Intensity...... 132
4.6.4 Effect of Eccentricity Ratio on Recirculation...... 134
4.7 A New Model for Transition Flow of Thin Films in Long Journal Bearings...... 135
4.7.1 The Geometry and Coordinates ...... 136
4.7.2 The Case with ε = 0 ...... 136
4.7.3 The Case with ε = 0.2 ...... 144
4.7.4 Proposed Transition Reynolds Equation Model ...... 146 viii 4.7.5 Coefficients Used in Transition Reynolds Equation Model ...... 148
4.7.6 Comparison between Our Model and Turbulence Models...... 158
V. NUMERICAL RESULTS (SHORT BEARING)...... 165
5.1 Introduction...... 165
5.2 Geometry and Coordinates...... 165
5.3 The Case with ε = 0...... 167
5.4 The Case with ε = 0.2...... 174
5.5 Proposed Transition Reynolds Equation Model for Short Bearings...... 182
5.6 Comparison between our Model and Turbulence Models for Short Bearing...183
VI. DESCRIPTION OF EXPERIMENTAL INSTALLATION ...... 186
6.1 Test Loop...... 186
6.2 Drive System...... 187
6.3 Test Section (Apparatus)...... 188
6.4 Torque Measuring System...... 191
6.5 Visualization System ...... 192
6.5.1 Creation of the Laser Sheet...... 192
6.5.2 Visualization of the Test Section ...... 192
6.5.3 The Tracer Particels ...... 195
6.6 Working Fluid...... 195
VII. CALIBRATIONS...... 197
7.1 Calibration of the Strain Gage ...... 197
7.2 Torque Measurements for the Case of Air between the Cylinders...... 203
VIII. EXPERIMENTAL PROCEDURE ...... 205 ix 8.1 Experimental Start-up ...... 205
8.1.1 Preparation of the Experimental Apparatus...... 205
8.1.2 Preparation of the Experiments...... 206
8.2 Experimental Procedure...... 207
8.2.1 Quick Pass...... 207
8.2.2 Individual Regime Study ...... 208
IX. EXPERIMENTAL RESULTS ...... 209
9.1 Results for the Case of Oil I [ = 0.006 kg/(m-s), ρ = 778 kg/m 3]...... 209
9.1.1 Torque Measurements for the Case of Oil I...... 209
9.1.2 Visualization of Flow Patterns for the Case of Oil I...... 213
9.2 Results for the Case of Oil II [ = 0.038 kg/(m-s), ρ = 961 kg/m 3]...... 231
9.2.1 Torque Measurements for the Case of Oil II ...... 231
9.2.2 Visualization of Flow Patterns for the Case of Oil II ...... 234
9.3 Results for the Case of Oil III [ = 0.099 kg/(m-s), ρ = 1048 kg/m 3] ...... 247
9.3.1 Torque Measurements for the Case of Oil III...... 248
9.3.2 Visualization of Flow Patterns for the Case of Oil III...... 251
X. COMPARISON BETWEEN NUMERICAL AND EXPERIMENTAL RESULTS..265
10.1 Comparison of Torque-speed Graphs ...... 265
10.2 Comparison of Flow Patterns for the Case of Oil I ...... 269
XI. CONCLUSIONS ...... 283
11.1 Conclusion to the Numerical Investigation...... 283
11.2 Conclusion to the Experimental Verification ...... 285
11.3 Future Work...... 286 x BIBLIOGRAPHY...... 288
APPENDICES...... 295
APPENDIX A: THE RESULTS OF CALIBRATION OF STRAIN GAGE...... 296
APPENDIX B: VOLTAGE MEASUREMENTS FOR THE CASE OF AIR...... 298
APPENDIX C: VOLTAGE MEASUREMENTS FOR THE CASE OF OIL I...... 300
APPENDIX D: VOLTAGE MEASUREMENTS FOR THE CASE OF OIL II....302
APPENDIX E: VOLTAGE MEASUREMENTS FOR THE CASE OF OIL III ...304
xi
LIST OF TABLES
Table Page
1.1 Critical Parameters for the Onset of Taylor Vortices for Various Values of η ...... 14
1.2 Comparison of Coefficients for the Calculations of Taylor Vortex Torques...... 19
1.3 Constants for Calculating k x and k z at Different Reynolds Numbers...... 39
4.1 Grid Convergence Experiments (C = 0.13 in., ε = 0.0 and ω = 1,550 rpm)...... 68
4.2 Time Step Convergence Experiments (C = 0.13 in., ε = 0.0 and ω = 4,000 rpm)...... 70
4.3 Torque and Flow Type vs. Speed for the Case of C = 0.01 in., ε = 0.0...... 89
4.4 Torque and Flow Type vs. Speed for the Case of C = 0.13 in., ε = 0.0...... 104
4.5 Positions of Points on a Typical Trajectory in Taylor Vortex Regime (Axial Direction)...... 121
4.6 Positions of Points on a Typical Trajectory in Taylor Vortex Regime (Radial Direction)...... 121
4.7 Positions of Points on a Typical Trajectory in Wavy Vortex Regime (Axial Direction)...... 122
4.8 Positions of Points on a Typical Trajectory in Wavy Vortex Regime (Radial Direction)...... 122
4.9 Recirculation Region with Different Speed when ε = 0.6...... 135
4.10 Order of Magnitude Analysis of the Navier-Stokes Equation in x-direction (C = 0.01 in., ε = 0.2)...... 162
xii
4.11 Order of Magnitude Analysis of the Navier-Stokes Equation in z-direction (C = 0.01 in., ε = 0.2)...... 163
6.1 Properties of the Working Fluid (Oil I)...... 195
6.2 Properties of the Working Fluid (Oil II)...... 196
6.3 Properties of the Working Fluid (Oil III)...... 196
7.1 Correlation Coefficients of the Curve Fit for the Calibration of the Strain Gage...... 200
9.1 Correlation Coefficients of the Curve Fit for Couette Flow (Oil I)...... 210
9.2 Correlation Coefficients of the Curve Fit for Pre-wavy Flow (Oil I)...... 211
9.3 Correlation Coefficients of the Curve Fit for Wavy Flow (Oil I)...... 211
9.4 Correlation Coefficients of the Curve Fit for Turbulent Flow (Oil I)...... 211
9.5 Correlation Coefficients of the Curve Fit for Couette Flow (Oil II)...... 233
9.6 Correlation Coefficients of the Curve Fit for Pre-wavy Flow (Oil II)...... 233
9.7 Correlation Coefficients of the Curve Fit for Wavy Flow (Oil II)...... 234
9.8 Correlation Coefficients of the Curve Fit for Couette Flow (Oil III)...... 250
9.9 Correlation Coefficients of the Curve Fit for Taylor Flow (Oil III)...... 250
9.10 Correlation Coefficients of the Curve Fit for Pre-wavy Flow (Oil III)...... 251
9.11 Correlation Coefficients of the Curve Fit for Wavy Flow (Oil III)...... 251
xiii
LIST OF FIGURES
Figure Page
1.1 Velocity Components in Cylindrical Coordinates...... 5
1.2 Rayleigh’s Stability Diagram [After Koschmieder (1993)]...... 7
1.3 Coordinates of Concentric Cylinders...... 9
1.4 The Relationship among Reynolds Number, Taylor Number, Curvature Effect and Flow Pattern...... 26
1.5 A Plot of Values of Rr C / R1 & RP C / R1 for Stable & Unstable Flow...... 27
1/2 1.6 Critical Value of Rr(C/R1) as a Function of ε for Various Values of θ...... 29
1.7 Critical Speed as a Function of ε for Various Values of Clearance Ratio...... 30
1.8 Lower and Upper Bounds for Instability in Bearing-like Clearance Ratios...... 32
1.9 Turbulence Coefficients G x and G z vs. Reynolds Number...... 43
1.10 Friction Factor vs. Reynolds Number...... 44
1.11 Operating Ranges for Typical Oil and Air Bearings...... 45
3.1 A Three-dimensional Computational Cell (Control Volume)...... 51
3.2 A Two-dimensional Cell (Control Volumes) ...... 52
3.3 Computational Boundary Cell ...... 57
3.4 Solution Flowchart...... 59
xiv 4.1 Geometric Description of the Cylinders (Not at scale)...... 63
4.2 Torque vs. Taylor Number for Various λs when C = 0.01 in. and ε = 0.0 ...... 64
4.3 Torque vs. Taylor Number for Different Aspect Ratio (L/C = 4 and 6)...... 64
4.4 Wavelength of Taylor Vortices for Different Aspect Ratios (L/C=2, 4 and 6) ...65
4.5 Torque vs. Time at Different CFLs (C = 0.13 in., ε = 0.0, ω = 4000 rpm, Re = 323.55, Ta = 120.65) ...... 70
4.6 Geometry for Studying the Influence of T b on the Number of Taylor Cells (Not at Scale) ...... 73
4.7 Velocity of Inner Cylinder vs. Time (T b is the Time Needed to Increase the Speed of Inner Cylinder from Rest to 263.33 m/s)...... 74
4.8 Number of Taylor Cells vs. Time...... 75
4.9 w - Velocity in Axial Direction vs. Time ...... 75
4.10 Maximum Vorticity vs. Time ...... 78
4.11 P – Generated Pressure vs. Time...... 78
4.12 The Process of Formation of Taylor Cells for the Case of Impulsive Start...... 79
4.13 The Process of Mergence of Taylor Cells for the Case of Impulsive Start ...... 80
4.14 Velocity of Inner Cylinder vs. Time...... 82
4.15 Wave Number in Circumferential Direction vs. Time...... 82
4.16 P – Generated Pressure vs. Time...... 83
4.17 The Development of Waves for the Case of T b = 0.002 s ...... 84
4.18 Torque vs. Ta and Re for the Case of C = 0.01 in., ε = 0.0 ...... 90
4.19 Velocity and Pressure Profiles (Re=283.22, Ta =28.46, C = 0.01 in.)...... 92
4.20 Trajectory of a Typical Particle in Couette Regime (Re=283.22, Ta =28.46, C = 0.01 in.) (Not at scale)...... 93
xv 4.21 Velocity and Pressure Profiles (Re=424.82, Ta =42.70, C = 0.01 in.)...... 94
4.22 Trajectory of a Typical Particle in Taylor Regime (Re=424.82, Ta =42.70, C = 0.01 in.) (Not at scale)...... 95
4.23 Velocity and Pressure Profiles (Re=708.04, Ta =71.16, C = 0.01 in.)...... 96
4.24 Velocity and Pressure Profiles (Re=708.04, Ta =71.16, C = 0.01 in.) (Not at scale)...... 97
4.25 Trajectory of a Typical Particle in Wavy Regime (Re=708.04, Ta =71.16, C = 0.01 in.) (Not at scale)...... 98
4.26 Velocity and Pressure Profiles of One Wave when Ta =71.16 [Not at scale in b) and c)]...... 103
4.27 Torque vs. Ta and Re for the Case of C = 0.13 in., ε = 0.0 ...... 105
4.28 Velocity and Pressure Profiles (C=0.13 in., Ta =30.16, A in Figure 4.25) ...... 108
4.29 Velocity and Pressure Profiles (C=0.13 in., Ta =46.75, B in Figure 4.25) ...... 109
4.30 Velocity and Pressure Profiles (C=0.13 in., Ta =66.35, C in Figure 4.25) ...... 112
4.31 w-velocity and Pressure Contours at Different z for the Wavy Vortex Regime (C=0.13 in., ω=4000 rpm, Re=323.55, Ta =120.65), D in Figure 4.25...... 113
4.32 w-velocity and Pressure Contours at Different r for the Wavy Vortex Regime (C=0.13 in., ω=4000 rpm, Re=323.55, Ta =120.65), D in Figure 4.25...... 114
4.33 w-velocity and Pressure Contours with Superimposed Velocity Vectors at Different θ for the Wavy Vortex Regime, D in Figure 4.25...... 115
4.34 Comparison of Velocity and Pressure Contours for Taylor Vortex Flow and Wavy Vortex Flow (C = 0.13 in.) ...... 117
4.35 Comparison of Iso-curves and Iso-surfaces of w and P for Taylor Vortex Flow and Wavy Vortex Flow (C = 0.13 in.) ...... 118
4.36 Trajectory of a Typical Particle in Taylor Vortex Regime (Re=125.38, Ta =46.75, C = 0.13 in.)...... 123 xvi 4.37 Trajectory of a Typical Particle in Wavy Vortex Regime (Re=323.55, Ta =120.65, C = 0.13 in.)...... 124
4.38 Comparison of Our Calculations and Experiments by Koschmieder ...... 125
4.39 Comparison of Velocity Profiles of Different Clearances...... 126
4.40 Torque vs. Taylor No. and Reynolds No. for Various Clearances ...... 127
4.41 First and Second Critical Taylor Numbers for Various Clearances ...... 128
4.42 Torque - Ta and Ta cr - Eccentricity Ratio Curves...... 130
4.43 Friction Factor vs. Reynolds Number when C = 0.01 in., ε = 0.0 – 0.8 ...... 131
4.44 Flow Pattern at Different Positions when ε = 0.2, ω = 65,000 rpm...... 132
4.45 Flow Pattern at Different Positions when ε = 0.6, ω = 110,000 rpm...... 133
4.46 Geometry and Coordinates of a Journal Bearing Film ...... 136
4.47 Torque vs. Ta When C = 0.01 in., ε = 0.0 ...... 137
4.48 Velocity and Pressure Profiles as Functions of y* during Couette Region ...... 138
4.49 Velocity and Pressure Profiles as Functions of y* during Taylor Region...... 139
4.50 Velocity and Pressure Profiles as Functions of y* during Wavy Vortex Region at a Given Time t = T (0) ...... 140
4.51 Pressure Contour in the x-z Plane during Wavy Vortex Regime ...... 144
4.52 Pressure Profiles as Functions of z* during Wavy Vortex Region at a Given Time t = T (0) ...... 146
4.53 Relationship between Velocity Gradient on Outer Cylinder and U*...... 149
4.54 Coefficients a 1 as Function of Reynolds Number for C/R = 0.01 ...... 150
4.55 Coefficients a 2 as Function of Reynolds Number for C/R = 0.01 ...... 150
4.56 Coefficients a 1 as Function of Reynolds Number for C/R = 0.004 ...... 151
4.57 Coefficients a 2 as Function of Reynolds Number for C/R = 0.004 ...... 151 xvii 4.58 Coefficients a 1 as Function of Reynolds Number for C/R = 0.001 ...... 152
4.59 Coefficients a 2 as Function of Reynolds Number for C/R = 0.001 ...... 152
4.60 Coefficients A 1 as Function of Reynolds Number for Different C/R...... 153
4.61 Coefficients A 2 as Function of Reynolds Number for Different C/R...... 153
4.62 Coefficients A 1 as Function of Ta for Different C/R...... 155
4.63 Coefficients A 2 as Function of Ta for Different C/R...... 155
4.64 Coefficients A 1 as Function of Inverse of Ta for Different C/R...... 156
4.65 Coefficients A 2 as Function of Inverse of Ta for Different C/R...... 156
4.66 Comparison of Torque on Bearing between Our Model and Turbulence Models as Functions of Reynolds Number...... 158
4.67 Comparison of Torque on Bearing between Our Model and Turbulence Models as Functions of Taylor Number ...... 158
5.1 Geometric Description of the Cylinders (Not at scale)...... 166
5.2 Geometry and Coordinates of a Journal Bearing Film ...... 166
5.3 Torque vs. Ta for Short Bearing When C = 0.01 in., ε = 0.0 ...... 167
5.4 Velocity and Pressure Contours during Couette Regime (Point A in Figure 5.3)...... 168
5.5 Velocity and Pressure Profiles as Functions of y* and z* during Couette Regime (Point A in Figure 5.3)...... 169
5.6 Velocity and Pressure Contours during Taylor Vortex Regime (Point B in Figure 5.3) ...... 171
5.7 Velocity and Pressure Profiles as Functions of y* and z* during Taylor Vortex Regime (Point B in Figure 5.3)...... 172
5.8 Torque vs. Ta for Short Bearing When C = 0.01 in., ε = 0.2 ...... 175
xviii 5.9 Velocity and Pressure Contours during Couette Regime (Point C in Figure 5.8) ...... 176
5.10 Velocity and Pressure Profiles as Functions of y* and z* during Couette Regime (Point C in Figure 5.8)...... 177
5.11 Velocity and Pressure Contours during Taylor Vortex Regime (Point D in Figure 5.8)...... 178
5.12 Velocity and Pressure Profiles as Functions of y* and z* during Taylor Vortex Regime (Point D in Figure 5.8) ...... 179
5.13 Comparison of Torque on Bearing between Our Model and Turbulence Models for Short Bearing...... 184
6.1 Schematic Drawing of Test Loop ...... 186
6.2 Driving System and Test Section...... 187
6.3 Schematic Drawing of Test Section...... 189
6.4 Torque Measuring System...... 190
6.5 Visualization of the Fluid in Longitudinal Cross Section View...... 193
6.6 Visualization of the Fluid in Front View...... 194
7.1 The Principle of the Torque Measurement ...... 197
7.2 Calibration of the Strain Gage ...... 198
7.3 Calibration Curve of Force-Voltage ...... 201
7.4 Curve-fit of the Calibration of the Strain Gage ...... 203
7.5 Torque Measurements for the Case of Air between the Cylinders...... 204
9.1 Voltage Measurements for the Case of Oil I ...... 212
9.2 Voltage due to Fluid Friction vs. Speed for the Case of Oil I ...... 212
9.3 Flow Pattern in Longitudinal Cross Section View in Couette Regime (Oil I) ....213
9.4 Flow Pattern in Longitudinal Cross Section View in Taylor Regime (Oil I) ...... 214
xix 9.5 The Collapse of Taylor Vortices for the Case of ω = 200 rpm (Oil I)...... 215
9.6 Flow Pattern in Longitudinal Cross Section View in Pre-wavy Regime (Oil I)...... 216
9.7 Flow Pattern in Longitudinal Cross Section View in Wavy Regime (Oil I) ...... 218
9.8 Flow Pattern in Longitudinal Cross Section View in Turbulent Regime (Oil I) ...... 219
9.9 Flow Pattern in Front View in Couette Regime (Oil I) ...... 219
9.10 Flow Pattern in Front View in Taylor Regime (Oil I) ...... 220
9.11 Flow Pattern in Front View in Pre-wavy Regime (Oil I)...... 222
9.12 Flow Pattern in Front View in Wavy Regime (Oil I) ...... 223
9.13 The Collapse of Waves for the Case of ω = 1,350 rpm (Oil I)...... 227
9.14 Flow Pattern in Front View in Turbulent Regime (Oil I) ...... 228
9.15 Voltage Measurements for the Case of Oil II...... 231
9.16 Voltage due to Fluid Friction vs. Speed for the Case of Oil II ...... 232
9.17 Flow Pattern in Longitudinal Cross Section View in Couette Regime (Oil II)...234
9.18 Flow Pattern in Longitudinal Cross Section View in Taylor Regime (Oil II).....236
9.19 The Collapse of Taylor Vortices for the Case of ω = 650 rpm (Oil II) ...... 237
9.20 Flow Pattern in Longitudinal Cross Section View in Pre-wavy Regime (Oil II) ...... 239
9.21 Flow Pattern in Longitudinal Cross Section View in Wavy Regime (Oil II)...... 240
9.22 Flow Pattern in Front View in Couette Regime (Oil II) ...... 241
9.23 Flow Pattern in Front View in Taylor Regime (Oil II)...... 242
9.24 Flow Pattern in Front View in Pre-wavy Regime (Oil II) ...... 243
9.25 Flow Pattern in Front View in Wavy Regime (Oil II)...... 244
xx 9.26 Flow Pattern in Front View in Turbulent Regime (Oil II)...... 247
9.27 Voltage Measurements for the Case of Oil III...... 248
9.28 Voltage due to Fluid Friction vs. Speed for the Case of Oil III...... 249
9.29 Flow Pattern in Longitudinal Cross Section View in Couette Regime (Oil III)..252
9.30 Flow Pattern in Longitudinal Cross Section View in Taylor Regime (Oil III) ...253
9.31 The Collapse of Taylor Vortices for the Case of ω = 1,200 rpm (Oil III) ...... 254
9.32 Flow Pattern in Longitudinal Cross Section View in Pre-wavy Regime (Oil III)...... 255
9.33 Flow Pattern in Longitudinal Cross Section View in Wavy Regime (Oil III).....256
9.34 Flow Pattern in Front View in Couette Regime (Oil III)...... 257
9.35 Flow Pattern in Front View in Taylor Regime (Oil III)...... 258
9.36 Flow Pattern in Front View in Pre-wavy Regime (Oil III)...... 261
9.37 Flow Pattern in Front View in Wavy Regime (Oil III)...... 262
10.1 Comparison of Torque-speed Graphs for the Case of Oil I...... 266
10.2 Comparison of Torque-speed Graphs for the Case of Oil II...... 267
10.3 Comparison of Torque-speed Graphs for the Case of Oil III ...... 268
10.4 Comparison of Flow Pattern in Longitudinal Cross Section View in Couette Regime (ω = 100 rpm) (Oil I)...... 270
10.5 Comparison of Flow Pattern in Front View in Couette Regime (ω = 100 rpm) (Oil I)...... 271
10.6 Comparison of Flow Pattern in Longitudinal Cross Section View in Taylor Regime (ω = 200 rpm) (Oil I) ...... 272
10.7 Comparison of Flow Pattern in Front View in Taylor Regime (ω = 200 rpm) (Oil I) ...... 273
10.8 Comparison of Flow Pattern in Longitudinal Cross Section View in Pre-wavy Regime (ω = 500 rpm) (Oil I)...... 275 xxi
10.9 Vorticity Comparison between Pre-wavy and Wavy Regimes (Oil I) ...... 276
10.10 Comparison of Flow Pattern in Front View in Pre-wavy Regime (ω = 500 rpm) (Oil I)...... 279
10.11 Comparison of Flow Pattern in Longitudinal Cross Section View in Wavy Regime (ω = 900 rpm) (Oil I) ...... 280
10.12 Comparison of Flow Pattern in Front View in Wavy Regime (ω = 900 rpm) (Oil I) ...... 281
10.13 Comparison of Flow Pattern in Longitudinal Cross Section View in Turbulent Regime (ω = 2,100 rpm) (Oil I) ...... 282
xxii
NOMENCLATURE
C = mean radial clearance, in. e = eccentric displacement of journal, in. h = film thickness, in.
L = length of cylinders, in.
R = radius of inner cylinder, in.
R+C = radius of outer cylinder, in.
RωC Re = Reynolds number = ν
t = time, sec
t* = dimensionless time = t/T
T = period that the waves travel in the circumferential direction, sec
2C Ta = Taylor number = Re 2 2R + C C Ta = Taylor number = Re2 for small value of clearance ratios R u = velocity in the circumferential direction, in./sec
v = velocity in the radial direction, in./sec
w = velocity in the axial direction, in./sec
U = average velocity in the circumferential direction, in./sec
V = average velocity in the radial direction, in./sec xxiii W = average velocity in the axial direction, in./sec x = circumferential coordinate, in. y = radial coordinate, in. z = axial coordinate, in. y* = dimensionless radial coordinate = y/h z* = dimensionless axial coordinate = z/L
ε = eccentricity ratio = e/C
θ = circumferential coordinate, in.
λ = dimensionless axial wavelength = L/C
ν = kinematic viscosity, in. 2/sec
ρ = fluid density, lb-sec 2/in. 4
ω = rotational speed, rad/sec
xxiv
CHAPTER I
THEORETICAL INTRODUCTION AND LITERATURE REVIEW
1.1 Introduction
People have been investigating the hydrodynamic stability of rotating fluids between two cylinders, both experimentally and analytically, for more than one hundred years.
Why did such an apparently “simple problem” become so popular? In fact, investigators were amazed to see the wealthy variety of flow patterns that occur, for example, when the speed of the inner cylinder is increased and the outer cylinder is at rest. One attractive feature of these patterns is their degree of symmetries, both spatially and temporally. For example, the Couette flow and Taylor vortex flow are axisymmetric along the axis of the cylinders and are time-independent when the speed of the inner cylinder is low.
However, as the speed of inner cylinder increases, the flow pattern becomes more and more complicated, breaking more and more symmetries both spatially and temporally.
With the increase of the inner cylinder speed, the Taylor vortex flow is replaced by wavy
Taylor vortex flow, which is non-axisymmetric and time-dependent. Eventually the flow becomes “turbulent”, containing large-scale structures with many degrees of symmetry.
This model problem then appears as an ideal example of a system that progressively approaches “turbulence”, which is still one of the challenging problems in the field of fluids. 1 The very first experimental investigator was Couette (1890), who designed an apparatus consisting of two concentric cylinders, the inner one was stationary and the outer one rotating, and the space between them was filled with water. The purpose of his experiment was to determine the viscosity of water by measuring the torque exerted by water on the inner cylinder. He found that when the angular velocity was not too large, water flow was laminar and the torque was proportional to the angular velocity.
However, when the angular velocity was increased to a large value, water flow became unstable and the slope of torque-speed diagram increased.
A few years later, Mallock (1896) designed a similar apparatus but allowed the inner cylinder to rotate. He was the first one who purposely did experiments to study the hydrodynamic stability problem. He stated that the object of the experiments was chiefly to examine the limits between which the motion of the fluid in the annulus was stable, and the manner in which the stability broke down. He noticed that instability occurred easily when the inner cylinder rotated and the outer one was stationary; and instability also occurred when the outer cylinder rotated at a large angular velocity and the inner one was at rest.
However, neither Couette nor Mallock established a definite criterion for the
hydrodynamic stability of the fluids. Rayleigh (1916) was the first one who found a
criterion for the hydrodynamic stability of inviscid fluid between two concentric rotating
cylinders. Taylor (1923) in his celebrated paper derived the linear theory of instability
for viscous fluids. His theory showed that there exists a critical Reynolds number, below
which all initially infinitesimal disturbances are damped and decay to zero with
increasing time and above which the disturbances will grow with time. Moreover, 2 Taylor’s experiments showed that this instability of Couette flow leads to a new steady secondary axisymmetric flow in the form of regularly spaced vortices in the axial direction, which are commonly called Taylor vortices. Since then, theories predicting the growth of the Taylor vortices and the onset of wavy Taylor vortices with further increase of speed were proposed by a lot of investigators, such as weakly nonlinear theory by
Stuart (1958) and Davey (1962), and theory of wavy vortices by DiPrima (1961) and
Davey et al. (1968).
Stability problem of fluid between two eccentric cylinders or a journal bearing is receiving more and more attention. The reason is that there is a trend in modern technology toward increasingly higher operational speed of journal bearings and toward the use of low viscosity fluids for lubrication, which results in situations where the fluid in a bearing film becomes unstable. Since the load capacity and frictional power loss of a bearing can increase significantly when the flow undergoes a transition from laminar to vortex flow and turbulence flow, it is very important to know when and how this transition occurs.
Wilcock (1950) and Smith and Fuller (1956) were the pioneers to recognize that flow transition occurs between eccentric cylinders in journal bearings. However, DiPrima
(1963) was the first one to establish a stability criterion of fluid between two eccentric cylinders, which is also called local theory of hydrodynamic stability. To account for the disagreement between the local theory and some experimental data, DiPrima and Stuart
(1972) established their nonlocal theory.
DiPrima’s local theory and DiPrima and Stuart’s nonlocal theory define the threshold for the onset of fluid instability, which usually appears in the form of Taylor vortices in 3 eccentric cylinders. With the further increase of inner cylinder speed, the flow becomes wavy vortices and eventually becomes fully developed “turbulence”. Constantinescu
(1959), Ng and Pan (1965), and Hirs (1973) established their own turbulence theories, which govern the fully developed “turbulence” in journal bearings. All the three theories are based on well-established empiricisms relating turbulent shear stress to mean velocity gradient, such as Prandtl mixing length concept or the eddy diffusivity concept.
During the next sections in this chapter, a) theories of fluid instability in concentric cylinders, i.e. Rayleigh’s stability criterion, linear theory, weakly nonlinear theory and theory of wavy Taylor vortices, b) theories of fluid instability in eccentric cylinders, i.e. local theory and nonlocal theory, c) theories of turbulence in bearings, i.e.
Constantinescu, Ng and Pan, and Hirs’ theories will be introduced in detail.
1.2 Theories of Fluid Instability in Concentric Cylinders and Literature Review
In this section, theories of fluid instability in concentric cylinders, i.e. Rayleigh’s stability criterion, linear theory, weakly nonlinear theory and theory of wavy Taylor vortices, will be introduced.
1.2.1 Rayleigh’s Stability Criterion
The first hydrodynamic stability criterion for inviscid fluids was derived by Rayleigh
(1916). His conclusion is that the flow is stable if the square of circulation increases in the radial direction for the case of inviscid axisymmetric flow; otherwise it is unstable.
Rayleigh reached that conclusion by starting from the Euler’s equations, Equation
(1.1), in cylindrical coordinates shown in Figure 1.1. 4
Figure 1.1 Velocity Components in Cylindrical Coordinates
∂v ∂v v ∂v v 2 ∂v ∂P r + v r + θ r − θ + v r = − (1.1a) ∂t r ∂r r ∂θ r z ∂z ∂r
∂v ∂v v ∂v v v ∂v 1 ∂P θ + v θ + θ θ + r θ + v θ = − (1.1b) ∂t r ∂r r ∂θ r z ∂z r ∂θ
∂v ∂v v ∂v ∂v ∂P z + v z + θ z + v z = − (1.1c) ∂t r ∂r r ∂θ z ∂z ∂z
where
dp P = −V (1.2) ∫ ρ
where V is the potential of extraneous forces.
With the assumption that vr, v θ, v z, and P are symmetric with respect to the axis of z, i.e., independent of θ, Equations (1.1a), (1.1b), and (1.1c) become:
5 ∂v ∂v v 2 ∂v ∂P r + v r − θ + v r = − (1.3a) ∂t r ∂r r z ∂z ∂r
∂v ∂v v v ∂v θ + v θ + r θ + v θ = 0 (1.3b) ∂t r ∂r r z ∂z
∂v ∂v ∂v ∂P z + v z + v z = − (1.3c) ∂t r ∂r z ∂z ∂z
Equation (1.3b) is equivalent to:
∂ ∂ ∂ ( + v + v )( rv θ ) = 0 (1.4) ∂t r ∂r z ∂z
Therefore, the angular momentum or circulation Γ(= rv θ ) of a fluid element, per unit mass, remains constant as we follow it with its motion. Then the force acting in the radial direction can be expressed as:
v 2 Γ 2 θ = (1.5) r r 3
The kinetic energy associated to a ring of fluid at a position r from the axis can be
expressed as:
πρ Γ 2 2 2 E = πρ vθ rdr = d (r ) (1.6) k ∫ ∫ 2r 2
Suppose now that we interchange the fluid contained in two elementary rings, of equal
2 2 areas d r1 or d r2 , at r = R1 and r = R2 (where R2 > R1). The corresponding increment in kinetic energy is represented by:
πρ Γ 2 Γ 2 Γ 2 Γ 2 ∆ = − = 2 2 + 1 − 1 − 2 Ek Ek 2 Ek1 d(r )( 2 2 2 2 ) 2 R1 R2 R1 R2
πρ = d(r 2 )( Γ 2 − Γ 2 )( R −2 − R −2 ) (1.7) 2 2 1 1 2
6 2 2 Remembering that R2 > R 1, we observed that Ek is positive if Γ2 > Γ1 , i.e., the circulation always increasing outwards makes kinetic energy a minimum and thus ensures stability. This can be expressed as the stability criterion:
> 2Ω > 2Ω (R2vθ 2 ) (R1vθ1 ) or (R2 2 ) (R1 1 ) (1.8)
If we denote that = 2/ 1 and η = R1/R 2, this relation can be illustrated with the
stability diagram in Figure 1.2.
Rayleigh’s stability criterion can only be applied to the case of inviscid flow or when
viscosity does not affect the occurrence of instability. However, viscosity is a very
important property for flows satisfying the Navier-Stokes equations. When the viscosity
of a fluid is greater than a critical value, all solutions tend monotonically to a basic flow,
Couette flow for the case of fluids between two concentric cylinders. When the viscosity
is less than the critical value, disturbances of the basic flow will grow. Therefore, next
section we will discuss linear theory, which deals with viscous fluid flow.
Figure 1.2 Rayleigh’s Stability Diagram [After Koschmieder (1993)]
7 1.2.2 Linear Theory of the Instability
The first linear theory on stability of viscous fluids was proposed by Taylor (1923).
He considered a viscous flow between two concentric rotating cylinders, which were assumed to be infinitely long and rotated about their common axis with constant angular velocities 1 at r = R 1, and 2 at r = R 2 (R2 > R 1). The governing equations are Navier-
Stokes equations in cylindrical coordinates (constant density and viscosity are assumed):
∂v ∂v v ∂v v 2 ∂v r + v r + θ r − θ + v r = ∂t r ∂r r ∂θ r z ∂z
2 2 2 1 ∂p ∂ v 1 ∂v v 1 ∂ v 2 ∂vθ ∂ v − +ν r + r − r + r − + r (1.9a) g r ρ ∂r ∂r 2 r ∂r r 2 r 2 ∂θ 2 r 2 ∂θ ∂z 2
∂v ∂v v ∂v v v ∂v θ + v θ + θ θ + r θ + v θ = ∂t r ∂r r ∂θ r z ∂z
∂ ∂ 2 v ∂v v ∂ 2v ∂v ∂ 2v − 1 p +ν θ + 1 θ − θ + 1 θ + 2 r + θ gθ (1.9b) ρr ∂θ ∂r 2 r ∂r r 2 r 2 ∂θ 2 r 2 ∂θ ∂z 2
∂v ∂v v ∂v ∂v z + v z + θ z + v z = ∂t r ∂r r ∂θ z ∂z
∂ ∂ 2v ∂v ∂ 2v ∂ 2v − 1 p +ν z + 1 z + 1 z + z g z (1.9c) ρ ∂z ∂r 2 r ∂r r 2 ∂θ 2 ∂z 2
The continuity equation in cylindrical coordinates for constant density is:
1 ∂(rv ) 1 ∂v ∂v r + θ + z = 0 (1.10) r ∂r r ∂θ ∂z
If the radial velocity component vr = 0, the vertical velocity component vz = 0, the
azimuthal velocity component vθ = vθ(r), the pressure p = p(r), and the gravity is negligible, then Equation (1.9b) reduces in the steady state to:
8 2 ∂ vθ 1 ∂vθ vθ + − = 0 (1.11) ∂r 2 r ∂r r 2
The solution of Equation (1.11) is:
vθ = Ar + B / r (1.12)
where
Ω η 2 1( − µ /η 2 ) A = − 1 (1.13) 1−η 2
Ω R 2 1( − µ) B = 1 1 (1.14) 1−η 2 where = 2/ 1 and η = R1/R 2.
Let u, v, and w be the radial, azimuthal, and axial velocity components of the
disturbance respectively as shown in Figure 1.3.
Figure 1.3 Coordinates of Concentric Cylinders
9
Assume that u, v, and w are small in comparison with vθ, and that the disturbance is symmetrical, so that they are functions of r, z and t only. Neglecting terms containing products of squares of u, v, and w, the equations of disturbance may be written as:
∂ 2 ∂ ∂ 2 ∂ ∂ 2 1 p − vθ = − u + + B +ν u + 1 u + u − u (2 A )v (1.15a) ρ ∂r r ∂t r 2 ∂r 2 r ∂r ∂z 2 r 2
∂ ∂ 2 ∂ ∂ 2 = − v − +ν v + 1 v + v − v (1.15b) 0 2Au 2 2 2 ∂t ∂r r ∂r ∂z r
∂ ∂ ∂ 2 ∂ ∂ 2 1 p = − w +ν w + 1 w + w (1.15c) ρ ∂z ∂t ∂r 2 r ∂r ∂z 2
The continuity equation is:
∂u u ∂w + + = 0 (1.16) ∂r r ∂z
The six boundary conditions are:
u = v = w = 0 at r = R 1 and r = R 2 (1.17)
A solution of the disturbance is assumed to be:
= λ σt u u1 (r) cos( z)e (1.18a)
= λ σt v v1 (r) cos( z)e (1.18b)
= λ σt w w1 (r)sin( z)e (1.18c)
If the pressure terms in Equations (1.15a) and (1.15c) are eliminated, then Equations
(1.15a), (1.15b), (1.15c), and (1.16) reduce to:
∂u u 1 + 1 + λw = 0 (1.19) ∂r r 1
10 ∂ 2 1 ∂ 1 σ ν + − − λ2 − = (1.20) 2 2 v1 2Au 1 ∂r r ∂r r ν
ν ∂ ∂ 2 1 ∂ σ B ∂ 2 1 ∂ 1 σ + − λ2 − = − + −ν + − − λ2 − (1.21) 2 w1 2 A 2 v1 2 2 u1 λ ∂r ∂r r ∂r ν r ∂r r ∂r r ν
Now the six boundary conditions become:
= = = u1 v1 w1 0 at r = R 1 and r = R 2 (1.22)
There are no terms containing z in these equations, which means that the normal modes of disturbance are simple harmonic with respect to z, the wave-length being 2π/λ. Note that σ is a quantity which determines the rate of increase in a normal disturbance. If σ is positive the disturbance increases and the motion is unstable. If σ is negative the disturbance decreases and the motion is stable. If σ is zero the motion is neutral.
The solution of Equations (1.19), (1.20) and (1.21) can be obtained by means of a type
of Bessel functions. Taylor assumed a Bessel series solution for u1:
∞ = κ (1.23) u1 ∑ am B1 ( m r) m=1
κ where B1 ( m r) is the Bessel function of order 1. Then he substituted this solution into
Equations (1.19), (1.20) and (1.21) and the solutions for v1 and w1 were obtained. Finally
after he used all the boundary conditions, Taylor obtained a system of linear homogenous
equations containing a1, a 2, , a m. The number of the unknowns is the same as that of the equations, so all the unknowns can be eliminated from the equations. The resulting equation takes the form of an infinite determinant equating to zero. The eigenvalue problem with σ = 0 can be solved by the Galerkin method, and the critical Taylor number
can be expressed by the following formula:
11 π 4 (1+ C 2/ R ) Ta = 1 (1.24) c ()()− + − −1 .0 0571 1 .0 652 C / R1 .0 00056 1 .0 652 C / R1
1.2.3 Literature Review on Linear Theory
After Taylor’s first success in the calculation of the critical Taylor number by using a linear stability theory, a lot of investigators improved the linear theory on Taylor instability by both analytical and numerical methods.
The first modification of the solution of Equations (1.19) through (1.22) was introduced by Jeffreys (1928). He reduced these equations to a single sixth-order differential equation, (1.25), for the case of narrow gap and marginal instability ( σ = 0).
3 ∂ 2 4A(λc) 2 c 4 B − ()λ 2 = + (1.25) 2 c v1 2 A 2 v1 ∂r ν r
Then by reasoning the analogy between the conditions in a layer of liquid heated below and in a liquid between two concentric cylinders rotating at different rates, Jeffreys showed that for the case of = 2/ 1 → 1, the critical Taylor number is equal to:
4Ac 4 Ta = Ω = 1709 (1.26) c ν 2 1 which is in good agreement with 1706, the value obtained by Taylor’s formula, Equation
(1.24).
Chandrasekhar (1961b) solved the nondimensional version of Equations (1.19) through
(1.22) for the case of narrow gap when the marginal state is stationary ( σ = 0). He expanded v1 in a sine series of the form:
∞ = πζ (1.27) v1 ∑Cm sin( m ) m=1 12 where ζ = (r-R1)/C . Then by manipulating the secular equation, the Taylor number in the
first approximation is obtained by the formula:
2 [π 2 +(λC)2 ]3 Ta = (1.28) c 1+µ (λC)2 {1−16 (λC)2 cosh 2 (λC [](/)2/ π 2 +(λC)2 )(sinh(λC)+λC)}
Equation (1.28) is valid for any , but it is too complicated. Therefore, Chandrasekhar gave two simpler formulas depending on the value of :
2 3416 − 1− µ Ta = 1− 61.7 ×10 3 (µ → ,1 η → )1 (1.29) c + µ + µ 1 1
= − µ 4 µ → −∞ η → Ta c Ta c0 1( ) ( , )1 (1.30) where Ta c0 ( = 1182) is comparable to the Taylor number when = 0.
Harris & Reid (1964) found the same asymptotic behavior of Ta c when → - ∞ as shown in Equation (1.30), and their value of Ta c0 was 1178.6. DiPrima & Swinney
(1981) verified that Equation (1.29) is quite satisfactory for -0.25 < < 1, and Equation
(1.30) holds with quite reasonable accuracy for < -1.
Chandrasekhar (1954) and DiPrima (1961) derived an expression of the critical Taylor number as a function of the clearance C and the inner radius R1 when C is much smaller than R1:
= ( + ) Ta c 1695 1 C 2/ R1 ( C << R 1) (1.31)
which is a simplified form of Taylor’s expression, Equation (1.24).
The stability problem in a wide gap was studied numerically by Roberts (1965) for the
case of = 0, and analytically by Sparrow et al. (1964) for the case in which both cylinders rotate. Their results were in good agreement with each other, and some results
13 of Robert are listed in Table 1.1. It is shown that both the critical Taylor number and wavelength increase with the decrease of radius ratio η.
Table 1.1 Critical Parameters for the Onset of Taylor vortices for Various Values of η
η Ta c λc
0.975 1723.89 3.1268
0.950 1754.76 3.1276
0.925 1787.93 3.1282
0.900 1823.37 3.1288
0.875 1861.48 3.1295
0.850 1902.40 3.1302
0.750 2102.17 3.1355
0.650 2383.96 3.1425
DiPrima and Hall (1984) studied the eigenvalue problem for the linear stability of
Couette flow between rotating concentric cylinders to axisymmetric disturbances. It was shown by numerical calculations and by formal perturbation methods that there exist complex eigenvalues corresponding to oscillatory damped disturbances when the outer cylinder is at rest. DiPrima and Hall (1984) were the first investigators to demonstrate the existence of complex eigenvalues for the boundary value problem for the stability of
Couette flow between rotating cylinders. Their results contradict an argument by Yih
(1972) that all the eigenvalues are real when the cylinders rotate in the same direction and the circulation of the basic flow decreases in the outward radial direction. However, 14 DiPrima and Hall (1984) also showed that all of the complex eigenvalues they found correspond to damped disturbances. Thus their results do not contradict the conjectured and widely believed principle of exchange of stabilities, i.e., for a fixed axial wave number, the first mode to become unstable as the speed of the inner cylinder is increased is non-oscillatory as the stability boundary is crossed.
