CONTROLLED DIFFUSION BLADE AXIAL FAN: FLUID MECHANICAL
MECHANISTIC EFFECTS ON FAN PERFORMANCE
By
David Jennings Barrent

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Mechanical Engineering – Master of Science
2015

ABSTRACT
CONTROLLED DIFFUSION BLADE AXIAL FAN: FLUID
MECHANICAL MECHANISTIC EFFECTS ON FAN PERFORMANCE
By
David Jennings Barrent
Axial fan configurations, with three-to-nine controlled diffusion (CD) blades, have
been examined using experimental methods. The initial measurements: pressure rise/flow
rate, revealed distinct and opposite patterns: discontinuous pressure ”jumps” for the 3-4
and 6-9 blade fans. The three, five and nine blade fans were selected for detailed examination using: i) time-mean surface pressure data from which the net lift force could
be obtained and ii) X array hot-wire data (stationary probe) in the near wakes of the
rotating blades. These data were processed to provide ensemble statistics. These data,
plus the input shaft power were interpreted using the First Law of Thermodynamics in its
control volume form. The results indicate that increasing the number of blades installed
on the fan suppresses turbulent fluctuations in the wake of the fan. Additionally, similar
phenomena in the flow field are observed at lower flow coefficient as the number of blades
on the fan is increased.

Copyright by
DAVID JENNINGS BARRENT
2015

Dedicated to my Mother and Father for supporting me when I have needed it the most.

iv

ACKNOWLEDGMENTS

I would like to thank Dr. John Foss for hiring me as an undergraduate research assistant
starting in 2012. I did not have graduate school on my radar before the experiences I
was exposed to in my first months within the lab. I have learned quite a bit the past few
years in the TSFL and I am honored to call Dr. Foss my mentor.
This research work would not be possible without students that have used the facility
and fan assembly before me. Andrew Cawood, Douglas Neal, and Behdad Davoudi have
always been available when I have questions concerning my research and their previous
research work. As my fluids lab TA Andrew told me about a research opportunity over
the summer and I am happy that I took him up on the offer. I am thankful for him
getting me into the lab that summer and sparking my interest in fluid mechanics during
spring with his enthusiasm in the classroom. The time spent working under him as he
finished his research work provided me a great foundation for the onset of my Master’s
Research. A special thanks to Doug for developing the experiment itself and providing
me with guidance to get the hot-wire wake velocity measurements up and running for
this research. The second half of this research effort would have been painful without his
help getting that experiment up and running. I have spent many hours processing and
interpreting data this semester and I am thankful to have had Behdad around to bounce
ideas off of when I process something new. I also greatly appreciate his guidance to some
of the aspects of his experiment as they pertain to mine.
Many thanks are necessary for the rest of the current group working with me in the
Turbulent Shear Flows Laboratory. Al, thank you for all of the work you have done to
help me finish this research work. I have truly enjoyed our political banter and lunches
v

over the years. Although, lunch has never been the same since Kyle Bade’s departure.
Miss having you around Kyle. Brian and Tyler, thanks for your help over the years with
various projects and I have enjoyed getting to know you in your time at the lab. Megan,
I am grateful for all of your help with my experiments the past two semesters. Running
the fan for pressure measurements and helping me make figures have been a tremendous
help to me the past two semesters.
The past semester has been a stressful whirlwind of finishing classes, running experiments, processing data and writing. To combat the stress I started practicing yoga this
semester and it has worked wonders on keeping me focused and level in these stressful
times. I would like to thank Jen Hayes of Yoga State for helping me on my journey to
reducing stress in my life, and all my friends I have made along the way at the studio.
Especially, Lyndsay, for getting on me every week asking how this document was going
and helping me fine tune diet this semester.
I am blessed to have made it this far and owe it to all of those who have supported
me. My family has been my rock for my entire life. I would not be writing this document
without the support from my Mother, Father, and Sister during my time in graduate
school; Thank you. Enjoy the read!

vi

TABLE OF CONTENTS
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x

KEY TO SYMBOLS AND ABBREVIATIONS . . . . . . . . . . . . . . . xxii
Chapter 1 INTRODUCTION . . . . . . . . . . . . . . . . . .
1.1 Overview for this Thesis . . . . . . . . . . . . . . . . . . . .
1.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 Previous Investigations . . . . . . . . . . . . . . . . .
1.1.2.1 Ph.D. Dissertation – Douglas Neal (2010) .
1.1.2.2 M.S. Thesis – Andrew Cawood (2012) . . .
1.1.2.3 M.S. Thesis – Behdad Davoudi (2014) . . .
1.1.3 The Present Investigation — Areas of Contributions .
1.2 Basic Fluid Mechanical Principals . . . . . . . . . . . . . . .
1.2.1 Control Volume Definition . . . . . . . . . . . . . . .
1.2.2 Fan Characteristic Curve Introduction . . . . . . . .
1.2.3 Characteristic Scales of the Subject Flow . . . . . . .

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1
1
1
1
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3
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3
4
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6
8

EXPERIMENTAL CONSIDERATIONS . . . . . .
Flow Research and Development Facility (AFRD) . . . . .
Overview . . . . . . . . . . . . . . . . . . . . . . . . . .
Fan Integral Data Experimental Features . . . . . . . . .
Near-Wake Velocity Measurements . . . . . . . . . . . .
2.1.3.1 Hot-Wire Calibration and Operating Procedure
2.1.3.2 Stationary Mount . . . . . . . . . . . . . . . . .
Controlled Diffusion Blade Axial Fan . . . . . . . . . . . . . . .
2.2.1 Incoming Velocity . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Fan Assembly . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Experimental Features . . . . . . . . . . . . . . . . . . .
2.2.3.1 Instrumented Blades . . . . . . . . . . . . . . .
2.2.3.2 Instrumentation in the Rotating Environment .
2.2.3.2.1 Mean Surface Pressure . . . . . . . . .
2.2.3.2.2 Fluctuating Surface Pressure . . . . .
2.2.3.2.3 Rotating Hot-Wire Wake Velocity . . .
2.2.3.2.4 Data Acquisition . . . . . . . . . . . .

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24

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26
26
26
28
33

Chapter 2
2.1 Axial
2.1.1
2.1.2
2.1.3

2.2

Chapter 3 RESULTS AND DISCUSSION . . .
3.1 Fan Integral Measurements . . . . . . . . . . .
3.1.1 Characteristic and Torque Curves . . . .
3.1.2 Operating Configurations and Conditions
3.2 Pressure Coefficients . . . . . . . . . . . . . . .
vii

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Selection
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3.3

3.4

3.2.1
3.2.2
Wake
3.3.1
3.3.2

First
3.4.1
3.4.2
3.4.3
3.4.4

Chapter 4

Lift Analysis . . . . . . . . . . . . . . . . .
Pressure Coefficient Profiles . . . . . . . .
Velocity . . . . . . . . . . . . . . . . . . . .
Mean Wake Velocities . . . . . . . . . . .
Reynolds Averaged Velocity Statistics . . .
3.3.2.1 Axial Turbulence Intensity . . . .
3.3.2.2 Azimuthal Turbulence Intensity .
3.3.2.3 Reynolds Shear Stress [z-θ plane]
Law of Thermodynamics Analysis . . . . . .
Input Shaft Power – Pshaf t ∗ . . . . . . . .
Net Kinetic Energy Flux – κ . . . . . . . .
Flow-Work Rate – χ . . . . . . . . . . . .
Turbulent Kinetic Energy Losses – ξ . . .

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34
34
44
48
55
56
60
63
67
67
68
75
78

SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . .

82

APPENDICES . . . . . . . . . . . . . . . .
A
PRESSURE COEFFICIENT FIGURES . . .
B
WAKE VELOCITY STATISTICS FIGURES
C
WORK RATE EVALUATION FIGURES . .
D
REYNOLDS AVERAGED FORMULATIONS

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. 85
. 86
. 95
. 136
. 153

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

viii

LIST OF TABLES
Table 1.1

Work rate equation terms . . . . . . . . . . . . . . . . . . .

6

Table 2.1

Pressure tap locations on the suction side of the blade . . .

19

Table 2.2

Pressure tap locations on the pressure side of the blade . . .

19

Table 2.3

Pressure tap locations on the suction side of the blade along
the trailing edge . . . . . . . . . . . . . . . . . . . . . . . .

20

Ensemble array used for phase averaging scheme

44

Table 3.1

ix

. . . . . .

LIST OF FIGURES
Figure 1.1

Control volumes for the present investigation . . . . . . . .

6

Figure 1.2

Sample axial fan characteristic curve [Lakshminarayana, B.
(1996)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

Axial Fan Research and Development Facility: (a) Facility
Exterior, (b) Schematic of Facility [Neal (2010)] . . . . . . .

9

Figure 2.2

Himmelstein Torque Sensor . . . . . . . . . . . . . . . . . .

11

Figure 2.3

Drawing of X-probe . . . . . . . . . . . . . . . . . . . . . .

12

Figure 2.4

Sample Mitchell probe utilized for measurements . . . . . .

13

Figure 2.5

Hot-wire calibration facility [Neal (2010)] . . . . . . . . . . .

14

Figure 2.6

Example of how the magnitude Qo and angle αo of the velocity are found relative to the probe from the output voltages
[Neal (2010)] . . . . . . . . . . . . . . . . . . . . . . . . . .

15

Figure 2.7

Mounted X-probe in fan wake . . . . . . . . . . . . . . . . .

16

Figure 2.8

Close up of probe in fan wake . . . . . . . . . . . . . . . . .

17

Figure 2.9

Incoming velocity profile . . . . . . . . . . . . . . . . . . . .

17

Figure 2.10

Fan Assembly: (a) five blade fan mid-assemby [Neal (2010)],
(b) five blade fan mounted and shroud . . . . . . . . . . . .

19

Suction and pressure side, left and right respectively, pressure
taps indicated x/c distances corresponding to Tables 2.1 and
2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

Figure 2.12

Streamwise distribution of pressure taps and η locations . .

21

Figure 2.13

Instrument cluster . . . . . . . . . . . . . . . . . . . . . . .

22

Figure 2.14

Pressure transducer scale [Neal (2010)] . . . . . . . . . . . .

22

Figure 2.15

Pressure measurement schematic [Neal (2010)] . . . . . . . .

23

Figure 2.16

Slip ring assembly at the base of the fan shaft [Neal (2010)]

25

Figure 2.1

Figure 2.11

x

Figure 3.1

Three to nine blade fan characteristic curves . . . . . . . . .

27

Figure 3.2

Three to nine blade fan torque curves . . . . . . . . . . . . .

28

Figure 3.3

Experimental operating conditions indicated on characteristic curves for Control Volume (1) . . . . . . . . . . . . . . .

29

Experimental operating conditions indicated on characteristic curves for Control Volume (2) . . . . . . . . . . . . . . .

30

Figure 3.4

Figure 3.5

Experimental operating conditions indicated on torque curves 31

Figure 3.6

Angle of incidence αi across the blade for all operating conditions and fan configurations . . . . . . . . . . . . . . . . .

32

Figure 3.7

Non-dimensional lift coefficient for single blade . . . . . . .

34

Figure 3.8

Non-dimensional lift coefficient for entire fan . . . . . . . . .

35

Figure 3.9

Pressure coefficients at Case I . . . . . . . . . . . . . . . . .

37

Figure 3.10

Pressure coefficients at Case II . . . . . . . . . . . . . . . .

38

Figure 3.11

Pressure coefficients at Case III . . . . . . . . . . . . . . . .

39

Figure 3.12

Pressure coefficients at Case IV . . . . . . . . . . . . . . . .

40

Figure 3.13

Pressure coefficients at Case V . . . . . . . . . . . . . . . .

41

Figure 3.14

Pressure coefficient along the trailing edge of the blade for all
cases and fan configurations . . . . . . . . . . . . . . . . . .

43

Figure 3.15

Hot-wire measurement schematic at an instant of time . . .

45

Figure 3.16

Complete fan wake velocity profile sample . . . . . . . . . .

47

Figure 3.17

Single blade fan wake velocity profile sample . . . . . . . . .

48

Figure 3.18

Case I Phase averaged axial velocity over λ9 . . . . . . . . .

49

Figure 3.19

Case II Phase averaged axial velocity over λ9 . . . . . . . .

50

Figure 3.20

Case III Phase averaged axial velocity over λ9 . . . . . . . .

50

Figure 3.21

Case IV Phase averaged axial velocity over λ9 . . . . . . . .

51

xi

Figure 3.22

Case V Phase averaged axial velocity over λ9 . . . . . . . .

51

Figure 3.23

Case I Phase averaged azimuthal velocity over λ9 . . . . . .

52

Figure 3.24

Case II Phase averaged azimuthal velocity over λ9

. . . . .

53

Figure 3.25

Case III Phase averaged azimuthal velocity over λ9 . . . . .

53

Figure 3.26

Case IV Phase averaged azimuthal velocity over λ9 . . . . .

54

Figure 3.27

Case V Phase averaged azimuthal velocity over λ9

. . . . .

54

Figure 3.28

Representative comparison in azimuthal velocity between fan
configurations . . . . . . . . . . . . . . . . . . . . . . . . . .

55

Figure 3.29

Case I Axial turbulence intensity over λ9 . . . . . . . . . . .

57

Figure 3.30

Case II Axial turbulence intensity over λ9 . . . . . . . . . .

58

Figure 3.31

Case III Axial turbulence intensity over λ9 . . . . . . . . . .

58

Figure 3.32

Case IV Axial turbulence intensity over λ9 . . . . . . . . . .

59

Figure 3.33

Case V Axial turbulence intensity over λ9 . . . . . . . . . .

59

Figure 3.34

Case I Azimuthal turbulence intensity over λ9 . . . . . . . .

60

Figure 3.35

Case II Azimuthal turbulence intensity over λ9 . . . . . . .

61

Figure 3.36

Case III Azimuthal turbulence intensity over λ9 . . . . . . .

61

Figure 3.37

Case IV Azimuthal turbulence intensity over λ9 . . . . . . .

62

Figure 3.38

Case V Azimuthal turbulence intensity over λ9 . . . . . . .

62

Figure 3.39

Case I Reynolds stress in z-θ – plane over λ9

. . . . . . . .

64

Figure 3.40

Case II Reynolds stress in z-θ – plane over λ9 . . . . . . . .

64

Figure 3.41

Case III Reynolds stress in z-θ – plane over λ9 . . . . . . . .

65

Figure 3.42

Case IV Reynolds stress in z-θ – plane over λ9 . . . . . . . .

65

Figure 3.43

Case V Reynolds stress in z-θ – plane over λ9 . . . . . . . .

66

xii

Figure 3.44

Non-dimensional input shaft power . . . . . . . . . . . . . .

68

Figure 3.45

Case I kinematic integrand exit kinetic energy flux over λ9 .

70

Figure 3.46

Case II kinematic integrand exit kinetic energy flux over λ9

70

Figure 3.47

Case III kinematic integrand exit kinetic energy flux over λ9

71

Figure 3.48

Case IV kinematic integrand exit kinetic energy flux over λ9

71

Figure 3.49

Case V kinematic integrand exit kinetic energy flux over λ9

72

Figure 3.50

Net kinetic energy flux Control Volume (1) . . . . . . . . . .

73

Figure 3.51

Flow-work rate Control Volume (1) . . . . . . . . . . . . . .

75

Figure 3.52

Flow-work rate Control Volume (2) . . . . . . . . . . . . . .

77

Figure 3.53

Case I turbulence kinetic energy at fan exit over λ9 . . . . .

79

Figure 3.54

Case II turbulence kinetic energy at fan exit over λ9 . . . .

79

Figure 3.55

Case III turbulence kinetic energy at fan exit over λ9 . . . .

80

Figure 3.56

Case IV turbulence kinetic energy at fan exit over λ9 . . . .

80

Figure 3.57

Case V turbulence kinetic energy at fan exit over λ9 . . . .

81

Figure 1

Case I pressure coefficients 3 blade fan . . . . . . . . . . . .

87

Figure 2

Case II pressure coefficients 3 blade fan . . . . . . . . . . . .

87

Figure 3

Case III pressure coefficients 3 blade fan . . . . . . . . . . .

88

Figure 4

Case IV pressure coefficients 3 blade fan . . . . . . . . . . .

88

Figure 5

Case V pressure coefficients 3 blade fan . . . . . . . . . . . .

89

Figure 6

Case I pressure coefficients 5 blade fan . . . . . . . . . . . .

90

Figure 7

Case II pressure coefficients 5 blade fan . . . . . . . . . . . .

90

Figure 8

Case III/IV pressure coefficients 5 blade fan . . . . . . . . .

91

Figure 9

Case V pressure coefficients 5 blade fan . . . . . . . . . . . .

91

xiii

Figure 10

Case I pressure coefficients 9 blade fan . . . . . . . . . . . .

92

Figure 11

Case II pressure coefficients 9 blade fan . . . . . . . . . . . .

92

Figure 12

Case III pressure coefficients 9 blade fan . . . . . . . . . . .

93

Figure 13

Case IV pressure coefficients 9 blade fan . . . . . . . . . . .

93

Figure 14

Case V pressure coefficients 9 blade fan . . . . . . . . . . . .

94

Figure 15

Case I Phase averaged axial velocity over 0 ≤ θ ≤ 2π . . . .

96

Figure 16

Case II Phase averaged axial velocity over 0 ≤ θ ≤ 2π . . . .

97

Figure 17

Case III Phase averaged axial velocity over 0 ≤ θ ≤ 2π . . .

98

Figure 18

Case IV Phase averaged axial velocity over 0 ≤ θ ≤ 2π . . .

99

Figure 19

Case V Phase averaged axial velocity over 0 ≤ θ ≤ 2π . . . . 100

Figure 20

Case I Phase averaged axial velocity over λ3 . . . . . . . . . 101

Figure 21

Case II Phase averaged axial velocity over λ3 . . . . . . . . 101

Figure 22

Case III Phase averaged axial velocity over λ3 . . . . . . . . 102

Figure 23

Case IV Phase averaged axial velocity over λ3 . . . . . . . . 102

Figure 24

Case V Phase averaged axial velocity over λ3 . . . . . . . . 103

Figure 25

Case I Phase averaged azimuthal velocity over 0 ≤ θ ≤ 2π . 104

Figure 26

Case II Phase averaged azimuthal velocity over 0 ≤ θ ≤ 2π . 105

Figure 27

Case III Phase averaged azimuthal velocity over 0 ≤ θ ≤ 2π

106

Figure 28

Case IV Phase averaged azimuthal velocity over 0 ≤ θ ≤ 2π

107

Figure 29

Case V Phase averaged azimuthal velocity over 0 ≤ θ ≤ 2π . 108

Figure 30

Case I Phase averaged azimuthal velocity over λ3 . . . . . . 109

Figure 31

Case II Phase averaged azimuthal velocity over λ3

Figure 32

Case III Phase averaged azimuthal velocity over λ3 . . . . . 110
xiv

. . . . . 109

Figure 33

Case IV Phase averaged azimuthal velocity over λ3 . . . . . 110

Figure 34

Case V Phase averaged azimuthal velocity over λ3

Figure 35

Case I Axial turbulence intensity over 0 ≤ θ ≤ 2π . . . . . . 112

Figure 36

Case II Axial turbulence intensity over 0 ≤ θ ≤ 2π . . . . . 113

Figure 37

Case III Axial turbulence intensity over 0 ≤ θ ≤ 2π . . . . . 114

Figure 38

Case IV Axial turbulence intensity over 0 ≤ θ ≤ 2π . . . . . 115

Figure 39

Case V Axial turbulence intensity over 0 ≤ θ ≤ 2π . . . . . 116

Figure 40

Case I Axial turbulence intensity over λ3 . . . . . . . . . . . 117

Figure 41

Case II Axial turbulence intensity over λ3 . . . . . . . . . . 117

Figure 42

Case III Axial turbulence intensity over λ3 . . . . . . . . . . 118

Figure 43

Case IV Axial turbulence intensity over λ3 . . . . . . . . . . 118

Figure 44

Case V Axial turbulence intensity over λ3 . . . . . . . . . . 119

Figure 45

Case I Azimuthal turbulence intensity over 0 ≤ θ ≤ 2π . . . 120

Figure 46

Case II Azimuthal turbulence intensity over 0 ≤ θ ≤ 2π . . . 121

Figure 47

Case III Azimuthal turbulence intensity over 0 ≤ θ ≤ 2π . . 122

Figure 48

Case IV Azimuthal turbulence intensity over 0 ≤ θ ≤ 2π . . 123

Figure 49

Case V Azimuthal turbulence intensity over 0 ≤ θ ≤ 2π . . . 124

Figure 50

Case I Azimuthal turbulence intensity over λ3 . . . . . . . . 125

Figure 51

Case II Azimuthal turbulence intensity over λ3 . . . . . . . 125

Figure 52

Case III Azimuthal turbulence intensity over λ3 . . . . . . . 126

Figure 53

Case IV Azimuthal turbulence intensity over λ3 . . . . . . . 126

Figure 54

Case V Azimuthal turbulence intensity over λ3 . . . . . . . 127

Figure 55

Case I Reynolds stress in z-θ – plane over 0 ≤ θ ≤ 2π . . . . 128
xv

. . . . . 111

Figure 56

Case II Reynolds stress in z-θ – plane over 0 ≤ θ ≤ 2π . . . 129

Figure 57

Case III Reynolds stress in z-θ – plane over 0 ≤ θ ≤ 2π . . . 130

Figure 58

Case IV Reynolds stress in z-θ – plane over 0 ≤ θ ≤ 2π . . . 131

Figure 59

Case V Reynolds stress in z-θ – plane over 0 ≤ θ ≤ 2π . . . 132

Figure 60

Case I Reynolds stress in z-θ – plane over λ3

Figure 61

Case II Reynolds stress in z-θ – plane over λ3 . . . . . . . . 133

Figure 62

Case III Reynolds stress in z-θ – plane over λ3 . . . . . . . . 134

Figure 63

Case IV Reynolds stress in z-θ – plane over λ3 . . . . . . . . 134

Figure 64

Case V Reynolds stress in z-θ – plane over λ3 . . . . . . . . 135

Figure 65

Case I kinematic integrand exit kinetic energy flux over 0 ≤
θ ≤ 2π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Figure 66

