Numerical and Experimental Investigation on the Flow in

Numerical and Experimental Investigation on

the Flow in Rotor-Stator Cavities

Von der Fakultät für Ingenieurwissenschaften, Abteilung Maschinenbau der

Universität Duisburg-Essen

zur Erlangung des akademischen Grades

DOKTOR-INGENIEUR

genehmigte Dissertation

von

Bo Hu

aus

Lanzhou, China

Referent: Prof. Dr.-Ing. F.-K. Benra

Korreferent: Prof. Marcello Manna

Tag der mündlichen Prüfung: 11. 07. 2018

II

Acknowledgement

I would like to express my gratitude to all, who helped me during my study in Germany.

My sincere gratitude goes first and foremost to my supervisor Prof. Dr. –Ing. F.–K. Benra for his

inspirational discussions, constant guidance and encouragement throughout this work. He offers

me the opportunity to start this interesting topic, and gives me great help by providing me with

important materials, advice and inspiration of new ideas during my research. His suggestion as well

as his own technical insights have improved my research work substantially. Furthermore, he has

offered me a lot of opportunities to take part in some good international conferences where I was

able to exchange new research ideas, results and innovations together with other participants

working in my research field.

I am deeply indebted to all my colleagues in the Chair of Turbomachinery at University of

Duisburg-Essen. Special thanks should go to Dr. –Ing. H. J. Dohmen and Detlev Weniger, who

have given me valuable suggestions and patient guidance during the design and construction of my

test rig.

High tribute is also paid to CSC (China Scholarship Council), which provide the funds to cover my

cost of living in Germany and Chair of Turbomachinery at University of Duisburg-Essen, which

provides the funds to build-up the test rig, laboratory site and experimental facilities.

Last but not the least, my gratitude also extends to my family who has been assisting, supporting

and caring for me throughout my life.

Duisburg, Germany

July, 2018

Bo Hu

III

Contents

Abstract ______________________________________________________________________ 1

1. Introduction _________________________________________________________________ 3

1.1 Significance of This Thesis __________________________________________________ 3

1.2 Important Variables and Limitations of Previous Studies ___________________________ 4

1.2.1 Core Swirl Ratio _______________________________________________________ 4

1.2.2 Axial Thrust __________________________________________________________ 4

1.2.3 Frictional Torque ______________________________________________________ 5

2. State of the Art ______________________________________________________________ 9

2.1 Basic Equations ___________________________________________________________ 9

2.2 Thickness of Boundary Layers ______________________________________________ 15

2.3 Core Swirl Ratio _________________________________________________________ 20

2.3.1 Enclosed Rotor-Stator Cavity ____________________________________________ 20

2.3.2 Impact of Through-Flow ________________________________________________ 21

2.3.2.1 With Centripetal Through-Flow (𝑄 < 0 m3/s) __________________________ 23

2.3.2.2 With Centrifugal Through-Flow (𝑄 > 0 m3/s) __________________________ 24

2.3.3 Impact of Surface Roughness in an Enclosed Rotor-Stator Cavity _______________ 26

2.3.4 Impact of Pre-Swirl ____________________________________________________ 26

2.3.4.1 Centripetal Pre-Swirl Through-Flow ___________________________________ 28

2.3.4.2 Centrifugal Pre-Swirl Through-Flow ___________________________________ 29

2.4 Axial Thrust _____________________________________________________________ 31

2.5 Moment Coefficient _______________________________________________________ 33

2.5.1 The Free Disk ________________________________________________________ 33

2.5.2 Enclosed Rotor-Stator Cavity ____________________________________________ 34

2.5.3 Rotor-Stator Cavity with Through-Flow ___________________________________ 36

2.5.4 Impact of Surface Roughness on the Moment Coefficient ______________________ 38

2.6 Flow Separation near the Entrance ___________________________________________ 41

2.7 Influence of the Sealing Gap Height __________________________________________ 44

2.7.1 Flow Structure inside the Sealing Gap _____________________________________ 44

2.7.2 Leakage Volumetric Flow Rate through the Sealing Gap ______________________ 48

2.8 Side Chamber Flow in a Centrifugal Pump _____________________________________ 49

3. Experimental Set-Up _________________________________________________________ 51

3.1 Mechanical Set-Up _______________________________________________________ 51

3.2 Uncertainty Analysis ______________________________________________________ 56

3.3 Experimental Validation ___________________________________________________ 57

4. Numerical Simulation Set-Up __________________________________________________ 60

4.1 Turbulence Model ________________________________________________________ 60

IV

4.2 Grid Generation __________________________________________________________ 61

4.3 Simulation Set-Up ________________________________________________________ 63

4.4 Validation for Numerical Simulation _________________________________________ 65

5. Results and Discussion _____________________________________________________ 67

5.1 Rotor-Stator Cavity with Centripetal Through-Flow _____________________________ 67

5.1.1 Simulation Results of Velocity Distributions ________________________________ 67

5.1.2 Core Swirl Ratio ______________________________________________________ 70

5.1.2.1 Impact of Through-Flow with a Smooth Disk ____________________________ 70

5.1.2.2 Impact of Surface Roughness of the Disk _______________________________ 71

5.1.2.3 Impact of Centripetal Pre-Swirl Through-Flow with a Smooth Disk __________ 73

5.1.3 Pressure Distribution ___________________________________________________ 74



5.1.3.3 Impact of Pre-Swirl on the Pressure Distribution with a Smooth Disk _________ 77

5.1.4 Thrust Coefficient _____________________________________________________ 78



5.1.4.3 Impact of Pre-Swirl on the Thrust Coefficient with a Smooth Disk ___________ 81

5.1.5 3D Diagram with Centripetal Through-Flow ________________________________ 83

5.1.6 Moment Coefficient ___________________________________________________ 84



5.1.6.3 Impact of Pre-Swirl on the Moment Coefficient with a Smooth Disk __________ 95

5.2 Rotor-Stator Cavity with Centrifugal Through-Flow _____________________________ 96

5.2.1 Simulation Results of Velocity Distributions ________________________________ 96

5.2.2 Core Swirl Ratio ______________________________________________________ 98



5.2.2.3 Impact of Centrifugal Pre-Swirl Through-Flow with a Smooth Disk _________ 100

5.2.3 Pressure Distribution __________________________________________________ 102

5.2.3.1 Impact of Through-Flow with a Smooth Disk ___________________________ 102

5.2.3.2 Impact of Surface Roughness of the Disk ______________________________ 103

5.2.3.3 Impact of Pre-Swirl on the Pressure Distribution with a Smooth Disk ________ 104

5.2.4 Thrust Coefficient ____________________________________________________ 104

5.2.4.1 Impact of Centrifugal Through-Flow with a Smooth Disk _________________ 104

5.2.4.2 Impact of Surface Roughness of the Disk ______________________________ 105

5.2.4.3 Impact of Pre-Swirl on the Thrust Coefficient with a Smooth Disk __________ 107

5.2.5 3D Diagram with Centrifugal Through-Flow _______________________________ 108

5.2.6 Moment Coefficient __________________________________________________ 108

5.2.6.1 Impact of Through-Flow with a Smooth Disk ___________________________ 108

5.2.6.2 Impact of Surface Roughness of the disk _______________________________ 110

5.2.6.3 Impact of Pre-Swirl on the Moment Coefficient with a Smooth Disk _________ 113

V

6. Applications of the Results in Radial Pumps _____________________________________ 115

6.1 Flow in the Rear Chamber of a Submersible Multi-Stage Slurry Pump (SMSP) _______ 115

6.1.1 Sand Discharge Groove Design _________________________________________ 115

6.1.2 Geometrical Parameters _______________________________________________ 116

6.1.3 Numerical Simulation _________________________________________________ 117

6.1.4 Results and Discussion ________________________________________________ 118

6.2 The Axial Thrust in a Deep Well Pump ______________________________________ 123

6.2.1 Main Geometric Parameters ____________________________________________ 123

6.2.2 Simulation Set-Up ____________________________________________________ 124

6.2.3 Results for Axial Thrust Coefficient ______________________________________ 124

7. Summary of the Results ____________________________________________________ 126

8. Outlook ________________________________________________________________ 128

References __________________________________________________________________ 129

VI

Nomenclature

Latin Symbols

A Area m2

a Hub radius m

𝑎𝑅 Velocity factors for the rotor in case of separated boundary layers −

𝑎𝑆 Velocity factors for the stator in case of separated boundary layers −

𝐵+ Constant in the expression for the boundary layer thickness −

b Outer radius of the disk m

𝑏2 Outlet width of impeller mm

𝐶𝑎𝑚 Coefficient of the inlet angular momentum −

𝐶𝐷 Through-flow coefficient −

𝐶𝐹 Axial thrust coefficient −

𝐶𝐹𝑓 𝐶𝐹 on the front surface −

𝐶𝐹𝑓−𝑎𝑐 𝐶𝐹𝑓 when disk rotates anti-clockwise −

𝐶𝐹𝑓−𝑐 𝐶𝐹𝑓 when disk rotates clockwise −

𝐶𝐹𝑏 𝐶𝐹 on the back surface −

𝐶𝐹𝑠 𝐶𝐹 when the shaft is rotating without the disk −

𝐶𝑓 Skin friction coefficient −

𝐶𝑀 Moment coefficient −

𝐶𝑀𝑎 𝐶𝑀 for the source region −

𝐶𝑀𝑏 𝐶𝑀 for the core region −

𝐶𝑀𝑐 𝐶𝑀 due to the cylindrical shroud −

𝐶𝑀𝑐𝑦𝑙 𝐶𝑀 on the cylinder surface of the disk −

𝐶𝑀𝑠 𝐶𝑀 of the shaft −

𝐶𝑀𝑠−𝑎𝑐 𝐶𝑀𝑠 when disk rotates anti-clockwise −

𝐶𝑀𝑠−𝑐 𝐶𝑀𝑠 when disk rotates clockwise −

𝐶𝑀1 𝐶𝑀 for regime I −

𝐶𝑀2 𝐶𝑀 for regime II −

𝐶𝑀3 𝐶𝑀 for regime III −

𝐶𝑀4 𝐶𝑀 for regime IV −

𝐶𝑝 Pressure coefficient −

𝐶𝑞 Through-flow rate coefficient −

𝐶𝑞𝑟 Local flow rate coefficient −

𝐶𝑅 Factor in the flow model formulation −

𝐶𝑆 Factor in the flow model formulation −

𝐶+ Constant in the equation for the boundary layer thickness on the stator −

𝐷1 Diameter of the impeller eye mm

VII

𝐷2 Outer diameter of the impeller mm

𝑑𝑔𝑎𝑝 Diameter of the sealing gap m

𝑑ℎ Diameter of the horizontal pipe m

𝐸𝑘 Ekman number −

e𝑇 Relative error of the transducer −

𝑐 Constant −

𝑐𝑠 Suitable dimensionless radial gap −

𝑐∗ Constant −

e𝐷 Relative error due to the data acquisition device −

Fa Axial thrust N

Faf Force on the front surface of the disk N

Fab Force on the back surface of the disk N

Fap Force on the impeller passage N

Fas Impulse fore at the impeller eye N

𝐹𝑎−𝑠 Axial thrust when the shaft is rotating without the disk N

𝑓 Body force N

𝑓𝑟 Body force in radial direction N

𝑓𝜑 Body force in tangential direction N

𝑓𝑧 Body force in axial direction N

𝑓∗ Correction function −

G Non-dimensional axial gap −

H Pressure head of a pump m

K Core swirl ratio at 휁 = 0.5 −

𝐾0 K for enclosed rotor-stator cavity −

𝐾𝑏 K at 𝑟 = 𝑏 −

𝐾𝑒 K at the entrance −

𝐾ℎ K at the radius of pre-swirl nozzle −

𝐾𝑝 K with pre-swirl −

𝐾ℎ,𝑒𝑓𝑓 Effective pre-swirl ratio at the radius of pre-swirl nozzle −

𝑘𝑠 Equivalent surface roughness 𝜇m

𝑘𝑠𝑙 Transition point between the hydraulic smooth disks and the disks in the transition zone 𝜇m

L Angular momentum in the centripetal through-flow −

l Streamwise coordinates m

𝑀 Frictional torque Nm

𝑀𝑐𝑦𝑙 Frictional resistance on the cylinder surface of the disk Nm

𝑀𝑟 Measured range −

𝑀𝑠 Frictional torque when the shaft is rotating without the disk Nm

m Constant −

VIII

�̇� Mass flow rate kg/s

𝑁𝐷 Uncertainty of the data acquisition system −

𝑁𝑇 Uncertainty of the transducer −

∆𝑁 Uncertainty of the measured results −

n Speed of rotation rpm

𝑛1 Constant −

𝑛𝑇 Number of transducers −

𝑛𝑀 Measuring times to obtain one result −

𝑛𝑠 Specific speed of pump −

p Pressure Pa

𝑝𝑏 Pressure at 𝑟 = 𝑏 Pa

𝑝𝑟 Pressure drop ratio −

𝑝𝑥 Pressure at 𝑡ℎ = 𝑥 Pa

𝑝0 Pressure at 𝑡ℎ = 0 Pa

𝑝∗ Non-dimensional pressure −

Q Volumetric through-flow rate m3/s

𝑄𝑒 Flow rate at the point of best efficiency L/s

Re Global circumferential Reynolds number −

Rep Perimeter Reynolds number for the sealing gap −

Reφ Local circumferential Reynolds number −

r Radial coordinate m

𝑟𝑎 Radius of the hub m

𝑟ℎ Radius of the horizontal pipe m

∆𝑟 Radial distance from the disk to the wall m

∆𝑟𝑠𝑒𝑎𝑙 Sealing gap height m

s Axial gap of the front chamber m

sb Axial gap of the back chamber m

𝑇𝑎 Taylor number −

t Thickness of the disk m

𝑡ℎ Time h

U Velocity of the free stream m/s

u Velocity along the flat plate m/s

𝑢∗ Friction velocity m/s

𝑉𝑟 Non-dimensional radial velocity −

𝑉𝑧 Non-dimensional axial velocity −

𝑉𝜑 Non-dimensional tangential velocity −

𝑉𝜑ℎ 𝑉𝜑 at 𝑥 = 𝑥ℎ −

v Velocity m/s

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IX

v𝑎𝑥𝑔𝑎𝑝 Sealing gap mean axial velocity m/s

v𝑔𝑎𝑝 Sealing gap circumferential velocity m/s

v𝑟 Radial velocity m/s

v𝑧 Axial velocity m/s

v𝜑 Tangential velocity m/s

v𝑚𝑒𝑎𝑛 Mean velocity m/s

x Non-dimensional radial coordinate −

𝑥ℎ Non-dimensional radius of the horizontal pipe −

𝑥𝑐 Radial location where source region ends −

𝑥∗ 𝑥∗ = 𝐾𝑝0.5 ∙ 𝑥𝑎 −

∆𝑥 Non-dimensional radial gap width −

𝑦 Spacing of the first layer node m

𝑦+ Non-dimensional wall distance −

𝑧 Axial coordinate m

𝑧𝑙 Normal coordinates m

Greek Symbols

𝛼 Index −

𝛽 Pre-swirl angle Deg

𝛽2 Outlet blade angle Deg

𝛾 Heat capacity ratio −

𝛶𝑅 Proportionality factor for the boundary layer thickness on the rotor −

𝛶𝑠 Proportionality factor for the boundary layer thickness on the stator −

𝛿 Thickness of the boundary layer m

𝜍 Distance from the buffer layer to the viscous sublayer m

𝛿𝑅 Thickness of the disk layer m

𝛿𝑠 Thickness of the wall layer m

휀 Diameter of spheres 𝜇m

휁 Non-dimensional axial coordinate m

휂 Pump efficiency −

휃 Angle in cylindrical coordinates Deg

휃1 Wrapping angle of blade Deg

𝜆𝑅 Friction factor for the rotor wall −

𝜆𝑆 Friction factor for the stator wall −

𝜆𝑇 Turbulent flow parameter −

𝜇 Dynamic viscosity of water N ∙ s m2⁄

𝑣 Kinematic viscosity of water m2 s⁄

𝜌 Density of water kg m3⁄

X

𝜏 Shear stress N m2⁄

𝜏𝑤 Wall shear stress N m2⁄

𝜑𝐺 Non-dimensional through-flow rate −

𝛹 Flow rate coefficient −

𝛺 Angular velocity of the disk rad s⁄

𝛺𝑓 Angular velocity of the fluid at 휁 = 0.5 rad s⁄

Abbreviations

DC Direct current

DNS Direct numerical simulation

DWP Deep well pump

FS Full scale

LDA Laser Doppler Anemometer

LDV Laser Doppler Velocimetry

LES Large eddy simulation

RANS Reynolds-averaged Navier-Stokes equations

RSM Reynolds Stress Models

rpm Revolution per minute

SMSP Submersible multi-stage slurry pump

SST Shear stress transport

TF Through-flow

SR-4 Simmons Ruge-4

XI

List of figures

Fig Name Page

1 Cross section of a centrifugal pump 3

2 Concerns during the design of a radial pump or turbine 4

3 Sources of the axial thrust for a radial pump 5

4 Typical velocity profiles for the four flow regimes 5

5 Distinguishing lines for flow regimes without through-flow (2D Daily&Nece diagram [28]) 6

6 Contents of this thesis 7

7 Main geometry of the test rig 7

8 Flow structure in an idealized rotor-stator cavity (left) and the velocity profiles for a wide

axial gap (replotted from Will [88]) 16

9 Radial velocity profiles in dependence on Eq. 36 and Eq. 37 (replotted from Will [88]) 18

10 Comparison of results from Eq. 51 and Eq. 52 24

11 Velocity profiles for Batchelor type flow, Couette type flow and Stewartson type flow 24

12 Results for K from different equations for centrifugal through-flow 25

13 Experimental results of K along the radius of disk for 𝐺 = 0.031, 𝑅𝑒 = 3.1 × 106 and

𝐶𝐷 = 0 by Kurokawa et al. [54] 26

14 Velocity triangles for the pre-swirl through-flow 27

15 Flow structure in the case of 𝐾 > 1 29

16 Sketch of the rotor-stator cavity (redrawn from Karabay et al. [49]) 30

17 Variation of 𝐾ℎ,𝑒𝑓𝑓 versus 𝐾ℎ by Karabay [49] 31

18 Flow structure around a free disk (According to Schlichting and Gersten [73]) 33

19 Torques on an annular volume element 33

20 Flow structure inside a rotor-stator cavity with (a) no through-flow, (b) with centrifugal

through-flow and 𝜆𝑇 < 0.219 and (c) with through−flow, 𝜆𝑇 > 0.219 37

21 Moment coefficient for a rotating disk 38

22 Schematic drawing of the test rig (redrawn from Daily and Nece [28]) 39

23 Rough disk torque data (replotted from Daily and Nece [28]) 40

24 Sketch of the test rig (replotted from Kurokawa et al. [54]) 41

25 Graphical representation of the velocity profile and the reverse flow which show the flow

separation [3] 41

26 Approximate separation line for centripetal through-flow 42

27 Solution procedure for the approximate separation line 43

28 Comparison of the results for the separation line (𝛺 = 0, 𝐶𝐷 = −5050, 𝐺 = 0.045) 43

29 One seventh segment of the back cavity (Will [88]) 44

30 Comparison between experimental and numerical results for the shroud side chamber (left)

and the hub side chamber (right) in case of 0.48 mm sealing gap height (Will [88]) 45

31 Comparison between experimental and numerical results for the hub side chamber in case of

0.24 mm sealing gap height (Will [88]) 45

XII

32 Flow in the sealing gap (∆𝑟 = 0.8 mm) for different leakage flow rates (replotted from Will

[88]) 48

33 Test rig design 52

34 Test rig set-up for each steps to measure 𝐹𝑎 53

35 Inlet swirlers in the horizontal pipe (for centrifugal through-flow) 53

36 Drawing of the centrifugal inlet swirler (Left: front view; Right: side view) 54

37 Geometry of the radial guide vanes and the flow in the rotor-stator cavities 54 to 55

38 Experimental results of 𝐶𝑀𝑠 and 𝐶𝐹𝑠 versus Re 57

39 Comparison of results when the shaft rotates in different direction 58

40 Comparison of 𝐶𝑝 along the radius for 𝑅𝑒 = 1.36 × 106 59

41 Domain for numerical simulation (𝐺 = 0.072) 64

42 Additional fluid domain for centrifugal pre-swirl through-flow 64

43 Mesh independence analysis 65

44 Comparison of radial pressure distribution for 𝑅𝑒 = 4.15 × 106 and 𝐺 = 0.036 66

45 Comparisons of radial pressure distribution for 𝑅𝑒 = 1.36 × 106 and 𝐺 = 0.0495 without

pre-swirl 66

46 Research procedure 67

47 Velocity profiles for 𝑅𝑒 = 1.9 × 106 and 𝐺 = 0.072 68


49 K (𝐶𝑞𝑟) curves for centripetal through-flow 71

50 K (x) curves for 𝐺 = 0.031, 𝑅𝑒 =3.1× 106 and 𝐶𝐷 = 0 by Kurokawa et al. [54] 72

51 Impact of 𝑘𝑠 on K when G=0.072 73

52 Inlet boundary conditions with centripetal through-flow 73

53 𝐾𝑝 (𝑥) curves for 𝐺 = 0.072, 𝐶𝐷 = −5050 and 𝑅𝑒 = 1.9 × 106 74

54 Influence of 𝐶𝐷 on 𝐶𝑝 in dependence of Re and G (𝑘𝑠 = 0.4 𝜇m) 75

55 𝐶𝑝 (x) curves along the radius of the disks 76 to 77

56 𝐶𝑝 (x) curves for 𝐺 = 0.072, 𝐶𝐷 = −5050 and 𝑅𝑒 = 1.9 × 106 with centripetal through-

flow 77

57 𝐶𝐹 (𝐶𝐷) curves for 𝑅𝑒 = 1.32 × 106 78

58 𝐶𝐹 (𝑅𝑒) curves in dependence of 𝐶𝐷 and G 79

59 𝐶𝐹 (𝑅𝑒) curves in dependence of 𝐶𝐷, G and 𝑘𝑠 80 to 81

60 Comparison of the results of 𝐶𝐹 for 𝐺 = 0.05 and 𝐶𝑎𝑚 𝐶𝑞⁄ = −0.619 [52] 82

61 𝐶𝐹 (𝐶𝑎𝑚) curves for 𝐶𝐷 = −5050 for various 𝑅𝑒 and G 82 to 83

62 3D diagram distinguishing regime III and regime IV with centripetal through-flow for 0.3 ×106 ≤ 𝑅𝑒 ≤ 3.3 × 106 84

63 Comparison of the results of 𝐶𝑀 for 𝐺 = 0.018 and 𝐺 = 0.072 at 𝐶𝐷 = 0 85

64 𝐶𝑀 (Re) curves with centripetal through-flow 85 to 86

65 Results of 𝐶𝑀3/𝐶𝑀4 at the distinguishing lines for centripetal through-flow 86

66 Comparison of 𝐶𝑀 with different values of 𝑘𝑠 in an enclosed rotor-stator cavity 87

XIII

67 Comparison of 𝐶𝑀 from different equations 88

68 𝐶𝑀 (𝑅𝑒) curves at 𝐺 = 0.012 and 𝐺 = 0.027 for different values of 𝑘𝑠 and |𝐶𝐷| 89 to 90

69 𝐶𝑀 (Re) curves at 𝐺 = 0.047 and 𝐺 = 0.065 for different values of 𝑘𝑠 and |𝐶𝐷| 91 to 93

70 𝐶𝑀 (Re) curves at various G for different values of 𝑘𝑠 and 𝐶𝐷 93 to 94

71 𝐶𝑀 (𝐶𝑎𝑚) curves for |𝐶𝐷| = 5050 at different 𝑅𝑒 95



74 𝐾 (𝐶𝑞𝑟) curves 98

75 Large differences of K attributed to the geometry near the outlet 99

76 𝐾 (𝐶𝑞𝑟) curves for various 𝑘𝑠 at 𝐺 = 0.072 100

77 𝐾𝑝 (𝐶𝑞𝑟) curves for 𝑅𝑒 = 1.9 × 106, 𝐶𝐷 = 5050 and 𝐺 = 0.072 101

78 Distribution of 𝐶𝑝 along the radius 102

79 Distribution of 𝐶𝑝 along the radius for the rough disks 103

80 𝐶𝑝 (x) curves for various 𝐶𝑎𝑚 at 𝐶𝐷 = 5050 and G=0.072 104

81 𝐶𝐹 (𝑅𝑒) curves (𝑘𝑠 = 0.4 𝜇m) 105

82 𝐶𝐹 (𝑅𝑒) curves in dependence of 𝐶𝐷, 𝐺 and 𝑘𝑠 106

83 Mean 𝐶𝐹 (𝑅𝑒) curves (𝑘𝑠 = 0.4 𝜇m) 107

84 3D diagram distinguishing regime III and regime IV with centrifugal through-flow for 0.3 ×106 ≤ 𝑅𝑒 ≤ 3.3 × 106 108

85 Curves for 𝐶𝑀 in dependence of 𝑅𝑒 for different values of 𝐶𝐷 and 𝐺 with centrifugal

through-flow 109

86 Results of 𝐶𝑚3/𝐶𝑚4 at the distinguishing lines for centrifugal through-flow 110

87 Curves for 𝐶𝑀 in dependence of 𝑅𝑒 for different values of 𝐶𝐷 and 𝐺 with centrifugal

through-flow 111 to 113

88 Experimental results of 𝐶𝑀 versus 𝑅𝑒 for 𝑘𝑠 = 0.4 𝜇m and 𝐶𝐷 = 5050 114

89 Leakage flow and flow pattern inside the rear chamber of a SMSP 115

90 Sand discharge groove 116

91 Leakage flow and flow pattern inside the rear chamber of a SMSP 118

92 Distribution of the radial velocity 118

93 Movement of particles on the meridian plane 119

94 Distribution of non-dimensional radial velocity 120

95 Distribution of non-dimensional tangential velocity 121

96 Distribution of solid volume fraction 122

97 Pump performance from experiments 122

98 Pressure drop during the abrasion test 123

99 Geometry of the impeller (left) and the guide vane (right) 124

100 Axial thrust in a centrifugal single stage well pump [84]: (a) Pressure distribution and (b)

Comparison of 𝐹𝑎𝑏 − 𝐹𝑎𝑓 125

XIV

List of Tables

Tab Name Page

1 Current research states on flow in rotor-stator cavities 6

2 Values of 𝐵+ and m from the literature 17

3 Values of 𝑐 and 𝑐∗ by Kurokawa and Toyokura [53] 17

4 Core swirl ratio 𝐾0 in the literature for an enclosed rotor-stator cavity (reorganized from Will

[88]) 21

5 Parameters of the experiments 44

6 Estimation of 𝑅𝑒𝑔𝑎𝑝 and 𝑇𝑎 in Will [88] 47

7 Parameters of the experiments 52

8 Parameters of the experiments conducted in the test rig 53

9 Pre-swirl angles of the radial guide vanes 56

10 Surface roughness of the disks 56

11 Uncertainty analysis for the measurements 57

12 Selections of turbulence model in the literature 60

13 Moment coefficients from different turbulence models (K. N. Volkov [51]) 61

14 Grid number and maximum values of 𝑦+ 63

15 Qualities of the meshes 63

16 Geometrical parameters of the impeller 116

17 Parameters of the 6 rotor-stator cavities 117

18 Uncertainties of results 122

19 Main geometric parameters of the pump 123

XV

List of Equations

𝜕𝑝

𝜕𝑡+

𝜕(𝜌 ∙ v𝑖)

𝜕𝑥𝑖

= 0 Eq. 1

𝜕(𝜌 ∙ v𝑖)

𝜕𝑡+

𝜕(𝜌 ∙ v𝑖 ∙ v𝑖)

𝜕𝑥𝑗

= 𝑓𝑖 −𝜕𝑝

𝜕𝑥𝑖

+𝜕𝜏𝑖𝑗

𝜕𝑥𝑗

Eq. 2

𝜏𝑖𝑗 = 𝜇 ∙ (𝜕v𝑖

𝜕𝑥𝑗

+𝜕v𝑗

𝜕𝑥𝑖

−2

3∙ 𝛿𝑖𝑗 ∙

𝜕v𝑘

𝜕v𝑘

) Eq. 3

𝜌 ∙ (𝜕v𝑟

𝜕𝑡+ v𝑟 ∙

𝜕v𝑟

𝜕𝑟+

v𝜑

𝑟∙

𝜕v𝑟

𝜕𝜑−

v𝜑2

𝑟+ v𝑧 ∙

𝜕v𝑟

𝜕𝑧)

= 𝑓𝑟 −𝜕𝑝

𝜕𝑟+

1

𝑟∙

𝜕(𝑟 ∙ 𝜏𝑟𝑟)

𝜕𝑟+

1

𝑟∙

𝜕(𝜏𝑟𝜑)

𝜕𝜑−

𝜏𝜑𝜑

𝑟+

𝜕(𝜏𝑟𝑧)

𝜕𝑧

Eq. 4

𝜌 ∙ (𝜕v𝜑

𝜕𝑡+ v𝑟 ∙

𝜕v𝜑𝜑

𝜕𝑟+

v𝜑

𝑟∙

𝜕v𝜑

𝜕𝜑+

v𝜑 ∙ v𝑟

𝑟+ v𝑧 ∙

𝜕v𝜑

𝜕𝑧)

= 𝑓𝜑 −1

𝑟∙

𝜕𝑝

𝜕𝜑+

1

𝑟2∙

𝜕(𝑟2 ∙ 𝜏𝜑𝑟)

𝜕𝑟+

1

𝑟∙

𝜕(𝜏𝜑𝜑)

𝜕𝜑+

𝜕(𝜏𝜑𝑧)

𝜕𝑧

Eq. 5

𝜌 ∙ (𝜕v𝑧

𝜕𝑡+ v𝑟 ∙

𝜕v𝑧

𝜕𝑟+

v𝜑

𝑟∙

𝜕v𝑧

𝜕𝜑+ v𝑧 ∙

𝜕v𝑧

𝜕𝑧) = 𝑓𝑧 −

𝜕𝑝

𝜕𝑧+

𝜕(𝜏𝑧𝑟)

𝜕𝑟+

𝜏𝑧𝑟

𝑟+

1

𝑟∙

𝜕(𝜏𝑧𝜑)

𝜕𝜑+

𝜕(𝜏𝑧𝑧)

𝜕𝑧

Eq. 6

1

𝑟∙

𝜕(𝑟 ∙ v𝑟)

𝜕𝑟+

1

𝑟∙

𝜕v𝜑

𝜕𝜑+

𝜕v𝑧

𝜕𝑧= 0

Eq. 7

𝜕(𝜌 ∙ v𝑖)

𝜕𝑡+

𝜕(𝜌 ∙ v𝑖 ∙ v𝑖)

𝜕𝑥𝑗

= 𝑓𝑖 −𝜕𝑝

𝜕𝑥𝑖

Eq. 8

𝜌 ∙ (v𝑟 ∙𝜕v𝑟

𝜕𝑟−

v𝜑2

𝑟+ v𝑧 ∙

𝜕v𝑧

𝜕𝑧) = −

𝜕𝑝

𝜕𝑟+

𝜕(𝜏𝑟𝑧)

𝜕𝑧

Eq. 9

𝜌 ∙ (v𝑟 ∙𝜕v𝜑

𝜕𝑟+

v𝑟 ∙ v𝜑

𝑟+ v𝑧 ∙

𝜕v𝜑

𝜕𝑧) =

𝜕(𝜏𝜑𝑧)

𝜕𝑧

Eq. 10

𝜌 ∙ (v𝑟 ∙𝜕v𝑧

𝜕𝑟+ v𝑧 ∙

𝜕v𝑧

𝜕𝑧) = −

𝜕𝑝

𝜕𝑧+

𝜏𝑟𝑧

𝑟+

𝜕(𝜏𝑟𝑧)

𝜕𝑟

Eq. 11

𝜕v𝑟

𝜕𝑟+

v𝑟

𝑟+

𝜕v𝑧

𝜕𝑧= 0

Eq. 12


𝜕𝑟+

v𝑟 ∙ v𝜑

𝑟) = 0

Eq. 13

v𝑟

𝑟∙

𝜕

𝜕𝑟(v𝜑 ∙ 𝑟) = 0

Eq. 14

𝜕𝑝

𝜕𝑟= 𝜌 ∙

v𝜑2

𝑟

Eq. 15

𝜕𝑝

𝜕𝑟= 𝜌(∙

v𝜑2

𝑟− v𝑟 ∙

𝜕v𝑟

𝜕𝑟)