1.2.4 Weakly Nonlinear Theory
In linear theory only the infinitesimal disturbances have been considered, so only the initial growth of the disturbance has been determined. In other words, the linear theory of hydrodynamic stability can predict correctly the critical Taylor number, but it cannot predict the establishment of a new equilibrium flow, Taylor vortex flow, above the critical Taylor number. It is obvious that the exponential growth of the disturbances considered in linear theory is not realistic. Therefore, it is necessary to solve the nonlinear equations, which means that the higher order terms cannot be neglected.
Stuart (1958) was the pioneer of nonlinear theory when he extended the linear theory to larger amplitude and studied the mechanics of disturbance growth taking the nonlinearity of the hydrodynamic system into account. He assumed that the flow had rotational symmetry and was therefore independent of θ, and then the Navier-Stokes and
continuity equations became:
∂v ∂v v 2 ∂v ∂ ∂ 2v ∂v v ∂ 2v r + r − θ + r = − 1 p +ν r + 1 r − r + r vr vz (1.32a) ∂t ∂r r ∂z ρ ∂r ∂r 2 r ∂r r 2 ∂z 2
∂v ∂v v v ∂v ∂ 2v ∂v v ∂ 2v θ + θ + r θ + θ = θ + 1 θ − θ + θ vr vz v (1.32b) ∂t ∂r r ∂z ∂r 2 r ∂r r 2 ∂z 2
15 ∂v ∂v ∂v ∂ ∂ 2v ∂v ∂ 2v z + z + z = − 1 p +ν z + 1 z + z vr vz (1.32c) ∂t ∂r ∂z ρ ∂z ∂r 2 r ∂r ∂z 2
1 ∂(rv ) ∂v r + z = 0 (1.33) r ∂r ∂z
Stuart developed the disturbances u, v and w in Fourier series, and then the series were
substituted into Equations (1.32a) through (1.32c). After some mathematic
manipulations and the application of the boundary conditions, then he obtained the
following disturbance energy equation:
∂v v ()− ρ θ − θ = µ ()ξ 2 +η 2 + ζ 2 ∫∫uv rdrdz ∫∫ rdrdz (1.34) ∂r r where the bars denote mean values and the vorticity components ξ, η and ζ are given by
∂v ∂u ∂w 1 ∂ ξ = − η = − ζ = (rv ) (1.35) ∂z ∂z ∂r r ∂r
Equation (1.34) means that the rate of transfer of kinetic energy from the mean flow of the disturbance is balanced by the rate of viscous dissipation of kinetic energy. If assumption of small gap is adopted and a few transformations are applied, Equation
(1.34) yields the formula of equilibrium amplitude:
.5 425 ×10 4 Ta A 2 = 1( − c ) (1.36) e Re 2 Ta where Ae is the amplitude of the disturbance, Re ( = 1R1C/ν) is the Reynolds number, Ta
2 3 2 (= 1 R1C /ν ) is the Taylor number, and Ta c (=1708) is the critical Taylor number.
−1 − Equation (1.36) states that the amplitude Ae is proportional to Re 1 Ta c /Ta , which is in agreement with the results of Landau (1944) for values of Ta close to Ta c, i.e. in the weakly nonlinear regime. 16 Stuart’s energy-balance method included the effect of distortion of the mean motion, but ignored the generation of harmonics of the fundamental mode and the distortion of the velocity associated with the fundamental mode. Davey (1962) found that those were not valid mathematical approximations, so he continued the investigation of weakly nonlinear Taylor vortex flow by a rigorous perturbation expansion. Davey assumed the velocity components of the disturbances in Fourier series in harmonics of the wave number λ:
∞ = λ u ∑un (r,t)cos( n z) (1.37a) n=1
∞ = λ v ∑vn (r,t)cos( n z) (1.37b) n=1
∞ = λ w ∑ wn (r,t)sin( n z) (1.37c) n=1
Davey eliminated p and vz from Equations (1.32) and (1.33) and tried a solution of the remaining two differential equations with the following forms:
∞ = n + 2m un (r,t) A un (r) ∑ A unm (r) (1.38a) m=1
∞ = n + 2m vn (r,t) A vn (r) ∑ A vnm (r) (1.38b) m=1
Then the time-dependent amplitudes are determined from the equation:
∞ 1 dA = 2m = σ ∑ am A a0 (1.39) A at m=0
Following the assumption that the amplitude was small, the amplitude equation
became:
17 = σ + 3 + 5 dA / dt A a1 A (0 A ) (1.40) and the solution of Equation (1.40) is:
2 = σ 2σt − 2σt A K e /( 1 a1 Ke ) (1.41) where K is a constant, which depends on the origin of time t. By using numerical
methods, Davey determined the following expression of the equilibrium amplitude for the
case of small gap:
Ta A 2 = .0 3257 1( − c ) (m = )0 (1.42) e Ta
1.2.5 Literature Review on Weakly Nonlinear Theory
The validity of the amplitude Equation (1.40), which is the major contribution by
Davey (1962), was supported by the torque measurements of some investigators. When the amplitude of the motion is known, the torque can be calculated by Davey’s theory,
Equation (1.43) for the case of small gap.
= − Ω −1 + Ω T0 906 6. 1 51 64. 1 (1.43)
Davey found a good agreement of the torque predicted by the theory and the one
measured by Donnelly (1958) over the range of the Taylor number above the critical
value for which the perturbation theory is expected to be valid. Later on an empirical
formula for calculating torque was given by Donnelly and Simon (1960), Equation
(1.44), which is very similar to Equation (1.43).
= Ω −1 + Ω 36.1 T0 a 1 b 1 (1.44)
where a and b are constants determined from the experimental data.
18 Direct verification of the amplitude of Taylor vortices as a function of the Taylor number was also done by some experimenters. The first such measurements were made by Donnelly and Schwarz (1965). By measuring the amplitude of the radial component
2 of the disturbance, they found that Ae , the square of the amplitude of disturbance, varies linearly with ( Ta-Ta c), the difference between the Taylor number and the critical Taylor number when Ta is close to Ta c. Their experimental data directly verified Davey’s law of
equilibrium amplitude, Equation (1.42).
Table 1.2 Comparison of Coefficients for the Calculation of Taylor Vortex Torques
Authors Η λ A B
DiPrima and Eagles (1977) 1 3.13 9.105 × 10 -4 (-8.3 ± 0.3) × 10 -7
Reynolds and Potter (1967) 1 3.13 9.047 × 10 -4 -8.44 × 10 -7
DiPrima and Eagles (1977) 0.95 3.0 7.884 × 10 -4 (-7.1 ± 0.3) × 10 -7
Kirchgassner and Sorger (1969) 0.95 3.0 7.882 × 10 -4 -6.7 × 10 -7
DiPrima and Eagles (1977) 0.5 3.0 1.421 × 10 -4 (-3.5 ± 0.3) × 10 -8
Kirchgassner and Sorger (1969) 0.5 3.0 1.422 × 10 -4 -4.12 × 10 -8
The quintic term in the amplitude Equation (1.40) was calculated by DiPrima and
Eagles (1977). Kirchgassner and Sorger (1969) also carried the calculations to the same order. According to the results of these two papers, the mean torque on the inner cylinder with the outer one at rest is given by:
π LρV 2 1( +η)ηR T = (G + G ) (1.45) 0 1(2 −η) 2 l t
19 2 where L is the height of the cylinders, Gl = -8/(1+ η) is the dimensionless torque due to
Couette flow, and Gt is the dimensionless Taylor vortex torque. The ratio of Gt to Gl can
be expressed in the form:
= η λ [ − η λ ] + η λ [ − η λ ]2 Gt / Gl A( , ) Ta Ta c ( , ) 2/ B( , ) Ta Ta c ( , ) 4/ (1.46) where A(η, λ) and B(η, λ) are coefficients. Values of A(η, λ) and B(η, λ) obtained by
DiPrima and Eagles (1977), by Kirchgassner and Sorger (1969), and by Reynolds and
Potter (1967) are given in Table 1.2.
1.2.6 Theory of Wavy Taylor Vortices and Literature Review
According to the experimental observations of Coles (1965), the axisymmetric time- independent Taylor vortex flow becomes unstable with the further increase of Taylor number, and the instability leads to non-axisymmetric time-dependent wavy Taylor vortex flow. DiPrima (1961) was the first one who analytically studied the linear problem for the stability to non-axisymmetric disturbances. He assumed the wavy azimuthal disturbances of the form:
u(r,θ, z,t) = u(r)cos( λz)ei(σt+mθ ) (1.47a)
v(r,θ, z,t) = v(r)cos( λz)ei(σt+mθ ) (1.47b)
w(r,θ, z,t) = w(r)sin( λz)ei(σt+mθ ) (1.47c) where m is the integer number of the azimuthal waves. Equations (1.47a) through (1.47c) are introduced into the linearized Navier-Stokes equations, and only the real part of the disturbances is considered. By using the narrow gap approximation and neglecting some small terms, eventually DiPrima derived the following equations:
20 1 [DLD − (λC) 2 L]u = − (λC) 2 Ta 1( + µ)v (1.48) 2
Lv = u (1.49)
= = = = = u v Du 0 at r R1 and r R2 (1.50)
where
L = [D 2 − (λC) 2 ] − i Re[ β + mf (r)] (1.51)
D = d / dr (1.52) and β is the real part of rotation speed and f(r) is a function of position r.
The eigenvalue problem defined by equations (1.48) – (1.52) was solved with a
variational method, which yields that the critical Taylor number for the onset of wavy
vortices increases slightly as the number of waves in the azimuthal direction increases
(from 0 to 3).
Roberts (1965) used a sixth-order system to solve the non-axisymmetric linear problem
for Couette flow, and his calculation supports DiPrima’s result, i.e. the critical Taylor
number is an increasing function of the wave number.
The stability of Couette flow with respect to non-axisymmetric disturbances was
studied by Krueger et al. (1966) in the narrow gap case when both of the cylinders
rotated. They solved the eigenvalue problem by a direct numerical procedure, and it is
found that there is a critical value of 2/ 1 of approximately -0.78 (counter-rotating
cylinders), above which the critical disturbance is axisymmetric and below which it is
non-axisymmetric. The result of Krueger et al. was confirmed by experimental
observations of Andereck et al. (1986).
21 Davey et al. (1968) attacked the problem of instability of the Taylor vortex flow against perturbations, which are periodic both in the axial and azimuthal directions, and moreover travel with some phase velocity in the latter. They made a number of assumptions, of which the most important one was the narrow gap assumption. Davey et al. assumed the azimuthal velocity component is periodic with period 2 π/λ in the
nondimensional vertical direction ζ and periodic with period 2 π/k in the azimuthal
direction φ:
∞ ∞ ζ ϕ τ = τ + τ λζ + τ λζ iqk ϕ v(x, , , ) ∑v0q (x, ) ∑vcnq (x, )cos( n ) vsnq (x, )sin( n )e (1.53) q=−∞ n=1 where x is the nondimensional radial coordinate, τ the nondimensional time, and k =
2 m 0C /ν.
Formulas similar to equation (1.53) were also applied to the u and w velocity
components. The velocities were then expanded in powers and products of the amplitude
Ac(τ), As(τ), Bc(τ), Bs(τ), the amplitudes B belonging to the non-axisymmetric motions.
These expansions should be consistent with the differential equations for u, v and w, and then the amplitude equations can be derived.
Davey et al. found that the Taylor vortex flow is stable against perturbations with the same axial wavelength and phase, but unstable against perturbations differing in phase by
π/2. The critical Taylor number for the onset of wavy vortex flow is found to be about
8% above the critical Taylor number for the onset of axisymmetric Taylor vortex flow.
The azimuthal wave number is found to be one by a slight preference.
The results of Davey et al. are in qualitative agreement with some experimenters’ data.
Cole (1976) observed that the second critical Taylor number (onset of wavy vortices) was
22 8.2% above the first critical Taylor number (onset of Taylor vortices) when the aspect ratio was large ( L/C = 107) and clearance ratio was small ( C/R1 = 0.0478). Schwarz et al.
(1964) observed that the second critical Taylor number was 5% above the first one, and
the azimuthal wave number was one.
However, the preference of the wave number to be one was disproved by some
investigators. Eagles (1974) calculated the torque corresponding wavy vortex flows with m = 1, 2, 3 and 4 for the case of η = 0.95. His numerical results and the experimental results of Donnelly (1958) and Debler et al. (1969) for the torque agree quite well with the theory of Davey et al. (1968) of wavy vortex flow with m = 4 (four azimuthal waves).
This suggests that the m = 4 nonaxisymmetric mode does indeed dominate the other
nonaxisymmetric modes to produce a wavy flow with four waves in the azimuthal
direction.
Experimental data show that the second critical Taylor number (onset of wavy
vortices) depends strongly on the radius ratio η. It is 5% - 10% above the first critical
Taylor number (onset of Taylor vortices) for η = 0.95, and is very much larger, ten times of the first Taylor number or greater, for η = 0.5 according to Snyder (1969) and Debler
et al. (1969).
Numerically, Eagles (1971) extended the application by dropping the narrow gap
approximation. He calculated the stability of Taylor vortex flow by using fifth-order
terms in amplitude, and he found that the instability to the Taylor vortex flow to non-
axisymmetric disturbances at about 10% above the first critical Taylor number when η =
0.95.
23 Dando et al. (2000) used a multiple scales method to derive an equation governing the stability of Taylor vortices in a high Reynolds number flow through a curved channel of small gap. They showed that this stability of Taylor vortices is governed by a modified
Burgers equation:
+ 2 = + λ U t U U x U U xx (1.54) where x is the circumferential direction coordinate and λ is the wave length in the axial
direction. Numerical solutions of this equation show that a family of periodic traveling
waves of period 2 π/N, where N is an integer, can be calculated from a primary 2 π- periodic traveling wave by a simple transformation. It was found that the primary traveling waves are the only stable ones emerging at large times for a given initial value problem. They also showed that the final stable steady state traveling waves become increasingly steeper as the length of the system is increased.
Riechelmann and Nanbu (1997) assumed that the smallest scale of turbulence, i.e. the size of the smallest vortices could be considered being still some orders larger than the molecular mean free path λ. Then they applied the direct simulation Monte Carlo
(DSMC) method, which solves flow problems on molecular level, thus avoiding turbulence models. Riechelmann and Nanbu (1997) found that the wavy vortices were seen to move in the azimuthal direction and the frequency of the waves was in agreement with the experimental observations. They were the first investigators to simulate the wavy Taylor flow by means of a molecular approach.
The flow between two cylinders usually experiences Couette, Taylor and wavy vortices as the speed of the inner cylinder increases. This is observed by most of the investigators both numerically and experimentally. However, Lim et al. (1998) reported 24 that there exists a new flow regime when the acceleration of the inner cylinder ( dRe/ dt ) is
higher than a critical value of about 2.2 s -1. In this regime the flow pattern shows
remarkable resemblance to regular Taylor vortex flow but is of shorter axial wavelength.
However, when the acceleration is lower than 2.2 s -1, a wavy flow is found to occur for
the same Reynolds number range as usually observed. This is the first time that such a
phenomenon has been observed.
1.3 Theories of Fluid Instability in Eccentric Cylinders and Literature Review
In this section, theories of fluid instability in eccentric cylinders, i.e. local and nonlocal
theories, and the corresponding literature review will be introduced.
1.3.1 DiPrima’s Local Theory and Literature Review
DiPrima (1963) investigated analytically both the concentric and eccentric cases for
1 ≤ C ≤ 1 small clearances that correspond to the range of 3 2 . He assumed that for the 10 R1 10 concentric case with the inner cylinder rotating and the outer one stationary, the Couette flow will make a transition to Taylor vortex flow first, if the Taylor number Ta
R Ω C R Ω C (= 1 1 C / R ) reaches 41.3 before the Reynolds number Re (= 1 1 ) attains ν 1 ν
2,000. However if Re = 2,000 is reached before Ta reaches its critical value of 41.3, then the flow will make a direct transition to turbulence. The dividing point would be a value of C/R1 of roughly 1/2500 (or 0.000426); for smaller values turbulence will develop directly and for larger values Taylor vortices will occur before the appearance of
25 turbulence as the velocity is increased. The relationship among Reynolds number, Taylor number, curvature effect and flow pattern is shown in Figure 1.4. For most journal bearings C/R1 will generally be greater than 1/2500 by a factor of 2 to 5, so the instability
will be in the form of Taylor vortices if the flow is unstable.
Laminar Flow:
Transition Flow:
Turbulent Flow:
0 131 400 1000 1306 2000 2920 3000 Re
Ta C/R=0.1 0 41.3 316.2 632.5 948.7 CurvatureEffect Ta C/R=0.01 0 41.3 100 200 300
Ta C/R=0.001
0 31.6 41.3 63.2 94.9
Ta C/R=0.000426 0 20.6 41.3 61.9
Ta C/R=0.0002 0 14.1 28.3 41.3 42.4
Figure 1.4 The Relationship among Reynolds Number, Taylor Number, Curvature Effect and Flow Pattern
26 For the eccentric case, DiPrima superimposed a circumferential pressure gradient on the basic Couette flow, and then the laminar velocity distribution is given by:
y 1 ∂P V (y,θ ) = V 1( − ) + (y 2 − yh ) (1.55) 1 µ ∂θ h 2 R1
where P/ θ denotes the circumferential pressure gradient, V1 is the velocity of inner cylinder, y is the radial coordinate measured from inner cylinder, and h = C (1 + ε cos θ) is the film thickness.
200
Rr C / R1 160
120
80 Unstable 40 Stable RP C / R1 0 -120 -80 -40 0 40 80 120 -40 Unstable -80
-120
-160
-200
Figure 1.5 A Plot of Values of Rr C / R1 & RP C / R1 for Stable & Unstable Flow
27 DiPrima (1963) shows that the stability of the flow to disturbances of the Taylor vortex
type is governed by the values of the two parameters, Rr C / R1 and RP C / R1 , where
Rr (= V1C/ν) is a Reynolds number based on the velocity of the inner cylinder, and
− C 2 ∂P C V C R = = P (1.56) P µ ∂θ ν ν 12 R1
is a Reynolds number based on the average velocity due to the circumferential pressure
gradient. The stability diagram is shown in Figure 1.5. Notice that for RP = 0, Taylor’s
criterion of Ta = 41.3 is observed.
By using the Sommerfeld solution of the Reynolds equation for the infinitely long journal bearing according to Pinkus and Sternlicht (1961), the pressure gradient can be expressed by:
∂P 2C 1( − ε 2 ) = 6µV R (h − h /) h3 h = (1.57) ∂θ 1 1 0 0 2 + ε 2
where h0 is the film thickness at the cross section where P/ θ = 0. Then DiPrima (1963)
calculated the local Reynolds numbers corresponding to the velocity and pressure
gradient in terms of ε and θ:
h = + ε θ /3 2 Rrh R 1( cos ) (1.58a) R1
h 1(2 − ε 2 ) 1( + ε cos θ ) 2/1 R = −R 1( + ε cos θ ) − (1.58b) Ph + ε 2 R1 2 2 where
V C C R = 1 (1.59) ν R1
28 Combining Equations (1.58) and Figure 1.5, the critical value of R as a function of ε
and θ can be derived, Figure 1.6, which is also called DiPrima’s local theory. In Figure
1.6, the variation of Rθ(ε) with ε is shown for several values of θ. It shows that the flow
to be the least stable at the position of maximum film thickness θ = 0˚. The critical
Taylor number Ta first decreases, as the eccentricity ratio ε increases from 0, and remains below its concentric value in the range of 0 < ε < 0.6. For ε < 0.6, Ta
increases rapidly as ε increases.
C Rr R1 100
90 θ = 0˚ θ = 60˚ 80 θ = 90˚ θ = 180˚ 70
60
50
40
30 0.0 0.2 0.4 0.6 0.8ε 1.0
Figure 1.6 Critical Value of R (C/R )1/2 as a Function of ε for Various Values of θ r 1
However, DiPrima’s local theory does not match with some experimental results. Both
Vohr’s (1968) and Cole’s (1957, 1965) experimental data show that the critical speed
increases monotonically with eccentricity, Figure 1.7. Based on both of the author’s
experimental results, Vohr also found that the critical speed for onset of vortices in flow 29 between eccentric cylinders does not depend significantly on clearance ratio when C/R1 is
greater than 0.1; whereas it depends significantly on clearance ratio when 0 ≤ C/R1 ≤ 0.1,
Figure 1.7. Concerning the application of the results to cylinders having clearance ratio less than 0.01, which is usually the case for bearings, Vohr suggested that the experimental curve for C/R1 = 0.0104 be used as an upper bound curve for flow stability
while DiPrima’s theoretical curve be used as a lower bound.
Vohr also found that the maximum intensity of the vortex motion is not at the position
θ = 0˚, but at a position θ = 50˚ downstream from the maximum clearance.
C R r R 140 1
130 DiPrima,C/R1 →0 Vohr, C/R1=0.0104 120 Vohr, C/R1=0.099 Cole, C/R1=0.27 110 Cole, C/R1=0.48
100
90
80 Upper bound 70 60
50
40 Lower bound 30 0.0 0.2 0.4 0.6 0.8ε 1.0
Figure 1.7 Critical Speed as a Function of ε for Various Values of Clearance Ratio
30 1.3.2 DiPrima and Stuart’s Non-local Theory and Literature Review
DiPrima’s local theory is based on the assumption that the effect of azimuthal variation of the tangential velocity is neglected. By considering the tangential velocity as a function of the radius ‘ r’ and θ, DiPrima and Stuart (1972) derived the linearized instability equations, which are partial differential equations rather than ordinary differential equations. By letting the two small parameters, the clearance ratio C/R1 and
1/2 the eccentricity ratio ε, tend to zero in such a way that ( C/R1) is proportional to ε, they
obtained a global solution to the stability problem. Their result, which is also called
linear nonlocal theory, shows that the critical speed increases monotonically with the
eccentricity ratio ε according to the relationship:
= + C ()+ ε 2 + C ε 2 C 2 ε 4 Ta c 1695 1 .1 162 1 .1 125 0 (, ) , (1.60) R1 R1 R1
The agreement between equation (1.60) and Vohr’s (1968) experimental data for the case of C/R1 = 0.0104 is excellent for values of ε up to about 0.5, which indicates that the term of order ε4 is needed in equation (1.60) for calculating the critical Taylor number
when ε is larger than 0.5. However, agreement between (1.60) and experimental data for
the case of large clearance ratio, Vohr (1968) for the case of C/R1 = 0.099 and Kamal
(1966) for the case of C/R1 = 0.0904, is good for values of ε up to about 0.2 only. This
indicates that the term of order ε4 in equation (1.60) is more significant for the case of
large clearance ratio.
DiPrima and Stuart’s linear nonlocal theory also indicates that the maximum intensity
of vortex motion is at θ = 90˚ downstream from the maximum clearance.
31 Later on, DiPrima and Stuart’s (1975) nonlinear nonlocal theory found that the angular position of maximum vortex intensity can be any value between θ = 0˚ and θ = 90˚,
depending on the value of the supercritical Taylor number T1, which is defined as T1 =
(Ta – Ta c)/ ε. Also they found good agreement with Vohr’s observation of θ = 50˚ for maximum vortex intensity for the case of C/R1 = 0.099 and ε = 0.475.
C Rr R1 70 DiPrima Vohr, C/R1=0.0104 60 DiPrima and Stuart Upper bound
50
40 Lower bound
30 0.0 0.2 0.4 0.6 0.8ε 1.0
Figure 1.8 Lower and Upper Bounds for Instability in Bearing-like Clearance Ratios
Many experimental studies for Taylor vortex transition of eccentric cylinders have been performed, but there are no conclusive results on the verification of the two theories, local and nonlocal. Measurements made with large clearance ratios, Vohr’s (1968) and
Cole’s (1957, 1965), appear to agree better with nonlocal theory. However, measurements made with bearing-like clearance ratios, Frene and Godet (1971, 1974)
(C/R1 = 0.005), appear to support local theory. From a fundamental point of view, the
problem remains unresolved. However, from a practical point of view, Gross et al. 32 (1980) concluded that the two theories of DiPrima (1963) and of DiPrima and Stuart
(1972) can safely be said to bracket the transition zone applied to the bearing-like clearance ratios. The conclusion of Gross et al. is very close to Vohr’s suggestion that the experimental curve for C/R1 = 0.0104 be used as an upper bound while DiPrima’s
theoretical curve be used as a lower bound, Figure 1.8.
1.4 Theories of Turbulence in Bearings and Literature Review
In this section, theories of turbulence in bearings, i.e. Constantinescu, Ng and Pan, and
Hirs’ theories and the corresponding literature review will be introduced.
1.4.1 Constantinescu’s Theory
Constantinescu’s (1959) turbulent lubrication theory starts from the simplified
turbulence version of Navier-Stokes equations:
∂ ∂ ∂ P = µ U − ρ u v'' (1.61a) ∂x ∂y ∂y
∂ ∂ ∂ P = µ W − ρ v'w' (1.61b) ∂z ∂y ∂y
Then Prandtl’s mixing length hypothesis is introduced by Equation (1.62), i.e. the
Reynolds stress component − ρu v'' is proportional to the square of the mixing length l, and Equation (1.63), i.e. the mixing length vanishes at the walls and to vary linearly with the distance from the nearest wall.
∂U ∂U − ρu v'' = ρl 2 (1.62) ∂y ∂y
33 l = ky (0 ≤ y ≤ h 2/ ) (1.63a)
l = ky ' (0 ≤ y'≤ h 2/ ) (1.63b) where y' = h – y. Introducing Equation (1.63) into Equation (1.62) and introducing
Equation (1.62) into Equation (1.61a), the governing equation for a long bearing
( ∂P = ,0 W = 0 ) is obtained. For 0 ≤ y ≤ h/2 this equation has the non-dimensional ∂z
form:
2 ∂ 2 ∂u ∂u ∂u h ∂P 2 + − = (1.64) k y Re h 0 ∂ y ∂ y ∂ y ∂ y µU ∂x where
= ρ µ Re h Uh / (1.65)
y = y / h (1.66)
u = U /U (1.67)
U = U2 is velocity of the rotating surface in the x direction relative to the stationary
bearing surface ( U1 = 0).
By integrating Equation (1.64) one has the following equation:
2 ∂u ∂u ∂u Ay + + B y − C = 0 (1.68) ∂ y ∂ y ∂ y where
= 2 A k Re h (1.69)
h 2 ∂P B = − (1.70) µU ∂x
34 The integration constant C in Equation (1.68) represents the dimensionless wall stress
(∂ ∂ ) = τ µ → [Equation (1.68) reduces to C = u / y y=0 h w / U as y 0 ]. Constantinescu chose
to follow Prandtl and divided the flow regime 0 ≤ y ≤ 2/1 into two layers. He assumed
≤ ≤ that the effect of the Reynolds stress is negligible in the viscous sublayer 0 y y L , and
then Equation (1.68) reduces to
∂u + B y − C = 0 (1.71) ∂ y
≤ ≤ whereas in the turbulent outer layer y L y 2/1 the effect of molecular viscosity is negligible and then Equation (1.68) becomes
2 ∂u ∂u Ay + B y − C = 0 (1.72) ∂ y ∂ y
Equations (1.71) and (1.72) need to be solved simultaneously, and for example, at the position of maximum film pressure ( B = 0) the velocity is:
u = 1− C y' 0 ≤ y'≤ (CA ) − 2/1 (1.73a)
C − u = 1− [1+ ln ()y' CA ] (CA ) 2/1 ≤ y'≤ 5.0 (1.73b) A
C − u = [1+ ln ()y CA ] (CA ) 2/1 ≤ y ≤ 5.0 (1.73c) A
u = C y' 0 ≤ y ≤ (CA ) − 2/1 (1.73d)
where C is given by the transcendental equation
C CA 1− 2 + ln = 0 (1.74) A 4
35 A similar analysis is performed to the axial flow, which is decoupled from the circumferential flow by Constantinescu. Finally the velocities in both the axial and circumferential directions are introduced to the continuity equation, and one gets the differential equation that governs the pressure distribution in a turbulent lubricant film,
Constantinescu (1967):
∂ h3 ∂P ∂ h3 ∂P U ∂h + = (1.75) ∂ µ ∂ ∂ µ ∂ ∂ x k x x z k z z 2 x where
= 1 = + .0 8265 k x 12 .0 0260 (Re h ) (1.76a) Gx
= 1 = + .0 741 k z 12 .0 0198 (Re h ) (1.76b) Gz
Another important parameter affected by turbulence is the shear stress τs acting on the rotating inner cylinder. The expression for τs under a laminar flow condition is
h ∂p τ = τ + (1.77) s c 2 ∂x where
µU τ = (1.78) c h is the Couette shear stress. Onset of turbulence increases both τc and p/ x. The increase of p/ x is calculated through the solution of the turbulence Reynolds equation (1.75),
and the increase of τc is calculated by the following expression:
µU τ = C (1.79) c f h
36 where Cf is a turbulence Couette shear stress factor. The expression of Cf is given by
Constantinescu and Galetuse (1965):
= + .0 855 C f 1 .0 0023 (Re h ) (1.80)
1.4.2 Ng-Pan Theory
Ng (1964) and Ng and Pan (1965) also started from the simplified turbulence version of Navier-Stokes Equations (1.61a) and (1.61b). Substituting Equations (1.81a) and
(1.81b) into Equations (1.61a) and (1.61b) respectively, and integrating twice with respect to y, then the velocity distributions in circumferential and axial directions are
obtained, Equations (1.82a) and (1.82b).
∂U ε ∂U τ = µ − ρu v''= 1+ m (1.81a) xy ∂y ν ∂y
∂W ε ∂W τ = µ − ρv'w' = 1+ m (1.81b) zy ∂y ν ∂y
1 h y dy ' ∂P y y'−h 2/ U = τ + dy ' (1.82a) µ xy ∫0 + ε ν ∂ ∫0 + ε ν 2 1 m / x 1 m /
1 h y dy ' ∂P y y'−h 2/ W = τ + dy ' (1.82b) µ zy ∫0 + ε ν ∂ ∫0 + ε ν 2 1 m / z 1 m /
where τxy (h/2) and τzy(h/2) are integration constants to be determined from the boundary
conditions at y = h.
Then Ng and Pan introduced Reichardt’s empirical formula of eddy viscosity profiles:
+ ε + + y m = k y − δ tanh (1.83) ν l δ + l
37 where
y + = y (τ / ρ) /ν (1.84)
δ + k is mixing length constant and l is a constant related to the thickness of the laminar
sublayer. Introducing Equation (1.83) into Equations (1.82a) and (1.82b), and then
substituting the velocity distributions into the continuity equation, the turbulent
lubrication equation is obtained. This equation is identical to Constantinescu’s Equation
(1.75), but the coefficients kx and kz are different:
1 1 y 2/1 −η g (η) = G = d y 1− c dη (1.85a) x ∫0 ∫ 0 ()η ()η k x f c f c
−η 1 = = 1 y 2/1 η Gz d y d (1.85b) ∫0 ∫ 0 ()η k z f c where
h + f ()y = 1+ k hy + − δ + tanh y c (1.86) c c l δ + l
+ 1 + h g ()y = k hy tanh 2 y c (1.87) c c δ + 2 l
y = y / h (1.88)
+ = (τ ρ ) ν hc h c / / (1.89)
The expressions of coefficients kx and kz are too complicated because of the double
integration in Equation (1.85). However, by least-squares fitting of polynomials to
Equation (1.85), Taylor (1970) obtained a much simpler expression:
1 = = + nx k x 12 K x (Re h ) (1.90a) Gx 38 1 = = + nz k z 12 K z (Re h ) (1.90b) Gz where the constants in Table 1.3 can be used over the appropriate range of Reynolds number.
Elrod and Ng (1967) also derived an expression for τs under turbulence flow condition:
µU h ∂p τ = C + (1.91) s f h 2 ∂x and the turbulence Couette shear stress factor Cf is curve-fitted by Taylor (1970) to give
the following expression:
= + 86.0 > C f 1 .0 00232 (Re h ) Re 10 , 000 (1.92a)
= + 96.0 ≤ C f 1 .0 00099 (Re h ) Re 10 , 000 (1.92b)
Table 1.3 Constants for Calculating k x and k z at Different Reynolds Numbers
Reynolds Number Kx nx Kz nz
50,000 < Re 0.0388 0.80 0.0213 0.80
10,000 ≤ Re < 50,000 0.0250 0.84 0.0136 0.84
5,000 ≤ Re < 10,000 0.0250 0.84 0.0088 0.88
Re < 5,000 0.0039 1.06 0.0021 1.06
1.4.3 Bulk Flow Theory of Hirs
The bulk flow theory of Hirs (1973) is primarily based on the empirical finding that the relationship between wall-shear stress and average velocity can be expressed by a simple formula for pressure flow, for drag flow, and for combinations of these two basic types of
39 flow. Thus two basic formulas, Equations (1.93a) and (1.93b), can be derived based on that finding.
τ u h m0 0 = n m (1.93a) ρ 2 0 ν 2/1 um
τ u h m1 1 = n m (1.93b) ρ 2 1 ν 2/1 um
where τ0 is shear stress at a surface due to flow under the influence of a pressure gradient,
τ1 is shear stress at a surface due to the sliding of a surface, um is the mean velocity, and n0, m0, n1 and m1 are empirical constants to be fitted to the available experimental results.
Experimental results by Hirs (1974) show that m0 = m1 = -0.25 and a = n0/n1 =1.2.
Then after he introduced a fictitious pressure gradient (d P1/d x) to account for the occurrence of the drag flow component, two formulas, one for the stationary surface,
Equation (1.94a), and another for the sliding surface, Equation (1.94b), can be obtained.
d − h ( p + p ) m0 1 u h dx = n m (1.94a) ρ 2 0 ν um
d − h ( p − p ) m0 1 (u −U )h dx = n m (1.94b) ρ − 2 0 ν (um U )
On eliminating the fictitious pressure gradient d P1/d x between Equations (1.94a) and
(1.94b), one can obtain the actual pressure gradient in terms of the average velocity um
and sliding speed U:
m m d p n ρu 2 u h 0 ρ(u −U ) 2 (u −U )h 0 = − 0 m m + m m (1.95) ν ν dx 2 h h
40 Equation (1.95) is valid for the flow in the direction of the representative pressure gradient, but this direction need not coincide with the direction of the sliding speed U [ x
direction in Equation (1.95)], in which case there will be two component equations, one
in the x direction and the other in the z direction. Similar analysis can be done to get the
actual pressure gradient in both x and z directions. Finally the dimensionless pressure
flow coefficients Gx and Gz can be obtained from Equations (1.96a) and (1.96b) by substitution.
2/1 − u /U G = mx (1.96a) x (h 2 / µU )( ∂P / ∂x)
− u /U G = mz (1.96b) z (h 2 / µU)( ∂P / ∂z)
where umx and umz are the mean velocities in the x and z directions respectively. If the
pressure flow component is much smaller than the drag flow component, which is the
case of self-acting bearings operating at moderate eccentricities, Equation (1.96) then
reduces to
1( +m ) 2 0 − + G = Re 1( m0 ) (1.97a) x + h n0 2( m0 )
1( +m ) 2 0 − + = 1( m0 ) Gz Re h (1.97b) n0
where n0 = 0.066 and m0 = -0.25 for smooth surfaces at Reynolds number smaller than
105. With these values Equation (1.97) gives the approximate formulas:
= 1 = 75.0 k x .0 0687 Re h (1.98a) Gx
41 = 1 = 75.0 k z .0 0392 Re h (1.98b) Gz
The values of kx and kz must not be allowed to fall below 12, which is the value appropriate to laminar flow. The limiting Reynolds numbers for this situation are:
= 1 = = k x 12 Re 977 (1.99a) Gx
= 1 = = k z 12 Re ,2 060 (1.99b) Gz
The turbulence Couette shear stress factor Cf is given by the following expression:
= + 75.0 C f 1 .0 00818 (Re h ) (1.100)
and Cf = 1, the value appropriate to laminar flow, when Re ≤ 607.
1.4.4 Literature Review on Theories of Turbulence in Bearings
Constantinescu’s (1959) theory is the first turbulent lubrication theory, but there are
some apparent limitations in his approach. The most obvious one is that the buffer zone
between the laminar sublayer and the fully turbulent region receives no attention. This
should cause a discontinuity in the shear stress, which is not shown in his theory.
Ng and Pan’s (1965) theory predicts the same form of turbulent Reynolds equation as
Constantinescu’s theory. However, the work of the former allows consideration of
Couette and Poiseuille flows and combinations of the two and is generally accepted to be
more accurate than that of Constantinescu. The theory obtained by Hirs (1973) is
completely different and is based solely on experimental observations. It does not give a
42 turbulent Reynolds equation directly but its results can be cast into the form used by
Constantinescu (1959) and Ng and Pan (1965).
As shown in Figures 1.9 and 1.10, the law of the wall approach by Ng and Pan is in excellent agreement with the predictions of the bulk flow theory by Hirs when the
Reynolds number is greater than 10 4. There is a discrepancy between the two approaches
in the transition region, which is after the onset of Taylor vortices and before the
Reynolds number reaches 10 4. Hirs claims that in this region the law of wall used by Ng and Pan is not in good agreement with experimental data. Since Hirs’ bulk flow theory is directly based on experimental evidence it is claimed that it is more applicable in this region and in general the most reliable turbulent lubrication theory.
0.1
Gx = 1/k x Gz
Gz = 1/k z G x
0.01
Constantinescu Ng-Pan Hirs Constantinescu Ng-Pan Hirs
0.001 100 1000 10000Re 100000
Figure 1.9 Turbulence Coefficients G x and G z vs. Reynolds Number
43
0.1
8Cf /Re Constantinescu Ng-Pan Hirs
0.01
0.001 100 1000 10000Re 100000
Figure 1.10 Friction Factor vs. Reynolds Number
In summary, the three turbulence models are based on well-established experimental results. None of these models, however, accounts for the inertia effects as well as the effect of the onset of the Taylor and wavy vortices. They use the Prandtl mixing length and the corresponding eddy viscosity concepts that imply the existence of a turbulent regime, which causes increased viscosity effects. The turbulent lubrication factors, Gx
and Gz, [used in Constantinescu (1959), Ng and Pan (1965) and Hirs (1973)] account for the apparent increased viscosity due to the eddy effects. This approach accurately predicts the flow behavior and the three models are relatively close to each other when
“turbulence” is truly fully developed, i.e. Reynolds number is larger than 2,000.
However, in the transition regime, i.e. after the onset of Taylor vortices and before the full development of turbulence, the discrepancy amongst these most accepted models is significant. Note that the transition regime is very important for both oil and air bearings, 44 because a majority of them operate in this regime, often mistaken for the turbulent one.
Figure 1.11 shows the relationship between the transition regime and the operating ranges for typical oil and air bearings. Thus, for oil bearings (R = 1 to 3 in., C = 0.001 to 0.003
in. and ν = 1×10 -5m2/s), operating in ranges from 3,000 rpm to 30,000 rpm, the Reynolds number varies from 20.3 to 1,824.1 and the Ta number varies from 0.6 to 57.5. For air
bearings ( R = 1 to 3 in., C = 0.001 to 0.003 in. and ν = 1.66×10 -5m2/s), operating in ranges from 30,000 rpm to 100,000 rpm, number Re varies from 122.1 to 3,662.9 while the Ta number varies from 3.9 to 115.8.
20.3 122.1 1824.1 2000 3662.9 Re
air bearings transition regime
oil bearings
0.6 3.9 41.3 57.5 115.8 Ta
Figure 1.11 Operating Ranges for Typical Oil and Air Bearings
Neither of these turbulence models [Constantinescu (1959), Ng and Pan (1965) and
Hirs (1973)] agrees completely with the calculations of Ho and Vohr (1974), who used
the Kolmogoroff-Prandtl energy model of turbulence, but Ng-Pan’s model (1965) comes
closest. Owing to the uncertainty of the flow behavior predicted by these turbulence
models in the transition regime, Gross et al. (1980) recommended the following
45 procedure: i) use Reynolds equations when Re < 41.3 R /C (before the onset of Taylor
vortices); ii) use Ng-Pan model when Re > 2,000 (fully developed turbulent flow); iii) in
between these critical values of Re, interpolate linearly using the Re number as the
interpolating variable. The lower limit for Gx and Gz is that Gx = Gz = 1/12 (laminar values) while the upper one is a function of Re, as Re reaches 2,000. Gross et al. didn’t justify their recommendation but rather forwarded it as a “common sense” approach.
Black et al. (1975) realized the significance of the presence of the Taylor (and wavy) vortices and they developed a theory covering both the vortex and turbulent regimes.
According to this model, the Reynolds equation is expressed in terms of effective viscosities, where the latter have different expressions in the vortex and turbulent regimes, respectively. Application of the theory of Black et al. (1975) to the calculation of load capacity, bearing friction, torque, and lubricant flow rate is compared with
experimental results for a Taylor number range between 2 Ta cr and 10 Ta cr , corresponding to Reynolds numbers between 1,620 and 8,102. For this range there is good agreement between theory and experiments. Constantinescu (1975) and
Constantinescu and Galetuse (1974) realized the importance of the inertia forces, and added an extra term I* to the turbulence model to account for their effect on the momentum equations. According to Constantinescu (1975) and Constantinescu and
Galetuse (1974), I* is a function of film thickness, average velocities, and turbulent lubrication factors and itself includes several correction coefficients. Constantinescu
(1975) concluded that “fluid films operating at large Reynolds numbers are subjected not only to transition and turbulence but also to inertia forces effects”.