Case II kinematic integrand exit kinetic energy flux over 0 ≤
θ ≤ 2π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Figure 67

Case III kinematic integrand exit kinetic energy flux over
0 ≤ θ ≤ 2π . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Figure 68

Case IV kinematic integrand exit kinetic energy flux over
0 ≤ θ ≤ 2π . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Figure 69

Case V kinematic integrand exit kinetic energy flux over 0 ≤
θ ≤ 2π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Figure 70

Case I kinematic integrand exit kinetic energy flux over λ3 . 142

Figure 71

Case II kinematic integrand exit kinetic energy flux over λ3

Figure 72

Case III kinematic integrand exit kinetic energy flux over λ3 143

Figure 73

Case IV kinematic integrand exit kinetic energy flux over λ3 143

Figure 74

Case V kinematic integrand exit kinetic energy flux over λ3

144

Figure 75

Case I turbulence kinetic energy at fan exit over 0 ≤ θ ≤ 2π

145

xvi

. . . . . . . . 133

142

Figure 76

Case II turbulence kinetic energy at fan exit over 0 ≤ θ ≤ 2π 146

Figure 77

Case III turbulence kinetic energy at fan exit over 0 ≤ θ ≤ 2π 147

Figure 78

Case IV turbulence kinetic energy at fan exit over 0 ≤ θ ≤ 2π 148

Figure 79

Case V turbulence kinetic energy at fan exit over 0 ≤ θ ≤ 2π 149

Figure 80

Case I turbulence kinetic energy at fan exit over λ3 . . . . . 150

Figure 81

Case II turbulence kinetic energy at fan exit over λ3 . . . . 150

Figure 82

Case III turbulence kinetic energy at fan exit over λ3 . . . . 151

Figure 83

Case IV turbulence kinetic energy at fan exit over λ3 . . . . 151

Figure 84

Case V turbulence kinetic energy at fan exit over λ3 . . . . 152

Figure 85

Case I advection of θ–component momentum per unit mass
over 0 ≤ θ ≤ 2π . . . . . . . . . . . . . . . . . . . . . . . . . 155

Figure 86

Case II advection of θ–component momentum per unit mass
over 0 ≤ θ ≤ 2π . . . . . . . . . . . . . . . . . . . . . . . . . 156

Figure 87

Case III advection of θ–component momentum per unit mass
over 0 ≤ θ ≤ 2π . . . . . . . . . . . . . . . . . . . . . . . . . 157

Figure 88

Case IV advection of θ–component momentum per unit mass
over 0 ≤ θ ≤ 2π . . . . . . . . . . . . . . . . . . . . . . . . . 158

Figure 89

Case V advection of θ–component momentum per unit mass
over 0 ≤ θ ≤ 2π . . . . . . . . . . . . . . . . . . . . . . . . . 159

Figure 90

Case I advection of θ–component momentum per unit mass
over λ3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

Figure 91

Case II advection of θ–component momentum per unit mass
over λ3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

Figure 92

Case III advection of θ–component momentum per unit mass
over λ3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

Figure 93

Case IV advection of θ–component momentum per unit mass
over λ3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

xvii

Figure 94

Case V advection of θ–component momentum per unit mass
over λ3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

Figure 95

Case I advection of θ–component momentum per unit mass
over λ9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

Figure 96

Case II advection of θ–component momentum per unit mass
over λ9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Figure 97

Case III advection of θ–component momentum per unit mass
over λ9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Figure 98

Case IV advection of θ–component momentum per unit mass
over λ9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

Figure 99

Case V advection of θ–component momentum per unit mass
over λ9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

Figure 100

Case I net Reynolds normal stress in θ over 0 ≤ θ ≤ 2π . . . 165

Figure 101

Case II net Reynolds normal stress in θ over 0 ≤ θ ≤ 2π . . 166

Figure 102

Case III net Reynolds normal stress in θ over 0 ≤ θ ≤ 2π . . 167

Figure 103

Case IV net Reynolds normal stress in θ over 0 ≤ θ ≤ 2π . . 168

Figure 104

Case V net Reynolds normal stress in θ over 0 ≤ θ ≤ 2π . . 169

Figure 105

Case I net Reynolds normal stress in θ over λ3 . . . . . . . . 170

Figure 106

Case II net Reynolds normal stress in θ over λ3 . . . . . . . 170

Figure 107

Case III net Reynolds normal stress in θ over λ3 . . . . . . . 171

Figure 108

Case IV net Reynolds normal stress in θ over λ3 . . . . . . . 171

Figure 109

Case V net Reynolds normal stress in θ over λ3 . . . . . . . 172

Figure 110

Case I net Reynolds normal stress in θ over λ9 . . . . . . . . 172

Figure 111

Case II net Reynolds normal stress in θ over λ9 . . . . . . . 173

Figure 112

Case III net Reynolds normal stress in θ over λ9 . . . . . . . 173

Figure 113

Case IV net Reynolds normal stress in θ over λ9 . . . . . . . 174
xviii

Figure 114

Case V net Reynolds normal stress in θ over λ9 . . . . . . . 174

Figure 115

Case I advection of z–component momentum per unit mass
over 0 ≤ θ ≤ 2π . . . . . . . . . . . . . . . . . . . . . . . . . 175

Figure 116

Case II advection of z–component momentum per unit mass
over 0 ≤ θ ≤ 2π . . . . . . . . . . . . . . . . . . . . . . . . . 176

Figure 117

Case III advection of z–component momentum per unit mass
over 0 ≤ θ ≤ 2π . . . . . . . . . . . . . . . . . . . . . . . . . 177

Figure 118

Case IV advection of z–component momentum per unit mass
over 0 ≤ θ ≤ 2π . . . . . . . . . . . . . . . . . . . . . . . . . 178

Figure 119

Case V advection of z–component momentum per unit mass
over 0 ≤ θ ≤ 2π . . . . . . . . . . . . . . . . . . . . . . . . . 179

Figure 120

Case I advection of z–component momentum per unit mass
over λ3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

Figure 121

Case II advection of z–component momentum per unit mass
over λ3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

Figure 122

Case III advection of z–component momentum per unit mass
over λ3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

Figure 123

Case IV advection of z–component momentum per unit mass
over λ3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

Figure 124

Case V advection of z–component momentum per unit mass
over λ3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

Figure 125

Case I advection of z–component momentum per unit mass
over λ9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

Figure 126

Case II advection of z–component momentum per unit mass
over λ9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Figure 127

Case III advection of z–component momentum per unit mass
over λ9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Figure 128

Case IV advection of z–component momentum per unit mass
over λ9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

xix

Figure 129

Case V advection of z–component momentum per unit mass
over λ9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

Figure 130

Case I net Reynolds shear stress in θ–z plane over 0 ≤ θ ≤ 2π 185

Figure 131

Case II net Reynolds shear stress in θ–z plane over 0 ≤ θ ≤ 2π186

Figure 132

Case III net Reynolds shear stress in θ–z plane over 0 ≤ θ ≤ 2π187

Figure 133

Case IV net Reynolds shear stress in θ–z plane over 0 ≤ θ ≤ 2π188

Figure 134

Case V net Reynolds shear stress in θ–z plane over 0 ≤ θ ≤ 2π189

Figure 135

Case I net Reynolds shear stress in θ–z plane over λ3 . . . . 190

Figure 136

Case II net Reynolds shear stress in θ–z plane over λ3 . . . 190

Figure 137

Case III net Reynolds shear stress in θ–z plane over λ3 . . . 191

Figure 138

Case IV net Reynolds shear stress in θ–z plane over λ3 . . . 191

Figure 139

Case V net Reynolds shear stress in θ–z plane over λ3 . . . 192

Figure 140

Case I net Reynolds shear stress in θ–z plane over λ9 . . . . 192

Figure 141

Case II net Reynolds shear stress in θ–z plane over λ9 . . . 193

Figure 142

Case III net Reynolds shear stress in θ–z plane over λ9 . . . 193

Figure 143

Case IV net Reynolds shear stress in θ–z plane over λ9 . . . 194

Figure 144

Case V net Reynolds shear stress in θ–z plane over λ9 . . . 194

Figure 145

Case I turbulence kinetic production by Reynolds shear stress
in θ–z plane over 0 ≤ θ ≤ 2π . . . . . . . . . . . . . . . . . 195

Figure 146

Case II turbulence kinetic production by Reynolds shear stress
in θ–z plane over 0 ≤ θ ≤ 2π . . . . . . . . . . . . . . . . . 196

Figure 147

Case III turbulence kinetic production by Reynolds shear
stress in θ–z plane over 0 ≤ θ ≤ 2π . . . . . . . . . . . . . . 197

Figure 148

Case IV turbulence kinetic production by Reynolds shear
stress in θ–z plane over 0 ≤ θ ≤ 2π . . . . . . . . . . . . . . 198
xx

Figure 149

Case V turbulence kinetic production by Reynolds shear stress
in θ–z plane over 0 ≤ θ ≤ 2π . . . . . . . . . . . . . . . . . 199

Figure 150

Case I turbulence kinetic production by Reynolds shear stress
in θ–z plane over λ3 . . . . . . . . . . . . . . . . . . . . . . 200

Figure 151

Case II turbulence kinetic production by Reynolds shear stress
in θ–z plane over λ3 . . . . . . . . . . . . . . . . . . . . . . 200

Figure 152

Case III turbulence kinetic production by Reynolds shear
stress in θ–z plane over λ3 . . . . . . . . . . . . . . . . . . . 201

Figure 153

Case IV turbulence kinetic production by Reynolds shear
stress in θ–z plane over λ3 . . . . . . . . . . . . . . . . . . . 201

Figure 154

Case V turbulence kinetic production by Reynolds shear stress
in θ–z plane over λ3 . . . . . . . . . . . . . . . . . . . . . . 202

Figure 155

Case I turbulence kinetic production by Reynolds shear stress
in θ–z plane over λ9 . . . . . . . . . . . . . . . . . . . . . . 202

Figure 156

Case II turbulence kinetic production by Reynolds shear stress
in θ–z plane over λ9 . . . . . . . . . . . . . . . . . . . . . . 203

Figure 157

Case III turbulence kinetic production by Reynolds shear
stress in θ–z plane over λ9 . . . . . . . . . . . . . . . . . . . 203

Figure 158

Case IV turbulence kinetic production by Reynolds shear
stress in θ–z plane over λ9 . . . . . . . . . . . . . . . . . . . 204

Figure 159

Case V turbulence kinetic production by Reynolds shear stress
in θ–z plane over λ9 . . . . . . . . . . . . . . . . . . . . . . 204

xxi

KEY TO SYMBOLS AND ABBREVIATIONS

Abbreviations

Description

AFRD

Axial Fan Research and Development Facility

CD

Controlled Diffusion Blade

RCDB

Rotating Controlled Diffusion Blade

X-probe

Axial Fan Research and Development Facility

Coordinates

Description

r–θ–z

Spherical coordinate system

ˆ zˆ
rˆ, θ,

Unit vectors in cylindrical coordinates

Greek Symbols

Description

αo

Measured velocity angle with respect to the probe axis

αAoI

Angle of incidence between incoming velocity and tangent to the leading edge of the fan blade

η

Spanwise position that ranges from [0,1] from hub to tip

κ

Non dimensional kinetic energy at the exit of the fan defined by c.s. 1/2 ρ(V · V ) V · n
ˆ dA [1/(rtip ×ρVtip 2 rtip 2 )]

λ

Axial velocity wake minimum centered azimuthal position of hot-wire probe in fan wake

µ

Dynamic viscosity of air

ν

Kinematic viscosity of air

ξ

Energy losses in the control volume due to dissipation
and viscous effects

ρ

Density of air

Σ

Representative comparison of azimuthal velocity defined
by Vθ (θ) dθ for the different fan configurations
xxii

Φ

Flow coefficient V /Vtip

χ

Non dimensional flow work rate term defined by
P
ˆ dA [1/(rtip × ρVtip 2 rtip 2 )]
c.s. /ρ ρ · V · n

Ψ

Head rise coefficient ∆P/(ρ ∗ Vtip 2 )

Ω

Angular velocity vector of the fan

Roman Symbols

Description

A

Flow area in r-θ – plane of the fan

Ai , Bi , ni

Hotwire calibration constants

c

Chord length of the RCDB (133.9 mm)

c.s.

Control Surface of Control Volume

Cp

Pressure coefficient (Ptap − Patm )/( 21 ρw∞

I-V

Operating conditions selected for further study from performance data (I, II, III, IV, V)

m
˙

Mass flow rate through the fan plane

n
ˆ

Outward facing normal of control volume surfaces

Pshaf t

Shaft power into the control volume

Patm

Atmospheric pressure in the laboratory

Prec

Receiver pressure

Pref

Common reference pressure for all pressure transducers

Ptap

Pressure at the blade surface

q

Turbulence kinetic energy

Qo

Measured velocity magnitude of processed X-probe

rˆ

Position vector in the radial direction

xxiii

Rec

Reynolds number based on airfoil chord length Rec =
(W (rcenter )c)/ν

rsensor

Radial distance to pressure transducer

rtap

Radial distance to surface pressure tap on an instrumented blade

rtip

Tip radius of the fan (0.366 meters)

j

si

Phase averaged sample for revolution: i and “Bin”: j

T

Input shaft torque measurement

t

Time during experimental measurement

V , Vi , V

Velocity vector V = V = Vi = Vr + Vθ + Vz

Vr , Vθ , Vz

Velocity components in cylindrical coordinates Vr , Vθ
measured in in the wake of the fan f = f (t)

Vr , Vθ , Vz

Velocity fluctuation components in cylindrical coordinates Vr , Vθ measured in the wake of the fan f = f (t)

Vtip

Tangential velocity at rtip of the fan

Vz∗ , Vθ∗

Phase averaged wake velocity in the axial and azimuthal
directions f = f (θ)

Vz , Vz

Phase averaged root mean square (RMS) of turbulence
intensity in the axial and azimuthal directions f = f (θ)

Vz Vθ

Phase averaged Reynolds stress in the θ-z–plane f = f (θ)

W

Apparent approach velocity which stagnates on the leading edge of the blade. W = W (r)

xxiv

Chapter 1
INTRODUCTION
1.1

Overview for this Thesis

1.1.1

Motivation

This experimental investigation explored fundamental fluid mechanical phenomena
in an axial fan flow field. More specifically, these experiments focused on how the flow
field changes for different operating conditions of the fan. In addition to providing insight
within the academic community, these results will be utilized by companies in industry to
improve axial fan design. Currently, members of the industrial community are challenged
with regulations and standards regarding noise pollution and mechanical efficiencies.
A Consortium was formed by a group of these companies to fund research that will
advance the field through fundamental investigations of axial fan flow fields. The axial
fan used in the present investigation has been examined in previous investigations in
the Turbulent Shear Flows Laboratory (TSFL) at Michigan State University. These
experimental investigations provide a diverse validation database for the computational
community. This database, to which the present investigation contributes, will support
to development of more accurate computational models for axial fans and improve fan
designs.
1.1.2
1.1.2.1

Previous Investigations
Ph.D. Dissertation – Douglas Neal (2010)

The axial fan used in this investigation was developed to detail the effects of rotation
in a flow field from a stationary reference frame to an equivalent rotating reference frame.
The original experiment, in the stationary reference frame, examined the development
of the boundary layer on a controlled diffusion (CD) airfoil. Controlled diffusion airfoils
1

have their camber and thickness distributions tailored to avoid boundary layer separation
on the suction surface by controlling diffusion (deceleration) from the peak velocity point
to the trailing edge [Gelder, T.F. et. al. (1987)] and [Johnson (1998)]. This airfoil shape
has been extensively studied in wind tunnel experiments and computational simulations
in a stationary reference frame and was selected for comparison in the rotating reference
frame.
A fan was designed with blades that have similar flow fields to the stationary experiment. The blade twist of the rotating controlled diffusion blade (RCDB) was designed
to ensure an identical angle of attack along the span of the blade for the selected design
operating condition. Changing the number of blades installed on the fan effected blade
loading and was compared with the blade loading of the CD blade in the stationary reference frame. Time averaged surface pressure measurements were collected and analyzed
in both experiments for this comparison. The three blade fan blade loading matched
closely with that of the CD blade in the wind tunnel experiment. Experimental and
computational results suggested that the solidity of the fan in the rotating experiment
was analogous to the jet width of the incoming flow to the stationary blade [Neal (2010)].
Additionally, Neal performed velocity hot-wire measurements in the wake of the fan, obtaining three components of the velocity vector, with a four wire method developed for
in his investigation. These data suggested that the radial velocity was small in the wake
region. Therefore, velocity measurements in the wake of the fan can be resolved in the
axial and azimuthal directions neglecting the radial velocity component. The present investigation used the surface pressure and hot-wire wake velocity experimental techniques
to explore the fan flow field.

2

1.1.2.2

M.S. Thesis – Andrew Cawood (2012)

This investigation developed a technique to measure the unsteady surface pressure on
the CD airfoil and RCDB. These measurements examined the development of the suction
side boundary layer these blades. Additionally, this investigation conducted experiments
at an off design operating condition with no pressure difference across the inlet and outlet
of the control volume considered for the fan. The original design operating condition
experienced a pressure drop across this control volume.
1.1.2.3

M.S. Thesis – Behdad Davoudi (2014)

This investigation examined the propagated far field aeroacoustic noise upstream
of the fan. A microphone array was mounted above the fan to record, isolate, and
characterize the fan noise. This investigation was conducted with the three and nine blade
fan configurations. Operating conditions studied in this investigation were restricted to
a pressure rise across the fan. Noise pollution from the downstream helper fan in the
experiment created this restriction. This investigation also obtained velocity magnitude
hot-wire data in the wake of the fan. These data were obtained at five different radial
locations and they were correlated with the turbulence intensity in the wake, propagated
far field noise, and operating condition of the fan.
1.1.3

The Present Investigation — Areas of Contributions

Fan performance and mechanical efficiency were the two main motivations of the
present investigation. The axial fan of this investigation can be assembled with two to
nine blades installed. The three to nine blade fan configurations were selected for study
in the present investigation. Fan integral data were obtained for all seven of these fan
configurations to provide a population of operating conditions to select from for further

3

investigation. Fan integral measurements include the pressure rise across the fan, mass
flow rate though the fan, and the fan shaft torque supplied by the driving motor. These
measurements were collected and analyzed. As a result, the three, five, and nine blade
fan configurations were selected for further/detailed investigations.
Time averaged surface pressure measurements and hot-wire velocity measurements in
the wake of the fan were two additional experimental methods utilized for this investigation. Previous investigations have studied surface pressure measurements on the RCDB;
however, this study expands these measurements to include different radial positions.
Four additional radial locations were available for study. Two component velocity hotwire measurements were also collected downstream in the near-wake of the fan. These
measurements were obtained at the center of the blade span to minimize tip and hub flow
effects. Examination of these data detailed the effect of blade loading on fan efficiency.
All of these measurements supplement the substantial experimental database for this
fan; they are made available as validation data to create more accurate computational
simulations of axial fans.
1.2

Basic Fluid Mechanical Principals
Three subject areas have been selected for this discussion: control volume analysis for

this axial fan; fan performance characteristic curves; characteristic lengths and velocities
of this investigation.
1.2.1

Control Volume Definition

A control volume approach was used to evaluate the global quantities including the
mechanical efficiency of this fan. Figure 1.1 presents the formulation of two control
volumes used to evaluate this fan. The red dashed line displays the boundaries of Control
Volume (1). The black double dot dashed line displays the boundaries of Control Volume
4

(2). The differences of inlet for these control volumes allow the fan to be considered in two
separate types of application. Control Volume (1) considers the fan in an environment
where atmospheric, negligible kinetic energy air is readily available to be moved through
the system. Control Volume (2) is similar to examining a fan in a duct where there will
be kinetic energy at the inlet control surface due to the fan bringing air into the duct
further upstream or supplied by an upstream fan within the duct.
Figure 1.1 also indicated various parameters important for the flow and each control
volume. A cylindrical coordinate system was used for the present investigation when in
the inertial reference frame with +z along the flow path through the fan. The fan rotates
in the −θ direction with a constant azimuthal velocity of 450 revolutions per minute or
7.5 Hertz. The pressure at the inlet control surface for Control Volume (2) is found by
calculating the dynamic pressure from the incoming flow velocity V . V is the average
incoming velocity in the contracted portion of the shroud from the mass flow rate of the
fan and assumed to be constant radially with thin boundary layers on either wall.
Equation 1.1 provides the work rate equation from the First Law of Thermodynamics
for these control volumes. Table 1.1 indicates the physical significance of each term in
the equation and provides the symbol each will be represented by in this document. The
research performed for this investigation produced representative values for (A), (B), and
(C) and allowed for quantitative comparison of the efficiency between fan configurations
and operating conditions. The input torque contributing to the shaft power, Pshaf t =
T orque|Ω|, was evaluated at the indicated location in Figure 1.1 by means of a torque
meter mounted on the shaft. κ will be evaluated as a surrogate quantity at the fan
plane exit surface because the velocity measurements were limited to one radial location

5

1
2

=

Control Volume (1)

=

Control Volume (2)

Torque

|⌦| = 450 RPM

Wake
Velocity
Measurement

r

Figure 1.1 Control volumes for the present investigation

Table 1.1 Work rate equation terms
Term
(A)
(B)
(C)
(D)

Physical Meaning
Input shaft power from motor
Kinetic energy of fluid at bottom control surface
Flow work rate from pressure
Dissipative losses

Notation
Pshaf t
κ
χ
ξ

indicated in Figure 1.1.