Eq. 16

v𝑟 ∙𝜕v𝑟

𝜕𝑟−

v𝜑2

𝜕𝑟+ v𝑧 ∙

𝜕v𝑟

𝜕𝑧=

1

𝑟∙ [

𝜕

𝜕𝑟∙ (𝑟 ∙ v𝑟

2) +𝜕

𝜕𝑧∙ (𝑟 ∙ v𝑟 ∙ v𝑧) − v𝜑

2] Eq. 17

v𝑟 ∙𝜕v𝜑

𝜕𝑟−

v𝑟 ∙ v𝜑

𝑟+ v𝑧 ∙

𝜕v𝜑

𝜕𝑧=

1

𝑟2∙ [

𝜕

𝜕𝑟∙ (𝑟 ∙ v𝑟 ∙ v𝜑) +

𝜕

𝜕𝑧∙ (𝑟2 ∙ v𝑧 ∙ v𝜑)]

Eq. 18

XVI

𝜕

𝜕𝑟(𝑟 ∙ v𝑟) + 𝑟 ∙

𝜕

𝜕𝑧∙ (v𝑟 ∙ v𝑧) − v𝜑

2 = −𝑟

𝜌∙

𝜕𝑝

𝜕𝑟+

𝑟

𝜌∙

𝜕(𝜏𝑟𝑧)

𝜕𝑧

Eq. 19

𝜕

𝜕𝑟(𝑟2 ∙ v𝑟 ∙ v𝜑) +

𝜕

𝜕𝑧∙ (𝑟2 ∙ v𝑧 ∙ v𝜑) = −

𝑟2

𝜌∙

𝜕(𝜏𝜑𝑧)

𝜕𝑟

Eq. 20

( ∫𝜕𝑋(𝑟, 𝑧)

𝜕𝑟

𝑧2

𝑧1

𝑑𝑧) =𝜕

𝜕𝑟( ∫ 𝑋(𝑟, 𝑧)

𝑧2

𝑧1

𝑑𝑧) +𝜕𝑧1

𝜕𝑟∙ 𝑋(𝑟, 𝑧1 ) −

𝜕𝑧2

𝜕𝑟∙ 𝑋(𝑟, 𝑧2)

Eq. 21

𝜕

𝜕𝑟(𝑟 ∙ ∫ v𝑟

2

𝑧2

𝑧1

𝑑𝑧) + 𝑟 ∙𝜕𝑧1

𝜕𝑟∙ v𝑟1

2 − 𝑟 ∙𝜕𝑧2

𝜕𝑟∙ v𝑟2

2 + 𝑟 ∙ v𝑟2 ∙ v𝑧2 − 𝑟 ∙ v𝑟1 ∙ v𝑧1 − ∫ v𝜑2𝑑𝑧

𝑧2

𝑧1

= −𝑟

𝜌∙ ∫

𝜕𝑝

𝜕𝑟𝑑𝑧 +

𝑟

𝜌

𝑧2

𝑧1

∙ ∫𝜕(𝜏𝑟𝑧)

𝜕𝑧𝑑𝑧

𝑧2

𝑧1

Eq. 22

𝜕

𝑟2∙

𝜕

𝜕𝑟(𝑟2 ∙ ∫ v𝑟 ∙ v𝜑

𝑧2

𝑧1

𝑑𝑧) +𝜕𝑧1

𝜕𝑟∙ v𝑟1 ∙ v𝜑1 −

𝜕𝑧2

𝜕𝑟∙ v𝑟2 ∙ v𝜑2 + 𝑟 ∙ v𝑧2 ∙ v𝜑2 − v𝑧1 ∙ v𝜑1

=1

𝜌∙ ∫

𝜕(𝜏𝜑𝑧)

𝜕𝑧𝑑𝑧

𝑧2

𝑧1

Eq. 23

1

𝑟∙

𝜕

𝜕𝑟(𝑟 ∙ ∫ v𝑟

2

𝑠

0

𝑑𝑧) −1

𝑟∙ ∫ v𝜑

2

𝑠

0

𝑑𝑧 = −1

𝜌∙ ∫

𝜕𝑝

𝜕𝑟𝑑𝑧

𝑠

0

+1

𝜌∙ (𝜏𝑟𝑧𝑆 − 𝜏𝑟𝑧𝑅)

Eq. 24

1

𝑟2∙

𝜕

𝜕𝑟(𝑟2 ∙ ∫ v𝑟v𝜑

𝑠

0

𝑑𝑧) =1

𝜌∙ (𝜏𝜑𝑧𝑆 − 𝜏𝜑𝑧𝑅)

Eq. 25

𝑑𝐾

𝑑𝑟=

2 ∙ 𝜋 ∙ 𝑏

�̇� ∙ 𝛺∙ (𝜏𝜑𝑧𝑆 − 𝜏𝜑𝑧𝑅) −

2 ∙ 𝐾

𝑅

Eq. 26

𝜏 = 𝜆 ∙𝜌

8∙ v𝑚𝑒𝑎𝑛

2

Where v𝑚𝑒𝑎𝑛𝑅 = 𝑟2 ∙ 𝛺2 ∙ (1 − 𝐾)2;

v𝑚𝑒𝑎𝑛𝑆 = 𝑟2 ∙ 𝛺2 ∙ 𝐾2. Eq. 27

𝑑𝐾

𝑑𝑟=

�̅�2

4 ∙ 𝜑𝐺

∙ (𝜆𝑠 ∙ 𝐾2 − 𝜆𝑅 ∙ (1 − 𝐾)2) −2 ∙ 𝐾

𝑅

Where 𝜑𝐺 =𝑄

𝜋∙𝛺∙𝑏3. Eq. 28

𝜑𝐺 → 0, 𝐾 = 0.5 Eq. 29

𝜑𝐺 → ∞, 𝐾 ∙ 𝑥2 = 0.5 Eq. 30

∫ v𝑟v𝜑𝑧=𝑠

𝑧=0𝑑𝑧 = ∫ v𝑟v𝜑

𝑧=𝛿𝑅

𝑧=0𝑑𝑧+∫ v𝑟v𝜑

𝑧=𝑠−𝛿𝑠

𝑧=𝛿𝑅𝑑𝑧+∫ v𝑟v𝜑

𝑧=𝑠

𝑧=𝑠−𝛿𝑠𝑑𝑧

Eq. 31

𝛿𝑅 = 𝐵+ ∙𝑟

𝑅𝑒𝜑

15

∙ (1 − 𝐾)𝑚

Eq. 32

XVII

𝛿𝑠 =𝑓 ∙ 𝑟

(𝑟2 ∙ 𝜔

𝜈)

15

Where 𝑓 =1

𝑐∙𝐾∙ [𝑐∗ ∙ 𝑏 ∙ (1 − 𝐾)3 −

120

49∙

𝑄

2∙𝜋∙𝜔∙𝑟3 ∙ (𝑟2∙𝜔

𝜈)

1

5]. Eq. 33

𝑄 = ∫ v𝑟

𝑧=𝑠

𝑧=0

𝑑𝑧

Eq. 34

∫ v𝑟𝑆

𝑧=𝛿𝑠

𝑧=0

𝑑𝑧𝑠 + ∫ v𝑟𝑅

𝑧=𝛿𝑅

𝑧=0

d𝑧𝑅 =𝑄

2 ∙ 𝜋 ∙ 𝑟

Eq. 35

v𝑟𝑅 = 𝑎𝑅 ∙ (1 − 𝐾) ∙ 𝑟 ∙ 𝛺 ∙ (1 −𝑧𝑅

𝛿𝑅

)𝑛1 ∙ (𝑧𝑅

𝛿𝑅

)1𝑚

Eq. 36

v𝑟𝑆 = −𝑎𝑆 ∙ 𝐾 ∙ 𝑟 ∙ 𝛺 ∙ (1 −𝑧𝑠

𝛿𝑠

)𝑛1 ∙ (𝑧𝑠

𝛿𝑠

)1𝑚

Eq. 37

𝑎𝑅 = 1.18 ∙ (𝑅𝑒𝜑

105+ 2)−0.49

Eq. 38

𝑎𝑆 = 1.03 ∙ (𝑅𝑒𝜑

105+ 2)−0.387

Eq. 39

𝛿𝑠 = 0.304 ∙𝑐∗

𝑐∙

(1 − 𝐾)125

𝐾∙

𝑟

𝑅𝑒𝜑

15

−𝑄

0.408 ∙ 𝑐 ∙ 2 ∙ 𝜋 ∙ 𝑟2 ∙ 𝛺 ∙ 𝐾

Eq. 40

𝛿𝑅 = 𝛶𝑅 ∙ 𝑟35 ∙ (

𝜈

𝛺)

15

Eq. 41

𝛿𝑠 = 𝛶𝑠 ∙ 𝑟35 ∙ (

𝜈

𝛺)

15

Eq. 42

𝜆𝑅 =0.18

𝐶𝑅

∙ 𝑅𝑒𝜑−

15 ∙ (

1

1 − 𝐾)

14

Eq. 43

𝜆𝑆 =0.18

𝐶𝑆

∙ 𝑅𝑒𝜑−

15 ∙ (

1

𝐾)

14

Eq. 44

𝐶𝑅 = 0.315 Eq. 45

𝐶𝑆 = 𝐶𝑅 ∙ (1 − 𝐾0

𝐾0

)74

Eq. 46

𝐾0 =1

1 + √1 + 5 ∙ 𝐺

Eq. 47

𝐾 = 0.25 ∙ [−1 + √5 − 4 ∙ 𝜑𝐺

√𝑅𝑒𝜑

𝑥2]

2

Eq. 48

(1 − 𝐾)85 ∙ (1 − 0.51 ∙ 𝐾) − 0.638 ∙ 𝐾

45 = 0.25 ∙ [−1 + 4 ∙ 𝜑𝐺√

𝑅𝑒𝜑

𝑥2]

2

Eq. 49

XVIII

𝑑𝐾

𝑑𝑅=

𝑅2

4 ∙ 𝜑𝐺

∙ (𝑓∗ ∙ 𝜆𝑠 ∙ 𝐾2 − 𝜆𝑅 ∙ (1 − 𝐾)2) −2 ∙ 𝐾

𝑅

Where 𝑓∗ = 1 + (𝑠

𝑏+𝑙1−𝑎+ 5 ∙ 𝑅4 ∙ |1 −

𝐾

0.58|

6

5).

Eq. 50

𝐾 = 2 ∙ (−5.9 ∙ 𝐶𝑞𝑟 + 0.63)5

7 − 1 , 𝐶𝑞𝑟 =𝑄∙𝑅𝑒𝜑

0.2

2∙𝜋∙𝛺∙𝑟3 Eq. 51

𝐾 = [−8.85 ∙ 𝐶𝑞𝑟 + 0.5

𝑒(−1.45𝐶𝑞𝑟)]

54

Eq. 52

𝐾 =𝐾0

12.74𝑄

𝛺 ∙ 𝑏3 ∙ 𝑅𝑒𝜑0.2 ∙ (

𝑏𝑟

)

135

+ 1

Eq. 53

𝐾 = 0.032 + 0.32 × 𝑒−𝐶𝑞𝑟

0.028 Eq. 54

𝑘𝑠 =𝜋∙

8 , 휀 = 0.978 ∙ 𝑅𝑧

Eq. 55

𝑘𝑠𝑙 =100 ∙ 𝜈

(1 − 𝐾) ∙ 𝑟 ∙ 𝛺

Eq. 56

tan(𝛽) =𝑉𝜑

𝑉𝑟

Eq. 57

tan(𝛽) =𝑉𝜑

𝑉𝑧

Eq. 58

49

720𝑎∗ ∙ 𝑏 ∙ (1 − 𝐾𝑏)3 +

5

6𝐶𝑞 ∙ 𝐾𝑏 =

0.0225 ∙ 𝐺

𝑏14

∙𝐾𝑏

74

(1 − 𝐾𝑏)12

∙ [(𝑎∗ ∙1 − 𝐾𝑏

𝐾𝑏

+ 1)]38

Eq. 59

[5

6∙

𝐶𝑞

𝑅135

−49

240∙ 𝑎∗ ∙ 𝑏 ∙ (1 − 𝐾)2] 𝑅

𝑑𝐾

𝑑𝑅= 0.0225 ∙ {

[(𝑎∗)2 + 1]38

𝑏14

∙ (1 − 𝐾)54 −

(𝑎2 + 1)38

𝑓14

∙ 𝐾74}

−5

3∙

𝐶𝑞

𝑅135

∙ 𝐾 −1127

3600∙ 𝑎∗ ∙ 𝑏 ∙ (1 − 𝐾)3

Eq. 60

49

720𝑎∗ ∙ 𝑏 ∙ (1 − 𝐾𝑒)3 +

5

6𝐶𝑞 ∙ 𝐾𝑒 = 0.0225 ∙ 𝐺 ∙ (𝑎2 + 1)

38 ∙ (

𝐾𝑒7

𝑓𝑒

)

14

− 𝐶𝑎𝑚

Eq. 61

𝐾𝑝 =𝑉𝜑

𝛺 ∙ 𝑟= 𝐾ℎ ∙ 𝑥ℎ

2 ∙ 𝑥−2 Eq. 62

𝜆𝑇 ≥ 0.437 ∙ [1 − (𝐾ℎ ∙ 𝑥ℎ2)1.175]1.656 Eq. 63

𝐾𝑝 =𝑉𝜃,∞

𝛺 ∙ 𝑟= 𝐾ℎ,𝑒𝑓𝑓 ∙ 𝑥ℎ

2 ∙ 𝑥−2 Eq. 64

𝐾ℎ,𝑒𝑓𝑓

𝐾ℎ

= 1.053 − 0.062 ∙ 𝐾ℎ Eq. 65


𝐾ℎ

= 1 − 0.056 ∙ 𝐾ℎ Eq. 66

𝐹𝑎 = 𝐹𝑎𝑏 − 𝐹𝑎𝑓 Eq. 67

𝐹𝑎𝑓 = 𝜋 ∙ 𝑝𝑏 ∙ 𝑏2 − 𝐶𝐹𝑓 ∙ 𝜌 ∙ 𝛺2 ∙ 𝑏4 Eq. 68

𝐹𝑎𝑏 = 𝜋 ∙ 𝑝𝑏 ∙ (𝑏2 − 𝑎2)−𝐶𝐹𝑏 ∙ 𝜌 ∙ 𝛺2 ∙ (𝑏4 − 𝑎4) Eq. 69

XIX

𝜕𝑝

𝜕𝑟= 𝜌 ∙ 𝛺2 ∙ 𝐾2 ∙ 𝑟

Eq. 70

𝜕𝑝

𝜕𝑟= 𝜌 ∙ (

𝑣𝜑2

𝑟− 𝑣𝑟

𝜕𝑣𝑟

𝜕𝑟) = 𝜌 ∙ 𝐾2 ∙ 𝛺2 ∙ 𝑟 +

𝜌∙𝑄2

4∙𝜋2∙𝑠2∙𝑟3 Eq. 71

𝐶𝐹 = 9.96 ∙ 𝐶𝑎𝑚 + 0.039; 𝐶𝑎𝑚 = (𝐿

2∙𝜋∙𝑏5∙𝛺2) ∙ 𝑅𝑒1

5

Where L is the angular momentum which centripetal through-flow brings into the flow field. Eq. 72

𝐶𝑎𝑚 = −[1 − 𝜙 ∙ cot(𝛽)] ∙ 𝐶𝑞; 𝐶𝑞 = (𝑄

2∙𝜋∙𝑏3∙𝛺) ∙ 𝑅𝑒

1

5

Where 𝛽 =arctan(𝑉𝜑

𝑉𝑟).

Eq. 73

𝐶𝑎𝑚 = −(1 + 5 ∙ 𝐾𝑏) ∙𝐶𝑞

6; 𝐶𝑞 = (

𝑄

2∙𝜋∙𝑏3∙𝛺) ∙ 𝑅𝑒

1

5


𝑉𝑧).

Eq. 74

𝐶𝑀 = 3.68 ∙ 𝑅𝑒−12

Eq. 75

𝐶𝑀 = 0.146 ∙ 𝑅𝑒−15

Eq. 76

𝐶𝑀 = 1.935 ∙ 𝑅𝑒−12

Eq. 77

𝐶𝑀 = 0.982 ∙ (log10𝑅𝑒)−2.58 Eq. 78

𝐶𝑀 = 0.131 ∙ 𝑅𝑒−0.186 Eq. 79

𝐶𝑀 =𝜋

𝐺 ∙ 𝑅𝑒

Eq. 80

𝐶𝑀 = 0.0308 ∙ 𝐺−14 ∙ 𝑅𝑒−

14

Eq. 81

𝐶𝑀1 =𝜋

𝐺 ∙ 𝑅𝑒

Eq. 82

𝐶𝑀2 = 1.85 ∙ 𝐺1

10 ∙ 𝑅𝑒−12

Eq. 83

𝐶𝑀3 = 0.04 ∙ 𝐺−16 ∙ 𝑅𝑒−

14

Eq. 84

𝐶𝑀4 = 0.0501 ∙ 𝐺1

10 ∙ 𝑅𝑒−15

Eq. 85

𝐶𝑀 = 𝐶𝑀0 ∙ (1 + 13.9 ∙ 𝐾0 ∙ 𝜆𝑇 ∙ 𝐺−18)

Eq. 86

𝐶𝑀 = 𝐶𝑀𝑎 + 𝑐𝑀𝑏 Eq. 87 (a)

𝐶𝑀𝑎 = 0.146 ∙ 𝑅𝑒(−15

) ∙ 𝑥𝑐

235

Eq. 87 (b)

𝐶𝑀𝑏 = 0.0796 ∙ 𝑅𝑒(−1

5) ∙ {(1 − 𝑥𝑐

23

5 ) + 14.7 ∙ 𝜆𝑇 ∙ (1 − 𝑥𝑐2) + 90.4 ∙ 𝜆𝑇

2 ∙ [1 − 𝑥𝑐(−

3

5)]}

Eq. 87 (c)

𝑥𝑐 = 1.79 ∙ 𝜆𝑇(

513

)

Eq. 87 (d)

XX

𝐶𝑀 = 0.666 ∙ 𝐶𝐷 ∙ 𝑅𝑒−1 Eq. 88

𝐶𝑀𝑐 = 0.36 ∙ 𝛾(−14

) ∙ 𝐾74 ∙ (1 − 𝐾)

320 ∙ 𝑅𝑒(−

15

)

Eq. 89

𝛾 = [81 ∙ (1 + 𝛼2)

38

49 ∙ (23 + 37 ∙ 𝐾) ∙ 𝛼]

45

𝛼 = [2300 ∙ (1 + 8𝐾)

49 ∙ (1789 − 409 ∙ 𝐾)]

12

𝐾 = 0.087 ∙ 𝐾0 ∙ 𝑒[5.2∙(0.486−𝜆𝑇)−1]

𝐾0 = 0.49 − 0.57 ∙𝑠

𝑏

𝐶𝑀 = 0.52 ∙ 𝐶𝐷0.37 ∙ 𝑅𝑒−0.57 + 0.0028 Eq. 90

𝐶𝑀 = 0.108 ∙ (𝑘𝑠

𝑏)0.272

Eq. 91

𝐶𝑀−

12 = −5.37 ∙ log10 (

𝑘𝑠

𝑏) − 3.4 ∙ 𝐺

14

Eq. 92

𝑅𝑒 ∙ 𝐶𝑀−

12 ≈ 16000 ∙ (

𝑘𝑠

𝑏)

−1

10

Eq. 93

v𝜕v

𝜕𝑙= −

1

𝜌

𝑑𝑝

𝑑𝑙+ 𝜈

𝜕2v

𝜕𝑧𝑙2

Eq. 94

𝑇𝑎 =v𝑔𝑎𝑝 ∙ ∆𝑟𝑠𝑒𝑎𝑙

𝜈∙ √

∆𝑟𝑠𝑒𝑎𝑙

𝑟𝑔𝑎𝑝

=𝑅𝑒𝑝

2∙ √


𝑟𝑔𝑎𝑝

Eq. 95

𝑅𝑒𝑝 =2 ∙ ∆𝑟𝑠𝑒𝑎𝑙 ∙ v𝑔𝑎𝑝

𝜈

Eq. 96

v𝑔𝑎𝑝 =2𝜋 ∙ 𝑟𝑔𝑎𝑝 ∙ 𝑛

60

Eq. 97

𝑅𝑒𝑔𝑎𝑝 = √(2 ∙ ∆𝑟𝑠𝑒𝑎𝑙 ∙ v𝑎𝑥𝑔𝑎𝑝

𝜈)2 + (

v𝑔𝑎𝑝 ∙ ∆𝑟𝑠𝑒𝑎𝑙

𝜈)2

Eq. 98

𝑄 = 𝛹 ∙ 𝐴 ∙ √2𝑔 ∙ ∆𝑝 Eq. 99

∆𝑁 = √𝑁𝑇2 + 𝑁𝐷

2; 𝑁𝑇 =√𝑛𝑇∙𝑛𝑀∙(𝑒𝑇∙𝑀𝑟)2

1.96∙√1000; 𝑁𝐷 =

√𝑛𝑇∙𝑛𝑀∙(𝑒𝐷∙𝑀𝑟)2

1.96∙√1000

Eq. 100

𝑦+ =𝑢∗ ∙ 𝑦

𝜈

Eq. 101

𝑢∗ = √𝜏𝑤

𝜌

Eq. 102

𝐶𝑓 =𝜏𝑤

12

∙ 𝜌 ∙ 𝑈2

Eq. 103

1

𝐶𝑓

12

= 1.7 + 4.15 ∙ log10(𝑅𝑒𝜑 ∙ 𝐶𝑓) Eq. 104

𝐶𝑓 = [2 ∙ log10((𝑅𝑒𝜑) − 0.65]−2.3 ; 𝑅𝑒𝜑 < 109 Eq. 105

𝐶𝑓 = (2.87 + 1.58 ∙ 𝑙𝑜𝑔10(𝑙/𝑘𝑠))−2.5 Eq. 106

XXI

𝑦 =𝑦+ ∙ 𝜈 ∙ √𝜌

√12

∙ 𝜌 ∙ 𝑈2 ∙ (2.87 + 1.58 ∙ log10(𝑙/𝑘𝑠))−2.5

Eq. 107

∆= |𝐶1.75 − 𝐶𝑥

𝐶1.75

| Eq. 108

𝐾𝑐~𝑐+1̅̅ ̅̅ ̅̅ ̅̅ ̅ = √

𝑝(𝑟𝑐) − 𝑝(𝑟𝑐+1) −𝜌 ∙ 𝑄2

8 ∙ 𝜋2 ∙ 𝑠2 (1

𝑟𝑐+12 −

1𝑟𝑐

2)

12

∙ 𝜌 ∙ 𝛺2 ∙ (𝑟𝑐2 − 𝑟𝑐+1

2)

Eq. 109

𝐾 = 0.97 ∙ [−8.5 ∙ 𝐶𝑞𝑟 + 0.5

𝑒(−1.45𝐶𝑞𝑟)]

54

Where −0.5 ≤ 𝐶𝑞𝑟 ≤ 0.03. Eq. 110

𝐾 = 0.97 ∙ 𝑒(600∙𝑘𝑠∙𝑟

𝑏2 )∙ [

−8.5 ∙ 𝐶𝑞𝑟 + 0.5

𝑒(−1.45𝐶𝑞𝑟)]

54

Where 0.018 ≤ 𝐺 ≤ 0.072;

𝑅𝑒 ≤ 3.17 × 106;

−5050 ≤ 𝐶𝐷 ≤ 0 and 𝑘𝑠 ≤ 58.9 𝜇m. Eq. 111

𝑝(𝑟) = 𝑝𝑏 + ∫ 𝜌 ∙ 𝐾2 ∙ 𝛺2 ∙ 𝑟𝑑𝑟𝑟

𝑏

+𝜌 ∙ 𝑄2

8 ∙ 𝜋2 ∙ 𝑠2(

1

𝑏2−

1

𝑟2)

Where ∫ 𝜌 ∙ 𝐾2 ∙ 𝛺2 ∙ 𝑟𝑑𝑟𝑟

𝑏≈

𝜌

2∙ 𝛺2 ∙ ∑ 𝐾𝑟𝑖+1

2 ∙ (𝑟𝑖2 − 𝑟𝑖+1

2)𝑐−1

0 ;

𝑟 = 𝑏 − 0.001 ∙ 𝑐 (m);

𝑟𝑖 − 𝑟𝑖+1 = −0.001 (m). Eq. 112

𝐶𝐹 = [6.6 ∙ 10−3 ∙ 𝑙𝑛 (𝑅𝑒) − 0.113] ∙ 𝑒(−1.2∙10−4∙𝐶𝐷) ∙ [0.122 ∙ 𝑙𝑛(𝐺) − 0.67] Eq. 113

𝐶𝐹 = [6.6 ∙ 10−3 ∙ ln(𝑅𝑒) − 0.113] ∙ 𝑒(−1.2∙10−4∙𝐶𝐷) ∙ [0.122 ∙ ln(𝐺) − 0.67] ∙ 𝑒(880∙𝑘𝑠𝑏

)

Where 0.018< 𝐺 ≤0.072;

𝑅𝑒 ≤ 3.17 × 106;

−5050 ≤ 𝐶𝐷 ≤ 0 and 𝑘𝑠 ≤ 58.9 𝜇m. Eq. 114

𝐶𝐹 =∫ 2𝜋 ∙ (𝑝𝑏 − 𝑝1) ∙ 𝑟𝑑𝑟

𝑟=𝑏

𝑟=𝑟𝑖1

𝜌 ∙ 𝜋 ∙ 𝛺2 ∙ 𝑏4

Eq. 115

𝐶𝑀𝑐𝑦𝑙 =2 ∙ |𝑀𝑐𝑦𝑙|

𝜌 ∙ 𝛺2 ∙ 𝑏5=

0.084 ∙ 𝜋 ∙ 𝑡

𝑏 ∙ (𝑙𝑔𝛺 ∙ 𝑏2

𝜐)1.5152

Eq. 116

𝐶𝑀3 = 0.011 ∙ 𝐺−16 ∙ 𝑅𝑒−

14 ∙ [𝑒(0.8∙10−4∙|𝐶𝐷|)]

Eq. 117

𝐶𝑀4 = 0.014 ∙ 𝐺1

10 ∙ 𝑅𝑒−15 ∙ [𝑒(0.46∙10−4∙|𝐶𝐷|)]

Eq. 118

𝐶𝑀3 = 0.32 ∙ 𝐺−16 ∙ 𝑅𝑒−

14 ∙ [𝑒(0.82∙10−4∙|𝐶𝐷|)] ∙ (

𝑘𝑠

𝑏)

0.272

Eq. 119

𝐶𝑀4 = 0.41 ∙ 𝐺1

10 ∙ 𝑅𝑒−15 ∙ [𝑒(0.46∙10−4∙|𝐶𝐷|)] ∙ (

𝑘𝑠

𝑏)

0.272

Eq. 120

𝐾 = 0.85 ∙ [0.032 + 0.32 × 𝑒(−𝐶𝑞𝑟

0.028)] Eq. 121

XXII

𝐾 = 0.85 ∙ 𝑒(600∙𝑘𝑠∙𝑟

𝑏2 )∙ [0.032 + 0.32 × 𝑒(

−𝐶𝑞𝑟

0.028)] Eq. 122

𝐾ℎ =𝑄

𝜋 ∙ 𝛺 ∙ (𝑑ℎ

2)3

∙ tan(90° − 𝛽)

Eq. 123

𝐾𝑝 =𝑄 ∙ 𝑥𝑎

2 ∙ 𝑡𝑎𝑛(90° − 𝛽)

𝜋 ∙ 𝑥2 ∙ 𝛺 ∙ (𝑑ℎ

2)3

Eq. 124

𝐶𝐹 = [6.6 ∙ 10−3 ∙ ln(𝑅𝑒) − 0.113] ∙ 𝑒(−1.6∙10−4∙𝐶𝐷) ∙ [0.122 ∙ ln(𝐺) − 0.67] Eq. 125

𝐶𝐹 = [6.6 ∙ 10−3 ∙ ln(𝑅𝑒) − 0.113] ∙ 𝑒(−1.6∙10−4∙𝐶𝐷) ∙ [0.122 ∙ ln(𝐺) − 0.67] ∙ 𝑒(880∙𝑘𝑠𝑏

)

Eq. 126

𝐶𝑀3 = 0.011 ∙ 𝐺−16 ∙ 𝑅𝑒−

14 ∙ [𝑒(10−4∙𝐶𝐷)]

Eq. 127

𝐶𝑀4 = 0.014 ∙ 𝐺1

10 ∙ 𝑅𝑒−15 ∙ [𝑒(0.6∙10−4∙𝐶𝐷)]

Eq. 128

𝐶𝑀3 = 0.32 ∙ 𝐺−16 ∙ 𝑅𝑒−

14 ∙ [𝑒(10−4∙𝐶𝐷)] ∙ (

𝑘𝑠

𝑏)

0.272

Eq. 129

𝐶𝑀4 = 0.41 ∙ 𝐺1

10 ∙ 𝑅𝑒−15 ∙ [𝑒(0.6∙10−4∙𝐶𝐷)] ∙ (

𝑘𝑠

𝑏)

0.272

Eq. 130

𝑝𝑟 =𝑃𝑥

𝑃0

Eq. 131

XXIII

Definitions of the Significant Non-dimensional Parameters

𝐶𝑎𝑚 = (𝐿

2 ∙ 𝜋 ∙ 𝑏5 ∙ 𝛺2) ∙ 𝑅𝑒

1

5

Centripetal through-flow (𝐶𝑞 < 0, 𝑄 < 0 m3/s):

𝐶𝑎𝑚 = −[1 − 𝜙 ∙ cot(𝛽)] ∙ (

𝑄

2 ∙ 𝜋 ∙ 𝑏3 ∙ 𝛺) ∙ 𝑅𝑒

1

5

Centrifugal through-flow (𝐶𝑞 > 0, 𝑄 > 0 m3/s):

𝐶𝑎𝑚 = −(1 + 5 ∙ 𝐾𝑏) ∙ (

𝑄

2 ∙ 𝜋 ∙ 𝑏3 ∙ 𝛺) ∙ 𝑅𝑒

15/6

𝐶𝐷 =�̇�

𝜇 ∙ 𝑏

𝐶𝐹 = ∫2 ∙ 𝜋 ∙ (𝑝

𝑏− 𝑝) ∙ 𝑟d𝑟

𝜌 ∙ 𝛺2 ∙ 𝑏4

𝑏

𝑎

𝐶𝑀 =2 ∙ |𝑀|

𝜌 ∙ 𝛺2 ∙ 𝑏5

𝐶𝑝 = 𝑝∗(𝑥 = 1) − 𝑝∗(𝑥); 𝑝∗ =𝑝

𝜌 ∙ 𝛺2 ∙ 𝑏2

𝐶𝑞 =𝑄

2 ∙ 𝜋 ∙ 𝛺 ∙ 𝑏3 ∙ 𝑅𝑒1

5

𝐶𝑞𝑟 =𝑄 ∙ 𝑅𝑒𝜑

0.2

2 ∙ 𝜋 ∙ 𝛺 ∙ 𝑟3

𝐸𝑘 =1

𝐺2 ∙ 𝑅𝑒

𝐺 =𝑠

𝑏

𝐾 =𝛺𝑓

𝛺

𝑛𝑠 =𝑛√𝑄

𝑒

𝐻3

4

𝑅𝑒 =𝛺 ∙ 𝑏2

𝑣

𝑅𝑒𝜑 =𝛺 ∙ 𝑟2

v

𝑉𝜑 =v𝜑

𝛺 ∙ 𝑏

𝑉𝑟 =v𝑟

𝛺 ∙ 𝑏

𝑉𝑧 =v𝑧

𝛺 ∙ 𝑏

𝑥 =𝑟

𝑏

∆𝑥 =∆𝑟

𝑏

𝜆𝑇 =𝐶𝐷

𝑅𝑒0.8

휁 =𝑧

𝑠

𝜑𝐺

=𝑄

𝜋 ∙ 𝛺 ∙ 𝑏3

1

Abstract

The leakage flow (centripetal or centrifugal through-flow) can be found in the side cavities between

the rotor and the stationary wall in nearly all kinds of radial pumps and turbines. The cavity flow

has a strong impact on the disk friction loss, leakage loss, which in this way influence the efficiency

of radial pumps and turbines. Many more effects are related to the side cavity flow, such as the

resulting axial force on the impeller and rotordynamic. To better understand the effects the flow in

rotor-stator cavities is investigated by means of analytical, numerical and experimental approaches

in this thesis.