46
CHAPTER II
SCOPE OF WORK
2.1 Numerical Simulations and Calculations of Long and Short Bearings
The numerical simulations and calculations of long and short bearings will be
performed by using a commercial code, CFD-ACE+ (2003), produced by ESI Group.
The work will focus on the structure of the fluid flow patterns in a section of fluid film
between two cylinders (finite axial length, 0˚ ≤ θ ≤ 360˚) where the inner one is rotating, and the outer one is stationary. To simulate the infinitely long bearings, periodic boundary conditions are applied to the two ends of the section of fluid film. To simulate the short bearings, zero pressure boundary conditions are applied to the two ends of the finite section of fluid film. The numerical simulations will concentrate on the flows in a micro-scale clearance (0.13 in. to 0.005 in.) with various eccentricity ratios (0.0 to 0.8) between the two cylinders. The work will concentrate on the flow pattern change, velocity profile change and torque generation under conditions when speed, clearance and eccentricity are changed on a parametric basis.
2.2 The Nature of “Instability” and “Turbulence” in Small Gap Journal Bearings
It will be shown from the results that there are two critical points for the onset of: (i) the first Taylor instability (appearance of Taylor vortices) and (ii) the second Taylor 47 instability (appearance of wavy Taylor vortices), as they are indicated by both the flow pattern changes and the inflections of the torque-speed curve. The inflection points in the torque-speed graphs coincide with the flow pattern changes. The Reynolds and Taylor numbers will be calculated for these inflection points and the onset of flow instabilities
(“turbulent” flow called by some literature) will be discussed versus the critical values of these dimensionless numbers. It will be shown that the slope change in the torque-speed graph is due to the change of the average velocity gradient on the outer cylinder wall.
This finding and not an increase in apparent viscosity is the cause for the inflection points on the torque-speed graph. Eventually, the question of “Are the flow formations observed in Taylor instability regimes the actual ‘turbulence’ as it is presently modeled in micro-scale clearance flow?” will be answered.
2.3 New Models for Transition Flow of Thin Films in Long and Short Bearings
The velocity profiles, pressure profiles across the radial direction and the average velocity gradients on the outer cylinder wall will be presented in a quantitative manner for both Taylor and wavy vortex regimes or the transition regime. Order of magnitude analyses of the Navier-Stokes equation component terms will be performed for the transition regime. Two new models (one for long bearing and the other for short bearings), which are fundamentally different from the three most accepted turbulence models [Constantinescu (1959), Ng and Pan (1965) and Hirs (1973)], for predicting flow behavior of thin films in transition regime of long and short journal bearings will be proposed and justified. A comparison will be made between the results of our models and the three turbulence models. 48 2.4 Experimental Verification of the Numerical Results
Experiments will be performed to verify the numerical results. The work will concentrate the flow visualizations and the torque measurements, which will be done simultaneously. It will be shown that numerical calculations and the experimental results appear to be in good agreement with each other. Applications of the new model of long bearings to the calculation of torque will be compared with experimental results in the range of Taylor and wavy vortex regimes.
49
CHAPTER III
NUMERICAL ALGORITHM
3.1 General Introduction to CFD-ACE+
CFD-ACE+ (2003) is a set of computer programs for multi-physics computational analysis. It introduces several modules, such as flow module, heat transfer module, and turbulence module etc, which are governed by different partial differential equations or
PDEs.
The numerical method to solve these PDEs consists of the discretization of the PDEs on a computational grid, the formation of a set of algebraic equations, and the solution of the algebraic equations. The numerical method yields a discrete solution of the field, which is comprised of the values of the variables at the cell centers. In this chapter, the numerical methodology adopted in CFD-ACE+ and flow module, the only module used in present work will be presented.
3.2 Numerical Methodology Adopted by CFD-ACE+
In this section, the numerical methodology adopted by CFD-ACE+, i.e. the discretization of the differential equations, the method of the velocity-pressure coupling, the application of the boundary conditions, and the solution method will be introduced in detail. 50 3.2.1 Discretization
To start the numerical solution process, discretization of the differential equations is introduced to produce a set of algebraic equations. In CFD-ACE+, the finite-volume approach is adopted due to its attractive capability of conserving solution quantities. The solution domain is divided into a number of cells known as control volumes. In the finite volume approach of CFD-ACE+, the governing equations are numerically integrated over each of these computational cells or control volumes. An example of one such control volume is shown in Figure 3.1. The geometric center of the control volume, which is denoted by P, is also often referred to as the cell center. CFD-ACE+ employs a collocated cell-centered variable arrangement, i.e. all dependent variables and material properties are stored at the cell center P. In other words, the average value of any quantity within a control volume is given by its value at the cell center.
Figure 3.1 A Three-dimensional Computational Cell (Control Volume)
Most of the governing equations can be expressed in the form of a generalized transport equation: 51 ∂(ρφ ) r + ∇ • (ρVφ) = ∇ • (Γ∇φ) + Sφ (3.1) ∂t 14243 14243 { 123 convection diffusion source transient
This equation is also known as the generic conservation equation for a quantity φ .
Integrating this equation over a control-volume cell, the equation becomes:
∂(ρφ ) r dϑ + ∇ • (ρVφ)dϑ = ∇ • (Γ∇φ)dϑ + S dϑ (3.2) ∫ ∂ ∫ϑ ∫ϑ ∫ φ ϑ t ϑ
The transient term in Equation (3.2) is discretized as follows,
∂(ρφ ) ρφϑ − ρ 0φ 0ϑ 0 dϑ = (3.3) ∫ ∂ ∆ ϑ t t
where the superscript “0” denotes an older time, while no superscript denotes the current
or the new time. The cell volume, represented by ϑ , may change with time, in particular
when moving grids are used.
Figure 3.2 A Two-dimensional Cell (Control Volumes)
The convection term is discretized as follows: r r ∇ • ρ φ ϑ = ρφ • r = ρ φ n = φ ( V )d V ndA ( e eVe )Ae Ce e (3.4) ∫ϑ ∫A ∑ ∑ e e
52 where subscript “ e” denotes one of the faces of the cell in question, Ae is the area of face
n e, Ve represents the velocity component in the direction that is normal to the face, Ce is thus the mass flux across the face. The evaluation of φ at control volume faces is expressed differently in different schemes.
For ease of illustration, let us consider a two-dimensional control volume as shown in
Figure 3.2. Because the solution variable φ is available only at the cell-centers, the cell-
face values of φ need to be interpolated. Various interpolation schemes with varying
levels of numerical accuracy and stability are used today. In CFD-ACE+ (2003) the user
has a choice of several popular schemes, including first-order upwind, central difference,
second-order upwind, second-order upwind with limiter, smart scheme, or third-order
scheme. After preliminary numerical experimentation third-order scheme was chosen in
present calculations because it took the shortest CPU time to get the converged solutions.
The diffusion term is discretized as follows:
r ∂φ ∇ • (Γ∇φ)dϑ = Γ∇φ • ndA = Γ A (3.5) ∫ϑ ∫ ∑ e ∂ e A e n e
With the three unit vectors defined in Figure 3.2, ∂φ / ∂n can be expressed:
∂φ 1 ∂φ r r ∂φ = r r − e •τ (3.6) ∂n n • e ∂e ∂τ
Then the diffusion term becomes: r r Γ ∂φ τ • eΓ ∂φ ∇ • (Γ∇φ)dϑ = r e r A − r r e A (3.7) ∫ϑ ∑ • ∂ e ∑ • ∂τ e e n e n e e n e e
where
53 ∂φ φ −φ = E P (3.8) ∂ δ e e P,E
∂φ φ − φ = C 2 C1 (3.9) ∂τ δ e C ,2 C1
δ δ where P,E and C ,2 C1 represent the distance between E and P, and C2 and C1, respectively.
If the source term is a function of φ itself, it can be linearized as the following:
U P Sφ = S + S φ (3.10) such that S P is negative. The linearized source term is integrated over the control volume,
which results in:
ϑ = + φ ∫ Sφ d SU S P P (3.11) ϑ
= Pϑ = Uϑ where S P S and SU S .
The numerically integrated transient, convection, diffusion and source terms are assembled together, which result in the following finite difference equation:
− φ = φ + (aP S P ) P ∑ anb ab SU (3.12) nb where the subscripts “ nb ” denote values at neighboring cells, anb are known as the link
coefficients. This finite difference equation is the discrete equivalent of the continuous
flow transport equation that we started with, Equation (3.2).
Equation (3.12), in general, is nonlinear because the link coefficients anb themselves
are functions of φ P, φ nb , etc. When this equation is formulated for each computational
cell, it results in a set of coupled nonlinear algebraic equations. No direct matrix
54 inversion method is available to solve a set of nonlinear algebraic equations. Therefore, an iterative procedure in employed in CFD-ACE+ at every time step.
3.2.2 Velocity-Pressure Coupling
The continuity equation, which governs mass conservation, requires special attention because it can not be written in the form of the general convection-diffusion equation.
Moreover, it is used to determine the pressure field in the pressure-based method, which is employed in CFD-ACE+. The continuity equation can be written in the form:
∂ρ + ∇ • (ρV ) = 0 (3.13) ∂t
Integrating the above equation over the cell in Figure 3.2, the equation becomes:
ρϑ − ρ 0ϑ 0 + ρ V n A = 0 (3.14) ∆ ∑ e e e t e
n where Ve is the face-normal component of the velocity at face e, which is obtained by the inner product of the velocity vector ( u, v, w) and the face-normal unit vector ( nx, ny, nz),
n = + + Ve ue nx ve n y we nz (3.15)
Solutions of the three momentum equations yield the three Cartesian components of
velocity. Even though pressure is an important flow variable, no governing PDE for
pressure is presented. Pressure-based methods use the continuity equation to formulate
an equation for pressure. In CFD-ACE+, the SIMPLEC (Semi-Implicit Method for
Pressure-Linked Equations Consistent) scheme, which was originally proposed by Van
Doormal and Raithby (1984), has been adopted.
The finite difference form of the x-momentum equation can be written as:
55 = + − aPuP ∑ anb uab SU ∑ Pe Ae nxe (3.16) nb P e P
with the subscript P again indicating that the equation is written for cell center P.
The pressure field should be provided to solve the above equation for u, but it is not
known in advance. If the equation is solved with a guessed pressure P*, it will yield
velocity u*, which satisfies the following equation:
* = * + − * aPuP ∑ anb unb SU ∑ Pe Ae nxe (3.17) nb P e P
In general, u* will not satisfy the continuity equation. The strategy is to find corrections to and so that an improved solution can be obtained. Let u' and P' stand for corrections, then:
u = u * + u′ (3.18)
P = P* + P′ (3.19)
An expression for u' p can be obtained by subtracting Equation (3.17) from Equation
(3.16):
′ = ′ − ′ aPuP ∑ anb uab ∑ Pe Ae nxe (3.20) nb P e P
which gives an expression for u' p if u' nb is approximated by u' p:
−1 u′ = P′A n (3.21) P − ∑ e e xe aP ∑ anb e P
Then substitute all the corrected velocity components, Equation (3.18) into Equation
(3.14), a pressure correction equation can be derived:
′ = ′ + aP PP ∑ anb Pab Sm (3.22) nb 56 where Sm represents the mass correction or mass source in the control volume:
ρ 0ϑ 0 − ρ *ϑ S = P − ρ V n A (3.23) m ∆ ∑ e e e t e
The SIMPLEC procedure can be summarized as follows:
1) Guess a pressure field P*.
2) Obtain u*, v*, and w* by solving discretized momentum equation (3.17).
3) Obtain P' by solving Equation (3.22).
4) Calculate P from Equation (3.19).
5) Calculate u, v, and w from Equation (3.18).
6) Solve the discretized equations for other flow variables, such as enthalpy, turbulent quantities etc.
7) Treat the corrected pressure P as a new guessed P*, return to step 2 and repeat the procedure until converged solution is obtained.
3.2.3 Boundary Conditions
Figure 3.3 Computational Boundary Cell
57 A control cell adjacent to the west boundary of the calculation domain is shown in
Figure 3.3. A fictitious boundary node labeled B is shown. The finite-volume equation for node P will be constructed as:
φ = φ + φ + φ + aP P aE E aN N aS S S (3.24)
where coefficient aW is set to zero after the links to the boundary node are incorporated into the source term S in its linearized form:
= + φ S SU S P P (3.25)
φ If the boundary value is fixed as B , the source term is modified as:
= + φ SU SU aW B (3.26)
= − S P S P aW (3.27)
At zero-flux boundaries, such as adiabatic walls for heat and symmetric boundaries for
any scalar variables, the boundary link coefficients are simply set to zero without
modifying source terms.
3.2.4 Solution Methods
CFD-ACE+ uses an iterative, segregated solution method wherein the equation sets for
each variable are solved sequentially and repeatedly until a converged solution is
obtained. The overall solution procedure is shown in Figure 3.4.
Note that all the parameters, which indicate how many times a procedure is repeated,
can be specified by the user. These are the number of iterations (NITER) and the number
of time step (NSTEP) in the case of a transient simulation. The number of iterations to be
performed is dictated by the overall residual reduction desired. At each iteration the
58 program will calculate a residual for each variable, which is the sum of the absolute value of the residual for that variable at each computational cell. In general, a five order of magnitude reduction in the residual is expected before declaring that convergence has been obtained.
Figure 3.4 Solution Flowchart
Under-relaxation of the dependent and auxiliary variables is used to constrain the change in the variable from iteration to iteration in order to prevent divergence of the solution procedure. For all dependent variables, this is done by modifying Equation
(3.12) in the following way,
59 + φ = φ + + φ * aP 1( I) P ∑ anb ab SU aP I P (3.28) nb
* where φ P is the current value of φ P. At convergence, when there is no change in φ P from one iteration to the next, the equation is not modified at all by the addition of these terms. Prior to convergence, however, they provide a link between the new value of φ P
* and the current value φ P . The larger the value of I is, the stronger the under-relaxation
will be. The values of I in the range of 0.2 to 0.8 are common in CFD-ACE+.
The type of linear equation solver is crucial for a solution method because CPU time
and memory requirements strongly depend on it. Two types of linear equation solvers
are available in CFD-ACE+. They are Conjugate Gradient Squared + Preconditioning
(CGS + Pre) Solver and Algebraic MultiGrid (AMG) Solver. However, only AMG
solver is adopted in our calculations due to its two major advantages: a) CPU time only
increase in proportion to the number of unknowns in the equations; b) faster convergence.
The basic idea of AMG is to use hierarchy of girds, from fine to coarse, to solve a set
of equations, with each grid being particularly effective for removing errors of
wavelength characteristic of the mesh spacing on that grid. The solution method can be
illustrated for a two-grid system:
1) On the fine grid (original grid) obtain residual after performing a few iterations,
2) Perform iterations on the coarse grid to obtain corrections, with the fine grid
residual being imposed as a source term,
3) Interpolate the corrections to the fine grid and update the fine grid solution,
4) Repeat the entire procedure until the residual is reduced to the desired level.
60 3.3 Flow Module
CFD-ACE+ offers several modules, such as flow module, heat transfer module, and turbulence module etc, but flow module is the most important one and is also the only module used for most of our numerical calculations.
The flow module allows CFD-ACE+ to simulate almost any fluid (gas or liquid) flow problem. Both internal and external flows can be simulated to obtain velocity and pressure fields.
The governing equations for the flow module represent mathematical statements of the conservation laws of physics for flow, namely, a) The mass of a fluid is conserved; b)
The time rate of change of momentum equals the sum of the forces on the fluid
(Newton’s second law). These two laws can be used to develop a set of equations
(known as the Navier-Stokes equations) for CFD-ACE+ to solve numerically.
The flow module can also be used in conjunction with many of the other modules, such as turbulence module, heat transfer module etc, in CFD-ACE+ to perform multi-physics analyses.
61
CHAPTER IV
NUMERICAL RESULTS (LONG BEARING)
4.1 Introduction
This chapter is concerned the relationship between the onset of Taylor instability and
appearance of what is commonly known as “turbulence” in narrow gaps between two
infinitely long cylinders with various eccentricity ratios. A question that we open to
debate is whether the flow formations observed during Taylor instability regimes are, or
are related to the actual “turbulence” as it is presently modeled in microscale clearance
flows. To start answering this question the viscous flow in small gaps (0.005 in. to 0.13
in.) between two cylinders with eccentricity ratios varying from 0.0 to 0.8, is investigated
using CFD-ACE+ as a computational engine.
On the problem of hydrodynamic stability of rotating fluids between two infinitely
long cylinders with the inner one rotating and the outer one at rest, there are four
controlling parameters: (i) the speed of the inner cylinder ω, (ii) the clearance between
the two cylinders C, (iii) the dimensionless axial wavelength of the instability λ (one can
not simulate the fluid between two infinitely long cylinders, instead one can study one
section of fluid with height L = λC due to the periodic behavior of the fluid instability), and (iv) the eccentricity ratio ε. Strategically and numerically it is impracticable to vary
all the parameters at one time. We have picked λ = 2 after a preliminary investigation 62 (see “The Geometry and Boundary Condition Applications” section in this chapter); the angular velocity ω needs to be changed as the main parameter of the problem. Therefore,
either C or ε needs to be kept constant at any given time. In the following sections the
effect of C and ω while ε = 0 (Section 4.5) and the effect of ε and ω while C = 0.01 in.
(Section 4.6) on the Taylor vortices induced instability will be presented. Finally a new
model for predicting the flow behavior in long journal bearing films in the transition
regime is proposed in Section 4.7.
4.2 Geometry and Boundary Condition Applications
In this section, the geometry adopted in the investigation of the fluid stability problem
and the application of the boundary conditions will be introduced.
4.2.1 Geometry
The viscous fluid flow in the gap between two cylinders with various eccentricities
will be investigated by using CFD-ACE+ as a computational engine. The geometry is
presented in Figure 4.1. Detail shown in c)
ω z R+C θ θ r L r e z R θ
r
a) b) c)
Figure 4.1 Geometric Description of the Cylinders (Not at scale) 63
0.1325 Torque λ = 2.2 (N.m) λ = 2.1 0.1300 λ = 2.0 λ = 1.9 λ = 1.8 0.1275
0.1250
0.1225 Ta 0.5 0.1200 41.0 41.5 42.0 42.5
Figure 4.2 Torque vs. Taylor Number for Various λs when C = 0.01 in. and ε = 0.0
Torque Torque λ = 2 (L/C = 6) (N.m) λ = 2 (L/C = 4) (N.m) 0.27 0.41
0.40
0.26 0.39
0.38 0.25
0.37 0.5 Ta 0.5 Ta 0.24 0.36 41.0 41.5 42.0 42.5 41.0 41.5 42.0 42.5
a) b)
Figure 4.3 Torque vs. Taylor Number for Different Aspect Ratio (L/C = 4 and 6)
64
L
L
L λC λC λC
C C C
a) b) c)
Figure 4.4 Wavelength of Taylor Vortices for Different Aspect Ratios (L/C = 2, 4 and 6)
65 The two cylinders have a finite length ofL = λ C , where λ represents the dimensionless
axial wavelength that we have chosen to be 2. After preliminary numerical
experimentation we chose this axial wavelength because it appears to be the wave length
at which the fluid is most unstable at the lowest rotational speed. For example as shown
in Figure 4.2 for the case of C = 0.01 in. and ε = 0.0, for λ ≠ 2 the critical Taylor number
is always larger than 41.52, which is the critical Taylor number for λ = 2 when C = 0.01 in. and ε = 0.0, therefore the disturbance with λ = 2 will grow first. Once the instability sets in at λ = 2, the other flow instabilities with λ ≠ 2 can not occur.
If the aspect ratio (L/C) is an even integer, the instability with λ = 2 will always set on when the critical speed is reached. The critical Taylor numbers for the cases of L/C = 4 and 6 are 41.52, Figures 4.3a and 4.3b, which is the same critical Taylor number when
L/C = 2. The wavelengths of Taylor vortices for different aspect ratios (L/C = 2, 4 and 6) are shown in Figure 4.4. For other clearances and eccentricities, similar numerical experimentation was also performed and it shows that the Taylor type instabilities with λ
= 2 always set on first.
The radius of the outer cylinder is fixed, R+C = 1.0 in., and the average clearance, C, between the two cylinders varies from 0.13 to 0.005 in., Figure 4.1. Therefore, the radius of the inner cylinder varies from 0.87 to 0.995 in. correspondingly. The centers of the cylinders are set at a distance ‘e’ apart. The corresponding eccentricity ratio, ε = e/C,
varies from 0.0 to 0.8. The angular speed of the inner cylinder ω, the average clearance
C, and the eccentricity ratio ε are the three controlling parameters.
66 4.2.2 Boundary Condition Applications
The top and bottom faces of fluid between the two cylinders are set as periodic boundary conditions, due to the λ axial wavelength characteristic. That means that the
values of the variables and material properties at the nodes on the top and bottom faces
are the same if their coordinates are the same in circumferential and radial directions but
at a distance λ apart in the axial direction. The outer cylinder is set as a non-slip wall boundary condition. The inner cylinder wall is set as a rotating non-slip wall, and the angular velocity vector in the z-direction is prescribed, as shown in Figure 4.1.
4.3 Convergence Criteria and Numerical Accuracy
In this section, the convergence criteria adopted in CFD-ACE+, grid convergence and
time step convergence will be introduced in detail.
4.3.1 Gridding and Grid Convergence
The three-dimensional grid for the fluid is obtained by extruding the two-dimensional
annulus-like x-y grid into the z-direction, Figure 4.1. The number of nodes varies in the
radial direction (12 – 20 nodes), circumferential direction (160 – 600 nodes), and axial
direction (12 – 18 nodes) as a function of clearance and eccentricity ratio. Also the
number of nodes in the circumferential direction for each circumferential quarter (Figure
4.1a shows that the fluid annulus is divided into four quarters) varies with the different
eccentricity ratio. The total number of cells for the entire annulus thus varies from
40,500 to 72,600.
67 CFD-ACE+ uses absolute convergence criteria, which for the pressure field usually,
require convergence of the residual of the order of 1.0E-4. For the cases considered here
we have used a convergence criterion of 1.0E-4 for each primitive variable (u, v, w, and
p). To improve convergence, the under-relaxation factors (URF) are adjusted
individually for each one of these variables. Thus, for velocities and pressure the typical
URFs vary between 0.1 and 0.8.
Table 4.1 Grid Convergence Experiments (C = 0.13 in., ε = 0.0, and ω = 1,550 rpm)
a) b) c)
Grid / Variables 13×108×12 20×160×18 30×240×26
Torque (N.m) 0.1513 0.1541 0.1552
Error (%) 1.85 0.71
Pressure (N/m 2) 679.8 731.8 757.6
Error (%) 7.65 3.53
Grid convergence experiments were also performed in order to ensure that the results presented below are fully converged and are grid independent. Since one had to choose between ultra accurate grid convergence and acceptable computation time for these parametric calculations, we decided that grid convergence within 5.0% is an acceptable threshold. Table 4.1 details the maximum error in torque on the outer cylinder and the generated pressure between three consecutive grids used for calibration of the grid convergence for the case of C = 0.13 in., ε = 0.0, and ω = 1,550 rpm. The baseline grid
was 13×108×12 as can be seen in column a) of Table 4.1. Two other denser grids were
68 also used for the same geometry. Each subsequent grid was increased by a factor of 1.5 per direction. For our computations we chose the grid of 20×160×18 detailed in column b), which yields results that are different at most by 3.53% from the results obtained with the densest grid in column c). Similar grid convergence experiments were also taken for the cases of other clearances, eccentricity ratios and angular speeds.
4.3.2 Time Step and Time Step Convergence
Each time step must be small enough and converged in order to get accurate and converged results for a transient case, which is needed for calculating the wavy vortices.
The time step can be calculated from the CFL [CFL stands for Courant, Friedrichs and
u∆t v∆t w∆t Lewy, Courant et al. (1967)] number: CFL = + + , where u, v, and w are the ∆x ∆y ∆z
characteristic speeds in the circumferential, radial, and axial directions respectively, t is the time step, x, y, and z are the sizes of the control volume in the circumferential,
radial, and axial directions respectively. Generally the allowable CFL numbers for
simple viscous fluid flow with implicit time integration range from 0.1 to 1.2, Courant et
al. (1967). However, for a specific problem one needs to do time step convergence
experiments to optimize the CFL number.
Figure 4.5 details the torque, wave numbers and CPU time at different CFL numbers
for the case of C = 0.13 in., ε = 0.0, and ω = 4,000 rpm ( Ta =120.65). There is no wave
at all or the wave number is 5 when CFL is larger than 0.5. However, the wave number
is always 6 when CFL is smaller or equal to 0.5. CFL of 0.4 is picked for our
calculations by the trade-off between the CPU time and the accuracy of the torque, which
69 is listed in both Figure 4.5 and Table 4.2. Similar time step convergence experiments were also taken for the transient cases of other clearances, eccentricities, and speeds.
0.020
0.018
0.016
0.014
CFL=2.0, No w aves, CPU=1(5.5hr.) 0.012 CFL=1.0, 5 w aves, CPU=4.4 Torque (N.m) Torque CFL=0.625, 5 w aves, CPU=13.1 0.010 CFL=0.5, 6 w aves, CPU=12 CFL=0.4, 6 w aves, CPU=5 0.008 CFL=0.3, 6 w aves, CPU=6.7
0.006 0.00 0.30 0.60 0.90 1.20 Time (s)
Figure 4.5 Torque vs. Time at Different CFLs
(C = 0.13 in., ε = 0.0, ω = 4000 rpm, Re = 323.55, Ta = 120.65)
Table 4.2 Time Step Convergence Experiments (C=0.13 in., ε=0.0, and ω=4,000 rpm)
a) b) c)
CFL/ Variable 0.5 0.4 0.3
Torque (N.m) 0.8306 0.8225 0.8155
Error (%) 0.98 0.85
CPU Time 12.0 5.0 6.7
70 4.4 Multiplicity and Transition of Taylor-Couette Flow
Multiplicity and transition of Taylor-Couette flow are two of the most attractive features in studying the hydrodynamic stability of rotating fluids between two cylinders.
Both of them will be presented in detail in this section.
4.4.1 Introduction
Multiplicity of Taylor-Couette flow is one of the most attractive features in studying the hydrodynamic stability of rotating fluids between two cylinders. Experimentally,
Coles (1965) observed as many as 20 or 25 different states at a given speed for the case of the inner cylinder rotating and the outer one at rest. According to Coles, each state is defined by the number of Taylor cells in the axial direction and the number of waves in circumferential direction.
Both analytically and experimentally, Benjamin (1978a, 1978b) attacked the problem of bifurcation phenomena in steady flows of a viscous fluid. He concluded that the flow strongly depended on two parameters, one the Reynolds number Re and the other the aspect ratio Г (length to clearance, L/C). For a given value of Г, a unique mode with an
even number of Taylor cells emerges as the primary mode as the speed of the inner
cylinder is gradually increased to a given value from rest. The other mode is also existed
as the secondary mode , which appears with sudden increase of the inner cylinder’s speed
to the desired value.
Benjamin and Mullin (1982) observed 15 different stable steady flows under the same
geometrical and dynamical boundary conditions ( η = 0.6, Г = 12.61, and Re = 359). The
12-cell primary mode was easily produced by slowly raising the speed of the inner 71 cylinder to the prescribed value. The other 14 secondary modes were realized by the manipulation of the acceleration rate of the inner cylinder speed and first at smaller or larger values of Г.
Blennerhassett and Hall (1979) investigated the stability of circular Couette flow in a
finite-length cylindrical annulus by using linear theory. Their results showed that the
instability changed from a two-cell to a four-cell primary flow at Г ≈ 1.3. This analytical work is close to Benjamin’s (1978b) experimental observation that the primary flow changed from a two-cell to a four-cell mode when Г ≈ 1.85.
Transition between those stable steady solutions is another attractive feature in
studying the hydrodynamic stability of rotating fluids between two cylinders. Benjamin
and Mullin (1982) concluded that there are at least N-1 additional unstable solutions if N
of distinct steady solutions are realizable. Then one question is raised: why and how the
transition happens between those steady solutions? Coles (1965) identified two distinct
kinds of transition in Taylor-Couette flow between concentric rotating cylinders, namely
spectral and catastrophic transitions. The spectral transition is characteristic of the
motion when the inner cylinder rotating and the outer one at rest. This kind of transition
may be viewed as a cascade process, in which energy is transferred through a discrete
spectrum to progressively higher frequencies in a two-dimensional wave-number space.
Ohmura et al. (1994) concluded that vorticity played an important role on the transition
between steady solutions. For the case of η = 0.615, Г = 3.0 and end wall boundaries,
Ohmura et al. found that higher acceleration rates of inner cylinder speed produce such strong vorticity for the Taylor cells on the end walls that a pair of cells is induced toward
72 the center and four cells are formed, while lower acceleration rates promote diffusion of vorticity into the center of the annulus and two cells are formed.
The next two sections will deal with the influence of T b, the time needed to increase the speed of the inner cylinder from rest to the prescribed value, on the number of Taylor cells formed in axial direction and waves formed in the circumferential direction. Section
4.4.2 will investigate the influence of T b on the number of Taylor cells, and the fluid in
finite length annulus ( η = 0.99, Г = 3.0 and end wall boundaries) will be studied with T b as a parameter. Section 4.4.3 will investigate the influence of T b on the number of waves
in the circumferential direction, and the fluid in finite length annulus ( η = 0.99, Г = 2.0
and periodic boundaries) will be studied with T b as a parameter.
4.4.2 The Influence of T b on the Number of Taylor Cells
Detail shown in c) ω R+C z θ r θ L r R z
θ r
a) b) c)
Figure 4.6 Geometry for Studying the Influence of T on the Number of Taylor Cells b (Not at Scale)
73 The geometry for studying the influence of Tb on the number of Taylor cells is
presented in Figure 4.6. The length of the two cylinders is L = 0.762 mm (0.03 in.) and
the clearance is C = 0.254 mm (0.01 in.) corresponding to Г = L/C = 3. The radii of the inner and the outer cylinders are R = 25.146 mm (0.99 in.) and R+C = 25.4 mm (1.0 in.).
The top and bottom faces of the annulus are set as no-slip wall boundary conditions.
The outer cylinder wall is set as a no-slip wall boundary condition. The inner cylinder wall is set as a rotating wall, and the angular velocity vector in the z-direction is prescribed, Figure 4.6.
The time (T b) needed to increase the speed of inner cylinder from rest to prescribed speed (263.33 m/s) is varied in a parametric manner. Figure 4.7 shows the velocity of the inner cylinder vs. time for different T b’s.
300
250
200
150
Impulsive Start 100 Tb = 0.0002 s Tb = 0.001 s 50 Tb = 0.002 s
Velocity of Velocity (m/s) Cylinder Inner 0 0 0.001 0.002t (s) 0.003
Figure 4.7 Velocity of Inner Cylinder vs. Time
(T b is the Time Needed to Increase the Speed of Inner Cylinder from Rest to 263.33 m/s)
74
C D E 5
4
3 B F
A 2 Impulsive Start 1 Tb = 0.0002 s
Number Number of Taylor Cells Tb = 0.001 s Tb = 0.002 s 0 0 0.001 0.002t (s) 0.003
Figure 4.8 Number of Taylor Cells vs. Time
D E F 7
6 C 5 2→4 4→2 4
3 w(m/s) Impulsive Start B 2 Tb = 0.0002 s 0→2 Tb = 0.001 s 1 Tb = 0.002 s A 0 0→2 0 0.001 0.002t (s) 0.003
Figure 4.9 w - Velocity in Axial Direction vs. Time
75 Figure 4.8 presents the number of Taylor cells formed in the axial direction as a function of time for different T b’s. It is shown that cell number becomes two, and then becomes four, and finally turns back to two as the time increases for the case of impulsive start (T b = 0 s) and T b = 0.0002 s. However, the cell number becomes two only as the time increases for the case of T b = 0.001 s and T b = 0.002 s.
The formation of Taylor cells is indicated by w, the velocity in axial direction. Figure
4.9 presents the velocity w as a function of time for different T b’s. It is shown that w, which can be used to denote the disturbance amplitude A(t), exhibits an initial exponential growth when the time t is small for all the cases (notice that this exponential growth is obvious only for the cases of T b = 0.001 s and T b = 0.002 s on the current time scale). Qualitatively our calculations reach a good agreement with the numerical computational results by Neitzel (1984), and experimental observations by Donnelly and
Schwarz (1965), i.e. the disturbance amplitude grows exponentially when t is small.
Figure 4.9 also shows that w is always increasing where there is a change of cell numbers. For example for the case of impulsive start, w always increases when cell number becomes two (“0 →2” is labeled to represent this moment), when cell number turns to four from two (“2 →4” is labeled to represent this moment), and when cell
number turns back to two from four (“4 →2” is labeled to represent this moment). Also for the case of T b = 0.002 s, w increases when cell number becomes two (“0 →2” is labeled to represent this moment).
There is only one stable solution (cell number is two) for all the T b’s, Figure 4.8.
However, there exists a transition solution (cell number is four) for the cases of impulsive start (T b = 0 s) and T b = 0.0002 s. How does T b cause the cell number to change? 76 Ohmura et al. (1994) concluded that vorticity played an important role on the formation of cells: higher acceleration rates of inner cylinder speed produce such strong vorticity on the end walls that a pair of cells is induced toward the center (four cells formed), while lower acceleration rates promote diffusion of vorticity into the center of the annulus (two cells formed).
Does vorticity really play an important role on the formation of the cells? Figure 4.10 presents the maximum vorticity generated as a function of time for different T b’s. The
maximum vorticity at where cell number becomes two (“0 →2”) is not much different from the one at where cell number turns to four from two (“2 →4”) and the one at where cell number turns back to two (“4 →2”) for the case of impulsive start. Therefore, the
conclusion can be made that at least the maximum vorticity doesn’t play a big role on the
formation or the mergence of Taylor cells. Instead the maximum vorticity is only the
consequence of the velocity of inner cylinder if one observes the similarity of Figures 4.7
and 4.10.
The formation of different cell numbers is not caused by the maximum vortictiy.
Instead, the generated pressure, P, plays an important role on the formation of cells.
Figure 4.11 show the generated pressure as a function of time for different T b’s. P at
where cell number becomes two (“0 →2”) is much smaller than the one at where cell number turns to four from two (“2 →4”), and P at where cell number turns back to two
(“4 →2”) is much larger than the former two ones for the case of impulsive start. The conclusion that P is the direct cause of w, the indication of the formation of cells can be
made if one observes the similarity of Figures 4.9 and 4.11.
77
2→4 4→2 A B C D E F
0→2 80 70 60
50
40
30 Impulsive Start Tb = 0.0005 s 20 Tb = 0.001 s Tb = 0.002 s 0→2
Maximum Vortictiy (E+5 1/s) 10
0 0 0.001 0.002t (s) 0.003
Figure 4.10 Maximum Vorticity vs. Time
F 5 E 4→2 D 2→4 4 C 3
2
P (E+5N/m^2)
Impulsive Start 0→2 Tb = 0.0002 s B 1 Tb = 0.001 s A Tb = 0.002 s
0→2 0 0 0.001 0.002t (s) 0.003
Figure 4.11 P – Generated Pressure vs. Time
78
a1) a2) a3) a) Point A in Figures 4.6 through 4.9
b1) b2) b3)
b) Point B in Figures 4.6 through 4.9
c1) c2) c3) c) Point C in Figures 4.6 through 4.9
Figure 4.12 The Process of Formation of Taylor Cells for the Case of Impulsive Start
79
a3) a1) a2) a) Point D in Figures 4.6 through 4.9
b1) b2) b3) b) Point E in Figures 4.6 through 4.9
c1) c2) c3) c) Point F in Figures 4.6 through 4.9
Figure 4.13 The Process of Mergence of Taylor Cells for the Case of Impulsive Start
80 Figure 4.12 shows the process of the formation of Taylor cells for the case of impulsive start. Two very weak cells first appear at the corners near the end walls, Figure
4.12a. As the time increases, the two cells grow and other two cells generate at the center of the annulus due to the increase of pressure, Figure 4.12b. Notice that the pressure in
Figure 4.12b is much larger than the one in Figure 4.12a, while the maximum vorticity in
Figure 4.12b is almost the same as the one in Figure 4.12a. Figure 4.12c shows that the four Taylor cells are fully developed but the two cells in the center are much smaller than those on the end walls. During the process from Figure 4.12b to Figure 4.12c one can notice that the pressure further increases significantly while the maximum vorticity does not increase much.
Figure 4.13 shows the process of the mergence of Taylor cells for the case of impulsive start. Figure 4.13a shows that the two cells at the center become smaller and the two cells on the end walls become larger. Then the two cells at the center disappear and the two cells on the end walls become further larger, Figure 4.13b. Finally the two cells on the end walls occupied the whole domain of the annulus, Figure 4.13c. Notice that pressure increase significantly, while the maximum vorticity doesn’t change much during this mergence process.
In summary, it is the generated pressure P, not the maximum vorticity, plays an
important role on the formation of Taylor cells. The pressure is the direct cause of the w,
the indication of the formation of cells, while the maximum vorticity is only the
consequence of the velocity of the inner cylinder.
81 4.4.3 The Influence of T b on the Number of Waves
The geometry for studying the influence of Tb on the number of waves is presented in
Figure 4.1 of Section 4.2.1.
300
250
200
150 Impulsive Start 100 Tb = 0.002 s Tb = 0.020 s 50 Tb = 0.040 s
Velocity of Velocity (m/s) Cylinder Inner 0 0 0.01 0.02 0.03 0.04 0.05 0.06 t (s)
Figure 4.14 Velocity of Inner Cylinder vs. Time
15
a b c 12
9
d e f 6 Impulsive Start Tb = 0.002 s 3 Tb = 0.020 s
Wave Number (Circumferential) Wave Tb = 0.040 s
0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 t (s)
Figure 4.15 Wave Number in Circumferential Direction vs. Time
82 The time T b needed to increase the speed of inner cylinder from rest to prescribed speed (263.33 m/s) is varied in a parametric manner. Figure 4.14 shows the velocity of the inner cylinder vs. time for different T b’s.
Figure 4.15 presents the number of waves formed in the circumferential direction as a function of time for different T b’s. It is shown that wave number becomes 13, and then
gradually drops to 12 (not happens for the case of impulsive start), 11, 10 and 9, and
eventually becomes 8 as the time increases for the cases of impulsive start (T b = 0 s) and
Tb = 0.002 s. For the case of T b = 0.02 s the wave number first appears as 14, and then
gradually drops to 10 and 8, and eventually becomes 7 with the increase of time. For the
case of T b = 0.04 s, however, the wave number first forms at 12, and it becomes and stays at 11 as the time increases.
18
a b Impulsive Start c Tb = 0.002 s 15 Tb = 0.020 s 1st peak Tb = 0.040 s f 12 d e 8→7 12 →11 9 0→13 9→8 2nd pressure zone (E+5N/m^2) P 6 9→8 1st pressure zone 0→13 0→14 st 3 1 peak 1st peak 0→12 1st peak 0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 t (s)
Figure 4.16 P – Generated Pressure vs. Time
83
2 1 10 2 1 10 3 3
a1) Point a in Figures 12 and 13) a2) Point a in Figures 12) and 13)
2 2 10 3 1 10 3 1
b1) Point b in Figures 12) and 13) b2) Point b in Figures 12) and 13)
3 2 1 10 3 2 1 10
c1) Point c in Figures 12) and 13) c2) Point c in Figures 12) and 13)
11 (were 1 & 2) 11 (were 1 & 2) 3 10 3 10
d1) Point d in Figures 12) and 13) d2) Point d in Figures 12) and 13)
Figure 4.17 The Development of Waves for the Case of T b = 0.002 s
84
11 (were 1 & 2) 11 (were 1 & 2) 3 10 3 10
e1) Point e in Figures 12) and 13) e2) Point e in Figures 12) and 13)
3 11 (were 1 & 2) 3 11 (were 1 & 2) 10 10
f1) Point f in Figures 12) and 13) f2) Point f in Figures 12) and 13)
Figure 4.17 The Development of Waves for the Case of T b = 0.002 s (Continued)
As it did on the formation of Taylor cells in the axial direction, the generated pressure,
P, also plays a significant role on the formation of waves in the circumferential direction. Figure 4.16 shows the generated pressure, P, as a function of time for
st different T b’s. As the time increases, a “1 peak” (shown in Figure 4.16) of the pressure appears for each case, which coincides with the formation of Taylor cells. The pressure increases significantly again, which coincides with the appearance of waves in the circumferential direction, after a slow pressure increase period. At this moment, the number of waves is different for different T b’s: 13 for the cases of impulsive start and T b
= 0.002 s, 14 for the case of T b = 0.02 s, and 12 for the case of T b = 0.004 s. “0 →13”,
“0 →13”, “0 →14” and “0 →12” are labeled to represent this moment for the cases of 85 impulsive start, T b = 0.002 s, T b = 0.02 s, and T b = 0.04 s, respectively, Figure 4.16.
Notice that at this moment the pressure is in the range of 3E+5 N/m2 to 5E+5 N/m 2,
which can be defined as the “1 st pressure zone” or the pressure zone for the formation of
waves. The “1 st pressure zone” acts as a threshold, i.e. the formation of waves in the circumferential direction will not occur until the pressure reaches the “1 st pressure zone”.
The number of waves then decreases gradually until it stays at a certain number of waves with the increase of time for each case. “9 →8”, “9 →8”, “8 →7” and “12 →11” are labeled to represent the moment of the wave number’s last decrease for the cases of impulsive start, T b = 0.002 s, T b = 0.02 s, and T b = 0.04 s, respectively, Figure 4.16.
Notice that at this moment the pressure is in the range of 7.5E+5 N/m 2 to 8E+5 N/m 2,
which can be defined as the “2 nd pressure zone” or the pressure zone for the last
mergence of waves. The “2 nd pressure zone” also acts as a threshold, i.e. the mergence of waves will occur before this moment when the pressure is above this “pressure zone”, while the mergence of waves will not occur after this moment when the pressure is below this “pressure zone”.