Pshaf t =
(A)

1.2.2

1
P
ρ(V · V ) V · n
ˆ dA +
ρ·V ·n
ˆ dA + Losses
c.s. 2
c.s. ρ
(

(1.1)

(D)

(B)

(C)

Fan Characteristic Curve Introduction

Characteristic curves of fan performance display the relationship between the pressure
increase across and the mass flux through the fan plane . An ideal fan would have a linear
6

Ψ
Φ

Figure 1.2 Sample axial fan characteristic curve [Lakshminarayana, B. (1996)]
relationship between these two parameters. Figure 1.2 [Lakshminarayana, B. (1996)]
plots the actual and ideal performance curves for a typical axial fan. The abscissa and
ordinate represent the flow coefficient, Φ, and the head rise coefficient, Ψ, respectively. Φ
is a non-dimensional representation of the average upstream axial velocity, V = m˙ /ρAinlet .
It is made non-dimensional by the tangential speed at the fan blade tip: Vtip = rˆtip × Ω;
Vtip = |Vtip |. Ω is the angular velocity vector of the fan and rtip
ˆ represents the position
vector to the tip of the blade. Ψ is a non-dimensional representation of the pressure
2 . In
difference control volume inlet and outlet: ∆P . It is made non-dimensional by: ρVtip

Figure 1.2 the actual curve reaches its max efficiency around the indicated design point.
Lakshminarayana notes the following two points about Figure 1.2: When the fan is
operated at lower flow coefficient than its design condition it will reach a maximum head
rise coefficient indicating the onset of blade stall; Blade stall will cause uncontrolled flow
separation along the blade surface and decreased efficiency. [Lakshminarayana, B. (1996)]
The present investigation examined various operating conditions along the characteristic
curve for the three, five, and nine blade fan configurations.
7

1.2.3

Characteristic Scales of the Subject Flow

The characteristic scales used in the present investigation are determined by the reference frame in which the measurements are obtained. Measurements obtained in the
inertial reference frame, integral data and hot-wire wake measurements, use the following
characteristic scales. The tangential velocity at the tip of the blade, Vtip , was used as
the characteristic velocity magnitude. The tip radius rtip = |ˆ
rtip | was selected as the
characteristic length. These scales were selected because they appropriately represent
the physical problem (i.e., they are independent variables). As added benefit, they remained constant between all operating conditions and blade configurations of the fan.
Measurements obtained in the rotating reference frame, time averaged surface pressure
measurements, used difference characteristic scales. The apparent velocity magnitude of
the approach flow at the leading edge of the blade, |W | = w∞ = w∞ (r), as a characteristic velocity magnitude that is a function of radial position. The length scale of this
reference frame is the chord length of the blade, c = 133.9 mm. This length and velocity
magnitude (at the center of the span of the blade) are used in the definition of a chord
based Reynolds number, Rec = (W (rcenter )c)/ν , for the flow over the fan. The Reynolds
number for these experiments was of the order ∼ O(105 )

8

Chapter 2
EXPERIMENTAL CONSIDERATIONS
2.1

Axial Flow Research and Development Facility (AFRD)

2.1.1

Overview

The present investigation was conducted using the Axial Fan Research and Development (AFRD) Facility within the TSFL at Michigan State University. This facility was
developed to obtain detailed velocity and performance data for axial fans. The inlet of
this flow facility was designed to accommodate various fans to be used in a variety of
research studies. Previous investigations of axial fans within this facility include: [Morris, S.C. et. al. (2001)], [Dusel (2005)], [Neal, D.R. and Foss, J.F. (2007)], [Neal (2010)],
[Cawood (2012)], [Davoudi (2014)].

(a)

(b)

Figure 2.1 Axial Fan Research and Development Facility: (a) Facility Exterior, (b)
Schematic of Facility [Neal (2010)]

9

Features of the AFRD are labeled in the schematic: Figure 2.1(b). This vertical wind
tunnel transported atmospheric air from above the facility into the upper receiver (D)
by the action of the fan assembly (A). (See Section 2.2) The prime mover (K) delivered
the air from (D) to the atmosphere. The shaft of the test fan was driven by a Reliance
Electric 11.2 kW (15 HP) electric D.C. motor with controlled angular velocity. Air
entering the facility at (A) is directed to the exit nozzles (E on either side of the facility)
into the lower receiver (H). Turning vanes (F) direct those flows, to the inlet of the prime
mover (I). These turning vanes measure the “net moment of momentum flux” via a force
transducer (G). That magnitude is proportional to the mass flow rate [Morris, S.C. et. al.
(2001)]. The “net moment of momentum flux” was measured by the moment of the force
transducer output about a pivot point. Previous investigations with this fan utilized one
turning vane and force transducer to measure the mass flow rate. Due to the higher mass
flow rates produced by the nine blade fan both turning vanes and force transducers were
used in the present investigation. The throttle plate (J) at the exit of the facility was
mechanically driven to adjust the mass flow rate of the system.
2.1.2

Fan Integral Data Experimental Features

Integral fan data were acquired within the AFRD for the present investigation. These
data include pressure rise across the fan, mass flow rate through the fan, input shaft
torque to the fan, and the angular velocity of the fan. The pressure difference between
the upper receiver and atmospheric pressure in the laboratory environment was measured
using a wall pressure tap (B) in Figure 2.1(b). The mass flow rate measurement was made
with the force transducer in the scheme detailed above (Section 2.1.1). A Himmelstein
torque meter, Figure 2.2, was installed on the fan shaft (L) to measure the input shaft
torque to the fan from the driving motor. An optical encoder was installed on the shaft of
10

Figure 2.2 Hammelstiein Torque Sensor
the fan to measure its angular position and the angular velocity of the fan. It records the
position once per revolution. All of these integral data were recorded using an IOTech
Inc. Wave-Book/516E analog to digital acquisition board. This data acquisition board
can sample at 1 MHz over 16 separate 16-bit channels.
2.1.3

Near-Wake Velocity Measurements

Downstream velocity measurements in the near-wake of the fan were obtained using
the traverse system (C) from Figure 2.1(b). A hot-wire probe was mounted in a stationary
position within the wake of the fan to obtain two velocity components as the fan rotated.
These data are referenced to azimuthal positions: θ, for a given radial location: r, and
downstream location: z. A two wire hot-wire probe, (X-probe), was utilized to obtain
these velocity measurements. Figure 2.3 displays a Computer Aided Design drawing of
an X-probe. This probe obtains two components of velocity with accuracy in time. An
approximate frequency response of 5000 Hz was used in this study.
The present investigation obtained measurements with the X-probe component of a
four wire hot-wire (Mitchell) probe, Figure 2.4. The X-probe components of this probe
are indicated by the red lines at the sensor location for visual aid. The Mitchell probes
used for the present investigation were designed and fabricated at the TSFL. They have
11

Figure 2.3 Drawing of X-probe
four 5µm diameter 1 mm active length tungsten sensors. Two of these sensors were
mounted at opposing 45◦ angles to the axis along the probe. The tungsten wire was
copper plated to assist in mounting the wire to the broaches and to provide aerodynamic
isolation of the active sensor. The entire length of each sensor is 3 mm. Two additional
wires were mounted normal to the plane made by the X-probe. (that were parallel to
each another) All four of these wires in tandem can measure the transverse vorticity in
the flow. Further detail in this experimental capability by this type of probe can be
found in [Wallace, J.M. and Foss, J.F. (1995)]. The probe used for these measurements
will be referred to interchangeably as a X-probe and hot-wire probe for the remainder of
this document.
2.1.3.1

Hot-Wire Calibration and Operating Procedure

A TSI IFA-300 constant temperature hot-wire anemometer was utilized for these
measurements. Operating hot-wires with this method maintains the tungsten sensor at
a (nominally) constant temperature and obtains velocity from the convection caused by
the oncoming flow. Therefore, a X-probe must be calibrated in order to be utilized for
experimental measurements to know the relationship between velocity and these volt-

12

Figure 2.4 Sample Mitchell probe utilized for measurements
age fluctuations. The calibration procedure used in the present investigation has been
developed from [Tropea and Foss (2007)]: see Section 5.2.8.
The velocity: Qi , and incoming angle of the flow: αi , are related to the output
voltage by Equation (2.1) where: Ei is the voltage from the sensor; αi is the angle of
the known calibration flow with respect to the probe axis; Qi is the velocity magnitude
of the incoming flow; and Ai , Bi , and ni are the calibration constants that relate the
anemometer output voltage to the flow velocity for each wire. A single wire probe will
evaluate this relationship for i = 1 αi = 0◦ only and not consider variations in α only
obtaining the magnitude of incoming velocity. The X-probe was calibrated at thirteen
angles i = 1–13 (−36◦ , −30◦ , −24◦ , −18◦ , −12◦ , −6◦ , −0◦ , 6◦ , 12◦ , 18◦ , 24◦ , 30◦ , 36◦ )
in order to accurately resolve all incoming velocities between the two bounding angles.
Corresponding Ai , Bi , and ni values are found for all of the different incoming angles for
each wire. These data are used to determine the incoming velocity magnitude and angle
with respect to the probe during the experiment.

Ei 2 (αi ) = Ai (αi ) + Bi (αi )Qi ni (αi )

13

(2.1)

Figure 2.5 Hot-wire calibration facility [Neal (2010)]
Figure 2.5 displays the calibration facility that was used to calibrate the X-probe before
and after experimental use. The probe was mounted in the probeholder: 1, with the
plane of the X (made by the two wires) parallel to the bottom of the calibration facility.
Pressure taps: 9, are used to calculate the velocity of the air exiting the inlet nozzle:
6, by the Bernoulli Equation. The inlet velocity to the probe was (slowly) reduced by
closing the exit throttle plate: 7, to allow for a quasi-steady state calibration. In this
investigation the calibration data were sampled for 60 second intervals at 20 kiloHertz
for each calibration angle. These thirteen data files were then processed to obtain the
corresponding Ai , Bi , and ni from Equation (2.1) for each wire.
Figure 2.6 displays the method that was used to determine the velocity magnitude
and angle for any given instantaneous output voltages from each hot-wire. Each voltage is
processed in Equation (2.1), with the appropriate constants for each wire. These velocities
and angles for the voltages are plotted. The intersection between the two curves made

14

Figure 2.6 Example of how the magnitude Qo and angle αo of the velocity are found
relative to the probe from the output voltages [Neal (2010)]
from the processed data give the velocity magnitude and angle of the oncoming flow with
respect to the probe axis. Further details regarding calibration methods for a X-probe
can be found in [Tropea and Foss (2007)]: see Section 5.2.8.
Proper experimental technique requires calibration of the probe before and after obtaining measurements in the wake of the fan. This ensures that the probe response did not
drift significantly over the course of the experiment. Typically a calibration is accurate
for three hours. The validity of the two calibrations was confirmed by processing a set of
calibration data with Ai , Bi , and ni from both calibrations and comparing the difference
in velocity magnitude and angle ensuring that they were within 5% of each other. A histogram of the angles obtained by the probe during each experimental measurement was
analyzed for each data set to ensure that maximum percentage of pitch angles: αi , at the
probe did not exceed the calibration range. The extremes of the calibration angles bound

15

(a)

(b)

Figure 2.7 Mounted X-probe in fan wake
the measurement capability of this device. Maximizing the measurement capability of
the probe in this regard was accounted for in these measurements.
2.1.3.2

Stationary Mount

Once calibrated, the X-probe was mounted on the traverse system in the wake of the
fan, Figure2.1(b)(C). Figure 2.7 displays the X-probe mounted in the wake of the fan
from two different perspectives. Figure 2.8 provides a close up of the probe mounted
downstream of the fan. The measurement portion of the X-probe was mounted 6 millimeters below the fan in z and at mid-span of the blade to minimize hub and tip flow
effects. Small variations in fabrication of the fan blades caused the distance in z to vary
slightly from blade to blade. Due to the fast response of the X-probe an accurate time
history of the velocity in the wake was obtained. These data were phase averaged to
calculate the means and statistics of the velocity in the wake for all θ using the optical
encoder signal on the shaft indicating fan position.

16

Blade
Trailing Edge

6mm

Figure 2.8 Close up of probe in fan wake

Figure 2.9 Incoming velocity profile

2.2

Controlled Diffusion Blade Axial Fan

2.2.1

Incoming Velocity

The apparent velocity to an observer on the leading edge of the fan blade can be
represented by Equation (2.2). The velocity of the air with respect to the blade: Va/b ,
is the vector sum of the velocity of the air with respect to the ground: Va/g , and the
velocity of the ground with respect to the blade: Vg/b .

Va/b = Va/g + Vg/b
17

(2.2)

This description is equivalent to Equation (2.3) presented with traditional turbomachinery nomenclature. Where: W represents the apparent incoming velocity to the blade
given by the vector sum of the axial inlet velocity V and tangential velocity at the tip
U . The definitions of V and U are provided in Equation (2.4).

W =V −U

U = U (r) = rˆ × Ω ; V =

(2.3)

m
˙
ρAinlet

(2.4)

The incoming velocity to a blade on the fan is displayed in Figure 2.9 in traditional
turbomachinery nomenclature. The angle of incidence αAoI is also noted in Figure 2.9
and is positive as indicated. αAoI is the angle between the apparent incoming velocity
W and the tangent to the leading edge of the blade.
2.2.2

Fan Assembly

This modular fan has nine blades available for installation and can be assembled
with two to nine blades. Figure 2.10(a) displays the five blade fan mid-assembly. Figure
2.10(b) displays the inner and outer shroud of the fan while the five blade fan is mounted.
2.2.3
2.2.3.1

Experimental Features
Instrumented Blades

Five of the nine available blades for the fan have pressure taps installed on both
surfaces of the airfoil. These blades have the same stream wise distribution of pressure
taps. There are eighteen pressure taps along this chord line at a given span position for
each blade and an additional three span wise taps located near the trailing edge of the
blade. A photo indicating the locations of the pressure taps on the suction and pressure
18

Blade

Outer
Shroud

(a)

Hub

(b)

Figure 2.10 Fan Assembly: (a) five blade fan mid-assemby [Neal (2010)], (b) five blade
fan mounted and shroud

Table 2.1 Pressure tap locations on the suction side of the blade
Tap
x/c

1
.013

2
.030

3
.052

5
.087

6
.149

7
.403

9
.534

11
.679

21
.858

22
.881

23
.899

24
.922

25
.978

side of the center span instrumented blade is provided in Figure 2.11.
The leading and trailing edge regions of the suction side of the blade have higher
concentrations of pressure taps to examine the boundary layer behavior in those regions
of the blade. The leading edge region will indicate the transition from a laminar to a
turbulent boundary layer. The trailing edge region indicates if the flow is still attached
leaving the trailing edge.
Tables 2.1 and 2.2 provide the chord position, x/c, of the eighteen stream wise pressure
Table 2.2 Pressure tap locations on the pressure side of the blade
Tap
x/c

4
.067

8
.400

10
.530

19

12
.675

29
.929

2

1

3

Leading Edge
5
4

6
7

8

9

10

11
12
21
22

29

23
25A

24

25
25B

Trailing Edge

25C

Figure 2.11 Suction and pressure side, left and right respectively, pressure taps indicated
x/c distances corresponding to Tables 2.1 and 2.2

Table 2.3 Pressure tap locations on the suction side of the blade along the trailing edge
Tap
Blade Color
Red
Blue
Green
Black
Yellow

25A

25B

25

25C

.119
.310
.500
.770
.873

.196
.387
.577
.847
.950

η
.073
.264
.454
.724
.827

.096
.287
.477
.747
.850

taps. Table 2.3 provides the radial position of the pressure taps that are represented nondimensionally as η defined by Equation (2.5). Note that η ranges from 0–1 from hub–tip.

η=

rtap − rhub
rtip − rhub

(2.5)

Figure 2.12 provides the five different radial locations that pressure measurements can
be collected with this equipment. The color of the line indicating each radial location will
be used to distinguish the different radial locations in plots of the experimental results.
20

𝜂𝜂 =0.302

𝜂𝜂 =0.119

𝜂𝜂 =0.603

𝜂𝜂 =0.500

𝜂𝜂 =0.865

Figure 2.12 Streamwise distribution of pressure taps and η locations

2.2.3.2

Instrumentation in the Rotating Environment

An instrument cluster was developed to mount within the hub of the fan to obtain
various experimental data. Figure 2.13 displays the instrument cluster indicating the
features of interest for the present investigation. The measurement capabilities of this
instrument cluster will be detailed in the following subsections.
2.2.3.2.1

Mean Surface Pressure

Time averaged pressure measurements were obtained in the rotating reference frame
utilizing 21 pressure transducers. Figure 2.14 displays the scale of the All Sensors model
D2-4V-MINI pressure transducers that were installed in the instrument cluster. Seven of
these transducers have ±5 inch H2 O measurement ranges and the other fourteen transducers have ±1 inch H2 O measurement ranges. To operate a pressure transducer, a
known reference pressure must be connected on one side of the measurement diaphragm.
21

Thermocouple
Common
Pressure
Reference
Pressure
Transducers
Data Acquisition Card

Figure 2.13 Instrument cluster

Figure 2.14 Pressure transducer scale [Neal (2010)]
The instrument cluster is equipped with a common reservoir for all of the pressure transducers; see Figure 2.13. This chamber is open to the environment inside of the hub which
is the pressure inside the receiver: Prec . The pressure differential measurement between
the atmosphere and the upper receiver detailed in Section 2.1.2 allows for the common
reference to be brought to atmospheric pressure: Patm .
Centrifugal forces from the rotating environment act on the air within the connecting
tubes from the pressure tap (hypodermic tubing) to the pressure transducer. A method
was developed [Neal (2010)] to account for these effects given in Equation (2.6). ∆Pcorr1
and ∆Pcorr2 are the centrifugal corrections for each tube on either side of the diagram

22

Figure 2.15 Pressure measurement schematic [Neal (2010)]
of the pressure transducer.

∆Pcorr1 =

2 − r2
ρ|Ω|2 (rtap
sensor )

2

; ∆Pcorr2 =

2 )
2
− rref
ρ|Ω|2 (rsensor

2

∆Pactual = ∆Pmeasured + ∆Pcorr1 + ∆Pcorr2

(2.6)

(2.7)

Figure 2.15 indicates the radial locations of the pressure taps and sensors respectively.
Equation (2.7) provides the formulation for the actual pressure difference between the
pressure tap and common reference chamber, ∆Pactual = Ptap − Pref . This equation can
be simplified to Equation (2.8) because the common reference chamber was installed at
the axis of rotation, rref = 0.

∆Pactual = ∆Pmeasured +

Ptap(gage) = ∆Pmeasured +

2
ρ|Ω|2 rtap

2
ρ|Ω|2 rtap

2

2

+ (Prec − Patm )

(2.8)

(2.9)

The pressure difference can be further adjusted to gage pressure by adding the pressure
difference between the upper receiver pressure and atmosphere, Prec − Patm , to the right
hand side of Equation (2.8) provided in Equation (2.9). The gage pressures at the taps
23

was used in the calculation of pressure coefficients for the present investigation.
2.2.3.2.2

Fluctuating Surface Pressure

Cawood (2012) developed a method using microphones to obtain the unsteady surface
pressure measurements at each pressure tap for the η = 0.5 instrumented blade. These
measurements were used to characterize the behavior of the boundary layer on the suction
surface of the blade for two different operating conditions. This capability was not
utilized for the present investigation. The details of the development, use and results of
this measurement capability are found in Surface Pressure Measurments on a Rotating
Controlled Diffusion Blade [Cawood (2012)].
2.2.3.2.3

Rotating Hot-Wire Wake Velocity

Neal (2010) and Cawood (2012) both obtained hot-wire data in the wake of a fan
blade with the probe moving with the fan. Similar measurements were not obtained
for any of the cases within the present investigation. The details of the development,
use, and results of this measurement capability are found in The Effects of Rotation on
the Flow Field Over a Controlled Diffusion Airfoil [Neal (2010)] and Surface Pressure
Measurments on a Rotating Controlled Diffusion Blade [Cawood (2012)].
2.2.3.2.4

Data Acquisition

Measurements within the hub assembly are obtained with the IOTech Inc. DaqBoard/
3035USB data acquisition board. This data acquisition board sampled at 1 MegaHertz
over 64 separate 16-bit channels. The position of this data acquisition board is indicated
in Figure 2.13. In order for this board to be controlled and its obtained data stored,
signals were transmitted to and from the instrument cluster via a slip ring (Figure 2.16)
installed on the shaft. Electrical connections on the rotating and stationary side of the
slip ring connected the power signal to the instrument cluster along with a data USB
24

connection. The digital signal minimized data loss using the slip ring assembly. This is in
contrast to an analogue signal that would be degraded from the electrical noise produced
by the slip ring assembly. USB cable length limits forced the computer controlling the
DaqBoard to be placed inside the AFRD. It was controlled remotely outside of the facility
via ethernet connection.