In chapter 1 and chapter 2, the research status and the progress for the core swirl ratio, the axial

thrust coefficient and the moment coefficient are introduced.

In chapter 3, the design of the test rig is described. The uncertainties of the experimental parameters

are estimated. The experimental results from the test rig are also compared with those from

literature to show that the results from the test rig are reliable.

In chapter 4, the numerical simulation set-up is illustrated. To minimize the error, the selection of

a turbulence model and the generation of the mesh are accomplished. The simulation results are in

good agreement with those from the literature, indicating that the numerical simulation set-up is

reasonable.

In chapter 5, the experimental results are presented for the core swirl ratio, the axial thrust

coefficient and the moment coefficient. The former correlations for the core swirl ratio are modified

based on the pressure measurements of the author and are extended by introducing the impact of

surface roughness. The values of core swirl ratio deduced from the pressure measurements are in

good agreement with the simulation results. Correlations for the axial thrust coefficient are

determined which cover the impact of global Reynolds number, axial gap width, through-flow

coefficient, surface roughness for both centripetal and centrifugal through-flow. The experimental

results for the moment coefficient are also compared with those from the correlations in the

literature according to the flow regimes, where a large gap occurs. The gap is explained by the

difference of surface roughness. The former correlations therefore are modified by introducing the

surface roughness based on the torque measurements with rough disks. Some experimental results

are also provided to understand how the pre-swirl impacts the above mentioned parameters.

In chapter 6, two examples are presented on the applications of the results in this thesis. The first

example is to accomplish the geometry optimization of the rear chamber of a submersible multi-

stage slurry pump based on the flow pattern. The service life of the pump is dramatically improved

2

by around 30%. The second example is to predict the axial thrust for a deep-well pump. The axial

thrust from the correlation is also in good agreement with the experimental results. The applications

indicate that the results in this thesis should be reasonable.

All the results will provide a database for the calculation of the axial thrust and the frictional loss

in order to better design radial pumps and turbines.

3

1. Introduction

1.1 Significance of This Thesis

Nowadays, some radial turbomachines can achieve the efficiency around 80% with well-designed

impellers. To further improve the service life and the efficiency, the flow in rotor-stator cavities

becomes one of the major concerns. The flow in such cavities (between the rotating impeller and

the stationary wall) can be either radially inward (in single-stage pumps and turbines) or radially

outward (between the two adjacent stages in multi-stage pumps). In Fig. 1, the cross section of a

radial pump is sketched. The effect of leakage flow whose volumetric flow rate is noted as Q

through the sealing gap (centripetal through-flow) can be better understood by investigating the

cavity flow.

Fig. 1: Cross section of a centrifugal pump

The concerns in the process of radial pumps or turbines design mainly include two parts: impeller

design and rotor-stator cavity design, depicted in Fig. 2. The design of a rotor-stator cavity, which

is closely related to many practical problems such as axial thrust, leakage flow, etc., is quite

important. In the light of the results from the literature, the leakage flow can excessively impact

the axial thrust, which significantly reduces the service life of a turbomachine. In addition, the

prediction of the frictional loss when the leakage flow occurs also attracts extensive attention to

minimize the energy consumption. To meet the demands of industry, the sources of the axial thrust

and the frictional loss of the disk are investigated. The issue becomes more complicated concerning

the surface roughness and the pre-swirl. To close the knowledge gaps, this thesis tries to quantify

how the cavity flow impacts the axial thrust and the frictional loss of the disk.

Impeller (rotor)

Q

Centripetal through-flow

(leakage flow)

Front cavity

Back cavity

4

Fig. 2: Concerns during the design of a radial pump or turbine

1.2 Important Variables and Limitations of Previous Studies

1.2.1 Core Swirl Ratio

The core swirl ratio K (referring to the ratio of the angular velocity of the fluid 𝛺𝑓 to that of the

disk 𝛺 at a position half of the axial gap width) is utilized to show the dominant tangential motion

of the fluid. The distribution of the pressure along the rotor can be approximately determined based

on the core swirl ratio K. From the pressure distribution, the axial thrust acting on the rotor can be

predicted. The amounts of K are very sensitive to the through-flow, the surface roughness of the

rotor and the angular momentum, which are still not sufficiently investigated.

1.2.2 Axial Thrust

The forces determining the axial thrust (𝐹𝑎) of the impeller are presented in Fig. 3. As commonly

understood, the direction of the axial thrust is towards the suction side. The axial thrust 𝐹𝑎 mainly

includes four parts: the force on the front surface 𝐹𝑎𝑓 , the force on the back surface 𝐹𝑎𝑏 , the

impulse force at the impeller eye 𝐹𝑎𝑠 and the force on the impeller passage 𝐹𝑎𝑝. The parameters

𝐹𝑎𝑓 and 𝐹𝑎𝑏 can be calculated by the radial pressure distributions. The values of Re (global

Reynolds number), G (non-dimensional axial gap width), 𝐶𝐷 (through-flow coefficient) and 𝑘𝑠

(equivalent surface roughness) impact the thrust coefficient 𝐶𝐹 in a certain manner, which is up

to now still not precisely predictable.

Radial turbomachine design

Impeller design

Pressure head

Efficiency

Rotor-stator cavity design

Axial thrust

Frictional losses

Leakage losses

Kn

ow

led

ge

gap

5

Fig. 3: Sources of the axial thrust for a radial pump

1.2.3 Frictional Torque

To improve the efficiency of a turbomachine, the geometry of the side chambers should be carefully

designed. Daily and Nece [28] find that the moment coefficient 𝐶𝑀 (on a single surface) can be

predicted according to the flow regimes by classifying the tangential velocity profiles. They

examine the flow regimes and distinguish between four flow regimes on the basis of the measured

tangential velocity profiles. The typical profiles of tangential velocity and radial velocity for all

four flow regimes are shown schematically in Fig. 4. In radial pumps or turbines, the turbulent flow

regimes (regime III and regime IV) are more likely to occur.

Laminar flow (𝑅𝑒 ≤ 1.5 × 105) Turbulent flow (𝑅𝑒 > 1.5 × 105)

Regime I

Small axial gap

Regime II

Large axial gap

Regime III

Small axial gap

Regime IV

Large axial gap

Tan

gen

tial

vel

oci

ty

Rad

ial

vel

oci

ty

Fig. 4: Typical velocity profiles for the four flow regimes

𝐹𝑎𝑓 𝐹𝑎𝑏

𝐹𝑎𝑠 𝐹𝑎𝑝

𝐹𝑎

Wall Disk

Wall Disk

6

The distinguishing lines among the flow regimes for enclosed rotor-stator cavities are depicted in

Fig. 5 (Daily and Nece [28]). In most of the former studies, the frictional losses are predicted with

the correlations in Daily and Nece [28] according to the flow regimes in the 2D Daily&Nece

diagram for enclosed rotor-stator cavities. Kurokawa et al. [53][55] illustrate the impact of

through-flow on 𝐶𝑀 without investigating the impact of through-flow on the distinguishing lines.

Known the impact, more precise correlations for 𝐶𝑀 are also demanded for a rotor-stator cavity

with both through-flow and rough disks.

G

Fig. 5: Distinguishing lines for flow regimes without through-flow (2D Daily&Nece diagram [28])

1.3 Research Goals and Proposed Approach

In Table 1, a selection of the current research states of the flow in a rotor-stator cavity are listed. To

provide more confidence for calculating the values of K, CF and CM, the contents of this thesis

are depicted in Fig. 6.

Parameters: 𝐶𝐷 Re G 𝑘𝑠 Pre-swirl

K S [30][52][68] S [27][28][29] N [27][28][29] N [51][54] N [50][52][53]

𝐶𝐹 S [52][53] S [52][53] N [52][53] N [54] N [52][53][79]

𝐶𝑀 N [12][27][29][53][66] S [27][53][55] S [27][53][55] N [32][54] N [49][50][53]

Table 1: Current research states on flow in rotor-stator cavities (S: Sufficient, N: Not sufficient)

0

0.02

0.04

0.06

0.08

0.1

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Regime II Regime IV

Regime III Regime I

Re 103 104 105 106 107

7

Fig. 6: Contents of this thesis

The main geometry of the rotor-stator cavity applied in this thesis is presented in Fig. 7. The front

chamber and the back chamber are separated by a disk in the middle. The front chamber is the test

region where the through-flow is imposed. The axial gap width of the front chamber s is changeable

while that of the back chamber 𝑠𝑏 is a fixed value (𝑠𝑏 = 8 mm).

Fig. 7: Main geometry of the test rig

The goals of this thesis are to:

1. Conduct the numerical simulation of the steady flow in a rotor-stator cavity with centripetal

or centrifugal through-flow;

2. Quantify the impacts of through-flow on K, 𝐶𝐹 and 𝐶𝑀 in a rotor-stator cavity with

through-flow for a smooth disk (𝑘𝑠 = 0.4 𝜇m);

3. Organize the 3D diagram distinguishing regime III and regime IV with a third axis through-

Co

nte

nts

Numerical simulation

Velocity distribution

3D Daily&Nece diagram

Pressure distribution

Axial thrust coefficient

Experiments

Pressure measurements

Core swirl ratio


Axial thrust measurements


Frictional torque measurements

Moment coefficient

Application of the results

Sand exclusion in a SMSP

Axial thrust in a DWP

New

dia

gra

ms

an

d c

orr

ela

tio

ns

Test region

Horizontal pipe

8

flow 𝐶𝐷;

4. Quantify the effect of surface roughness on K, 𝐶𝐹 and 𝐶𝑀 in a rotor-stator cavity with

centripetal or centrifugal through-flow;

5. Provide more results on how the pre-swirl impacts 𝐶𝐹 and 𝐶𝑀 in a rotor-stator cavity

with centripetal or centrifugal through-flow.

9

2. State of the Art

2.1 Basic Equations

The fundamental equations for any analytical analyses are the conservation laws for mass,

momentum and energy. In Will et al. [85][88], Eq. 1 is determined as the continuity equation for the

compressible, unsteady flow.

𝜕𝑝

𝜕𝑡+

𝜕(𝜌 ∙ v𝑖)

𝜕𝑥𝑖= 0

Eq. 1

The momentum equations predict the acceleration of a fluid particle to the surface and body forces

of the flow. As commonly understood, the change rate of the momentum per unit volume plus the

outflowing minus the inflowing momentum over the surface equals the sum of the forces acting on

a control volume element. The universal momentum equation for the compressible unsteady flow

is written in Eq. 2. The left hand side shows the inertia terms while the right hand side includes the

impact of external forces, pressure and friction [88].

𝜕(𝜌 ∙ v𝑖)

𝜕𝑡+

𝜕(𝜌 ∙ v𝑖 ∙ v𝑖)

𝜕𝑥𝑗= 𝑓𝑖 −

𝜕𝑝

𝜕𝑥𝑖+

𝜕𝜏𝑖𝑗

𝜕𝑥𝑗

Eq. 2

For compressible Newtonian fluids, the stress tensor is written based on the hypothesis of Stokes.

It is basically a generalization of Newton’s one dimensional shear stress approach (Schlichting

[81]):

𝜏𝑖𝑗 = 𝜇 ∙ (𝜕v𝑖

𝜕𝑥𝑗+

𝜕v𝑗

𝜕𝑥𝑖−

2

3∙ 𝛿𝑖𝑗 ∙

𝜕v𝑘

𝜕v𝑘)

Eq. 3

Due to the geometry of rotor-stator cavities, the correlations are better expressed with cylindrical

coordinates. For isothermal and incompressible flow with constant density, the energy equation can

be omitted. Without making any closure assumptions for the shear stresses, the full equations of

momentum with cylindrical coordinates are determined in Eq. 4~Eq. 6 by Will [88].

10

Radial momentum:

𝜌 ∙ (𝜕v𝑟

𝜕𝑡+ v𝑟 ∙

𝜕v𝑟

𝜕𝑟+

v𝜑

𝑟∙

𝜕v𝑟

𝜕𝜑−

v𝜑2

𝑟+ v𝑧 ∙

𝜕v𝑟

𝜕𝑧)

= 𝑓𝑟 −𝜕𝑝

𝜕𝑟+

1

𝑟∙

𝜕(𝑟 ∙ 𝜏𝑟𝑟)

𝜕𝑟+

1

𝑟∙

𝜕(𝜏𝑟𝜑)

𝜕𝜑−

𝜏𝜑𝜑

𝑟+

𝜕(𝜏𝑟𝑧)

𝜕𝑧

Eq. 4

Tangential momentum:

𝜌 ∙ (𝜕v𝜑

𝜕𝑡+ v𝑟 ∙

𝜕v𝜑𝜑

𝜕𝑟+

v𝜑

𝑟∙

𝜕v𝜑

𝜕𝜑+

v𝜑 ∙ v𝑟

𝑟+ v𝑧 ∙

𝜕v𝜑

𝜕𝑧)

= 𝑓𝜑 −1

𝑟∙

𝜕𝑝

𝜕𝜑+

1

𝑟2∙

𝜕(𝑟2 ∙ 𝜏𝜑𝑟)

𝜕𝑟+

1

𝑟∙

𝜕(𝜏𝜑𝜑)

𝜕𝜑+

𝜕(𝜏𝜑𝑧)

𝜕𝑧

Eq. 5

Axial momentum:

𝜌 ∙ (𝜕v𝑧

𝜕𝑡+ v𝑟 ∙

𝜕v𝑧

𝜕𝑟+

v𝜑

𝑟∙

𝜕v𝑧

𝜕𝜑+ v𝑧 ∙

𝜕v𝑧

𝜕𝑧)

= 𝑓𝑧 −𝜕𝑝

𝜕𝑧+

𝜕(𝜏𝑧𝑟)

𝜕𝑟+

𝜏𝑧𝑟

𝑟+

1

𝑟∙

𝜕(𝜏𝑧𝜑)

𝜕𝜑+

𝜕(𝜏𝑧𝑧)

𝜕𝑧

Eq. 6

The continuity equation for a rotor-stator cavity is given in Eq. 7 with cylindrical coordinates.

1

𝑟∙

𝜕(𝑟 ∙ v𝑟)

𝜕𝑟+

1

𝑟∙

𝜕v𝜑

𝜕𝜑+

𝜕v𝑧

𝜕𝑧= 0

Eq. 7

Several assumptions are made in order to simplify Eq. 4~Eq.7 by Will [88].

(1) Steady flow (𝜕

𝜕𝑡= 0);

(2) Axisymmetric flow (𝜕

𝜕𝜑= 0);

(3) No body force;

(4) No normal stresses due to viscosity;

(5) No shear stress due to gradients of the tangential velocity.

With above simplification, Eq. 4~Eq. 6 are valid for general flows even when they do not have

Newtonian viscosity. Neglecting the viscous terms 𝜕𝜏𝑖𝑗

𝜕𝑥𝑗 in Eq. 2, the equation is still non-linear since

it has the convective terms (Will [88]). Now Eq. 2 can be written as:

𝜕(𝜌 ∙ v𝑖)

𝜕𝑡+

𝜕(𝜌 ∙ v𝑖 ∙ v𝑖)

𝜕𝑥𝑗= 𝑓𝑖 −

𝜕𝑝

𝜕𝑥𝑖

Eq. 8

11

Likewise, the simplified momentum equations (Eq. 9~Eq. 11) and the continuity equation (Eq. 12) are

conducted by Will [88] for steady, axisymmetric flow in rotor-stator cavities.

Radial momentum:

𝜌 ∙ (v𝑟 ∙𝜕v𝑟

𝜕𝑟−

v𝜑2

𝑟+ v𝑧 ∙

𝜕v𝑧

𝜕𝑧) = −

𝜕𝑝

𝜕𝑟+

𝜕(𝜏𝑟𝑧)

𝜕𝑧

Eq. 9

Tangential momentum:


𝜕𝑟+

v𝑟 ∙ v𝜑

𝑟+ v𝑧 ∙

𝜕v𝜑

𝜕𝑧) =

𝜕(𝜏𝜑𝑧)

𝜕𝑧

Eq. 10

Axial momentum:

𝜌 ∙ (v𝑟 ∙𝜕v𝑧

𝜕𝑟+ v𝑧 ∙

𝜕v𝑧

𝜕𝑧) = −

𝜕𝑝

𝜕𝑧+

𝜏𝑟𝑧

𝑟+

𝜕(𝜏𝑟𝑧)

𝜕𝑟

Eq. 11

Continuity:

𝜕v𝑟

𝜕𝑟+

v𝑟

𝑟+

𝜕v𝑧

𝜕𝑧= 0

Eq. 12

Neglecting the axial velocity, Will [88] conducts Eq. 13 as the tangential momentum equation for

the inviscid core region.


𝜕𝑟+

v𝑟 ∙ v𝜑

𝑟) = 0

Eq. 13

In Will [88], Eq. 13 is also written in the following form:

v𝑟

𝑟∙

𝜕

𝜕𝑟(v𝜑 ∙ 𝑟) = 0

Eq. 14

Assuming that v𝑟 equals to the radial velocity in the central core, the radial pressure distribution

can be evaluated using the radial balance between the centrifugal force and the pressure forces,

written in Eq. 15:

𝜕𝑝

𝜕𝑟= 𝜌 ∙

v𝜑2

𝑟

Eq. 15

For a rotor-stator cavity with through-flow, the radial velocity conducts a radial transport of angular

momentum. The radial velocity v𝑟 should be therefore introduced in Eq. 15. The solution as a

12

potential swirl is assumed. Hence, the determination of the pressure distribution requires an

additional term taking into account, namely the radial convection (Will [88]), written in Eq. 16.

𝜕𝑝

𝜕𝑟= 𝜌(∙

v𝜑2

𝑟− v𝑟 ∙

𝜕v𝑟

𝜕𝑟)

Eq. 16

To predict the flow in both the front and the back cavities (see Fig. 1), a common method is found

on the basis of the integral boundary layer theory. The equations therefore have to be integrated in

the axial direction. In this approach, the integral relations are only fulfilled for integrated values

across the boundary layer thicknesses. Senoo and Hayami [75] assume the thickness of the disk

boundary layer to be twice that of the wall boundary layer. The solution is reported to be little

affected by this assumption.

The integral boundary layer method requires the specification of velocity profiles. The profiles are

assumed instead of being found as a part of the solution. Accordingly, the quality of the solution

depends on the assumed profiles. Nevertheless, the integral quantities such as the frictional torque

or the axial thrust can be well predicted by the method because the main velocity component is in

the circumferential direction. The following identities are valid for the simplified radial and

tangential momentum equations, given in Eq. 17 and Eq. 18 (Owen and Rogers [66]):

v𝑟 ∙𝜕v𝑟

𝜕𝑟−

v𝜑2

𝜕𝑟+ v𝑧 ∙

𝜕v𝑟

𝜕𝑧=

1

𝑟∙ [

𝜕

𝜕𝑟∙ (𝑟 ∙ v𝑟

2) +𝜕

𝜕𝑧∙ (𝑟 ∙ v𝑟 ∙ v𝑧) − v𝜑

2] Eq. 17

v𝑟 ∙𝜕v𝜑

𝜕𝑟−

v𝑟 ∙ v𝜑

𝑟+ v𝑧 ∙

𝜕v𝜑

𝜕𝑧=

1

𝑟2∙ [

𝜕

𝜕𝑟∙ (𝑟 ∙ v𝑟 ∙ v𝜑) +

𝜕

𝜕𝑧∙ (𝑟2 ∙ v𝑧 ∙ v𝜑)]

Eq. 18

The radial and the tangential momentum now can be written as:

𝜕

𝜕𝑟(𝑟 ∙ v𝑟) + 𝑟 ∙

𝜕

𝜕𝑧∙ (v𝑟 ∙ v𝑧) − v𝜑

2 = −𝑟

𝜌∙

𝜕𝑝

𝜕𝑟+

𝑟

𝜌∙

𝜕(𝜏𝑟𝑧)

𝜕𝑧

Eq. 19

𝜕

𝜕𝑟(𝑟2 ∙ v𝑟 ∙ v𝜑) +

𝜕

𝜕𝑧∙ (𝑟2 ∙ v𝑧 ∙ v𝜑) = −

𝑟2

𝜌∙

𝜕(𝜏𝜑𝑧)

𝜕𝑟

Eq. 20

These equations are now integrated in the axial direction for a control volume from the axial

coordinates 𝑧1 to 𝑧2. Considering a possible variation of the limits 𝑧1 and 𝑧2 with radius, the

application of the Leibniz rule for a general variable X yields Eq. 21 (Will [88]).

13

( ∫

𝜕𝑋(𝑟, 𝑧)

𝜕𝑟

𝑧2

𝑧1

𝑑𝑧) =𝜕

𝜕𝑟( ∫ 𝑋(𝑟, 𝑧)

𝑧2

𝑧1

𝑑𝑧) +𝜕𝑧1

𝜕𝑟∙ 𝑋(𝑟, 𝑧1 ) −

𝜕𝑧2

𝜕𝑟∙ 𝑋(𝑟, 𝑧2)

Eq. 21

Combining Eq. 21 with Eq. 19 and Eq. 20, the conducted integrated forms of the radial and the

tangential momentum equations are given in Eq. 22 and Eq. 23.

𝜕

𝜕𝑟(𝑟 ∙ ∫ v𝑟

2

𝑧2

𝑧1

𝑑𝑧) + 𝑟 ∙𝜕𝑧1

𝜕𝑟∙ v𝑟1

2 − 𝑟 ∙𝜕𝑧2

𝜕𝑟∙ v𝑟2

2 + 𝑟 ∙ v𝑟2 ∙ v𝑧2 − 𝑟 ∙ v𝑟1 ∙ v𝑧1

− ∫ v𝜑2𝑑𝑧 = −

𝑟

𝜌∙

𝑧2

𝑧1

∫𝜕𝑝

𝜕𝑟𝑑𝑧 +

𝑟

𝜌

𝑧2

𝑧1

∙ ∫𝜕(𝜏𝑟𝑧)

𝜕𝑧𝑑𝑧

𝑧2

𝑧1

Eq. 22

𝜕

𝑟2∙

𝜕

𝜕𝑟(𝑟2 ∙ ∫ v𝑟 ∙ v𝜑

𝑧2

𝑧1

𝑑𝑧) +𝜕𝑧1

𝜕𝑟∙ v𝑟1 ∙ v𝜑1 −

𝜕𝑧2

𝜕𝑟∙ v𝑟2 ∙ v𝜑2 + 𝑟 ∙ v𝑧2 ∙ v𝜑2 − v𝑧1 ∙ v𝜑1

=1

𝜌∙ ∫

𝜕(𝜏𝜑𝑧)

𝜕𝑧𝑑𝑧

𝑧2

𝑧1

Eq. 23

In the following equations, a plane rotor-stator cavity is assumed. In Zilling [90], it is mentioned

that the conical walls can be treated as the parallel walls for the inclination angles smaller than 12

degrees. For a plane rotor-stator system with 𝑧1 = 0 and 𝑧2 = 𝑠, the following two correlations

are conducted since the velocity components v𝑧 and v𝑟 are zero at the walls (no-slip wall

condition):

1

𝑟∙

𝜕

𝜕𝑟(𝑟 ∙ ∫ v𝑟

2

𝑠

0

𝑑𝑧) −1

𝑟∙ ∫ v𝜑

2

𝑠

0

𝑑𝑧 = −1

𝜌∙ ∫

𝜕𝑝

𝜕𝑟𝑑𝑧

𝑠

0

+1

𝜌∙ (𝜏𝑟𝑧𝑆 − 𝜏𝑟𝑧𝑅)

Eq. 24

1

𝑟2∙

𝜕

𝜕𝑟(𝑟2 ∙ ∫ v𝑟v𝜑

𝑠

0

𝑑𝑧) =1

𝜌∙ (𝜏𝜑𝑧𝑆 − 𝜏𝜑𝑧𝑅)

Eq. 25

Eq. 24 and Eq. 25 are the integral momentum relations expressed in the cylindrical coordinates for a

steady, incompressible flow field. They are crucial for most analytical flow models in the literature

(Kurokawa et al. [52][53][54], Baibikov and Karakhan’yan [7], Baibikov [8]). A further

solution requires the specification of velocity profiles and suitable expressions for the wall shear

stresses.

14

To predict the radial gradient of the core rotation, Will [88] correlates Eq. 26 by expanding Eq. 25

with 2𝜋 and by introducing K. The wall shear stress is predicted with Eq. 27 in Will [88].

𝑑𝐾

𝑑𝑟=

2 ∙ 𝜋 ∙ 𝑏

�̇� ∙ 𝛺∙ (𝜏𝜑𝑧𝑆 − 𝜏𝜑𝑧𝑅) −

2 ∙ 𝐾

𝑅

Eq. 26

𝜏 = 𝜆 ∙𝜌

8∙ v𝑚𝑒𝑎𝑛

2 Eq. 27

Where v𝑚𝑒𝑎𝑛𝑅 = 𝑟2 ∙ 𝛺2 ∙ (1 − 𝐾)2;

v𝑚𝑒𝑎𝑛𝑆 = 𝑟2 ∙ 𝛺2 ∙ 𝐾2.

Thus, Eq. 26 becomes:

𝑑𝐾

𝑑𝑟=

�̅�2

4 ∙ 𝜑𝐺∙ (𝜆𝑠 ∙ 𝐾2 − 𝜆𝑅 ∙ (1 − 𝐾)2) −

2 ∙ 𝐾

𝑅

Eq. 28

Where 𝜑𝐺 =𝑄

𝜋∙𝛺∙𝑏3.

For 𝜆𝑅 = 𝜆𝑆, the limits of Eq. 28 can be obtained by considering a vanishing small through-flow or

leakage (𝜑𝐺 → 0) and the opposite case of an infinite high leakage (𝜑𝐺 → ∞). The values of K are

given in Eq. 29 and Eq. 30 in Will [88].

Solid body rotation (forced vortex):

𝜑𝐺 → 0, 𝐾 = 0.5 Eq. 29

Potential swirl (free vortex):

𝜑𝐺 → ∞, 𝐾 ∙ 𝑥2 = 0.5 Eq. 30

According to Lomakin [62], a simplified flow model for the radial distribution of K is deduced

from Eq. 25 assuming a constant value along the axial gap width. The parameter v𝜑 is therefore

taken outside the integral sign. The idea is that the circumferential velocity is almost constant in

the core region. A significant change appears only within the thin boundary layers. The axial

distance with a significant change of v𝜑, compared to the region with an almost constant value,

can thus be neglected. This assumption restricts the model validity in case of stronger variations in

the velocity profiles, for example, in case of flow regime III. Numerical simulations as well as a

plenty of experimental investigations, however, show that this is an admissible assumption in most

cases. Velocity profiles have to be assumed in order to accomplish the integration across the axial

15

gap. In principle, variable limits are used to account for the geometrical changes of the cavity in

the radial direction. For example, Will [88] fragments the integral into:

∫ v𝑟v𝜑𝑧=𝑠

𝑧=0𝑑𝑧 = ∫ v𝑟v𝜑

𝑧=𝛿𝑅

𝑧=0𝑑𝑧+∫ v𝑟v𝜑

𝑧=𝑠−𝛿𝑠

𝑧=𝛿𝑅𝑑𝑧+∫ v𝑟v𝜑

𝑧=𝑠

𝑧=𝑠−𝛿𝑠𝑑𝑧

Eq. 31

Will [88] solves Eq. 31 by assuming that the value of v𝑟 in the central core is zero. From the results

of numerical simulation by Will [88], the largest value of 𝑉𝑟 is 0.08 at 𝑅𝑒 = 0.38 × 106 (𝑛 =

300 rpm), 𝐺 = 0.018 ( 𝑠 = 0.002 mm ) and 𝐶𝐷 = 5050 ( 𝑄 = 2 m3/h ) at 𝑥 = 0.4 . This

value is much smaller than 𝑉𝜑. The outer radius is found to be the dominant region with respect to

the frictional torque and the axial thrust. Hence, the simplification will not result in large errors. If

the radial velocity in the core is considered zero, the second term on the right hand side of Eq. 31

becomes zero.

2.2 Thickness of Boundary Layers

Almost all the flow models that emerge from the integral boundary layer theory require information

about the radial evolution of the boundary layer thickness along the radius of the rotating disk and

the stationary wall. The boundary layer thickness essentially impacts the rotation of the core and

the moment coefficient. In principle, smaller boundary layer thicknesses result in higher frictional

resistances for larger velocity gradients. Several different correlations for the boundary layer

thickness are determined in the literature. Some of them are mentioned in this thesis. Unfortunately,

no experimental data are available to validate these theoretical expressions.

Kurokawa and Sakuma [55] mention that a centrifugal through-flow favors the merging of the

boundary layers. More on, transition from the laminar flow to the turbulent flow is additionally

influenced by the through-flow. The reason is that the externally applied leakage is in a turbulent

state in general. Consequently, in this case, the whole flow in the cavity is likely to become entirely

turbulent. For a wide axial gap, an intermediate velocity establishes between the walls, which is

referred to as “core region”. Due to the dominant tangential movement, a radial pressure gradient

establishes with the maximum pressure value at the outer radius of the disk. In the core, the

centrifugal force and the pressure force can balance each other without any additional forces. This

state is usually known as radial equilibrium. At the stationary wall, the tangential velocity is

reduced to zero and therefore a radial inflow develops in the stator boundary layer caused by the

dominant pressure forces. Near the rotor surface, the dominant centrifugal forces result in a radial

outflow. Since the main fluid motion is in the circumferential direction, the radial velocity

16

component is noted as secondary flow. In Fig. 8, when the axial gap is large enough to allow a

formation of two boundary layers on the rotor and the stator (Will [88]), the most frequently

encountered flow pattern is depicted. The boundary layers can either be merged or not; the flow

can be laminar or turbulent. In case of flow regimes II or IV (separated disk boundary layer and

wall boundary layer), which is supposed to apply in the majority of practical applications (Gülich

[34], Hamkins [41]). The basic flow structure is indicated by Senoo and Hayami [75]: A boundary

layer on the rotor (IV) and the stator (I), a core region (III) and an intermediate layer (II) between

the stator boundary layer and the central core with a small radial outflow. Due to mass conservation,

an axial convection of fluid from the stator to the rotor takes place at small radius. Although the

axial velocity is very small, the axially convected angular momentum is not (Hamkins [41]). To

investigate the cavity flow, the thickness of both the disk boundary layer and the wall boundary

layer is one of the major concerns.

Fig. 8: Flow structure in an idealized rotor-stator cavity (left) and the velocity profiles for a wide

axial gap (replotted from Will [88])

In Daily et al. [27], the velocity profiles in the case of a centrifugal leakage flow are measured.

Based on the experimental results, an empirical correlation for the thickness of the disk boundary

layer is determined in Eq. 32.

𝛿𝑅 = 𝐵+ ∙𝑟

𝑅𝑒𝜑

15

∙ (1 − 𝐾)𝑚

Eq. 32

휁

𝑥

Roto

r

Sta

tor

𝑉𝑟

𝑉𝜑

I II III IV

I II III IV

17

In Eq. 32, 𝐵+ and m are two constants to fit the experimental results. This correlation is used with

slight modifications in all the flow models proposed by Kurokawa et al. [54][55]. Some of the

parameter combinations for the constants 𝐵+ and m are given in Table 2.

𝐵+ m

Daily et al. [27] 0.4 2

Kurokawa and Toyokura [54] 0.526 2

Kurokawa and Sakuma [55] 0.54 2.5

Table 2: Values of 𝑩+ and m from the literature

The thickness of the wall boundary layer can be predicted with Eq. 33 by Kurokawa et al. [52].

They also give the values of 𝑐 and 𝑐∗ based on the experimental results in Table 3.

𝛿𝑠 =𝑓 ∙ 𝑟

(𝑟2 ∙ 𝜔

𝜈 )15

Eq. 33

Where 𝑓 =1

𝑐∙𝐾∙ [𝑐∗ ∙ 𝑏 ∙ (1 − 𝐾)3 −

120

49∙

𝑄

2∙𝜋∙𝜔∙𝑟3 ∙ (𝑟2∙𝜔

𝜈)

1

5].