There are three stable steady solutions, which have wave numbers of 7, 8 and 11 respectively, Figure 4.15. There are at least five unstable solutions, which have wave numbers of 14, 13, 12, 10 and 9 respectively. How does the transition happen between these solutions, stable or unstable? Figure 4.17 presents the contours of axial direction velocity and pressure on the mid-plane in the radial direction for the case of T b = 0.002 s, which shows the process of the decrease of wave number from 10 to 9 (waves No. 1 and
No. 2 are going to be merged).
86 Axial velocity – w contour indicates that there are 10 waves in the circumferential direction (No. 1, 2, 3 and 10 are labeled), Figure 4.17 a1. The pressure at the position of wave No. 1 is increased, which starts the mergence process of the waves, Figure 4.17 a2.
Figure 4.17 b1 shows that the wavelengths of both of the waves (No. 1 and No. 2) become shorter, and the generated pressure P further increases (notice that P is the difference between the highest and lowest pressure in the whole domain of the annulus),
Figure 4.17 b2. Figure 4.17 c1 shows that wave No. 1 and wave No. 2’s wavelengths become further shorter and are ready to merge. Figure 4.17 d1 indicates that wave No. 1 and wave No. 2 merged to be a new wave with a long wavelength, which is labeled as
No. 11. Then the wavelength of the new wave (No. 11) becomes shorter to be regular,
Figure 4.17 e1. This process matches Coles’ (1965) experimental observation that “the tangential waves first become gradually unequal in wavelength” and “a short wave may disappear and the pattern then becomes once more very regular”. During this mergence process (b → c → d → e) the pressure drops monotonically, Figures 4.17 b2 through
Figure 4.17 e2. Therefore, the pressure first increasing to start the process and then decreasing monotonically in the whole process characterize the process of the wave mergence, (a → b → c → d → e) in Figure 4.16. Figure 4.17f shows that the pressure increases again indicating a new wave mergence process (No. 11 and No. 3 will be merged).
In summary, the pressure plays a significant role on both the formation and mergence of the waves in the circumferential direction. The formation of the waves will not start until the pressure is higher than the “1 st pressure zone” and the mergence of the waves will stop once the pressure is lower than the “2 nd pressure zone”. The pressure first 87 increasing to start the process and then decreasing monotonically in the whole process characterize the process of the wave mergence.
4.4.4 Summary on Multiplicity and Transition
Multiplicity and transition are the two of the most attractive features in studying the
hydrodynamic stability of rotating fluids between two cylinders. A lot of solutions,
stable or unstable, could realize in the same fluid subject to the same dynamical boundary
condition. The history of the boundary condition, or the time (T b) needed to increase the
speed of inner cylinder from rest to prescribed speed, plays a significant role on the
solution of the problem. It is impracticable to study all the solutions, so all the results
presented hereinafter will be obtained in the fluid subject to the boundary condition with
impulsive start, i.e. T b = 0.
4.5 Effect of Clearance on Taylor Vortices Induced Instability
In this section we will study the effect of clearance on Taylor vortices induced instability by considering the viscous fluid flow of light silicone oil in clearances varying from 0.005 in to 0.13 in between two concentric cylinders. The flow patterns, velocity and pressure profiles, and torque-speed graphs will be presented for the concentric cylinder cases with clearances of 0.01 in. (a typical small clearance) and 0.13 in. (a typical large clearance). Then the onset of two types of instabilities: i) the Taylor vortices and ii) the wavy vortices, and their relationship with “turbulent” flow will be discussed versus the critical values of Reynolds and Taylor numbers on a parametric basis. 88 4.5.1 Concentric Cylinder Case with Clearance of 0.01 in.
For the case of C = 0.01 in. and ε = 0.0, the torque exerted on the outer cylinder, Tout ,
and the mutually corresponding Reynolds and Taylor numbers have been calculated for
each angular speed. The results are listed in Table 4.3.
Table 4.3 Torque and Flow Type vs. Speed for the Case of C = 0.01 in., ε = 0.0 Re Ta ω (rpm) RωC T (N.m) Flow Type (= ) RωC C out ν (= ) ν R 1 40,000 283.22 28.46 54.67 2 50,000 354.02 35.58 68.34 Couette Flow 3 55,000 389.42 39.14 75.18 4 58,000 410.66 41.27 79.28 1st Critical 5 60,000 424.82 42.70 87.16 Taylor Vortex 6 62,000 438.98 44.12 96.65 2nd Critical 7 80,000 566.43 56.93 159.16 Wavy Vortex 8 100,000 708.04 71.16 246.53
It is well accepted that Tacr = 41.3 for the onset of the first Taylor instability when
the clearances is as low as 0.01 in. According to Pinkus and Sternlicht (1961) the
0.5 corresponding Reynolds number is Re Ta,cr = 41.3 (R/C) . For the case presented herein,
that leads to a Re Ta,cr = 410.66, which is well below the clearance based, critical turbulence Reynolds number reported to be between 1,000 and 2,000 according to Gross et al. (1980) and Szeri (1979). At Re Ta,cr = 410.66 the inflection point in slope is clearly visible, Figure 4.18. From the flow patterns that will be presented later we have determined that the axisymmetric cell Taylor vortex flow is replaced by a time-dependent 89 wavy vortex flow that starts to occur when Tawave = 44.12. The corresponding
Re Ta,wave = 438.98, which is also well below the accepted norm for the onset of turbulence. At Re Ta,cr = 438.98 the inflection point in slope is also clearly visible, while
numerical experiments show no inflection in the Torque- Ta at Reynolds numbers beyond 438.98, Figure 4.18.
T (N.m) Tout (N.m) out 200 400 600 Re 400 450 Re 500 300 125 Couette Couette 115 Taylor Vortex 250 Taylor Vortex Wavy Taylor Vortex Wavy Taylor Vortex 200 105 Supercritical Couette (3) Supercritical Couette (3) Supercritical Taylor Supercritical Taylor 150 95
(2) 100 (2) 85 50 (1) 75 (1) 0 Detail shown in b) 65 0 20 40 60 80 Ta 35 40 45Ta 50 a) b)
Figure 4.18 Torque vs. Ta and Re for the Case of C = 0.01 in., ε = 0.0
Figure 4.18a is associated with the numbers presented in Table 4.3. Note that the plot
shows a straight line when the angular velocity is low (laminar Couette flow) and up until
the angular velocity, ω, reaches 58,000 rpm; the first inflection occurs when the critical
Ta reaches 41.27, Figure 4.18b. This region is associated with the classical Couette flow. Beyond this inflection point there is a second straight line that ends when Ta 90 reaches 44.12, Figure 4.18b, as ω reaches 62,000 rpm. This second region is associated with the steady periodic Taylor cells. At this velocity, one can observe the formation of a second inflection point where the Torque- Ta curve changes slope again, but continues
to exhibit a linear behavior. This third region characterizes the wavy vortex region. All
these changes in slope may be associated with an apparent change in viscosity.
Nevertheless, the corresponding critical Reynolds numbers remain below the accepted
values for the onset of the turbulence regime, 1,000 or 2,000.
Note also that the critical Taylor numbers for the appearance of Taylor vortices and
wavy vortices are 41.3 [DiPrima (1963)] and 44.03 [the case of four waves, Cole (1976)],
respectively. The differences between our results (41.27 and 44.12, Table 4.3) and theirs
are 0.07% and 0.20%, respectively. The inflection points shown in Figure 4.18 coincide
with the flow pattern changes as shown in Figures 4.19 through 4.26. In these figures we
have developed three typical cases: (a) when Ta = 28.46 < Tacr for the first line (1),
(b) when Ta = 42.70, which is slightly larger than Tacr , for the second line (2), and
(c) when Ta = 71.16 >> Tacr for the third line (3). The lines (1), (2), and (3) belong
to the three regimes delineated in Figure 4.18.
Figures 4.19a through 4.19d present the contours of u (velocity in the circumferential
direction), v (velocity in the radial direction), w (velocity in the axial direction) and P
(pressure) on the ‘r-z’ plane cross section across the radial direction, respectively, for the
case of Re = 283.22, Ta = 28.46, C = 0.01 in. Note that the total velocity vectors are
also projected on the ‘r-z’ plane in each figure.
91
Outer Cylinder Inner Cylinder C C C
L=2C L=2C L=2C
z z z
r r r a) b) c)
C ω
1
L=2C
z
z r
r d) e)
θ r
1
f)
Figure 4.19 Velocity and Pressure Profiles (Re=283.22, Ta =28.46, C = 0.01 in.)
92 Figure 4.19a shows that the velocity u doesn’t vary in the axial direction and it is linear across the radial direction, which indicates that the flow is Couette in this regime.
Figures 4.19b and 4.19c show that v and w are equal to zero, so no Taylor vortices are possible in this regime. The pressure contours shown in Figure 4.19d are straight lines parallel with the axial z-direction. This means that there are no pressure gradients in the axial direction, which is consistent with w = 0, Figure 4.19c. Figure 4.19e presents in a three-dimensional manner stacked velocity profiles across the radial direction, which are also shown in 2-D space in section 1 of Figure 4.19f. It is shown that the velocity profiles across the clearance are linear, which characterizes a two-dimensional Couette flow irrelevant of the axial elevation. A particle released and tracked, would reveal the flow microstructure to be a Couette flow, where the particle is describing a perfect circle,
Figure 4.20. Thus no Taylor vortices are visible or possible at this stage.
z View A θ r
a)
View A
b)
Figure 4.20 Trajectory of a Typical Particle in Couette Regime (Re=283.22, Ta =28.46, C = 0.01 in.) (Not at scale) 93
C Outer Cylinder Inner Cylinder C
L=2C L=2C
z z z
r r r Planes limiting a) the cell b) c)
ω C
1
2 3 L=2C 4 5 z 6 z 7 r 8 r 9 d) e)
θ θ θ θ
r r r r
1 2 3 4
θ θ θ θ θ r r r r r
5 6 7 8 9
f)
Figure 4.21 Velocity and Pressure Profiles (Re=424.82, Ta =42.70, C = 0.01 in.) 94
z θ r
View A
View B
View A
View B
Figure 4.22 Trajectory of a Typical Particle in Taylor Regime (Re=424.82, Ta =42.70, C = 0.01 in.) (Not at scale)
95
C Outer Cylinder Inner Cylinder C
L=2C L=2C
z z z
r r r a) b) c)
C ω
1 2 3 L=2C 4 5 6 z 7 z 8 r 9 r d) e)
θ θ θ θ r r r r
1 2 3 4
θ θ θ θ θ r r r r r
5 6 7 8 9
f)
Figure 4.23 Velocity and Pressure Profiles (Re=708.04, Ta =71.16, C = 0.01 in.)
96
Detail A
a)
Detail A 1
1 b)
1 -- 1
z θ c)
Figure 4.24 Velocity Profiles (Re=708.04, Ta =71.16, C = 0.01 in.) (Not at scale)
97
z θ r
Detail A a)
Detail A
View B View B
b) c)
Figure 4.25 Trajectory of a Typical Particle in Wavy Regime (Re=708.04, Ta =71.16, C = 0.01 in.) (Not at scale)
Figures 4.21a through 4.21d present the contours of u, v, w and P on the ‘r-z’ plane cross section across the radial direction, respectively, for the case of Re = 424.82, Ta =
42.70, C = 0.01 in. The total velocity vectors are also projected on the ‘r-z’ plane in each figure.
98 Figure 4.21a shows that the velocity u varies in the axial direction and it is not linear
across the radial direction, which indicates that the flow is not Couette in this regime.
Figures 4.21b and 4.21c show the v and w velocity contours that are indicative of the
formation of the Taylor cells. Consistent with the wave number chosen in Section 4.2.1 λ
= 2, there are two Taylor cells formed along the length L = 0.02 in. These vortices, which occur in the ‘r-z’ plane, occupy the entire clearance, are positioned horizontally and parallel to the ‘r-θ’ plane, and their structure is independent of θ, as they run around
the circumference.
Figure 4.21d shows the pressure contours, which are not anymore straight lines, unlike those in Figure 4.19d. They are indicative of non-zero axial velocities and thus predict the formation of the recirculating Taylor cells. Figure 4.21e presents in a three- dimensional manner stacked along the cylinder generator the velocity profiles that are shown in 2-D space in sections 1 through 9 of Figure 4.21f. These velocity profiles are changing, as the axial elevation z decreases, from a convex- (sections 1 through 5) to concave-type (sections 6 through 8), and then back to convex-type (section 9). The velocity profiles become complicated 3-D shapes, which can be seen both in the ‘r-z’ and
‘r-θ’ planes.
Figure 4.22 shows that, in a Taylor vortex flow, a typical particle travels like a 3-D wave, up and down in the axial direction (View A), and inward and outward in radial direction (View B). However, it is contained in a vortex cell and never goes through the symmetry planes limiting the vortex cell (note that there is a ½ cell-1 full cell- ½ cell sequence strung along the axial direction, Figure 4.21a). It is important to note that at
99 this stage the particle gets enough energy to travel like a wave but not enough to penetrate through the limiting planes.
The coincidence between the occurrence of the inflection in the Torque- Ta graph
and the appearance of the Taylor vortex means that the centrifugal instability overcomes
some of the natural fluid viscosity effects. The new flow patterns cause the average
velocity gradient along the outer cylinder wall to increase, and thus cause the torque on
the outer cylinder to increase.
With a further increase in angular speed, as shown in Figures 4.23 through 4.26 for the
case of Re = 708.04, Ta = 71.16 and C = 0.01 in., the velocity profiles become even
more complicated than those of Figures 4.21 and 4.22 for the case of Re = 424.82, Ta =
42.70 and C = 0.01 in.
Figures 4.23a through 4.23d present the contours of u, v, w and P on the ‘r-z’ plane cross section across the radial direction, respectively, for the case of Re = 708.04, Ta =
71.16, C = 0.01 in. The total velocity vectors are also projected on the ‘r-z’ plane in each figure.
Figure 4.23a shows that the velocity u is not linear across the radial direction, and its nonlinearity is much stronger than the one in Taylor regime, Figure 4.21a. Figures 4.23b and 4.23c show the v and w velocity contours that indicate the disappearance of the organized flow structure of Figure 4.21 and its replacement with velocity contours that are indicative of the wavy vortices regime (see also Figure 4.24c). The structure of two, almost equal size Taylor cells, consistent with λ = 2, is not present anymore, even though
we can still find two cells, one of them being the dominant structure, while the other
100 appears to exist in order to respect flow continuity, Figures 4.23a through 4.23d. While the cells are still fully occupying the clearance and rotate in the ‘r-z’ plane, just like the regular Taylor cells, they are not anymore parallel with the ‘r-θ’ plane, since they have a
3-D wavy motion, and the structure of the cell is now dependent on its θ-circumferential position.
Figure 4.23d shows the pressure contours, which are now showing a strong variation
(strong pressure gradient) along the axial direction of the cylinder. These gradients are responsible for the increase in the velocities both in the +z and –z directions, which in turn are instrumental in the destruction of the Taylor vortex structure. Figure 4.23e presents in a three-dimensional manner, stacked along the cylinder generator the velocity profiles that are shown in 2-D space in sections 1 through 9 of Figure 4.23f. The velocity profiles are changing significantly both in the radial direction and with the decrease in the z-axial elevation. The velocity profiles shape changes now also within the same cross section from concave- to convex-type. The velocity profiles have acquired complicated
3-D shapes, (Figures 4.23e and 4.23f), of which some exhibit sharp and quickly changing gradients like the ones shown in sections 1, 6, 7, 8, and 9.
The consequence of this situation is a further change in torque, and the appearance of the second inflection point that signals the start of the wavy regime, graphically seen in
Figure 4.24. Figure 4.24c shows the velocity vectors projected on the mid-plane in the radial direction (z-θ plane), which graphically indicate “one wave” on that section.
Figure 4.25 shows the trajectory of a typical particle, in a wavy vortex flow, traveling
up and down at the basic wave frequency with minor harmonic frequencies
superimposed. Its trajectory turns are more drastic than the ones in the Taylor vortex 101 flow and now the particle gets enough energy to push the limiting planes to be wavy or even go through the planes. The particle travels with a larger amplitude in the axial direction and almost touches the outer wall, as shown in Detail A and View B of Figure
4.25.
The wavy regime that develops when Ta =71.16, (presented in Figures 4.23 through
4.25) is further analyzed in Figure 4.26.
The reader can now see a cross section in the ‘r-θ’ and ‘z-θ’ planes of Figure 4.24,
rendered in Figures 4.26a and 4.26b, respectively. These figures continue to prove the
wavy nature of the flow in the circumferential direction while at the same time there is a
Taylor-like recirculation flow cell across the clearance in the ‘r-z’ plane. One can
visualize this structure if one thinks of an undulating rope around the circumference.
Figure 4.26c shows the pressure gradients in the ‘z-θ’ plane; they are responsible for the formation and travel of the wave in the circumferential direction. For the entire cylinder circumference there are 8 waves that travel in the circumferential direction. In
Figure 4.26a we have chosen one arbitrary wave, which covers the angle of 2 π/8
containing sections 1 to 5 as indicated. Figures 4.26d and 4.26e present the development
of the pressure contours and w-axial velocity contours, respectively. The angle between
each one of the five sections is 2 π/32. In Figure 4.26d one can see the development of significant axial pressure gradients in each one of the sections (1 through 5). They are responsible for the development of the Taylor-like circulating flow where one of the two
Taylor vortices is diminished, while the other is enlarged.
102
1 2 6 6 – 6
3 4 z θ b) 5 θ = 2 π/8
6 6 – 6
a)
z θ c)
z r 1 2 3 4 5 d)
z r 1 2 3 4 5 e)
Figure 4.26 Velocity and Pressure Profiles of One Wave when Ta =71.16 [Not at scale in b) and c)]
103 4.5.2 Concentric Cylinder Case with Clearance of 0.13 in.
For the case of C = 0.13 in. and ε = 0.0, the torque exerted on the outer cylinder, Tout ,
and the mutually corresponding Reynolds and Taylor numbers have been calculated for
each angular speed. The results are listed in Table 4.4. Note that the expression of Ta for the case of C = 0.13 in. is slightly different from the one for the case of C = 0.01 in., in that the former contains the radii ratio η [=R/(R+C)] rather than clearance ratio (=
C/R).
Table 4.4 Torque and Flow Type vs. Speed for the Case of C = 0.13 in., ε = 0.0
Re Ta 5.0 ω (rpm) RωC 2 4 2 T (N.m) Flow Type (= ) 2η C ω out ν = 1−η 2 ν
1 1,000 80.89 30.16 0.00173 Couette 2 1,425 115.27 42.98 0.00246 3 1,440 116.48 43.43 0.00249 1st Critical 4 1,550 125.38 46.75 0.00308 Taylor Vortex 5 1,600 129.42 48.26 0.00335 2nd Critical 6 2,000 161.78 60.32 0.00527 7 3,000 242.66 90.48 0.01045 Wavy Vortex 8 4,000 323.55 120.65 0.01661 9 5,000 404.44 150.81 0.02393
The first critical Taylor number is Tacr = 41.3 for a small clearance and the
0.5 corresponding Reynolds number is Re Ta,cr = 41.3 (R/C) = 410.66. However, for a large
clearance, Tacr and Re Ta,cr are different from those of a small clearance due to the 104 larger curvature effect. Our calculations show that the first critical Taylor number occurs
at Tacr = 43.43 for the case of C = 0.13 in., and its corresponding Reynolds number is
0.5 Re Ta,cr = 43.43 [(2R+C)/(2C)] = 116.48 according to Roberts (1965). Again, the critical
Reynolds number for onset of the 1 st Taylor instability, 116.48, is well below the
clearance based, critical turbulence Reynolds number reported to be between 1,000 and
2,000 according to Gross et al. (1980) and Szeri (1979). At Re Ta,cr = 116.48 the inflection
point in slope is also clearly visible. The axisymmetric Taylor vortex flow is replaced by
a time-dependent wavy vortex flow when the Taylor number reaches its second critical
value of Tawave = 48.26 for the case of C = 0.13 in. The corresponding Re Ta,wave =
129.42 is also well below the accepted norm for the onset of turbulence. At Re Ta,cr =
129.42 the inflection point in slope is also clearly visible.
Torque-Re Diagram (C=0.13 in.) Torque-Re Diagram (C=0.13 in.)
30 60 90 120 150 180 Ta 35 40 45 50 55 Ta 0.030 CFL ≥2 Taylor vortex 0.0050 Couette False solution Couette (3’) Taylor Vortex 0.025 0.0045 Taylor Vortex Supercritical Taylor Vortex Supercritical Taylor Vortex Wavy Taylor Vortex 0.020 0.0040 Wavy Taylor Vortex Supercritical Couette CFL =0.4 Supercritical Couette (3) Wavy vortex 0.015 Correct solution 0.0035 nd C 2 Inflection B B Point Torque (N.m) Torque (N.m) 0.010 D 0.0030 (2) A Couette flow (1) 0.005 0.0025 False solution 1st Inflection Point 0.000 Detail shown in b) 0.0020 0 100 200 300 400 500 80 100 120 140 160 Re Re
a) b)
Figure 4.27 Torque vs. Ta and Re for the Case of C = 0.13 in., ε = 0.0
105 Figure 4.27a is associated with the values presented in Table 4.4 for the case of C =
0.13 in. In Figure 4.27b one can see the three straight lines of the curve which represent the three typical regimes, i.e. the Couette flow (1), the Taylor vortex flow (2), and the wavy vortex flow (3). The confines of three regions are defined by the two inflection points, which represent the critical values for the 1 st and 2 nd Taylor instabilities.
Note that in Figure 4.27a three possible paths are potentially possible after the 2 nd inflection point: (i) the supercritical Taylor vortex is marked by the dashed line with open circles, (ii) the supercritical Couette flow is shown by the dashed line with open triangles, and (iii) the wavy vortex flow is marked by the continuous line with solid circles. The supercritical Couette flow would set beyond the first inflection point in a linear extrapolation of line (1) in Figure 4.27b. That solution doesn’t exist. Instead, the solution moves along line (2), which is the Taylor vortex regime, until it reaches the 2 nd inflection point, where one transits to the wavy vortex flow. However, again, attention has to be paid not to choose line (3’), which is the path of a false solution that computationally exists if the CFL number is wrongly chosen. Thus, the only correct solution is that for the wavy vortex flow, which emerges only if the CFL number is less than or equal to 0.5 (see discussion on the CFL number in Section 4.3.2). The torque and flow patterns could be different if one picks a large CFL number, which under circumstances can yields the wrong though converged solution that can be mistaken for the correct solution.
Figure 4.28 presents a typical point in the Couette regime for the case of Ta = 30.16 and 0.13 in. (Point A in Figure 4.27). Figures 4.28a through 4.28d present the contours
106 of u, v, w and P on the ‘r-z’ plane cross section across the radial direction, respectively.
The total velocity vectors are also projected on the ‘r-z’ plane in each figure.
Again Figure 4.28a shows that the velocity u doesn’t vary in the axial direction and it
is linear across the radial direction, which indicates that the flow is Couette in this
regime. Like the ones in the Couette regime for the case of 0.01 in., Figures 4.28b and
4.28c show that for the case of 0.13 in., v and w are equal to zero, so no Taylor vortices
are possible in Couette regime.
Similar to the case of C = 0.01 in., the pressure gradients in the axial direction are zero
for the case of C = 0.13 in. in Couette regime, Figure 4.28d. However, for the case of C
= 0.13 in. the pressure potential in the radial direction is only 238.55 N/m 2, and thus much smaller than the one for the case of C = 0.01 in., which was 3.397E+4 N/m 2. Note that this occurs because centrifugal force is balanced by the radial pressure gradient for
2 Couette flow, i.e. ρVθ /r = P/ r. However, the crossover occurs at very different speed
(1,440 rpm for the case of C = 0.13 in. vs. 58,000 rpm for the case of C = 0.01 in.) even though the median radii are almost the same.
Figures 4.28e and 4.28f characterize the nature of Couette flow, where velocity
profiles across the clearance are always linear.
Figure 4.29 represents a typical point B on line (2) in the Taylor vortex regime for the
case of C = 0.13 in., Figure 4.27. Figures 4.29a through 4.29d present the contours of u,
v, w and P on the ‘r-z’ plane cross section across the radial direction, respectively. Again
the total velocity vectors are projected on the ‘r-z’ plane in each figure.
107
C Outer Cylinder Inner Cylinder C
L=2C L=2C
z z z r r
a) b) c)
C Z = L ω
1
L=2C
z
z r r Z = 0
d) e)
θ r
1
f)
Figure 4.28 Velocity and Pressure Profiles (C=0.13 in., Ta =30.16, A in Figure 4.25)
108
C Outer Cylinder Inner Cylinder C
L=2C L=2C
z z z r r r
a) b) c)
Z = L ω C
1 2 3 L=2C 4 5 z 6 z 7 r 8 Z = 0 r 9 d) e)
θ θ θ θ r r r r
1 2 3 4
θ θ θ θ θ r r r r r
5 6 7 8 9
f)
Figure 4.29 Velocity and Pressure Profiles (C=0.13 in, Ta =46.75, B in Figure 4.25) 109 Like the one in Taylor vortex regime for the case of 0.01 in., Figure 4.29a shows that the velocity u varies in the axial direction and it is not linear across the radial direction.
Figures 4.29b and 4.29c show the v and w velocity contours that are indicative of the formation of the Taylor vortices. However, the w-axial velocity is only w = 0.1292 m/s, and by comparison much smaller than the one calculated for the case of C = 0.01 in. where w = 0.8869 m/s, Figure 4.21c.
Figure 4.29d shows the same velocity vectors projected on the r-z plane superimposed
over the pressure contours. A study of this figure reveals that the pressure variation in
the axial direction becomes the primary cause for the formation of the Taylor vortices.
The pressure potentials in the radial and axial directions are about 730 N/m 2 and 120
N/m 2, respectively, and thus much smaller than those calculated for the case of C = 0.01 in. (9.1E+4 N/m 2 and 1.0E+4 N/m 2 respectively, Figure 4.21d. Again this is due to the larger centrifugal forces occurring for the case of C = 0.01 in. Due to larger rotational velocity necessary to reach the critical crossover point for the case of C = 0.01 in., the
Reynolds number is also much larger than that of the case of C = 0.13 in. (424.82 vs.
125.38). The Taylor numbers, however, are closer in magnitudes, (42.70 vs. 46.75).
Figures 4.29e presents the velocity profiles at different positions along the axial
direction in a three-dimensional ‘stacked’ manner. These velocities can be viewed in
detail in a two-dimensional format in Figure 4.29f. On inspection one can see the
velocity profiles changing from a linear configuration (section 1), to a convex-type
(sections 2 through 4), and then back to linear (section 5), to change again to concave-
type (sections 6 through 8), and finally back to linear (section 9). For purpose of
comparison one has to mention that the velocity profile changes for the case of C = 0.13 110 in. are qualitatively similar to those of the case of C = 0.01 in., Figure 4.21f, even though the latter are less curved than those shown in Figure 4.29d due to the smaller Taylor number.
Figure 4.30 presents a typical point in the wavy Taylor vortex regime (Point C, Figure
4.27) for the case of C = 0.13 in. Figures 4.30a through 4.30d present the contours of u, v, w and P on the ‘r-z’ plane cross section across the radial direction, respectively.
Again Figure 4.30a shows that the velocity u is not linear across the radial direction, and its nonlinearity is much stronger than the one in Taylor regime, Figure 4.29a.
Figures 4.30b and 4.30c show the v and w velocity contours, which indicate the disappearance of the organized Taylor vortex structure and its replacement by the wavy vortices regime.
Figure 4.30d shows the pressure contours, which vary strongly in both the axial and radial directions. The axial velocity engendered by this pressure potential is much smaller than that of the case of C = 0.01 in., Figure 4.23 (0.365 m/s vs. 6.397 m/s). This can be explained by the difference in the axial pressure potentials between the two cases
(450 N/m 2 vs. 10E+4 N/m 2, respectively).
Figure 4.30e presents the velocity profiles in a ‘stacked’ 3-dimensional configuration while Figure 4.30f presents the same velocity profiles detailed in a two-dimensional manner. The velocity profiles are changing significantly in both the radial and axial directions as the elevation decrease from z = L to z = 0.
111
C Outer Cylinder Inner Cylinder C
L=2C L=2C
z z z
r r r a) b) c)
Z = L C ω
1 2
3 L=2C 4 5 z 6 z 7 r r 8 9 Z = 0 d) e)
θ θ θ θ r r r r
1 2 3 4
θ θ θ θ θ r r r r r
5 6 7 8 9
f)
Figure 4.30 Velocity and Pressure Profiles (C=0.13 in., Ta =66.35, C in Figure 4.25)
112
a) w contours at z = L/4 b) Pressure contours at z = L/4
c) w contours at z = L/2 d) Pressure contours at z = L/2
e) w contours at z = 3L/4 f) Pressure contours at z = 3L/4
g) w contours at z = L h) Pressure contours at z = L
Figure 4.31 w-velocity and Pressure Contours at Different z for the Wavy Vortex Regime (C = 0.13 in., ω = 4000 rpm, Re = 323.55, Ta = 120.65), D in Figure 4.25 113
r θ R
R+C a)
one wave one wave
b) w contours at r = R+C/4 c) Pressure contours at r = R+C/4 one wave one wave
d) w contours at r = R+C/2 e) Pressure contours at r = R+C/2 one wave one wave
f) w contours at r = R+3C/4 g) Pressure contours at r = R+3C/4
Figure 4.32 w-velocity and Pressure Contours at Different r for the Wavy Vortex Regime (C = 0.13 in., ω = 4000 rpm, Re = 323.55, Ta = 120.65), D in Figure 4.25 114
z z z z z
Cell 1 Cell 1 Cell 1
Cell 2 Cell 2 Cell 2
r r r r r
θ = π/12 θ = 2π/12 θ = 3π/12 θ = 4π/12 θ = 5π/12
a) w-velocity Contours with Superimposed total velocity vectors at different θ
z z z z z
Cell 1 Cell 1 Cell 1
Cell 2 Cell 2 Cell 2
r r r r r
θ = π/12 θ = 2π/12 θ = 3π/12 θ = 4π/12 θ = 5π/12
b) Pressure Contours with Superimposed total velocity vectors at θ
Figure 4.33 w-velocity and Pressure Contours with Superimposed Velocity Vectors at Different θ for the Wavy Vortex Regime (C = 0.13 in., ω = 4000 rpm, Re = 323.55, Ta = 120.65), D in Figure 4.25
115 Figure 4.31 shows the velocity vectors superimposed over the w-axial velocity and pressure contours on the r-θ planes at different z when the wavy vortices are fully
developed at ω = 4,000 rpm (Re = 323.55 and Ta = 120.65), Point D in Figure 4.27. On
inspection, one can distinguish clearly the six waves around the circumference. Note that
the shape of w-velocity and pressure contours changes significantly with the change in
the axial position. This, importantly, indicates a strong three-dimensional flow associated
with this wavy regime and the corresponding strong pressure gradient in both the radial
and axial directions.
Figure 4.32 shows the velocity vectors superimposed over the w-axial velocity and
pressure contours on the z-θ planes at different r when the wavy vortices are fully
developed at ω = 4,000 rpm (Re = 323.55 and Ta = 120.65), Point D in Figure 4.27.
Note that one wave in the circumferential is shown in each figure. The shape of w- velocity and pressure contours also changes significantly with the change in the radial position. Again this indicates a strong three-dimensional flow associated with this wavy regime and the corresponding strong pressure gradient in both the radial and axial directions.
Figure 4.33 shows, for one wave, five typical axial (r-z plane) cross sections at different angles θ along the circumferential direction. The velocity vectors are
superimposed over the w-velocity and pressure contours. The development and change
in the vortex cells in the r-z plane, as one moves around the circumference, is obvious: (i)
at θ = π/12, the sizes of the two cells are almost equal, (ii) at θ = 2 π/12, cell 2 is enlarged
at the expense of cell 1, (iii) at θ = 3 π/12 and θ = 4 π/12, cell 1 has grown to occupy most
116 of the space while cell 2 becomes practically extinct, and (iv) finally at θ = 5 π/12, the sizes of the two cells become almost equal again.
a) w Contours, Re=125.38, Ta = 46.75 b) w Contours, Re=323.55, Ta = 120.65
c) Pressure Contours, Re=125.38, Ta = 46.75 d) Pressure Contours, Re=323.55, Ta = 120.65
Figure 4.34 Comparison of Velocity and Pressure Contours for Taylor Vortex Flow
(Left Column, Re=125.38, Ta =46.75) and Wavy Vortex Flow (Right Column, Re=323.55, Ta =120.65) (C = 0.13 in.)
117
a) Iso-curves and Iso-surfaces of w b) Iso-curves and Iso-surfaces of w (m/s) , Re=323.55, Ta =120.65 (m/s), Re=125.38, Ta =46.75
c) Iso-curves and Iso-surfaces of P d) Iso-curves and Iso-surfaces of P 2 (N/m 2), Re=125.38, Ta =46.75 (N/m ), Re=323.55, Ta =120.65
Figure 4.35 Comparison of Iso-curves and Iso-surfaces of w and P for Taylor Vortex Flow (Left Column, Re=125.38, Ta =46.75) and Wavy Vortex Flow (Right Column, Re=323.55, Ta =120.65) (C = 0.13 in. )
A comparison between the Taylor vortex and wavy vortex flows for the w-axial velocity and pressure contours for the case of C = 0.13 in. is shown in Figure 4.34.
Figures 4.34a and 4.34c show that the nature of the Taylor vortex flow [Re = 125.38,
Ta = 46.75, point B on line (2) in Figure 4.27] for both the velocity w and the pressure 118 contours is axisymmetric and time-independent. The axial pressure variation, shown in
Figure 4.34c, is responsible for the formation of Taylor vortices depicted in Figure 4.34a.
It is important to note that in this regime there is no circumferential variation in pressure.
Figures 4.34b and 4.34d show that the nature of the Taylor wavy flow [Re = 323.55,
Ta = 120.65, point D on line (3), Figure 4.27] for both the velocity w and the pressure
contours is non-axisymmetric and time-dependent. The lack of circumferential symmetry
is responsible for the apparition of the circumferential pressure variations shown in
Figure 4.34d. In turn these variations will destruct the regular symmetry of the Taylor
vortex and be responsible for the formation of wavy regime, shown in Figure 4.34b.
Figure 4.35 presents a comparison of iso-curves and iso-surfaces of w-axial direction
velocity and pressure between Taylor and wavy vortex flows for the case of C = 0.13 in.
Figures 4.35a and 4.35c present the iso-curves and iso-surfaces for w and P, respectively,
for the Taylor vortex flow [Re = 125.38, Ta = 46.75, point B on line (2) in Figure
4.27]. The iso-curves of w indicate the shape of the Taylor vortices, Figure 4.35a, and the iso-curves of P indicate the mechanism for the formation of Taylor vortices, which is the weak pressure variation in both the axial and radial direction, Figure 4.35c. The iso- surfaces of w and P indicate the axisymmetric and time-independent nature of Taylor vortices.
Figures 4.35b and 4.35d present the iso-curves and iso-surfaces for w and P,
respectively, for the wavy vortex flow [Re = 323.55, Ta = 120.65, point D on line (3)
in Figure 4.27]. The iso-curves of w indicate the breakdown of the Taylor vortices,
Figure 4.35b, and the iso-curves of P indicate the mechanism for the breakdown of
119 Taylor vortices, which is the strong pressure variation in both the axial and radial directions, Figure 4.35d. The iso-surfaces of w and P indicate the non-axisymmetric and time-dependent nature of wavy vortices.
In Section 4.5.1 it was shown that, for the case of C = 0.01 in., a typical particle travels like a 3-D wave and it is contained in a vortex cell in the Taylor vortex regime, while the particle gets enough energy to go through the limiting planes in the wavy vortex regime.
However, only part of the top view of the particle track was shown and its 3-D view was not clear due to the small clearance.
Figure 4.36 clearly presents the overall picture of a typical particle traveling in the
Taylor vortex regime for the case of C = 0.13 in. Like the one for the case of C = 0.01 in., this particle travels like a 3-D wave, up and down in the axial direction and inward and outward in the radial direction, Figure 4.36a. However, it is contained in the domain limited by the two planes and it never goes through those planes, Figure 4.36b. Figure
4.36c presents the top view of the particle trajectory, which shows the way the particle moves: it moves counterclockwise from 1 to 2, then from 2 to 3, and all the way to 12, and finally back to 1, thus finishing one period.
The way the particle moves is very interesting and regular. For example, point 1 is low
in axial direction and middle in radial direction and point 2 is middle in axial direction
and inner in radial direction, thus the positions change from low to middle in axial
direction and from middle to inner in radial direction as the particle moves from point 1
to point 2. The locations of all the points from 1 to 12 are listed in Tables 4.5 and 4.6.
Figure 4.36d presents the front view of the particle trajectory, which also shows that the
120 particle is contained in the domain formed by the two limiting planes. It doesn’t get enough energy to go through the planes at this stage.
Table 4.5 Positions of Points on a Typical Trajectory in Taylor Vortex Regime (Axial Direction) High 3 7 11
Middle 2 4 6 8 10 12
Low 1 5 9
Table 4.6 Positions of Points on a Typical Trajectory in Taylor Vortex Regime (Radial Direction) Inner 2 6 10
Middle 1 3 5 7 9 11
Outer 4 8 12
Figure 4.37 clearly presents the overall picture of a typical particle traveling in the wavy vortex regime for the case of C = 0.13 in. In this regime, the particle also travels like a 3-D wave, up and down in the axial direction and inward and outward in the radial direction, Figure 4.37a. However, its trajectory turns are more drastic than the ones in the Taylor vortex regime and it gets enough energy to go through the two planes, Figure
4.37b. Figure 4.37c presents the top view of the particle trajectory, which shows the way the particle moves: it moves counterclockwise from 1 to 2, then from 2 to 3, and all the way to 10, and it eventually goes out of this domain.
121 The way the particle moves is similar to the one in Taylor vortex regime. For example, point 1 is middle in axial direction and outer in radial direction and point 2 is high in axial direction and middle in radial direction, thus the positions change from middle to high in axial direction and from outer to middle in radial direction as the particle moves from point 1 to point 2. The locations of all the points from 1 to 10 are listed in Tables 4.7 and 4.8. Notice that the points 1 through 12 cover a full circle and the particle moves periodically in Taylor vortex regime, Tables 4.5 and 4.6. However, points
1 through 10 cover only one third of a circle and the particle eventually goes out of the domain, Tables 4.7 and 4.8. Figure 4.37d presents the front view of the particle trajectory, which also shows that the particle gets enough energy to go through the planes at this stage.
Table 4.7 Positions of Points on a Typical Trajectory in Wavy Vortex Regime (Axial Direction) High 2 6 10
Middle 1 3 5 7 9
Low 4 8
Table 4.8 Positions of Points on a Typical Trajectory in Wavy Vortex Regime (Radial Direction) Inner 3 7
Middle 2 4 6 8 10
Outer 1 5 9
122
a)
Upper Plane
b) Lower Plane 12
7 6 1
11 5 2 8
4 3 c) 10 9 Upper Plane
Lower Plane d)
Figure 4.36 Trajectory of a Typical Particle in Taylor Vortex Regime (Re=125.38, Ta =46.75, C = 0.13 in.) 123
a)
Upper Plane
b) Lower Plane 5 3 7 1 4 9 6 8 2
10
c)
Upper Plane
Lower Plane d)
Figure 4.37 Trajectory of a Typical Particle in Wavy Vortex Regime (Re=323.55, Ta =120.65, C = 0.13 in.) 124
a) Velocity Vectors for Taylor Vortex Flow b) Velocity Vectors for Wavy Vortex Flow
( Ta =46.75, η = 0.87) ( Ta =120.65, η = 0.87)
c) Contours of Velocity Magnitudes for Taylor d) Contours of Velocity Magnitudes for Wavy Vortex Flow ( Ta =46.75, η = 0.87) Vortex Flow ( Ta =120.65, η = 0.87)
e) Picture of Taylor Vortex Flow ( Ta =46.10, η f) Picture of Wavy Vortex Flow ( Ta =124.71, η
= 0.896). After Koschmieder. = 0.896). After Koschmieder.
Figure 4.38 Comparison of Our Calculations and Experiments by Koschmieder
125 A comparison of our calculations for velocity vectors and contours with the experimental results of Koschmieder (1993) is presented in Figure 4.38. Taylor numbers and clearances are slightly different but of the same order of magnitude ( Ta = 46.75 vs. 46.10 for Taylor vortex flow, Ta = 120.65 vs. 124.71 for wavy vortex flow, and η =
0.87 vs. 0.896), but one can see qualitatively the similarities in the structures of the
Taylor and wavy vortices between the numerical and experimental results.
4.5.3 Effect of Clearance on Taylor Vortices Induced Instability
Figures 4.39a and 4.39b show the normalized circumferential velocity profiles (u*
u = ) across the radial direction (y* = y/C) at different elevation in the axial direction Rω
(z* = z/L) for the two cases of C = 0.01 in. and C = 0.13 in. The average velocity gradients on the outer cylinder, of 2.3798 and 1.944 respectively, are greater than 1, which is the reference normalized velocity gradient for Couette flow.
C = 0.01 in., Ta^0.5 = 71.16 C = 0.13 in., Ta^0.5 = 66.35
1.0 z* = 1 (0) 1.0 z* = 1 (0) z* = 7/8 z* = 7/8 0.9 0.9 z* = 3/4 z* = 3/4 0.8 z* = 5/8 0.8 z* = 5/8 z* = 1/2 z* = 1/2 0.7 z* = 3/8 0.7 z* = 3/8 z* = 1/4 z* = 1/4 0.6 z* = 1/8 0.6 z* = 1/8 Average Average 0.5 0.5 u* u*
0.4 0.4 0.3 0.3 0.2 0.2 y = 2.7093x 3 - 4.0778x 2 + 2.3798x - 0.0072 y = 2.1339x 3 - 3.0678x 2 + 1.944x - 0.0055 0.1 R2 = 0.9986 0.1 R2 = 0.999
0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 y* y*
a) Velocity Profiles when C = 0.01 in. b) Velocity Profiles when C = 0.13 in.