Figure 2.16 Slip ring assembly at the base of the fan shaft [Neal (2010)]

25

Chapter 3
RESULTS AND DISCUSSION
3.1

Fan Integral Measurements
Initial experiments for the present investigation focused on obtaining integral data

for a variety of fan configurations. The seven separate configurations with three to nine
blades were selected for this comparison. Obtained integral data measurements consisted
of the pressure rise across the fan plane, the mass flow rate though the fan plane, and
the input shaft torque to the fan supplied by the driving motor, as discussed in Section
2.1.2.
3.1.1

Characteristic and Torque Curves

The characteristic curve examines the behavior of the pressure rise from inlet to exit
of the selected control volume of interest via the head rise coefficient: Ψ, as a function
of the flow coefficient: Φ. Control Volume (1) was selected to compare the the head rise
coefficient and flow coefficient for the selected fans. Ψ1 and Φ are defined in Equation
(3.1). The data collected from the three to nine blade fans with Control Volume (1) is
displayed in Figure 3.1. These data were collected for a low flow coefficient up to the
zero head rise coefficient operating condition.

Ψ1 =

V
Prec − Patm
; Φ=
2
Vtip
ρVtip

(3.1)

Each of the fans exhibit similar behavior as the flow coefficient is decreased from the zero
head rise coefficient (right to left on Figure 3.1). The trajectory of these curves is different
between fan configurations and corresponds to higher magnitude slope as the number
of blades is increased moving from left to right. One major difference occurs midway

26

3
4
5
6
7
8
9

0.16

Ψ1

0.12

Blade
Blade
Blade
Blade
Blade
Blade
Blade

0.08

0.04

0

0

0.1

0.2

0.3

0.4

0.5

0.6

Φ
Figure 3.1 Three to nine blade fan characteristic curves
through the span of mass flow rates considered for these fans. All fan configurations,
except for the five blade fan, experience a discontinuity in the performance curve. This
discontinuity manifests itself as a head rise coefficient jump for three and four blade fans,
a smooth curve for the five blade fan, and a head rise coefficient drop for the six through
nine blade fans with decreasing flow coefficient. This behavior of the performance curve
suggests that the flow is separating through aerodynamic stall. The different behavior at
these discontinuities was selected as an area of interest for further study in the present
investigation
Torque measurements were also obtained for these operating conditions of the seven
fan configurations. These data are presented in Figure 3.2. The non-dimensional torque,
T∗ , is defined by Equation (3.2). These curves have been plotted over the range of low
flow coefficient, 0.5, to the flow coefficient of zero pressure rise for each individual fan.
These curves show a similar trend between number of blades and the amount of torque

27

0.12
0.1

T

∗

0.08
0.06
0.04

3
4
5
6
7
8
9

0.02
0

0

0.1

0.2

0.3

0.4

Blade
Blade
Blade
Blade
Blade
Blade
Blade
0.5

Φ
Figure 3.2 Three to nine blade fan torque curves
required to drive the fan at the same flow coefficient. The maximum torque for these
fans is located on the right side of the discontinuity of the characteristic curves from
Figure 3.1. It is important to note that the minimum operating torque is found on the
low flow coefficient side of the characteristic curve and suggests that separated flow on
the suction side of the blades is decreasing the necessary torque to drive the fan. The
torque will eventually increase as flow coefficient moves to zero from the fan pressurizing
the receiver.
T∗ =

3.1.2

T orque
2 r2
rtip × ρVtip
tip

(3.2)

Operating Configurations and Conditions Selection

Operating conditions were selected from the characteristic curve for further study in
the present investigation. The three, five, and nine blade fans were selected to examine
the differences between the three characteristic curves at the “stall” condition. Additionally, five operating conditions were selected for these fans. These operating conditions
28

0.16

3 Blade Cases
5 Blade Cases
9 Blade Cases

III
V

0.12

IV

Ψ1

V
0.08
V

II

IV/III
II

IV
III II

0.04

0

0

0.1

0.2

Φ

0.3

I

I

I
0.4

0.5

Figure 3.3 Experimental operating conditions indicated on characteristic curves for
Control Volume (1)
have been marked on Figure 3.3 with roman numerals to represent Cases: I, II, III, IV,
and V. These operating conditions represent five distinct points on the three and nine
blade fan configurations and four operating points on the five blade configuration. Case I
was selected to compare the zero head rise coefficient condition (Control Volume (1)) for
each fan. Case II was selected to represent a point before the onset of the discontinuity
for the three and nine blade fans. Case III was selected as the right hand side of the
discontinuity on the characteristic curve for the three and nine blade fans. Case IV was
selected as the low flow coefficient side of the discontinuity on the characteristic curve
for the three and nine blade fans. The five blade fan considers these points as one with
paired III and IV operating condition since its characteristic curve is continuous. Case
V is a low flow coefficient operating condition for the three, five, and nine blade fans.
The characteristic curve was also evaluated with Control Volume (2) for the fan
configurations of interest in the present investigation. These characteristic curves are
29

III

II

IV

0.16

I

V
0.12

Ψ2

V

0.08

V

IV

IV/III

II

I
I

II
III

0.04
0

3 Blade Cases
5 Blade Cases
9 Blade Cases
0

0.1

0.2

Φ

0.3

0.4

0.5

Figure 3.4 Experimental operating conditions indicated on characteristic curves for
Control Volume (2)
provided in Figure 3.4. The head rise coefficient for Control Volume (2) is given by
Equation (3.3). These data continue to have the discontinuity in the performance curve
but they are noticeably flattened. There is less variation between the different conditions
in the head rise coefficient as defined for Control Volume (2). The number of blades on
the fan has a direct relationship with how much variation in pressure is present as Φ is
varied. This is a direct effect of the flow coefficient increasing as the number of blades is
increased for similar head rise coefficients in Control Volume (1), where the inlet kinetic
energy is negligible into the control volume.

Ψ2 =

Prec − Patm + 1/2ρV
2
ρVtip

2

(3.3)

The the torque curves for the selected fan configurations and operating conditions have
been indicated on Figure 3.5.

30

0.12
IV/III

0.1
V

T∗

0.08

V

V

0.04

IV/III

I

II

IV/III

0.06

II

II

I

I

0.02
0

3 Blade Cases
5 Blade Cases
9 Blade Cases
0

0.1

0.2

0.3

0.4

0.5

Φ
Figure 3.5 Experimental operating conditions indicated on torque curves
Understanding the oncoming flow to the fan blades is paramount in interpreting the
results from the present investigation. Figure 3.6, provides the angle of incidence as a
function of span location: η for the different fan configurations and operating conditions.
As the conditions move up the characteristic curve from I to V there is a distinct increase
in angle of incidence. This is caused the change in mass flow rate from condition I to V.
Figure 2.9 displays the relationship between the mass flow rate and the angular velocity of
the fan that determined the apparent approach velocity and its angle with respect to the
tangent of the leading edge of the blade. For all of these experiments the angular velocity
was held constant. Therefore, as the mass flow rate is decreased the angle between the
apparent incoming velocity, W , and the leading edge tangent direction will increase.
Thus, the angle of incidence will increase in the positive direction. As the incidence
angle of the flow is increased it will become difficult for the CD blade to control the flow
on the suction side of the blade. The angles of incidence for these operating conditions

31

3 Blade Fan

60

I
II
III
IV
V

αAoI

40
20
0
−20

0

0.2

0.4

0.6

0.8

5 Blade Fan

60

I
II
III,IV
V

40

αAoI

1

20
0

−20

0

0.2

0.4

0.6

0.8

9 Blade Fan

60

I
II
III
IV
V

40

αAoI

1

20
0

−20

0

0.2

0.4

0.6

0.8

1

η

Figure 3.6 Angle of incidence αAoI across the blade for all operating conditions and fan
configurations

32

suggest that there will be flow separation on the suction side of the blade at low flow
coefficient. The angle of incidence becomes more favorable at the tip of the blade. This
is a product of U (r) component of the approach velocity from Figure 2.9. The effects of
flow separation from high the angle of incidence occur at a higher flow coefficient as the
number of blades installed on the fan is decreased.
3.2

Pressure Coefficients
Time averaged surface pressure measurements were obtained for the selected operating

conditions and fan configurations for the present investigation. These measurements
provide the pressure difference between the surface pressure and laboratory atmospheric
pressure which can be expressed non-dimensionally as the pressure coefficient. Equation
(3.4) provides the definition of the pressure coefficient. The pressure coefficients obtained
in the present investigation were used to evaluate a non-dimensional lift coefficient for
the blade. Equation (3.5) provides the integral that was evaluated across the blade
surface to provide a representative value of lift. This lift coefficient is representative of
one blade within the system. It is also useful to consider the lift generated by all of the
blades installed on the fan to compare each fan configuration. These pressure data were
sampled at 5 kHz for a duration of 10 seconds and averaged over five separate samples.

CP =
1

Cl =

0

(Ptap − Patm )
2
1/2ρw∞

(3.4)

1
0

(CP,pressure − CP,suction ) d(x/c) dη

33

(3.5)

1.75
1.5

Cl Per Blade

1.25
1
0.75
0.5
3 Blade Fan
5 Blade Fan
9 Blade Fan

0.25
0

0

0.1

0.2

0.3

0.4

0.5

0.6

Φ
Figure 3.7 Non-dimensional lift coefficient for single blade

3.2.1

Lift Analysis

Figure 3.7 provides the non-dimensional lift coefficient for an individual blade on the
three, five, and nine blade fans. An opposite trend between the overall lift coefficient for
an individual blade is apparent as the number of blades is increased. The non-dimensional
lift per blade is maximized with the three blade fan configuration. Figure 3.8 provides the
overall lift for the entire fan assembly which provides greater overall lift with increasing
number of blades. This increase in lift correlates with the increase in torque necessary to
drive the fan from Figure 3.5. The integral measurements correlate with the measured
surface pressure providing validation that these measurement techniques are responding
appropriately.
3.2.2

Pressure Coefficient Profiles

Figures 3.9–3.13 provide the complete pressure coefficient profiles for the three, five,
and nine blade fan configurations. Each of these figures represents a single operating

34

10

Cl Entire Fan

8

6

4

2
3 Blade Fan
5 Blade Fan
9 Blade Fan
0

0

0.1

0.2

0.3

0.4

0.5

0.6

Φ
Figure 3.8 Non-dimensional lift coefficient for entire fan
condition (Case I, Case II, etc.) and will be referred to by operating condition in this
section of the document. The legend for each of these figures is located in the nine blade
fan section of the figure. The ordinates of these axes were reversed to visualize the suction
side and pressure side of the blade as they are perceived when viewing the profile of the
blade. These distributions were used in Equation (3.5) to evaluate Cl . These figures are
provided by Case and fan configuration in Appendix A.
Overall, the magnitude of the pressure coefficients near the leading edge of the suction
surface of the blade shows an opposite trend to the number of the installed fan blades.
The pressure coefficients are much smaller in magnitude on the nine blade fan compared
to the three and five blade fans for all Cases I–V. This is interpreted as a of a blade
to blade interaction suppressing the suction side pressure on the nine blade fan. The
proximity of the adjacent pressure side of the next blade would have the ability to be
the causal factor for these differences. The magnitude of the pressure coefficients is

35

comparable for the three and five blade fans for Cases II–V. Case I shows distinctly
different pressure coefficient distributions for the three different fan configurations. The
angles of incidence for Case I across the blade for all three fans are more favorable. The
boundary layer develops differently from leading to trailing edge at different η locations
across the blade surface. The negligible variations between Case IV and V suggest that
the flow is separated on the suction surface of the blade at Case IV. Moving to a lower
flow coefficient does not significantly impact these distributions. The three and five blade
fans do not experience much change between Case II–V. This may indicate that the flow
has separated for these fan configurations at Case II. This suggests that the discontinuity
in the characteristic curve for the six to nine blade configurations may be influenced by
the onset of stall. It is important to note that the fan was designed to operate at a flow
coefficient that would create a pressure drop across the fan when considering Control
Volume (1). The present investigation was purposefully limited to Ψ1 ≥ 0.

36

3 Blade Fan

3

- Cp

2
1
0
−1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.6

0.7

0.8

0.9

1

5 Blade Fan

3

- Cp

2
1
0
−1

0

0.1

0.2

0.3

0.4

0.5

9 Blade Fan

3

η=0.119
η=0.302
η=0.500
η=0.603
η=0.865

- Cp

2
1
0
−1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x/c

Figure 3.9 Pressure coefficients at Case I

37

0.9

1

3 Blade Fan

3

- Cp

2
1
0
−1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.6

0.7

0.8

0.9

1

5 Blade Fan

3

- Cp

2
1
0
−1

0

0.1

0.2

0.3

0.4

0.5

9 Blade Fan

3

η=0.119
η=0.302
η=0.500
η=0.603
η=0.865

- Cp

2
1
0
−1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x/c

Figure 3.10 Pressure coefficients at Case II

38

0.9

1

3 Blade Fan

3

- Cp

2
1
0
−1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.6

0.7

0.8

0.9

1

5 Blade Fan

3

- Cp

2
1
0
−1

0

0.1

0.2

0.3

0.4

0.5

9 Blade Fan

3

η=0.119
η=0.302
η=0.500
η=0.603
η=0.865

- Cp

2
1
0
−1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x/c

Figure 3.11 Pressure coefficients at Case III

39

0.9

1

3 Blade Fan

3

- Cp

2
1
0
−1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.6

0.7

0.8

0.9

1

5 Blade Fan

3

- Cp

2
1
0
−1

0

0.1

0.2

0.3

0.4

0.5

9 Blade Fan

3

η=0.119
η=0.302
η=0.500
η=0.603
η=0.865

- Cp

2
1
0
−1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x/c

Figure 3.12 Pressure coefficients at Case IV

40

0.9

1

3 Blade Fan

3

- Cp

2
1
0
−1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.6

0.7

0.8

0.9

1

5 Blade Fan

3

- Cp

2
1
0
−1

0

0.1

0.2

0.3

0.4

0.5

9 Blade Fan

3

η=0.119
η=0.302
η=0.500
η=0.603
η=0.865

- Cp

2
1
0
−1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x/c

Figure 3.13 Pressure coefficients at Case V

41

0.9

1

The pressure coefficients from the spanwise trailing edge taps (Table 2.3) were also
obtained during these experiments, Figure 3.14. These data show a general trend of
increasingly negative pressure as the flow coefficient is decreased.

42

3 Blade Fan

2

- Cp

1.5
1
0.5
0
−0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.6

0.7

0.8

0.9

1

5 Blade Fan

2

- Cp

1.5
1
0.5
0
−0.5

0

0.1

0.2

0.3

0.4

0.5

9 Blade Fan

2

I
II
III
IV
V

- Cp

1.5
1
0.5
0
−0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

Figure 3.14 Pressure coefficient along the trailing edge of the blade for all cases and fan
configurations

43

3.3

Wake Velocity
Hot-wire velocity measurements were obtained in the near-wake of the fan for the

present investigation. These measurements were obtained 6 mm below the fan exit plane
at the η = 0.5 span wise location. The stationary probe was able to resolve velocity
components: Vθ , Vz in time with an approximate frequency response of 5 kHz. Velocity
statistics, mean and the turbulent fluctuations, were derived from these data using a
phase averaging scheme. Figure 3.15 shows a schematic hot-wire measurements in the
near-wake of the fan. The right hand coordinate system: r, θ, z, and positive z aligned
with V , results in Ω < 0 rotational velocity for the blade. Vmeas. is the measured
velocity at the probe that is composed of the axial and azimuthal components. The
selected coordinate system produces velocity components at the probe of opposite sign
+Vz and −Vθ . λ in this figure is detailed later in the document in reference to Figures
3.16 and 3.17.
Table 3.1 Ensemble array used for phase averaging scheme
Revolution

1

Spatial “Bins” in θ
2
3
...

1

s11

s21

s31

s...
1

s6000
1

2

s12

s22

s32

s...
2

s6000
2

3

s13

s23

s33

s...
3

s6000
3

...

s1...

s2...

s3...

s...
...

s6000
...

∼2600

s1∼2600

s2∼2600

s3∼2600

s...
∼2600

s6000
∼2600

6000

These data were obtained sampling eight 16-bit channels sequentially at 50 kHz for six
44

Figure 3.15 Hot-wire measurement schematic at an instant of time
minutes. The constant fan angular velocity of 7.5 Hz allowed for ∼2600 revolutions to be
obtained for each fan configuration and operating condition. The optical encoder on the
shaft, indicating position, allowed for the time signal to decomposed the velocity signal
into individual segments that corresponded with a single revolution. These selected data
were then filtered into “Bins” that were selected to represent 0.06◦ in the θ direction.
Figure 3.15 provides a visual representations these “Bins”. Each “Bin” was filled with
selected values from the 2600 total revolutions to build an ensemble array to obtain
velocity statistics at 6000 individual points in space. Table 3.1 provides an example of
the ensemble array that was created from the time history. Each “Bin” was filled with
j

samples from the time series si . Subscript i references the revolution the sample was
measured in and superscript j references the “bin” in which this value is stored. Note
that the “Bin” size was selected to include one sample at minimum from the original
signal to ensure that every “Bin” was contributed to per revolution. The average value

45

from multiple points was used in each “Bin” with more than one individual sample. The
time series the probe observed begins at s11 traversing left to right and moving down the
table until the last sample s6000
∼2600 . The vector from the columns in this array represent
the “time” history generated for each individual “Bin”. Evaluating velocity statistics for
each “Bin” is similar to velocity statistics from measurements in a non-rotating system
such as a hot-wire in a turbulent jet.
A representative sample of these data is provided in Figure 3.16. These mean axial
velocity data show the blade traces for each fan configuration. Previously, it was noted
that the trailing edges of the fan blades are not at a uniform height above the probe.
The lowest blade on the fan in the z direction was recorded for each experimental run
and has been marked in Figure 3.16. The marked angular segment λi : i = 3, 5, 9,
is the segment that each individual fan blade occupies on the respective fan assembly.
These individual segments will be used for comparison between the different velocity
measurement values and operating conditions. All of the plotted data from the hot-wire
velocity wake measurements from Section 3.3 have been presented in a format similar to
Figure 3.16 within Appendix B.
Figure 3.17 provides a view of Figure 3.16 with all three blades centered on the wake
minima. This presentation allows for better comparison between the fan configurations
at different operating conditions. Additionally, centering the wake minima at λ = 0
allows for a simple definition of the pressure and suction side of the wake. The probe
measures velocities in time and is first exposed to the pressure side of the fan blade as
the fan rotates towards it, as indicated in Figure 3.15. The abscissa for Figures 3.16 and
3.17 are plotted with increasing time. Hence, the pressure side of the wake is indicated
in −λ and the suction side is indicated in +λ. Figure 3.15 provides a representation of
46

3 Blade Fan
0.8
0.7

λ3	


V z∗

0.6
0.5
0.4
0

π/2

π

3π/2

2π

5 Blade Fan
0.8
λ5	


0.7

V z∗

0.6
0.5
0.4
0

π/2

π

3π/2

2π

3π/2

2π

9 Blade Fan
0.8
0.7

V z∗

0.6
0.5
λ9	


0.4
0

π/2

π

θ

Figure 3.16 Complete fan wake velocity profile sample

47

0.7
0.6
0.5

V z∗

0.4
λ9	


0.3
0.2

3 Blade Fan
5 Blade Fan
9 Blade Fan

0.1
0
−π/3

−π/6

0

π/6

π/3

λ

Figure 3.17 Single blade fan wake velocity profile sample
λ with respect to the blade. These designations will be important as they will be used
to interpret these results. However, it is important to indicate that the probe observed
the effects of the entire blade passing above from suction side to pressure side in time.
Therefore, the driving cause for the probe’s observations cannot be isolated to the suction
or pressure side individually, but rather considered for the entire blade. The indicated
λ9 on Figure 3.17 is the range of lambda in which all hot-wire velocity measurements
will be displayed within the body of this document. A complete collection of the figures
presented in Section 3.3 are available in a format similar to Figure 3.17 within Appendix
B.
3.3.1

Mean Wake Velocities

Mean velocities were evaluated using the phase averaging scheme detailed in the
previous section. These phase averages provided 6000 data points in θ with distinct
velocities in the θ and z directions. Figures 3.18-3.22 provide the mean axial velocity for

48

the three, five, and nine blade fans for Cases I–V. Equation (3.6) provides the definition
of the non-dimensional axial velocity Vz∗ . These data are presented in this section and
will be discussed in conjunction with evaluation of the kinetic energy flux at the common
exit of both control volumes.