Kurokawa and Toyokura [53] 𝑐 = 0.374 𝑐∗ = 0.220

Table 3: Values of 𝒄 and 𝒄∗ by Kurokawa and Toyokura [53]

The volume flow rate passing through the cavity is defined as follows:

𝑄 = ∫ v𝑟

𝑧=𝑠

𝑧=0

𝑑𝑧

Eq. 34

The distribution of the stator boundary layer thickness can also be predicted from the continuity

equation, assuming a zero radial velocity in the core (Will [88]):

∫ v𝑟𝑆

𝑧=𝛿𝑠

𝑧=0

𝑑𝑧𝑠 + ∫ v𝑟𝑅

𝑧=𝛿𝑅

𝑧=0

d𝑧𝑅 =𝑄

2 ∙ 𝜋 ∙ 𝑟

Eq. 35

The radial velocity in the disk boundary layer and the wall boundary layer are noted as v𝑟𝑅 and

v𝑟𝑆, respectively. According to the assumption of the velocity distributions by Eq. 36 and Eq. 37 in

Kurokawa et al. [54][55], the profiles of radial velocity are shown in Fig. 9. For the successive

considerations, 𝑛1=2 is used since the profile appears to be closer to the actual flow physics in

Kurokawa et al. [54], while 𝑛1 equals to 1. The value of m is determined as 7 [54][55].

18

v𝑟𝑅 = 𝑎𝑅 ∙ (1 − 𝐾) ∙ 𝑟 ∙ 𝛺 ∙ (1 −

𝑧𝑅

𝛿𝑅)𝑛1 ∙ (

𝑧𝑅

𝛿𝑅)

1𝑚

Eq. 36

v𝑟𝑆 = −𝑎𝑆 ∙ 𝐾 ∙ 𝑟 ∙ 𝛺 ∙ (1 −

𝑧𝑠

𝛿𝑠)𝑛1 ∙ (

𝑧𝑠

𝛿𝑠)

1𝑚

Eq. 37

𝑉𝑟𝑅 𝑉𝑟𝑆

Fig. 9: Radial velocity profiles in dependence on Eq. 36 and Eq. 37 (replotted from Will [88])

The velocity factors 𝑎𝑅 and 𝑎𝑠 are determined from the empirical correlations given in

Kurokawa and Sakuma [55] based on the flow angle measurements for flow regime IV:

𝑎𝑅 = 1.18 ∙ (𝑅𝑒𝜑

105+ 2)−0.49

Eq. 38

𝑎𝑆 = 1.03 ∙ (𝑅𝑒𝜑

105+ 2)−0.387

Eq. 39

Will [88] determines Eq. 40 to predict the impact of through-flow on the thickness of the wall

boundary layer. Two constants of 0.304 and 0.408 are used in the equation. The amounts of c and

𝑐∗ are in Table 3. For centripetal through-flow, the volume flow rate is negative and the thickness

of the stator boundary layer therefore increases since the flow passes the cavity mostly in the

vicinity of the stator.

𝛿𝑠 = 0.304 ∙𝑐∗

𝑐∙

(1 − 𝐾)125

𝐾∙

𝑟

𝑅𝑒𝜑

15

−𝑄

0.408 ∙ 𝑐 ∙ 2 ∙ 𝜋 ∙ 𝑟2 ∙ 𝛺 ∙ 𝐾

Eq. 40

0

0.02

0.04

0.06

0.08

0.1

0.12

0 0.2 0.4 0.6 0.8 1

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0 0.2 0.4 0.6 0.8 1

𝑧𝑅 𝛿𝑅⁄

𝑛1 = 1 𝑛1 = 2

1 − 𝑧𝑠 𝛿𝑠⁄

19

A common approach (e.g. Schultz-Grunow [74], Daily and Nece [27][28][29], Zilling [90],

Möhring [59], Senoo and Hayami [75], Kurokawa et al. [52] 55], Lauer [58]) is the

implementation of the Blasius law of the velocity distributions for pipe flow. Further evaluations

require information about the boundary layer thickness. Commonly, the dependencies for the free

disk are used. In contrast, in the laminar case, the boundary layer thickness of both the disk

boundary layer and the wall boundary layer under turbulent conditions are found to increase with

the 3/5 power of the radial coordinate (Dorfman [32][33]):

𝛿𝑅 = 𝛶𝑅 ∙ 𝑟35 ∙ (

𝜈

𝛺)

15

Eq. 41

𝛿𝑠 = 𝛶𝑠 ∙ 𝑟35 ∙ (

𝜈

𝛺)

15

Eq. 42

The proportionality factors (𝛶𝑅 and 𝛶𝑠 ) are now different from the free disk value (𝛶𝑅 = 𝛶𝑠 =

0.526). The friction factors for the rotor (𝜆𝑅) and the stator (𝜆𝑠) are in dependency upon the local

Reynolds number (Will [88]):

𝜆𝑅 =

0.18

𝐶𝑅∙ 𝑅𝑒𝜑

−15 ∙ (

1

1 − 𝐾)

14

Eq. 43

𝜆𝑆 =

0.18

𝐶𝑆∙ 𝑅𝑒𝜑

−15 ∙ (

1

𝐾)

14

Eq. 44

In Möhring [63], the constant 𝐶𝑅 is determined by comparing the impeller torque with the results

of Schultz-Grunow [74] and adjustment with his own results to:

𝐶𝑅 = 0.315 Eq. 45

The momentum of the secondary flow can be estimated by assuming equilibrium between the

torque on the walls and the momentum difference of the secondary flow in case of zero leakage

(Will [88]). An implementation is reasonable by increasing the frictional resistance of the casing

𝐶𝑆 (in Eq. 44) in Möhring [63]:

𝐶𝑆 = 𝐶𝑅 ∙ (

1 − 𝐾0

𝐾0)

74

Eq. 46

20

2.3 Core Swirl Ratio

2.3.1 Enclosed Rotor-Stator Cavity

In the case of solid body rotation (forced vortex) in an enclosed rotor-stator cavity, the core swirl

ratio only depends on the geometric magnitudes. Will [88] implements Eq. 47 to predict the impact

of G on 𝐾0.

𝐾0 =

1

1 + √1 + 5 ∙ 𝐺

Eq. 47

A lot of researches are accomplished to determine the values of 𝐾0. Some of them are listed below.

Radtke and Ziemann [78] show that the core swirl ratio decreases because of the higher ratio of

decelerating to accelerating surfaces with increasing radial coordinate.

Itoh et al. [40] measure the distributions of both the radial and the tangential velocities in an

enclosed rotor-stator cavity (𝐺 = 0.08) with a stationary shroud at 𝑅𝑒 = 106 using a hot-wire

anemometry. Cheah et al. [19] present the LDA and the hot-film velocimetry measurements in a

similar configuration for a wider cavity (𝐺 = 0.127 ). The local Reynolds number 𝑅𝑒𝜑 varies

from 0.3 × 106 to 1.6 × 106. In contrast to the results of Itoh et al. [40], they argue that the

turning at the outer radial shroud is mainly responsible for the entirely turbulent flow in the stator

boundary layer by causing upstream disturbances which destabilize the flow. Although the local

Reynolds number is confirmed to be the major parameter for the local velocity profiles, the authors

suggest that both the global Reynolds number and the axial gap width have to be taken into account

as well. At the lowest Reynolds number investigated, the flow along the rotor is almost completely

laminar while the wall boundary layer remains turbulent. As a consequence, the fluid which flows

in the axial direction from the stator to the rotor must relaminarize.

Watanabe et al. [83] study the influence of the fine spiral grooves on the frictional resistance of

an enclosed rotating disk in case of both the merged (flow regime III) and the separated (flow

regime IV) boundary layers. In practical cases, the bounding walls of rotor-stator cavities are

frequently inclined. The inclination angle influences the effective area in the formulation of the

torques on the rotating and the stationary wall.

Some of the typical values of 𝐾0 found in the literature are summarized in Table 4 for enclosed

rotor-stator cavities. The parameter 𝐺 is close to 0 for the disks with nearly infinite radius.

21

Authors Year 𝐾0 G Flow state From

Schultz-Grunow [74] 1935 0.512 ≈ 0

Turbulent

Theoretical

0.357 0.3 Experimental

Daily and Nece [28] 1960

0.46 0.0637

Experimental

0.454 Theoretical

0.444 0.102

Experimental

0.432 Theoretical

0.412 0.217

Experimental

0.388 Theoretical

0.46 0.051

Laminar

Experimental

0.504 Theoretical

0.44 0.102

Experimental

0.46 Theoretical

0.36 0.217

Experimental

0.386 Theoretical

Lance and Rogers [56] 1962 0.3

≈ 0

Simulation Cooper and Reshotko [20] 1975

0.3135

0.5

Turbulent Zilling [90] 1973 0.5 Theoretical

Möhring [63] 1979 0.5

Kurokawa [54] 1978 0.43 0.078 Laminar Experimental

Dijkstra and Van Heijst [27] 1983 0.313 ≈ 0 Turbulent

Simulation

Radtke and Ziemann [78] 1983 0.41 0.125 Experimental

Owen [63] 1989 0.382

0.069 Laminar Theoretical

0.426 Turbulent

Theoretical

Itoh et al. [40] 1992 0.42

0.08

Experimental 0.31 Laminar

Cheah et al. [19] 1994 0.4

0.127

Turbulent 0.35

Andersson and Lygren [5] 2006 0.4 0.1

Simulation 0.47 0.0632

Table 4: Core swirl ratio 𝑲𝟎 in the literature for an enclosed rotor-stator cavity (reorganized

from Will [88])

2.3.2 Impact of Through-Flow

Owen [63] develops a flow model to calculate the core swirl ratio and the frictional torque by

solving the Ekman equations for flow regime IV. In his configuration, the flow direction is usually

radially outward (e.g. Rabs et al. [70], Da Soghe et al. [26]). The Ekman equations are simplified,

linearized forms of the equations of motion. The functional relation for the core swirl ratio in case

of the laminar flow is finally obtained by using the continuity equation disregarding the outer

cylindrical wall:

𝐾 = 0.25 ∙ [−1 + √5 − 4 ∙ 𝜑𝐺

√𝑅𝑒𝜑

𝑥2]

2

Eq. 48

22

For an enclosed rotor-stator cavity, Eq. 48 implies 𝐾 = 𝐾0 = 0.382 , which is in very good

agreement with the measurements by Daily et al. [29]. The core swirl ratio is completely

suppressed for a critical value of the non-dimensional through-flow rate (𝜑𝐺 ∙ √𝑅𝑒𝜑

𝑥2 = 1 ) which

is approximately 16% bigger than the entrainment rate for the laminar flow over the free disk. It

appears when a leakage flow rate is sufficient to completely suppress the core rotation. For larger

leakage flow rates, a flow structure according to Stewartson [77] occurs. For the turbulent flow,

Owen [63] uses the integral method according to Van Kármán [47] to solve the non-linear

equations of motion for the disk boundary layer. This solution is combined with a simplified (linear)

approach on the stator side. Balancing the flow rates finally yields Eq. 49 for moderate through-flow

rates (𝜋 ∙ 𝜑𝐺 ∙ 𝑅𝑒𝜑2 ∙ 𝑥2.6 < 0.1 ). In the case of zero leakage, Eq. 49 implies 𝐾 = 𝐾0 = 0.426 ,

which is again in good agreement with the experiments by Daily et al. [29].

(1 − 𝐾)85 ∙ (1 − 0.51 ∙ 𝐾) − 0.638 ∙ 𝐾

45 = 0.25 ∙ [−1 + 4 ∙ 𝜑𝐺√

𝑅𝑒𝜑

𝑥2]

2

Eq. 49

Tests with the flow model in its original form indicate that in the case of lower entrance rotation

and large axial gap width, the computed curves show a slightly strong increase in the core rotation

compared to the available measurements performed by Lauer [58]. The decelerating effect of a

radial shroud onto the core rotation, especially at the outer radius, has yet not been accounted for

in the present flow model. The basic formulation of the model is derived both from the general

Navier-Stokes equations as well as from the principle of conservation of the angular momentum

for a small cylindrical volume element, given in Eq. 50 (see next page). In the latter case, it is obvious

that the influence of an outer casing is not taken into consideration. The overall decelerating

influence of an outer (stationary) shroud on the core rotation can be observed for the enclosed

rotating disk where usually both the measured and the computed values are markedly below the

theoretical maximum of 0.5. If a centripetal through-flow is present, the shear stress created by the

interaction of the entering leakage with the shroud gives rise to an additional “dynamic stress”

whose influence decreases at small radius. To include both effects in the flow model, a correction

function 𝑓∗ for the friction factor of the stator is introduced.

𝑑𝐾

𝑑𝑅=

𝑅2

4 ∙ 𝜑𝐺∙ (𝑓∗ ∙ 𝜆𝑠 ∙ 𝐾2 − 𝜆𝑅 ∙ (1 − 𝐾)2) −

2 ∙ 𝐾

𝑅

Eq. 50

Where 𝑓∗ = 1 + (𝑠

𝑏+𝑙1−𝑎+ 5 ∙ 𝑅4 ∙ |1 −

𝐾

0.58|

6

5).

23

Zilling [90] and Lauer [58] account for a torque resulting from a turbulent shear stress acting on

the lateral area of the control volume. Möhring [63] argues that this effect can be neglected because

the velocity gradient within the boundary layers in the axial direction (which determines for

example the wall frictional torque) is much greater than the velocity gradient in the radial direction

responsible for the turbulent shear stress.

2.3.2.1 With Centripetal Through-Flow (𝑸 < 𝟎 𝒎𝟑/𝒔)

In centrifugal pumps, the wear-ring leakage is driven by the pressure difference through the sealing

gap. The through-flow can also affect the boundaries between the different flow regimes by Daily

and Nece [24] because of its influence on the boundary layer thickness. For a rotor-stator cavity

with through-flow, the values of K are greatly influenced by the amount of through-flow. In this

thesis, the amounts of Q, 𝐶𝐷 and 𝐶𝑞𝑟 are negative for centripetal through-flow.

Altmann [1][2] measures the radial pressure distribution and determines the corresponding core

rotation distribution from the assumption of radial equilibrium between the pressure and the

centrifugal forces in the core region. In the case of a centripetal through-flow, a strong

magnification of the core rotation towards the center is observed. In the opposite flow direction,

the contrary situation prevails. The values of K decrease with increasing radial coordinate.

Using a two-component LDA system, Poncet et al. (2005) determine Eq. 51 to evaluate the core

swirl ratio K with centripetal through-flow when −0.2 ≤ 𝐶𝑞𝑟 ≤ 0.035 . Debuchy et al. [30]

determine Eq. 52 to calculate the amounts of K for a wider range −0.5 ≤ 𝐶𝑞𝑟 ≤ 0.035. The results

are compared in Fig. 10.

𝐾 = 2 ∙ (−5.9 ∙ 𝐶𝑞𝑟 + 0.63)5

7 − 1 , 𝐶𝑞𝑟 =𝑄∙𝑅𝑒𝜑

0.2

2∙𝜋∙𝛺∙𝑟3 Eq. 51

𝐾 = [

−8.85 ∙ 𝐶𝑞𝑟 + 0.5

𝑒(−1.45𝐶𝑞𝑟)]

54

Eq. 52

24

Fig. 10: Comparison of results from Eq. 51 and Eq. 52

2.3.2.2 With Centrifugal Through-Flow (𝑸 > 𝟎 𝒎𝟑/𝒔)

In a rotor-stator cavity with centrifugal through-flow, the flow can be classified into three flow

types: Batchelor type flow (regime IV), Couette type flow (regime III) and Stewartson type flow.

The main profiles of the non-dimensional tangential velocity 𝑉𝜑 and the non-dimensional radial

velocity 𝑉𝑟 for Batchelor type flow (Batchelor [10]), Couette type flow (Batchelor [10]) and

Stewartson type flow (Poncet et al. [68]) are shown in Fig. 11. Based on the experimental results

from Poncet et al. [68], the converting point of the rotation dominant flow (Batchelor or Couette

type flow) and the through-flow dominant flow (Stewartson type flow) is at 𝐶𝑞𝑟 ≈ 0.035.

Rotation dominant 𝐶𝑞𝑟 ≤ 0.035 Through-flow dominant 𝐶𝑞𝑟 > 0.035

Batchelor type (Regime IV) Couette type (Regime III) Stewartson type

Fig. 11: Velocity profiles for Batchelor type flow, Couette type flow and Stewartson type flow

Daily et al. [29] correlate Eq. 53 for the radial evolution of the core rotation in case of centrifugal

through-flow with zero entrance rotation. The equation is in good agreement with the

measurements of tangential velocity. 𝐾0 can be assumed to be 0.5 since Daily et al. [29] do not

find a unique dependence on the axial gap width G.

0

1

2

3

-0.5 -0.4 -0.3 -0.2 -0.1 0

𝐶𝑞𝑟

Eq. 51

Eq. 52

Disk Wall

x

𝑉𝜑

Disk

x

Wall

𝑉𝜑

𝑉𝑟

휁

x 𝑉𝑟

휁

휁

Disk Wall

𝑉𝑟

𝑉𝜑

K

25

𝐾 =𝐾0

12.74𝑄

𝛺 ∙ 𝑏3 ∙ 𝑅𝑒𝜑0.2 ∙ (

𝑏𝑟)

135

+ 1

Eq. 53

Kurokawa and Sakuma [55] mention that a radially outward through-flow favors the merging of

the boundary layers. Transition from the laminar to the turbulent flow is additionally influenced by

the through-flow because in general, the externally applied leakage is in a turbulent state.

Consequently, in this case, the whole flow in the cavity likely becomes entirely turbulent.

Bayley and Owen [13] investigate the influence of a centrifugal through-flow on the flow structure

between a rotating and a stationary disk (𝐺 = 0.008, 𝐺 = 0.03). The system is open towards the

atmosphere at the outer radius. The boundary layer equations are solved numerically using a finite

difference method and the flow is treated as turbulent for 𝑅𝑒𝜑 > 3 × 105. The externally applied

flow is without swirl. Consequently, the measured torque of the disk increases with increasing flow

rate because the core swirl ratio decreases. This agrees with the principal relations found by

Truckenbrodt [82] in case of a free disk subjected to an impinging external flow. From the results

of a two-component LDA system, Poncet et al. (2005) derive Eq. 54 for Stewartson type flow. The

results from the Eq. 51, Eq. 52 and Eq. 54 are depicted in Fig. 12.

Stewartson type flow (𝐶𝑞𝑟 > 0.035):

𝐾 = 0.032 + 0.32 × 𝑒−𝐶𝑞𝑟

0.028 Eq. 54

K K (Magnification of abscissa in the frame in the

left figure)

(a) Comparison of the results (b) Near the converting point

Eq. 51 Eq. 52 Eq. 54

Fig. 12: Results for K from different equations for centrifugal through-flow

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.20

0.1

0.2

0.3

0.4

0.5

0 0.02 0.04 0.06 0.08 0.1

𝐶𝑞𝑟 𝐶𝑞𝑟

26

2.3.3 Impact of Surface Roughness in an Enclosed Rotor-Stator Cavity

The measured surface roughness 𝑅𝑧 is the average value of five peaks and five valleys. It is a

challenging work to introduce the measured 𝑅𝑧 into Eq. 51, Eq. 52 and Eq. 54. Based on the results

of Adams et al. [81], a directly use of the measured surface roughness is not adequate. A conversion

of the measured surface roughness 𝑅𝑧 to the equivalent sand-grain roughness 𝑘𝑠 can be done

using Eq. 55.

𝑘𝑠 =𝜋∙

8 , 휀 = 0.978 ∙ 𝑅𝑧

Eq. 55

Schlichting [22] determines Eq. 56 to predict the transition point between a hydraulic smooth disk

and a disk in the transition zone, noted as 𝑘𝑠𝑙. In this thesis, 𝑘𝑠𝑙 ≈ 6 𝜇m is applied.

𝑘𝑠𝑙 =

100 ∙ 𝜈

(1 − 𝐾) ∙ 𝑟 ∙ 𝛺

Eq. 56

In Kurokawa et al. [54], the impact of surface roughness on the values of K is investigated. The

results show that K becomes larger towards the outer radius with rougher disks when 𝑘𝑠𝑙 < 𝑘𝑠 ≤

56 𝜇m (the disks are in the transition zone). Based on their velocity measurements, the variations

of K towards the outer radius of the shaft are depicted in Fig. 13.

Fig. 13: Experimental results of K along the radius of disk for 𝑮 = 𝟎. 𝟎𝟑𝟏, 𝑹𝒆 = 𝟑. 𝟏 × 𝟏𝟎𝟔 and

𝑪𝑫 = 𝟎 by Kurokawa et al. [54]

2.3.4 Impact of Pre-Swirl

In a turbomachine, the entering leakage flow normally contains a significant amount of angular

momentum (from the impeller outflow). Kurokawa and Toyokura [53], for example, show that

the angular momentum flux (instead of the volume flow rate) has the most significant effect on the

𝑘𝑠

=0

.6 𝜇

m

0.4

0.5

0.6

0.7

0.8

0.9

1

0.4 0.5 0.6

K

x

27

axial thrust for centripetal through-flow.

One major issue in this thesis is the specification of proper boundary conditions, especially with

respect to the swirl component of the leakage flow. Guide vanes or swirlers, oriented either in the

radial direction (centripetal through-flow) or in the axial direction (centrifugal through-flow) are

used in the experiments in this thesis to deflect the fluid in the circumferential direction. Since

insufficient information about the respective flow magnitudes are given in the literature, the

circumferential velocity component must be estimated from the geometrical outlet angle of the

guide vanes (assuming that the flow angle equals to the geometrical vane outlet angle). The

meridional component is calculated from the continuity equation disregarding the obstruction by

the guide vanes. However, the flow angle does not correspond to the geometrical angle of the

profile and it further varies in the axial direction. The velocity triangles when the fluid leaves the

inlet swirlers are plotted in Fig. 14, for centripetal (Fig. 14 (a)) and centrifugal (Fig. 14 (b)) through-

flow.

(a) Centripetal (24 passages) (b) Centrifugal (four swirlers)

Fig. 14: Velocity triangles for the pre-swirl through-flow

𝛽

View along

“A” direction

(Ignoring the

thickness of

the swirler)

v𝜑

𝛽

“A” direction

Swirler

Horizontal pipe

v𝑧

28

According to Fig. 14, the tangential velocities are calculated from the following relationships:

Centripetal through-flow:

tan(𝛽) =

𝑉𝜑

𝑉𝑟

Eq. 57

Centrifugal through-flow:

tan(𝛽) =

𝑉𝜑

𝑉𝑧

Eq. 58

Kurokawa et al. [52] derive Eq. 59 to predict the values of 𝐾𝑏 (K at 𝑟 = 𝑏). 𝐶𝑞 is the through-

flow rate coefficient.

49

720𝑎∗ ∙ 𝑏 ∙ (1 − 𝐾𝑏)3 +

5

6𝐶𝑞 ∙ 𝐾𝑏 =

0.0225 ∙ 𝐺

𝑏14

∙𝐾𝑏

74

(1 − 𝐾𝑏)12

∙ [(𝑎∗ ∙1 − 𝐾𝑏

𝐾𝑏+ 1)]

38

Eq. 59

In Kurokawa et al. [52], the distribution of K along the radius is determined with Eq. 60 (for

definition of f see Eq. 33).

[5

6∙

𝐶𝑞

𝑅135

−49

240∙ 𝑎∗ ∙ 𝑏 ∙ (1 − 𝐾)2] 𝑅

𝑑𝐾

𝑑𝑅= 0.0225 ∙ {

[(𝑎∗)2 + 1]38

𝑏14

∙ (1 − 𝐾)54 −

(𝑎2 + 1)38

𝑓14

∙ 𝐾74}

−5

3∙

𝐶𝑞

𝑅135

∙ 𝐾 −1127

3600∙ 𝑎∗ ∙ 𝑏 ∙ (1 − 𝐾)3

Eq. 60

2.3.4.1 Centripetal Pre-Swirl Through-Flow

In a rotor-stator cavity with centripetal pre-swirl through-flow, the core rotates faster than the disk

at lower radius which results in a creation of an attachment line where the fluid and the rotor rotate

at the same speed. Outgoing from this point, the dominant pressure gradient drives the fluid radially

inwards even in the rotor boundary layer for decreasing radius.

Based on the measured velocity profiles, Radtke and Ziemann [71] give a qualitative streamline

pattern of the flow structure when the core rotates faster than the disk (𝐾 > 1). Two counter rotating

vortices are formed by radial outflow in the core region, shown in Fig. 15 (a). Owen and Rogers

[66] also discuss this issue. Their theoretical streamline pattern dose not show any sign of positive

radial outflow. The fluid is directed radially inwards along the complete gap, illustrated in Fig. 15

(b). This is also confirmed by the LDA measurements by Poncet et al. [72].

29

(a) (b)

Fig. 15: Flow structure in the case of 𝑲 > 𝟏

Kurokawa et al. [52] correlate Eq. 61 to associate the value of 𝐾𝑒 (value of K at the entrance) with

𝐶𝑎𝑚 and 𝐶𝑞. 𝐶𝑎𝑚 is the coefficient of the inlet angular momentum in the through-flow.

49

720𝑎∗ ∙ 𝑏 ∙ (1 − 𝐾𝑒)3 +

5

6𝐶𝑞 ∙ 𝐾𝑒 = 0.0225 ∙ 𝐺 ∙ (𝑎2 + 1)

38 ∙ (

𝐾𝑒7

𝑓𝑒)

14

− 𝐶𝑎𝑚 Eq. 61

2.3.4.2 Centrifugal Pre-Swirl Through-Flow

In a rotor-stator cavity with centrifugal pre-swirl through-flow, the flow structure depends

principally on the turbulent flow parameters 𝜆𝑇 and 𝐾𝑒 (Owen and Rogers [66]). Karabay et

al. [49] show that, at sufficiently large amounts of 𝜆𝑇, free vortex flow occurs in the core outside

the boundary layer on the rotating surface. Under these conditions, 𝑉𝜑 is in proportion to 𝑟−1,

where 𝑉𝜑 is the tangential component of the velocity. 𝐾𝑝 follows Eq. 62 for an ideal system with

no losses. The test rig is depicted in Fig. 16. The pre-swirl nozzles instead of inlet swirlers at the

inlet are designed to generate the pre-swirl through-flow. The parameter 𝑥ℎ is the non-

dimensional radial coordinate at the radius of pre-swirl nozzle.

𝐾𝑝 =

𝑉𝜑

𝛺 ∙ 𝑟= 𝐾ℎ ∙ 𝑥ℎ

2 ∙ 𝑥−2 Eq. 62

𝐾 > 1

Pre-swirl

vanes

Q Q

Cas

ing

Cas

ing

Dis

k

Shaft

𝐾 > 1 Q

Q

Dis

k

Cas

ing

30

Fig. 16: Sketch of the rotor-stator cavity (redrawn from Karabay et al. [49])

In the cover-plate system where 𝐾ℎ > 1, Karabay et al. [49] show that free vortex flow will take

place throughout the cavity when:

𝜆𝑇 ≥ 0.437 ∙ [1 − (𝐾ℎ ∙ 𝑥ℎ2)1.175]1.656 Eq. 63

In practical cover-plate systems, there are losses between the outlet of the pre-swirl nozzles and the

inlet to the rotor-stator cavity. If 𝐾ℎ > 1, viscous effects cause a loss of angular momentum of the

fluid, reducing the effective swirl ratio. Karabay et al. [49] determine Eq. 64 to correlate 𝐾𝑝 with

𝐾ℎ,𝑒𝑓𝑓 (effective pre-swirl ratio at 𝑥 = 𝑥ℎ).

𝐾𝑝 =

𝑉𝜃,∞

𝛺 ∙ 𝑟= 𝐾ℎ,𝑒𝑓𝑓 ∙ 𝑥ℎ

2 ∙ 𝑥−2 Eq. 64

Karabay et al. [49] find that the effects of 𝑅𝑒𝜑 and 𝜆𝑇 on 𝐾ℎ,𝑒𝑓𝑓 are negligible. They proposed

Eq. 65 for the computed values of 𝐾ℎ,𝑒𝑓𝑓. Based on the experimental results, they modify Eq. 65 into

Eq. 66, compared in Fig. 17.


𝐾ℎ= 1.053 − 0.062 ∙ 𝐾ℎ

Eq. 65


𝐾ℎ= 1 − 0.056 ∙ 𝐾ℎ

Eq. 66

Blade-cooling passage

Pre-swirl nozzles

Q

Dis

k

Cas

ing

Cas

ing

Shaft

Q

31


Fig. 17: Variation of 𝑲𝒉,𝒆𝒇𝒇 versus 𝑲𝒉 by Karabay [49]: Solid line: Eq. 65; Dashed line: Eq.

66; Point: Exp

2.4 Axial Thrust

The difference of the forces on both sides of the disk is the main source for the axial thrust (𝐹𝑎),

calculated with Eq. 67 (direction see Fig. 3). 𝐹𝑎𝑓 (calculated with Eq. 68) and 𝐶𝐹𝑓 respectively

represent the force and the thrust coefficient on the front surface of the disk (in the front chamber,

shown in Fig. 7), while 𝐹𝑎𝑏 (calculated with Eq. 69) and 𝐶𝐹𝑏 are those on the back surface of the

disk (in the back chamber). The parameters a and 𝑝𝑏 represent the radius of the hub (see Fig. 7)

and the pressure at 𝑥 = 1 , respectively. The back chamber (𝐺 = 0.072 ), shown in Fig. 7, is

supposed to be an enclosed cavity. The values of 𝐶𝐹𝑏 are obtained when 𝐶𝐷 = 0 and the axial

gaps of both cavities have the same width for different Re (under that condition 𝐶𝐹𝑓 = 𝐶𝐹𝑏). After

obtaining those amounts, the values of 𝐶𝐹𝑓 with different amounts of 𝐶𝐷 can be calculated.

𝐹𝑎 = 𝐹𝑎𝑏 − 𝐹𝑎𝑓 Eq. 67

𝐹𝑎𝑓 = 𝜋 ∙ 𝑝𝑏 ∙ 𝑏2 − 𝐶𝐹𝑓 ∙ 𝜌 ∙ 𝛺2 ∙ 𝑏4 Eq. 68

𝐹𝑎𝑏 = 𝜋 ∙ 𝑝𝑏 ∙ (𝑏2 − 𝑎2)−𝐶𝐹𝑏 ∙ 𝜌 ∙ 𝛺2 ∙ (𝑏4 − 𝑎4) Eq. 69

To get an estimate of the expected forces, the simplified radial equilibrium, presented in Eq. 70, is

used in Will [88].

𝜕𝑝

𝜕𝑟= 𝜌 ∙ 𝛺2 ∙ 𝐾2 ∙ 𝑟

Eq. 70

0

0.5

1

1.5

2

2.5

3

3.5

4

0 1 2 3 4 5 6

𝐾ℎ

32

Will et al. [85][86][87] determine Eq. 71 to evaluate the pressure distribution along the radius of the

disk for the incompressible, steady flow. It is obtained directly from the radial momentum equation

when the turbulent shear stress is neglected. In a rotor-stator cavity, the cross sectional area changes

along the radius. Consequently, the pressure must also change since the velocity changes in the

radial direction according to the continuity equation with a known pressure distribution along the

radius, the axial thrust at the surface of the rotor can be estimated by integrating the following

equation:

𝜕𝑝

𝜕𝑟= 𝜌 ∙ (

𝑣𝜑2

𝑟− 𝑣𝑟

𝜕𝑣𝑟

𝜕𝑟) = 𝜌 ∙ 𝐾2 ∙ 𝛺2 ∙ 𝑟 +

𝜌∙𝑄2

4∙𝜋2∙𝑠2∙𝑟3 Eq. 71

Stepanoff [79] investigates the resulting axial force in centrifugal pumps. For theoretical analysis,

it is assumed that the fluid rotates as a solid body with a fixed value for the core rotation factor

(𝐾 = 0.5). The assumption of a constant core rotation is nowadays proved as inefficient in a rotor-

stator cavity with through-flow.

Kurokawa et al. [52] derive Eq. 72 to estimate the thrust coefficient 𝐶𝐹 for the case of 𝐺 = 0.05.

𝐶𝐹 = 9.96 ∙ 𝐶𝑎𝑚 + 0.039; 𝐶𝑎𝑚 = (𝐿

2∙𝜋∙𝑏5∙𝛺2) ∙ 𝑅𝑒1

5 Eq. 72

Where L is the angular momentum which centripetal through-flow brings into the flow field.

Kurokawa et al. [53] determine Eq. 73 and Eq. 74 to calculate 𝐶𝑎𝑚 in a radial pump. The parameter

𝛽 is shown in Fig. 14. According to their results, the values of 𝜙 range from 0.1 to 0.15.