Figure 4.39 Comparison of Velocity Profiles of Different Clearances
126 Generally speaking, the calculations show that these stronger nonlinearities in the velocity profiles for the case of C = 0.01 in. are due to the pressure gradients that are stronger than those of the case of C = 0.13 in. Consequently we can also explain why the critical inflection points on the Torque- Ta curve are sharper for the case of C = 0.01 in.
(also see Figure 4.40b).
It is apparent that the larger velocity gradients (than Couette flow) exerted on the outer cylinder are responsible for the inflection in the Torque- Ta graph, rather than a change
in viscosity due to the appearance of turbulent eddies (µ tot = µ lam + µ turb ). One has to
mention also that this large change (22.5%) in the normalized velocity gradients between
the two cases is associated with only a modest difference (7.3%) in the respective Taylor
numbers of 71.16 and 66.35.
Taylor Vortex Taylor Vortex 0.50 0.50 C=0.13 in. C=0.13 in. 0.45 C=0.065 in. 0.45 C=0.065 in. C=0.0325 in. C=0.0325 in. 0.40 C=0.01 in. 0.40 C=0.01 in. C=0.005 in. 0.35 0.35 C=0.005 in. 1st Crit. 1st Crit. 2nd Crit. 0.30 0.30 2nd Crit.
0.25 0.25
0.20 Couette Flow 0.20 Torque(N.m) Wavy Vortex Torque(N.m) Couette Flow Wavy Vortex 0.15 0.15
0.10 0.10 0.05 0.05 0.00 0.00 0 40 80 120 160 0 100 200 300 400 500 600 700 800 Ta Re a) b)
Figure 4.40 Torque vs. Taylor No. and Reynolds No. for Various Clearances
Figure 4.40 shows the Torque versus Ta and Reynolds number for different
clearances. Thus one can now precisely identify and evaluate the effect of the change in
127 clearance on the onset of the 1 st and 2 nd inflection points for the critical Taylor and the
critical Reynolds numbers. It can be seen that as the critical Taylor numbers decrease,
the critical Reynolds numbers increase with the decrease of clearance. The critical
Taylor numbers prove to be much less dependent on clearance, Figure 4.40a, than the
corresponding critical Reynolds numbers, Figure 4.40b. Again, one has to stress that the
critical Reynolds numbers for all the clearances considered in this paper range from 116
to 587, and thus are much below the generally accepted norm for the critical Reynolds
number for the onset of turbulence (1,000 - 2,000).
49.0
48.0
1st Crit., Roberts 47.0 1st Crit., Walow it et al. 1st Crit., Cole 46.0 1st Crit., Our Cal. 2nd Crit, Weinstein 45.0 2nd Crit., Cole Ta 2nd Crit., Our cal. 44.0
43.0
42.0
41.0
40.0 0.80 0.85 0.90 0.95 1.00 η [=R/(R+C)]
Figure 4.41 First and Second Critical Taylor Numbers for Various Clearances
Figure 4.41 plots the 1 st and 2 nd critical Taylor numbers for our calculations in comparison with those of other investigators: (i) the experimental values of Cole (1976) 128 and (ii) the analytical values of Roberts (1965), Walowit et al. (1964) and Weinstein
(1975). All the results show that both the 1 st and 2 nd critical Taylor numbers decrease
with the decrease in clearance. The numerical and experimental results for the 1 st critical
Taylor number match better with each other than those of the 2 nd one. The reason is
probably that the inflection for the 2 nd instability in less pronounced than the one for the
1st instability and thus harder to isolate.
4.6 Effect of Eccentricity Ratio on Taylor Vortices Induced Instability
In this section, the effect of the eccentricity ratio on the Taylor type instability, including the effect of eccentricity ratio on the torque and magnitude of critical Taylor number, friction factor, maximum vortex intensity and recirculation, will be discussed in detail.
4.6.1 Effect of Eccentricity Ratio on Torque and Magnitude of Critical Taylor Number
The concentric clearance is 0.01 in. and the eccentricity ratio ε is varied from 0.0 to
0.8. The torque at the outer cylinder T out has been calculated for each speed and at each
eccentricity ratio. The resulting Torque - Ta graph is shown in Figure 4.42a. One can see that the change in slope of the curves is clearly discernable when the ε ≤ 0.4.
However, for the cases where ε > 0.4, the critical inflection point becomes less
noticeable. This tendency matches well with the results of the experiments presented by
Cole (1967). In Figure 4.42a we established the variation of the torque with Tacr when
eccentricity ratio was used as a parameter. 129
Torque (N.m) Ta 300 100 DiPrima & Stuart ε = 0.0 Our Calculations 250 ε = 0.2 Vohr 80 ε = 0.4 DiPrima θ = 0 200 ε = 0.6 DiPrima θ = 60 DiPrima θ = 90 ε = 0.8 60 150
40 100
50 20
0 Ta Eccentricity Ratio 0 0 10 20 30 40 50 60 70 80 90 0.0 0.2 0.4 0.6 0.8 1.0
a) Torque vs. Ta When C=0.01 in., b) Critical Value of Ta vs. ε = 0.0 – 0.8 Eccentricity Ratio
Figure 4.42 Torque- Ta and Ta cr - Eccentricity Ratio Curves
For purpose of comparison with other published data it is useful to use the information
yielded by this figure to plot directly Tacr versus ε, Figure 4.42b. The calculations performed in this paper match well with the non-local theory of DiPrima and Stuart
(1972), which indicate that the critical speed increases monotonically with eccentricity as
C given by: Ta = 1695 1( + .1 162 )(1+ .2 624 ε2 ). Our calculations are also close to the cr R
experimental data of Vohr (1968).
For comparison, we also listed the results of the local theory of DiPrima (1963), which
indicates the flow to be least stable at the position of maximum film thickness θ = 0˚.
However, our results are close to the critical values of Tacr when θ = 90˚ obtained by
DiPrima (1963).
130 4.6.2 Effect of Eccentricity Ratio on the Friction Factor
1.0E-02
ε = 0.0 1.0E-03 ε = 0.2 ε = 0.4 FrictionFactor ε = 0.6
ε = 0.8 Onset of instability
1.0E-04 100 1000 10000 Reynolds Number
Figure 4.43 Friction Factor vs. Reynolds Number when C = 0.01 in., ε = 0.0 – 0.8
τ As shown in Figure 4.43, the friction factor = variation is plotted as a 1 ρ 2 2 V function of the Reynolds number. At lower Reynolds number, the friction factors are different for different eccentricity ratios. However, these lines tend to collapse into one single line as the Reynolds number approaches the region of 900 - 1,000. That means that friction factor doesn’t depend on, or has become a very weak function of the eccentricity ratio once the wavy Taylor instability has set in. The merging of the friction curves shows that for the geometry and working fluid used here, there is a trend for the instability to set in at the same value of the friction factor of 5E-4 no matter what the eccentricity has become. This trend matches well with the experimental data and conclusions of Cole (1967). However, our friction factor and that of Cole’s differ in
131 magnitudes. It is probable that such a difference can be attributed to a difference in the fluid viscosity between the two experiments.
4.6.3 Effect of Eccentricity Ratio on Maximum Vortex Intensity
θ r
z r θ = 0˚
θ = 15˚ θ = 30˚ θ = 45˚ θ = 60˚ θ = 75˚
θ = 90˚ θ = 120˚ θ = 150˚ θ = 180˚ θ = 240˚ θ = 300˚
Figure 4.44 Flow Pattern at Different Positions when ε = 0.2, ω = 65,000 rpm 132
θ r
z r θ = 0˚
θ = 30˚ θ = 60˚ θ = 90˚ θ = 105˚ θ = 120˚
θ = 150˚ θ = 180˚ θ = 240˚ θ = 270˚ θ = 300˚ θ = 330˚
Figure 4.45 Flow Pattern at Different Positions when ε = 0.6, ω = 110,000 rpm
In Figure 4.44 the flow patterns are displayed at different circumferential angular positions for the case of ε = 0.2 and ω = 65,000 rpm, while in Figure 4.45 the flow
patterns are presented for the case of ε = 0.6 and ω=110,000 rpm.
It was found that the position of maximum axial velocity w, and thus maximum vortex intensity occurs at about 45˚ for ε = 0.2 in Figure 4.44 and at 105˚ for ε = 0.6 in Figure 133 4.45, downstream of the position of the maximum film thickness. The results in Figure
4.44 match well with Vohr’s (1968) finding that the maximum intensity of vortex motion is positioned at θ = 50˚ downstream of maximum clearance. The local theory of DiPrima
(1963) showed the flow to be least stable (maximum intensity) at the position of maximum film thickness θ = 0˚. However, according to the nonlocal theory of DiPrima and Stuart (1972), the position of maximum vortex intensity is located at θ = 90˚. Our calculations indicate that the angular position of the maximum vortex intensity is not fixed. It varies with speed and the eccentricity ratio and may be located between θ = 0˚
and θ = 120˚. Figures 4.44 and 4.45 show that velocity component, w, in the axial
direction is almost zero from θ = 180˚ to θ = 360˚, which means that the flow is most stable in this zone for both ε = 0.2 and ε = 0.6.
4.6.4 Effect of Eccentricity Ratio on Recirculation
Figure 4.45 presents the tangential velocity vectors superimposed on the w-axial
velocity contours. The areas showing the vectors indicate flow that moves in the same
direction of inner cylinder rotation. The areas where the vectors are ‘missing’ indicate
that the flow has turned around and is now oriented against the direction of rotation. The
study of all the panels in this figure indicates that a three-dimensional recirculation
pattern is taking place. No recirculation is found when the eccentricity ratio is 0.2, Figure
4.44. In Table 4.9 one can find the recirculation range at different speeds when ε = 0.6.
We found that the recirculation region is symmetric about θ = 0˚ when the speed is low
while the symmetry begins to break when the speed reaches 100,000 rpm.
134 Table 4.9 Recirculation Region with Different Speeds when ε = 0.6
Re Ta
ω (rpm) RωC Recirculation Region (= ) RωC C ν (= ) ν R
1 40,000 354.02 35.58 Θ = - 103˚ to 103˚
2 60,000 495.63 49.81 Θ = - 103˚ to 103˚
3 80,000 566.43 56.93 Θ = - 103˚ to 103˚
4 90,000 637.24 64.04 Θ = - 103˚ to 103˚
5 100,000 637.24 64.04 Θ = - 103˚ to 104˚
6 110,000 708.04 71.16 Θ = - 102˚ to 106˚
4.7 A New Model for Transition Flow of Thin Films in Long Journal Bearings
The two previous sections, 4.5 and 4.6, presented the flow patterns, velocity and pressure profiles in a qualitative manner, for the flow in various size gaps (0.005 to 0.13 in.) between two cylinders; the inner cylinder was rotating and various eccentricity ratios
(0.0 to 0.8) were used. It was found that the inflection points of the torque-speed graphs coincide with significant qualitative and quantitative flow pattern changes, and the corresponding slope changes were actually caused by changes in the average velocity gradient on the outer cylinder wall. This finding, and not an increase in the apparent viscosity, is found to be the cause for the appearance of the inflection points known as the
First and respectively Second Taylor instability thresholds.
135 The findings of sections 4.5 and 4.6 are further extended in this section through new
numerical experiments performed in the transition region (the Re number varies from
410.7 to 3,944.9 and Ta number varies from 41.3 to 250.0 depending on the clearance
ratios). At the end of this section a new model for characterizing the flow behavior in
long journal bearing films (L/D > 2) in the transition regime is proposed and justified.
4.7.1 The Geometry and Coordinates
The geometry considered herein is presented in Figure 4.46a (also see Figure 4.1). The
system of coordinates shown in Figure 4.46b and at the top of Figure 4.46a sets y = 0 at
the journal surface along the radial direction, the positive x-axis points in the direction of
rotation and the axial direction runs along the z-coordinate as shown.
y C+e
x L y z ω L e R+C h R e x C-e z Rω C
a) b)
Figure 4.46 Geometry and Coordinates of a Journal Bearing Film
4.7.2 The Case with ε = 0
For the geometry introduced above the flow before the onset of instability is a
superposition of Couette and Poiseuille flows due to the effects of the rotating inner
136 cylinder and the circumferential pressure gradient (for long bearings), respectively. The pressure gradient is caused by the eccentricity ratio ε, which is usually greater than 0 for a journal bearing. However, as it will be shown later on, and somewhat surprisingly, the eccentricity ratio ε has no significant effect on the onset of Taylor instability in a bearing as long as ε is low ( ≤ 0.2) (also see sections 4.5 and 4.6). The first case to be presented considers concentric cylinders (ε = 0), and thus isolates out the effect of the pressure gradient, and as such can serve as a baseline case.
T (N.m) Tout (N.m) out 200 400 600 Re 400 450 Re 500 300 125 Couette Couette 115 Taylor Vortex 250 Taylor Vortex Wavy Taylor Vortex Wavy Taylor Vortex 200 105 Supercritical Couette (3) Supercritical Couette (3) Supercritical Taylor Supercritical Taylor C 150 95 B (2) 100 (2) 85 nd A 2 Critical Point 50 (1) Added torque 75 (1) 0 Detail shown in b) 1st Critical Point 65 0 20 40 60 80 Ta 35 40 45Ta 50 a) b)
Figure 4.47 Torque vs. Ta When C = 0.01 in., ε = 0.0
As shown in sections 4.5 and 4.6, Couette, Taylor and wavy vortex regimes are defined by the appearance of two inflection points in the torque-speed graph, Figure 4.47. Each one of the regimes is represented by a straight line, while the First and Second critical
Taylor threshold inflection points, indicate the onset of Taylor and wavy vortex instabilities, respectively, Figure 4.47b. 137
y y
C = 0.01 in., ε = 0, Ta^0.5 = 35.58 C = 0.01 in., ε = 0, Ta^0.5 = 35.58 z 140 z 1.0 z* = 0 (1) 120 z* = 1/8 z* = 1/4 z* = 3/8 0.5 z* = 1/2 100 z* = 5/8 z* = 3/4 z* = 7/8 Average 80 0.0 u(m/s) 60 z* = 0 (1) (m/s) v 0.0 0.2 0.4 0.6 0.8 1.0 z* = 1/8 z* = 1/4 40 z* = 3/8 z* = 1/2 z* = 5/8 -0.5 z* = 3/4 20 z* = 7/8 Average z* = 0 (1) 0 -1.0 0.0 0.2 0.4y* 0.6 0.8 1.0 y*
a) b)
C = 0.01 in., ε = 0, Ta^0.5 = 35.58 C = 0.01 in., ε = 0, Ta^0.5 = 35.58
1.0 6.0 z* = 0 (1) z* = 1/8 z* = 1/4 z* = 3/8 5.0 z* = 1/2 0.5 z* = 5/8 y z* = 3/4 z* = 7/8 4.0 z Average
3.0 z* = 1 (0) 0.0 z* = 7/8 z* = 3/4 w(m/s) 0.0 0.2 0.4 0.6 0.8 1.0 y z* = 5/8 P (E+4 N/m^2) (E+4 P 2.0 z* = 1/2 z* = 3/8 z* = 1/4 -0.5 z* = 1/8 z 1.0 Average
0.0 -1.0 0 0.2 0.4y* 0.6 0.8 1 y*
c) d)
Figure 4.48 Velocity and Pressure Profiles as Functions of y* during Couette Region
138
C = 0.01 in., ε = 0, Ta^0.5 = 42.70 C = 0.01 in., ε = 0, Ta^0.5 = 42.70 z* = 0 (1) y 1.0 z* = 1/8 160 z* = 1/4 z* = 3/8 z* = 1/2 140 z* = 5/8 z* = 3/4 z z* = 7/8 120 0.5 Average
100 y
80 0.0 v (m/s) v u(m/s) 0.0 0.2 0.4 0.6 0.8 1.0 z* = 0 (1) z 60 z* = 1/8 z* = 1/4 z* = 3/8 40 z* = 1/2 -0.5 z* = 5/8 z* = 3/4 20 z* = 7/8 Average Detail shown in f) 0 -1.0 0.0 0.2 0.4y* 0.6 0.8 1.0 y* a) b)
C = 0.01 in., ε = 0, Ta^0.5 = 42.70 y C = 0.01 in., ε = 0, Ta^0.5 = 42.70 1.0 z* = 0 (1) z* = 1/8 1.2 z* = 1/4 z* = 3/8 z* = 1/2 z* = 5/8 z z* = 3/4 0.5 z* = 7/8 0.9 y Average
z 0.0 0.6 z* = 1 (0) z* = 7/8 w(m/s) 0.0 0.2 0.4 0.6 0.8 1.0 z* = 3/4
P (E+5 N/m^2) (E+5 P z* = 5/8 z* = 1/2 z* = 3/8 0.3 -0.5 z* = 1/4 z* = 1/8 Average
0.0 -1.0 y* y* 0 0.2 0.4 0.6 0.8 1 c) d)
C = 0.01 in., ε = 0, Ta^0.5 = 42.70 z* = 0 (1) C = 0.01 in., ε = 0, Ta^0.5 = 42.70 30 z* = 1/8 z* = 1/4 40 z* = 3/8 z* = 1/2 Couette 20 z* = 5/8 Average z* = 3/4 u(z*=1/4) z* = 7/8 30 z* = 0(1) Average z* = 1/4 10
20 0 u(m/s) u(m/s)
0.0 0.2 0.4 0.6 0.8 1.0 -10 10
-20 0 0.80 0.85 0.90 0.95 1.00 -30 y* y* e) f)
Figure 4.49 Velocity and Pressure Profiles as Functions of y* during Taylor Vortex Region
139
C = 0.01 in., ε = 0, Ta^0.5 = 71.16, t = T (0) C = 0.01 in., ε = 0, Ta^0.5 = 71.16, t = T (0) y 5 z* = 0 (1) y 280 z* = 1/8 4 z* = 1/4 z* = 3/8 240 z* = 1/2 3 z* = 5/8 z z* = 3/4 z 2 z* = 7/8 200 Average 1 160 0
v (m/s) v 0.0 0.2 0.4 0.6 0.8 1.0 u(m/s) 120 z* = 0 (1) -1 z* = 1/8 z* = 1/4 -2 80 z* = 3/8 z* = 1/2 z* = 5/8 -3 40 z* = 3/4 z* = 7/8 Average -4 0 -5 0.0 0.2 0.4y* 0.6 0.8 1.0 y* a) b) y C = 0.01 in., ε = 0, Ta^0.5 = 71.16, t = T (0) C = 0.01 in., ε = 0, Ta^0.5 = 71.16, t = T (0) 5 6.5 4 z 6.0 3
2 5.5 y
1 5.0 0 z
w(m/s) 0.0 0.2 0.4 0.6 0.8 1.0 -1 4.5 z* = 1 (0) z* = 7/8 z* = 0 (1) N/m^2) (E+5 P z* = 3/4 -2 z* = 1/8 z* = 1/4 4.0 z* = 5/8 z* = 3/8 z* = 1/2 -3 z* = 1/2 z* = 3/8 z* = 5/8 3.5 z* = 1/4 z* = 3/4 z* = 1/8 -4 z* = 7/8 Average Average -5 3.0 y* 0 0.2 0.4y* 0.6 0.8 1 c) d)
C = 0.01 in., ε = 0, Ta^0.5 = 71.16, t = T (0)
80 z* = 0 (1) z* = 1/8 z* = 1/4 60 z* = 3/8 z* = 1/2 z* = 5/8 40 z* = 3/4 z* = 7/8 Average 20
0
u(m/s) 0.0 0.2 0.4 0.6 0.8 1.0 -20
-40
-60
-80 e) y*
Figure 4.50 Velocity and Pressure Profiles as Functions of y* during Wavy Vortex Region at a Given Time t = T (0)
140 Figure 4.48 presents a typical point (point A in Figure 4.47) in the Couette regime for the case of C = 0.01 in. and ε = 0. For a long bearing, the local velocity uc (in the
circumferential direction) is linear, and the local velocities vc (in the radial direction) and wc (in the axial direction) are practically 0, Figures 4.48a, 4.48b, and 4.48c. Thus in the coordinate system shown in Figure 4.46b one can write
=( − ) uc 1 y*U (4.1a)
= vc 0 (4.1b)
= wc 0 (4.1c) where y* = y/h and U (= R ω) is the velocity of the inner cylinder.
The radial pressure gradient exists in order to balance the centrifugal force in this regime, Figure 4.48d. Note that the velocity, or pressure contours in the y-z plane contain also the projected velocity vectors; the contours’ gray scale indicates the magnitudes of the respective variables. It can be seen from the contours’ study that the velocity and pressure do not vary in the axial z-direction and thus no flow is possible in that direction.
Figure 4.49 presents a typical point (point B in Figure 4.47) in the Taylor vortex regime. In this regime, the local velocities ut, v t and wt are non-linear, and their profiles
differ at different z-axial position, Figures 4.49a, 4.49b, and 4.49c. The velocities in this
regime can be expressed as
=−+=−+( ) ∆( ) ( π) + ( π ) ut 1 y*U u 1 y*U a(Re,z*)sin1c y* a(Re,z*)sin 2 c 2 y* U (4.2a)
=+=+∆ ( π) + ( π ) vt0 v 0 b1 (Re c ,z*)sin y* b 2 (Re c ,z*)sin 3 y* U (4.2b)
=+=+∆ ( π) + ( π ) wt0 w 0 c(Re,z*)sin1 c y* c(Re,z*)sin 2 c 2 y* U (4.2c) 141 where z* = z/L , and the clearance, C, based Reynolds number is Rec (= ρRωC/ ), and for i = 1, 2 the coefficients a i(Re c, z*),, b i(Re c, z*), and c i(Re c, z*) vary with Re c and the axial
position in the domain.
The ‘added velocity’ u represents the velocity difference between the curvilinear shaped u-velocity profile in the Taylor vortex regime and what would be its hypothetical value if there were a linear Couette profile at the same angular velocity ( u at the
elevation of z* = ¼ is shown in Figure 4.49f); similarly v and w are the velocity differences between the non-linear shaped v- and w- velocity profiles, in the Taylor vortex regime and what would be their hypothetical values (0) in the Couette regime.
These ‘added velocities’ represent the effect of the onset of the Taylor vortices.
Graphically u, v and w are shown in Figures 4.49e, 4.49b and 4.49c respectively
(Note that v and v have the same magnitude in this regime, and so do w and w).
As evidenced by Figure 4.49d in the Taylor vortex regime, pressure gradients have
now appeared in both the y-radial and z-axial directions. Just like in Figure 4.48d, here,
the radial pressure gradient balances the centrifugal force, while the axial one accounts
for the formation of the Taylor vortices. Note that both the velocities and pressure are
axisymmetric, and their profiles are time-independent in this regime.
Figure 4.50 presents a typical point (point C in Figure 4.47) at a certain arbitrary time
in the wavy vortex regime. In this regime, the local velocities uw, v w and ww are non- linear, curvilinear, just like those in the Taylor regime, but their non-linearity is stronger and they are also time-dependent as shown in Figures 4.50a, 4.50b, and 4.50c (velocity profiles are shown only at a given arbitrary point in time). Thus, u, v and w must
now be expressed as a function of Z( z*)Y( y*)T( t*). From the numerical simulations for 142 the Taylor vortex we have already shown that the product Z( z*)Y( y*) can be expressed as a sinusoidal function of space. By extension, we will propose expressions for the local velocities that include a trigonometric temporal variation as well:
= ( − ) + ∆ = ( − ) + uw 1 y * U u 1 y * U
(π) +( π) ( π) + ( π ) U a1 (Rec ,z*)sin y* a 2 (Re c ,z*)sin2 y* a 3 (Re c ,z*)sin 2 t* a 4 (Re c ,z*)cos 2 t* (4.3a)
= + ∆ = + vw 0 v 0
(π) +( π) ( π) + ( π ) U b(Re1c ,z*)sin y* b 2 (Re c ,z*)sin3 y* b 3 (Re c ,z*)sin 2 t* b 4 (Re c ,z*)cos 2 t* (4.3b)
= + ∆ = + ww 0 v 0
(π) + ( π) ( π) + ( π ) U c1 (Rec ,z*)sin y* c 2 (Re c ,z*)sin2 y* c 3 (Re c ,z*)sin 2 t* c 4 (Re c ,z*)cos 2 t* (4.3c)
where t* = t/T ( T being the period of the waves traveling in the circumferential direction), and for i = 3, 4 the coefficients a i(Re c, z*), b i(Re c, z*), and c i(Re c, z*) are dependent on
Re c and the axial position.
The ‘added velocities’ u, v and w, shown in Figures 4.50e, 4.50b and 4.50c,
respectively represent the effect of the onset of the wavy Taylor vortices. (Note that v
and v have the same magnitude in this regime, and so do w and w). Further as Figures
4.50d and 4.51 show pressure gradients are now present in all three directions. The radial
pressure gradient balances the centrifugal force, while the axial pressure gradient
accounts for the formation of Taylor vortices, and the circumferential pressure gradient is
responsible for the formation of the superimposed waves. The corresponding time mean
velocities of uw, v w and ww can be expressed
+ =t* 1 =−+() ()π + () π uw udt* w 1 y*U a(Re,z*)sin1c y* a(Re,z*)sin 2 c 2 y* U (4.4a) ∫t*
143 + =t* 1 = ()π + () π vwwc vdt* b(Re,z*)sin1 y* b(Re,z*)sin 2 c 3 y* U (4.4b) ∫t*
+ =t* 1 = ()π + () π wwwc wdt* c(Re,z*)sin1 y* c(Re,z*)sin 2 c 2 y* U (4.4c) ∫t*
On inspection, Equations (4.2) and (4.4) have the same composition of the right hand
side, except that the coefficients have different values. Thus one can generalize and
apply the same equation for these two regimes (Taylor and wavy regimes) as long as the
appropriate different constants are introduced,
=−+( ) (π) + ( π ) ut or uw1 y*U a(Re,z*)sin1 c y* a(Re,z*)sin 2 c 2 y* U (4.5a)
= (π) + ( π ) vt or vw b(Re,z*)sin1 c y* b(Re,z*)sin 2 c 3 y* U (4.5b)
= (π) + ( π ) wt or ww c(Re,z*)sin1 c y* c(Re,z*)sin 2 c 2 y* U (4.5c) where ai, b i and c i have local values [(Re c, z*)] for the Taylor vortex regime, while for the
wavy vortex regime a i(Re c, z*) = ai (Re c, z*), b i(Re c, z*) = bi (Re c, z*) and c i(Re c, z*) =
ci (Re c, z*).
z x
Figure 4.51 Pressure Contour in the x-z Plane during Wavy Vortex Regime
4.7.3 The Case with ε = 0.2
The effect of the eccentricity ratio on the flow stability manifests through the addition of a circumferential pressure gradient over the flow of the concentric case. This pressure
144 gradient will stabilize (or destabilize) the flow depending on the relationship of its direction with respect to the rotation direction of the inner cylinder (journal). According to DiPrima (1963) and Szeri (1979) these conditions are: (i) stabilizing, when Ta × Re p <
− 2 ∂ = C P C 0 and (ii) destabilizing, when Ta × Re p > 0, where Re p is the mean 12 µR ∂θ ν pressure based Reynolds number. The stabilization effect depends on the sign of the
1 ∂P circumferential pressure gradient . The local Reynolds number Re h (= ρRωh/ ) R ∂θ
varies circumferentially directly proportional with h. In this context, the eccentricity ratio
becomes a cause of instability through the change in vortex intensity due to the increase
in the local Reynolds number. Thus, the velocities for the eccentricities ε ≤ 0.2 can be
expressed as
∂ π π =−+yP1 2 −+ y + 2 y ut or uw 1 U() y yh a(Re,z)sin1h a(Re,z)sin 2 h U (4.6a) hx2µ ∂ h h
π π =y + 3 y vt or vw b(Re,z)sin1 h b(Re,z)sin 2 h U (4.6b) h h
π π =y + 2 y wt or ww c(Re,z)sin1 h c(Re,z)sin 2 h U (4.6c) h h
where a i(Re h, z), b i(Re h, z), and c i(Re h, z) are coefficients depending on the local
Reynolds number, Re h, and the axial positions. By comparison with Equations (4.5),
Equations (4.6) have the pressure gradient term due to eccentricity and the coefficients in
Equation (4.6) depend on the local Reynolds number Re h rather than the mean Reynolds number Re c.
145 4.7.4 Proposed Transition Reynolds Equation Model
The transition Reynolds equation derivation starts with the integration of the classical continuity equation across the film thickness:
h ∂(ρv) h ∂(ρu) h ∂(ρw) ∫ dy = −∫ dy − ∫ dy (4.7) 0 ∂y 0 ∂x 0 ∂z
By replacing u and w with their respective expressions from Equation (4.6) and
performing the integration for the boundary conditions v = 0 at y = 0 and y = h, we obtain
the initial form of the transition Reynolds equation as
∂∂∂ρ 3 ∂ρ ∂ ρ h P a(Re,z)1h h c(Re,z) 1 h h =6(ρ Uh ) + 24 U + (4.8) ∂∂∂xxxµ ∂ x π ∂ z π
Equation (4.8) is valid for an infinitely long bearing with low eccentricities ( ε ≤ 0.2), and its corresponding velocity profiles are expressed in Equation (4.6).
C = 0.01 in., ε = 0, Ta^0.5 = 71.16, t = T (0) 6.5
6.0 y 5.5
5.0 z
4.5
P (E+5 N/m^2) (E+5 P y* = 0 y* = 1/8 4.0 y* = 1/4 y* = 3/8 y* = 1/2 y* = 5/8 3.5 y* = 3/4 y* = 7/8 y* = 1 3.0 0 0.2 0.4z* 0.6 0.8 1
Figure 4.52 Pressure Profiles as Functions of z* during Wavy Vortex Region at a Given Time t = T (0)
146 The pressure P is a function of x and z , which is evident from the term containing
∂ ∂ P . There is no term containing P in Equation (4.8), but P is a function of z ∂x ∂z
through the two terms containing a 1(Re h, z) and c 1(Re h, z) respectively, which are functions of z. The fact that P is a function of z is evident in Figure 4.52 for the case of C
= 0.01in., ε = 0 and Ta = 71.16.
This formulation of the Reynolds equation is applicable in the transition regime, i.e.
after the onset of Taylor vortices ( Ta > 41.3) and before the full development of
turbulence (Re < 2,000). For comparison, the laminar Reynolds Equation (4.9), which is
applicable before the onset of Taylor vortices, and turbulent Reynolds equation (4.10),
which is good after the full development of turbulence (Re < 2,000), are listed below.
∂ ρh 3 ∂P ∂ = 6 (ρUh ) (4.9) ∂x µ ∂x ∂x
∂ ρh 3 ∂P ∂ = 6 (ρUh ) (4.10) ∂ µ ∂ ∂ x k x (Re h ) x x
The classical turbulent Reynolds equation (4.10) uses the eddy viscosity model to
account for the “turbulence” effect through the coefficient k x(Re h). In the transition
Reynolds equation (4.8), however, the viscosity keeps its laminar value (due to the nature
of a fluid); but it adds the terms a i(Re h, z) and c i(Re h, z) to represent the effect of Taylor, or wavy vortices. For an infinitely long bearing, the pressure and velocities predicted by the classical Equations (4.9) and (4.10) are one and two dimensional respectively, while the velocities predicted by Equation (4.6) are three-dimensional (x, y and z directions) and the pressure predicted by Equation (4.8) is two-dimensional (both x and z directions). 147 4.7.5 Coefficients Used in Transition Reynolds Equation Model
Like the empirical coefficient k x(Re h) in Equation (4.10), the coefficients a i(Re h, z),
bi(Re h, z), and c i(Re h, z) in Equations (4.6) and (4.8) can be obtained numerically and verified experimentally.
To get the overall average effect of ut, v t and wt in the Taylor vortex regime, one can
integrate Equation (4.6) over the axial domain z∈(0, L), and obtain
1 L y 1 ∂P = = − + ()2 − + ∆ = U t ∫ ut dz 1 U y yh U L 0 h 2µ ∂x
y 1 ∂P πy 2πy 1− U + ()y 2 − yh + A (Re )sin U + A (Re )sin U (4.11a) h 2µ ∂x 1 h h 2 h h
= 1 L = Vt ∫ vt dz 0 (4.11b) L 0
= 1 L = Wt ∫ wt dz 0 (4.11c) L 0
= 1 L = 1 L where A1 (Re h ) ∫ a1 (Re h , z)dz and A2 (Re h ) ∫ a2 (Re h , z)dz . L 0 L 0
Note that A1(Re h) and A2(Re h) are coefficients depending on the local Reynolds
number (Re h), Ut, Vt and Wt are the average velocities in the circumferential, radial and
axial directions respectively. They are represented by the continuous lines labeled
“Average” in the legends of Figures 4.49a, 4.49b and 4.49c respectively.
For the Taylor vortex flow, U = A1(Re h)sin( πy/h )] U + A2(Re h)sin(2πy/h )] U is the
average ‘added velocity’ in the circumferential direction due to the onset of the Taylor
instability, and is represented by the continuous line labeled “Average” in the legend of
Figure 4.49e. The ‘added torque’, due to U, exerted on the outer cylinder wall (at y* =
1 or y = h) accounts for the first inflection point in the torque-speed graphs presented in 148 Section 4.5 and 4.6 (also see Figure 4.47). Graphically this ‘added torque’ can be visualized by looking at the difference between the gradients of the velocities U(Taylor) and U(Couette) in the Taylor and Couette flows respectively, Figure 4.53. Similarly, one can also integrate Equation (4.6) to get the overall effect of uw, v w and ww in the wavy vortex regime.
C = 0.01 in., ε = 0, Ta^0.5 = 42.70 160
U Couette 120 Average ∂∆ ∂ ∂ U = U (Taylor ) − U(Couette ) ∂ * ∂ y * ∂ y * y y*=1 y*=1
U 80
∂U (Taylor ) 40 ∂ * y Y *=1 ∂U (Couette ) = 1 ∂ y * 0 0.00 0.20 0.40 0.60y* 0.80 1.00
Figure 4.53 Relationship between Velocity Gradient on Outer Cylinder and U*
One can also integrate Equation (4.8) over the axial domain z∈(0, L), and obtain:
∂ ρ 3 ∂ ∂ ∂ ρ h P = ρ + 24 ( hU ) 6 ( Uh ) A1 (Re h ) (4.12) ∂x µ ∂x ∂x π ∂x
L ∂ ∂ = 1 P = 1 L P where A1 (Re h ) ∫ a1 (Re h , z)dz and ∫ dz . L 0 ∂x L 0 ∂x
L ∂ 1 c1 (Re h , z) Note that the term with c 1(Re h, z) disappears because ∫ dz = 0 due to the L 0 ∂z periodic nature of c1(Re h, z) in the z direction.
Therefore, the coefficients that really matter are A1(Re h) and A2(Re h) from the overall point of view. A1(Re h) and A2(Re h) play an important role on the velocity profiles 149 through Equation(4.11), and A1(Re h) is significant to the pressure profile through
Equation (4.12). To study the nature of coefficients A1(Re h) and A2(Re h), three typical
cases with clearance ratios of C/R = 0.01, 0.004 and 0.001 will be presented.
C/R = 0.01 0.32 0.24 a1(z* = 0) 0.16 a1(z* = 0.125) a1(z* = 0.25) 0.08 a1(z* = 0.375) 0.00 a1(z* = 0.5) a1 a1(z* = 0.625) -0.08 a1(z* = 0.75) a1(z* = 0.875) -0.16 -0.24
-0.32 400 600 800 1000 1200 1400
Reynolds Number
Figure 4.54 Coefficients a 1 as Function of Reynolds Number (Re h) for C/R = 0.01
C/R = 0.01 0.00 a2(z* = 0) a2(z* = 0.125) a2(z* = 0.25) -0.05 a2(z* = 0.375) a2(z* = 0.5) a2(z* = 0.625) -0.10 a2(z* = 0.75) a2 a2(z* = 0.875)
-0.15
-0.20 400 600 800 1000 1200 1400 Reynolds Number
Figure 4.55 Coefficients a as Function of Reynolds Number (Re ) for C/R = 0.01 2 h 150 Figures 4.54 and 4.55 present the coefficients a1(Re h, z) and a 2(Re h, z) as functions of
Reynolds number at various z* for the case of C/R = 0.01. It is shown that a1(Re h, z)
increases or decreases with the increase of Reynolds number depending on the position of
z*, while a2(Re h, z) always decreases with the increase of Reynolds number. All the
coefficients a1(Re h, z) and a 2(Re h, z) behavior nonlinearly, and their magnitudes increase
monotonically.
C/R = 0.004 0.50
a1(z* = 0) 0.25 a1(z* = 0.125) a1(z* = 0.25) a1(z* = 0.375) 0.00 a1(z* = 0.5) a1 a1(z* = 0.625) a1(z* = 0.75) a1(z* = 0.875) -0.25
-0.50 600 800 1000 1200 1400 1600 1800 2000 2200 Reynolds Number
Figure 4.56 Coefficients a 1 as Function of Reynolds Number (Re h) for C/R = 0.004
C/R = 0.004 0.00 a2(z* = 0) a2(z* = 0.125) a2(z* = 0.25) -0.05 a2(z* = 0.375) a2(z* = 0.5) a2(z* = 0.625) -0.10 a2(z* = 0.75)
a2 a2(z* = 0.875) -0.15
-0.20
-0.25 600 800 1000 1200 1400 1600 1800 2000 2200
Reynolds Number
Figure 4.57 Coefficients a 2 as Function of Reynolds Number (Re h) for C/R = 0.004 151
C/R = 0.001 0.50
a1(z* = 0) 0.25 a1(z* = 0.125) a1(z* = 0.25) a1(z* = 0.375) 0.00 a1 a1(z* = 0.5) a1(z* = 0.625) a1(z* = 0.75) -0.25 a1(z* = 0.875)
-0.50 1000 1500 2000 2500 3000 3500 Reynolds Number
Figure 4.58 Coefficients a 1 as Function of Reynolds Number (Re h) for C/R = 0.001
C/R = 0.001 0.00
-0.05
-0.10 a2(z* = 0) -0.15 a2(z* = 0.125) a2 a2(z* = 0.25)
-0.20 a2(z* = 0.375) a2(z* = 0.5) a2(z* = 0.625) -0.25 a2(z* = 0.75) a2(z* = 0.875) -0.30 1000 1500 2000 2500 3000 3500 Reynolds Number
Figure 4.59 Coefficients a 1 as Function of Reynolds Number (Re h) for C/R = 0.001
The coefficients a1(Re h, z) and a 2(Re h, z) behavior similarly for the cases of C/R =
0.004, Figures 4.56 and 4.57, and C/R = 0.001, Figures 4.58 and 4.59. That is, a1(Re h, z) increases or decreases with the increase of Reynolds number depending on the position of 152 z*, while a2(Re h, z) always decreases with the increase of Reynolds number. All the
coefficients a1(Re h, z) and a 2(Re h, z) behavior nonlinearly, and their magnitudes increase
monotonically.
0.010 C/R = 0.01 C/R = 0.004 0.008 C/R = 0.001
A1 0.006
0.004
0.002
0.000 0 500 1000 1500 2000 2500 3000 3500 Reynolds Number
Figure 4.60 Coefficients A 1 as Function of Reynolds Number for Different C/R
0.00
-0.05
-0.10 -0.15 A2 -0.20 C/R = 0.01 -0.25 C/R = 0.004 C/R = 0.001
-0.30 0 500 1000 1500 2000 2500 3000 3500
Reynolds Number
Figure 4.61 Coefficients A 2 as Function of Reynolds Number for Different C/R
153 L It is convenient to get the average coefficients = 1 and A1 (Re h ) ∫ a1 (Re h , z)dz L 0
L = 1 when Equations (4.11) and (4.12) are applied. Figures 4.60 A2 (Re h ) ∫ a2 (Re h , z)dz L 0
and 4.61 present the average coefficients A1(Re h) and A2(Re h) as functions of Reynolds
number for the cases of C/R = 0.01, 0.004 and 0.001.
It is shown that A1(Re h) increases with the increase of Reynolds number, while A2(Re h)
decreases with the increase of Reynolds number. A1(Re h) is always positive and A2(Re h)
is always negative. The magnitude of both A1(Re h) and A2(Re h) is larger for the larger clearance ratio at the same Reynolds number. The curves of A1(Re h) or A2(Re h) for
different clearance ratios are separated from each other when Reynolds number is used as
the abscissa.
However, the curves of A1(Re h) or A2(Re h) for different clearance ratios tend to
collapse to one when Taylor number is used as the abscissa, particularly when Taylor
number is less than 70, Figures 4.62 and 4.63. This can be explained that the magnitude
of A1(Re h) or A2(Re h) is directly related to the strength of Taylor or wavy Taylor vortices, which is represented by the Taylor number. The fact that Taylor number includes the curvature effect or clearance ratio can be seen from the mathematic expression of Taylor number ( Ta = Re C / R ). Therefore, Taylor number used as the dependent variable is more appropriate than Reynolds number during the transition regime, particularly in the low range of this regime (say Ta < 70).