Vz∗ =

Vz
Vtip

(3.6)

0.7
0.6
0.5

V z∗

0.4
0.3
0.2
3 Blade Fan
5 Blade Fan
9 Blade Fan

0.1
0
−π/9

−π/18

0

π/18

λ

Figure 3.18 Case I Phase averaged axial velocity over λ9

49

π/9

0.7
0.6
0.5

V z∗

0.4
0.3
0.2
3 Blade Fan
5 Blade Fan
9 Blade Fan

0.1
0
−π/9

−π/18

0

π/18

π/9

λ

Figure 3.19 Case II Phase averaged axial velocity over λ9

0.7
0.6
0.5

V z∗

0.4
0.3
0.2
3 Blade Fan
5 Blade Fan
9 Blade Fan

0.1
0
−π/9

−π/18

0

π/18

λ

Figure 3.20 Case III Phase averaged axial velocity over λ9

50

π/9

0.7
0.6
0.5

V z∗

0.4
0.3
0.2
3 Blade Fan
5 Blade Fan
9 Blade Fan

0.1
0
−π/9

−π/18

0

π/18

π/9

λ

Figure 3.21 Case IV Phase averaged axial velocity over λ9

0.7
0.6
0.5

V z∗

0.4
0.3
0.2
3 Blade Fan
5 Blade Fan
9 Blade Fan

0.1
0
−π/9

−π/18

0

π/18

λ

Figure 3.22 Case V Phase averaged axial velocity over λ9

51

π/9

Figures 3.23-3.27 provide the mean azimuthal velocity for the three five and nine
blade fans for Cases I–V. Equation (3.7) provides the definition of the non-dimensional
azimuthal velocity Vθ∗ . These data are presented in this section and will be discussed in
conjunction with evaluation of the kinetic energy leaving the fan plane in both Control
Volumes.
Vθ∗ =

Vz
Vtip

(3.7)

0.7
3 Blade Fan
5 Blade Fan
9 Blade Fan

0.6

− V θ∗

0.5
0.4
0.3
0.2
0.1
0
−π/9

−π/18

0

π/18

λ

Figure 3.23 Case I Phase averaged azimuthal velocity over λ9

52

π/9

0.7
3 Blade Fan
5 Blade Fan
9 Blade Fan

0.6

− V θ∗

0.5
0.4
0.3
0.2
0.1
0
−π/9

−π/18

0

π/18

π/9

λ

Figure 3.24 Case II Phase averaged azimuthal velocity over λ9

0.7
3 Blade Fan
5 Blade Fan
9 Blade Fan

0.6

− V θ∗

0.5
0.4
0.3
0.2
0.1
0
−π/9

−π/18

0

π/18

λ

Figure 3.25 Case III Phase averaged azimuthal velocity over λ9

53

π/9

0.7
3 Blade Fan
5 Blade Fan
9 Blade Fan

0.6

− V θ∗

0.5
0.4
0.3
0.2
0.1
0
−π/9

−π/18

0

π/18

π/9

λ

Figure 3.26 Case IV Phase averaged azimuthal velocity over λ9

0.7
3 Blade Fan
5 Blade Fan
9 Blade Fan

0.6

− V θ∗

0.5
0.4
0.3
0.2
0.1
0
−π/9

−π/18

0

π/18

λ

Figure 3.27 Case V Phase averaged azimuthal velocity over λ9

54

π/9

A representative swirl number has been formulated to express the relative magnitudes
of the azimuthal velocities for the different configurations. The swirl number Σ is given
by Equation (3.8). Results, provided in Figure 3.28, indicate that there is a direct
relationship between the number of blades installed on the fan and the magnitude of the
azimuthal velocity at the fan exit plane.

Σ = (N umber of Blades) ∗

λi /2
−λi /2

Vθ∗ (s) ds

(3.8)

3.5
3 Blade Fan
5 Blade Fan
9 Blade Fan

3
2.5

Σ

2
1.5
1
0.5
0

0

0.1

0.2

0.3

0.4

0.5

0.6

Φ
Figure 3.28 Representative comparison in azimuthal velocity between fan configurations
3.3.2

Reynolds Averaged Velocity Statistics

The observed velocity in the near-wake of the fan has been decomposed into the
average velocity components and the fluctuating velocity components. The fluctuations
of the flow are found from the phase averaging ensemble described in Table 3.1. The

55

velocity component fluctuations contribute to the Reynolds stress tensor evaluated in
the Reynolds averaged momentum equations. The two velocity component measurements
obtained in the present investigation provide three terms from the Reynolds stress tensor.
These stresses exhibit an effect similar to viscous stress on the mean velocity field and
are diffusive. Equation (3.9) provides the three terms available from the measurements
obtained. These terms are the axial turbulence intensity, azimuthal turbulence intensity,
and Reynolds shear stress in the θ–z–plane respectively. A more detailed analysis of the
Reynolds averaging process can be found in [Pope (2000)] or [Hinze (1975)] and is also
outlined in Appendix D.

Vθ 2 ; Vz 2 ; Vθ Vz
3.3.2.1

(3.9)

Axial Turbulence Intensity

Figures 3.29-3.33 provide the turbulence intensity in the axial direction at the exit
of the fan plane. The formulation for the value in these figures is provided in Equation
(3.10). Case I (Figure 3.29) shows that the three blade fan takes much longer to return
to a low axial turbulence condition after the blade passes by the probe. The angle of
incidence for this operating condition and fan configuration may generate this phenomena. Additionally, increasing the number of blades decreased the time taken for the axial
turbulence intensity to recover to a magnitude before the blade passes.

Vz = RMS(Vz ) =

Vz 2

(3.10)

Case II (Figure 3.30) shows that the entire region between the blades of the three blade
fan is experiencing greater axial turbulence than in Case I. Note that the nine blade fan
56

now exhibits the shape of the five blade fan at Case I. This suggests that axial turbulence
for increased number of blades is similar to that of a fan with less blades operated at a
higher flow coefficient. Case III (Figure 3.31) is similar to Case II with the distinction
that the five blade fan now has axial turbulence in for the entire region between blades.
Case IV (Figure 3.32) now has matching behavior for the three fan configurations as
the blade approaches the probe. The magnitude of the axial turbulence intensity has an
inverse relationship with the number of blades installed on the fan near the wake region.
Case V (Figure 3.33) shows increased fluctuations for the nine blade fan on the pressure
side of the wake and that the entire span of λ exhibits axial turbulence for all three fans.

0.24
3 Blade Fan
5 Blade Fan
9 Blade Fan

Vz

0.16

0.08

0
−π/9

−π/18

0

π/18

λ

Figure 3.29 Case I Axial turbulence intensity over λ9

57

π/9

0.24
3 Blade Fan
5 Blade Fan
9 Blade Fan

Vz

0.16

0.08

0
−π/9

−π/18

0

π/18

π/9

λ

Figure 3.30 Case II Axial turbulence intensity over λ9

0.24
3 Blade Fan
5 Blade Fan
9 Blade Fan

Vz

0.16

0.08

0
−π/9

−π/18

0

π/18

λ

Figure 3.31 Case III Axial turbulence intensity over λ9

58

π/9

0.24
3 Blade Fan
5 Blade Fan
9 Blade Fan

Vz

0.16

0.08

0
−π/9

−π/18

0

π/18

π/9

λ

Figure 3.32 Case IV Axial turbulence intensity over λ9

0.24
3 Blade Fan
5 Blade Fan
9 Blade Fan

Vz

0.16

0.08

0
−π/9

−π/18

0

π/18

λ

Figure 3.33 Case V Axial turbulence intensity over λ9

59

π/9

3.3.2.2

Azimuthal Turbulence Intensity

Figures 3.34-3.38 provide the turbulence intensity in the aziumthal direction at the
exit of the fan plane. The formulation for the value in these figures is provided in Equation
(3.11). These results are similar to the axial turbulent fluctuations. Additionally, the
nine blade fan shares similar behavior to the three and five blade fans beginning at Case
IV. At Case III the nine blade fan also has the same behavior of the five blade fan at Case
II. This suggests that azimuthal turbulence, for increased number of blades, is similar to
that of a fan with less blades operated at a higher flow coefficient. The observed axial
and azimuthal turbulence intensities provide similar results.

Vθ = RMS(Vθ ) =

Vθ 2

(3.11)

0.24
3 Blade Fan
5 Blade Fan
9 Blade Fan

Vθ

0.16

0.08

0
−π/9

−π/18

0

π/18

λ

Figure 3.34 Case I Azimuthal turbulence intensity over λ9

60

π/9

0.24
3 Blade Fan
5 Blade Fan
9 Blade Fan

Vθ

0.16

0.08

0
−π/9

−π/18

0

π/18

π/9

λ

Figure 3.35 Case II Azimuthal turbulence intensity over λ9

0.24
3 Blade Fan
5 Blade Fan
9 Blade Fan

Vθ

0.16

0.08

0
−π/9

−π/18

0

π/18

λ

Figure 3.36 Case III Azimuthal turbulence intensity over λ9

61

π/9

0.24
3 Blade Fan
5 Blade Fan
9 Blade Fan

Vθ

0.16

0.08

0
−π/9

−π/18

0

π/18

π/9

λ

Figure 3.37 Case IV Azimuthal turbulence intensity over λ9

0.24
3 Blade Fan
5 Blade Fan
9 Blade Fan

Vθ

0.16

0.08

0
−π/9

−π/18

0

π/18

λ

Figure 3.38 Case V Azimuthal turbulence intensity over λ9

62

π/9

3.3.2.3

Reynolds Shear Stress [z-θ plane]

Figures 3.39-3.43 provide the Reynolds shear stress of the velocity fluctuations. Case
I (Figure 3.39) shows that the fluctuations for higher blade fans are greater in magnitude
near the wake. However, the effect of the blade passing above the probe is short in λ.
The three blade fan wake continues to be disturbed for a longer period of time after the
fan blade passes above in Case I. Case II (Figure 3.40) shows that the disturbances of
the blade passing propagated much further in λ for the five and blade fans. Additionally
the disturbance is carrying over to the pressure side of the next blade. This response is
then displayed by the five blade fan at Case III, Figure 3.41, and the nine blade fan at
Case IV, Figure 3.42. These observations suggest that a fan with less blades installed will
experience similar Reynolds shear stress in the wake at lower flow coefficient. Generally,
increasing the number of blades for a given operating condition suppresses turbulent
fluctuations in the fan wake. Each of the Reynolds stress tensor terms analyzed provided
this observation.

63

1.5
3 Blade Fan
5 Blade Fan
9 Blade Fan

2
Vz Vθ /Vtip
∗ 10 2

1
0.5
0

−0.5
−1
−1.5
−π/9

−π/18

0

π/18

π/9

λ

Figure 3.39 Case I Reynolds stress in z-θ – plane over λ9

1.5
3 Blade Fan
5 Blade Fan
9 Blade Fan

2
Vz Vθ /Vtip
∗ 10 2

1
0.5
0

−0.5
−1
−1.5
−π/9

−π/18

0

π/18

λ

Figure 3.40 Case II Reynolds stress in z-θ – plane over λ9

64

π/9

1.5
3 Blade Fan
5 Blade Fan
9 Blade Fan

2
Vz Vθ /Vtip
∗ 10 2

1
0.5
0

−0.5
−1
−1.5
−π/9

−π/18

0

π/18

π/9

λ

Figure 3.41 Case III Reynolds stress in z-θ – plane over λ9

1.5
3 Blade Fan
5 Blade Fan
9 Blade Fan

2
Vz Vθ /Vtip
∗ 10 2

1
0.5
0

−0.5
−1
−1.5
−π/9

−π/18

0

π/18

λ

Figure 3.42 Case IV Reynolds stress in z-θ – plane over λ9

65

π/9

1.5
3 Blade Fan
5 Blade Fan
9 Blade Fan

2
Vz Vθ /Vtip
∗ 10 2

1
0.5
0

−0.5
−1
−1.5
−π/9

−π/18

0

π/18

λ

Figure 3.43 Case V Reynolds stress in z-θ – plane over λ9

66

π/9

3.4

First Law of Thermodynamics Analysis
Section 1.2.1 outlined the work rate equation for the two control volumes that have

been identified for the present investigation. Equation (3.12) states the First Law of Thermodynamics in non-dimensional form. The present measurements: input shaft torque ,
pressure rise across the control volume, the mass flow rate through the fan, and the exit
kinetic energy in the wake of the fan at the blade mid-span allow a partial evaluation of
the terms in Equation (3.12). A qualitative comparison between the three different fan
configurations can be made using the structure of Equation (3.12).

Pshaf t ∗ =

1
P
ρ(V · V ) V · n
ρ·V ·n
ˆ dA + losses
ˆ dA +
c.s. ρ
c.s. 2
ξ
κ

χ

1
2 r2
rtip × ρVtip
tip
(3.12)

3.4.1

Input Shaft Power – Pshaf t ∗

The left hand side of Equation (3.12) was evaluated by Equation (3.13). Figure 3.44
provides the input shaft power as function of flow coefficient and marks the operating
conditions of interest for the three, five, and nine blade fans. These data are the same
as the non-dimensional torque presented in Figure 3.5, scaled with the angular velocity
of the fan. These figures indicate that more power is required to operate the fan with
a higher number of blades. That observation is consistent with the higher lift force
generated by the fan blades from the non-dimensional lift coefficient analysis performed
in Section 3.2.1.

Pshaf t ∗ = T∗ |Ω|
67

(3.13)

5

IV/III
V

4

P∗shaft

V

IV/III

3
V

2

IV/III

II

II

I

II

I

I

1
0

3 Blade Cases
5 Blade Cases
9 Blade Cases
0

0.1

0.2

Φ 0.3

0.4

0.5

Figure 3.44 Non-dimensional input shaft power

3.4.2

Net Kinetic Energy Flux – κ

The first term on the right hand side of Equation (3.12), κ, can be evaluated for
Control Volume (1) and (2) with the velocity data provided in the wake. This integral
can only be evaluated as a representative value because the measurements were only taken
at one radial location. Therefore, the integral will be approximately evaluated with exit
wake kinetic energy that is constant in the radial direction. This assumption allows this
value to provide a comparison between the three different fan configurations. However,
it does not give an absolute assessment of the net kinetic energy flux through the control
volume. Before evaluating this integral the non-dimensional kinematic integrand will be
presented in Figures 3.45–3.49. These data show the kinetic energy flux values of the
fluid for the pressure and suction sides of the blade. For Case I (Figure 3.45) the trend
shows that increasing the number of blades increases the overall kinetic energy exiting
the control volume. Additionally, the magnitude kinetic energy produced by the blade

68

passing above the probe near the suction side overtakes the pressure side in magnitude
as the number of blades is increased. Cases II–V (Figures 3.46–3.49) provide distinctly
different and consistent observations of the exit kinetic energy regardless from Case I.
There is a trend in these figures where the three blade fan produces the highest kinetic
energy flux out of the control volume as the blade passes above the probe and the nine
the least. The pressure side near the wake collapses onto a curve for all four of these
operating conditions. Case II–V show that the exit kinetic energy produced by the blade
passing above the probe shows an opposite trend to the number of blades installed on the
fan. This is self consistent with the swirl number results that indicate that the lower mass
flow rate operating conditions have a direct relationship with azimuthal mean velocity
and number of fan blades installed. A complete collection of the figures presented in
Section 3.4.2 are available in a format similar to Figures 3.16 and 3.17 within Appendix
C.

69

3
(1/2)[(Vθ2 + Vz2 )Vz ]/Vtip

0.2

3 Blade Fan
5 Blade Fan
9 Blade Fan

0.15

0.1

0.05

0
−π/9

−π/18

0

π/18

π/9

λ

Figure 3.45 Case I kinematic integrand exit kinetic energy flux over λ9

3
(1/2)[(Vθ2 + Vz2 )Vz ]/Vtip

0.2

3 Blade Fan
5 Blade Fan
9 Blade Fan

0.15

0.1

0.05

0
−π/9

−π/18

0

π/18

π/9

λ

Figure 3.46 Case II kinematic integrand exit kinetic energy flux over λ9

70

3
(1/2)[(Vθ2 + Vz2 )Vz ]/Vtip

0.2

3 Blade Fan
5 Blade Fan
9 Blade Fan

0.15

0.1

0.05

0
−π/9

−π/18

0

π/18

π/9

λ

Figure 3.47 Case III kinematic integrand exit kinetic energy flux over λ9

3
(1/2)[(Vθ2 + Vz2 )Vz ]/Vtip

0.2

3 Blade Fan
5 Blade Fan
9 Blade Fan

0.15

0.1

0.05

0
−π/9

−π/18

0

π/18

π/9

λ

Figure 3.48 Case IV kinematic integrand exit kinetic energy flux over λ9

71

3
(1/2)[(Vθ2 + Vz2 )Vz ]/Vtip

0.2

3 Blade Fan
5 Blade Fan
9 Blade Fan

0.15

0.1

0.05

0
−π/9

−π/18

0

π/18

π/9

λ

Figure 3.49 Case V kinematic integrand exit kinetic energy flux over λ9

72

7
6
5

κ1

4
3
2
3 Blade Fan
5 Blade Fan
9 Blade Fan

1
0

0

0.1

0.2

0.3

0.4

0.5

0.6

Φ
Figure 3.50 Net kinetic energy flux Control Volume (1)
The integrands in Figures 3.45–3.49 were evaluated for Control Volume (1) and (2).
The inlet air to Control Volume (1) had negligible kinetic energy; therefore, the integrands
from Figures 3.45–3.49 can be evaluated directly from the obtained dimensional values.
Equation (3.14) provides the integral evaluated to find κ1 . Figure 3.50 provides the
results from evaluating this integral. For this control volume the kinetic energy is high
for all three fan configurations at high flow coefficient. Case I displays a similar trend
between κ1 and the number of blades installed on the fan. This trend disappears as
the flow coefficient is reduced from Case I. These results are consistent with the plotted
integrands. At lower flow coefficients the net kinetic energy flux through Control Volume
(1) is significantly lower than the net kinetic energy flux at Case I. These results are
supported by the increased influence of lower flow coefficient on the azimuthal velocity
at the fan exit plane.

73

κ1 =

rtip
2π 1
1
ρ(Vθ 2 + Vz 2 )Vz rdθ dr
2
2
2
rtip × ρVtip rtip rhub 0

(3.14)

The kinetic energy at the inlet of Control Volume (2) is given by the average velocity
profile at the inlet of the fan by the mass flow rate through the fan. This velocity, V , is
assumed to be uniform over the inlet. Equation (3.15) provides the integral to find the
net kinetic energy flux through Control Volume (2). This integral cannot be properly
evaluated given the limitation of one radial position for the wake velocity measurements.

κ2 =

rtip
2π 1
1
3
ρ[(Vθ 2 + Vz 2 )Vz − V ] rdθ dr
2
2
rtip × ρVtip rtip rhub 0 2

74

(3.15)

4
3 Blade Fan
5 Blade Fan
9 Blade Fan

χ1

3

2

1

0

0

0.1

0.2

0.3

0.4

0.5

0.6

Φ
Figure 3.51 Flow-work rate Control Volume (1)

3.4.3

Flow-Work Rate – χ

The second term on the right hand side of Equation (3.12) was evaluated from the
integral measurements obtained from experiment. The present analysis assumes that the
pressure at the control volume exit plane is uniform and equal to the receiver pressure. In
reality, the curved streamlines at the exit fan plane indicate an increase in pressure with
increasing radius. Equation (3.16) provides the evaluation of the integral from Equation
(3.12) for Control Volume (1). Figure 3.51 provides the flow-work rate as a function of
flow coefficient for Control Volume (1). Each of these curves reach a maximum around
the discontinuity region of the characteristic curve.

χ1 =

[Prec − Patm ]
m
˙
ρ

(3.16)

Equation (3.17) provides the evaluation of the integral from Equation (3.12) for Con75

trol Volume (2). Figure 3.52 provides the flow-work rate as a function of flow coefficient
for Control Volume (2). This formulation for χ2 shows higher flow-work rate for the
higher flow coefficient operating conditions. This is due to the dynamic pressure of the
inlet velocity being added to the previous pressure used for the definition of χ1 . The
comparison for Control Volume (1) and (2) arises from observing where energy is transferred between terms on the right hand side of Equation (3.12). In Control Volume (1),
a majority of the power transferred from the fan to the fluid is through kinetic energy.
This is due to the negligible kinetic energy that the fluid has entering the control volume. Control Volume (2) shifts this energy into the flow-work rate term which is given
by the pressure difference across the fan. This pressure difference is increased because
the pressure at the inlet surface of the control volume is lower from the dynamic pressure
of the moving air entering the control volume.

χ2 =

[Prec − Patm ] + 1/2ρV
ρ

76

2

m
˙

(3.17)

8

χ2

6

4

2
3 Blade Fan
5 Blade Fan
9 Blade Fan
0

0

0.1

0.2

0.3

0.4

0.5

Φ
Figure 3.52 Flow-work rate Control Volume (2)

77

0.6

3.4.4

Turbulent Kinetic Energy Losses – ξ

The last term to consider in Equation (3.12) is the loss term ξ. The mechanical
losses in this system are considered to be a summation with multiple causes. This term
cannot be evaluated as the others have previously in this document. An indirect way
to observe how this term is changing between operating conditions is to look at the
turbulence kinetic energy. Figures 3.53-3.57 provide the turbulence kinetic energy of the
fluid at the exit plane in the wake of the fan. These measurements are not directly loss
terms; however, this energy is not recoverable in the system and will take away from the
available input shaft power. As the flow coefficient is decreased the turbulence kinetic
energy magnitude is increased for all the fan configurations. Again, the response of the
probe to the passing blade overhead is similar for a fan with less blades and higher flow
coefficient. This suggests that similar flow conditions occur for the different fans in a
inverse relationship with flow coefficient and number of blades. For example, the nine
blade fan will experience similar flow conditions at a lower flow coefficient as the five
blade fan does. This may be influenced by the increased mass flow rate through the fan
when blades are added. A complete collection of the figures presented in Section 3.4.4
are available in a format similar to Figures 3.16 and 3.17 within Appendix C.