Centripetal through-flow (𝐶𝑞 < 0, 𝑄 < 0 m3/s):

𝐶𝑎𝑚 = −[1 − 𝜙 ∙ cot(𝛽)] ∙ 𝐶𝑞; 𝐶𝑞 = (𝑄

2∙𝜋∙𝑏3∙𝛺) ∙ 𝑅𝑒

1

5 Eq. 73


𝑉𝑟).

Centrifugal through-flow (𝐶𝑞 > 0, 𝑄 > 0 m3/s):

𝐶𝑎𝑚 = −(1 + 5 ∙ 𝐾𝑏) ∙ 𝐶𝑞/6; 𝐶𝑞 = (𝑄

2∙𝜋∙𝑏3∙𝛺) ∙ 𝑅𝑒

1

5 Eq. 74


𝑉𝑧).

33

2.5 Moment Coefficient

2.5.1 The Free Disk

To study the flow in a rotor-stator cavity, a starting point is the flow near the free disk. The general

flow structure of the free disk problem is shown schematically in Fig. 18.

Fig. 18: Flow structure around a free disk (According to Schlichting and Gersten [73])

The frictional torque and the axial thrust are essentially influenced by the flow conditions at the

outer radius. Zilling [90] and Lauer [58] account for a torque resulting from a turbulent shear stress

acting on the lateral area of the control volume, shown in Fig. 19.

Fig. 19: Torques on an annular volume element

𝑉𝑍

𝑉𝑟

𝑉𝜑

𝑧

𝑥

𝑦

Disk

𝑧 r x

y

𝑟 + 𝑑𝑟

𝛺

𝛺

Disk Disk Fluid

34

The flow created by a rotating disk is three dimensional from inception and is first theoretically

investigated by Von Kármán [47] in 1921. Two analytical expressions are developed for 𝐶𝑀 for

a free disk under both the laminar and the turbulent flow conditions (for a single surface of the

disk):

Laminar flow: 𝐶𝑀 = 3.68 ∙ 𝑅𝑒−12 Eq. 75

Turbulent flow: 𝐶𝑀 = 0.146 ∙ 𝑅𝑒−15 Eq. 76

Cochran [22] give a more exact solution of the ordinary differential equations derived by Von

Kármán [47]. He solves the equations using a power series expansion in the near wall vicinity

which is combined with an asymptotic approach in the outer region. The results of Von Kármán

[47] and Cochran [19] are principally valid for the case of a disk with infinite radius. Eq. 77 is

derived from the integration of the shear stress across the radius for 𝐶𝑀.

𝐶𝑀 = 1.935 ∙ 𝑅𝑒−

12 Eq. 77

Dorfman [32][33] uses analytical methods to calculate the moment coefficient of a free disk using

a logarithmic boundary layer velocity profile with Eq. 78 (for a single surface of the disk).

𝐶𝑀 = 0.982 ∙ (log10𝑅𝑒)−2.58 Eq. 78

Experiments on a single disk are performed by Bayley and Owen [13], with a radius 𝑏 =

0.381 m, rotating at speeds of up to 4500 rpm . The frictional torque is measured using an

optical torquemeter built into the drive shaft. Based on the experimental results, they correlate Eq.

79 for the moment coefficient (for a single surface of the disk).

𝐶𝑀 = 0.0655 ∙ 𝑅𝑒−0.186 Eq. 79

2.5.2 Enclosed Rotor-Stator Cavity

Apart from the fundamental free disk case, a disk rotating in a casing more closely resembles the

configurations commonly encountered in actual turbomachinery applications.

Schultz-Grunow [74] is one of the first to analyze the flow caused by a rotating disk in a stationary

housing. For small gap sizes and circumferential Reynolds numbers, the tangential velocity varies

35

linearly with the axial coordinate. Neglecting both the radial and the axial velocities in the gap, a

simple relationship for the moment coefficient is found analytically (laminar flow, for a single

surface of the disk):

𝐶𝑀 =𝜋

𝐺 ∙ 𝑅𝑒

Eq. 80

Soo [78] theoretically investigates the laminar flow of a rotor-stator system for small through-flow

and gap widths in order to resolve some inconsistencies from the older reports. The basic solution

is similar to the one obtained by Schultz-Grunow [74] (for a single surface of the disk), written in

Eq. 81.

𝐶𝑀 = 0.0308 ∙ 𝐺−

14 ∙ 𝑅𝑒−

14 Eq. 81

Daily and Nece [28] examine the flow of an enclosed rotating disk both analytically and

experimentally. They distinguish the four flow regimes (see Fig. 5) by correlating different

empirical equations of the moment coefficients, written in Eq. 82~Eq. 85 (for a single surface of the

disk). The turbulent flow regimes III and IV are the most likely to occur in practice. The small gap

cases (regime I and regime III) essentially represent a viscous Couette-type of flow while the other

two regimes correspond to the Batchelor-flow solution characterized by an inviscid core region.

For merged boundary layers (regime I and III), the frictional resistance decreases with increasing

axial gap width as a consequence of the reduced velocity gradient. In contrast, the frictional

resistance increases with increasing axial gap width in case of separated boundary layers (regime

II and IV) due to the braking influence of the outer cylindrical shroud.

Regime I 𝐶𝑀1 =

𝜋

𝐺 ∙ 𝑅𝑒

Eq. 82

Regime II 𝐶𝑀2 = 1.85 ∙ 𝐺1

10 ∙ 𝑅𝑒−12 Eq. 83

Regime III 𝐶𝑀3 = 0.04 ∙ 𝐺−16 ∙ 𝑅𝑒−

14 Eq. 84

Regime IV 𝐶𝑀4 = 0.0501 ∙ 𝐺1

10 ∙ 𝑅𝑒−15 Eq. 85

36

2.5.3 Rotor-Stator Cavity with Through-Flow

According to Daily et al. [29], the moment coefficient increases with increasing through-flow rate

(for centripetal through-flow). The correlation given in Eq. 86 allows the calculation of the moment

coefficient. The parameter 𝐶𝑀0 is the moment coefficient without through-flow, calculated by

Daily and Nece [28] (from Eq. 82 to Eq. 85 according to the flow regime).

𝐶𝑀 = 𝐶𝑀0 ∙ (1 + 13.9 ∙ 𝐾0 ∙ 𝜆𝑇 ∙ 𝐺−18) Eq. 86

Bayley and Owen [13] investigate the effects of a superimposed radial outflow on both windage

and pressure distribution in a rotor-stator system. They use a disk of radius 𝑏 = 0.381 𝑚, rotating

up to speeds of 4500 rpm, with a stationary shroud. The tests cover the range of rotational

Reynolds numbers 4 × 105 ≤ 𝑅𝑒 ≤ 4 × 106. Three values of axial gap width G are used: 𝐺 =

0.06, 0.12 and 0.18. The windage torque on the disk is measured using a shaft mounted optical

torque meter. They observe that, as G decreases, the sensitivity of the torque to variations in G also

decreases.

Dibelius et al. [31] carry out tests on a shrouded rotor-stator system with a disk (𝑏 = 0.4 m). It

has the capacity for both centripetal and centrifugal through-flow ( |𝐶𝐷| ≤ 1.4 × 105 ).

Measurements are obtained for the static pressure, the velocity distribution and the disk frictional

torque. The range of Re tested is 2 × 106 ≤ 𝑅𝑒 ≤ 3 × 107. The torque measurements show that

the moment coefficient for centrifugal through-flow exceeds that for centripetal through-flow.

Owen [65] distinguishes between the source region close to the axis of rotation and the core region

radially outward of the source region. He distinguishes the Stewartson type flow from the Batchelor

type based on the amounts of 𝜆𝑇. He also correlates an analytical approach to predict the moment

coefficient in a shrouded rotor-stator cavity for Batchelor type flow. The correlations for 𝜆𝑇 <

0.219 are given in Eq. 87 (a~d). It can be seen that Eq. 87 (b) is the same as that from von Kármán

[48] for the free disk solution. The flow structures are depicted in Fig. 20 depending on the values

of 𝜆𝑇. The flow structure for an enclosed rotor-stator cavity is sketched in Fig. 20 (a). When the

centrifugal through-flow is imposed and 𝜆𝑇 < 0.219, the centripetal radial velocity near the wall

can still be found, depicted in Fig. 20 (b). In Fig. 20 (c), the case with through-flow for 𝜆𝑇 >

0.219 is depicted. All the radial velocity in the cavity seems to be centrifugal. The parameter 𝑥𝑐

is defined as the radial location at which the source region ends and the core rotation begins. In the

case where 𝜆𝑇 > 0.219, the source region is assumed to apply for the whole rotor-stator cavity

(𝑥𝑐 > 1).

37

𝐶𝑀 = 𝐶𝑀𝑎 + 𝑐𝑀𝑏 Eq. 87 (a)

𝐶𝑀𝑎 = 0.146 ∙ 𝑅𝑒(−15

) ∙ 𝑥𝑐

235 Eq. 87 (b)

𝐶𝑀𝑏 = 0.0796 ∙ 𝑅𝑒(−

1

5)

∙ {(1 − 𝑥𝑐

23

5 ) + 14.7 ∙ 𝜆𝑇 ∙ (1 − 𝑥𝑐2) + 90.4 ∙ 𝜆𝑇

2 ∙ [1 − 𝑥𝑐(−

3

5)]}

Eq. 87 (c)

𝑥𝑐 = 1.79 ∙ 𝜆𝑇(

513

) Eq. 87 (d)

Fig. 20: Flow structure inside a rotor-stator cavity with (a) no through-flow, (b) with centrifugal

through-flow and 𝝀𝑻 < 𝟎. 𝟐𝟏𝟗 and (c) with through−flow, 𝝀𝑻 > 𝟎. 𝟐𝟏𝟗

It can be seen that 𝐶𝑀𝑎 is the same as the free disk solution from von Kármán [48]. In the case

where 𝜆𝑇 > 0.219, the source region is assumed to apply for the whole rotor-stator cavity (𝑥𝑐 >

1). However, the free disk solution is inapplicable therefore, in this case, the moment coefficient

𝐶𝑀 is calculated with Eq. 88.

𝐶𝑀 = 0.666 ∙ 𝐶𝐷 ∙ 𝑅𝑒−1 Eq. 88

Gartner [37][38] uses an analytical approach to extend Eq. 87 (a) for the frictional torque, in a

shrouded rotor-stator system with radial outflow, to include the effects of the cylindrical shrouding.

In the case where 𝜆𝑇 < 0.219 , a third term 𝐶𝑀𝑐 is added into Eq. 87 (a), which represents the

moment coefficient due to the cylindrical shroud. The expression for 𝐶𝑀𝑐 is defined in Eq. 89. This

correlation shows a small improvement on Eq. 87 in Owen [65], when compared with the measured

torque. Coren et al. [23] produce a correlation giving the moment coefficient in terms of Reand

𝐶𝐷, given in Eq. 90.

s

b

𝛺

Dis

k D

isk

Dis

k

Wal

l Wal

l

Wal

l

(a) (b) (c)

𝑥𝑐

𝑥𝑐

𝑥𝑐

38

𝐶𝑀𝑐 = 0.36 ∙ 𝛾(−

14

)∙ 𝐾

74 ∙ (1 − 𝐾)

320 ∙ 𝑅𝑒(−

15

)

𝛾 = [81 ∙ (1 + 𝛼2)

38

49 ∙ (23 + 37 ∙ 𝐾) ∙ 𝛼]

45

𝛼 = [

2300 ∙ (1 + 8𝐾)

49 ∙ (1789 − 409 ∙ 𝐾)]

12

𝐾 = 0.087 ∙ 𝐾0 ∙ 𝑒[5.2∙(0.486−𝜆𝑇)−1]

Eq. 89 𝐾0 = 0.49 − 0.57 ∙

𝑠

𝑏

𝐶𝑀 = 0.52 ∙ 𝐶𝐷0.37 ∙ 𝑅𝑒−0.57 + 0.0028 Eq. 90

2.5.4 Impact of Surface Roughness on the Moment Coefficient

Dorfman [32] calculates the frictional torque of a rough disk and develops Eq. 91 for the moment

coefficient for an enclosed rotor-stator cavity. The results are plotted in Fig. 21. With the increase

of 𝑏

𝑘𝑠, the values of 𝐶𝑀 from his analytical correlation drop and are in very good agreement with

those from Eq. 91.

𝐶𝑀 = 0.108 ∙ (

𝑘𝑠

𝑏)0.272

Eq. 91

Fig. 21: Moment coefficient for a rotating disk (circles: from an analytical correlation; dashed

line: Eq. 91; triangle: test)

Daily and Nece [28] study the effects of disk roughness in enclosed rotor-stator systems. The test

fluids are water and three solvent-refined, paraffin-based commercial lubricating oils. The test rig

is illustrated in Fig. 22. During the experiments, there is no through-flow, despite a recirculation

system equipped with a heat exchanger is provided for cooling the test fluid. Torques are measured

by means of four SR-4 (Simmons Ruge-4) bonded strain gages connected to a battery-powered

0

4

8

12

16

20

24

28

32

36

40

1 10 100

10

00

𝐶𝑀

𝑘𝑠

𝑏/100𝑘𝑠

39

bridge circuit and mounted to the inside wall of a recess in the shaft. Voltage differences from the

circuit are taken off through slip rings mounted on the shaft. A DC galvanometer serves as the

indicating device. As the shaft recess containing the strain gages is situated on the fluid side of the

shaft seal, the only torque measured is due to the fluid friction on the disk surface. No deductions

are necessary for bearing and sealing friction.

Fig. 22: Schematic drawing of the test rig (redrawn from Daily and Nece [28])

The commercial grid papers are bonded to both the rotor and the stator. The experiments are

conducted for relative roughnesses of 𝑘𝑠

𝑏= 10−3, 5 × 10−4 and 3.2 × 10−4, for G=0.023, 0.061

and 0.112, and for 4 × 103 < 𝑅𝑒 < 6 × 106. It is found that roughness had no significant effect

on the moment coefficient in the laminar regimes I and II; for 𝑅𝑒 > 2 × 105, in the turbulent

regime III and IV, 𝐶𝑀 increases as the roughness increases. From results obtained at the largest

values of Re, Daily and Nece [28] determine the correlation:

𝐶𝑀

−12 = −5.37 ∙ log10 (

𝑘𝑠

𝑏) − 3.4 ∙ 𝐺

14

Eq. 92

They also conclude that roughness effects start to become significant for:

𝑅𝑒 ∙ 𝐶𝑀

−12 ≈ 16000 ∙ (

𝑘𝑠

𝑏)

−1

10

Eq. 93

Shaft

Disk

Wall

Strain gage

To motor

Bearing

Slip rings Seal

40

Within the Reynolds number regions of significant roughness effect, roughness changes produce

much larger changes in 𝐶𝑀 than changes in axial spacing do (Daily and Nece[28]), plotted in Fig.

23.

𝐶𝑀

Fig. 23: Rough disk torque data (replotted from Daily and Nece [28])

Kurokawa et al. [53] study the moment coefficient in a rotor-stator cavity with different through-

flow coefficients, noted as 𝐶𝐷. The sketch of their test rig is given in Fig. 24. Kurokawa et al. [54]

also investigate the roughness effects on the flow in an enclosed rotor-stator cavity. Their analytical

results are in a satisfactory agreement with the measured data in both the hydraulic smooth and the

complete rough regions. In their experiments, a disk of 164 mm radius is rotated in a water-filled

casing with the radial and the axial gaps of 2 mm (𝐺 = 0.012 ) and 12.8 mm (𝐺 = 0.078 )

respectively.

0.001

0.01

0.1

10000 100000 1000000 10000000

𝑅𝑒

𝑘𝑠

𝑏=

1

3200, 𝐺 = 0.0609

104 105 106 107

Smooth disk and wall: 𝐺 = 0.0255

Smooth disk and wall: 𝐺 = 0.0609

𝑘𝑠

𝑏=

1

1000, 𝐺 = 0.0609

𝑘𝑠

𝑏=

1

2000, 𝐺 = 0.0609

41

Fig. 24: Sketch of the test rig (replotted from Kurokawa et al. [54])

2.6 Flow Separation near the Entrance

When the velocity at the inlet is large enough, the flow is through-flow dominant. Since there is an

area change when the flow enters the cavity from the guide vane, flow separation may occur when

the boundary layer travels far enough against an adverse pressure gradient that the speed of the

boundary layer relative to the object falls almost to zero (Anderson, John D. [3]). The fluid flow

becomes detached from the surface of the object, and instead takes the forms of eddies and vortices,

depicted in Fig. 25. The flow reversal is primarily caused by an adverse pressure gradient imposed

on the boundary layer by the outer potential flow. The direction 𝑙 and 𝑧𝑙 are also shown in Fig.

25. The approximately streamwise momentum equation inside the boundary layer is stated in Eq.

94.

Fig. 25: Graphical representation of the velocity profile and the reverse flow which show the

flow separation [3]

𝑄

𝑧𝑙

𝑙

𝑄

I III IV

∆𝑟

𝑏

𝑟

I: Wall boundary layer

III: Core

IV: Disk boundary layer

𝑉𝑟

𝑉𝜑

𝛺

Disk

Wall

Surface streamline Wall

v

https://en.wikipedia.org/wiki/Adverse_pressure_gradient

https://en.wikipedia.org/wiki/Eddy_(fluid_dynamics)

https://en.wikipedia.org/wiki/Vortex

https://en.wikipedia.org/wiki/Adverse_pressure_gradient

https://en.wikipedia.org/wiki/Potential_flow

42

v

𝜕v

𝜕𝑙= −

1

𝜌

𝑑𝑝

𝑑𝑙+ 𝜈

𝜕2v

𝜕𝑧𝑙2

Eq. 94

In the case with centripetal through-flow, flow separation may occur when there is a jet flow

through each guide vane passage. The approximate separation line is depicted with green dotted

line in Fig. 26.

Fig. 26: Approximate separation line for centripetal through-flow

The solution procedure for the approximate separation line is shown in Fig. 27. When the jet flow

is very strong through the guide vane channel and the disk rotates slowly, the tangential velocity

component can be neglected compared with the radial velocity. The cavity flow is therefore

through-flow dominant. The pressure and the velocity at each radius should be calculated to

understand the cavity flow. The position of the separation line is very important for the calculation

of above parameters. The case when 𝛺 = 0 , 𝐶𝐷 = −5050 (centripetal through-flow, 𝑄 =

−2 m3/h ) and 𝐺 = 0.045 (𝑠 = 0.005 m ) is taken as an example, shown in Fig. 28. The

approximate separation line (dashed line) from the results of a numerical simulation is in relative

good agreement with those from the solution procedure in Fig. 27. Known the position of the

separation line, the cross section 𝐴𝑛 at each radius can be calculated. Then, the average values of

p and v𝑛 can be estimated. The shape of the separation lines from the solution procedure and the

numerical simulation are in relative good agreement with each other, depicted in Fig. 28.

Separation line:

Green dotted line

𝑝𝑛, v𝑛, 𝐴𝑛

𝑝𝑛+1, v𝑛+1, 𝐴n+1

𝑙

𝑧𝑙

Disk

Wall Wall

Control

volume

1, 2, … … , 𝑎

1,2

,……

,𝑑

43

Starting parameter:

𝑎 = 1, 𝑑 = 1, n=1

𝑙𝑑, 𝑛

𝑧𝑙𝑎

𝐴𝑛+1

𝐴𝑛v𝑛 = 𝐴𝑛+1 ∙ v𝑛+1, 𝑉𝑛𝜕v

𝜕𝑙= −

1

𝜌

𝑑𝑝

𝑑𝑙+ 𝜈

𝜕2v

𝜕𝑍2

𝑝𝑛+1 = 𝑝𝑛 +𝜌v𝑛

2

2−

𝜌 ∙ 𝐴𝑛2v𝑛

2

2 ∙ 𝐴𝑛+12 −

𝜌 ∙ 𝐴𝑛2v𝑛

2

2 ∙ 𝐴𝑛+12 (

𝐴𝑛+1

𝐴𝑛− 1)2

Get velocity profile along 𝑧𝑙: ν𝜕2v

𝜕𝑍2 = 𝑉𝑛𝜕v

𝜕𝑙+

1

𝜌

𝑑𝑝

𝑑𝑙

Get Average velocity: v̇𝑛+1

0.999 ∙ v̇𝑛+1 < v𝑛+1 < 1.001 ∙ v̇𝑛+1

Save 𝑙𝑑 , 𝑧𝑙𝑎, 𝑝𝑛+1, , v𝑛+1, 𝑛

Fig. 27: Solution procedure for the approximate separation line

𝑉𝑟 from numerical simulation 𝑉𝑟 from Matlab (Program see Fig. 27)

Blue: Radially inward velocity; Red: Radially outward velocity.

Fig. 28: Comparison of the results for the separation line (𝜴 = 𝟎, 𝑪𝑫 = −𝟓𝟎𝟓𝟎, 𝑮 = 𝟎. 𝟎𝟒𝟓)

𝑎 = 𝑎 + 1 𝑑 = 𝑑 + 1

𝑛 = 𝑛 + 1

Numerical

simulation From

solution

procedure

𝑥 = 0.83

Disk Wall 0.005 − 𝑧 (mm)

𝑥 = 1.01 Inlet

Wal

l

Wall

Dis

k

Inlet

𝑥 = 0.83

𝑥 = 1.01

𝑙 (m

)

Dis

k

Wall

Wal

l

44

2.7 Influence of the Sealing Gap Height

2.7.1 Flow Structure inside the Sealing Gap

The sealing gaps in centrifugal pumps are usually very narrow in order to keep the leakage flow

rate as low as possible. During the operation, the gap height will change due to wear or variations

in the tangential direction because of an eccentricity of the rotor.

Will [88] investigates the effect of the sealing gap height in the back cavity of a radial pump. The

domain for numerical simulation is depicted in Fig. 29. Two sealing gap heights (given in Table 5)

are tested.

Fig. 29: One seventh segment of the back cavity (Will [88])

Config. 1 Config. 2

Gap height shroud side chamber 0.5 mm 0.5 mm

Gap height hub side chamber 0.48 mm 0.24 mm

Table 5: Parameters of the experiments

Will [88] compares the measured pressure with those from numerical simulation, depicted in Fig.

30. The results for the front cavity are satisfactory while they have bigger deviations in the back

cavity.

Q

Front view Cross section

Sealing gap

Q

Disk/impeller

Casing

Balance hole

Hub

45

𝑥 𝑥

Points: Pressure measurements; Dashed lines: Numerical simulation.

Fig. 30: Comparison between experimental and numerical results for the shroud side chamber

(left) and the hub side chamber (right) in case of 0.48 mm sealing gap height (Will [88])

When the sealing gap height decreases to 0.24 mm, smaller deviations between the numerical and

the experimental results are apparent across the sealing gap in the back cavity (hub side), depicted

in Fig. 31 (compared with those in the right figure of Fig. 30). The sealing gap controls the amount

of leakage that passes through the cavity and vice versa determines the pressure difference across

the gap. Mechanical energy conveyed to the fluid in the impeller is dissipated into heat by the

choking effect of the sealing which directly influences the efficiency of the pump. The values of

the 𝑝∗ drop more tremendously across the sealing gap from the results of both numerical

simulation and pressure measurements, compared with those in Fig. 30 for wider sealing gap at the

hub side.

𝑥

Points: Pressure measurements; Dasheded lines: Numerical simulation.

Fig. 31: Comparison between experimental and numerical results for the hub side chamber in

case of 0.24 mm sealing gap height (Will [88])

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.5 0.7 0.9 1.1 1.3 1.5

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.5 0.7 0.9 1.1 1.3 1.5

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.5 0.7 0.9 1.1 1.3 1.5

𝑝∗

𝑝∗

𝑝∗

46

One major parameter for the pressure drop is the gap height which is also pointed out by Iino et al.

[44] for multistage radial pumps. In order to reduce the leakage losses, the gap height is generally

very small and therefore difficult to determine under operating conditions. As expected, the

pressure drop considerably increases and the agreement with the experimental data improves.

Unfortunately, some discrepancies are still apparent behind the sealing gap which might be

attributed to an insufficient resolution of the flow in the gap itself. Currently, the flow is only

resolved by 8 cells in the radial direction in the shroud side cavity and by 13 cells in the hub side

cavity. For example, Tamm [80] uses 6 cells and notes that doubling the number does not change

the results.

According to Gülich [38], the flow in the gap is unstable because both the circumferential velocity

and the centrifugal forces decrease from the rotating shaft to the stationary casing. A similar

situation is presented in the radial gap between the shroud of the rotating disk and the radial casing

in the numerical study of the enclosed rotating disk. The stability criterion for the formation of

Taylor vortices in the gap resulting from this instability is the Taylor number 𝑇𝑎, given in Eq. 95.

The perimeter Reynolds number for the sealing gap 𝑅𝑒𝑝 is calculated with Eq. 96. The

circumferential velocity in the sealing gap v𝑔𝑎𝑝 is predicted with Eq. 97.

𝑇𝑎 =v𝑔𝑎𝑝 ∙ ∆𝑟𝑠𝑒𝑎𝑙

𝜈∙ √


𝑟𝑔𝑎𝑝=

𝑅𝑒𝑝

2∙ √


𝑟𝑔𝑎𝑝

Eq. 95

𝑅𝑒𝑝 =

2 ∙ ∆𝑟𝑠𝑒𝑎𝑙 ∙ v𝑔𝑎𝑝

𝜈

Eq. 96

v𝑔𝑎𝑝 =

2𝜋 ∙ 𝑟𝑔𝑎𝑝 ∙ 𝑛

60 Eq. 97

For 𝑇𝑎 ≥ 41.3, counter-rotating Taylor vortices are expected in the sealing between the rotating

shaft and the stationary casing which strongly influence the pressure drop and energy dissipation

in the gap. To determine whether the flow is turbulent or not, both v𝑔𝑎𝑝 and v𝑎𝑥𝑔𝑎𝑝 (mean axial

velocity in the sealing gap) have to be taken into account considering the gap Reynolds number

𝑅𝑒𝑔𝑎𝑝, given in Eq. 98. The characteristic linear dimension for a fluid moving between two plane

parallel surfaces is 2 ∙ ∆𝑟𝑠𝑒𝑎𝑙.

𝑅𝑒𝑔𝑎𝑝 = √(2 ∙ ∆𝑟𝑠𝑒𝑎𝑙 ∙ v𝑎𝑥𝑔𝑎𝑝

𝜈)2 + (

v𝑔𝑎𝑝 ∙ ∆𝑟𝑠𝑒𝑎𝑙

𝜈)2

Eq. 98

47

Transition to the turbulent flow takes place at 𝑅𝑒𝑔𝑎𝑝 = 2000 while the flow will always be

turbulent for 𝑅𝑒𝑔𝑎𝑝 = 4000 (Gülich [38]). For the turbulent flow in the gap, the friction factor

for the sealing additionally depends strongly on the wall roughness due to the very small gap height

(the relative roughness is comparatively large).

An estimation of the Taylor number and the gap Reynolds number 𝑅𝑒𝑔𝑎𝑝 of both cavities indicates

that the flow is turbulent. Taylor vortices are to be expected (see Table 6) (Will [88]).

Shroud disk cavity

Hub disk cavity

𝑅𝑒𝑔𝑎𝑝 12841 4450

𝑇𝑎 457 147

Table 6: Estimation of 𝑹𝒆𝒈𝒂𝒑 and 𝑻𝒂 in Will [88]

Will [88] states that the flow in the sealing gap is unstable and favors the formation of Taylor

vortices. An important parameter influencing the flow in the small sealing gap is the through-flow.

To further investigate this detail, a parametric study is accomplished. The through-flow rate is first

set to zero and afterwards successively increased by varying the axial velocity component at the

cavity inlet. These simulations are performed with a sealing gap height ∆𝑟 = 0.8 mm to ensure

that the resulting Taylor number of 𝑇𝑎 = 1003 is far above the critical number of 𝑇𝑎 = 41.3. The

results in Fig. 32 clearly prove the expected dependency of the flow structure on the externally

applied through-flow rate. For a very small through-flow rate, several vortex pairs are clearly

visible in the gap and the flow is dominated by the instabilities rather than the through-flow. If the

through-flow rate increases, the vortex pairs gradually diminish. This trend continues with

increasing through-flow until the streamlines are perfectly aligned in the axial direction and the

flow structure is controlled by the through-flow. The critical through-flow rate in this thesis

amounts approximately 16% of the nominal leakage flow. For higher values, no more vortices

occur (Will [88]).

48

𝑄 = 0.0064 m3/h, v𝑎𝑥𝑔𝑎𝑝 = 0.003 m s⁄

𝑄 = 0.0139 m3/h, v𝑎𝑥𝑔𝑎𝑝 = 0.0065 m s⁄

𝑄 = 0.0642 m3/h, v𝑎𝑥𝑔𝑎𝑝 = 0.03 m s⁄

Fig. 32: Flow in the sealing gap (∆𝒓 = 𝟎. 𝟖 𝐦𝐦) for different leakage flow rates (replotted from

Will [88])

2.7.2 Leakage Volumetric Flow Rate through the Sealing Gap

The through-flow coefficient 𝐶𝐷 , calculated with the leakage volumetric flow rate, is very

important during the calculations of K, 𝐶𝐹 and 𝐶𝑀. Baskharone et al. [9] estimate the leakage

flow between pump stages based on finite-element analyses. Zhao et al. [91] investigate the sealing

gap height on the pump performance based on a CFD method. Wu and Squires [89] correlate Eq.

Q

Q

Q

Q

Q

Q

Rotor (Impeller)

Casing

Casing

Rotor (Impeller)

Rotor (Impeller)

Casing

∆𝑟

∆𝑥 =∆𝑟

𝑏

49

99 to predict the volumetric leakage flow rate through the sealing gap of a multi-stage centrifugal

pump. They mention that for the radial pump, the pressure drops can be estimated with ∆𝑝 = 0.6𝐻

(H is the pump pressure head in m) and 𝛹 = 0.5~0.6.

𝑄 = 𝛹 ∙ 𝐴 ∙ √2𝑔 ∙ ∆𝑝 Eq. 99

2.8 Side Chamber Flow in a Centrifugal Pump

The design process of radial pumps requires reliable information about the fluid dynamics in every

single component of the machine. Aside from the flows which are responsible for the energy

transmission in the main parts of a machine, such as the blade passages in the impeller or the volute

casing, the flow in regions resulting from construction conversion is of great importance. These

parts, such as the side cavities between the rotating impeller and the stationary casing, do not serve

the purpose of increasing the energy level of the fluid. They lead to a reduction of the machine

efficiency and can affect the operational reliability. In the side chambers, the frictional torque of

the side walls has a great impact on the required power of the machine. Especially for low specific

speed impellers, this is an inherent problem. Gülich [38] mentions that for low specific speed

machines, disk friction losses can amount up to 50% of the useful power and the leakage losses can

make up to 12% of the nominal flow rate (Bahm [6]). Furthermore, in general, the leakage flow,

which passes the sealing gaps, leads to a reduction of the hydraulic efficiency.

The flow in the side chambers directly influences the mechanical design of a radial pump because

the resulting axial force, whose knowledge is necessary for the design of the bearing, is essentially

determined by the static pressure distribution on the outer side walls of the impeller. The pressure

distribution depends on the rotation of the fluid in the side chamber which is strongly influenced

by the incoming swirling leakage flow. The leakage again is a function of the operating point of

the machine. Besides, the central role of the leakage flow with respect to the axial thrust has already

been noticed in Bahm [6].

The flow structure in the volute depends on the operating point and an axisymmetric pressure

distribution is usually only presented in design flow conditions (Pavesi [67]). According to Bahm

[6], the volute acts like a diffuser at part load. Consequently, the flow in the volute is retarded and

the pressure increases in tangential direction. For overflow conditions, the volute acts like a nozzle

accelerating the fluid in the tangential direction. This leads to a pressure decrease in the

circumferential direction. Further, the authors observe an inward directed flow close to the wall

whose velocity and direction depend on the circumferential position. This imbalance results in a

50

radial load which deflects the impeller into an eccentric position. As a result, the line of influence

of the axial force does not coincide with the axis of rotation leading to an additional torque. In

centrifugal pumps, part load recirculation is a well-known problem.

Gülich [38] notices that especially low energy backflow fluid entering the side chambers

significantly dampens the rotation of fluid and thus affects the axial force. A reduced fluid rotation

leads to a smaller pressure drop which results in a larger pressure force. In dependence on the axial

displacement of the impeller, the recirculation region can occur close to either of the side chambers

at part load conditions. Under certain circumstances, this can even cause a reversal of the resulting

axial force in multi-stage pumps.