154
0.010
C/R = 0.01 0.008 C/R = 0.004 C/R = 0.001 0.006 A1
0.004
0.002
0.000 0 20 40 60 80 100 120 140 Taylor Number Ta
Figure 4.62 Coefficients A 1 as Function of Ta for Different C/R
0.00
-0.05
-0.10
-0.15 A2 C/R = 0.01 -0.20 C/R = 0.004 C/R = 0.001 -0.25
-0.30 0 20 40 60 80 100 120 140 Ta
Figure 4.63 Coefficients A as Function of Ta for Different C/R 2
155
0.008 y = -0.351739x + 0.008677 C/R=0.001 2 R = 0.998056 C/R=0.004 C/R=0.01 0.006
y = -0.415668x + 0.010151 2 0.004 R = 0.998435 A1
0.002 y = -0.338344x + 0.008244 2 R = 0.998440 0.000 0.000 0.005 0.010 0.015 0.020 0.025 0.030 1/ Ta
Figure 4.64 Coefficients A 1 as Function of Inverse of Ta for Different C/R
1/ Ta 0.00 0.00 0.01 0.02 0.03 -0.05 y = 10.578983x - 0.268713 R2 = 0.989518 C/R=0.001 -0.10 C/R=0.004 C/R=0.01 -0.15 A2 y = 12.076184x - 0.306934 -0.20 R2 = 0.997970
-0.25 y = 11.017909x - 0.280079 R2 = 0.993804 -0.30
Figure 4.65 Coefficients A 2 as Function of Inverse of Ta for Different C/R
156 To get the expressions for coefficients A 1 and A 2 as functions of Ta and C/R, first one can get them as functions of inverse of Ta for different C/R, Figures 4.64 and 4.65.
It is shown that relationship between A 1 (or A 2) and inverse of Ta is linear. Further
from Figures 4.64 and 4.65, the expressions for coefficients A 1 and A 2 can be obtained as
follows:
α = 1 + β A1 1 (4.13a) Ta
where
C 2 C α = − .0 01368 *ln − .0 1239 *ln − .0 6188 (4.13b) 1 R R
and
C 2 C β = .0 000257 *ln + .0 002125 *ln + .0 01259 (4.13c) 1 R R
α = 2 + β A2 2 (4.14a) Ta
where
C 2 C α = .0 1235 *ln + .0 7716 *ln +11 .5131 (4.14b) 2 R R
and
C 2 C β = − .0 003026 *ln − .0 01824 *ln − .0 2885 (4.14c) 2 R R
157 4.7.6 Comparison between Our Model and Turbulence Models
3.5 Constantinescu Ng-Pan 3.0 Hirs Gross et al. Our Model (C/R=0.01) 2.5 Our Model (C/R=0.004) Our Model (C/R=0.001)
2.0 M* 1.5
1.0
0.5 0 500 1000 1500 2000 2500 3000 3500 Reynolds Number
Figure 4.66 Comparison of Torque on Bearing between Our Model and Turbulence Models as Functions of Reynolds Number
3.0 Constantinescu Ng-Pan Hirs 2.5 Gross et al. Our Model (C/R=0.01) Our Model (C/R=0.004) 2.0 Our Model (C/R=0.001)
M* 1.5
1.0
0.5 0 20 40 60 80 100 120 140 Taylor Number
Figure 4.67 Comparison of Torque on Bearing between Our Model and Turbulence Models as Functions of Taylor Number
158 Once the coefficients of A1(Re h) and A2(Re h) in Equation (4.11) are calculated
according to Equations (4.13) and (4.14), the dimensionless ‘added torque’ exerted on the
outer cylinder, M*, due to the onset of instability can also be calculated. The calculation of M* allows further analysis and insight into the results of the four
previously introduced turbulent models. Figure 4.66 presents the torque (M*) exerted on
the outer stationary cylinder as it is calculated by our model for three cases, C/R = 0.01,
0.004 and 0.001. For comparison purposes, these results are then superimposed over the
results of the four turbulence models of Constantinescu (1959), Ng-Pan (1965), Hirs
(1973), and Gross et al. (1980). Note that Reynolds number is used as the abscissa in
Figure 4.66.
For the case of C/R = 0.001, our curve collapses over the one of Hirs when Reynolds
number is less than 1,000, and our result is very close to that of Hirs when Reynolds
number is larger than 1,700 and smaller than 2,500. The critical Reynolds number of our
model is 1,300 and the one of Hirs is about 1,000. The difference between our results
and those of Hirs start to be large when Reynolds number is over 2,500. Our results are
not even close to other turbulence models for the case of C/R = 0.001.
For the case of C/R = 0.004, our threshold Reynolds number for the onset of the Taylor
instability is about 700 and thus different from that of Hirs’, which is about 1,000. The
results obtained with the transition model are close to those of Constantinescu when
Reynolds number reaches 1,000, but they predict differently from all the other models in
the transition regime.
For the case of C/R = 0.01, our results are not even close to any turbulence model. The
reason is that all the turbulence models assume bearing-like clearance (C/R is in the 159 magnitude of 0.001), but our model consider the effect of clearance ratio. Therefore, the results calculated by our model are different for different clearance ratios.
The fact that Taylor number is more appropriate to be the independent variable than
Reynolds number during the transition regime is also shown in Figure 4.67. Figure 4.67 presents the torque (M*) calculated by our model and the turbulence models with Taylor number being the abscissa for three cases, C/R = 0.01, 0.004 and 0.001. Note that the
Taylor number used for the turbulence models corresponds to C/R = 0.004. One can notice that the torque curves tend to collapse to one when Taylor number less than 70.
Finally one has to offer an explanation regarding the significant difference between our model and the four turbulence models in the transition regime. To answer this question, one has to take a look at the origins of the derivations of these turbulence models. One can state that these models neglect the inertia effects and the effect of the presence of the
Taylor vortices. Instead, they use the Prandtl mixing length, or the eddy viscosity concept. They all start from the simplified (no inertia effects) “turbulent” Navier-Stokes equations shown below:
∂ ∂ ∂ P = µ U − ρ u v'' (4.15a) ∂x ∂y ∂y
∂ ∂ ∂ P = µ W − ρ v'w' (4.15b) ∂z ∂y ∂y
However, in the wavy regime an order of magnitude analysis of each one of the terms
in the full “turbulent” Navier-Stokes equations shows that the inertia terms are as
significant as the pressure, viscous, and the Reynolds stress terms. Table 4.10 details the
results of order of magnitude analysis for each one of the terms in the Navier-Stokes
160 equation in the x-direction for the different flow regimes (the case of C = 0.01 in. and ε =
0.2). For a typical point before the onset of Taylor instability ( Ta = 35.5, Figure 4.47),
1 ∂P µ ∂ 2 u the pressure term ( − ) is balanced by the viscous term ( ). In the Taylor ρ ∂x ρ ∂y 2
vortex regime (a typical point when Ta = 44.0, Figure 4.47), however, those two terms
∂u ∂u are challenged by the magnitude of the two inertia terms ( u and v ) and the other ∂x ∂y
µ ∂ 2 u 1 ∂P µ ∂ 2 u viscous term ( ), although ( − ) and ( ) dominate. In the Taylor wavy ρ ∂z 2 ρ ∂x ρ ∂y 2 regime (two typical points when Ta = 56.8 and 71.0, Figure 4.47), the magnitudes of
∂u ∂u ∂u 1 ∂P the inertia terms ( u , v and w ) are as significant as the pressure term ( − ), ∂x ∂y ∂z ρ ∂x
µ ∂ 2 u µ ∂ 2 u ∂ the two viscous terms ( and ), and the Reynolds stress term ( − (u v'' )). A ρ ∂y 2 ρ ∂z 2 ∂y
further order of magnitude analysis of each term in the Navier-Stokes equation in the
axial z-direction is also performed and the same conclusion can be drawn, Table 4.11.
Based on these findings and supported by the numerical results presented above on a
comparative basis (Figures 4.66 and 4.67), the use of the simplified “turbulence” Navier-
Stokes equations, Equation (4.15), is not appropriate in the Taylor and wavy vortex
regime, and thus the four turbulence models proposed for the use in the classical
Reynolds equation are not applicable to the transition regime. Instead we propose the
transition Reynolds equation for the Taylor and wavy vortex regime; it incorporates
implicitly the effects of inertia through their modified expressions for velocities and is
161 clearly supported by the results presented above. Thus, if the long bearings are operating in the transition regime the following form of the Navier-Stokes equations has to be used
∂u ∂u ∂u 1 ∂P µ ∂ 2 u µ ∂ 2 u ∂ u + v + w = − + + − ()u v'' (4.16a) ∂x ∂y ∂z ρ ∂x ρ ∂y 2 ρ ∂z 2 ∂y
∂w ∂w ∂w 1 ∂P µ ∂ 2 w µ ∂ 2 w ∂ u + v + w = − + + − ()v'w' (4.16b) ∂x ∂y ∂z ρ ∂z ρ ∂y 2 ρ ∂z 2 ∂y
Based on results yielded by these equations, one can use instead the simpler transition
form of the Reynolds equation proposed through Equation (4.8) with its corresponding
velocity profiles described by Equation (4.6). Specifically this can be done because the
effects of Equation (4.16) are already incorporated in the content of Equation (4.6).
Table 4.10 Order of Magnitude Analysis of the Navier-Stokes Equation in x-direction (C = 0.01 in., ε = 0.2)
Ta ∂u ∂u ∂u 1 ∂P µ∂2u µ ∂2u µ∂2u ∂ u v w − − (u v'' ) ∂x ∂y ∂z ρ ∂x ρ ∂x2 ρ ∂y2 ρ ∂z2 ∂y
35.5 A 0 3 0 -164 0 161 0 0
B -1 9 0 -181 0 173 0 0
44.0 A 5 -21 0 -193 0 190 -13 0
B -13 57 0 -200 0 229 15 0
56.8 A 944 -739 -574 -352 -1 -473 -386 843
B 1,426 1,226 680 -330 0 1,736 189 1,737
71.0 A 892 -1,630 -205 -514 -2 -457 -722 752
B 2,262 831 1,780 -483 0 2,592 169 2,595
162 Notes: 1) A and B represent two typical points, (0, 0.5, 0) and (0, 0.25, 0.75) respectively.
2) Units are 10 3m/s 2.
Table 4.11 Order of Magnitude Analysis of the Navier-Stokes Equation in z-direction (C = 0.01 in., ε = 0.2)
Ta ∂w ∂w ∂w ∂ µ ∂2w µ ∂2 w µ∂2 ∂ −1 P w − (v w'' ) u v w 2 ∂ ∂x ∂y ∂z ρ ∂z ρ ∂x ρ ∂y2 ρ ∂z2 y
35.5 A 0 0 0 0 0 0 0 0
B 0 0 0 0 0 0 0 0
44.0 A 0 0 0 1 0 -1 0 0
B 0 0 0 -5 0 1 4 0
56.8 A 34 74 -11 154 0 -83 -26 52
B 30 -32 -11 -190 0 118 22 37
71.0 A 23 172 10 320 0 -109 -54 48
B 70 -36 39 -240 0 200 36 77
Notes: 1) A and B represent two typical points, (0, 0.5, 0) and (0, 0.25, 0.75) respectively.
2) Units are 10 3m/s 2.
Finally, to correctly predict the flow behavior in a long bearing (L/D > 2), we
recommend the following procedure: i) use laminar Reynolds equation (4.9) when Re <
41.3 R /C (before the onset of Taylor vortices); ii) use turbulent Reynolds equation
(4.10) when Re > 2,000 (fully developed turbulent flow); iii) use transition Reynolds
163 equation (4.8) in the transition regime, i.e. after the onset of Taylor vortices ( Ta >
41.3) and before the full development of turbulence (Re < 2,000).
164
CHAPTER V
NUMERICAL RESULTS (SHORT BEARING)
5.1 Introduction
Chapter IV studied the relationship between the onset of Taylor instability and appearance of “turbulence” in narrow gaps between two “infinitely” long cylinders.
Flow patterns, velocity and pressure profiles were presented. A transition Reynolds equation for predicting the flow behavior of “long” journal bearings in transition regime was proposed and justified. This chapter will deal with the flow of a viscous flow in narrow gaps between two “finitely” long cylinders with various eccentricity ratios. Flow patterns, velocity and pressure profiles will be presented in a quantitative manner. A transition Reynolds equation for predicting the flow behavior of “short” journal bearings in transition regime will be proposed and justified.
5.2 The Geometry and Coordinates
The geometry is presented in Figure 5.1. The two cylinders have a finite length of L =
0.2 in. (L/D = 0.1 < 2), which qualifies the assumption of short bearings. The radius of the outer cylinder is fixed, R+C = 1.0 in., and the average clearance, C, between the two cylinders varies from 0.01 to 0.002 in. Therefore, the radius of the inner cylinder varies from 0.99 to 0.998 in. correspondingly. The centers of the cylinders are set at a distance 165 ‘e’ apart. The corresponding eccentricity ratio, ε = e/C, varies from 0.0 to 0.4. The
angular speed of the inner cylinder ω, the average clearance C, and the eccentricity ratio ε
are the three controlling parameters.
Detail shown in c)
ω R+C θ L r
z z R e θ θ r r
a) b) c)
Figure 5.1 Geometric Description of the Cylinders (Not at scale)
y
x l ω z y R+C R e l e h x z U=R ω C
b) a)
Figure 5.2 Geometry and Coordinates of a Journal Bearing Film
166 The top and bottom faces of the fluid between the two cylinders are set as outlet boundary conditions with fixed pressure of 0 N/m 2. The outer cylinder is set as a non-
slip wall boundary condition. The inner cylinder wall is set as a rotating non-slip wall,
and the angular velocity vector in the z-direction is prescribed, as shown in Figure 5.1.
The system of coordinates sets y = 0 at the journal surface along the radial direction;
the positive x-axis points in the direction of rotation and the axial direction runs along the
z-coordinate as shown, Figure 5.2. The axial length of the domain of concern is set to be
l such that this domain (from z = 0 to z = l) covers one wave in the axial direction.
5.3 The Case with ε = 0
For the geometry introduced above the flow before the onset of instability is a superposition of Couette and Poiseuille flows due to the effects of the rotating inner cylinder and the axial pressure gradient (for short bearings), respectively. Like the one presented for long bearings in chapter IV, the first case to be presented in this chapter considers concentric cylinders (ε = 0), and thus isolates out the effect of the axial pressure gradient, and as such can serve as a baseline case.
Tout (N.m) 4.0
Couette Taylor Vortex 3.0 Wavy Vortex
2.0 B A 2nd Critical Point 1.0
1st Critical Point 0.0 0 20 40 60 80 Ta
Figure 5.3 Torque vs. Ta for Short Bearing When C = 0.01 in., ε = 0.0 167
End effect zone End effect zone End effect zone End effect zone
Detail shown in e) Detail shown in f) Detail shown in g) Detail shown in h)
End effect zone End effect zone End effect zone End effect zone a) b) c) d)
y y y y
z = 0 z = 0 z = 0 z = 0
z z z z
z = l z = l z = l z = l
e) f) g) h)
Figure 5.4 Velocity and Pressure Contours during Couette Regime (Point A in Figure 5.3) 168
Short Bearing, C = 0.01 in., ε = 0, Ta^0.5 = 28.46 Short Bearing, C = 0.01 in., ε = 0, Ta^0.5 = 28.46 120 1.5 z* = 0 (1) z* = 1/8 z* = 1/4 100 1.0 z* = 3/8 z* = 1/2 z* = 5/8 z* = 3/4 z* = 7/8 80 0.5 Average 60 0.0 v (m/s) v u(m/s) z* = 0 (1) 0.0 0.2 0.4 0.6 0.8 1.0 z* = 1/8 40 z* = 1/4 z* = 3/8 -0.5 z* = 1/2 z* = 5/8 20 z* = 3/4 z* = 7/8 -1.0 Average
0 -1.5 0.0 0.2 0.4y* 0.6 0.8 1.0 y*
a) b)
Short Bearing, C = 0.01 in., ε = 0, Ta^0.5 = 28.46 Short Bearing, C = 0.01 in., ε = 0, Ta^0.5 = 28.46
1.5 7.5 z* = 0 (1) z* = 1/8 End effect zone End effect zone 1.0 z* = 1/4 z* = 3/8 5 z* = 1/2 z* = 5/8 unaffected zone z* = 3/4 0.5 z* = 7/8 Average 2.5
0.0
w(m/s) 0.0 0.2 0.4 0.6 0.8 1.0 0 P(E+3 N/m^2) P(E+3 0 0.2 0.4 0.6 0.8 1 -0.5
-2.5 -1.0
-5 -1.5 y* z* c) d)
Figure 5.5 Velocity and Pressure Profiles as Functions of y* and z* during Couette Regime (Point A in Figure 5.3)
Like the ones for long bearings, Couette, Taylor and wavy vortex regimes are defined by the appearance of two inflection points in the torque-speed graph for the case of short bearing, Figure 5.3. Each one of the regimes is represented by a straight line, while the 169 First and Second critical Taylor threshold inflection points, indicate the onset of Taylor and wavy vortex instabilities, respectively, Figure 5.3.
Figure 5.4 presents the velocity and pressure contours of a typical point (point A in
Figure 5.3) in the Couette regime. Note that the “end effect zones” shown in Figures 5.4a through 5.4d are affected by the fixed-pressure outlet boundary conditions. The “end effect zone” is quite different from the one unaffected by the end boundary conditions or the “unaffected zone”. However, the “end effect zone” is only a small portion for a typical short bearing. Therefore, only the “unaffected zone” will be studied hereinafter.
A typical section of the “unaffected zone” (from z = 0 to z = l) is shown in Figures 5.4e
through 5.4h.
Figure 5.5 presents the velocity and pressure profiles of a typical point (point A in
Figure 5.3) in the Couette regime corresponding to Figures 5.4e through 5.4h. For a short bearing in this regime, the local velocity uc (in the circumferential direction) is linear, and the local velocities vc (in the radial direction) and wc (in the axial direction) are practically
0, Figures 5.5a, 5.5b, and 5.5c. Thus in the coordinate system shown in Figure 5.2 one can write
=( − ) uc 1 y*U (5.1a)
= vc 0 (5.1b)
= wc 0 (5.1c) where y* = y/h and U (= R ω) is the velocity of the inner cylinder. The axial pressure
profile of the mid-plane (y* = 1/2) is shown Figure 5.5d, which indicates that ∂P /∂z is
zero in the “unaffected zone” for the concentric case ( ε = 0).
170
Detail shown in e) Detail shown in f) Detail shown in g) Detail shown in h)
a) b) c) d)
y y y y
z = 0 z = 0 z = 0 z = 0
z z z z
z = l z = l z = l z = l
f) e) g) h)
Figure 5.6 Velocity and Pressure Contours during Taylor Vortex Regime (Point B in Figure 5.3) 171
Short Bearing, C = 0.01 in., ε = 0, Ta^0.5 = 42.70 Short Bearing, C = 0.01 in., ε = 0, Ta^0.5 = 42.70
1.5 z* = 0 (1) 160 z* = 1/8 z* = 1/4 z* = 3/8 140 z* = 1/2 1.0 z* = 5/8 z* = 3/4 120 z* = 7/8 Average 0.5 100
80 0.0 u(m/s) z* = 0 (1) (m/s) v 0.0 0.2 0.4 0.6 0.8 1.0 60 z* = 1/8 z* = 1/4 z* = 3/8 -0.5 40 z* = 1/2 z* = 5/8 z* = 3/4 20 z* = 7/8 -1.0 Average 0 -1.5 0.0 0.2 0.4y* 0.6 0.8 1.0 y*
a) b)
Short Bearing, C = 0.01 in., ε = 0, Ta^0.5 = 42.70 Short Bearing, C = 0.01 in., ε = 0, Ta^0.5 = 42.70
1.5 z* = 0 (1) 25 z* = 1/8 z* = 0 (1) z* = 1/4 z* = 1/8 z* = 3/8 20 z* = 1/4 z* = 1/2 z* = 3/8 1.0 z* = 5/8 z* = 1/2 z* = 3/4 15 z* = 5/8 z* = 7/8 z* = 3/4 Average z* = 7/8 10 Average 0.5 5
0.0 0 u(m/s)
w(m/s) 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 -5 -0.5 -10
-15 -1.0 -20 -1.5 -25 y* y*
c) d)
Short Bearing, C = 0.01 in., ε = 0, Ta^0.5 = 42.70
7.5 End effect zone End effect zone
5
2.5
unaffecte d zone 0 P(E+3 N/m^2) P(E+3 0 0.2 0.4 0.6 0.8 1
-2.5
-5 z* e)
Figure 5.7 Velocity and Pressure Profiles as Functions of y* and z* during Taylor Vortex Regime (Point B in Figure 5.3) 172 Figure 5.6 presents the velocity and pressure contours of a typical point (point B in
Figure 5.3) in the Taylor vortex regime. Figure 5.7 presents the velocity and pressure profiles corresponding to Figures 5.6e through 5.6h. In this regime, the local velocities ut, v t and wt are non-linear, and their profiles differ at different z-axial position, Figures
5.7a, 5.7b, and 5.7c. The velocities in this regime can be expressed as
=−+=−+( ) ∆( ) ( π) + ( π ) ut 1 y*U u 1 y*U a(Re,z*)sin1c y* a(Re,z*)sin 2 c 2 y* U (5.2a)
=+=+∆ ( π) + ( π ) vt0 v 0 b1 (Re c ,z*)sin y* b 2 (Re c ,z*)sin 3 y* U (5.2b)
=+=+∆ ( π) + ( π ) wt0 w 0 c(Re,z*)sin1 c y* c(Re,z*)sin 2 c 2 y* U (5.2c)
where z* = z/l, and the clearance, C, based Reynolds number is Rec (= ρRωC/ ), for i = 1,
2 the coefficients a i(Re c, z*),, b i(Re c, z*), and c i(Re c, z*) vary with Re c and the axial position z* in the domain.
The ‘added velocity’ u represents the velocity difference between the curvilinear shaped u-velocity profile in the Taylor vortex regime and what would be its hypothetical value if there were a linear Couette profile at the same angular velocity; similarly v and
w are the velocity differences between the non-linear shaped v- and w- velocity profiles, in the Taylor vortex regime and what would be their hypothetical values (0) in the
Couette regime. Like the one for long bearing, the ‘added velocities’ for short bearing represent the effect of the onset of the Taylor vortices. Graphically u, v and w are
shown in Figures 5.7d, 5.7b and 5.7c respectively (Note that v and v have the same
magnitude in this regime, and so do w and w).
173 Like the one for long bearing, pressure gradient in Taylor vortex regime for short bearing exists in both the y-radial and z-axial directions, Figure 5.6h. The radial pressure gradient balances the centrifugal force, and the axial pressure gradient accounts for the formation of the Taylor vortices. Its profile of the mid-plane (y* = 1/2) is shown Figure
5.7e, which indicates that ∂P /∂z is not zero and is periodic in the axial direction in the
“unaffected zone” for the concentric case ( ε = 0).
Like those for long bearings, the velocity expressions for short bearings in wavy vortex
regime have the same composition as those in Taylor vortex regime, except that the
coefficients have different values. Thus one can generalize and apply the same equation
for these two regimes (Taylor and wavy regimes) as long as the appropriate different
constants are introduced,
=−+( ) (π) + ( π ) ut or uw1 y*U a(Re,z*)sin1 c y* a(Re,z*)sin 2 c 2 y* U (5.3a)
= (π) + ( π ) vt or vw b1 (Re c ,z*)sin y* b 2 (Re c ,z*)sin3 y* U (5.3b)
= (π) + ( π ) wt or ww c1 (Re c ,z*)sin y* c 2 (Re c ,z*)sin2 y* U (5.3c)
where ai, b i and c i have local values [(Re c, z*)] for the Taylor vortex regime, while for the
wavy vortex regime a i(Re c, z*) = ai (Re c, z*), b i(Re c, z*) = bi (Re c, z*) and c i(Re c, z*) =
ci (Re c, z*).
5.4 The Case with ε = 0.2
One effect of the eccentricity ratio on the flow stability for short bearings manifests through the addition of an axial pressure gradient over the flow of the concentric case. 174 This axial pressure gradient will cause an axial flow, thus stabilizing the flow in the circumferential direction. Chandrasekhar (1960, 1961a) used a perturbation method to predict the onset of instability in a narrow gap and found a relationship between the critical Taylor number Ta c and the axial Reynolds number Re a, which is
= + 2 Ta c (Re a ) Ta c )0( 26 5. Re a . DiPrima (1960) and Krueger and DiPrima (1964) also found that the critical Taylor number increases as axial Reynolds number increases using
Galerkin method. Our calculations indicate that axial pressure gradient caused by eccentricity ratio stabilizes the flow. The critical Taylor numbers for the onset of Taylor vortex instability are 39.14 and 41.27 for the cases of ε = 0 and ε = 0.2 respectively,
Figures 5.3 and 5.8.
Tout (N.m) 4.0
Couette 3.0 Taylor Vortex Wavy Vortex
2.0 C D 2nd Critical Point 1.0 1st Critical Point 0.0 0 20 40 60Ta 80
Figure 5.8 Torque vs. Ta for Short Bearing When C = 0.01 in., ε = 0.2
Like Figure 5.3 for the case of ε = 0, Figure 5.8 shows that Couette, Taylor and wavy
vortex regimes are defined by the appearance of two inflection points in the torque-speed
graph for the case of ε = 0.2. The Couette regime is represented by a straight line, but the regimes of Taylor vortex and wavy vortex are represented by curved lines other than straight lines, Figure 5.8. 175
Detail shown in e) Detail shown in f) Detail shown in g) Detail shown in h)
a) b) c) d)
y y y y
z = 0 z = 0 z = 0 z = 0
z z z z
z = l z = l z = l z = l
e) f) g) h)
Figure 5.9 Velocity and Pressure Contours during Couette Regime (Point C in Figure 5.8) 176
Short Bearing, C = 0.01 in., ε = 0.2, Ta^0.5 = 28.46 Short Bearing, C = 0.01 in., ε = 0.2, Ta^0.5 = 28.46
120 1.0 z* = 0 (1) z* = 1/8 100 z* = 1/4 z* = 3/8 z* = 1/2 0.5 z* = 5/8 z* = 3/4 80 z* = 7/8 Average
60 0.0 u(m/s) z* = 0 (1) (m/s) v 0.0 0.2 0.4 0.6 0.8 1.0 z* = 1/8 40 z* = 1/4 z* = 3/8 z* = 1/2 z* = 5/8 -0.5 20 z* = 3/4 z* = 7/8 Average 0 -1.0 0.0 0.2 0.4 y* 0.6 0.8 1.0 y*
a) b)
Short Bearing, C = 0.01 in., ε = 0.2, Ta^0.5 = 28.46 Short Bearing, C = 0.01 in., ε = 0.2, Ta^0.5 = 28.46 0.6 z* = 0 (1) 50 z* = 1/8 z* = 1/4 z* = 3/8 z* = 1/2 40 z* = 5/8 0.4 z* = 3/4 z* = 7/8 30 w(m/s) 20 0.2 N/m^2) P(E+3 10
0.0 0 0.0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1 y* z*
c) d)
Figure 5.10 Velocity and Pressure Profiles as Functions of y* and z* during Couette Regime (Point C in Figure 5.8)
177
Detail shown in e) Detail shown in f) Detail shown in g) Detail shown in h)
a) b) c) d)
y y y y
z = 0 z = 0 z = 0 z = 0
z z z z
z = l z = l z = l z = l
e) f) g) h)
Figure 5.11 Velocity and Pressure Contours during Taylor Vortex Regime (Point D in Figure 5.8) 178 Short Bearing, C = 0.01 in., ε = 0.2, Ta^0.5 = 42.70 Short Bearing, C = 0.01 in., ε = 0.2, Ta^0.5 = 42.70
2.0 z* = 0 (1) 160 z* = 1/8 z* = 1/4 z* = 3/8 140 1.5 z* = 1/2 z* = 5/8 z* = 3/4 120 1.0 z* = 7/8 Average 100 0.5
80 0.0 u(m/s) z* = 0 (1) (m/s) v 0.0 0.2 0.4 0.6 0.8 1.0 60 z* = 1/8 -0.5 z* = 1/4 z* = 3/8 40 z* = 1/2 z* = 5/8 -1.0 z* = 3/4 20 z* = 7/8 Average -1.5 0 -2.0 0.0 0.2 0.4y* 0.6 0.8 1.0 y*
a) b)
Short Bearing, C = 0.01 in., ε = 0.2, Ta^0.5 = 42.70 Short Bearing, C = 0.01 in., ε = 0.2, Ta^0.5 = 42.70
3.0 z* = 0 (1) 40 z* = 1/8 z* = 0 (1) z* = 1/4 z* = 1/8 z* = 3/8 z* = 1/4 z* = 1/2 30 z* = 3/8 z* = 5/8 z* = 1/2 2.0 z* = 3/4 z* = 5/8 z* = 7/8 20 z* = 3/4 Average z* = 7/8 Average 10 1.0 0 u(m/s)
w(m/s) 0.0 0.2 0.4 0.6 0.8 1.0 0.0 -10 0.0 0.2 0.4 0.6 0.8 1.0 -20 -1.0 -30
-2.0 -40 y* y*
c) d)
Short Bearing, C = 0.01 in., ε = 0.2, Ta^0.5 = 42.70 Short Bearing, C = 0.01 in., ε = 0.2, Ta^0.5 = 42.70 2.0 z* = 0 (1) z* = 1/8 50 z* = 1/4 z* = 3/8 z* = 1/2 z* = 5/8 z* = 3/4 40 1.0 z* = 7/8 Average
30
0.0 w (m/s) 0.0 0.2 0.4 0.6 0.8 1.0
20 P(E+3 N/m^2) P(E+3
-1.0 10
0 -2.0 0 0.2 0.4 0.6 0.8 1 y* z* e) f)
Figure 5.12 Velocity and Pressure Profiles as Functions of y* and z* during Taylor Vortex Regime (Point D in Figure 5.8) 179 Figure 5.9 presents the velocity and pressure contours of a typical point (point C in
Figure 5.8) in the Couette regime for the case of ε = 0.2. Figure 5.10 presents the velocity and pressure profiles corresponding to Figures 5.9e through 5.9h. For a short bearing in this regime, the local velocity uc (in the circumferential direction) is linear, and the local velocities vc (in the radial direction) is 0, but wc (in the axial direction) is
parabolic due to the axial pressure gradient, Figures 5.10a, 5.10b, and 5.10c. Thus in the
coordinate system shown in Figure 5.2 one can write
=( − ) uc 1 y*U (5.4a)
= vc 0 (5.4b)
1 ∂P w = (y 2 − yh ) (5.4c) c 2µ ∂z
where y* = y/h and U (= R ω) is the velocity of the inner cylinder. The axial pressure
profile of the mid-plane (y* = 1/2) is shown Figure 5.10d, which indicates that ∂P /∂z is
parabolic in the “unaffected zone” for the eccentric case ( ε = 0.2).
Figure 5.11 presents the velocity and pressure contours of a typical point (point D in
Figure 5.8) in the Taylor vortex regime for the case of ε = 0.2. Figure 5.12 presents the
velocity and pressure profiles corresponding to Figures 5.11e through 5.11h. In this
regime, the local velocities ut, v t and wt are non-linear, and their profiles differ at different
z-axial position, Figures 5.12a, 5.12b, and 5.12c. Note that another effect of the
eccentricity ratio on the flow stability is that the local Reynolds number Re h (= ρRωh/ )
varies circumferentially directly proportional with h. In this context, the eccentricity ratio
becomes a cause of instability through the change in vortex intensity due to the increase
180 in the local Reynolds number. Thus, the velocities for the eccentricities ε ≤ 0.2 can be
expressed as
y u or w = 1− U + ∆u = t w h
π π − y + () y + () 2 y 1 U a1 Re h , z sin a2 Re h , z sin U (5.5a) h h h
π π = + ∆ = + () y + () 3 y vt or vw 0 v 0 b1 Re h , z sin b2 Re h , z sin U (5.5b) h h
1 ∂P w or w = (y 2 − yh )+ ∆w = t w 2µ ∂z
∂ π π 1 P ()2 − + () y + () 2 y y yh c1 Re h , z sin c2 Re h , z sin U (5.5c) 2µ ∂z h h
where a i(Re h, z), b i(Re h, z), and c i(Re h, z) are coefficients depending on the local
Reynolds number, Re h, and the axial positions.
The ‘added velocity’ u represents the velocity difference between the curvilinear shaped u-velocity profile in the Taylor or wavy vortex regime and what would be its hypothetical value if there were a linear Couette profile at the same angular velocity; v is the velocity difference between the non-linear shaped v- velocity profile in the Taylor
or wavy vortex regime and what would be its hypothetical value (0) in the Couette
regime; w is the velocity difference between the non-linear shaped w- velocity profile in
the Taylor or wavy vortex regime and what would be its parabolic velocity profile in the
Couette regime. By comparison with Equation (5.3), Equation (5.5) has the axial
pressure gradient term due to the eccentricity and the coefficients in Equation (5.5)
depend on the local Reynolds number Re h rather than the mean Reynolds number Re c. 181 The axial pressure profile of the mid-plane (y* = 1/2) is shown Figure 5.12d, which indicates that ∂P /∂z is the summation of the parabolic shape due to eccentricity and periodic shape due to the onset of Taylor instability.
5.5 Proposed Transition Reynolds Equation Model for Short Bearing
The transition Reynolds equation derivation starts with the integration of the classical continuity equation across the film thickness:
h ∂(ρv) h ∂(ρu) h ∂(ρw) ∫ dy = −∫ dy − ∫ dy (5.6) 0 ∂y 0 ∂x 0 ∂z
By replacing u and w with their respective expressions from Equation (5.5) and
performing the integration for the boundary conditions v = 0 at y = 0 and y = h, we obtain
the initial form of the transition Reynolds equation for short bearing as
∂ ρ 3 ∂ ∂ ∂ ρ ∂ ρ h P a1 (Re h , z) h c1 (Re h , z) h = 6 (ρUh ) + 24 U + (5.7) ∂z µ ∂z ∂x ∂x π ∂z π
Calculations show that Equation (5.7) is valid for a short bearing with low
eccentricities ( ε ≤ 0.4), and its corresponding velocity profiles are expressed in Equation
(5.5).
This formulation of the Reynolds equation is applicable in the transition regime, i.e. after the onset of Taylor vortices ( Ta > 41.3) and before the full development of
turbulence (Re < 2,000). For comparison, the laminar Reynolds Equation (5.8), which is
applicable before the onset of Taylor vortices, and turbulent Reynolds equation (5.9),
which is good after the full development of turbulence (Re < 2,000), are listed below.
182 ∂ ρh 3 ∂P ∂ = 6 (ρUh ) (5.8) ∂z µ ∂z ∂x
∂ ρh3 ∂P ∂ = 6 (ρUh ) (5.9) ∂ µ ∂ ∂ z k z (Re h ) z x
Like the one for long bearing, in the transition Reynolds equation (5.7) for short
bearing the viscosity keeps its laminar value (due to the nature of a fluid); but it adds the
terms a i(Re h, z) and c i(Re h, z) to represent the effect of Taylor, or wavy vortices. Like the empirical coefficient k z(Re h) in Equation (5.9), the coefficients a 1(Re h, z) and c 1(Re h, z) in Equation (5.7) can be obtained numerically and verified experimentally.
5.6 Comparison between Our Model and Turbulence Models for Short Bearings
To get the overall average effect of ut, v t and wt in the Taylor vortex regime, one can
integrate Equation (5.5) over the axial domain z∈(0, l), and obtain
l π = 1 = − y + ∆ = − y + 2 y U t ∫ ut dz 1 U U 1 U A(Re h )sin( )U (5.10a) l 0 h h h
= 1 l = + ∆ = + Vt ∫ vt dz 0 V 0 0 (5.10b) l 0
l ∂ ∂ = 1 = 1 P ( 2 − )+ ∆ = 1 P ( 2 − )+ Wt ∫ wt dz y yh W y yh 0 (5.10c) l 0 2µ ∂x 2µ ∂x
π l π π 2 y = U () y + () 2 y Note that in Eqn (10a) A(Re h )sin( )U ∫ a1 Re h , z sin a2 Re h , z sin dz h l 0 h h where A(Re h) is a coefficient depending on the local Reynolds number (Re h), Ut, Vt and
Wt are the average velocities in the circumferential, radial and axial directions respectively. They are represented by the continuous lines labeled “Average” in the 183 legends of Figures 5.12a, 5.12b and 5.12c respectively. U = A(Re h)sin(2πy/h )] U is the
average ‘added velocity’ in the circumferential direction due to the onset of the Taylor
instability, and is represented by the continuous line labeled “Average” in the legend of
Figure 5.12d. V and W are the average ‘added velocity’ in the radial and axial
directions, respectively. They are represented by the continuous lines labeled “Average”
in the legend of Figures 5.12b and 5.12e. It is shown that their values are zero
practically.
Short Bearing, C/R = 0.002, ε = 0.2 Short Bearing, C/R = 0.004, ε = 0.2 2.25 2.00 Our Model Our Model Contantinescu Contantinescu 2.00 Ng-Pan Ng-Pan Hirs 1.75 Hirs M* Gross et al. Gross et al. 1.75 M* 1.50 1.50
1.25 1.25
1.00 1.00 0 500 1000 1500 2000 2500 3000 3500 4000 0 500 1000 1500 2000 2500 3000 3500 Reynolds Number Reynolds Number
a) b)
Figure 5.13 Comparison of Torque on Bearing between Our Model and Turbulence Models for Short Bearing
Once the average ‘added velocity’ U is calculated, the dimensionless ‘added torque’ exerted on the outer cylinder, M*, due to the onset of instability can also be calculated.
The calculation of M* allows comparison with the results of the four previously introduced turbulent models. Figures 5.13a and 5.13b present the torque (M*) exerted on the outer stationary cylinder as it is calculated by our model for two cases: i) C = 0.002 184 in, (C/R = 0.002) and ε = 0.2, and ii) C = 0.004 in, (C/R = 0.004) and ε = 0.2. For
comparison purposes, these results are then superimposed over the results of the four
turbulence models of Constantinescu (4), Ng-Pan (5), Hirs (6), and Gross et al. (8).
For both of the cases, our curves collapse over those of Hirs’ and Gross et al.’s in the
region before the onset of Taylor vortices (Re ≈ 900 and 650 for the cases of C/R = 0.002 and 0.004 respectively). Our result approaches to those of Constantinescu and Hirs as the
Reynolds number reaches 4,000 and 3,500 for the cases of C/R = 0.002 and 0.004 respectively. Overall, for the case of C/R = 0.002, the results yielded by the proposed transition model are closest to those of Gross et al. when Reynolds number is below
2,500, and they are closest to those of Constantinescu when Reynolds number is higher than 2,500. For the case of C/R = 0.004, our results are way off from all the turbulence models during the transition regime, and they are closest to those of Constantinescu when
Reynolds number higher than 3,000, Figure 5.13b.
185
CHAPTER VI
DESCRIPTION OF EXPERIMENTAL INSTALLATION
6.1 Test Loop
Speed Drive Motor
Coupling Drive System
Test Section Shaft Plexiglass Box
Lens Visualization System Torque Measuring Mirror System Lens Digital Multimeter Lens
Camera Laser
BAM-1
Figure 6.1 Schematic Drawing of Test Loop
186
The test loop consists of four main sections: (1) the drive system, (2) the test section,
(3) the torque measuring system and (4) the visualization system, Figure 6.1. The detailed descriptions of each section of the test loop will be shown as follows.
6.2 Drive System
Tube Motor
Coupling
Bracket (Bearing)
Shaft Speed Drive
Air Trapper
Plug
Plexiglass Box Thermometer
Aluminum Housing (Bearing)
Figure 6.2 Drive System and Test Section
187 The drive system consists of three parts: the adjustable speed drive, the motor and the flexible coupling, Figure 6.2. The Baldor IDNM 3542 motor (0.75 H. P. and 6,000 max rpm) is controlled by the Saftronics CV10 AC drive. The motor shaft is connected to the test shaft by the Bellows flexible coupling.
6.3 Test Section (Apparatus)
The test section consists of the bracket (including a bearing inside), the shaft, the air trapper, the Plexiglass box, the plug, the aluminum housing (including a bearing inside) and the tube, Figure 6.2. For the better understanding of the function of each part in the test section, the schematic drawing of the test section is shown in Figure 6.3. The shaft, which is driven by the motor, rotates as the inner cylinder. The internal shape of the
Plexiglass box is cylindrical so it acts as the outer cylinder, while the external shape of the Plexiglass box is square so it can be parallelepiped with the square aluminum housing. The aluminum housing sits on the metal base and it could rotate freely through the thrust bearing, but its rotation is stopped by a stopper and a metal cantilever beam,
Figure 6.4. Therefore, the Plexiglass box acts as the stationary outer cylinder. The annulus between the two cylinders is filled with silicone oil. The air would get into the oil annulus when the speed of the inner cylinder is large, so air trapper is placed on the top of the Plexiglass box to keep the air out of the oil annulus. One end of the tube is connected to the oil annulus and the other end is connected to the air. The purpose of the tube is to dissipate the energy generated by the motor and to get rid of the air from the oil by the higher oil level of the tube.
188
Bearing Bracket
Air Trapper
Shaft
Particle Access Tube
Plug
Plexiglass Box
Bearing
Aluminum Housing
Thrust Bearing
Metal Base
Figure 6.3 Schematic Drawing of Test Section
189
a) BAM b) Digital Multimeter
Cantilever Beam
Plexiglass Box
Aluminum Strain Gage Housing Stopper
c) Beam with the Strain Gage
Figure 6.4 Torque Measuring System
190 The diameter of the inner cylinder is 1.740 ± 0.001 in., while the diameter of the outer cylinder is 2.000 ± 0.001 in., which gives the clearance of the annulus C = 0.13 in. The length of the annulus is 13.00 in. Therefore, the ratio of the inner cylinder and outer cylinders is η = 0.87 and the aspect ratio is Γ = 100, which makes the assumption of
infinitely long annulus valid.
6.4 Torque Measuring System
The torque measuring system consists of the bridge amplifier & meter or BAM, the digital multimeter, the cantilever beam, the strain gage and the stopper, Figure 6.4. The cantilever beam with the strain gage is attached to the aluminum housing on one end and it touches the stopper on the other end. The torque exerted on the Plexiglass box tends to rotate it due to the fluid friction when the shaft rotates. Due to the trend of rotation the cantilever beam deflects and changes the resistance of the strain gage, which is wired in half bridge configuration. The stain gage is connected to the BAM in a Wheatstone bridge, so the voltage observed on the scale of the BAM represents the voltage needed to balance the resistance change caused by the deflection of the cantilever beam. This voltage can be precisely read from the Hewlett Packard 3466A digital multimeter. The
DC voltage range is set to 2V and the precision is as accurate as 0.1 mV.