78

0.08
3 Blade Fan
5 Blade Fan
9 Blade Fan

Vθ + V z

2

0.06

2

0.04

0.02

0
−π/9

−π/18

0

π/18

π/9

λ

Figure 3.53 Case I turbulence kinetic energy at fan exit over λ9

0.08
3 Blade Fan
5 Blade Fan
9 Blade Fan

Vθ + V z

2

0.06

2

0.04

0.02

0
−π/9

−π/18

0

π/18

λ

Figure 3.54 Case II turbulence kinetic energy at fan exit over λ9

79

π/9

0.08
3 Blade Fan
5 Blade Fan
9 Blade Fan

Vθ + V z

2

0.06

2

0.04

0.02

0
−π/9

−π/18

0

π/18

π/9

λ

Figure 3.55 Case III turbulence kinetic energy at fan exit over λ9

0.08
3 Blade Fan
5 Blade Fan
9 Blade Fan

Vθ + V z

2

0.06

2

0.04

0.02

0
−π/9

−π/18

0

π/18

λ

Figure 3.56 Case IV turbulence kinetic energy at fan exit over λ9

80

π/9

0.08
3 Blade Fan
5 Blade Fan
9 Blade Fan

Vθ + V z

2

0.06

2

0.04

0.02

0
−π/9

−π/18

0

π/18

λ

Figure 3.57 Case V turbulence kinetic energy at fan exit over λ9

81

π/9

Chapter 4
SUMMARY AND CONCLUSIONS
The present investigation examined differences in flow behavior over fan blade varying
the number of blades installed on the fan. This fan had nine identical blade sections that
were installed in configurations with 3–9 blades. These fans were compared using integral
fan measurements including: pressure rise across the fan, mass flow rate through the fan,
and the input shaft torque to the fan. The characteristic curves (Figure 3.1) for these
fans displayed a trend with a discontinuous portion of the performance curve for the 3–4
and 6–9 blade fans. The discontinuity on the 3 and 4 blade fans experienced a decrease in
the head-rise coefficient with increasing flow coefficient. The 6–9 blade fans experienced
an increase in the head-rise coefficient with increasing flow coefficient. The 5 blade fan
did not have a discontinuity in its characteristic curve. The extreme cases: 3 and 9
blade fans, and the 5 blade fan were selected to be examined with other experimental
techniques. These data were also used to select operating conditions to be investigated
in detail using surface pressure measurements on the blade as well as near-wake hotwire velocity measurements. Five operating conditions (Cases I–V) were selected for
experimentation; they are defined in Figure 3.3. These data were used to interpret two
different control volumes.
Surface pressure measurements were obtained across the span of the blade for the
selected fan configurations and operating conditions. These data were used to evaluate
a non-dimensional lift coefficient for an individual blade for each configuration as well as
an overall value for the entire fan. Figure 3.7 provided that there was an opposite trend
with the non-dimensional lift coefficient per blade and the number of blades installed
on the fan. Figure 3.8 showed that there was a similar trend with the overall fan non82

dimensional lift coefficient and the number of blades installed on the fan. The latter result
indicates that the pressure measurements are self consistent with the input shaft torque
measurements; see Figure 3.5. The surface pressure measurements have been provided for
each operating condition in Figures 3.9–3.13. Additionally, surface pressures obtained
from twenty taps along the trailing edge of the blade were obtained and presented in
Figure 3.14.
Wake velocity measurements were obtained with a two velocity component X-probe
in the present investigation. These measurements were evaluated as a phase ensemble
to find the statistics of the measured velocity. This method provided these data at 6000
discrete points in the azimuthal direction in the near-wake of the fan for one revolution of
the fan. These data provide another way to compare the three fan configurations and five
operating conditions in the present investigation. The statistics from the velocity measurements suggest that the blade to blade interactions suppress turbulent fluctuations
at a given operating condition in the fan wake. The obtained mean axial and azimuthal
velocities in the fan wake are presented in Figures 3.18–3.27. A representative value for
the azimuthal portion of the velocity in the fan wake was presented, as a swirl number,
in Figure 3.28. These data indicate that the magnitude of the azimuthal component of
velocity increases as the number of blades installed on the fan increases. Also the swirl
number decreases as the with flow coefficient is increased. The fluctuating velocity components from these data were evaluated as the terms in the Reynolds stress tensor from
the Reynolds averaged momentum equations. (Section D) The obtained axial and azimuthal turbulence intensities are presented in Figures 3.29–3.38. The obtained Reynolds
shear stress values are presented in Figures 3.39-3.43. These data suggest that increasing
the number of blades suppresses turbulence at a given value of the flow coefficient.
83

The integral and wake velocity measurements were used to evaluate the First Law
of Thermodynamics for the two control volumes considered in the present investigation.
The input shaft power is presented in Figure 3.44. A surrogate for the net kinetic energy
flux was evaluated for each control volume. These results are presented in Figures 3.50
and ??. The flow work rate was also evaluated for these control volume analyses. These
results are presented in Figures 3.51 and 3.52. The turbulence kinetic energy was also
considered as a contributor to energy lost in the control volume. These data, provided
in Figures 3.53–3.57, indicate that the turbulence kinetic energy increases as function of
flow coefficient. Additionally, there is an inverse relationship between the magnitude of
turbulence kinetic energy and number of blades installed on the fan.

The following conclusions are supported by the results of the present investigation:
• Non-dimensional lift coefficient per blade decreases as the number of blades installed
on the fan is increased
• Non-dimensional lift coefficient for the entire fan increases as the number of blades
installed on the fan is increased and self consistent with the measured input shaft
torque to the fan
• Magnitude of azimuthal velocity in the wake of the fan increases as the number of
blades installed on the fan is increased
• Turbulent fluctuations are suppressed as the number of blades installed on the fan
increases.

84

APPENDICES

85

A

PRESSURE COEFFICIENT FIGURES
Complete collection of pressure coefficient data for every operating condition (Case
I–V) and fan configuration (3–9 Blades). These data are represented in Figures 3.9-3.10
for a given operating condition.

86

2.5
η=0.119
η=0.302
η=0.500
η=0.603
η=0.865

2

- Cp

1.5
1
0.5
0
−0.5
−1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x/c
Figure 1 Case I pressure coefficients 3 blade fan

2.5
η=0.119
η=0.302
η=0.500
η=0.603
η=0.865

2

- Cp

1.5
1
0.5
0
−0.5
−1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x/c
Figure 2 Case II pressure coefficients 3 blade fan

87

0.9

1

2.5
η=0.119
η=0.302
η=0.500
η=0.603
η=0.865

2

- Cp

1.5
1
0.5
0
−0.5
−1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x/c
Figure 3 Case III pressure coefficients 3 blade fan

2.5
η=0.119
η=0.302
η=0.500
η=0.603
η=0.865

2

- Cp

1.5
1
0.5
0
−0.5
−1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x/c
Figure 4 Case IV pressure coefficients 3 blade fan

88

0.9

1

2.5
η=0.119
η=0.302
η=0.500
η=0.603
η=0.865

2

- Cp

1.5
1
0.5
0
−0.5
−1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x/c
Figure 5 Case V pressure coefficients 3 blade fan

89

0.9

1

2.5
η=0.119
η=0.302
η=0.500
η=0.603
η=0.865

2

- Cp

1.5
1
0.5
0
−0.5
−1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x/c
Figure 6 Case I pressure coefficients 5 blade fan

2.5
η=0.119
η=0.302
η=0.500
η=0.603
η=0.865

2

- Cp

1.5
1
0.5
0
−0.5
−1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x/c
Figure 7 Case II pressure coefficients 5 blade fan

90

0.9

1

2.5
η=0.119
η=0.302
η=0.500
η=0.603
η=0.865

2

- Cp

1.5
1
0.5
0
−0.5
−1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x/c
Figure 8 Case III/IV pressure coefficients 5 blade fan

2.5
η=0.119
η=0.302
η=0.500
η=0.603
η=0.865

2

- Cp

1.5
1
0.5
0
−0.5
−1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x/c
Figure 9 Case V pressure coefficients 5 blade fan

91

0.9

1

2.5
η=0.119
η=0.302
η=0.500
η=0.603
η=0.865

2

- Cp

1.5
1
0.5
0
−0.5
−1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x/c
Figure 10 Case I pressure coefficients 9 blade fan

2.5
η=0.119
η=0.302
η=0.500
η=0.603
η=0.865

2

- Cp

1.5
1
0.5
0
−0.5
−1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x/c
Figure 11 Case II pressure coefficients 9 blade fan

92

0.9

1

2.5
η=0.119
η=0.302
η=0.500
η=0.603
η=0.865

2

- Cp

1.5
1
0.5
0
−0.5
−1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x/c
Figure 12 Case III pressure coefficients 9 blade fan

2.5
η=0.119
η=0.302
η=0.500
η=0.603
η=0.865

2

- Cp

1.5
1
0.5
0
−0.5
−1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x/c
Figure 13 Case IV pressure coefficients 9 blade fan

93

0.9

1

2.5
η=0.119
η=0.302
η=0.500
η=0.603
η=0.865

2

- Cp

1.5
1
0.5
0
−0.5
−1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x/c
Figure 14 Case V pressure coefficients 9 blade fan

94

0.9

1

B

WAKE VELOCITY STATISTICS FIGURES
This appendix contains the entire collection of figures that were used to generate
the wake velocity statistics with the method presented in Section 3.3. These figures are
presented in the same sequence as they are in Section 3.3 and include these statistics:
Vz , Vθ , Vz , Vθ , and Vz Vθ .