The pressure distribution at the impeller outlet is non-stationary (Gülich [38]) due to the complex

interaction of the rotating impeller and the stationary casing or the guide vanes. This causes a

fluctuating behavior of the flow magnitudes that even extends into the side cavities.

Before investigating a centrifugal pump, the flow in a rotor-stator cavity should be investigated

carefully. The assumption of the steady, axisymmetric flow conditions is fundamental to nearly all

publications concerning rotor-stator cavities. A lot of correlations are, however, still not satisfactory

even for the rotor-stator cavity model. Therefore, the following study tries to provide more results

and determine some correlations.

51

3. Experimental Set-Up

3.1 Mechanical Set-Up

The design of both radial pumps and turbines demands more accurate values of 𝐶𝐹 and 𝐶𝑀 .

Above two parameters are still not precisely predictable based on the results and the correlations

from the literature. Measurements are still required to provide more results aiming to determine

some correlations. In order to investigate these relationships experimentally, a test rig is designed

at the University of Duisburg-Essen. The cross section of the test rig is depicted in Fig. 33 (a). The

through-flow (volumetric through-flow rate 𝑄) is supplied with water by a pump system. A tank

with 3 m3 pure water at 20°C is used to feed the water in the pump and also to recollect the water

out of the test rig. The view along the “A” direction is sketched in Fig. 33 (b). In Fig. 33 (c), the

shaft sealing with the radius of 10 mm at the back cavity is depicted. A picture of the test rig is

shown in Fig. 33 (d). The shaft is driven by an electrical motor. A frequency converter is utilized

to adjust the speed of rotation (0 to 2500 rpm) with the absolute uncertainty of 7.5 rpm. The speed

of rotation is increased softly and gradually. In this thesis, only the axial gap of front chamber is

changeable (by installing six sleeves (I) with different length). There are 24 channels in the radial

guide vane (II) instead of entirely open at the periphery, shown in Fig. 33 (b). With centripetal

through-flow, one of the guide vanes with different blade angle is installed to produce pre-swirl.

For centrifugal through-flow, there are four inlet swirlers (XI) in the horizontal pipe. A radial

directed guide vane (II) with wide passages is installed at the centrifugal outlet. Other parameters

of the experiments are given in Table 7. The transducers in the test rig include 12 pressure tubes, a

torque meter and three tension compression transducers. A thrust plate (VIII) is fixed by a ball

bearing and a nut from each side to convey the axial thrust to the tension compression transducers.

A linear bearing (VI) is utilized to minimize the frictional resistance during the axial thrust

measurements. In the process of the axial thrust measurements, the axial thrust transducers are

calibrated when the axial gap width of the front chamber is changed.

52

(V)

(I)

(VI) (II)

(VII) (III)

(VIII) (IV)

(XI) (IX)

(X)

(I). Sleeves (to change the axial gap), (II). Guide vane (24 channels), (III). Front chamber, (IV). Disk, (V). Back cover,

(VI). Linear bearing, (VII). Tension compression sensor, (VIII). Thrust plate, (IX). Nut, (X). Shaft, (XI) Swirler

(a) Cross section of the test rig

(b) View along “A” direction (c) Shaft sealing (d) Test rig

Fig. 33: Test rig design

b (mm) n (rpm) |𝑄| (m3/s) s (mm) sb (mm) a (mm) t (mm)

110 0 to 2500 0 to 5.56 × 10−4 2 to 8 8 23 10

Table 7: Parameters of the experiments

The measurements of the axial thrust coefficient 𝐶𝐹 include two steps. The test rig set-up for each

step is drawn in Fig. 34. The first step, plotted in Fig. 34 (a), is to measure the results imposed by

the drive end of the motor when the shaft without the disk is rotating at different speeds of rotation

in the air. For the second step, the measurements of 𝐹𝑎 are performed at different Re, G, 𝐶𝐷 and

𝐶𝑎𝑚. All the measured results are modified by subtracting above values obtained in the first step

according to the speed of rotation. The shaft head in Fig. 34 (b) is considered as a part of the front

surface (the area of front surface is 𝛺 ∙ 𝑏2). The values of 𝐶𝐹𝑏 are obtained when 𝐶𝐷 = 0 and

both the cavities have the same axial gap width for different Re (under that condition 𝐶𝐹𝑓 = 𝐶𝐹𝑏)

according to the area difference between the front surface and the back surface.

Direction of 𝐹𝑎

“A” direction

Shaft connected

to an electric

motor Shaft sealing

𝑟𝑠𝑒𝑎𝑙 = 10 mm

Shaft

Shaft sealing

53

(a) Step 1 (b) Step 2

Fig. 34: Test rig set-up for each steps to measure 𝑭𝒂

To compute the torque precisely, the friction due to the shaft needs to be subtracted. Therefore,

the measurements without the disk and at different rotating speed are accomplished. The values of

𝐶𝑀 at the back surface are obtained when 𝐶𝐷 = 0 and the axial gaps of the both cavities are of

the same size for different Reynolds numbers Re.

The positions for the pressure measurement are presented in Table 8.

Type of measurement Location Number of positions X

min max

p Front chamber 12 0.454 0.955

Table 8: Parameters of the experiments conducted in the test rig

The inlet swirler system is designed in the horizontal pipe, depicted in Fig. 35. In the test rig, four

swirlers, fixed with screws, is angled every 15° from −90° to 90°. The component drawing of

the centrifugal inlet swirler is illustrated in Fig. 36.

Top view

Fig. 35: Inlet swirlers in the horizontal pipe (for centrifugal through-flow)

𝛺

Dis

k

Wat

er a

t 2

0

°C

Air

Shaft Shaft 𝛺

Wal

l

Wal

l

Wal

l

Wal

l

Screw to adjust

pre-swirl angle

54

Fig. 36: Drawing of the centrifugal inlet swirler (Left: front view; Right: side view)

The geometry of the radial guide vanes (for centripetal pre-swirl through-flow) and the flow in the

rotor-stator cavities are depicted in Fig. 37. Every single part in green color refers to the flow in

each guide vane passage. The guide vanes can be categorized into two groups: with small passages

and with large passages. The through-flow is depicted by the arrows. A jet flow, which becomes

stronger for the guide vanes with smaller passages, exists near each cavity inlet in the guide vane.

Guide vane Flow

No.1

(sm

all

pas

sages

)

Blade

Q

Screw

Unit: mm

Flow in one passage 24 Passages

55

No

.2 (

smal

l pas

sag

es)

No.3

(la

rge

pas

sages

)

No.

4(l

arge

pas

sages

)

No

. 5

(la

rge

pas

sages

)

Fig. 37: Geometry of the radial guide vanes and the flow in the rotor-stator cavities

Q

Q

Q

24 Passages

24 Passages

24 Passages

24 Passages

Flow in one passage

Flow in one passage

Flow in one passage

Flow in one passage

Q

56

The pre-swirl angles (angle between the vane and the tangential direction) of the radial guide vanes

in Fig. 37 are given in Table 9.

Guide vane No.1 No.2 No.3 No.4 No.5

Passage Small Small large large large

Pre-swirl angle 0° 60° 0° 52° 26°

Table 9: Pre-swirl angles of the radial guide vanes

Six disks are manufactured with different surface roughness for comparisons. In Table 10, the

values of measured 𝑅𝑧 and the amounts of 𝑘𝑠 are given. The values of 𝑘𝑠 for the disks ranges

from 0.4 𝜇m to 81.3 𝜇m, which covers the parameter range of a majority of turbomachines.

The values of 𝑅𝑧 on all other wet surfaces of the test rig are below 1.6 μm.

Disk No. 0 1 2 3 4 5

𝑅𝑧 (𝜇m) 1.0 9 29.6 69 153 211

𝑘𝑠 (𝜇m) 0.4 3.5 11.4 26.5 58.9 81.3

Table 10: Surface roughness of the disks

3.2 Uncertainty Analysis

In Table 11, the relative error (e𝑇) and the measuring range ((𝑀𝑟)) of the pressure, the torque and

the axial thrust transducers are listed. All the experimental results are an ensemble average of 1000

samples. The uncertainties of the measured results ∆𝑁 , which is estimated with the root sum

squared method with Eq.100, are the differences between the real values and the measured values.

The parameter 𝑁𝑇 is the uncertainty due to the transducers while the parameter 𝑁𝐷 is that due to

the data acquisition system. The input voltage signals are the following ranges: 0 to 10 V for the

pressure transducers and the torque transducer, −10 V to 10 V for the axial thrust transducers. The

absolute accuracy of the data acquisition system is 4.28 mV. The random noise and the zero order

uncertainty are neglected because they are considered to be very small. The distributions of the

results are considered as normal distributions and the normal distribution coefficient is selected as

1.96 (95% confidence level). The parameter 𝑛𝑇 represents the number of transducers used to

obtain one result together. To evaluate 𝐹𝑎 and M on the front surface, the total results, the results

on the back surface (obtained when 𝐶𝐷 = 0 and 𝑠 = 𝑠𝑏) and the results when the shaft is rotating

without a disk in the air are measured with the transducers. Hence, the measuring times to obtain

one result 𝑛𝑀 is 3 for the axial thrust, the frictional torque and Re (by measuring n). The

uncertainties of the measurements are also given in Table. 11.

57

∆𝑁 = √𝑁𝑇

2 + 𝑁𝐷2; 𝑁𝑇 =

√𝑛𝑇∙𝑛𝑀∙(𝑒𝑇∙𝑀𝑟)2

1.96∙√1000; 𝑁𝐷 =

√𝑛𝑇∙𝑛𝑀∙(𝑒𝐷∙𝑀𝑟)2

1.96∙√1000

Eq. 100

p (bar) 𝐹𝑎 (N) M (Nm) Re 𝐶𝐷

e𝑇 1% (FS) 0.5% (FS) 0.1% (FS) − −

𝑀𝑟 0 to 2.5 bar −100 to 100 N 0 to 10 Nm − −

∆𝑁 4.04× 10−4 2.43× 10−2 3.00× 10−4 9.01× 104 4.1

𝑛𝑇 1 3 1 1 1

𝑛𝑀 1 3 3 3 1

Table 11: Uncertainty analysis for the measurements (FS: full scale)

3.3 Experimental Validation

In the experiments, the values of M and 𝐹𝑎 when the shaft without the disk is rotating at different

speeds of rotation are subtracted. The parameters 𝑀𝑠 and 𝐹𝑎−𝑠 respectively are the measured

torque and the measured axial thrust without the disk. The values of 𝐶𝑀𝑠 and 𝐶𝐹𝑠 respectively

are plotted in Fig. 38 (a) and Fig. 38 (b) at different Re.

𝐶𝑀

𝑠=

2∙|

𝑀𝑠|

𝜌∙𝛺

2∙𝑏

5

𝐶𝐹

𝑠=

2∙|

𝐹 𝑎−

𝑠|

𝜌𝜋

𝛺2

𝑏4

(a) 𝐶𝑀𝑠 versus Re (b) 𝐶𝐹𝑠 versus Re

Fig. 38: Experimental results of 𝑪𝑴𝒔 and 𝑪𝑭𝒔 versus Re

In some experiments of 𝐶𝐹 and 𝐶𝑀 concerning pre-swirl, the shaft has to counter-rotate to get a

negative pre-swirl angle. It is therefore important to check whether the axial thrust is sensitive to

the direction of rotation. The ratio of 𝐶𝑀𝑠−𝑐/𝐶𝑀𝑠−𝑎𝑐 versus Re are shown in Fig. 39 (a). The

subscript “c” refers to a clockwise rotation of the disk, while the subscript “ac” represents an anti-

clockwise rotation of the disk. The values of both 𝐶𝑀𝑠−𝑐/𝐶𝑀𝑠−𝑎𝑐 and 𝐶𝐹𝑠−𝑐/𝐶𝐹𝑠−𝑎𝑐 range from

0.96 to 1.04, shown in Fig. 39. Hence, the experimental results for 𝐶𝑀 and 𝐶𝐹 remain almost

unaffected by the direction of rotation.

0

0.005

0.01

0.015

0.02

0.025

0.03

0 1 2 3 4

0

0.005

0.01

0.015

0.02

0.025

0.03

0 1 2 3 4

Re

(106)

Re

(106)

58

𝐶𝑀

𝑠−

𝑐

𝐶𝑀

𝑠−

𝑎𝑐

(a) 𝐶𝑀𝑠−𝑐/𝐶𝑀𝑠−𝑎𝑐

𝐶𝐹

𝑠−

𝑐

𝐶𝐹

𝑠−

𝑎𝑐

𝐶𝐹𝑠−𝑐/𝐶𝐹𝑠−𝑎𝑐

Fig. 39: Comparison of results when the shaft rotates in different direction

In Fig. 40, the measured pressure coefficients at 𝑅𝑒 = 1.36 × 106, 𝐶𝐷 = 0 and 𝐺 = 0.0454 are

compared with those in Kurokawa et al. [52] (𝐺 = 0.046). They are in good accordance with each

other both using smooth and plane disks. The results from the pressure transducers therefore are

considered to be reliable.

0.95

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

0 1 2 3 4

0.95

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

0 1 2 3 4

Re

(106)

Re

(106)

59

𝑥

Fig. 40: Comparison of 𝑪𝒑 along the radius for 𝑹𝒆 = 𝟏. 𝟑𝟔 × 𝟏𝟎𝟔

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Measurements G=0.0454

𝐶𝑝

Measurements in Kurokawa et al. [52]

G=0.046

60

4. Numerical Simulation Set-Up

In this thesis, the velocity profiles are predicted using the ANSYS CFX 14.0 code. The distributions

of K from numerical simulations (varying G, Re, 𝐶𝐷 and 𝐶𝑎𝑚) are compared with those calculated

based on the pressure measurements in Chapter 5. This can improve the reliability of the correlation

of K derived based on the pressure measurements. The profiles of tangential velocity from

numerical simulation are also analyzed to estimate the impact of 𝐶𝐷 on the distinguishing lines

between flow regime III and flow regime IV for various G and Re.

4.1 Turbulence Model

During the past 60 years, numerous numerical studies are devoted to the flow in a rotor-stator cavity.

The turbulence models turn out to be a key feature. Besides, these flows are very well suited to

study the effects of rotation on turbulence. Part of the selected turbulence models in the literature

are listed in Table 12.

Authors Through-flow Shroud Models Findings

Morse [64] Yes Yes Low Re 𝑘 − 휀 Under-predicted 𝐶𝑀

Chew and Vaughan [20] No Yes Mixing length

formulation Succeed to predict 𝐶𝑀

Iacovides and Theofanopoulus [42] No Yes 𝑘 − 휀 and

algebraic stress

models

No single model is satisfactory in all

cases

Iacovides and Toumpanakis [43] No Yes 𝑘 − 휀 and 𝑘 − 𝜔

Both fail to predict the size of the

laminar region in the rotor boundary

layer

Elena and Schiestel [34] Yes Yes RSM Succeed to predict the cavity flow

Launder et al. [60] No Yes Second moment

closure approach Embedded pressure reflection impacts

Randriamampianina et al. [72] No Yes RSM More accurate predictions

Haddadi and Poncet [39] Yes Yes

Low Re second-

order full stress

closure

Different flow regimes coexist

Owen [65] No Yes RANS Asymmetrical and unsteady flow

Wu and Squires [89] No No LES

Promoting sweeps and ejections

contribute to the shear stress

production

Anderson and Lygren [4][5] No Yes DNS Isotropic eddy viscosity models

produce reliable results

Anderson and Lygren [5] No No LES Idealized model of an unshrouded

rotor-stator system

Séverac et al. [86] No Yes LES Can not predict the Large scale

rotating structures

Craft et al. [21] No Yes RANS No large scale structures in the core

region

Poncet et al. [69] Yes Yes RSM Fail to predict the influence of G

Itoh et al. [45] No Yes Realizable 𝑘 − 휀 Over prediction of the core rotation

Poncet et al. [69] Yes Yes RSM Fail to predict the influence of G

Will [88] No Yes SST 𝑘 − 𝜔 Closest to the pressure measurements

Barabas et al. [11] No Yes SST 𝑘 − 𝜔 Closest to the pressure measurements

Table 12: Selections of turbulence model in the literature

61

Will [88] compares the simulation results of both the tangential velocity and the radial velocity

from the realizable 𝑘 − 휀 turbulence model and the SST 𝑘 − 𝜔 turbulence model. He finds that

the velocity distributions obtained with the SST 𝑘 − 𝜔 turbulence model are closest to the

velocity measurements in rotor-stator cavities from the literature.

Barabas et al. [11] and Hu et al. [16] show that the simulation results of pressure from the SST

𝑘 − 𝜔 turbulence model in combination with the scalable wall functions is in good agreement with

those from pressure measurements in rotor-stator cavities with air and water.

The study by Volkov [51] focuses on the moment coefficients of the disk in an enclosed rotor-stator

cavity both numerically and experimentally. The moment coefficient from the SST 𝑘 − 𝜔

turbulence model shows good agreement with that from torque measurements, compared in Table

13. Hence, in this thesis, the SST 𝑘 − 𝜔 turbulence model is selected.

Models M (Nm) 𝐶𝑀 ∙ 103

𝑘 − 휀 2.3026 3.5115

𝑘 − 휀 2.4138 3.7125

SST 𝑘 − 𝜔 2.2350 3.4376

RSM 2.3729 3.6496

Table 13: Moment coefficients from different turbulence models (K. N. Volkov [51])

4.2 Grid Generation

One of the most time-consuming procedures in the CFD simulation is the generation of meshes for

the computational domains. The convergence of the numerical simulation and the accuracy of the

results from numerical simulation are strongly dependent on the quality of the meshes. In this thesis,

structured meshes are generated for all the simulation models with ICEM 14.0 and Turbogrid 14.0.

The wall laws of the SST 𝑘 − 𝜔 turbulence model have restrictions on the values of 𝑦+ (the non-

dimensional wall distance). The values of 𝑦+, used to describe how fine a mesh is for a particular

flow pattern, are important for determining the proper size of the elements near the walls. In the

fluid dynamics, the law of the wall shows that according to different flow, the boundary layer can

be divided into three parts from the wall to the free-stream flow region, namely the viscous sublayer,

the buffer layer and the log-law region. If the distance from the buffer layer to the viscous sublayer

is 𝜍, then the log-law region will extend to about 100𝜍 away from the wall. In both the viscous

sublayer and the buffer layer, the viscous force plays a dominant role, while the inertial force can

be ignored. The viscous force is linearly related to the velocity gradient. The first node should be

placed in the log-law region. The non-dimensional wall distance 𝑦+ can be predicted with the

friction velocity 𝑢∗, written in Eq.101.

62

𝑦+ =

𝑢∗ ∙ 𝑦

𝜈

Eq. 101

Where y is the distance from the first node to the wall.

The values of 𝑦+ are controlled below 10 in the grid generation processes for all the simulation

models. To estimate the values of y, the friction velocity 𝑢∗ should be estimated with Eq. 102. The

parameter 𝜏𝑤 is the wall shear stress, defined as 𝜏𝑤 = 𝜇 (𝜕𝑢

𝜕𝑦)

𝑦=0. The velocity gradient, however,

is not available. Hence, a new parameter, namely the skin friction coefficient 𝐶𝑓, is introduced to

predict the wall shear stress, written in Eq. 103.

𝑢∗ = √𝜏𝑤

𝜌

Eq. 102

𝐶𝑓 =𝜏𝑤

12 ∙ 𝜌 ∙ 𝑈2

Eq. 103

Where U is the velocity of the free stream.

Von Kármán [48] determines Eq. 104 to predict the values of 𝐶𝑓.

1

𝐶𝑓

12

= 1.7 + 4.15 ∙ log10(𝑅𝑒𝜑 ∙ 𝐶𝑓)

Eq. 104

Schlichting and Gersten [73] deduce Eq. 105 to predict the values of 𝐶𝑓 . The parameter 𝐶𝑓

increases with increasing roughness. As a consequence, the friction velocity 𝑢∗ increases.

𝐶𝑓 = [2 ∙ log10((𝑅𝑒𝜑) − 0.65]−2.3 ; 𝑅𝑒𝜑 < 109 Eq. 105

Above two equations, however, fail to consider the roughness effect, which results in large errors

during the predictions of 𝐶𝑓 and 𝑦+ . Schlichting and Gersten [73] propose Eq. 106 for 𝐶𝑓

including the roughness effect. With the increase of roughness, 𝐶𝑓 increases. As a consequence,

the friction velocity 𝑢∗ will also increase. The parameter 𝑙 is the streamwise coordinate.

𝐶𝑓 = (2.87 + 1.58 ∙ 𝑙𝑜𝑔10(𝑙/𝑘𝑠))−2.5 Eq. 106

63

Combining Eq. 101, Eq. 102, Eq. 103 and Eq. 106, the values of y during the generation of the meshes

can be estimated with Eq. 107 for rough walls.

𝑦 =𝑦+ ∙ 𝜈 ∙ √𝜌

√12

∙ 𝜌 ∙ 𝑈2 ∙ (2.87 + 1.58 ∙ log10(𝑙/𝑘𝑠))−2.5

Eq. 107

The simulation models are generated with structured meshes with different element numbers. When

𝑅𝑧 = 211 𝜇𝑚 ( 𝑘𝑠 = 81.3 𝜇𝑚 ), 𝐶𝐷 = −5050 and 𝑅𝑒 = 1.9 × 106 , the variation of the

maximum value of 𝑦+ versus the total element numbers from the numerical simulations are given

in Table 14. The qualities of the meshes are given in Table 15.

Total element numbers (106) 0.18 0.4 0.47 0.796 0.97 1.4 1.75

Maximal 𝑦+ 89 50 41 26 21 13.4 11.2

Table 14: Grid number and maximum values of 𝒚+

Criterion Determinant 3 × 3 × 3 Aspect ratio Angle

Values ≥ 0.89 ≤ 41 ≥ 27°

Table 15: Qualities of the meshes

4.3 Simulation Set-Up

To predict the cavity flow, numerical simulations are carried out using the ANSYS CFX 14.0 code.

There are 24 channels in the guide vane with small or large passages (see Fig. 37). The domain for

numerical simulation for 𝐺 = 0.072 is depicted with yellow color in Fig. 41. Considering the

axial symmetry of the problem, a segment of 15 degrees (with one radial guide vane passage) of

the whole domain is modeled and a rotational periodic boundary condition is applied. The

discretization scheme is set as second order upwind. The simulation type is set as steady state. The

convergence criteria are set as 10−5 in maximum type. The no-slip wall condition in relative

frame is set for all the walls.

For non-pre-swirl through-flow, the boundary conditions at the inlet and at the outlet respectively

are pressure inlet and mass flow outlet. The values of pressure at the inlet are set according to the

pressure sensor at the pump outlet.

For centripetal pre-swirl through-flow, the boundary condition at the inlet is velocity inlet (with

both the radial and the tangential velocity components) and at the outlet is pressure outlet (1 bar

reference pressure).

64

Centripetal through-flow

Centrifugal through-flow

Fig. 41: Domain for numerical simulation (𝑮 = 𝟎. 𝟎𝟕𝟐)

For centrifugal pre-swirl through-flow, the geometry of the swirlers is considered when generating

the simulation models and the meshes. Additional fluid domains with different pre-swirl angles are

connected to the inlet (at the horizontal pipe), depicted in Fig. 42. The length of the domain is five

times the inner diameter of the horizontal pipe.

𝛽 = 15° 𝛽 = 45°

Fig. 42: Additional fluid domain for centrifugal pre-swirl through-flow

Mesh sensitivity analyses are accomplished for all the simulation models. The pressure and the

tangential velocity at 𝑥 = 0.955 and 휁 = 0.5 for 𝐺 = 0.072 , 𝐶𝐷 = −5050 and 𝑅𝑒 = 1.9 ×

106 are compared in Fig. 43. The parameter Δ is a non-dimensional parameter which shows the

relative change rate of the results for different total element numbers, defined in Eq. 108. The

parameter 𝐶𝑥 represents the value of a certain variable when the total element number is x. The

parameter 𝐶1.75 is the value of the variable when the total element number is 1.75 million. When

the total element numbers exceed 1.4 million, the results are independent of the total element

numbers. Hence, the total element numbers should exceed 1.4 million to improve the accuracy of

Disk

Domain

Casing

Shaft

Disk

Domain

Casing

Shaft

Q

Q

Q

𝛽

65

the numerical simulations.

∆= |

𝐶1.75 − 𝐶𝑥

𝐶1.75|

Eq. 108

∆

Fig. 43: Mesh independence analysis

4.4 Validation for Numerical Simulation

In Fig. 44, the results from numerical simulations for 𝐶𝑝 along the radius are compared with those

from pressure measurements by Poncet et al. [68] at 𝑅𝑒 = 4.15 × 106 and 𝐺 = 0.036 in a

rotor-stator cavity with centrifugal through-flow. The reference pressures are taken at 𝑥 = 0.92

instead of 𝑥 = 1 to compare the results with those from the measurements by Poncet at al. [68].

The simulation results of the pressure are in very good agreement with the measurements in Poncet

at al. [68].

0

0.02

0.04

0.06

0.08

0.1

0.12

0 0.5 1 1.5 2

Total grid number (106)

Pressure

Tangential velocity

66

X

Colors: Black: 𝐶𝐷 = 0, Red: 𝐶𝐷 = 2579, Green: 𝐶𝐷 = 5159; Solid lines: RSM from Poncet et al. [68]; Dashed

lines: Simulation results (SST 𝑘 − 𝜔); Points: Measurements by Poncet et al. [68].

Fig. 44: Comparison of radial pressure distribution for 𝑹𝒆 = 𝟒. 𝟏𝟓 × 𝟏𝟎𝟔 and 𝑮 = 𝟎. 𝟎𝟑𝟔

In Fig. 45, the results from numerical simulation for 𝐶𝑝 along the radius are compared with those

from the pressure measurements by Kurokawa et al. [52]. The values of 𝐶𝑞 are negative for

centripetal through-flow. The results of the simulations and the measurements are in a very good

accordance.

X

Dashed lines: Simulation results (SST 𝑘 − 𝜔); Points: Measurements by Kurokawa et al. [52].

Fig. 45: Comparisons of radial pressure distribution for 𝑹𝒆 = 𝟏. 𝟑𝟔 × 𝟏𝟎𝟔 and 𝑮 = 0.0495

without pre-swirl

The comparison of the simulation results with the experimental results from the literature shows

that the simulation set-up should be reasonable.

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.02 0.04 0.06 0.08 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3

𝐶𝑝

𝐶𝑝

𝐶𝑞 = 0

𝐶𝑞 = −0.00273

𝐶𝑞 = −0.00829

𝑄

𝛺

𝑄 𝛺

67

5. Results and Discussion

According to the results from the literature, the most influential parameters in a rotor-stator cavity

are 𝐶𝐷, G, Re, 𝑘𝑠 and 𝐶𝑎𝑚. This section is divided into two parts on the basis of through-flow

direction. In each part, first the through-flow without pre-swirl is considered in a rotor-stator cavity

varying the values of G, Re and 𝐶𝐷 using a disk with a smooth surface (𝑘𝑠 = 0.4 𝜇m). Afterwards,

the influence of the disk surface roughness is taken into consideration. In the third step, the impact

of centripetal or centrifugal pre-swirl through-flow on K, 𝐶𝑝 , 𝐶𝐹 and 𝐶𝑀 is investigated.

According to the results from both the numerical simulations and the measurements, the flow in a

rotor-stator cavity is analyzed with both centripetal (Hu et al. [16][17]) and centrifugal (Hu et al.

[18]) through-flow. The research procedure of this chapter is depicted in Fig. 46.

Fig. 46: Research procedure

5.1 Rotor-Stator Cavity with Centripetal Through-Flow

5.1.1 Simulation Results of Velocity Distributions

The inlet boundary condition greatly affects the velocity profiles. The non-dimensional velocities

in the front chamber therefore are discussed in this part. There is a small jet flow through each

channel in the radial guide vane (see Fig. 37). All presented velocities are made non-dimensional

with 𝛺 ∙ 𝑏 . The values of 𝑉𝑧 (non-dimensional axial velocity) are positive when they have a

direction from the disk to the wall. The velocity profiles at three radial positions for 𝑅𝑒 = 1.9 ×

106 and 𝐺 = 0.072 (small passages in the guide vane) are shown in Fig. 47. The non-

dimensional radial velocities 𝑉𝑟 are not exactly zero in the central cores, shown in Fig. 47 (a to c).

The distribution of 𝑉𝜑 (non-dimensional tangential velocity) shows that there are central cores at

all the investigated radial positions with nearly zero velocity gradient, shown in Fig. 47 (d to f). At

𝑥 = 0.955 and 𝑥 = 0.79 , the values of 𝑉𝜑 are smaller at 휁 = 0.5 when |𝐶𝐷| increases from

Par

amet

ers:

K ,

𝐶𝑝

, 𝐶

𝐹,

𝐶𝑀

Introduce through-flow (Smooth disk)

Introduce disk roughness

Introduce pre-swirl (Smooth disk)

Introduce through-flow (Smooth disk)

Introduce disk roughness

Introduce pre-swirl (Smooth disk)

5.1: Centripetal

5.2: Centrifugal

Step 1

Step 1

Step 2

Step 2

Step 3

Step 3

68

1262 to 3787 and 5050, depicted in Fig. 47 (d, e). The profiles contrast with the predictable pattern

according to the measurements from Poncet et al. [68] and Debuchy et al. [30]. Presumably, the

increase of |𝐶𝐷| should lead to in an increase of K for centripetal through-flow. As a consequence,

the tangential velocity should increase rather than decrease. Probably this disagreement can be

attributed to the jet flow through the guide vane channel at the inlet, which is stronger for large

|𝐶𝐷|. At 𝑥 = 0.955, the values of |𝑉𝑧| become smaller for larger |𝐶𝐷| in general, shown in Fig.

47 (g). The direction of 𝑉𝑧 is from the disk towards the wall for |𝐶𝐷| = 0 and |𝐶𝐷| = 1262 at

𝑥 = 0.955, while it is from the wall towards the disk at |𝐶𝐷| = 3787 and |𝐶𝐷| = 5050, but with

a very small amount. From the axial velocity profiles, the axial circulations of the fluid exist in the

front chamber.

𝑥 = 0.955 𝑥 = 0.79 𝑥 = 0.57

𝑉𝑟 (a) 𝑉𝑟 (b) 𝑉𝑟 (c)

𝑉𝜑 (d) 𝑉𝜑 (e) 𝑉𝜑 (f)

𝑉𝑧 (g) 𝑉𝑧 (h) 𝑉𝑧 (i)

Colors: Black: |𝐶𝐷| = 0; Blue: |𝐶𝐷| = 1262; Green: |𝐶𝐷| = 3787; Red: |𝐶𝐷| = 5050.

Fig. 47: Velocity profiles for 𝑹𝒆 = 𝟏. 𝟗 × 𝟏𝟎𝟔 and 𝑮 = 𝟎. 𝟎𝟕𝟐

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 0.5 1-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 0.5 1-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 0.5 1

0

0.2

0.4

0.6

0.8

1

0 0.5 1

0

0.2

0.4

0.6

0.8

1

0 0.5 10

0.2

0.4

0.6

0.8

1

0 0.5 1

-0.03

-0.01

0.01

0.03

0 0.5 1

-0.03

-0.01

0.01

0.03

0 0.5 1

-0.03

-0.01

0.01

0.03

0 0.5 1

휁 휁 휁

휁 휁 휁

Disk Wall

휁 휁 휁

69

The velocity profiles at three radial coordinates for 𝑅𝑒 = 1.9 × 106 and 𝐺 = 0.018 (small gap)

are depicted in Fig. 48. The values of 𝑉𝑟 vary along 휁, shown in Fig. 48 (a to c). Near the wall,

they mainly increase with increasing |𝐶𝐷| and become negative all along 휁 for high through-flow

rate. The tangential velocity 𝑉𝜑 decreases constantly along 휁, shown in Fig. 48 (d to f), which is

the characteristic of flow regime III. At 𝑥 = 0.955 and 𝑥 = 0.79 , the values of tangential

velocity are much smaller for large |𝐶𝐷|, shown in Fig. 48 (d). The reason is that the impact of the

jet flow at the inlet becomes stronger for smaller G. The profiles of 𝑉𝑧 are quite different at 𝑥 =

0.955 in Fig. 48 (g), compared with those in Fig. 47 (g). Most of the values of |𝑉𝑧| are smaller

than those from Fig. 48 (h, i), which indicates the weaker axial circulations of the fluid for smaller

G.