In order to get accurate measurements, the cantilever beam is grounded, thus reducing the electrical noise picked by the strain gages and the wires. For the most stable operation at high sensitivity, the BAM is maintained at a reasonably ambient temperature and is warmed-up for 30 minutes.
191 6.5 Visualization System
In this section, the creation of the laser sheet, the visualization of the test section and the tracer particles will be introduced in detail.
6.5.1 Creation of the Laser Sheet
The visualization system consists of the laser, the lenses, the mirror and the camera,
Figure 6.1. The Spectra-Physics Merlin laser provides an intensive light source for the
flow visualization. The lenses and mirror create a thin sheet of light, which is shone on
the fluid annulus. This thin sheet of light illuminates the particles in the fluid flowing
through it. The purpose of the lenses is to transform the beam into a sheet of light, and
they can also control the collimation, height and width of the light sheet. The mirror can
change the direction of the light sheet.
The images are obtained by a Pulnix TM-740 CCD-TV Camera, which offers a CCD array with a resolution of 756 (Horizontal) by 581 (Vertical) pixels. This camera is connected to a computer so the images can be videotaped.
6.5.2 Visualization of the Test Section
The test section is visualized in two ways:
(1) Visualization of the fluid in longitudinal cross section view;
(2) Visualization of the fluid in front view.
The visualization of the fluid in longitudinal cross section and front views is shown in
Figures 6.5 and 6.6. In order to avoid the end effect from the upper and lower
192 boundaries, the flow is visualized in the middle of the fluid annulus (6 in. form the upper boundary).
z = 6.0 in.
Laser Beam
Outer Cylinder
Inner Cylinder
Laser Beam
Figure 6.5 Visualization of the Fluid in Longitudinal Cross Section View
193
z = 6.0 in.
Laser Beam
Outer Cylinder Inner Cylinder
Laser Beam
Figure 6.6 Visualization of the Fluid in Front View
194 6.5.3 The Tracer Particles
The flow visualization experiments need seeding the fluids with particles small enough to accurately represent the flow and large enough to reflect the light necessary to be seen by the camera.
Initially magnesium oxide (MgO 2) flakes with an average dimension of 51
micrometers were tried, but the flow images were not good. Then polymer micro-spheres
with an average dimension of 91 micrometers were used and the results were very good.
Therefore, polymer micro-spheres with 91 micrometers were used as the tracer particles
for all the experiments.
6.6 Working Fluids
The working fluids for the experiments performed are three kinds of oil. The
properties of these oils are given in Tables 6.1, 6.2 and 6.3. The temperature of the lab
does not change much (T = 74 ± 1F˚) during the performance of the experiments.
Therefore, all the properties are given for that temperature.
Table 6.1 Properties of the Working Fluid (Oil I)
Property Symbol Value
Viscosity 0.006 kg/(m-s)
Density ρ 778 kg/m 3
195 Table 6.2 Properties of the Working Fluid (Oil II)
Property Symbol Value
Viscosity 0.038 kg/(m-s)
Density ρ 961 kg/m 3
Table 6.3 Properties of the Working Fluid (Oil III)
Property Symbol Value
Viscosity 0.099 kg/(m-s)
Density ρ 1048 kg/m 3
196
CHAPTER VII
CALIBRATIONS
7.1 Calibration of the Strain Gage
The simplest way by which the torque can be measured is on the basis of the change of
the strain in such a setup where the dependence of the force on the displacement is linear.
The principle of the deformation of a cantilever beam is used for the torque
measurements in the experiments, Figure 7.1.
Inner Cylinder
Outer Cylinder
Lt = 4.875” Cantilever Beam
Strain Gage
Stopper
a) b)
Figure 7.1 The Principle of the Torque Measurement 197
Strain Gage
Cantilever Beam
Weights
Figure 7.2 Calibration of the Strain Gage
The cantilever beam is considered as an elastic element, which has a property that the force exerted on its free end varies linearly with its displacement. The voltage recorded due to the imbalance of the strain gage, due to the bending of the cantilever beam, also varies linearly with the deformation of the gage. Thus the recorded voltage allows the calculation of the applied force once it has been correlated.
In our configuration, the cantilever beam is fixed to the outer cylinder while the free end is held by the pin, Figure 7.1. There is a trend for the outer cylinder to rotate due to the viscous force induced by the rotation of the inner cylinder. Due to the trend of rotation of the outer cylinder, the cantilever beam bends and changes the resistance of the strain gage. The strain gage bridge goes out of equilibrium and a voltage is generated. 198 During the calibration procedure the voltage is correlated to the force (created by the
application of weights) applied to the free end of the cantilever beam, Figure 7.2, thus the
calibration curve of voltage-torque is obtained.
The measurements are repeated four times in order to evaluate the repeatability and to allow the uncertainty and error analysis, Figure 7.3. It is shown in Figure 7.3 that the cantilever beam is loaded in such a way that the weight is increasing until 2 lb at the increment of 1/8 lb. Then the cantilever beam is unloaded to 0 lb at the same increment.
The purpose of loading-unloading is to check the hysteresis characteristics. The beam would be permanently bent and the force-voltage curve could not be reproduced if there was a hysteresis. This is not the case according to the measurements shown in Figure
7.3, which means that the cantilever beam has a good elastic property and the permanent bending is avoided. The raw data of the strain gage calibration are presented in the
Appendix A.
In order to obtain the correlation of the force and voltage, all trails are taken into consideration. By least-squares curve fitting, the relationship between the force F (lbf) and the voltage U (V) can be expressed as (Figure 7.4):
F = .7 8393 U + .0 0008 (7.1)
To evaluate how good this curve-fit is, the correlation coefficient r is calculated.
According to Holman (1994), the correlation coefficient r is defined by:
σ 2 2/1 r = 1− y,x (7.2) σ 2 y
where σy is the standard deviation of the forces, given as
199 n /1 2 − 2 ∑ ( y i y m ) σ = i =1 (7.3) y n − 1
and σy,x is the deviation related to the curve fitted values of the forces, given as
n /1 2 − 2 ∑ ( y i y ic ) σ = i=1 (7.4) y , x n − 2 where n is the number of observations for one measurement, yi is the measured value of
the force, yic is the value of the force computed from the correlation equation (7.1) for the same value of x (voltage), and ym is the arithmetic mean of the forces, given as
n = 1 y m ∑ y i (7.5) n i=1
Then the correlation coefficients for all the four trials are calculated, Table 7.1. It is
shown that the correlation coefficients are very close to 1, which means that the Equation
(7.1) represents a very good curve fit.
Table 7.1 Correlation Coefficients of the Curve Fit for the Calibration of the Strain Gage
Trial I II III IV
Fm (lbf) 1.00 1.00 1.00 1.00
σy (lbf) 0.6312 0.6312 0.6312 0.6312
σy,x (lbf) 0.00413 0.00504 0.00545 0.00276
r 0.99998 0.99997 0.99996 0.99999
200
Trial I 2.0
1.5
1.0
Force(lbf) Load
0.5 Unload
0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30
Voltage (V)
a)
Trial II
2.0
1.5
1.0
Force(lbf) Load 0.5 Unload
0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Voltage (V)
b)
Figure 7.3 Calibration Curve of Force-Voltage
201
Trial III 2.0
1.5
1.0
Force (lbf) Force Load 0.5 Unload
0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Voltage (V)
c)
Trial IV
2.0
1.5
1.0
Force(lbf) Load 0.5 Unload
0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Voltage (V)
d)
Figure 7.3 Calibration Curve of Force-Voltage (Continued)
202
2.0
y = 7.8393x + 0.0008 R2 = 1 1.5
Trial I - Load 1.0 Trial I - Unload Trial II - Load (lbf) Force Trial II - Unload Trial III - Load 0.5 Trial III - Unload Trial IV - Load Trial IV - Unload 0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Voltage (V)
Figure 7.4 Curve-fit for the Calibration of the Strain Gage
7.2 Torque Measurements for the Case of Air between the Cylinders
The torque exerted on the outer cylinder consists of two parts: (1) the torque due to the
fluid friction and (2) the torque due to the friction in the bearings and seal, Figure 6.3.
The torque due to the friction in the bearings and seal always exists and it is necessary to
be isolated in order to get the torque due to the fluid friction only. This is the major
reason why the torque measurements for the case of air between the cylinders are made.
The torque measurements for the case of air between the cylinders are presented in
Figure 7.5. It is shown that the torque-speed graph behaviors linearly in a piecewise way.
That is, it behaviors linearly in the speed ranges of 0 - 200 rpm, 250 – 400 rpm, 500 –
700 rpm, and 800 – 1,300 rpm. The torque drops at the speed of 250, 500, and 800 rpm, respectively. The torque drop coincides with the flow pattern change of the oil in the 203 aluminum housing. The aluminum housing is filled with oil I [ = 0.006 kg/(m-s), ρ =
778 kg/m 3] so that the bearing is submerged, thus reducing the friction in the bearing,
Figure 6.3. The raw data of the voltage measurements for the case of air between the
cylinders are presented in Appendix B.
The calibration shows that the amount of the torque produced by the friction in the bearings and seal is hard to be isolated from the torque due to the fluid friction. The reason is that the torque measurements done for the case of air between cylinders show the transition points. These transition points are due to the flow pattern changes of the oil in the aluminum housing, but the transition points are different when the space between cylinders is filled with oil. Thus the torque measured for the case of oil between cylinders will simply be offset until the torque-speed curve reaches the origin.
Torque (N.m) 0.40
0.35
0.30 Trial I Trial II Trial III 0.25 Trial IV
0.20 0 500 1000 1500 Angular Speed (rpm)
Figure 7.5 Torque Measurements for the Case of Air between the Cylinders
204
CHAPTER VIII
EXPERIMENTAL PROCEDURE
8.1 Experimental Start-up
The experimental start-up can be divided into two phases: (i) Preparation of the experimental apparatus and (ii) Preparation of the experiment.
8.1.1 Preparation of the Experimental Apparatus
Preparation of the experimental apparatus includes alignment of the motor shaft with the experimental shaft (inner cylinder), calibration of the strain gage, torque measurements for the case of air between the two cylinders, and preparation of the working fluids [Oil I: = 0.006 kg/(m-s), ρ = 778 kg/m 3; Oil II: = 0.038 kg/(m-s), ρ =
961 kg/m 3; Oil III: = 0.099 kg/(m-s), ρ = 1048 kg/m 3]
The alignment of the motor shaft with the inner cylinder, calibration of the strain gage
and torque measurements for the case of air between the two cylinders can be done days
before the experiments for the case of oil between the two cylinders. No major problems
encountered during the performance of those preparations. The two major problems
encountered during the preparation of the silicone oil are: (i) to separate the air from the
oil and (ii) to mix the oil and the visualization particles so that the best images will be
presented. 205 Preliminary experiments show that the air was sucked into the oil when the speed of the inner cylinder is larger than 1,500 rpm. The mixing of the air and the oil creates two- phase fluid which has different properties from those of the initially used oil. The torque measurements dropped significantly corresponding to the mixture of the air with the oil during the preliminary experiments. The air trapper with oil and the tube with high level oil are used to drive the air out of the oil, Figure 6.3. Results show that these measures work well during the experiments.
Since the polymer micro-spheres with an average dimension of 91 micrometers mixed in the fluids gave the best images, they were used as the tracer particles for all the experiments. Usually the particles will stay homogeneously in the fluids for several hours after they are mixed. Therefore, they are mixed on the same day when the experiments will be done.
8.1.2 Preparation of the Experiment
The typical procedure of the experimental start-up can be summarized as following:
1) Turn on all the instrumentation: BAM, Digital Multimeter, Camera and Laser. The
BAM needs 30 minutes to warm up, while the Laser needs a few minutes to release the beam of light.
2) Prepare the visualization system. The first step is to check the positions of lenses and mirror so that a sheet of light is produced and shined on the test section. The second step is to adjust the position of the Camera so that a sharp image will be shown on the computer.
206 3) Injected the particle-mixed oil in the space between the two cylinders, and then run the inner cylinder with a low speed so that the particles are mixed homogeneously.
4) Balance the BAM. Set the reading on the Multimeter to zero by adjusting the AMP balance and bridge balance on the BAM.
5) Start the Motor.
8.2 Experimental Procedure
Experimental procedure can be divided into two steps: (i) Quick Pass and (ii)
Individual Regime Study.
8.2.1 Quick Pass
A quick pass is used to identify roughly the ranges of the three different flow regimes, i.e. Couette, Taylor and wavy vortex regimes. The first critical speed (the onset of the
Taylor vortices) and the second critical speed (the onset of wavy vortices) were calculated numerically and analytically. These two critical speeds are now verified experimentally. The experiments are performed at the speed increment of 50 rpm for the whole range up to 1,300 rpm for Oil I [ = 0.006 kg/(m-s), ρ = 778 kg/m 3], at the speed
increment of 100 rpm for the whole range up to 2,600 rpm for Oil II [ = 0.038 kg/(m-s),
ρ = 961 kg/m 3], and at the speed increment of 200 rpm for the whole range up to 3,600 rpm for Oil III [ = 0.099 kg/(m-s), ρ = 1048 kg/m 3].
Once the ranges of the flow patterns are roughly identified, individual regime study
will be performed.
207 8.2.2 Individual Regime Study
For the case of individual regime study, the speed of the inner cylinder is increased the
same way for the case of the quick pass. This was done in order to get the same
conditions in all experiments since each flow regime should obtain a steady state that can
be examined.
For the case of Oil I, the speed of the inner cylinder is increased at the step of 50 rpm until it reaches 150 rpm. The speed increment is changed to 10 rpm when the speed is larger than 150 rpm and less than 200 rpm. The speed range between 150 rpm and 200 rpm covers the transition regimes from Couette to Taylor vortex and from Taylor to wavy vortex flow. The speed increment is resumed to 50 rpm beyond 200 rpm.
For the case of Oil II, the speed of the inner cylinder is increased at the step of 100 rpm until it reaches 400 rpm. The speed increment is changed to 25 rpm when the speed is larger than 400 rpm and less than 700 rpm. The speed range between 400 rpm and 700 rpm covers the transition regimes from Couette to Taylor vortex and from Taylor to wavy vortex flow. The speed increment is resumed to 100 rpm beyond 700 rpm.
For the case of Oil III, the speed of the inner cylinder is increased at the step of 200 rpm until it reaches 600 rpm. The speed increment is changed to 50 rpm when the speed is larger than 600 rpm and less than 1,300 rpm. The speed range between 600 rpm and
1,300 rpm covers the transition regimes from Couette to Taylor vortex and from Taylor to wavy vortex flow. The speed increment is resumed to 200 rpm beyond 1,300 rpm.
208
CHAPTER IX
EXPERIMENTAL RESULTS
9.1 Results for the Case of Oil I [ = 0.006 kg/(m-s), ρ = 778 kg/m 3]
This section will present the results of the torque measurements and the visualization
of flow patterns at different regimes for the case of Oil I.
9.1.1 Torque Measurements for the Case of Oil I
For the case of oil I, the voltages measured on the BAM for each speed can be plotted in Figure 9.1. The measurements are repeated four times in order to evaluate the repeatability and to allow uncertainty and error analysis. The raw data of the voltage measurements for this case are presented in the Appendix C.
Since the voltages generated by the friction in the bearings and seal were hard to be
separated from the torque due to fluid friction, the voltages measured were offset until the
voltage-speed curves reach the origin, Figure 9.2.
Figure 9.2 and its corresponding flow pattern visualizations, which will be presented in
next section, show that the flow patterns can be divided into five types, i.e. Couette (0 –
150 rpm), Taylor (150 – 200 rpm), Pre-wavy (200 – 750 rpm), Wavy (750 – 1,500 rpm)
and Turbulent (> 1,500 rpm) flows. Pre-wavy flow is defined here because it is different
than both the Taylor and wavy vortex flows as its flow pattern will be shown later. The 209 points representing the Couette, Pre-wavy, Wavy and Turbulent flows can be curve-fitted by four straight lines. The range of Taylor flow is short and the corresponding points connect the lines representing Couette and Pre-wavy flows.
By least-square curve fitting, the relationship between the voltage U (V) and the speed
ω (rpm) for each regime can be expressed as (Figure 9.2):
U = .0 000153 ω − .0 002961 (Couette) (9.1)
U = .0 000177 ω − .0 017409 (Pre-wavy) (9.2)
U = .0 000230 ω − .0 070351 (Wavy) (9.3)
U = .0 000168 ω − .0 119115 (Turbulence) (9.4)
To evaluate how good these curve-fits are, the coefficient r is calculated for each
regime by using Equations (7.2), (7.3), (7.4) and (7.5). The results are listed in Tables
9.1 – 9.4. It is shown that the correlation coefficients range from 0.91 to 0.995 except
those for turbulent flow, which means that Equations (9.1) - (9.3) represent good curve
fits for the Couette, Pre-wavy and Wavy regimes. The reason that the curve fit for the
turbulent regime is not so good may be the large vibrations occurred during this regime.
Table 9.1 Correlation Coefficients of the Curve Fit for Couette Flow (Oil I)
Trial I II III IV
Um (V) 0.01213 0.01388 0.01438 0.01438
σy (V) 0.00742 0.00950 0.00711 0.00685
σy,x (V) 0.00253 0.00375 0.00211 0.00243
r 0.94026 0.91894 0.95490 0.93514
210 Table 9.2 Correlation Coefficients of the Curve Fit for Pre-wavy Flow (Oil I)
Trial I II III IV
Um (V) 0.06157 0.05179 0.05936 0.05686
σy (V) 0.03271 0.03589 0.03294 0.03218
σy,x (V) 0.00594 0.00949 0.00536 0.00447
r 0.98339 0.96443 0.98668 0.99031
Table 9.3 Correlation Coefficients of the Curve Fit for Wavy Flow (Oil I)
Trial I II III IV
Um (V) 0.18371 0.18529 0.18536 0.18936
σy (V) 0.05574 0.06165 0.05103 0.05410
σy,x (V) 0.00579 0.00954 0.00676 0.00545
r 0.99459 0.98796 0.99119 0.99491
Table 9.4 Correlation Coefficients of the Curve Fit for Turbulent Flow (Oil I)
Trial I II III IV
Um (V) 0.20550 0.20983 0.19050 0.19783
σy (V) 0.03192 0.03591 0.03404 0.03600
σy,x (V) 0.01354 0.02350 0.01869 0.00519
r 0.90553 0.75618 0.83592 0.98957
211 Voltage (V) 0.8 Trial I Trial II 0.7 Trial III Trial IV
0.6
0.5
0.4
0.3 0 500 1000 1500 2000 2500
Speed (rpm)
Figure 9.1 Voltage Measurements for the Case of Oil I
Wavy Turbulence Voltage (V)
0.30 y = 0.000230x - 0.070351 R2 = 0.998078 0.25
0.20
y = 0.000153x - 0.002961 y = 0.000168x - 0.119115 0.15 R2 = 0.980895 R2 = 0.968270
0.10 Trial I Trial II 0.05 y = 0.000177x - 0.017409 Trial III R2 = 0.992251 Trial IV Couette 0.00 Pre -wavy 0 500 1000 1500 2000 2500 Taylor Speed (rpm)
Figure 9.2 Voltage due to Fluid Friction vs. Speed for the Case of Oil I
212 9.1.2 Visualization of Flow Patterns for the Case of Oil I
As explained in Chapter 6, the fluid was visualized in the longitudinal cross section view and front view, Figures 6.5 and 6.6. Figures 9.3 – 9.8 visualize the flow patterns in the longitudinal cross section view for each regime corresponding to Figure 9.2.
0.15 in.
Inner Cylinder Outer Cylinder
0.15 in.
ω = 100 rpm
Figure 9.3 Flow Pattern in Longitudinal Cross Section View in Couette Regime (Oil I)
Figure 9.3 presents the flow pattern in longitudinal cross section view in Couette regime ( ω = 100 rpm). Here, as in all subsequent figures in longitudinal cross section views, the rotating inner cylinder boundary in on the left, while the stationary outer cylinder is on the right. In all the figures in both longitudinal cross section view and front view, the dimension of the grid in both horizontal and vertical directions is 0.15 in. It is shown in Figure 9.3 that some of the particles are accumulated near the inner cylinder, 213 which is due to the difference between the centrifugal forces of the oil and the particles
(The density of the particles is smaller than the one of the oil). All the particles in other area are homogeneous, which means that there is no pressure variation in the axial direction and the velocities in both radial and axial directions are zero. The zero pressure gradients in the axial direction and zero velocities in radial and axial directions characterize the Couette flow.
a) ω = 180 rpm b) ω = 190 rpm c) ω = 200 rpm
Figure 9.4 Flow Pattern in Longitudinal Cross Section View in Taylor Regime (Oil I)
Figure 9.4 presents the flow pattern in longitudinal cross section view in Taylor regime
(ω = 180, 190 and 200 rpm). Taylor vortex cells start to form when the speed reaches
180 rpm. The boundaries of the Taylor cells become more and more discernible with the
increase of speed, and they are very clearly defined when the speed reaches 200 rpm. 214 The formation of the Taylor cells indicates that the velocities in both radial and axial directions are not zero. It means that the pressure gradient in the axial direction is not zero, either.
a) t = 0 s b) t = 6 s c) t = 11s d) t = 17s
e) t = 24 s f) t = 32 s g) t = 40s
Figure 9.5 The Collapse of Taylor Vortices for the Case of ω = 200 rpm (Oil I)
215 The speed of ω = 200 rpm actually is the critical speed for the disappearance of Taylor vortices and the appearance of Pre-wavy flow. Figure 9.5 presents the process of the collapse of the Taylor vortices with the increase of time. At the beginning of this process the Taylor vortices are perfect and clearly defined, Figure 9.5a. However, the shear force between the vortices is large enough to deform them with the increase of time, Figure
9.5b. Then the vortices are distorted by the shear force, Figures 9.5c and 9.5d. With further increase of time, the particles are able to enter the center areas of the vortices,
Figures 9.5e and 9.5f. Finally the vortices disappeared and they are replaced with Pre- wavy flow, Figure 9.5g. The whole process lasts for about 40 seconds.
Particle-accumulated area
a) ω = 300 rpm b) ω = 500 rpm c) ω = 700 rpm
Figure 9.6 Flow Pattern in Longitudinal Cross Section View in Pre-wavy Regime (Oil I)
216 With further increase of speed, the Taylor vortex flow is replaced by Pre-wavy flow.
Figure 9.6 presents the flow pattern in longitudinal cross section view in Pre-wavy regime ( ω = 300, 500 and 700 rpm). Taylor vortex cells are disappeared but the wavy vortices are not formed yet in this regime. The pattern of Pre-wavy flow is also different from the one of turbulent flow because there is a particle-accumulated area in this regime,
Figure 9.6. Pre-wavy is defined because it is a new equilibrium flow, different from all the other flows (Couette, Taylor, wavy and turbulent flows), and it occurs before the appearance of the wavy flow. In this regime, the velocities in both radial and axial directions are not zero, either. It also means that the pressure gradient in the axial direction is not zero.
Figure 9.7 presents the flow pattern in longitudinal cross section view in Wavy regime
(ω = 750, 800, 900, 1,000, 1,100 and 1,200 rpm). The wavy vortices begin to form when
the speed reaches 750 rpm, Figure 9.7a. More and more particles are accumulated at the
centers of the wavy vortices with the increase of speed, Figures 9.7b, 9.7c and 9.7d.
Nearly all the particles are accumulated at the centers of the vortices as the speed reaches
1,100 rpm, Figure 9.7e. Then the particles begin to disperse with the further increase of
the speed, Figure 9.7f. In this regime, the shape of the wavy vortices is distorted, i.e. they
are not aligned with the vertical and horizontal directions. The size of the wavy vortex
alternates between large and small along the axial direction. From the observation of the
shape of the wavy vortices, it is obvious that in this regime the velocity and pressure vary
strongly in all the radial, axial and circumferential directions. The wavy flow will be
replaced by turbulent flow with further increase of speed.
217
a) ω = 750 rpm b) ω = 800 rpm c) ω = 900 rpm
d) ω = 1000 rpm e) ω = 1100 rpm f) ω = 1200 rpm
Figure 9.7 Flow Pattern in Longitudinal Cross Section View in Wavy Regime (Oil I)
218
a) ω = 1,700 rpm b) ω = 1,900 rpm c) ω = 2,100 rpm
Figure 9.8 Flow Pattern in Longitudinal Cross Section View in Turbulent Regime (Oil I)
ω = 100 rpm
Figure 9.9 Flow Pattern in Front View in Couette Regime (Oil I)
219
a) ω = 180 rpm
b) ω = 190 rpm
Figure 9.10 Flow Pattern in Front View in Taylor Regime (Oil I)
220
c) ω = 200 rpm
Figure 9.10 Flow Pattern in Front View in Taylor Regime (Oil I) (Continued)
Figure 9.8 presents the flow pattern in longitudinal cross section view in Turbulent regime ( ω = 1,700, 1,900 and 2,100 rpm). It is shown in Figure 9.8 that all the particles are homogeneous in the whole area like the case of Couette flow. The difference is that some of the particles accumulated near the inner cylinder in Couette regime and the particles are blurred due to the larger velocity in turbulent regime. The particles are more and more blurred with the increase of speed due to the large particle velocity in both the radial and axial directions.
Figures 9.9 – 9.14 visualize the flow patterns in the front view for each regime corresponding to Figure 9.2. Figure 9.9 presents the flow pattern in front view in Couette 221 regime ( ω = 100 rpm). It is shown that the velocity in the axial direction is zero and the flow is laminar in the circumferential direction, which characterizes the Couette flow.
Figure 9.10 presents the flow pattern in front view in Taylor regime ( ω = 180, 190 and
200 rpm). The boundaries of the Taylor cells, which are indicated by the accumulated
particles in horizontal lines, become more and more discernible with the increase of
speed. The boundaries are very clearly defined when the speed reaches 200 rpm. The
front view shows the nature of the flow pattern in Taylor vortex regime, axisymmetric
and time-independent. This also means that the velocities and pressure do not vary in the
circumferential direction.
ω = 300 rpm
Figure 9.11 Flow Pattern in Front View in Pre-wavy Regime (Oil I)
222
a) ω = 700 rpm
b) ω = 750 rpm
Figure 9.12 Flow Pattern in Front View in Wavy Regime (Oil I)
223
c) ω = 800 rpm
d) ω = 900 rpm
Figure 9.12 Flow Pattern in Front View in Wavy Regime (Oil I) (Continued)
224
e) ω = 1000 rpm
f) ω = 1100 rpm
Figure 9.12 Flow Pattern in Front View in Wavy Regime (Oil) (Continued)
225
g) ω = 1200 rpm
h) ω = 1300 rpm
Figure 9.12 Flow Pattern in Front View in Wavy Regime (Oil) (Continued)
226
a) t = 0s b) t = 2s
c) t = 2.5 s d) t = 3s
e) t = 4s
Figure 9.13 The Collapse of Waves for the Case of ω = 1,350 rpm (Oil I)
227
a) ω = 1700 rpm
b) ω = 2100 rpm
Figure 9.14 Flow Pattern in Front View in Turbulent Regime (Oil I)
228 Figure 9.11 presents the flow pattern in front view in Pre-wavy regime ( ω = 300 rpm).
The horizontal lines, which indicate the boundaries of Taylor cells, are disappeared in this regime. The flow pattern looks like homogenous, but the particles move in both circumferential and axial directions.
Figure 9.12 presents the flow pattern in front view in Wavy regime ( ω = 700, 750, 800,
900, 1,000, 1,100, 1,200 and 1,300 rpm). The waves in the circumferential direction
begin to form when the speed reaches 700 rpm, Figure 9.12a. Notice that the wavy
vortices did not form until the speed reached 750 rpm in the longitudinal cross section
view, Figure 9.7a. This was due to the fact that the experiments videotaping the front
views were done at a different time when those videotaping the longitudinal cross section
views were done. The critical speed may vary a little at different experiments due to the
different environment. Another reason is the multiplicity nature of Taylor type flow, i.e.
there is more than one stable steady flow under the same geometrical and dynamical
boundary conditions, depending on the history of the flow. With the increase of speed,
more and more particles are accumulated on the waves, Figures 9.12b and 9.12c. Nearly
all the particles are accumulated on the waves as the speed reaches 900 rpm, Figure
9.12d. Then the particles begin to disperse with the further increase of the speed, Figure
9.12e. Then the waves become non-continuous in the circumferential direction, which
means that the waves are also undulating in the radial direction, Figures 9.12f, 9.12g and
9.12h. All the waves move in the circumferential direction in Figure 9.12, which
indicates the nature of the flow pattern of Wavy vortex flow, non-axisymmetric and time-
dependent. This also means that the velocities and pressure do vary strongly in all the
229 radial, axial and circumferential directions. The wavy vortices will collapse with the further increase of speed.
The speed of ω = 1,350 rpm is the critical speed for the disappearance of waves in the circumferential direction. Figure 9.13 presents the process of the collapse of the waves with the increase of time. At the beginning of this process the waves are perfect and clearly defined, Figure 9.13a. However, the energy of the waves is large enough to deform them with the increase of time, Figure 9.13b. Then portions of the waves are disappeared due to the strong undulation, Figure 9.13c. With further increase of time, the particles are further dispersed and only traces of the waves can be observed, Figures
9.13d. Finally waves are totally disappeared, Figure 9.13e. The whole process lasts for about 4 seconds, which is much shorter than 40 seconds, the time it takes for the disappearance of the Taylor vortices, Figure 9.5.
Notice that the waves collapsed at the speed of ω = 1,350 rpm, but the turbulent regime will not occur until the speed is larger than 1,500 rpm according to voltage – speed graph,
Figure 9.2. Again this was due to the different experimental environment and the multiplicity nature of Taylor type flow.
Figure 9.14 presents the flow pattern in front view in Turbulent regime ( ω = 1,700 and
2,100 rpm). It is shown in Figure 9.14 that all the particles are homogeneous in the
whole area like the case of Couette flow. The difference is that the particles are blurred
due to the larger velocity in turbulent regime. The particles are more and more blurred
with the increase of speed due to the large particle velocity in both the circumferential
and axial directions.
230 9.2 Results for the Case of Oil II [ = 0.038 kg/(m-s), ρ = 961 kg/m 3]
This section will present the results of the torque measurements and the visualization of flow patterns at different regimes for the case of Oil II.
9.2.1 Torque Measurements for the Case of Oil II
For the case of oil II, the voltages measured on the BAM for each speed are plotted in
Figure 9.15. The measurements are repeated four times in order to evaluate the
repeatability and to allow uncertainty and error analysis. The raw data of the voltage
measurements for this case are presented in the Appendix D.
Voltage (V)
1.7
1.5
1.3
1.1
0.9 Trial I 0.7 Trial II Trial III 0.5 Trial IV 0.3 0 500 1000 1500 2000 2500 3000
Speed (rpm)
Figure 9.15 Voltage Measurements for the Case of Oil II
Like the case of Oil I, the voltages measured were offset until the voltage-speed curves reach the origin for the case of Oil II, Figure 9.16. 231 Figure 9.16 and its corresponding flow pattern visualizations, which will be presented
in next section, show that the flow patterns can be divided into four types, i.e. Couette (0
– 500 rpm), Taylor (500 – 650 rpm), Pre-wavy (650 – 1,700 rpm) and Wavy (> 1,700
rpm) flows. Notice the turbulent flow regime was not presented for the case of Oil II due
to the large vibrations and bad images. The points representing the Couette, Pre-wavy
and Wavy flows can be curve-fitted by three straight lines. The range of Taylor flow is
short and the corresponding points connect the lines representing Couette and Pre-wavy
flows.
Voltage (V) Pre -wavy Wavy
1.5
y = 0.000560x - 0.193675 R2 = 0.997327
1.0
Trial I y = 0.000336x - 0.020718 R2 = 0.994117 Trial II 0.5 Trial III
y = 0.000656x - 0.223833 Trial IV R2 = 0.998797
Couette 0.0 0 500 1000 1500 2000 2500 3000 Taylor Speed (rpm)
Figure 9.16 Voltage due to Fluid Friction vs. Speed for the Case of Oil II
By least-square curve fitting, the relationship between the voltage U (V) and the speed
ω (rpm) for each regime can be expressed as (Figure 9.16): 232 U = .0 000336 ω − .0 020718 (Couette) (9.5)
U = .0 000656 ω − .0 223833 (Pre-wavy) (9.6)
U = .0 000560 ω − .0 193675 (Wavy) (9.7)
Like the one for the case of Oil I, the coefficient r for the case of Oil II is also
calculated for each regime by using Equations (7.2), (7.3), (7.4) and (7.5). The results are
listed in Tables 9.5 – 9.7. It is shown that the correlation coefficients range from 0.96 to
0.999, which means that Equations (9.5) - (9.7) represent good curve fits.
Table 9.5 Correlation Coefficients of the Curve Fit for Couette Flow (Oil II)
Trial I II III IV
Um (V) 0.06960 0.07830 0.08110 0.07740
σy (V) 0.04598 0.05104 0.05002 0.04613
σy,x (V) 0.01235 0.00600 0.00681 0.00626
R 0.96323 0.99306 0.99070 0.99075
Table 9.6 Correlation Coefficients of the Curve Fit for Pre-wavy Flow (Oil II)
Trial I II III IV
Um (V) 0.45475 0.46058 0.46608 0.45550
σy (V) 0.24677 0.24184 0.23659 0.23710
σy,x (V) 0.01322 0.00969 0.01501 0.01051
R 0.99856 0.99920 0.99799 0.99902
233 Table 9.7 Correlation Coefficients of the Curve Fit for Wavy Flow (Oil II)
Trial I II III IV
Um (V) 1.0403 1.0383 1.0403 1.0322
σy (V) 0.17907 0.17706 0.17782 0.17532
σy,x (V) 0.01657 0.00997 0.00953 0.01432
R 0.99571 0.99841 0.99856 0.99666
9.2.2 Visualization of Flow Patterns for the Case of Oil II
Figures 9.17 – 9.21 visualize the flow patterns in the longitudinal cross section view for each regime for the case of Oil II corresponding to Figure 9.16.
ω = 300 rpm
Figure 9.17 Flow Pattern in Longitudinal Cross Section View in Couette Regime (Oil II)
234 Figure 9.17 presents the flow pattern in longitudinal cross section view in Couette
regime ( ω = 300 rpm). Like the one for the case of Oil I, it is shown in Figure 9.17 that
for the case of Oil II, some of the particles are accumulated near the inner cylinder and all
the particles in other area are homogeneous, which indicates that there is no pressure in
the axial direction and the velocities in both radial and axial directions are zero.
Figure 9.18 presents the flow pattern in longitudinal cross section view in Taylor
regime ( ω = 400, 450, 500, 550, 600 and 650 rpm). Note that the points of ω = 400, 450 and 500 rpm belong to Couette regime in the voltage – speed graph, Figure 9.16, i.e. there is a discrepancy between the first appearance of Taylor vortices ( ω = 400) and the
critical speed ( ω = 500) based on the torque measurements. This discrepancy is due to
the fact that the mean shear stress in the flow between rotating cylinders, when averaged
over the length of the cylinders, will be influenced by vortex disturbances only through
quadratic terms in the disturbance velocities, DiPrima (1964). Therefore, the influence of
vortices on the torque is a second-order effect, while the visual appearance of vortices in
the flow is a first-order effect, Vohr (1968).
The larger viscosity of Oil II makes the process of Taylor vortex formation clearer than the one for the case of Oil I. First, two cells with small area initiate near the inner cylinder, Figure 9.18a. With the increase of speed, the boundaries of the cells become more and more discernible, and their areas are also extending toward the outer cylinder,
Figure 9.18b. Then the cells fully occupy the space between the inner and outer cylinders, Figures 9.18c and 9.18d. Eventually the boundaries of Taylor cells are clearly defined by the particles, which mean that they are fully formed, Figures 9.18e and 9.18f.
235
a) ω = 400 rpm b) ω = 450 rpm c) ω = 500 rpm
d) ω = 550 rpm e) ω = 600 rpm f) ω = 650 rpm
Figure 9.18 Flow Pattern in Longitudinal Cross Section View in Taylor Regime (Oil II)
236
a) t = 0 s b) t = 2 s c) t = 3 s
d) t = 4 s e) t = 5 s f) t = 7 s
Figure 9.19 The Collapse of Taylor Vortices for the Case of ω = 650 rpm (Oil II)
237 For the case of Oil II, the speed of ω = 650 rpm is the critical speed for the
disappearance of Taylor vortices and the appearance of Pre-wavy flow. Figure 9.19
presents the process of the collapse of the Taylor vortices with the increase of time. The
process is similar to the one for the case of Oil I, i.e. first the Taylor vortices are perfect
and clearly defined, Figure 9.19a, then the shear force between the vortices is large
enough to deform and distort them with the increase of time, Figures 9.19b and 9.19c,
and then the particles are able to enter the center areas of the vortices, Figures 9.19d and
9.19e, and finally the vortices disappeared and they are replaced with Pre-wavy flow,
Figure 9.19f. The difference is that the whole process lasts for about 7 seconds for the
case of Oil II, which is much shorter for the case of Oil I, 40 seconds. The collapse time
for each of the cases, Oil I ( ν = 7.7 × 10 -6 m 2/s) and Oil II ( ν = 4.0 × 10 -5 m 2/s), is
inversely proportional to the kinematic viscosity of the oil. This corresponds to the fact
observed by Snyder (1969), who experimentally showed that the relaxation time is
approximately L 2/ν, where L is the length of the vortex column and ν is the kinematic viscosity.
The flow pattern in longitudinal cross section view in Pre-wavy regime for the case of
Oil II is shown in Figure 9.20. Like the case of Oil I, there are particle-accumulated areas
in this regime for the case of Oil II. The difference is that the distance in the axial
direction between the particle-accumulated areas is shorter for the case of Oil II than the
one for the case of Oil I. Note that there are two particle-accumulated areas for the case
of Oil II, Figure 9.20, while there is only one particle-accumulated area for the case of
Oil I shown in Figure 9.6. Also note that the distance between the particle-accumulated
238 areas is larger for the larger speed ( ω = 1,700 rpm) than the one for the smaller speed ( ω
= 700 rpm), Figure 9.20.
Particle-accumulated area
L L
a) ω = 700 rpm b) ω = 1,700 rpm
Figure 9.20 Flow Pattern in Longitudinal Cross Section View in Pre-wavy Regime (Oil II)
Figure 9.21 presents the flow pattern in longitudinal cross section view in Wavy regime ( ω = 1,750, 1,800, 1,900, 2,000 and 2,300 rpm). The process of the formation of
wavy vortices is similar to the one for the case of Oil I, i.e. the wavy vortices begin to
form when the speed reaches 1,750 rpm, Figure 9.21a, and then more and more particles
are accumulated at the centers of the wavy vortices with the increase of speed, Figures
9.21b, 9.21c and 9.21d, and eventually nearly all the particles are accumulated at the
centers of the vortices as the speed reaches 2,300 rpm, Figure 9.21e. The particles will
disperse and the flow pattern becomes turbulence with the further increase of the speed. 239
a) ω = 1,750 rpm b) ω = 1,800 rpm c) ω = 1,900 rpm
d) ω = 2,000 rpm e) ω = 2,300 rpm
Figure 9.21 Flow Pattern in Longitudinal Cross Section View in Wavy Regime (Oil II)
240 For the case of Oil II, the flow pattern in longitudinal cross section view in turbulent regime is very similar to the one for the case of Oil I, i.e. the particles are homogeneous and they are blurred due to the large velocity in this regime.
Figures 9.22 – 9.26 visualize the flow patterns in the front view for each regime corresponding to Figure 9.16. Figure 9.22 presents the flow pattern in front view in
Couette regime ( ω = 400 rpm). Like the one for the case of Oil I, Figure 9.22 shows that
the velocity in the axial direction is zero and the flow is laminar in the circumferential
direction, which characterizes the Couette flow.
ω = 400 rpm
Figure 9.22 Flow Pattern in Front View in Couette Regime (Oil II)
241
a) ω = 450 rpm
b) ω = 500 rpm
Figure 9.23 Flow Pattern in Front View in Taylor Regime (Oil II)
242 Figure 9.23 presents the flow pattern in front view in Taylor regime ( ω = 450 and 500 rpm). The boundaries of the Taylor cells, which are indicated by the accumulated particles in horizontal lines, begin to form at the speed of ω = 450 rpm, Figure 9.23a, and
they become come more discernible with the increase of speed, Figure 9.23b. Again, the
front view shows the nature of the flow pattern of Taylor vortex flow, axisymmetric and
time-independent.
With the further increase of the speed, the flow pattern becomes Pre-wavy, Figure
9.24. In this regime the horizontal lines disappeared and the particles become homogenous, which indicate the collapse of Taylor vortices and the appearance of Pre- wavy flow.
ω = 700 rpm
Figure 9.24 Flow Pattern in Front View in Pre-wavy Regime (Oil II)
243
a) ω = 1,700 rpm
b) ω = 1,800 rpm
Figure 9.25 Flow Pattern in Front View in Wavy Regime (Oil II)
244
c) ω = 2,000 rpm
d) ω = 2,100 rpm
Figure 9.25 Flow Pattern in Front View in Wavy Regime (Oil II) (Continued)
245
e) ω = 2,200 rpm
Figure 9.25 Flow Pattern in Front View in Wavy Regime (Oil II) (Continued)
Figure 9.25 presents the flow pattern in front view in Wavy regime ( ω = 1,700, 1,800,
2,000, 2,100 and 2,200 rpm). Like the one for the case of Oil I, the waves in the circumferential direction begin to form when the speed reaches 1,700 rpm, Figure 9.25a, and then more and more particles are accumulated on the waves with the increase of speed, Figures 9.25b through 9.25e. It is clearly shown in this case that the waves look like a three-strand rope, and each strand undulates in both the axial and radial directions.