95

3 Blade Fan
0.8
0.7

V z∗

0.6
0.5
0.4
0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
0.8
0.7

V z∗

0.6
0.5
0.4
0

π/2

π
9 Blade Fan

0.8
0.7

V z∗

0.6
0.5
0.4
0

π/2

π

θ

Figure 15 Case I Phase averaged axial velocity over 0 ≤ θ ≤ 2π

96

3 Blade Fan
0.7

V z∗

0.6
0.5
0.4
0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
0.7

V z∗

0.6
0.5
0.4
0

π/2

π
9 Blade Fan

0.7

V z∗

0.6
0.5
0.4
0

π/2

π

θ

Figure 16 Case II Phase averaged axial velocity over 0 ≤ θ ≤ 2π

97

3 Blade Fan
0.7

V z∗

0.6
0.5
0.4
0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
0.7

V z∗

0.6
0.5
0.4
0

π/2

π
9 Blade Fan

0.7

V z∗

0.6
0.5
0.4
0

π/2

π

θ

Figure 17 Case III Phase averaged axial velocity over 0 ≤ θ ≤ 2π

98

3 Blade Fan
0.7

V z∗

0.6
0.5
0.4
0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
0.7

V z∗

0.6
0.5
0.4
0

π/2

π
9 Blade Fan

0.7

V z∗

0.6
0.5
0.4
0

π/2

π

θ

Figure 18 Case IV Phase averaged axial velocity over 0 ≤ θ ≤ 2π

99

3 Blade Fan
0.45

V z∗

0.4
0.35
0.3
0.25

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
0.45

V z∗

0.4
0.35
0.3
0.25

0

π/2

π
9 Blade Fan

0.45

V z∗

0.4
0.35
0.3
0.25

0

π/2

π

θ
Figure 19 Case V Phase averaged axial velocity over 0 ≤ θ ≤ 2π

100

0.7
0.6
0.5

V z∗

0.4
0.3
0.2
3 Blade Fan
5 Blade Fan
9 Blade Fan

0.1
0
−π/3

−π/6

0

π/6

π/3

λ

Figure 20 Case I Phase averaged axial velocity over λ3

0.7
0.6
0.5

V z∗

0.4
0.3
0.2
3 Blade Fan
5 Blade Fan
9 Blade Fan

0.1
0
−π/3

−π/6

0

π/6

λ

Figure 21 Case II Phase averaged axial velocity over λ3

101

π/3

0.7
0.6
0.5

V z∗

0.4
0.3
0.2
3 Blade Fan
5 Blade Fan
9 Blade Fan

0.1
0
−π/3

−π/6

0

π/6

π/3

λ

Figure 22 Case III Phase averaged axial velocity over λ3

0.7
0.6
0.5

V z∗

0.4
0.3
0.2
3 Blade Fan
5 Blade Fan
9 Blade Fan

0.1
0
−π/3

−π/6

0

π/6

λ

Figure 23 Case IV Phase averaged axial velocity over λ3

102

π/3

0.7
0.6
0.5

V z∗

0.4
0.3
0.2
3 Blade Fan
5 Blade Fan
9 Blade Fan

0.1
0
−π/3

−π/6

0

π/6

λ

Figure 24 Case V Phase averaged axial velocity over λ3

103

π/3

3 Blade Fan
0.8

− V θ∗

0.6
0.4
0.2
0

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
0.8

− V θ∗

0.6
0.4
0.2
0

0

π/2

π
9 Blade Fan

0.8

− V θ∗

0.6
0.4
0.2
0

0

π/2

π

θ

Figure 25 Case I Phase averaged azimuthal velocity over 0 ≤ θ ≤ 2π

104

3 Blade Fan
0.8

− V θ∗

0.6
0.4
0.2
0

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
0.8

− V θ∗

0.6
0.4
0.2
0

0

π/2

π
9 Blade Fan

0.8

− V θ∗

0.6
0.4
0.2
0

0

π/2

π

θ

Figure 26 Case II Phase averaged azimuthal velocity over 0 ≤ θ ≤ 2π

105

3 Blade Fan
0.8

− V θ∗

0.6
0.4
0.2
0

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
0.8

− V θ∗

0.6
0.4
0.2
0

0

π/2

π
9 Blade Fan

0.8

− V θ∗

0.6
0.4
0.2
0

0

π/2

π

θ

Figure 27 Case III Phase averaged azimuthal velocity over 0 ≤ θ ≤ 2π

106

3 Blade Fan
0.8

− V θ∗

0.6
0.4
0.2
0

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
0.8

− V θ∗

0.6
0.4
0.2
0

0

π/2

π
9 Blade Fan

0.8

− V θ∗

0.6
0.4
0.2
0

0

π/2

π

θ

Figure 28 Case IV Phase averaged azimuthal velocity over 0 ≤ θ ≤ 2π

107

3 Blade Fan
0.5

− V θ∗

0.4
0.3
0.2
0.1

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
0.5

− V θ∗

0.4
0.3
0.2
0.1

0

π/2

π
9 Blade Fan

0.5

− V θ∗

0.4
0.3
0.2
0.1

0

π/2

π

θ

Figure 29 Case V Phase averaged azimuthal velocity over 0 ≤ θ ≤ 2π

108

0.7
3 Blade Fan
5 Blade Fan
9 Blade Fan

0.6

− V θ∗

0.5
0.4
0.3
0.2
0.1
0
−π/3

−π/6

0

π/6

π/3

λ

Figure 30 Case I Phase averaged azimuthal velocity over λ3

0.7
3 Blade Fan
5 Blade Fan
9 Blade Fan

0.6

− V θ∗

0.5
0.4
0.3
0.2
0.1
0
−π/3

−π/6

0

π/6

λ

Figure 31 Case II Phase averaged azimuthal velocity over λ3

109

π/3

0.7
3 Blade Fan
5 Blade Fan
9 Blade Fan

0.6

− V θ∗

0.5
0.4
0.3
0.2
0.1
0
−π/3

−π/6

0

π/6

π/3

λ

Figure 32 Case III Phase averaged azimuthal velocity over λ3

0.7
3 Blade Fan
5 Blade Fan
9 Blade Fan

0.6

− V θ∗

0.5
0.4
0.3
0.2
0.1
0
−π/3

−π/6

0

π/6

λ

Figure 33 Case IV Phase averaged azimuthal velocity over λ3

110

π/3

0.7
3 Blade Fan
5 Blade Fan
9 Blade Fan

0.6

− V θ∗

0.5
0.4
0.3
0.2
0.1
0
−π/3

−π/6

0

π/6

λ

Figure 34 Case V Phase averaged azimuthal velocity over λ3

111

π/3

3 Blade Fan
0.2

Vz

0.15
0.1

0.05
0

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
0.2

Vz

0.15
0.1

0.05
0

0

π/2

π
9 Blade Fan

0.2

Vz

0.15
0.1

0.05
0

0

π/2

π

θ

Figure 35 Case I Axial turbulence intensity over 0 ≤ θ ≤ 2π

112

3 Blade Fan
0.2

Vz

0.15
0.1

0.05
0

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
0.2

Vz

0.15
0.1

0.05
0

0

π/2

π
9 Blade Fan

0.2

Vz

0.15
0.1

0.05
0

0

π/2

π

θ

Figure 36 Case II Axial turbulence intensity over 0 ≤ θ ≤ 2π

113

3 Blade Fan
0.2

Vz

0.15
0.1

0.05
0

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
0.2

Vz

0.15
0.1

0.05
0

0

π/2

π
9 Blade Fan

0.2

Vz

0.15
0.1

0.05
0

0

π/2

π

θ

Figure 37 Case III Axial turbulence intensity over 0 ≤ θ ≤ 2π

114

3 Blade Fan
0.25

Vz

0.2
0.15
0.1
0.05

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
0.25

Vz

0.2
0.15
0.1
0.05

0

π/2

π
9 Blade Fan

0.25

Vz

0.2
0.15
0.1
0.05

0

π/2

π

θ

Figure 38 Case IV Axial turbulence intensity over 0 ≤ θ ≤ 2π

115

3 Blade Fan
0.25

Vz

0.2
0.15
0.1
0.05

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
0.25

Vz

0.2
0.15
0.1
0.05

0

π/2

π
9 Blade Fan

0.25

Vz

0.2
0.15
0.1
0.05

0

π/2

π

θ

Figure 39 Case V Axial turbulence intensity over 0 ≤ θ ≤ 2π

116

0.24
3 Blade Fan
5 Blade Fan
9 Blade Fan

Vz

0.16

0.08

0
−π/3

−π/6

0

π/6

π/3

λ

Figure 40 Case I Axial turbulence intensity over λ3

0.24
3 Blade Fan
5 Blade Fan
9 Blade Fan

Vz

0.16

0.08

0
−π/3

−π/6

0

π/6

λ

Figure 41 Case II Axial turbulence intensity over λ3

117

π/3

0.24
3 Blade Fan
5 Blade Fan
9 Blade Fan

Vz

0.16

0.08

0
−π/3

−π/6

0

π/6

π/3

λ

Figure 42 Case III Axial turbulence intensity over λ3

0.24
3 Blade Fan
5 Blade Fan
9 Blade Fan

Vz

0.16

0.08

0
−π/3

−π/6

0

π/6

λ

Figure 43 Case IV Axial turbulence intensity over λ3

118

π/3

0.24
3 Blade Fan
5 Blade Fan
9 Blade Fan

Vz

0.16

0.08

0
−π/3

−π/6

0

π/6

λ

Figure 44 Case V Axial turbulence intensity over λ3

119

π/3

3 Blade Fan
0.2

Vθ

0.15
0.1
0.05
0

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
0.2

Vθ

0.15
0.1
0.05
0

0

π/2

π
9 Blade Fan

0.2

Vθ

0.15
0.1
0.05
0

0

π/2

π

θ

Figure 45 Case I Azimuthal turbulence intensity over 0 ≤ θ ≤ 2π

120

3 Blade Fan
0.2

Vθ

0.15
0.1
0.05
0

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
0.2

Vθ

0.15
0.1
0.05
0

0

π/2

π
9 Blade Fan

0.2

Vθ

0.15
0.1
0.05
0

0

π/2

π

θ

Figure 46 Case II Azimuthal turbulence intensity over 0 ≤ θ ≤ 2π

121

3 Blade Fan
0.2

Vθ

0.15
0.1
0.05
0

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
0.2

Vθ

0.15
0.1
0.05
0

0

π/2

π
9 Blade Fan

0.2

Vθ

0.15
0.1
0.05
0

0

π/2

π

θ

Figure 47 Case III Azimuthal turbulence intensity over 0 ≤ θ ≤ 2π

122

3 Blade Fan
0.25

Vθ

0.2
0.15
0.1
0.05

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
0.25

Vθ

0.2
0.15
0.1
0.05

0

π/2

π
9 Blade Fan

0.25

Vθ

0.2
0.15
0.1
0.05

0

π/2

π

θ

Figure 48 Case IV Azimuthal turbulence intensity over 0 ≤ θ ≤ 2π

123

3 Blade Fan
0.25

Vθ

0.2
0.15
0.1
0.05

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
0.25

Vθ

0.2
0.15
0.1
0.05

0

π/2

π
9 Blade Fan

0.25

Vθ

0.2
0.15
0.1
0.05

0

π/2

π

θ

Figure 49 Case V Azimuthal turbulence intensity over 0 ≤ θ ≤ 2π

124

0.24
3 Blade Fan
5 Blade Fan
9 Blade Fan

Vθ

0.16

0.08

0
−π/3

−π/6

0

π/6

π/3

λ

Figure 50 Case I Azimuthal turbulence intensity over λ3

0.24
3 Blade Fan
5 Blade Fan
9 Blade Fan

Vθ

0.16

0.08

0
−π/3

−π/6

0

π/6

λ

Figure 51 Case II Azimuthal turbulence intensity over λ3

125

π/3

0.24
3 Blade Fan
5 Blade Fan
9 Blade Fan

Vθ

0.16

0.08

0
−π/3

−π/6

0

π/6

π/3

λ

Figure 52 Case III Azimuthal turbulence intensity over λ3

0.24
3 Blade Fan
5 Blade Fan
9 Blade Fan

Vθ

0.16

0.08

0
−π/3

−π/6

0

π/6

λ

Figure 53 Case IV Azimuthal turbulence intensity over λ3

126

π/3

0.24
3 Blade Fan
5 Blade Fan
9 Blade Fan

Vθ

0.16

0.08

0
−π/3

−π/6

0

π/6

λ

Figure 54 Case V Azimuthal turbulence intensity over λ3

127

π/3

3 Blade Fan
2
Vz Vθ /Vtip
∗ 10 2

1
0.5
0

−0.5
−1

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
2
Vz Vθ /Vtip
∗ 10 2

1
0.5
0

−0.5
−1

0

π/2

π
9 Blade Fan

2
Vz Vθ /Vtip
∗ 10 2

1
0.5
0

−0.5
−1

0

π/2

π

θ

Figure 55 Case I Reynolds stress in z-θ – plane over 0 ≤ θ ≤ 2π

128

3 Blade Fan
2
Vz Vθ /Vtip
∗ 10 2

1
0.5
0

−0.5
−1

−1.5

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
2
Vz Vθ /Vtip
∗ 10 2

1
0.5
0

−0.5
−1

−1.5

0

π/2

π
9 Blade Fan

2
Vz Vθ /Vtip
∗ 10 2

1
0.5
0

−0.5
−1

−1.5

0

π/2

π

θ

Figure 56 Case II Reynolds stress in z-θ – plane over 0 ≤ θ ≤ 2π

129

3 Blade Fan
2
Vz Vθ /Vtip
∗ 10 2

1
0

−1
−2

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
2
Vz Vθ /Vtip
∗ 10 2

1
0

−1
−2

0

π/2

π
9 Blade Fan

2
Vz Vθ /Vtip
∗ 10 2

1
0

−1
−2

0

π/2

π

θ

Figure 57 Case III Reynolds stress in z-θ – plane over 0 ≤ θ ≤ 2π

130

3 Blade Fan
2
Vz Vθ /Vtip
∗ 10 2

0.5
0

−0.5
−1

−1.5

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
2
Vz Vθ /Vtip
∗ 10 2

0.5
0

−0.5
−1

−1.5

0

π/2

π
9 Blade Fan

2
Vz Vθ /Vtip
∗ 10 2

0.5
0

−0.5
−1

−1.5

0

π/2

π

θ

Figure 58 Case IV Reynolds stress in z-θ – plane over 0 ≤ θ ≤ 2π

131

3 Blade Fan
2
Vz Vθ /Vtip
∗ 10 2

0.5
0

−0.5
−1

−1.5
−2

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
2
Vz Vθ /Vtip
∗ 10 2

0.5
0

−0.5
−1

−1.5
−2

0

π/2

π
9 Blade Fan

2
Vz Vθ /Vtip
∗ 10 2

0.5
0

−0.5
−1

−1.5
−2

0

π/2

π

θ

Figure 59 Case V Reynolds stress in z-θ – plane over 0 ≤ θ ≤ 2π

132

1.5
3 Blade Fan
5 Blade Fan
9 Blade Fan

2
Vz Vθ /Vtip
∗ 10 2

1
0.5
0

−0.5
−1
−1.5
−π/3

−π/6

0

π/6

π/3

λ

Figure 60 Case I Reynolds stress in z-θ – plane over λ3

1.5
3 Blade Fan
5 Blade Fan
9 Blade Fan

2
Vz Vθ /Vtip
∗ 10 2

1
0.5
0

−0.5
−1
−1.5
−π/3

−π/6

0

π/6

λ

Figure 61 Case II Reynolds stress in z-θ – plane over λ3

133

π/3

1.5
3 Blade Fan
5 Blade Fan
9 Blade Fan

2
Vz Vθ /Vtip
∗ 10 2

1
0.5
0

−0.5
−1
−1.5
−π/3

−π/6

0

π/6

π/3

λ

Figure 62 Case III Reynolds stress in z-θ – plane over λ3

1.5
3 Blade Fan
5 Blade Fan
9 Blade Fan

2
Vz Vθ /Vtip
∗ 10 2

1
0.5
0

−0.5
−1
−1.5
−π/3

−π/6

0

π/6

λ

Figure 63 Case IV Reynolds stress in z-θ – plane over λ3

134

π/3

1.5
3 Blade Fan
5 Blade Fan
9 Blade Fan

2
Vz Vθ /Vtip
∗ 10 2

1
0.5
0

−0.5
−1
−1.5
−π/3

−π/6

0

π/6

λ

Figure 64 Case V Reynolds stress in z-θ – plane over λ3

135

π/3

C

WORK RATE EVALUATION FIGURES
This appendix contains the entire collection of figures that were used to generate the
kinematic kinetic energy integrand and the turbulence kinetic energy from Sections 3.4.2
and 3.4.4, respectively. (Method in Section 3.3) These figures are presented in the same
3 and
sequence as they are in Sections 3.4.2 and 3.4.4 and include: (1/2)[(Vz2 + Vθ2 )Vz ]/Vtip
Vz2 + Vθ2 .

136

3
(1/2)[(Vθ2 + Vz2 )Vz ]/Vtip
3
(1/2)[(Vθ2 + Vz2 )Vz ]/Vtip
3
(1/2)[(Vθ2 + Vz2 )Vz ]/Vtip

3 Blade Fan
0.2
0.15
0.1
0.05
0

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
0.2
0.15
0.1
0.05
0

0

π/2

π
9 Blade Fan

0.2
0.15
0.1
0.05
0

0

π/2

π

θ

Figure 65 Case I kinematic integrand exit kinetic energy flux over 0 ≤ θ ≤ 2π

137

3
(1/2)[(Vθ2 + Vz2 )Vz ]/Vtip
3
(1/2)[(Vθ2 + Vz2 )Vz ]/Vtip
3
(1/2)[(Vθ2 + Vz2 )Vz ]/Vtip

3 Blade Fan

0.2
0.15
0.1
0.05
0

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan

0.2
0.15
0.1
0.05
0

0

π/2

π
9 Blade Fan

0.2
0.15
0.1
0.05
0

0

π/2

π

θ

Figure 66 Case II kinematic integrand exit kinetic energy flux over 0 ≤ θ ≤ 2π

138

3
(1/2)[(Vθ2 + Vz2 )Vz ]/Vtip
3
(1/2)[(Vθ2 + Vz2 )Vz ]/Vtip
3
(1/2)[(Vθ2 + Vz2 )Vz ]/Vtip

3 Blade Fan

0.2
0.15
0.1
0.05
0

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan

0.2
0.15
0.1
0.05
0

0

π/2

π
9 Blade Fan

0.2
0.15
0.1
0.05
0

0

π/2

π

θ

Figure 67 Case III kinematic integrand exit kinetic energy flux over 0 ≤ θ ≤ 2π

139

3
(1/2)[(Vθ2 + Vz2 )Vz ]/Vtip
3
(1/2)[(Vθ2 + Vz2 )Vz ]/Vtip
3
(1/2)[(Vθ2 + Vz2 )Vz ]/Vtip

3 Blade Fan

0.2
0.15
0.1
0.05
0

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan

0.2
0.15
0.1
0.05
0

0

π/2

π
9 Blade Fan

0.2
0.15
0.1
0.05
0

0

π/2

π

θ

Figure 68 Case IV kinematic integrand exit kinetic energy flux over 0 ≤ θ ≤ 2π

140

3
(1/2)[(Vθ2 + Vz2 )Vz ]/Vtip
3
(1/2)[(Vθ2 + Vz2 )Vz ]/Vtip
3
(1/2)[(Vθ2 + Vz2 )Vz ]/Vtip

3 Blade Fan
0.1
0.08
0.06
0.04
0.02

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
0.1
0.08
0.06
0.04
0.02

0

π/2

π
9 Blade Fan

0.1
0.08
0.06
0.04
0.02

0

π/2

π

θ

Figure 69 Case V kinematic integrand exit kinetic energy flux over 0 ≤ θ ≤ 2π

141

3
(1/2)[(Vθ2 + Vz2 )Vz ]/Vtip

0.2

3 Blade Fan
5 Blade Fan
9 Blade Fan

0.15

0.1

0.05

0
−π/3

−π/6

0

π/6

π/3

λ

Figure 70 Case I kinematic integrand exit kinetic energy flux over λ3

3
(1/2)[(Vθ2 + Vz2 )Vz ]/Vtip

0.2

3 Blade Fan
5 Blade Fan
9 Blade Fan

0.15

0.1

0.05

0
−π/3

−π/6

0

π/6

π/3

λ

Figure 71 Case II kinematic integrand exit kinetic energy flux over λ3

142

3
(1/2)[(Vθ2 + Vz2 )Vz ]/Vtip

0.2

3 Blade Fan
5 Blade Fan
9 Blade Fan

0.15

0.1

0.05

0
−π/3

−π/6

0

π/6

π/3

λ

Figure 72 Case III kinematic integrand exit kinetic energy flux over λ3

3
(1/2)[(Vθ2 + Vz2 )Vz ]/Vtip

0.2

3 Blade Fan
5 Blade Fan
9 Blade Fan

0.15

0.1

0.05

0
−π/3

−π/6

0

π/6

π/3

λ

Figure 73 Case IV kinematic integrand exit kinetic energy flux over λ3

143

3
(1/2)[(Vθ2 + Vz2 )Vz ]/Vtip

0.2

3 Blade Fan
5 Blade Fan
9 Blade Fan

0.15

0.1

0.05

0
−π/3

−π/6

0

π/6

π/3

λ

Figure 74 Case V kinematic integrand exit kinetic energy flux over λ3

144

3 Blade Fan

0.04

2

Vθ + Vz

2

0.06

0.02
0

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan

0.04

2

Vθ + V z

2

0.06

0.02
0

0

π/2

π
9 Blade Fan

0.04

2

Vθ + Vz

2

0.06

0.02
0

0

π/2

π

θ

Figure 75 Case I turbulence kinetic energy at fan exit over 0 ≤ θ ≤ 2π

145

3 Blade Fan
0.06

Vθ + Vz

2

0.08

2

0.04
0.02
0

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
0.06

Vθ + V z

2

0.08

2

0.04
0.02
0

0

π/2

π
9 Blade Fan

0.06

Vθ + Vz

2

0.08

2

0.04
0.02
0

0

π/2

π

θ

Figure 76 Case II turbulence kinetic energy at fan exit over 0 ≤ θ ≤ 2π

146

3 Blade Fan
0.06

Vθ + Vz

2

0.08

2

0.04
0.02
0

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
0.06

Vθ + V z

2

0.08

2

0.04
0.02
0

0

π/2

π
9 Blade Fan

0.06

Vθ + Vz

2

0.08

2

0.04
0.02
0

0

π/2

π

θ

Figure 77 Case III turbulence kinetic energy at fan exit over 0 ≤ θ ≤ 2π

147

3 Blade Fan
0.06

Vθ + Vz

2

0.08

2

0.04
0.02
0

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
0.06

Vθ + V z

2

0.08

2

0.04
0.02
0

0

π/2

π
9 Blade Fan

0.06

Vθ + Vz

2

0.08

2

0.04
0.02
0

0

π/2

π

θ

Figure 78 Case IV turbulence kinetic energy at fan exit over 0 ≤ θ ≤ 2π

148

3 Blade Fan
0.06

Vθ + Vz

2

0.08

2

0.04
0.02
0

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
0.06

Vθ + V z

2

0.08

2

0.04
0.02
0

0

π/2

π
9 Blade Fan

0.06

Vθ + Vz

2

0.08

2

0.04
0.02
0

0

π/2

π

θ

Figure 79 Case V turbulence kinetic energy at fan exit over 0 ≤ θ ≤ 2π

149

0.08
3 Blade Fan
5 Blade Fan
9 Blade Fan

Vθ + V z

2

0.06

2

0.04

0.02

0
−π/3

−π/6

0

π/6

π/3

λ

Figure 80 Case I turbulence kinetic energy at fan exit over λ3

0.08
3 Blade Fan
5 Blade Fan
9 Blade Fan

Vθ + V z

2

0.06

2

0.04

0.02

0
−π/3

−π/6

0

π/6

λ

Figure 81 Case II turbulence kinetic energy at fan exit over λ3

150

π/3

0.08
3 Blade Fan
5 Blade Fan
9 Blade Fan

Vθ + V z

2

0.06

2

0.04

0.02

0
−π/3

−π/6

0

π/6

π/3

λ

Figure 82 Case III turbulence kinetic energy at fan exit over λ3

0.08
3 Blade Fan
5 Blade Fan
9 Blade Fan

Vθ + V z

2

0.06

2

0.04

0.02

0
−π/3

−π/6

0

π/6

λ

Figure 83 Case IV turbulence kinetic energy at fan exit over λ3

151

π/3

0.08
3 Blade Fan
5 Blade Fan
9 Blade Fan

Vθ + V z

2

0.06

2

0.04

0.02

0
−π/3

−π/6

0

π/6

λ

Figure 84 Case V turbulence kinetic energy at fan exit over λ3

152

π/3

D

REYNOLDS AVERAGED FORMULATIONS
The velocity data obtained in the present investigation have been processed to provide
representative values for available terms in the Reynolds averaged momentum equations
in cylindrical coordinates and the turbulence kinetic energy balance. A short introduction
to these formulations is provided in the following section.
Measurements obtained in turbulent flows will have fluctuating and average statistical values that can be derived from the total sample. Equation (1) provides how the
velocity vector and pressure field are separated into an averaged component, indicated
by over-bars, and a fluctuating component, in lowercase. Traditionally experimental
measurements for the average component for these quantities are obtained over time in a
fixed position. As discussed in Section 3.3 a phase averaging scheme was utilized for these
measurements to provide velocity statistics for all θ locations in the fan wake. This was
found by building a time history for each discrete position from each revolution. It is convenient to consider these new equations in tensor notation for an intuitive understanding
of the terms.
(1)
Vi = Vi + vi ; P = P + p
(B)

(A)

Substituting (1)(A) and (1)(B) into the momentum equation for constant viscosity
and incompressible flow provides Equation (2) [Hinze (1975)]. Equations (3) and (4)
are the θ and z momentum equations, respectively, derived from Equation (2). These
equations have omitted radial velocity components which are negligible in the near wake
of this fan [Neal (2010)]. Terms A, B, C, and D from Equations (3) and (4) have been
evaluated with the measured from the wake velocity measurements. and are presented
in the following section. A represents the advection of the θ–component momentum per
unit mass. B represents the net normal Reynolds stress in θ. C represents the advection
of the z–component momentum per unit mass. D represents the net Reynolds shear
stress in the z–θ–plane.
∂Vi
∂P
∂
∂V
∂Vi
+ Vj ·
)=−
+
(µ i − ρvi vj ) − Fi
∂t
∂xj
∂xi ∂xj ∂xj

(2)

∂V θ
∂V θ
∂P
∂
∂
+V z
=−
+ τviscous − ρ(
Vθ 2 + Vz Vθ )
r∂θ
∂z
r∂θ
r∂θ
∂z

(3)

ρ(

Vθ

A

Vθ

B

∂V z
∂V z
∂P
∂
∂
+V z
=−
+ τviscous − ρ(
Vθ 2 + Vz Vθ )
r∂θ
∂z
∂z
r∂θ
∂z
C

(4)

D

Equation (5) provides the balance of turbulence kinetic energy [Hinze (1975)] ; where
q represents turbulent kinetic energy. This steady flow will set the left hand side of
Equation (5) to zero. The turbulence kinetic energy production term: E, has been
evaluated with the wake velocity measurements considering xi = rˆ + θˆ + zˆ. These results
are provided in the following section.

153

∂Vj
∂vj
∂vj ∂vj
∂q 2
∂q 2
∂
p
q2
∂
∂v
∂v
= −V j
+−
vi ( + ) − vi vj
+ν
vi ( i +
) − ν( i +
)
∂t
∂xj
∂xi
ρ
2
∂xi
∂xi ∂xj
∂xi
∂xj
∂xi ∂xi
E

(5)
This section contains the entire collection of figures that were generated for the
Reynolds averaged terms identified in Equations (3), (4), and (5). The derivatives
in the azimuthal direction were evaluated with a central differencing scheme provided
by Equation (6). ∆θ is given by the “bin” size from the phase averaging scheme
(0.06◦ ∗ π/180) radians. (Section 3.3) All of these terms were evaluated non-dimensionally.
Therefore C, and D from Equations (3) and (4), are kinematic in the data presented.
The radius at mid-span used in the derivative in the theta direction was made nondimensional by the tip radius: r = rmid−span/rtip . The following pages include all of the
evaluated values for the parameters in Equation (7) from Equations (3–5).
f (θ + ∆θ) − f (θ − ∆θ)
∂
f (θ) =
r∂θ
r2∆θ
Vθ∗

(6)

∗
∂Vθ∗
∂ 2
∂
∂V ∗
2 ; (V V /V 2 ) ∂Vz
;
Vθ ; Vθ∗ z ;
Vθ Vz /Vtip
z θ tip
r∂θ r∂θ
r∂θ r∂θ
r¯∂θ
A

B

C

D

154

E

(7)

3 Blade Fan
15

Vθ∗ r∂θθ

∂V ∗

10
5
0
−5

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
15

Vθ∗ r∂θθ

∂V ∗

10
5
0
−5

0

π/2

π
9 Blade Fan

15

Vθ∗ r∂θθ

∂V ∗

10
5
0
−5

0

π/2

π

θ

Figure 85 Case I advection of θ–component momentum per unit mass over 0 ≤ θ ≤ 2π

155

3 Blade Fan
15

Vθ∗ r∂θθ

∂V ∗

10
5
0
−5

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
15

Vθ∗ r∂θθ

∂V ∗

10
5
0
−5

0

π/2

π
9 Blade Fan

15

Vθ∗ r∂θθ

∂V ∗

10
5
0
−5

0

π/2

π

θ

Figure 86 Case II advection of θ–component momentum per unit mass over 0 ≤ θ ≤ 2π

156

3 Blade Fan
15

Vθ∗ r∂θθ

∂V ∗

10
5
0
−5

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
15

Vθ∗ r∂θθ

∂V ∗

10
5
0
−5

0

π/2

π
9 Blade Fan

15

Vθ∗ r∂θθ

∂V ∗

10
5
0
−5

0

π/2

π

θ

Figure 87 Case III advection of θ–component momentum per unit mass over 0 ≤ θ ≤ 2π

157

3 Blade Fan

Vθ∗ r∂θθ

∂V ∗

10
5
0

−5

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan

Vθ∗ r∂θθ

∂V ∗

10
5
0

−5

0

π/2

π
9 Blade Fan

Vθ∗ r∂θθ

∂V ∗

10
5
0

−5

0

π/2

π

θ

Figure 88 Case IV advection of θ–component momentum per unit mass over 0 ≤ θ ≤ 2π