𝑥 = 0.955 𝑥 = 0.79 𝑥 = 0.57





Fig. 48: Velocity profiles for 𝑹𝒆 = 𝟏. 𝟗 × 𝟏𝟎𝟔 and 𝑮 = 𝟎. 𝟎𝟏𝟖

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 0.5 1-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 0.5 1

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 0.5 1

0

0.2

0.4

0.6

0.8

1

0 0.5 1

0

0.2

0.4

0.6

0.8

1

0 0.5 10

0.2

0.4

0.6

0.8

1

0 0.5 1

-0.05

-0.03

-0.01

0.01

0.03

0.05

0 0.5 1-0.05

-0.03

-0.01

0.01

0.03

0.05

0 0.5 1

-0.05

-0.03

-0.01

0.01

0.03

0.05

0 0.5 1

휁 휁 휁

Disk Wall

휁 휁 휁

휁 휁 휁

70


5.1.2.1 Impact of Through-Flow with a Smooth Disk

To predict the axial thrust, the distribution of K along the radius should be estimated. In a majority

of the rotor-stator cavities in radial pumps or turbines, measuring the velocities is quite expensive

and complicated. By derivation of equations, Will et al. [85][86][87][88] deduce Eq. 71 to associate

K with the measured pressure. The average values of K between each two adjacent pressure tubes

(12 tubes from 𝑟 = 0.05 m (𝑥 = 0.455 ) to 𝑟 = 0.105 m (𝑥 = 0.954 )), noted as 𝐾𝑐~c+1̅̅ ̅̅ ̅̅ ̅̅ ̅ , are

calculated according to the pressure measurements with Eq. 109. Since the radial distances between

the adjacent pressure tubes are small, the application of the average values of K results a small error

only. The values of 𝐶𝑞𝑟 for the experimental results are calculated with 𝑟 =𝑟𝑐+𝑟𝑐+1

2.

𝐾𝑐~𝑐+1̅̅ ̅̅ ̅̅ ̅̅ ̅ = √

𝑝(𝑟𝑐) − 𝑝(𝑟𝑐+1) −𝜌 ∙ 𝑄2

8 ∙ 𝜋2 ∙ 𝑠2 (1

𝑟𝑐+12 −

1𝑟𝑐

2)

12 ∙ 𝜌 ∙ 𝛺2 ∙ (𝑟𝑐

2 − 𝑟𝑐+12)

Eq. 109

Poncet et al. [68] and Debuchy et al. [30] neglect the effect of G on K according to the LDA

measurements. In most of the radial pumps or turbines, the wall is not parallel to the disk, resulting

in a variable G. A simplified correlation with good accuracy can help to easily account for varying

G. Based on the pressure measurements of the author, Eq. 110 is correlated to predict the values of

K when G ranges from 0.018 to 0.072 (Batchelor or Couette type flow) in a rotor-stator cavity with

centripetal through-flow. The results from Eq. 110 are in good accordance with the simulation results

and those from Eq. 51 and Eq. 52, depicted in Fig. 49.

𝐾 = 0.97 ∙ [

−8.5∙𝐶𝑞𝑟+0.5

𝑒(−1.45𝐶𝑞𝑟)]

5

4 Eq. 110

Where −0.5 ≤ 𝐶𝑞𝑟 ≤ 0.03.

71

Hollow points: Simulation; Solid points: By pressure measurements (Eq. 109); Dashed lines: Equations.

Colors (points): Black: 𝐺 = 0.018; Green: 𝐺 = 0.036; Yellow: 𝐺 = 0.054; Red: 𝐺 = 0.072.

Colors (Dashed lines): Red: Eq. 51; Green: Eq. 52; Purple: Eq. 110.

Fig. 49: K (𝑪𝒒𝒓) curves for centripetal through-flow

5.1.2.2 Impact of Surface Roughness of the Disk

The distributions of K along a rough disk are estimated with Eq. 109 in a rotor-stator cavity with

centripetal through-flow. Based on the pressure measurements, Eq. 110 is extended by introducing

𝑘𝑠, written in Eq. 111.

𝐾 = 0.97 ∙ 𝑒

(600∙𝑘𝑠∙𝑟

𝑏2 )∙ [

−8.5 ∙ 𝐶𝑞𝑟 + 0.5

𝑒(−1.45𝐶𝑞𝑟)]

54

Eq. 111

Where 0.018 ≤ 𝐺 ≤ 0.072;

𝑅𝑒 ≤ 3.17 × 106;

−5050 ≤ 𝐶𝐷 ≤ 0 and 𝑘𝑠 ≤ 58.9 𝜇m.

0

0.5

1

1.5

2

2.5

3

-0.5 -0.4 -0.3 -0.2 -0.1 0

𝐶𝑞𝑟

𝐾

72

Most of the experimental results of K by Kurokawa et al. [54] are in good agreement with those

from Eq. 111, depicted in Fig. 50. Relatively large errors only occur when 𝑘𝑠 = 56 𝜇m. Near the

outer radius of the disk (where the values of |𝐶𝑞𝑟| are of small values), K increases faster for larger

𝑘𝑠 , which is in consistent with the trend of the measurements by Kurokawa et al. [54] in an

enclosed rotor-stator cavity.

Solid points: By pressure measurements (Eq. 109); Dashed lines: Eq. 111.

Colors : Blue: 𝑘𝑠 = 0.6 𝜇m; Red: 𝑘𝑠 = 22 𝜇m; Green: 𝑘𝑠 = 24 𝜇m; Red: 𝑘𝑠 = 56 𝜇m.

Fig. 50: K (x) curves for 𝑮 = 𝟎. 𝟎𝟑𝟏, 𝑹𝒆 =3.1× 𝟏𝟎𝟔 and 𝑪𝑫 = 𝟎 by Kurokawa et al. [54]

The impact of the surface roughness is analyzed according to the results of 𝐾/𝐾𝑘𝑠=0.4 𝜇m. Since

the results of K are almost not affected by changing G, only the results when 𝐺 = 0.072 are

compared, shown in Fig. 51. The values of K increase with increasing 𝑘𝑠 and |𝐶𝑞𝑟|. The results

indicate that the impact of surface roughness can be ignored for the two hydraulic smooth disks

(𝐾𝑘𝑠=3.5 𝜇m ≈ 𝐾𝑘𝑠=0.4 𝜇m with given 𝐶𝑞𝑟). For a disk in the transition zone (11.4 𝜇m ≤ 𝑘𝑠 ≤

58.9 𝜇m), the results of K can be predicted using Eq. 111 with good accuracy.

0.4

0.5

0.6

0.7

0.8

0.9

1

0.4 0.45 0.5 0.55 0.6

𝐾

𝑥

73

Hollow points: Simulation; Solid points: By pressure measurements (Eq. 109); Dashed lines: Eq. 111.

Colors : Black: 𝑘𝑠 = 3.5 𝜇m; Green: 𝑘𝑠 = 11.4 𝜇m; Yellow: 𝑘𝑠 = 26.5 𝜇m; Red: 𝑘𝑠 = 58.9 𝜇m.

Fig. 51: Impact of 𝒌𝒔 on K when G=0.072

5.1.2.3 Impact of Centripetal Pre-Swirl Through-Flow with a Smooth Disk

The inlet boundary conditions for the small and the large guide vane passages are depicted in Fig.

52. The simulation results of 𝐾𝑝 are compared with those by pressure measurements with Eq. 109

for 𝐺 = 0.072 , 𝑅𝑒 = 1.9 × 106 , 𝐶𝐷 = −5050 , depicted in Fig. 53. Based on the results, the

boundary conditions at the centripetal inlet (see Fig. 37) play an important role in the distributions

of 𝐾𝑝. The shaft rotates in the opposite direction to get negative values of pre-swirl angle 𝛽 (flow

counter-rotates to the shaft when entering the cavity). For small guide vane passages where the jet

flow occurs, the values of 𝐾𝑝 are much smaller compared with those with large guide vane

passages. The jet flow is weakening the rotation of the fluid. Currently, it is quite hard to quantify

the impact of 𝐶𝑎𝑚 on 𝐾𝑝, which deserves further investigations.

Small guide vane passages Large guide vane passages

Fig. 52: Inlet boundary conditions with centripetal through-flow

1

1.05

1.1

1.15

1.2

1.25

-0.5 -0.4 -0.3 -0.2 -0.1 0

𝐶𝑞𝑟

𝐾/𝐾

𝑘𝑠=

0.4

𝜇

m

Disk Disk + −

𝛺 − +

𝛺

74

x Small guide vane passages x Large guide vane passages

Solid points: By pressure measurements (Eq. 109); Solid lines: Numerical simulation.

Colors: Gray: 𝐶𝑎𝑚 = −0.0078 (𝛽 = 60°); Red: 𝐶𝑎𝑚 = −0.0075 (𝛽 = 52°); Blue: 𝐶𝑎𝑚 =

−0.0069 (𝛽 = 26°); Green: 𝐶𝑎𝑚 = 0 (without pre-swirl); Purple: 𝐶𝑎𝑚 = 0.0083 (𝛽 = −26°);

Orange: 𝐶𝑎𝑚 = 0.0089 (𝛽 = −52°); Dark blue: 𝐶𝑎𝑚 = 0.0093 (𝛽 = −60°).

Fig. 53: 𝑲𝒑 (𝒙) curves for 𝑮 = 𝟎. 𝟎𝟕𝟐, 𝑪𝑫 = −𝟓𝟎𝟓𝟎 and 𝑹𝒆 = 𝟏. 𝟗 × 𝟏𝟎𝟔

5.1.3 Pressure Distribution


The pressure measurements are performed to analyze the impact of centripetal through-flow on 𝐶𝐹

with a smooth disk (𝑘𝑠 = 0.4 𝜇m). Based on Eq. 71, the pressure along the radius can be calculated

with Eq. 112 based on the values of K from Eq. 110. K is a variable along the radius of the disk. A

simplification is made as follows: K is a fixed value every 1 mm in the radial direction. The

parameter 𝑝𝑏 represents the pressure at 𝑥 = 1. Due to the construction of the geometry, however,

there is no pressure tube at 𝑥 = 1 . The values of 𝑝𝑏 are calculated combining the measured

pressure at 𝑥 = 0.955 (𝑟 = 0.105 m) with Eq. 112.

𝑝(𝑟) = 𝑝𝑏 + ∫ 𝜌 ∙ 𝐾2 ∙ 𝛺2 ∙ 𝑟𝑑𝑟

𝑟

𝑏

+𝜌 ∙ 𝑄2

8 ∙ 𝜋2 ∙ 𝑠2(

1

𝑏2−

1

𝑟2)

Eq. 112

Where ∫ 𝜌 ∙ 𝐾2 ∙ 𝛺2 ∙ 𝑟𝑑𝑟𝑟

𝑏≈

𝜌

2∙ 𝛺2 ∙ ∑ 𝐾𝑟𝑖+1

2 ∙ (𝑟𝑖2 − 𝑟𝑖+1

2)𝑐−1

0 ;

𝑟 = 𝑏 − 0.001 ∙ 𝑐 (m);

𝑟𝑖 − 𝑟𝑖+1 = −0.001 (m).

Since the pressure drops towards the shaft, the values of the pressure coefficient 𝐶𝑝 are positive

values. In Fig. 54, 𝐶𝑝 is plotted versus x for three values of Re and G (with small passages in the

guide vane). The pressure measurements show that the values of 𝐶𝑝 increase with increasing |𝐶𝐷|.

At a given |𝐶𝐷|, 𝐶𝑝 decrease with the increase of Re and G. When 𝑅𝑒 = 1.9 × 106 and 𝑅𝑒 =

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.2 0 0.2 0.4 0.6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.2 0 0.2 0.4 0.6 0.8 1𝐾𝑝

𝑥ℎ 𝑥ℎ

𝐾𝑝

75

2.79 × 106 , the uncertainties of the 𝐶𝑝 are 2.7 × 10−4 and 1.3 × 10−4 , respectively. This is

very small compared with the measured results. Hence, they are not plotted in Fig. 54 (d~i).

G=0.018 G=0.036 G=0.072

Re=

0.76

×1

06

x (a) x (b) x (c)

Re=

1.9×

10

6

x (d) x (e) x (f)

Re=

2.79

×1

06

x (g) x (h) x (i)

Solid points: Pressure measurements; Solid lines: Numerical simulation.


Fig. 54: Influence of 𝑪𝑫 on 𝑪𝒑 in dependence of Re and G (𝒌𝒔 = 𝟎. 𝟒 𝝁𝐦)


When |𝐶𝐷|, Re and G are given, the amounts of K increase with increasing 𝑘𝑠 at a given radial

position. This is in accordance with the velocity measurements by Kurokawa et al. [54]. Hence,

the pressure drops faster towards the shaft with rougher disks. The values of 𝐶𝑝 when 𝐺 = 0.072

from Eq. 112 (K is calculated with Eq. 111) are in good accordance with those by pressure

measurements with large passages in the guide vane, presented in Fig. 55. All the results indicate

that Eq. 112 can predict the effect of 𝑘𝑠 on K with good accuracy when 𝑘𝑠 ≤ 58.9 𝜇m.

0.4

0.6

0.8

1

0 0.05 0.1 0.150.4

0.6

0.8

1

0 0.05 0.1 0.150.4

0.6

0.8

1

0 0.05 0.1 0.15

0.4

0.6

0.8

1

0 0.05 0.1 0.150.4

0.6

0.8

1

0 0.05 0.1 0.15

0.4

0.6

0.8

1

0 0.05 0.1 0.15

0.4

0.6

0.8

1

0 0.05 0.1 0.15

0.4

0.6

0.8

1

0 0.05 0.1 0.150.4

0.6

0.8

1

0 0.05 0.1 0.15

𝐶𝑝

|𝐶𝐷|

𝐶𝑝 𝐶𝑝 𝐶𝑝

𝐶𝑝 𝐶𝑝


|𝐶𝐷| |𝐶𝐷|

|𝐶𝐷| |𝐶𝐷| |𝐶𝐷|


76

|𝐶𝐷| =0 |𝐶𝐷| =2525 |𝐶𝐷| =5050 G

=0

.072，

𝑘

𝑠=

0.4

𝜇m

x x x

x x x

G=

0.0

72，

𝑘

𝑠=

3.5

𝜇m

G=

0.0

72，

𝑘

𝑠=

11.4

𝜇m

x x x

G=

0.0

72，

𝑘

𝑠=

26.5

𝜇m

x x x

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.10.5

0.6

0.7

0.8

0.9

1

0 0.1 0.20.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.10.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.10.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.20.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3

𝐶𝑝

Re

𝐶𝑝

𝐶𝑝 𝐶𝑝



𝐶𝑝

𝐶𝑝

Re

Re

Re

Re Re

Re Re Re

Re Re Re

77

G=

0.0

72，

𝑘

𝑠=

58

.9 𝜇

m x x x

Solid points: Pressure measurements; Dashed lines: Numerical simulation.

Colors: Red: 𝑅𝑒 = 0.76 × 106; Purple: 𝑅𝑒 = 1.14 × 106; Green: 1.90 × 106; Yellow: 𝑅𝑒 = 3.17 × 106.

Fig. 55: 𝑪𝒑 (x) curves along the radius of the disks

5.1.3.3 Impact of Pre-Swirl on the Pressure Distribution with a Smooth Disk

In Fig. 56, the impacts of pre-swirl on the pressure distribution for 𝐺 = 0.072, 𝐶𝐷 = −5050 and

𝑅𝑒 = 1.9 × 106 with small or large guide vane passages are presented. With increasing 𝛽

(−𝐶𝑎𝑚), the centripetal through-flow brings larger tangential velocity component into the cavity

(the values of 𝐾 become larger). Hence, greater values of 𝛽 result in a larger pressure drop

towards the shaft. The results of 𝐶𝑝 from numerical simulation are in relative good agreement

with those from pressure measurements.

x Small guide vane passage x Large guide vane passage

Solid points: Pressure measurements; Solid lines: Numerical simulation.

Colors: Gray: 𝐶𝑎𝑚 = −0.0078 (𝛽 = 60°); Red: 𝐶𝑎𝑚 = −0.0075 (𝛽 = 52°); Blue: 𝐶𝑎𝑚 = −0.0069 (𝛽 =

26°); Green: 𝐶𝑎𝑚 = 0 (without pre-swirl); Purple: 𝐶𝑎𝑚 = 0.0083 (𝛽 = −26°); Orange: 𝐶𝑎𝑚 =

0.0089 (𝛽 = −52°); Dark blue: 𝐶𝑎𝑚 = 0.0093 (𝛽 = −60°).

Fig. 56: 𝑪𝒑 (x) curves for 𝑮 = 𝟎. 𝟎𝟕𝟐, 𝑪𝑫 = −𝟓𝟎𝟓𝟎 and 𝑹𝒆 = 𝟏. 𝟗 × 𝟏𝟎𝟔 with centripetal

through-flow

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.20.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.12 0.24 0.36

𝐶𝑝 𝐶𝑝

𝐶𝑝

Disk

𝐶𝑝

Disk

𝐶𝑝

Re

Re Re

78

5.1.4 Thrust Coefficient


According to the axial thrust measurements, a correlation for the thrust coefficient 𝐶𝐹 is

determined for a smooth disk (𝑘𝑠 = 0.4 𝜇m), given in Eq. 113. The through-flow will result in a

significant decrease of the pressure towards the shaft, compared with the case for an enclosed rotor-

stator cavity. Kurokawa et al. [53] mention that 𝐶𝐹 is not affected so much by the through-flow

rate without inlet angular momentum. However, according to their experimental results, shown in

Fig. 57, the values of 𝐶𝐹 decrease by around 30% with the increase of 𝐶𝐷. The conclusion is

therefore considered not precise. Although the results from Eq. 113 are in line with the trend of the

experimental results in Kurokawa et al. [53], there is a large gap between them at a given set of

𝐶𝐷 and 𝐺. Possibly the differences can be attributed to the uncertainty of the surface roughness of

the disks.

𝐶𝐹 = [6.6 ∙ 10−3 ∙ 𝑙𝑛 (𝑅𝑒) − 0.113] ∙ 𝑒(−1.2∙10−4∙𝐶𝐷) ∙ [0.122 ∙ 𝑙𝑛(𝐺) − 0.67] Eq. 113

𝐶𝐹

Solid points: Pressure measurements in Kurokawa et al. [53]; Solid lines: Eq. 113; Dashed lines: Theoretical analyses in

Kurokawa et al. [53].

Colors: Blue: 𝐺 = 0.024; Yellow: 𝐺 = 0.048; Red: 𝐺 = 0.097.

Fig. 57: 𝑪𝑭 (𝑪𝑫) curves for 𝑹𝒆 = 𝟏. 𝟑𝟐 × 𝟏𝟎𝟔

The values of 𝐶𝐹 increase with increasing |𝐶𝐷|, while decrease with increasing Re, plotted in Fig.

58. For increasing |𝐶𝐷| , the values of 𝐶𝐹 increase, which can be attributed to the drop of the

pressure. The values of 𝐶𝐹 are smaller for higher values of G and Re.

0

0.01

0.02

0.03

0.04

0.05

-3000 -2000 -1000 0

𝐶𝐷

79

𝐶𝐹 𝐺 = 0.018 𝐶𝐹 𝐺 = 0.036

𝐶𝐹 𝐺 = 0.054 𝐶𝐹 𝐺 = 0.072

Solid points: Axial thrust measurements; Dashed lines: Eq. 113.

Colors: Black: |𝐶𝐷| = 0; Blue: |𝐶𝐷| = 1262; Yellow: |𝐶𝐷| = 2525; Green: |𝐶𝐷| = 3787; Red: |𝐶𝐷| = 5050.

Fig. 58: 𝑪𝑭 (𝑹𝒆) curves in dependence of 𝑪𝑫 and G


Based on the experimental results, an empirical correlation for the thrust coefficient is determined

to predict the thrust coefficient for rough disks, given in Eq. 114.

𝐶𝐹 = [6.6 ∙ 10−3 ∙ ln(𝑅𝑒) − 0.113] ∙ 𝑒(−1.2∙10−4∙𝐶𝐷) ∙ [0.122 ∙ ln(𝐺) − 0.67] ∙ 𝑒(880∙

𝑘𝑠𝑏

) Eq. 114

Where 0.018< 𝐺 ≤0.072;

𝑅𝑒 ≤ 3.17 × 106;

−5050 ≤ 𝐶𝐷 ≤ 0 and 𝑘𝑠 ≤ 58.9 𝜇m.

The values of 𝐶𝐹 can also be directly calculated according to the pressure distribution with Eq. 115.

𝐶𝐹 =

∫ 2𝜋 ∙ (𝑝𝑏 − 𝑝1) ∙ 𝑟𝑑𝑟𝑟=𝑏

𝑟=𝑟𝑖1

𝜌 ∙ 𝜋 ∙ 𝛺2 ∙ 𝑏4

Eq. 115

0.01

0.03

0.05

0.07

0.3 0.8 1.3 1.8 2.3 2.8 3.30.01

0.03

0.05

0.07

0.3 0.8 1.3 1.8 2.3 2.8 3.3

0.01

0.03

0.05

0.07

0.3 0.8 1.3 1.8 2.3 2.8 3.3

0.01

0.03

0.05

0.07

0.3 0.8 1.3 1.8 2.3 2.8 3.3

Re (106)

Re (106) Re (106)

Re (106)



80

The results of 𝐶𝐹 from Eq. 114 are in good agreement with those from Eq. 115 and thrust

measurements for 𝑘𝑠 ≤ 58.9 𝜇m, depicted in Fig. 59. The impact of 𝑘𝑠 on 𝐶𝐹 is strong. With

the increase of |𝐶𝐷|, the values of 𝐶𝐹 increase, which can be attributed to the drop of the pressure.

The values of 𝐶𝐹 increase with increasing 𝑘𝑠 at a given Re. The results also indicate that Eq. 111

is capable to describe the impact of 𝑘𝑠 on K in a rotor-stator cavity with centripetal through-flow

when 𝑘𝑠 ≤ 58.9 𝜇m.

𝐺 = 0.018 𝐺 = 0.036 𝐺 = 0.072

𝑘𝑠

=0

.4

𝜇m

𝐶𝐹 𝐶𝐹 𝐶𝐹


𝑘𝑠

=3

.5

𝜇m


𝑘𝑠

=1

1.4

𝜇

m

0

0.02

0.04

0.06

0.08

0.1

0.3 1.3 2.3 3.3

0

0.02

0.04

0.06

0.08

0.1

0.3 1.3 2.3 3.3

百万

0

0.02

0.04

0.06

0.08

0.1

0.3 1.3 2.3 3.3

0

0.02

0.04

0.06

0.08

0.1

0.3 1.3 2.3 3.3

0

0.02

0.04

0.06

0.08

0.1

0.3 1.3 2.3 3.3

百万

0

0.02

0.04

0.06

0.08

0.1

0.3 1.3 2.3 3.3

0

0.02

0.04

0.06

0.08

0.1

0.3 1.3 2.3 3.3

0

0.02

0.04

0.06

0.08

0.1

0.3 1.3 2.3 3.30

0.02

0.04

0.06

0.08

0.1

0.3 1.3 2.3 3.3

|𝐶𝐷|

Re

(106)

Re

(106)

Re

(106)

Re

(106)

Re

(106)



|𝐶𝐷|

Re

(106)

Re

(106)

Re

(106)

Re

(106)


81


𝑘𝑠

=2

6.5

𝜇m


𝑘𝑠

=5

8.9

𝜇

m

Solid points: Axial thrust measurements; Hollow points: Eq. 115; Dashed lines: Eq. 114.


Fig. 59: 𝑪𝑭 (𝑹𝒆) curves in dependence of 𝑪𝑫, G and 𝒌𝒔

5.1.4.3 Impact of Pre-Swirl on the Thrust Coefficient with a Smooth Disk

Kurokawa et al. [52] state that when the through-flow contains angular momentum, 𝐶𝐹 increases

with the increase of |𝐶𝑎𝑚| . 𝐶𝐹 shows little change so long as 𝐶𝑎𝑚 is a constant, even if 𝐶𝑞

varies. In Fig. 60, the impact of the pre-swirl on 𝐶𝐹 with different hub radius ratio (a/b) is depicted

[52]. It seems that the hub radius also plays an important role in calculating the axial thrust, which

deserves further investigation in the future work.

0

0.02

0.04

0.06

0.08

0.1

0.3 1.3 2.3 3.30

0.02

0.04

0.06

0.08

0.1

0.3 1.3 2.3 3.3

0

0.02

0.04

0.06

0.08

0.1

0.3 1.3 2.3 3.3

0

0.02

0.04

0.06

0.08

0.1

0.3 1.3 2.3 3.3

0

0.02

0.04

0.06

0.08

0.1

0.3 1.3 2.3 3.30

0.02

0.04

0.06

0.08

0.1

0.3 1.3 2.3 3.3

Re

(106)

Re

(106)

Re

(106)

Re

(106)


Re

(106)

Re

(106)


82

𝐶𝐹

Fig. 60: Comparison of the results of 𝑪𝑭 for 𝑮 = 𝟎. 𝟎𝟓 and 𝑪𝒂𝒎 𝑪𝒒⁄ = −𝟎. 𝟔𝟏𝟗 [52]

The experimental results of 𝐶𝐹 by axial thrust measurements are depicted versus 𝐶𝑎𝑚 in Fig. 61

for 𝐶𝐷 = −5050 (𝑘𝑠 = 0.4 𝜇m) at various Re and G. Due to the drop of 𝐾𝑝, 𝐶𝐹 decreases with

increasing 𝐶𝑎𝑚 at a fixed Re. For wider axial gap, the impact of 𝐶𝑎𝑚 on 𝐶𝐹 becomes lesser.

According to the experimental results, it is recognized that the axial thrust is greatly influenced by

the angular momentum at the guide vane entrance. The impact of 𝐶𝑎𝑚 on 𝐶𝐹 therefore should be

carefully considered, which is worthy of further investigation.

𝐶𝐹 G = 0.018 𝐶𝐹 G = 0.036

0

0.02

0.04

0.06

0.08

0.1

0.12

-0.006 -0.005 -0.004 -0.003 -0.002 -0.001 0

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

0.065

0.07

-0.002 -0.001 0 0.001 0.0020.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

0.065

0.07

-0.002 -0.001 0 0.001 0.002

𝐶𝑎𝑚 𝐶𝑎𝑚

𝛽=

0°

Co-rotating Counter-rotating

𝛽=

0°

𝐶𝑎𝑚

83

𝐶𝐹 G = 0.054 𝐶𝐹 G = 0.072

Colors of points: Red: 𝑅𝑒 = 0.76 × 106; Purple: 𝑅𝑒 = 1.14 × 106; Blue: 𝑅𝑒 = 1.9 × 106; Green: 𝑅𝑒 = 3.17 × 106.

Fig. 61: 𝑪𝑭 (𝑪𝒂𝒎) curves for 𝑪𝑫 = −𝟓𝟎𝟓𝟎 for various 𝑹𝒆 and G

5.1.5 3D Diagram with Centripetal Through-Flow

The typical tangential velocity profiles for regime III (merged disk boundary layer and wall

boundary layer) and regime IV (separated disk boundary layer and wall boundary layer) are given

in Fig. 4. According to the tangential velocity profiles by numerical simulations, the 2D

Daily&Nece diagram (see Fig. 5) is extended into 3D by distinguishing the tangential velocity

profiles at 𝑥 = 0.955, 𝑥 = 0.79 and 𝑥 = 0.57, depicted in Fig. 62. The scope of this thesis is

the following parameter ranges: |𝐶𝐷| ≤ 5050 , 0.38 × 106 ≤ 𝑅𝑒 ≤ 3.17 × 107 and 0.018 ≤

𝐺 ≤ 0.072. They are categorized into two regimes, namely regime III (below the distinguishing

lines) and regime IV (above the distinguishing lines). Near the distinguishing surface, there is a

mixing zone where regime III and regime IV coexist in the front chamber (the flow regimes at the

above three radial positions differ). In this thesis, it is not plotted in Fig. 62 for its small size.

Currently, five distinguishing lines are found for different |𝐶𝐷| , shown in Fig. 62 (b). The

distinguishing line at 𝐶𝐷 = 0 is almost equal to that from Daily and Nece [28]. The distinguishing

lines become steeper for larger |𝐶𝐷|. The approximate distinguishing surface is drawn through the

lines, shown in Fig. 62 (a). Below and above the surface respectively are regime III and regime IV.

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

0.065

0.07

-0.002 -0.001 0 0.001 0.0020.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

0.065

0.07

-0.002 -0.001 0 0.001 0.002

𝛽=

0°

𝐶𝑎𝑚

𝛽=

0°

𝐶𝑎𝑚

84

G G

Fig. 62: 3D diagram distinguishing regime III and regime IV with centripetal through-flow for

𝟎. 𝟑 × 𝟏𝟎𝟔 ≤ 𝑹𝒆 ≤ 𝟑. 𝟑 × 𝟏𝟎𝟔

By Kurokawa et al. [54], the surface roughness of the disk has almost no influence on the thickness

of the disk layer when 𝑘𝑠 ≤ 95 𝜇m . Hence, the impact of 𝑘𝑠 on the distinguishing lines is

neglected in this thesis.

5.1.6 Moment Coefficient


According to the experimental results from Han et al. [40], the moment coefficient on the cylinder

surface of the disk 𝐶𝑀𝑐𝑦𝑙 can be estimated with Eq. 116 for smooth disks. For 𝐶𝐷 = 0, the values

of 𝐶𝑀 for 𝐺 = 0.018 (regime III) and 𝐺 = 0.072 (regime IV) are compared in Fig. 63 (a) and

Fig. 63 (b), respectively. The differences between the experimental results and those from the

correlations by Daily and Nece [28] for both regime III and regime IV, are colossal. Hence, Eq. 117

and Eq. 118 are determined to satisfy the own experimental results (𝑘𝑠 = 0.4 μm for the disk).

𝐶𝑀𝑐𝑦𝑙 =

2 ∙ |𝑀𝑐𝑦𝑙|

𝜌 ∙ 𝛺2 ∙ 𝑏5=

0.084 ∙ 𝜋 ∙ 𝑡

𝑏 ∙ (𝑙𝑔𝛺 ∙ 𝑏2

𝜐 )1.5152

Eq. 116

Regime III: 𝐶𝑀3 = 0.011 ∙ 𝐺−16 ∙ 𝑅𝑒−

14 ∙ [𝑒(0.8∙10−4∙|𝐶𝐷|)] Eq. 117

Regime IV: 𝐶𝑀4 = 0.014 ∙ 𝐺1

10 ∙ 𝑅𝑒−15 ∙ [𝑒(0.46∙10−4∙|𝐶𝐷|)] Eq. 118

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.1 1.0 10.0

百万

𝐺 = 0.072

𝐺 = 0.054

𝐺 = 0.036

𝐺 = 0.018

Re (106)

85

𝐶𝑀 𝐶𝑀

(a) 𝐺 = 0.018, regime III (b) 𝐺 = 0.072, regime IV

Exp Eq. 117 Eq. 84 ( Daily and Nece) Eq. 118 Eq. 85 (Daily and Nece)

Fig. 63: Comparison of the results of 𝑪𝑴 for 𝑮 = 𝟎. 𝟎𝟏𝟖 and 𝑮 = 𝟎. 𝟎𝟕𝟐 at 𝑪𝑫 = 𝟎

To introduce the influence of 𝐶𝐷 on the moment coefficient, the results of 𝐶𝑀 from both the

torque measurements and above two correlations are plotted versus Re in Fig. 64. With increasing

Re, the flow regime may change from regime III to regime IV (see the distinguishing lines in Fig.

62). For 𝐺 = 0.018 and 𝐺 = 0.036, most of the flow is within regime III and the results are close

to those from Eq. 117, shown in Fig. 64 (a) and Fig. 64 (b). With increasing G, the flow regime shifts

from regime III to regime IV a given Re and |𝐶𝐷|. The regime changes can also be found according

to the experimental results of 𝐶𝑀. For example, in Fig. 64 (b) for G=0.036 and 𝐶𝐷 = 1262, the

results of 𝐶𝑀 from Eq. 118 rather than from Eq. 117 are approaching those from torque measurements

when 𝑅𝑒 ≥ 0.9 × 106. For 𝐺 = 0.054 and 𝐺 = 0.072, most of the flow regime is regime IV

and the results are close to those from Eq. 118, shown in Fig. 64 (c) and in Fig. 64 (d). The results

of 𝐶𝑀 from the equations are in good agreement with those by torque measurements. The values

of 𝐶𝑀 increase with increasing |𝐶𝐷|, while they decrease with the increase of Re. For large values

of Re, the impact of 𝐶𝐷 on 𝐶𝑀 becomes lesser.