It is also shown that the size of the strand is large when the speed is small, Figure 9.25b, and the size of the strand becomes smaller and then larger with the increase of the speed,
Figures 9.25c, 9.25d and 9.25e. The movement of the waves can be described as a three- strand rope, spinning around itself and also moving in the circumferential direction. 246 Figure 9.26 presents the flow pattern in front view in Turbulent regime ( ω = 2,400 rpm). Like the one for the case of Oil I, it is shown in Figure 9.26 that all the particles are homogeneous in the whole area and the particles are blurred due to the large velocity in turbulent regime. Notice that the point ω = 2,400 rpm belongs to the wavy regime in the torque-speed graph, Figure 9.16. Again this discrepancy is due to the fact that the influence of vortices on the torque is second-order effect, while the visual appearance of vortices in the flow is a first-order effect.
ω = 2,400 rpm
Figure 9.26 Flow Pattern in Front View in Turbulent Regime (Oil II)
9.3 Results for the Case of Oil III [ = 0.099 kg/(m-s), ρ = 1048 kg/m 3]
This section will present the results of the torque measurements and the visualization of flow patterns at different regimes for the case of Oil III. 247 9.3.1 Torque Measurements for the Case of Oil III
For the case of oil III, the voltages measured on the BAM for each speed are plotted in
Figure 9.27. The measurements are repeated four times in order to evaluate the
repeatability and to allow uncertainty and error analysis. The raw data of the voltage
measurements for this case are presented in the Appendix E. Again, the voltages
measured were offset until the voltage-speed curves reach the origin, Figure 9.28.
Voltage (V) 3.5
3.0
2.5
2.0 Trial I 1.5 Trial II Trial III
1.0 Trial IV
0.5 0 1000 2000 3000 4000
Speed (rpm)
Figure 9.27 Voltage Measurements for the Case of Oil III
Figure 9.28 and its corresponding flow pattern visualizations, which will be presented
in next section, show that the flow patterns can be divided into four types, i.e. Couette (0
– 800 rpm), Taylor (800 – 1,600 rpm), Pre-wavy (1,600 – 3,000 rpm) and Wavy (> 3,000
rpm) flows. Again the turbulent flow regime was not presented for the case of Oil III due 248 to the large vibrations and bad images. The points representing the Couette, Taylor, Pre- wavy and Wavy flows can be curve-fitted by four straight lines. Note that the range of
Taylor flow is long enough to have a curve-fit for the points in this regime for the case of
Oil III.
Couette Tay lor Pre -wavy Wavy Voltage (V)
3.0 y = 0.000370x + 1.325100 R2 = 0.965418
Trial I 2.5 Trial II Trial III 2.0 Trial IV y = 0.000750x + 0.260456 R2 = 0.999188 1.5
1.0 y = 0.001104x - 0.231625 R2 = 0.999900
0.5 y = 0.000912x - 0.013188 2 R = 0.999139 0.0 0 1000 2000 3000 4000 Speed (rpm)
Figure 9.28 Voltage due to Fluid Friction vs. Speed for the Case of Oil III
By least-square curve fitting, the relationship between the voltage U (V) and the speed
ω (rpm) for each regime can be expressed as (Figure 9.28):
U = .0 000912 ω − .0 013188 (Couette) (9.8)
U = .0 001104 ω − .0 231625 (Taylor) (9.9)
U = .0 000750 ω + .0 260456 (Pre-wavy) (9.10) 249 U = .0 000370 ω + .1 325100 (Wavy) (9.11)
Again the coefficient r for the case of Oil III can be calculated for each regime by using Equations (7.2), (7.3), (7.4) and (7.5). The results are listed in Tables 9.8 – 9.11. It is shown that the correlation coefficients range from 0.97 to 0.998 for Couette, Taylor and Pre-wavy regimes, which means that Equations (9.8) - (9.10) represent good curve fits. The correlation coefficients range from 0.73 to 0.90 for Wavy flow, which means that the curve fit for wavy flow is not so good. The reason for that probably is due to the large vibrations occurred during this regime.
Table 9.8 Correlation Coefficients of the Curve Fit for Couette Flow (Oil III)
Trial I II III IV
Um (V) 0.45100 0.44788 0.45475 0.41688
σy (V) 0.24015 0.24263 0.23092 0.22841
σy,x (V) 0.01670 0.01448 0.01943 0.03978
R 0.99758 0.99822 0.99645 0.98472
Table 9.9 Correlation Coefficients of the Curve Fit for Taylor Flow (Oil III)
Trial I II III IV
Um (V) 1.22000 1.19488 1.20075 1.20063
σy (V) 0.28446 0.30401 0.28715 0.26634
σy,x (V) 0.02634 0.02902 0.02454 0.02875
R 0.99570 0.99543 0.99634 0.99416
250 Table 9.10 Correlation Coefficients of the Curve Fit for Pre-wavy Flow (Oil III)
Trial I II III IV
Um (V) 2.00000 1.94825 1.92475 1.91825
σy (V) 0.33882 0.32847 0.32529 0.33871
σy,x (V) 0.07436 0.01907 0.04116 0.04336
R 0.97562 0.99831 0.99196 0.99177
Table 9.11 Correlation Coefficients of the Curve Fit for Wavy Flow (Oil III)
Trial I II III IV
Um (V) 2.58438 2.57688 2.50513 2.52463
σy (V) 0.09388 0.11654 0.09155 0.08861
σy,x (V) 0.06051 0.05172 0.06235 0.04377
R 0.76454 0.89614 0.73229 0.86951
9.3.2 Visualization of Flow Patterns for the Case of Oil III
Figures 9.29 – 9.33 visualize the flow patterns in the longitudinal cross section view for each regime for the case of Oil III corresponding to Figure 9.18.
Figure 9.29 presents the flow pattern in longitudinal cross section view in Couette
regime ( ω = 400 rpm). Like the one for the cases of Oil I and Oil II, it is shown in Figure
9.29 that for the case of Oil III, some of the particles are accumulated near the inner cylinder and all the particles in other area are homogeneous. It is shown that the larger
251 the viscosity and the density of the oil, the more particles near the inner cylinder, Figures
9.3, 9.17 and 9.29. Note that there is a difference between the centrifugal forces of the oil and the particles due to their different densities.
Figure 9.30 presents the flow pattern in longitudinal cross section view in Taylor
regime ( ω = 500, 600, 700, 800, 900, 1,000, 1,100 and 1,200 rpm). The process of
Taylor vortex formation is the similar to those for Oil I and Oil II, i.e. first two cells with
small area initiate near the inner cylinder, Figures 9.30a and 9.30b, then the boundaries of
the cells become more and more discernible, and they are also extending toward the outer
cylinder, Figures 9.30c, 9.30d and 9.30e, and eventually the cells fully occupy the space
between the inner and outer cylinders and the boundaries of Taylor cells are clearly
defined by the particles, Figures 9.30f, 9.30g and 9.30h.
ω = 400 rpm
Figure 9.29 Flow Pattern in Longitudinal Cross Section View in Couette Regime (Oil III)
252
a) ω = 500 rpm b) ω = 600 rpm c) ω = 700 rpm d) ω = 800 rpm
e) ω = 900 rpm f) ω = 1,000 rpm g) ω = 1,100 rpm h) ω = 1,200 rpm
Figure 9.30 Flow Pattern in Longitudinal Cross Section View in Taylor Regime (Oil III)
253
a) t = 0 s b) t = 1 s c) t = 2 s
d) t = 3 s e) t = 3.5 s
Figure 9.31 The Collapse of Taylor Vortices for the Case of ω = 1,200 rpm (Oil III)
254 For the case of Oil III, the speed of ω = 1,200 rpm is the critical speed for the disappearance of Taylor vortices and the appearance of Pre-wavy flow. Figure 9.31 presents the process of the collapse of the Taylor vortices with the increase of time. The process is similar to the those for the cases of Oil I and Oil II, i.e. first the Taylor vortices are perfect and clearly defined, Figure 9.31a, then the shear force between the vortices is large enough to deform and distort them, Figure 9.31b, and then the particles are able to enter the center areas of the vortices, Figures 9.31c and 9.31d, and finally the vortices disappeared and they are replaced with Pre-wavy flow, Figure 9.31e. The whole process lasts for about 3.5 seconds for the case of Oil III. Again, the collapse time for each of the cases, 40 seconds for Oil I ( ν = 7.7 × 10 -6 m 2/s), 7 seconds for Oil II ( ν = 4.0 × 10 -5 m 2/s), and 3.5 seconds for Oil III ( ν = 9.4 × 10 -5 m 2/s), is inversely proportional to the kinematic
viscosity of the oil. Particle-accumulated area
L
L
a) ω = 1,600 rpm b) ω = 2,900 rpm
Figure 9.32 Flow Pattern in Longitudinal Cross Section View in Pre-wavy Regime (Oil III) 255 Figure 9.32 shows the flow pattern in longitudinal cross section view in Pre-wavy regime for the case of Oil III. Like the cases of Oil I and Oil II, there are particle- accumulated areas in this regime for the case of Oil III. Like the one for the case of Oil
II, the distance between the particle-accumulated areas is larger for the larger speed ( ω =
2,900 rpm) than the one for the smaller speed ( ω = 1,600 rpm), Figure 9.32. It is also shown that the larger the viscosity and the density of the oil, the more particles in the particle-accumulated areas, Figures 9.6, 9.20 and 9.32.
a) ω = 3,000 rpm b) ω = 3,100 rpm c) ω = 3,300 rpm
Figure 9.33 Flow Pattern in Longitudinal Cross Section View in Wavy Regime (Oil III)
Figure 9.33 presents the flow pattern in longitudinal cross section view in Wavy regime ( ω = 3,000, 3,100 and 3,300 rpm). The process of the formation of wavy vortices
is not as clear as those for the cases of Oil I and Oil II due to the large speed for this case. 256 However, it still shows the typical process, i.e. first the wavy vortices begin to form,
Figure 9.33a, then more and more particles are accumulated at the centers of the wavy vortices with the increase of speed, Figure 9.33b, and then the particles begin to disperse,
Figure 9.33c, and eventually the flow pattern will become turbulence with the further increase of the speed.
For the case of Oil III, the flow pattern in longitudinal cross section view in turbulent regime is very similar to those for the cases of Oil I and Oil II, i.e. the particles are homogeneous and they are blurred due to the large velocity in this regime.
Figures 9.34 – 9.38 visualize the flow patterns in the front view for each regime corresponding to Figure 9.28. Figure 9.34 presents the flow pattern in front view in
Couette regime ( ω = 400 rpm). Like those for the cases of Oil I and Oil II, Figure 9.34
shows that the velocity in the axial direction is zero and the flow is laminar in the
circumferential direction.
ω = 400 rpm
Figure 9.34 Flow Pattern in Front View in Couette Regime (Oil III) 257
a) ω = 600 rpm
b) ω = 800 rpm
Figure 9.35 Flow Pattern in Front View in Taylor Regime (Oil III)
258
c) ω = 900 rpm
d) ω = 1,100 rpm
Figure 9.35 Flow Pattern in Front View in Taylor Regime (Oil III) (Continued)
259
e) ω = 1,200 rpm
f) ω = 1,300 rpm
Figure 9.35 Flow Pattern in Front View in Taylor Regime (Oil III) (Continued)
260 Figure 9.35 presents the flow pattern in front view in Taylor regime ( ω = 600, 800,
900, 1,100, 1,200 and 1,300 rpm). The process for the formation of Taylor cells is clearer in this case than the previous ones. First the particles begin to accumulate, Figure 9.35a, then more particles are accumulated to form the horizontal lines, Figure 9.35b, and then the horizontal lines become come more discernible, which indicate the formation of the boundaries of the Taylor cells, Figures 9.35c, 9.35d and 9.35e, and eventually the horizontal lines become the clearest and thinnest when the speed reaches 1,300 rpm,
Figure 9.35f. The particles will disperse and the horizontal lines will disappear with the further increase of the speed, which indicate the end of Taylor regime.
ω = 2,400 rpm
Figure 9.36 Flow Pattern in Front View in Pre-wavy Regime (Oil III)
261
a) ω = 3,100 rpm
b) ω = 3,200 rpm
Figure 9.37 Flow Pattern in Front View in Wavy Regime (Oil III)
262
c) ω = 3,400 rpm
d) ω = 3,600 rpm
Figure 9.37 Flow Pattern in Front View in Wavy Regime (Oil III) (Continued)
263 Figure 9.36 presents the flow pattern in front view in Pre-wavy regime for the case of
Oil III. Like those for the cases of Oil I and Oil II, the horizontal lines disappeared and
the particles become homogenous, which indicate the collapse of Taylor vortices and the
appearance of Pre-wavy flow. It is also shown that the larger the speed, the more blurred
the particles due to their larger velocities, Figures 9.11, 9.24 and 9.36.
Figure 9.37 presents the flow pattern in front view in Wavy regime ( ω = 3,100, 3,200,
3,400 and 3,600 rpm). The formation of the waves in the circumferential direction for this case is quite different than those for the cases of Oil I and Oil II. First the particles start to accumulate in the right hand side, Figure 9.37a, then more particles accumulate and bands begin to appear, Figure 9.37b, which indicate the formation of waves, and then more and more particles join the bands, Figure 9.37c, and finally most of the particles accumulate in the right hand side and there are only few particles in the left hand side,
Figure 9.37d. This means that only portions of the waves are shown in the figures and the waves are much longer than those shown for the cases of Oil I and Oil II.
264
CHAPTER X
COMPARISON BETWEEN NUMERICAL AND EXPERIMENTAL RESULTS
The comparison between numerical and experimental results of torque-speed graphs
will be made for all the three kinds of oils, i.e. Oil I [ = 0.006 kg/(m-s), ρ = 778 kg/m 3],
Oil II [ = 0.038 kg/(m-s), ρ = 961 kg/m 3] and Oil III [ = 0.099 kg/(m-s), ρ = 1048 kg/m 3]. However, the comparison between numerical and experimental results of flow patterns will be made for Oil I only because the flow patterns are similar for all the three kinds of oils as shown in the previous chapter.
10.1 Comparison of Torque – speed Graphs
Figure 10.1 presents the comparison between the numerical and experimental results of the torque – speed graphs for the case of Oil I. Figure 10.1a shows that in general there is a good agreement between the numerical and experimental data except those in turbulent regime. The torque measurements by experiments are well below those calculated by numerical simulation. The reason for the big difference between the numerical and experimental results in turbulent regime is due to the different flow patterns in this regime, which will be presented later in this section. Figure 10.1b shows that the numerical critical speed for the onset of Taylor vortices is 125 rpm, which is lower than the measured critical speed, 150 rpm, while the critical speed for the onset of Pre-wavy regime is 200 rpm for both numerical and experimental results. 265
Torque (N.m) Wavy Turbulence
2.0 CFD EXP I 1.5 EXP II EXP III EXP IV 1.0
Detail shown in b) 0.5
0.0 Couette 0 500 1000 1500 2000 2500
Speed (rpm) Taylor Pre -wavy a)
Experimental Taylor Torque (N.m) Couette Pre -wavy 0.20 CFD EXP I 0.15 EXP II EXP III EXP IV
0.10
0.05
0.00 0 100 200 300 Couette Taylor Pre -wavy Speed (rpm) CFD b)
Figure 10.1 Comparison of Torque – Speed Graphs for the Case of Oil I 266
Torque (N.m) Wavy 6 CFD 5 EXP I EXP II 4 EXP III EXP IV 3 Detail shown in b) 2
1
0 Couette 0 500 1000 1500 2000 2500 3000
Pre -wavy Speed (rpm) Taylor a)
Experimental Taylor Torque (N.m) Couette Pre -wavy 2.5 CFD EXP I 2.0 EXP II EXP III 1.5 EXP IV
1.0
0.5
0.0 0 200 400 600 800 1000 1200 Couette Taylor Pre -wavy Speed (rpm)
CFD b)
Figure 10.2 Comparison of Torque – Speed Graphs for the Case of Oil II 267
Torque (N.m) Experimental Couette Taylor Pre -wavy Wavy
16 CFD EXP I 12 EXP II EXP III EXP IV
8
4
0 0 500 1000 1500 2000 2500 3000 3500 4000 Couette Pre -wavy Wavy Speed (rpm) Taylor CFD
Figure 10.3 Comparison of Torque – Speed Graphs for the Case of Oil III
Figure 10.2 presents the comparison between the numerical and experimental results of the torque – speed graphs for the case of Oil II. Figure 10.2a shows that there is a good agreement between the numerical and experimental data in the Couette, Taylor and Pre- wavy regimes. In the wavy regime, the numerical data are larger than the corresponding measured torques but the difference is not larger than 14%. Figure 10.2b shows that the numerical critical speeds for the onset of Taylor vortices and Pre-wavy are 600 and 800 rpm, respectively, while the measured critical speeds are 450 and 650 rpm, respectively. 268 Figure 10.3 presents the comparison between the numerical and experimental results of the torque – speed graphs for the case of Oil III. In general, the agreement between the numerical and experimental data is as good as the ones for Oil I and Oil II. The biggest difference between numerical and experimental data is as high as 37%. The reason is probably the large vibrations occurred when the speed is large in this case. Figure 10.3 also shows that the numerical critical speed for the onset of Taylor vortices is 1,425 rpm, which is higher than the measured critical speed, 800 rpm, while the critical speed for the onset of Pre-wavy regime is 1,600 rpm for both numerical and experimental results.
10.2 Comparison of Flow Patterns for the Case of Oil I
In this section, comparison between experimental and numerical flow pattern will be
made for each regime for the case of Oil I. The flow pattern of one typical speed will be
presented for each flow regime, i.e. 100 rpm for Couette regime, 200 rpm for Taylor
regime, 500 rpm for Pre-wavy regime, 900 rpm for Wavy regime and 2,100 rpm for
Turbulent regime.
Figure 10.4 presents the comparison between experimental and numerical flow
patterns in longitudinal cross section view in Coutte regime ( ω = 100 rpm). It is shown in
Figure 10.4a that the particles are homogeneous, which means that the velocities in both
axial and radial directions are zero and there is no pressure variation in the axial
direction. The zero velocities in the axial and radial directions are verified numerically in
Figures 10.4b and 10.4c, while the zero pressure variation in the axial direction is verified
in Figure 10.4e. The linear characteristic of Couette flow is also indicated by the linear
variation of speed u in the radial direction, Figure 10.4d. 269
a) Observed flow b) w contour c) v contour pattern
d) u contour e) P contour
Figure 10.4 Comparison of Flow Pattern in Longitudinal Cross Section View in Couette Regime ( ω = 100 rpm) (Oil I)
270
a) Observed flow pattern b) w contour
c) P contour
Figure 10.5 Comparison of Flow Pattern in Front View in Couette Regime (ω = 100 rpm) (Oil I)
271
a) Observed b) w contour c) v contour flow pattern
d) u contour e) P contour
Figure 10.6 Comparison of Flow Pattern in Longitudinal Cross Section View in
Taylor Regime ( ω = 200 rpm) (Oil I)
272
a) Observed flow pattern b) w contour
c) P contour
Figure 10.7 Comparison of Flow Pattern in Front View in Taylor Regime (ω = 200 rpm) (Oil I)
273 Figure 10.5 presents the comparison between experimental and numerical flow patterns in front view in Coutte regime ( ω = 100 rpm). It is shown in Figure 10.5a that the particles do not move in the axial direction and the tracks of the particles are laminar in the circumferential direction. The zero velocity in the axial direction and laminar characteristic of Couette flow are verified numerically in Figure 10.5b. This also means that there is no pressure variation in the axial and circumferential directions, which is verified in Figure 10.5c.
Figure 10.6 presents the comparison between experimental and numerical flow patterns in longitudinal cross section view in Taylor regime ( ω = 200 rpm). It is shown in
Figure 10.6a that Taylor vortices are clearly defined by the particles. The formation of
the Taylor vortices indicates that the velocities in both axial and radial directions are not
zero and the pressure variation in the axial direction is not zero, either. The non-zero
velocities and non-zero pressure variation are verified numerically in Figures 10.6b,
10.6c and 10.6e. Figure 10.6d shows that the velocity u varies in the axial direction and
it is not linear across the radial direction, which indicates that the disappearance of
Couette flow and appearance of Taylor vortices.
Figure 10.7 presents the comparison between experimental and numerical flow patterns in front view in Taylor regime ( ω = 200 rpm). Figure 10.7a shows that the
boundaries of the Taylor vortices are clearly defined by the horizontal particle-
accumulated lines. Those lines are accumulated on the boundaries between large and
small velocity zones, which can be verified numerically in Figures 10.7b and 10.7c.
Figures 10.7a also shows the nature of the flow pattern in Taylor vortex regime,
274 axisymmetric and time-independent. This nature is also verified numerically in Figures
10.7b and 10.7c.
Particle-accumulated area
a) Observed b) w contour c) v contour flow pattern
Low-pressure area
d) u contour e) P contour
Figure 10.8 Comparison of Flow Pattern in Longitudinal Cross Section View in Pre-wavy Regime ( ω = 500 rpm) (Oil I)
275
a) Pre-wavy ( ω = 500 rpm) b) Wavy ( ω = 900 rpm)
Figure 10.9 Vorticity Comparison between Pre-wavy and Wavy Regimes (Oil I)
Figure 10.8 presents the comparison between experimental and numerical flow patterns in longitudinal cross section view in Pre-wavy regime ( ω = 500 rpm). It is
shown in Figure 10.8a that the Taylor vortices are disappeared but the wavy vortices are
not formed yet in this regime. However, it is shown numerically that the Taylor vortices
are distorted and the wavy vortices are formed in this regime, Figures 10.8b, 10.8c, 10.8d
and 10.8e. Why is there a discrepancy between the experimental and numerical flow
patterns? It is because that the Taylor vortices are destroyed by the fluid flowing
between the vortices, Figure 10.8b, but the vorticity is not strong enough to hold the
particles in the centers of the vortices in this regime (The particles are accumulated in the
centers of the wavy vortices in wavy regime, which will be shown in Figure 10.11a). 276 Figure 10.9 presents the vorticity contours in Pre-wavy ( ω = 500 rpm) and Wavy ( ω =
900 rpm) regimes. It is shown that the vorticity in the center area of the wavy vortices in
Wavy regime ranges from 150 to 500 1/s, while the one in Pre-wavy regime only ranges from 70 to 200 1/s, much smaller than the ones in Wavy regime. Note that there is a particle-accumulated area near the inner cylinder, Figure 10.8a, which is corresponding to the low-pressure area, Figure 10.8e. Therefore, the particles are accumulated near the inner cylinder by the pressure variation in this regime.
Figure 10.10 presents the comparison between experimental and numerical flow patterns in front view in Pre-wavy regime ( ω = 500 rpm). Figure 10.10a shows that the
particles look like homogeneous, but they move in both circumferential and axial
directions. However, there are waves in the circumferential direction, numerically shown
in Figures 10.10b, 10.10c and 10.10d. Again, the discrepancy between the experimental
and numerical flow patterns is due to the fact that the vorticity in the vortex center is not
strong enough to hold the particles in this regime.
Figure 10.11 presents the comparison between experimental and numerical flow
patterns in longitudinal cross section view in Wavy regime ( ω = 900 rpm). It is shown in
Figure 10.11a that the particles are accumulated on the center areas of the wavy vortices.
The corresponding numerical results show that the Taylor vortices are distorted and the
wavy vortices are formed, Figures 10.11b, 10.11c, 10.11d and 10.11e. Note that the
particles are accumulated on the Taylor vortex boundaries in Taylor regime, Figure
10.6a, while the particles are accumulated on the center areas of the wavy vortices,
Figure 10.11a. This is because that in Taylor regime the fluid is moving toward the
boundaries, thus forcing the particles accumulate on the Taylor vortex boundaries, Figure 277 10.6b, while in Wavy regime the Taylor vortices are destroyed and the fluid is flowing between the vortices by the strong vorticity, thus forcing the particles accumulate on the center areas of the wavy vortices, Figure 10.11b.
Figure 10.12 presents the comparison between experimental and numerical flow patterns in front view in Wavy regime ( ω = 900 rpm). Figure 10.12a shows that the particles are accumulated on the waves in the circumferential direction. There are two whole waves shown in Figure 10.12a, the particles on the upper wave dispersing due to the diverging velocity vectors shown in Figure 10.12b, while the particles on the lower one increasing due to the converging velocity vectors shown in Figure 10.12b. The corresponding contours of axial direction velocity and pressure are presented in Figures
10.12c and 10.12d, which show the nature of the flow pattern in wavy vortex regime, non-axisymmetric and time-dependent.
Figure 10.13 presents the comparison between experimental and numerical flow patterns in longitudinal cross section view in Turbulent regime ( ω = 2,100 rpm). It is
shown in Figure 10.13a that the particles are homogeneous in the whole area but they
move in all the directions. The particles are also blurred due to the large velocity in this
regime. However, the corresponding numerical results show that the wavy vortices are
still well-organized, Figures 10.13b, 10.13c, 10.13d and 10.13e. There is a discrepancy
between experimental and numerical flow patterns in this regime. Note that the torque
measurements by experiments are well below those calculated by numerical simulation in
turbulent regime, Figure10.1a. This discrepancy between experimental and numerical
flow patterns accounts for the big difference between the numerical and experimental
data. Also Note that the numerical flow pattern is actually supercritical due to the perfect 278 geometric and dynamic boundary conditions, which can not be observed during the experiments.
a) Observed b) velocity vectors flow pattern
c) P contour d) w contour
Figure 10.10 Comparison of Flow Pattern in Front View in Pre-wavy Regime ( ω = 500 rpm) (Oil I)
279
a) Observed b) w contour c) v contour flow pattern
d) u contour e) P contour
Figure 10.11 Comparison of Flow Pattern in Longitudinal Cross Section View in
Wavy Regime ( ω = 900 rpm) (Oil I)
280
a) Observed b) velocity vectors flow pattern
c) w contour d) P contour
Figure 10.12 Comparison of Flow Pattern in Front View in Wavy
Regime ( ω = 900 rpm) (Oil I)
281
a) Observed b) w contour c) v contour flow pattern
d) u contour e) P contour f) vorticity contour
Figure 10.13 Comparison of Flow Pattern in Longitudinal Cross Section View in Turbulent Regime ( ω = 2,100 rpm) (Oil I)
282
CHAPTER XI
CONCLUSIONS
11.1 Conclusion to the Numerical Investigation
This dissertation presents the relationship between the onset of Taylor type instability and appearance of what is commonly known as “turbulence” in narrow gaps between two cylinders. The most accepted turbulence models, Constantinescu, Ng-Pan, Hirs and
Gross et al., can accurately predict the flow behavior and the results are relatively close to each other when “turbulence” is truly fully developed. However, in the transition regime, i.e. after the onset of Taylor vortices and before the full development of turbulence, the discrepancy amongst these most accepted models is significant.
To solve the problem of calculating the flow in the transition regime, a new model, or transition Reynolds model, for predicting the flow behavior in long and short journal bearing films in the transition regime was proposed.
The viscous fluid flow in narrow gaps between two cylinders with eccentricity ratios was calculated. The computational engine was provided by CFD-ACE+, a commercial multi-physics software. The flow patterns, velocity profiles and torques on the outer cylinder were determined when the speed of inner cylinder, clearance size and eccentricity ratio were changed on a parametric basis.
283 Calculations showed that there are two critical points for the onset of: (i) the first
Taylor instability (appearance of Taylor vortices) and (ii) the second Taylor instability
(appearance of wavy Taylor vortices), as they are indicated by both the flow pattern
changes and the inflections of the torque-speed curve. The inflection points in the
torque-speed graphs coincide with the flow pattern changes. The Reynolds and Taylor
numbers were calculated for these inflection points and the onset of flow instabilities
(“turbulent” flow called by some authors in the literature) was discussed versus the
critical values of these dimensionless numbers. It was shown that the Reynolds numbers
corresponding to the first and second critical Taylor transitions are well below the
accepted norm for the onset of the turbulence, Re = 1,000 and 2,000, for all the
clearances studied.
It was shown that the slope change in the torque-speed graph is due to the change of the average velocity gradient on the outer cylinder wall. This finding and not an increase in apparent viscosity is the cause for the inflection points on the torque-speed graph.
Calculations also showed that during the Taylor vortex regime velocity profiles in the radial direction are sinusoidal, and that pressure does vary in the axial direction even for the case of the “long journal bearing” (L/D>2). For the concentric case ( ε = 0), both
velocity and pressure profiles are axisymmetric and time-independent during the Taylor
vortex regime. During the wavy vortex regime the velocities maintain their sinusoidal
profiles, while pressure varies in both axial and circumferential directions. Both velocity
and pressure profiles are non-axisymmetric and time-dependent. An order of magnitude
analysis of the Navier-Stokes equation terms showed that the inertia, viscous terms,
284 pressure and the Reynolds stress terms are equally significant during the transition regime
(Taylor and wavy vortex regimes).
Based on these findings, transition Reynolds model for predicting the flow behavior in
long and short journal bearing films in the transition regime was proposed. This
transition Reynolds model indicated that the velocity profiles are sinusoidal and depend
on the local Reynolds number (or Taylor number) and the position in the axial direction.
Unlike the modified turbulent viscosity of the most accepted models (Constantinescu,
Ng-Pan, Hirs and Gross et al.), the viscosity used in the new model is kept at its laminar
value. A comparison was made between the transition Reynolds model and the four most
accepted turbulence models mentioned above. The effect of Taylor or wavy vortices is
considered in the transition Reynolds model, while it is neglected in the turbulence
models. The velocity and pressure fields are three-dimensional and two-dimensional
respectively for the transition Reynolds model, while they are two-dimensional and one-
dimensional respectively for the turbulence models.
11.2 Conclusion to the Experimental Verification
Experimental torque measurements and flow visualization were performed for three kinds of oils with different viscosities. It was shown that in general there is a good agreement between the numerical and experimental torques except those in turbulent regime. The different flow patterns in the turbulent regime accounts for the difference between the numerical and experimental torques.
Comparison between numerical and experimental flow patterns was also made and it
showed that they match well in the Couette, Taylor and wavy regimes. There is a 285 discrepancy between numerical and experimental flow patterns in the pre-wavy regime.
The reason is that the Taylor vortices are destroyed but the vorticity is not strong enough to hold the particles in the center areas of the vortices during experiments in this regime.
There is also a discrepancy between numerical and experimental flow patterns in turbulent regime. The reason is that the numerical flow pattern is actually supercritical due to the prefect geometric and dynamic boundaries conditions, which can’t be observed during the experiments.
In general there is a good agreement between the numerical and experimental results including torque measurements and flow patterns. Therefore, the transition Reynolds model for predicting the flow behavior in journal bearing films in the transition regime is justified.
11.3 Future Work
The transition Reynolds model has been derived based on the numerical calculations and verified by the experimental torque measurements and flow visualization. Based on the conclusions drawn above, a few recommendations for the future work are listed as follows.
1) Present work could be extended to the field of heat transfer, which is very
important for high speed journal bearings.
2) The comparison between numerical and experimental flow patterns is only made in
a qualitative manner, i.e. the velocities of the particles could be measured directly
so they could be compared with the numerical ones.
286 3) The current clearance between the inner and outer cylinders is 0.13 in., which is
much larger than the thickness of a typical journal bearing film. A smaller
clearance and a motor with higher speed would allow direct experimental
verification of the transition Reynolds model.
4) A lower guide bearing could be added so the test section would not be floated, thus
reducing the vibrations occurred when the speed is large.
5) A longer air trapper could be added so it could prevent the air entering the oil.
287
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294
APPENDICES
295
APPENDIX A
THE RESULTS OF CALIBRATION OF THE STRAIN GAGE
Trial I Trial II
Force (lbf) Voltage (mV) Force (lbf) Voltage (mV) Load Unload Load Unload
0.000 0.0 0.5 0.000 0.0 0.4
0.125 15.8 16.4 0.125 15.8 16.1
0.250 31.7 32.2 0.250 31.7 31.9
0.375 47.6 48.1 0.375 47.5 47.8
0.500 63.5 64.1 0.500 63.4 63.6
0.625 79.3 80.0 0.625 79.2 79.4
0.750 95.4 96.2 0.750 95.0 95.3
0.875 111.6 112.2 0.875 110.8 111.0
1.000 127.7 128.4 1.000 126.6 126.8
1.125 143.8 144.4 1.125 142.3 142.6
1.250 159.9 160.5 1.250 158.3 158.6
1.375 176.0 176.2 1.375 174.4 174.5
1.500 192.0 192.3 1.500 190.6 190.8
1.625 208.0 208.2 1.625 206.6 206.8
1.750 223.9 224.2 1.750 222.6 222.7
1.875 239.8 240.1 1.875 238.8 238.8
2.000 255.8 255.8 2.000 254.8 254.8
296 THE RESULTS OF CALIBRATION OF THE STRAIN GAGE (CONTINUED)
Trial III Trial IV
Force (lbf) Voltage (mV) Force (lbf) Voltage (mV) Load Unload Load Unload 0.000 0.0 0.3 0.000 0.0 1.0
0.125 15.8 16.0 0.125 15.7 16.7
0.250 31.6 31.8 0.250 31.5 32.6
0.375 47.4 47.6 0.375 47.3 48.3
0.500 63.2 63.5 0.500 63.1 64.2
0.625 79.1 79.3 0.625 79.0 80.1
0.750 95.0 95.1 0.750 95.1 96.2
0.875 110.7 110.9 0.875 111.0 112.3
1.000 126.7 126.7 1.000 127.3 128.5
1.125 142.3 142.6 1.125 143.4 144.4
1.250 158.3 158.4 1.250 159.4 160.3
1.375 174.3 174.6 1.375 175.4 176.2
1.500 190.4 190.9 1.500 191.5 192.1
1.625 206.4 206.8 1.625 207.4 207.9
1.750 222.6 223.0 1.750 223.5 223.7
1.875 238.8 239.0 1.875 239.3 239.5
2.000 254.9 254.9 2.000 255.1 255.1
297
APPENDIX B
VOLTAGE MEASUREMENTS FOR THE CASE OF AIR Trial I Trial II Speed Voltage (mV) Speed Voltage (mV) (rpm) Min. Max. Average (rpm) Min. Max. Average
50 306 348 327 50 302 357 330
100 307 366 337 100 305 358 332
125 309 367 338 125 308 364 336
150 319 367 343 150 317 366 342
200 318 374 346 200 311 364 338
250 324 375 350 250 319 366 343
300 329 367 348 300 324 360 342
400 338 385 362 400 330 369 350
500 319 361 340 500 311 355 333
600 343 360 352 600 338 352 345
700 348 383 366 700 340 380 360
800 332 378 355 800 329 375 352
900 335 370 353 900 337 376 357
1000 344 365 355 1000 352 380 366
1100 352 364 358 1100 360 373 367
1200 359 368 364 1200 364 376 370
1300 362 397 380 1300 369 395 382
298 VOLTAGE MEASUREMENTS FOR THE CASE OF AIR (CONTINUED) Trial III Trial IV Speed Voltage (mV) Speed Voltage (mV) (rpm) Min. Max. Average (rpm) Min. Max. Average 50 306 360 333 50 299 369 334
100 309 361 335 100 300 363 332
125 308 370 339 125 303 365 334
150 319 367 343 150 319 368 344
200 314 370 342 200 311 368 340
250 319 367 343 250 314 365 340
300 327 363 345 300 325 360 343
400 328 372 350 400 326 369 348
500 310 348 329 500 308 349 329
600 339 353 346 600 334 360 347
700 338 373 356 700 330 376 353
800 327 370 349 800 325 368 347
900 332 365 349 900 335 362 349
1000 348 373 361 1000 342 368 355
1100 350 368 359 1100 355 369 362
1200 362 373 368 1200 368 380 374
1300 359 401 380 1300 360 406 383
299
APPENDIX C
VOLTAGE MEASUREMENTS FOR THE CASE OF OIL I Trial I Trial II Speed Voltage (mV) Speed Voltage (mV) (rpm) Min. Max. Average (rpm) Min. Max. Average
50 370 434 402 50 365 436 401
100 380 440 410 100 376 446 411
125 385 440 413 125 380 455 418
150 389 451 420 150 385 460 423
200 391 458 425 200 393 438 416
250 401 462 432 250 395 446 421
300 409 467 438 300 398 453 426
400 437 473 455 400 418 463 441
500 454 486 470 500 443 471 457
600 485 504 495 600 478 495 487
700 459 562 511 700 462 558 510
800 477 558 518 800 472 548 510
900 497 558 528 900 494 562 528
1000 530 571 551 1000 533 579 556
1100 571 592 582 1100 572 590 581
1200 599 611 605 1200 596 606 601
1300 607 642 625 1300 605 639 622
300 VOLTAGE MEASUREMENTS FOR THE CASE OF OIL I (CONTINUED) Trial III Trial IV Speed Voltage (mV) Speed Voltage (mV) (rpm) Min. Max. Average (rpm) Min. Max. Average 50 376 440 408 50 383 429 406
100 380 436 408 100 389 433 411
125 387 450 419 125 393 437 415
150 390 448 419 150 388 441 415
200 397 443 420 200 397 446 422
250 408 446 427 250 397 452 425
300 422 464 443 300 402 463 433
400 428 478 453 400 428 474 451
500 455 478 467 500 453 484 469
600 480 493 487 600 483 504 494
700 467 558 513 700 459 540 500
800 479 550 515 800 478 556 517
900 517 560 539 900 503 576 540
1000 544 594 569 1000 539 582 561
1100 569 590 580 1100 584 596 590
1200 595 608 602 1200 604 618 611
1300 608 634 621 1300 609 644 627
301
APPENDIX D
VOLTAGE MEASUREMENTS FOR THE CASE OF OIL II Trial I Trial II Speed Voltage (mV) Speed Voltage (mV) (rpm) Min. Max. Average (rpm) Min. Max. Average
100 368 424 396 100 359 418 389 200 384 438 411 200 389 441 415 300 414 503 459 300 428 494 461 400 467 505 486 400 475 510 493 450 476 527 502 450 489 530 510 500 501 531 516 500 503 543 523 550 520 545 533 550 525 543 534 600 545 566 556 600 552 575 564 650 558 611 585 650 561 615 588 700 569 635 602 700 578 643 611 900 693 780 737 900 698 768 733 1100 859 880 870 1100 862 881 872 1300 984 1018 1001 1300 978 1017 998 1600 1207 1235 1221 1600 1197 1229 1213 1800 1186 1192 1189 1800 1182 1195 1189 2000 1299 1310 1305 2000 1289 1298 1294 2200 1422 1440 1431 2200 1418 1430 1424 2400 1543 1562 1553 2400 1524 1540 1532 2600 1619 1640 1630 2600 1621 1636 1629
302 VOLTAGE MEASUREMENTS FOR THE CASE OF OIL II (CONTINUED) Trial III Trial IV Speed Voltage (mV) Speed Voltage (mV) (rpm) Min. Max. Average (rpm) Min. Max. Average
100 346 408 377 100 380 426 403
200 381 434 408 200 404 446 425
300 398 486 442 300 420 492 456
400 468 498 483 400 480 510 495
450 470 522 496 450 496 530 513
500 493 530 512 500 505 543 524
550 510 530 520 550 515 540 528
600 540 566 553 600 545 570 558
650 559 609 584 650 568 604 586
700 576 642 609 700 589 650 620
900 692 762 727 900 709 782 746
1100 844 868 856 1100 856 880 868
1300 960 1002 981 1300 970 1015 993
1600 1191 1208 1200 1600 1200 1215 1208
1800 1170 1190 1180 1800 1180 1209 1195
2000 1267 1286 1277 2000 1280 1298 1289
2200 1400 1410 1405 2200 1420 1434 1427
2400 1516 1524 1520 2400 1518 1534 1526
2600 1610 1630 1620 2600 1619 1640 1630
303
APPENDIX E
VOLTAGE MEASUREMENTS FOR THE CASE OF OIL III
Trial I Trial II
Speed Voltage (mV) Speed Voltage (mV) (rpm) (rpm) Min. Max. Average Min. Max. Average
200 624 690 657 200 531 609 570
400 803 860 832 400 711 775 743
600 1029 1046 1038 600 922 946 934
800 1169 1247 1208 800 1049 1216 1133
1000 1356 1406 1381 1000 1218 1240 1229
1200 1562 1605 1584 1200 1464 1511 1488
1400 1784 1824 1804 1400 1694 1731 1713
1600 2015 2068 2042 1600 1919 1958 1939
1800 2116 2150 2133 1800 1997 2021 2009
2000 2282 2309 2296 2000 2139 2168 2154
2400 2586 2621 2604 2400 2464 2492 2478
2800 2880 2916 2898 2800 2713 2768 2741
3000 2922 2980 2951 3000 2826 2840 2833
3200 3015 3047 3031 3200 2921 2942 2932
3400 3106 3159 3133 3400 3021 3042 3032
3600 3126 3180 3153 3600 3085 3114 3100
304 VOLTAGE MEASUREMENTS FOR THE CASE OF OIL III (CONTINUED)
Trial III Trial IV
Speed Voltage (mV) Speed Voltage (mV) (rpm) (rpm) Min. Max. Average Min. Max. Average
200 520 588 554 200 470 525 498
400 677 769 723 400 624 686 655
600 908 925 917 600 840 870 855
800 1018 1153 1086 800 976 1064 1020
1000 1208 1230 1219 1000 1230 1258 1244
1200 1442 1486 1464 1200 1403 1436 1420
1400 1680 1720 1700 1400 1621 1664 1643
1600 1864 1896 1880 1600 1844 1869 1857
1800 1962 1974 1968 1800 1896 1920 1908
2000 2076 2096 2086 2000 2062 2084 2073
2400 2403 2440 2422 2400 2362 2394 2378
2800 2666 2701 2684 2800 2658 2690 2674
3000 2734 2776 2755 3000 2738 2760 2749
3200 2820 2860 2840 3200 2824 2860 2842
3400 2913 2943 2928 3400 2915 2937 2926
3600 2945 2970 2958 3600 2923 2960 2942
305