158

3 Blade Fan

Vθ∗ r∂θθ

∂V ∗

10
5
0

−5

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan

Vθ∗ r∂θθ

∂V ∗

10
5
0

−5

0

π/2

π
9 Blade Fan

Vθ∗ r∂θθ

∂V ∗

10
5
0

−5

0

π/2

π

θ

Figure 89 Case V advection of θ–component momentum per unit mass over 0 ≤ θ ≤ 2π

159

3 Blade Fan
5 Blade Fan
9 Blade Fan

12

Vθ∗ r∂θθ

∂V ∗

8

4

0

−4
−π/3

−π/6

0

π/6

π/3

λ

Figure 90 Case I advection of θ–component momentum per unit mass over λ3

3 Blade Fan
5 Blade Fan
9 Blade Fan

12

Vθ∗ r∂θθ

∂V ∗

8

4

0

−4
−π/3

−π/6

0

π/6

π/3

λ

Figure 91 Case II advection of θ–component momentum per unit mass over λ3

160

3 Blade Fan
5 Blade Fan
9 Blade Fan

12

Vθ∗ r∂θθ

∂V ∗

8

4

0

−4
−π/3

−π/6

0

π/6

π/3

λ

Figure 92 Case III advection of θ–component momentum per unit mass over λ3

3 Blade Fan
5 Blade Fan
9 Blade Fan

12

Vθ∗ r∂θθ

∂V ∗

8

4

0

−4
−π/3

−π/6

0

π/6

π/3

λ

Figure 93 Case IV advection of θ–component momentum per unit mass over λ3

161

3 Blade Fan
5 Blade Fan
9 Blade Fan

12

Vθ∗ r∂θθ

∂V ∗

8

4

0

−4
−π/3

−π/6

0

π/6

π/3

λ

Figure 94 Case V advection of θ–component momentum per unit mass over λ3

3 Blade Fan
5 Blade Fan
9 Blade Fan

12

Vθ∗ r∂θθ

∂V ∗

8

4

0

−4
−π/9

−π/18

0

π/18

π/9

λ

Figure 95 Case I advection of θ–component momentum per unit mass over λ9

162

3 Blade Fan
5 Blade Fan
9 Blade Fan

12

Vθ∗ r∂θθ

∂V ∗

8

4

0

−4
−π/9

−π/18

0

π/18

π/9

λ

Figure 96 Case II advection of θ–component momentum per unit mass over λ9

3 Blade Fan
5 Blade Fan
9 Blade Fan

12

Vθ∗ r∂θθ

∂V ∗

8

4

0

−4
−π/9

−π/18

0

π/18

π/9

λ

Figure 97 Case III advection of θ–component momentum per unit mass over λ9

163

3 Blade Fan
5 Blade Fan
9 Blade Fan

12

Vθ∗ r∂θθ

∂V ∗

8

4

0

−4
−π/9

−π/18

0

π/18

π/9

λ

Figure 98 Case IV advection of θ–component momentum per unit mass over λ9

3 Blade Fan
5 Blade Fan
9 Blade Fan

12

Vθ∗ r∂θθ

∂V ∗

8

4

0

−4
−π/9

−π/18

0

π/18

π/9

λ

Figure 99 Case V advection of θ–component momentum per unit mass over λ9

164

3 Blade Fan
30

2
∂
r∂θ Vθ

20
10
0
−10

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
30

2
∂
r∂θ Vθ

20
10
0
−10

0

π/2

π
9 Blade Fan

30

2
∂
r∂θ Vθ

20
10
0
−10

0

π/2

π

θ

Figure 100 Case I net Reynolds normal stress in θ over 0 ≤ θ ≤ 2π

165

3 Blade Fan
30

2
∂
r∂θ Vθ

20
10
0
−10

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
30

2
∂
r∂θ Vθ

20
10
0
−10

0

π/2

π
9 Blade Fan

30

2
∂
r∂θ Vθ

20
10
0
−10

0

π/2

π

θ

Figure 101 Case II net Reynolds normal stress in θ over 0 ≤ θ ≤ 2π

166

3 Blade Fan
30

2
∂
r∂θ Vθ

20
10
0
−10

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
30

2
∂
r∂θ Vθ

20
10
0
−10

0

π/2

π
9 Blade Fan

30

2
∂
r∂θ Vθ

20
10
0
−10

0

π/2

π

θ

Figure 102 Case III net Reynolds normal stress in θ over 0 ≤ θ ≤ 2π

167

3 Blade Fan

2
∂
r∂θ Vθ

20
10
0

−10

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan

2
∂
r∂θ Vθ

20
10
0

−10

0

π/2

π
9 Blade Fan

2
∂
r∂θ Vθ

20
10
0

−10

0

π/2

π

θ

Figure 103 Case IV net Reynolds normal stress in θ over 0 ≤ θ ≤ 2π

168

3 Blade Fan

2
∂
r∂θ Vθ

20
10
0

−10

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan

2
∂
r∂θ Vθ

20
10
0

−10

0

π/2

π
9 Blade Fan

2
∂
r∂θ Vθ

20
10
0

−10

0

π/2

π

θ

Figure 104 Case V net Reynolds normal stress in θ over 0 ≤ θ ≤ 2π

169

30
3 Blade Fan
5 Blade Fan
9 Blade Fan

25
20
2
∂
r∂θ Vθ

15
10
5
0
−5
−10
−π/3

−π/6

0

π/6

π/3

λ

Figure 105 Case I net Reynolds normal stress in θ over λ3

30
3 Blade Fan
5 Blade Fan
9 Blade Fan

25
20
2
∂
r∂θ Vθ

15
10
5
0
−5
−10
−π/3

−π/6

0

π/6

λ

Figure 106 Case II net Reynolds normal stress in θ over λ3

170

π/3

30
3 Blade Fan
5 Blade Fan
9 Blade Fan

25
20
2
∂
r∂θ Vθ

15
10
5
0
−5
−10
−π/3

−π/6

0

π/6

π/3

λ

Figure 107 Case III net Reynolds normal stress in θ over λ3

30
3 Blade Fan
5 Blade Fan
9 Blade Fan

25
20
2
∂
r∂θ Vθ

15
10
5
0
−5
−10
−π/3

−π/6

0

π/6

λ

Figure 108 Case IV net Reynolds normal stress in θ over λ3

171

π/3

30
3 Blade Fan
5 Blade Fan
9 Blade Fan

25
20
2
∂
r∂θ Vθ

15
10
5
0
−5
−10
−π/3

−π/6

0

π/6

π/3

λ

Figure 109 Case V net Reynolds normal stress in θ over λ3

30
3 Blade Fan
5 Blade Fan
9 Blade Fan

25
20
2
∂
r∂θ Vθ

15
10
5
0
−5
−10
−π/9

−π/18

0

π/18

λ

Figure 110 Case I net Reynolds normal stress in θ over λ9

172

π/9

30
3 Blade Fan
5 Blade Fan
9 Blade Fan

25
20
2
∂
r∂θ Vθ

15
10
5
0
−5
−10
−π/9

−π/18

0

π/18

π/9

λ

Figure 111 Case II net Reynolds normal stress in θ over λ9

30
3 Blade Fan
5 Blade Fan
9 Blade Fan

25
20
2
∂
r∂θ Vθ

15
10
5
0
−5
−10
−π/9

−π/18

0

π/18

λ

Figure 112 Case III net Reynolds normal stress in θ over λ9

173

π/9

30
3 Blade Fan
5 Blade Fan
9 Blade Fan

25
20
2
∂
r∂θ Vθ

15
10
5
0
−5
−10
−π/9

−π/18

0

π/18

π/9

λ

Figure 113 Case IV net Reynolds normal stress in θ over λ9

30
3 Blade Fan
5 Blade Fan
9 Blade Fan

25
20
2
∂
r∂θ Vθ

15
10
5
0
−5
−10
−π/9

−π/18

0

π/18

λ

Figure 114 Case V net Reynolds normal stress in θ over λ9

174

π/9

3 Blade Fan
10

z
Vθ∗ ∂V
r∂θ

∗

5
0
−5
−10

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
10

z
Vθ∗ ∂V
r∂θ

∗

5
0
−5
−10

0

π/2

π
9 Blade Fan

10

z
Vθ∗ ∂V
r∂θ

∗

5
0
−5
−10

0

π/2

π

θ

Figure 115 Case I advection of z–component momentum per unit mass over 0 ≤ θ ≤ 2π

175

3 Blade Fan

z
Vθ∗ ∂V
r∂θ

∗

10
5
0

−5

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan

z
Vθ∗ ∂V
r∂θ

∗

10
5
0

−5

0

π/2

π
9 Blade Fan

z
Vθ∗ ∂V
r∂θ

∗

10
5
0

−5

0

π/2

π

θ

Figure 116 Case II advection of z–component momentum per unit mass over 0 ≤ θ ≤ 2π

176

3 Blade Fan

z
Vθ∗ ∂V
r∂θ

∗

10
5
0

−5

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan

z
Vθ∗ ∂V
r∂θ

∗

10
5
0

−5

0

π/2

π
9 Blade Fan

z
Vθ∗ ∂V
r∂θ

∗

10
5
0

−5

0

π/2

π

θ

Figure 117 Case III advection of z–component momentum per unit mass over 0 ≤ θ ≤
2π

177

3 Blade Fan

z
Vθ∗ ∂V
r∂θ

∗

10
5
0

−5

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan

z
Vθ∗ ∂V
r∂θ

∗

10
5
0

−5

0

π/2

π
9 Blade Fan

z
Vθ∗ ∂V
r∂θ

∗

10
5
0

−5

0

π/2

π

θ

Figure 118 Case IV advection of z–component momentum per unit mass over 0 ≤ θ ≤
2π

178

3 Blade Fan

z
Vθ∗ ∂V
r∂θ

∗

4
2
0

−2

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan

z
Vθ∗ ∂V
r∂θ

∗

4
2
0

−2

0

π/2

π
9 Blade Fan

z
Vθ∗ ∂V
r∂θ

∗

4
2
0

−2

0

π/2

π

θ

Figure 119 Case V advection of z–component momentum per unit mass over 0 ≤ θ ≤ 2π

179

3 Blade Fan
5 Blade Fan
9 Blade Fan

6

z
Vθ∗ ∂V
r∂θ

∗

3

0

−3

−6
−π/3

−π/6

0

π/6

π/3

λ

Figure 120 Case I advection of z–component momentum per unit mass over λ3

3 Blade Fan
5 Blade Fan
9 Blade Fan

6

z
Vθ∗ ∂V
r∂θ

∗

3

0

−3

−6
−π/3

−π/6

0

π/6

π/3

λ

Figure 121 Case II advection of z–component momentum per unit mass over λ3

180

3 Blade Fan
5 Blade Fan
9 Blade Fan

6

z
Vθ∗ ∂V
r∂θ

∗

3

0

−3

−6
−π/3

−π/6

0

π/6

π/3

λ

Figure 122 Case III advection of z–component momentum per unit mass over λ3

3 Blade Fan
5 Blade Fan
9 Blade Fan

6

z
Vθ∗ ∂V
r∂θ

∗

3

0

−3

−6
−π/3

−π/6

0

π/6

π/3

λ

Figure 123 Case IV advection of z–component momentum per unit mass over λ3

181

3 Blade Fan
5 Blade Fan
9 Blade Fan

6

z
Vθ∗ ∂V
r∂θ

∗

3

0

−3

−6
−π/3

−π/6

0

π/6

π/3

λ

Figure 124 Case V advection of z–component momentum per unit mass over λ3

3 Blade Fan
5 Blade Fan
9 Blade Fan

6

z
Vθ∗ ∂V
r∂θ

∗

3

0

−3

−6
−π/9

−π/18

0

π/18

π/9

λ

Figure 125 Case I advection of z–component momentum per unit mass over λ9

182

3 Blade Fan
5 Blade Fan
9 Blade Fan

6

z
Vθ∗ ∂V
r∂θ

∗

3

0

−3

−6
−π/9

−π/18

0

π/18

π/9

λ

Figure 126 Case II advection of z–component momentum per unit mass over λ9

3 Blade Fan
5 Blade Fan
9 Blade Fan

6

z
Vθ∗ ∂V
r∂θ

∗

3

0

−3

−6
−π/9

−π/18

0

π/18

π/9

λ

Figure 127 Case III advection of z–component momentum per unit mass over λ9

183

3 Blade Fan
5 Blade Fan
9 Blade Fan

6

z
Vθ∗ ∂V
r∂θ

∗

3

0

−3

−6
−π/9

−π/18

0

π/18

π/9

λ

Figure 128 Case IV advection of z–component momentum per unit mass over λ9

3 Blade Fan
5 Blade Fan
9 Blade Fan

6

z
Vθ∗ ∂V
r∂θ

∗

3

0

−3

−6
−π/9

−π/18

0

π/18

π/9

λ

Figure 129 Case V advection of z–component momentum per unit mass over λ9

184

3 Blade Fan
∂
2
r∂θ Vθ Vz /Vtip

2
0

−2

−4

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
∂
2
r∂θ Vθ Vz /Vtip

2
0

−2

−4

0

π/2

π
9 Blade Fan

∂
2
r∂θ Vθ Vz /Vtip

2
0

−2

−4

0

π/2

π

θ

Figure 130 Case I net Reynolds shear stress in θ–z plane over 0 ≤ θ ≤ 2π

185

3 Blade Fan
∂
2
r∂θ Vθ Vz /Vtip

1
0

−1

−2

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
∂
2
r∂θ Vθ Vz /Vtip

1
0

−1

−2

0

π/2

π
9 Blade Fan

∂
2
r∂θ Vθ Vz /Vtip

1
0

−1

−2

0

π/2

π

θ

Figure 131 Case II net Reynolds shear stress in θ–z plane over 0 ≤ θ ≤ 2π

186

3 Blade Fan
∂
2
r∂θ Vθ Vz /Vtip

2
1
0
−1
−2

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
∂
2
r∂θ Vθ Vz /Vtip

2
1
0
−1
−2

0

π/2

π
9 Blade Fan

∂
2
r∂θ Vθ Vz /Vtip

2
1
0
−1
−2

0

π/2

π

θ

Figure 132 Case III net Reynolds shear stress in θ–z plane over 0 ≤ θ ≤ 2π

187

3 Blade Fan
∂
2
r∂θ Vθ Vz /Vtip

1
0

−1

−2

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
∂
2
r∂θ Vθ Vz /Vtip

1
0

−1

−2

0

π/2

π
9 Blade Fan

∂
2
r∂θ Vθ Vz /Vtip

1
0

−1

−2

0

π/2

π

θ

Figure 133 Case IV net Reynolds shear stress in θ–z plane over 0 ≤ θ ≤ 2π

188

3 Blade Fan
∂
2
r∂θ Vθ Vz /Vtip

1
0

−1

−2

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan
∂
2
r∂θ Vθ Vz /Vtip

1
0

−1

−2

0

π/2

π
9 Blade Fan

∂
2
r∂θ Vθ Vz /Vtip

1
0

−1

−2

0

π/2

π

θ

Figure 134 Case V net Reynolds shear stress in θ–z plane over 0 ≤ θ ≤ 2π

189

2
3 Blade Fan
5 Blade Fan
9 Blade Fan

∂
2
r∂θ Vθ Vz /Vtip

1

0

−1

−2

−3
−π/3

−π/6

0

π/6

π/3

λ

Figure 135 Case I net Reynolds shear stress in θ–z plane over λ3

2
3 Blade Fan
5 Blade Fan
9 Blade Fan

∂
2
r∂θ Vθ Vz /Vtip

1

0

−1

−2

−3
−π/3

−π/6

0

π/6

λ

Figure 136 Case II net Reynolds shear stress in θ–z plane over λ3

190

π/3

2
3 Blade Fan
5 Blade Fan
9 Blade Fan

∂
2
r∂θ Vθ Vz /Vtip

1

0

−1

−2

−3
−π/3

−π/6

0

π/6

π/3

λ

Figure 137 Case III net Reynolds shear stress in θ–z plane over λ3

2
3 Blade Fan
5 Blade Fan
9 Blade Fan

∂
2
r∂θ Vθ Vz /Vtip

1

0

−1

−2

−3
−π/3

−π/6

0

π/6

π/3

λ

Figure 138 Case IV net Reynolds shear stress in θ–z plane over λ3

191

2
3 Blade Fan
5 Blade Fan
9 Blade Fan

∂
2
r∂θ Vθ Vz /Vtip

1

0

−1

−2

−3
−π/3

−π/6

0

π/6

π/3

λ

Figure 139 Case V net Reynolds shear stress in θ–z plane over λ3

2
3 Blade Fan
5 Blade Fan
9 Blade Fan

∂
2
r∂θ Vθ Vz /Vtip

1

0

−1

−2

−3
−π/9

−π/18

0

π/18

λ

Figure 140 Case I net Reynolds shear stress in θ–z plane over λ9

192

π/9

2
3 Blade Fan
5 Blade Fan
9 Blade Fan

∂
2
r∂θ Vθ Vz /Vtip

1

0

−1

−2

−3
−π/9

−π/18

0

π/18

π/9

λ

Figure 141 Case II net Reynolds shear stress in θ–z plane over λ9

2
3 Blade Fan
5 Blade Fan
9 Blade Fan

∂
2
r∂θ Vθ Vz /Vtip

1

0

−1

−2

−3
−π/9

−π/18

0

π/18

π/9

λ

Figure 142 Case III net Reynolds shear stress in θ–z plane over λ9

193

2
3 Blade Fan
5 Blade Fan
9 Blade Fan

∂
2
r∂θ Vθ Vz /Vtip

1

0

−1

−2

−3
−π/9

−π/18

0

π/18

π/9

λ

Figure 143 Case IV net Reynolds shear stress in θ–z plane over λ9

2
3 Blade Fan
5 Blade Fan
9 Blade Fan

∂
2
r∂θ Vθ Vz /Vtip

1

0

−1

−2

−3
−π/9

−π/18

0

π/18

λ

Figure 144 Case V net Reynolds shear stress in θ–z plane over λ9

194

π/9

2 ∂Vz
(Vz Vθ /Vtip
) r¯∂θ

∗

3 Blade Fan
0.1
0

−0.1
0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

2 ∂Vz
(Vz Vθ /Vtip
) r¯∂θ

∗

5 Blade Fan
0.1
0

−0.1
0

π/2

π

2 ∂Vz
(Vz Vθ /Vtip
) r¯∂θ

∗

9 Blade Fan
0.1
0

−0.1
0

π/2

π

θ

Figure 145 Case I turbulence kinetic production by Reynolds shear stress in θ–z plane
over 0 ≤ θ ≤ 2π

195

3 Blade Fan

2 ∂Vz
(Vz Vθ /Vtip
) r¯∂θ

∗

0.1
0.05
0

−0.05
−0.1

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan

2 ∂Vz
(Vz Vθ /Vtip
) r¯∂θ

∗

0.1
0.05
0

−0.05
−0.1

0

π/2

π
9 Blade Fan

2 ∂Vz
(Vz Vθ /Vtip
) r¯∂θ

∗

0.1
0.05
0

−0.05
−0.1

0

π/2

π

θ

Figure 146 Case II turbulence kinetic production by Reynolds shear stress in θ–z plane
over 0 ≤ θ ≤ 2π

196

3 Blade Fan

2 ∂Vz
(Vz Vθ /Vtip
) r¯∂θ

∗

0.1
0.05
0

−0.05
−0.1

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan

2 ∂Vz
(Vz Vθ /Vtip
) r¯∂θ

∗

0.1
0.05
0

−0.05
−0.1

0

π/2

π
9 Blade Fan

2 ∂Vz
(Vz Vθ /Vtip
) r¯∂θ

∗

0.1
0.05
0

−0.05
−0.1

0

π/2

π

θ

Figure 147 Case III turbulence kinetic production by Reynolds shear stress in θ–z plane
over 0 ≤ θ ≤ 2π

197

3 Blade Fan

2 ∂Vz
(Vz Vθ /Vtip
) r¯∂θ

∗

0.1
0.05
0

−0.05
−0.1

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

5 Blade Fan

2 ∂Vz
(Vz Vθ /Vtip
) r¯∂θ

∗

0.1
0.05
0

−0.05
−0.1

0

π/2

π
9 Blade Fan

2 ∂Vz
(Vz Vθ /Vtip
) r¯∂θ

∗

0.1
0.05
0

−0.05
−0.1

0

π/2

π

θ

Figure 148 Case IV turbulence kinetic production by Reynolds shear stress in θ–z plane
over 0 ≤ θ ≤ 2π

198

2 ∂Vz
(Vz Vθ /Vtip
) r¯∂θ

∗

3 Blade Fan
0.1

0

−0.1

0

π/2

π

3π/2

2π

3π/2

2π

3π/2

2π

2 ∂Vz
(Vz Vθ /Vtip
) r¯∂θ

∗

5 Blade Fan
0.1

0

−0.1

0

π/2

π

2 ∂Vz
(Vz Vθ /Vtip
) r¯∂θ

∗

9 Blade Fan
0.1

0

−0.1

0

π/2

π

θ

Figure 149 Case V turbulence kinetic production by Reynolds shear stress in θ–z plane
over 0 ≤ θ ≤ 2π

199

3 Blade Fan
5 Blade Fan
9 Blade Fan

0.02

2 ∂Vz
(Vz Vθ /Vtip
) r¯∂θ

∗

0.06

−0.02
−0.06
−0.1
−0.14
−π/3

−π/6

0

π/6

π/3

λ

Figure 150 Case I turbulence kinetic production by Reynolds shear stress in θ–z plane
over λ3

3 Blade Fan
5 Blade Fan
9 Blade Fan

0.02

2 ∂Vz
(Vz Vθ /Vtip
) r¯∂θ

∗

0.06

−0.02
−0.06
−0.1
−0.14
−π/3

−π/6

0

π/6

π/3

λ

Figure 151 Case II turbulence kinetic production by Reynolds shear stress in θ–z plane
over λ3

200

3 Blade Fan
5 Blade Fan
9 Blade Fan

0.02

2 ∂Vz
(Vz Vθ /Vtip
) r¯∂θ

∗

0.06

−0.02
−0.06
−0.1
−0.14
−π/3

−π/6

0

π/6

π/3

λ

Figure 152 Case III turbulence kinetic production by Reynolds shear stress in θ–z plane
over λ3

3 Blade Fan
5 Blade Fan
9 Blade Fan

0.02

2 ∂Vz
(Vz Vθ /Vtip
) r¯∂θ

∗

0.06

−0.02
−0.06
−0.1
−0.14
−π/3

−π/6

0

π/6

π/3

λ

Figure 153 Case IV turbulence kinetic production by Reynolds shear stress in θ–z plane
over λ3

201

3 Blade Fan
5 Blade Fan
9 Blade Fan

0.02

2 ∂Vz
(Vz Vθ /Vtip
) r¯∂θ

∗

0.06

−0.02
−0.06
−0.1
−0.14
−π/3

−π/6

0

π/6

π/3

λ

Figure 154 Case V turbulence kinetic production by Reynolds shear stress in θ–z plane
over λ3

3 Blade Fan
5 Blade Fan
9 Blade Fan

0.02

2 ∂Vz
(Vz Vθ /Vtip
) r¯∂θ

∗

0.06

−0.02
−0.06
−0.1
−0.14
−π/9

−π/18

0

π/18

π/9

λ

Figure 155 Case I turbulence kinetic production by Reynolds shear stress in θ–z plane
over λ9

202

3 Blade Fan
5 Blade Fan
9 Blade Fan

0.02

2 ∂Vz
(Vz Vθ /Vtip
) r¯∂θ

∗

0.06

−0.02
−0.06
−0.1
−0.14
−π/9

−π/18

0

π/18

π/9

λ

Figure 156 Case II turbulence kinetic production by Reynolds shear stress in θ–z plane
over λ9

3 Blade Fan
5 Blade Fan
9 Blade Fan

0.02

2 ∂Vz
(Vz Vθ /Vtip
) r¯∂θ

∗

0.06

−0.02
−0.06
−0.1
−0.14
−π/9

−π/18

0

π/18

π/9

λ

Figure 157 Case III turbulence kinetic production by Reynolds shear stress in θ–z plane
over λ9

203

3 Blade Fan
5 Blade Fan
9 Blade Fan

0.02

2 ∂Vz
(Vz Vθ /Vtip
) r¯∂θ

∗

0.06

−0.02
−0.06
−0.1
−0.14
−π/9

−π/18

0

π/18

π/9

λ

Figure 158 Case IV turbulence kinetic production by Reynolds shear stress in θ–z plane
over λ9

3 Blade Fan
5 Blade Fan
9 Blade Fan

0.02

2 ∂Vz
(Vz Vθ /Vtip
) r¯∂θ

∗

0.06

−0.02
−0.06
−0.1
−0.14
−π/9

−π/18

0

π/18

π/9

λ

Figure 159 Case V turbulence kinetic production by Reynolds shear stress in θ–z plane
over λ9

204

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205

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