𝐶𝑀 (a) 𝐺 = 0.018 𝐶𝑀 (b) 𝐺 = 0.036

0

0.001

0.002

0.003

0.004

0.3 1.3 2.3 3.3

0

0.001

0.002

0.003

0.004

0.3 1.3 2.3 3.3

百万

0.0004

0.0009

0.0014

0.3 1.3 2.3 3.3

百万

0.0004

0.0009

0.0014

0.3 1.3 2.3 3.3

Re (106)

|𝐶𝐷|

Re (106)

|𝐶𝐷|

Re (106) Re (106)

86

𝐶𝑀 (c) 𝐺 = 0.054 𝐶𝑀 (d) 𝐺 = 0.072

Solid points: Measurements; Dashed lines: Eq. 117; Solid lines: Eq. 118.


Fig. 64: 𝑪𝑴 (Re) curves with centripetal through-flow

The results from Eq. 117 and Eq. 118 at the lines distinguishing regime III and regime IV should be

equal. The ratios of 𝐶𝑀3/𝐶𝑀4 at the distinguishing lines (see Fig. 62) are presented in Fig. 65. The

differences, attributed to the existence of the mixing zone, cover an amount less than ±4%. The

results indicate that the lines distinguishing regime III and regime IV are reasonable.

𝐶𝑀3/𝐶𝑀4


Fig. 65: Results of 𝑪𝑴𝟑/𝑪𝑴𝟒 at the distinguishing lines for centripetal through-flow


Combining the torque measurements with the results from Daily and Nece [28] and Dorfman

[32][33], two correlations can be determined to predict the influence of surface roughness on the

moment coefficient (for a single surface), given in Eq. 119 and Eq. 120.

0.0004

0.0009

0.0014

0.3 1.3 2.3 3.3

0.0004

0.0009

0.0014

0.3 1.3 2.3 3.3

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

0.1 1 10

Re (106) Re (106)

Re (106)


87

Regime III: 𝐶𝑀3 = 0.32 ∙ 𝐺−

16 ∙ 𝑅𝑒−

14 ∙ [𝑒(0.82∙10−4∙|𝐶𝐷|)] ∙ (

𝑘𝑠

𝑏)

0.272

Eq. 119

Regime IV: 𝐶𝑀4 = 0.41 ∙ 𝐺

110 ∙ 𝑅𝑒−

15 ∙ [𝑒(0.46∙10−4∙|𝐶𝐷|)] ∙ (

𝑘𝑠

𝑏)

0.272

Eq. 120

The first step is to introduce the influence of surface roughness 𝑘𝑠 in an enclosed rotor-stator

cavity. The results from Eq. 119 and Eq. 120 (with the geometry from Kurokawa et al. [54]) are

compared with the analytical solutions from Kurokawa et al. [54] for a single surface, depicted in

Fig. 66. For 𝐺 = 0.115 and 𝑅𝑒 = 106, the flow regime is regime IV in Fig. 62. The results from

Eq. 120 show the same trend as those of analytical solutions by Kurokawa et al. [54][54]. For 𝐺 =

0.0255 and 𝑅𝑒 = 106 , the flow regime is regime III and the results from Eq. 119 are also in

acceptable agreement with the analytical solutions. More important, Eq. 119 and Eq. 120 cover a wider

parameter range of 𝑘𝑠.

Colors: Green: Analytical solutions by Kurokawa et al. [54]; Blue: Eq. 119 or Eq. 120 according to G.

Lines: Solid: G=0.115; Dashed: G=0.0255.

Fig. 66: Comparison of 𝑪𝑴 with different values of 𝒌𝒔 in an enclosed rotor-stator cavity

The second step is to compare the results from Eq. 119 and Eq. 120 with those from Eq. 87 by Owen

[65]. Since the surface roughness is not available in the work of Owen [68], only the results for

disk 3 (𝑘𝑠 = 26.5 𝜇m, which is considered reasonable) are compared in Fig. 67. In Fig. 62, the

flow regime is regime IV when 𝑅𝑒 ≥ 0.7 × 106 for |𝐶𝐷| = 2525, 𝐺 = 0.047 and when 𝑅𝑒 ≥

2.3 × 106 for |𝐶𝐷| = 5050 , 𝐺 = 0.047 . At 𝐺 = 0.012 , the flow regime is regime III for all

through-flow coefficients. The experimental results are in good agreement with those from Eq. 119

and Eq. 120 according to the flow regime, shown in Fig. 67. The trend of the results by Eq. 119 and

Eq. 120 is similar with those from Eq. 87. The difference can be attributed to the uncertainty of surface

roughness.

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0 2000 4000 6000 8000 10000

𝑏

𝑘𝑠

𝐶𝑀

88

𝐶𝑀 |𝐶𝐷|=2525, G=0.047 𝐶𝑀 |𝐶𝐷|=5050, G=0.047

𝐶𝑀 |𝐶𝐷| =2525, G=0.012 𝐶𝑀 |𝐶𝐷| =5050, G=0.012

Triangles: Exp; Dot dash line: Eq. 87; Dashed line: Eq. 119; Solid line: Eq. 120.

Fig. 67: Comparison of 𝑪𝑴 from different equations

The third step is to compare the experimental results of 𝐶𝑀 for a rotor-stator cavity with centripetal

through-flow and rough disks with those from Eq. 119 and Eq. 120. For 𝐺 = 0.012 and 𝐺 = 0.027,

most of the flow regimes are regime III with increasing |𝐶𝐷| (see Fig. 62). The results of 𝐶𝑀 by

torque measurements for various 𝑘𝑠 are in very good consistent with those from Eq. 119, depicted

in Fig. 68. The values of 𝐶𝑀 increase as 𝑘𝑠 increases. The values of 𝐶𝑀 also increase with the

increase of |𝐶𝐷|. The values of 𝐶𝑀 drop faster with the increase of Re for smaller axial gap. When

G increases from 0.012 to 0.027 (right column of diagrams), the values of 𝐶𝑀 decrease, which is

the characteristic for regime III.

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

0.005

0.3 2.3 4.3 6.30.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

0.005

0.3 2.3 4.3 6.3

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

0.005

0.3 2.3 4.3 6.3

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

0.005

0.3 2.3 4.3 6.3

From regime III

to regime IV

From regime III

to regime IV

Regime III

Re

(106) Re

(106)

Re

(106)

Re

(106)

Regime III

89

𝐶𝑀 𝐺 = 0.012, 𝑘𝑠 = 0.4 𝜇m 𝐶𝑀 𝐺 = 0.027, 𝑘𝑠 = 0.4 𝜇m

𝐶𝑀 𝐺 = 0.012, 𝑘𝑠 = 3.5 𝜇m 𝐶𝑀 𝐺 = 0.027, 𝑘𝑠 = 3.5 𝜇m

𝐶𝑀 𝐺 = 0.012, 𝑘𝑠 = 11.4 𝜇m 𝐶𝑀 𝐺 = 0.027, 𝑘𝑠 = 11.4 𝜇m

0.0003

0.0008

0.0013

0.0018

0.30 1.30 2.30 3.30

百万

0.0003

0.0008

0.0013

0.0018

0.30 1.30 2.30 3.30

百万

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.30 1.30 2.30 3.30

百万

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.30 1.30 2.30 3.30

百万

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.30 1.30 2.30 3.30

百万

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.30 1.30 2.30 3.30

百万

Re

(106)

Disk 0

|𝐶𝐷|

Disk 0

Re

(106)

Disk 1

Re

(106)

Disk 2

Re

(106)

Disk 2

Re

(106)

Re

(106)

Disk 1



|𝐶𝐷|

90

𝐶𝑀 𝐺 = 0.012, 𝑘𝑠 = 26.5 𝜇m 𝐶𝑀 𝐺 = 0.027, 𝑘𝑠 = 26.5 𝜇m

𝐶𝑀 𝐺 = 0.012, 𝑘𝑠 = 58.9 𝜇m 𝐶𝑀 𝐺 = 0.027, 𝑘𝑠 = 58.9 𝜇m

𝐶𝑀 𝐺 = 0.012, 𝑘𝑠 = 81.3 𝜇m 𝐶𝑀 𝐺 = 0.027, 𝑘𝑠 = 81.3 𝜇m



Fig. 68: 𝑪𝑴 (𝑹𝒆) curves at 𝑮 = 𝟎. 𝟎𝟏𝟐 and 𝑮 = 𝟎. 𝟎𝟐𝟕 for different values of 𝒌𝒔 and |𝑪𝑫|

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

0.005

0.30 1.30 2.30 3.30

百万

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

0.005

0.30 1.30 2.30 3.30

百万

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

0.005

0.0055

0.006

0.30 1.30 2.30 3.30

百万

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

0.005

0.0055

0.006

0.30 1.30 2.30 3.30

百万

0.002

0.0025

0.003

0.0035

0.004

0.0045

0.005

0.0055

0.006

0.0065

0.007

0.30 1.30 2.30 3.30

百万

0.002

0.0025

0.003

0.0035

0.004

0.0045

0.005

0.0055

0.006

0.0065

0.007

0.30 1.30 2.30 3.30

百万

Re

(106)

Disk 3

|𝐶𝐷|

Disk 4

Disk 5

Re

(106)

Disk 5

Re

(106)

Disk 4

Re

(106)

|𝐶𝐷|

|𝐶𝐷|

Re

(106)

Re

(106)

|𝐶𝐷|

|𝐶𝐷|

|𝐶𝐷|

Disk 3

91

For 𝐺 = 0.047 and 𝐺 = 0.065, the results of 𝐶𝑀 by torque measurements are also in very good

agreement with those from Eq. 119 and Eq. 120 according to the flow regimes in Fig. 62, depicted in

Fig. 69. The impact of 𝑘𝑠 and 𝐶𝐷 on 𝐶𝑀 is similar to those from 𝐺 = 0.012 and 𝐺 = 0.027.

The flow regime may change from regime III to regime IV with the increase of Re (see Fig. 62).

For example, when |𝐶𝐷| = 5050 and 𝐺 = 0.047, the experimental results of 𝐶𝑀 are closer to

those from Eq. 119 until 𝑅𝑒 ≈ 2.8 × 106 (in the 3D Daily&Nece diagram, the transition point is at

𝑅𝑒 ≈ 2.3 × 106 in Fig. 62). When |𝐶𝐷| = 5050 and 𝐺 = 0.065, the values of 𝐶𝑀 are closer to

those from Eq. 120 when 𝑅𝑒 ≥ 0.8 × 106, while in Fig. 62, the transition point is at 𝑅𝑒 ≈ 1 × 106.

Considering the existence of a mixing zone, where the regime III and regime IV coexist, the

difference is acceptable. With centripetal through-flow, the moment coefficient for rough disks can

be predicted with sufficient accuracy with Eq. 119 and Eq. 120 according to the 3D diagram in Fig.

62.

𝐶𝑀 𝐺 = 0.047, 𝑘𝑠 = 0.4 𝜇m 𝐶𝑀 𝐺 = 0.065, 𝑘𝑠 = 0.4 𝜇m

𝐶𝑀 𝐺 = 0.047, 𝑘𝑠 = 3.5 𝜇m 𝐶𝑀 𝐺 = 0.065, 𝑘𝑠 = 3.5 𝜇m

0.0004

0.0009

0.0014

0.3 1.3 2.3 3.3

百万

0.0004

0.0009

0.0014

0.3 1.3 2.3 3.3

百万

0.0005

0.001

0.0015

0.002

0.0025

0.3 1.3 2.3 3.3

百万

0.0005

0.001

0.0015

0.002

0.0025

0.3 1.3 2.3 3.3

百万

Re

(106)

Disk 0

|𝐶𝐷|

Re

(106)

Disk 0

Re

(106)

Disk 1

Re

(106)

Disk 1


|𝐶𝐷|

92

𝐶𝑀 𝐺 = 0.047, 𝑘𝑠 = 11.4 𝜇m 𝐶𝑀 𝐺 = 0.065, 𝑘𝑠 = 11.4 𝜇m

𝐶𝑀 𝐺 = 0.047, 𝑘𝑠 = 26.5 𝜇m 𝐶𝑀 𝐺 = 0.065, 𝑘𝑠 = 26.5 𝜇m

𝐶𝑀 𝐺 = 0.047, 𝑘𝑠 = 58.9 𝜇m 𝐶𝑀 𝐺 = 0.065, 𝑘𝑠 = 58.9 𝜇m

0.001

0.0015

0.002

0.0025

0.003

0.3 1.3 2.3 3.3

百万

0.001

0.0015

0.002

0.0025

0.003

0.3 1.3 2.3 3.3

百万

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.3 1.3 2.3 3.3

百万

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.3 1.3 2.3 3.3

百万

0.002

0.0025

0.003

0.0035

0.004

0.0045

0.005

0.3 1.3 2.3 3.3

百万

0.002

0.0025

0.003

0.0035

0.004

0.0045

0.005

0.3 1.3 2.3 3.3

百万

Re

(106)

Disk 4

Re

(106)

Disk 4

Re

(106)

Disk 2

Re

(106)

Disk 3

Re

(106)

Disk 3

Re

(106)

Disk 2

|𝐶𝐷|


|𝐶𝐷|


93

𝐶𝑀 𝐺 = 0.047, 𝑘𝑠 = 81.3 𝜇m 𝐶𝑀 𝐺 = 0.065, 𝑘𝑠 = 81.3 𝜇m



Fig. 69: 𝑪𝑴 (Re) curves at 𝑮 = 𝟎. 𝟎𝟒𝟕 and 𝑮 = 𝟎. 𝟎𝟔𝟓 for different values of 𝒌𝒔 and |𝑪𝑫|

Some more experimental results are presented in Fig. 70. All the results show that the impact of

𝑘𝑠 can be predicted with Eq. 119 and Eq. 120 with good accuracy when 𝑘𝑠 ≤ 81.3 𝜇m.

𝐺 = 0.018 𝐺 = 0.036 𝐺 = 0.072

𝑘𝑠

=0

.4 𝜇

m

𝐶𝑀 𝐶𝑀 𝐶𝑀

𝑘𝑠

=3

.5 𝜇

m


0.002

0.0025

0.003

0.0035

0.004

0.0045

0.005

0.3 1.3 2.3 3.3

百万

0.002

0.0025

0.003

0.0035

0.004

0.0045

0.005

0.3 1.3 2.3 3.3

百万

0.0004

0.0009

0.0014

0.3 1.3 2.3 3.3

百万

0.0004

0.0009

0.0014

0.3 1.3 2.3 3.30.0004

0.0009

0.0014

0.3 1.3 2.3 3.3

0.0005

0.001

0.0015

0.002

0.0025

0.3 1.3 2.3 3.3

百万

0.0005

0.001

0.0015

0.002

0.0025

0.3 1.3 2.3 3.3

0.0005

0.001

0.0015

0.002

0.0025

0.3 1.3 2.3 3.3

Re

(106)

Disk 5

Re

(106)

Re

(106)

|𝐶𝐷|

Re

(106)

Re

(106)

Re

(106)

Re

(106)

Disk 5

|𝐶𝐷|

|𝐶𝐷|

Re

(106)



94

𝑘𝑠

=2

6.5

𝜇m


𝑘𝑠

=5

8.9

𝜇m



𝑘𝑠

=8

1.3

𝜇m



Fig. 70: 𝑪𝑴 (Re) curves at various G for different values of 𝒌𝒔 and 𝑪𝑫

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.3 1.3 2.3 3.3

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.3 1.3 2.3 3.3

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.3 1.3 2.3 3.3

百万

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

0.005

0.0055

0.3 1.3 2.3 3.3

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

0.005

0.0055

0.3 1.3 2.3 3.3

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

0.005

0.0055

0.3 1.3 2.3 3.3

百万

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

0.005

0.0055

0.006

0.0065

0.3 1.3 2.3 3.3

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

0.005

0.0055

0.006

0.0065

0.3 1.3 2.3 3.3

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

0.005

0.0055

0.006

0.0065

0.3 1.3 2.3 3.3

百万

Re

(106)

Re

(106)

Re

(106)

Re

(106)

Re

(106)

Re

(106)

Re

(106)

|𝐶𝐷|

Re

(106)

Re

(106)

|𝐶𝐷|

|𝐶𝐷|

|𝐶𝐷|

|𝐶𝐷|

|𝐶𝐷|

|𝐶𝐷|

|𝐶𝐷|

|𝐶𝐷|

95

5.1.6.3 Impact of Pre-Swirl on the Moment Coefficient with a Smooth Disk

In Fig. 71, the 𝐶𝑀 (𝐶𝑎𝑚 ) curves for |𝐶𝐷| = 5050 varying the pre-swirl angle 𝛽 and Re are

plotted. The values of 𝐶𝑎𝑚 are calculated with Eq. 73 (𝛽 ranges from −60° to 60°, see Table 9).

As commonly understood, near the external diameter of the disk, the tangential velocity gradient

on the disk surface has enormous impact on 𝐶𝑀 (Kurokawa et al [53]). By torque measurements,

the pre-swirl leads to a decrease of 𝐶𝑀 when the fluid co-rotates with the disk at the entrance. For

the counter-rotating flow when entering the cavity (with a large tangential velocity gradient), 𝐶𝑀

increases significantly, especially for small Re. The values of 𝐶𝑀 are dominant by the angular

momentum at the inlet (𝑥 = 1.01) rather than G. It seems that the experimental results of 𝐶𝑀 vary

versus 𝐶𝑎𝑚 following an exponent law for the inlet boundary conditions.

𝐶𝑀 𝐺 = 0.018 𝐶𝑀 𝐺 = 0.036

𝐶𝑀 𝐺 = 0.054 𝐶𝑀 𝐺 = 0.072

Colors: Red: 𝑅𝑒 = 0.76 × 106; Purple: 𝑅𝑒 = 1.14 × 106; Blue: 𝑅𝑒 = 1.9 × 106; Green: 𝑅𝑒 = 3.17 × 106.

Fig. 71: 𝑪𝑴 (𝑪𝒂𝒎) curves for |𝑪𝑫| = 𝟓𝟎𝟓𝟎 at different 𝑹𝒆

0

0.002

0.004

0.006

0.008

0.01

0.012

-0.02 -0.01 0 0.01 0.02

0

0.002

0.004

0.006

0.008

0.01

0.012

-0.02 -0.01 0 0.01 0.02

0

0.002

0.004

0.006

0.008

0.01

0.012

-0.02 -0.01 0 0.01 0.020

0.002

0.004

0.006

0.008

0.01

0.012

-0.02 -0.01 0 0.01 0.02

𝐶𝑎𝑚

𝐶𝑎𝑚 𝐶𝑎𝑚

Counter-rotating Co-rotating

𝐶𝑎𝑚

96

5.2 Rotor-Stator Cavity with Centrifugal Through-Flow

5.2.1 Simulation Results of Velocity Distributions

The velocity profiles at three chosen radial positions for 𝑅𝑒 = 1.9 × 106 and 𝐺 = 0.072 (small

passages in the radial guide vane) are presented in Fig. 72. The non-dimensional radial velocities

are not exactly zero in the central cores (휁 = 0.5), shown in Fig. 72 (a to c). From the distribution

of tangential velocity, central cores are identified at all the investigated radial positions by a

constant value of 𝑉𝜑 along 휁, shown in Fig. 72 (d to f). The values of the tangential velocity

decline at 휁 = 0.5 when 𝐶𝐷 increases, depicted in Fig. 72 (d, e). The trends of the tangential

velocity are in good agreement with the measured tangential velocity in the literature (such as by

Poncet et al. [68] and by Debuchy et al. [30]). The values of |𝑉𝑧| become smaller towards the

shaft.

𝑥 = 0.955 𝑥 = 0.79 𝑥 = 0.57 𝑉𝑟 (a) 𝑉𝑟 (b) 𝑉𝑟 (c)



Colors: Black: 𝐶𝐷 = 0; Blue: 𝐶𝐷 = 1262; Green: 𝐶𝐷 = 3787; Red: 𝐶𝐷 = 5050.

Fig. 72: Velocity profiles for 𝑹𝒆 = 𝟏. 𝟗 × 𝟏𝟎𝟔 and 𝑮 = 𝟎. 𝟎𝟕𝟐

-0.1

-0.05

0

0.05

0.1

0 0.5 1

-0.1

-0.05

0

0.05

0.1

0 0.5 1

-0.1

-0.05

0

0.05

0.1

0 0.5 1

0

0.2

0.4

0.6

0.8

1

0 0.5 1

0

0.2

0.4

0.6

0.8

1

0 0.5 10

0.2

0.4

0.6

0.8

1

0 0.5 1

-0.03

-0.01

0.01

0.03

0 0.5 1

-0.03

-0.01

0.01

0.03

0 0.5 1

-0.03

-0.01

0.01

0.03

0 0.5 1

휁 휁 휁

Disk Wall

휁 휁 휁

휁 휁 휁

97

The velocity profiles at the three selected radial coordinates for 𝑅𝑒 = 1.9 × 106 and 𝐺 = 0.018

(small gap) are presented in Fig. 73. The values of 𝑉𝑟 vary along 휁, shown in Fig. 73 (a to c). The

values of 𝑉𝑟 increase with the increase of 𝐶𝐷. At 𝐶𝐷 = 3787 and 𝐶𝐷 = 5050, all the values of

𝑉𝑟 are positive (all the flow in the boundary layers is centrifugal). The flow type is therefore of the

Stewartson type. From the disk to the wall, 𝑉𝜑 decreases constantly, which is the characteristic of

the regime III, shown in Fig. 73 (d to f). The values of |𝑉𝑧| are very small, compared with those

in Fig. 72 (g~i). This indicates that the axial circulation of the fluid is weaker for a small axial gap

width.

𝑥 = 0.955 𝑥 = 0.79 𝑥 = 0.57





Fig. 73: Velocity profiles for 𝑹𝒆 = 𝟏. 𝟗 × 𝟏𝟎𝟔 and 𝑮 = 𝟎. 𝟎𝟏𝟖

-0.1

-0.05

0

0.05

0.1

0.15

0 0.5 1

-0.05

0

0.05

0.1

0 0.5 1

-0.1

-0.05

0

0.05

0.1

0 0.5 1

0

0.2

0.4

0.6

0.8

1

0 0.5 1

0

0.2

0.4

0.6

0.8

1

0 0.5 1

0

0.2

0.4

0.6

0.8

1

0 0.5 1

-0.01

-0.005

0

0.005

0.01

0 0.5 1

-0.01

-0.005

0

0.005

0.01

0 0.5 1

-0.01

-0.005

0

0.005

0.01

0 0.5 1

휁 휁 휁

Disk Wall

휁 휁 휁

휁 휁 휁

98



In most of the radial pumps or turbines, the parameter G is a variable along the radius. A simplified

correlation is required with good accuracy over the whole range of G. By pressure measurements

and numerical simulations of core swirl ratio, Eq. 121 is correlated to predict the values of K when

G ranges from 0.018 to 0.072 for Stewartson type flow. The results are compared in Fig. 74. The

results from Eq. 110 and Eq. 121 are in good accordance with results of 𝐾 both from numerical

simulations and from the pressure measurements with Eq. 109.

Stewartson type flow (𝐶𝑞𝑟 ≥ 0.04):

𝐾 = 0.85 ∙ [0.032 + 0.32 × 𝑒(

−𝐶𝑞𝑟

0.028)] Eq. 121

K

Hollow points: Simulation; Solid points: By pressure measurements (Eq. 109); Dashed lines: Eq. 110 or Eq. 121.

Colors: Black: 𝐺 = 0.018; Green: 𝐺 = 0.036; Yellow: 𝐺 = 0.054; Red: 𝐺 = 0.072.

Fig. 74: 𝑲 (𝑪𝒒𝒓) curves

0

0.1

0.2

0.3

0.4

0.001 0.01 0.1

Batchelor/Couette type flow

Stewartson type flow

Co

nver

tin

g r

egio

n

𝐶𝑞𝑟

Eq. 121 Eq. 110

99

Some results of K, however, do not fit to the results from Eq. 110 and Eq. 121, especially at 𝑥 = 0.955

for wider axial gaps. A selection of the results is shown in Fig. 73. Near the outlet, an area change

from the front cavity to the channel in the guide vane for 𝐺 = 0.036, 0.054 and 0.072 occurs. The

measured pressure at 𝑥 = 0.955 is strongly influenced by the geometry at the outlet of the test rig.

Based on the simulation results (𝐺 = 0.072, 𝐶𝐷 = 5050), small vortexes near the outer radius of

the disk exist and therefore the measured values at 𝑥 = 0.955 are only partly used during the

calculation of K.

K

(a) Part of the results do not fit Eq. 110 and Eq. 122 (b) Surface streamlines near the outlet

Hollow points: Simulation; Solid points: By pressure measurements; Dashed lines: Eq. 110 or Eq. 121.

Colors: Black: 𝐺 = 0.018; Green: 𝐺 = 0.036; Yellow: 𝐺 = 0.054; Red: 𝐺 = 0.072.

Fig. 75: Large differences of K attributed to the geometry near the outlet


By pressure measurements, Eq. 122 is determined to describe the impact of surface roughness on K

for Stewartson type flow.

Stewartson type flow (𝐶𝑞𝑟 ≥ 0.04):

𝐾 = 0.85 ∙ 𝑒

(600∙𝑘𝑠∙𝑟

𝑏2 )∙ [0.032 + 0.32 × 𝑒(

−𝐶𝑞𝑟

0.028)] Eq. 122

For Batchelor or Couette type flow, the impact of 𝑘𝑠 on K is the same as the case for centripetal

through-flow. The amounts of K by pressure measurements are compared with those from Eq. 122

for 𝐺 = 0.072, presented in Fig. 72. When 𝐶𝑞𝑟 ≤ 0.03, the experimental results of K are close to

0

0.1

0.2

0.3

0.4

0.5

0.001 0.01 0.1

𝐶𝑞𝑟

𝑥 = 0.955

Disk Wall

Guide vane

Eq. 110 Eq. 121 C

on

ver

tin

g r

egio

n

100

those from Eq. 111 for Batchelor or Couette type flow. For 𝐶𝑞𝑟 ≥ 0.04 , the results are in good

agreement with those from Eq. 122 for Stewartson type flow. For rougher surfaces, the K (𝐶𝑞𝑟) curves

become steeper. A faster increase of K can be spotted with the decrease of 𝐶𝑞𝑟 (with the increase

of radial coordinates) for rougher surfaces, which is in accordance with the trend in Kurokawa et

al. [54] for an enclosed rotor-stator cavity. In the transition zone, the values of K can be predicted

with both Eq. 111 and Eq. 122.

K

Solid points: By pressure measurements; Solid lines: Eq. 111; Dashed lines: Eq. 122.

Colors: Blue: 𝑘𝑠 = 0.4 𝜇m; Orange: 𝑘𝑠 = 11.4 𝜇m; Green: 𝑘𝑠 = 26.5 𝜇m; Red: 𝑘𝑠 = 58.9 𝜇m.

Fig. 76: 𝑲 (𝑪𝒒𝒓) curves for various 𝒌𝒔 at 𝑮 = 𝟎. 𝟎𝟕𝟐

5.2.2.3 Impact of Centrifugal Pre-Swirl Through-Flow with a Smooth Disk

Four inlet guide vanes exist in the horizontal pipe to generate the pre-swirl. When the flow passes

the guide vane, it will obtain a tangential velocity component. When introducing the through-flow,

Karabay et al. [49][50] mention that the values of 𝐾ℎ can be calculated with Eq. 123 at 𝑟 =𝑑ℎ

2

(with inlet pre-swirl nozzles, see Fig. 16, instead of swirlers).

𝐾ℎ =

𝑄

𝜋 ∙ 𝛺 ∙ (𝑑ℎ2 )3

∙ 𝑡𝑎𝑛(90° − 𝛽)

Eq. 123

0

0.1

0.2

0.3

0.4

0.5

0.6

0.01 0.02 0.03 0.04 0.05 0.06 0.07

Stewartson type flow

Bat

chel

or/

Couet

te

type

flo

w

Co

nver

tin

g r

egio

n

𝐶𝑞𝑟

101

Based on Eq. 62 from Karabay et al. [49][50], the values of K along the radius can be calculated

with Eq. 124.

𝐾𝑝 =

𝑄 ∙ 𝑥𝑎2 ∙ 𝑡𝑎𝑛(90° − 𝛽)

𝜋 ∙ 𝑥2 ∙ 𝛺 ∙ (𝑑ℎ2 )3

Eq. 124

The results of 𝐾𝑝 from numerical simulations for 𝐶𝐷 = 5050, 𝑅𝑒 = 1.9 × 106 and 𝐺 = 0.072

are depicted in Fig. 77. The simulation models include the geometry of the inlet swirler (see Fig.

35). The fluid domain near the swirler can be seen in Fig. 42. When compared with those from Eq.

124 by Karabay et al. [49][50], large differences can be observed. This indicates that Eq. 124 can not

be used to predict the value of 𝐾𝑝 with the boundary conditions used in this thesis. The values of

𝛽 are positive when the fluid co-rotates with the disk when entering the cavity. The results from

four pre-swirl angles (𝛽 = ±15° and 𝛽 = ±45°) are selected for comparison. Overall, for the

co-rotating flow, the amounts of 𝐾𝑝 will firstly decrease, then increase towards the outer radius of

the disk. For the counter-rotating flow, the values of K increase gradually by the disk from 𝑥 = 𝑥ℎ

to 𝑥 = 1. How to quantify the values of angular momentum introduced by the centrifugal through-

flow deserves further investigation.

Solid points: By pressure measurements; Solid lines: Numerical simulations; Dashed lines: Eq. 124.

Colors: Red: 𝛽 = 45°; Blue: 𝛽 = 15°; Green: no pre-swirl; Purple: 𝛽 = −15°; Orange: 𝛽 = −45°.

Fig. 77: 𝑲𝒑 (𝑪𝒒𝒓) curves for 𝑹𝒆 = 𝟏. 𝟗 × 𝟏𝟎𝟔, 𝑪𝑫 = 𝟓𝟎𝟓𝟎 and 𝑮 = 𝟎. 𝟎𝟕𝟐

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

𝑉𝑧

𝛽

𝑉𝜑 𝑄

𝑥

𝐾𝑝 𝑥 = 𝑥ℎ

Swirler at the inlet

102

5.2.3 Pressure Distribution


Like the case for centripetal through-flow, a reference pressure is taken at the non-dimensional

radial coordinate 𝑥 = 1 . The pressure values at 𝑥 = 1 are calculated by Eq. 112. The pressure

coefficient 𝐶𝑝 are positive values because the pressure drops towards the shaft. In Fig. 78, the

values of 𝐶𝑝 are plotted versus 𝐶𝐷. The experimental results show that 𝐶𝑝 decreases with the

increasing 𝐶𝐷, Re and G in general. The experimental results are in good agreement with those

from Eq. 112 based on the equations for K (Eq. 121 or Eq. 110 by distinguishing 𝐶𝑞𝑟 every 1 mm).

When 𝑅𝑒 = 2.79 × 106, the uncertainty of 𝐶𝑝 is 1.3 × 10−4, which is very small compared with

the measured results. Hence, it is neglected in Fig. 78 (d~f).

x x x

(a) 𝑅𝑒 = 0.76 × 106, 𝐺 = 0.072 (b) 𝑅𝑒 = 0.76 × 106, 𝐺 = 0.036 (c) 𝑅𝑒 = 0.76 × 106, 𝐺 = 0.018

x x x

(d) 𝑅𝑒 = 2.79 × 106, 𝐺 = 0.072 (e) 𝑅𝑒 =2.79× 106, 𝐺 = 0.036 (f) 𝑅𝑒 =2.79× 106, 𝐺 = 0.018

Solid points: By pressure measurements; Solid lines: Eq. 112.


Fig. 78: Distribution of 𝑪𝒑 along the radius

0.4

0.6

0.8

1

0 0.01 0.02 0.03 0.04 0.05

0.4

0.6

0.8

1

0 0.01 0.02 0.03 0.04 0.050.4

0.6

0.8

1

0 0.01 0.02 0.03 0.04 0.05

0.4

0.6

0.8

1

0 0.01 0.02 0.03 0.04 0.05

0.4

0.6

0.8

1

0 0.01 0.02 0.03 0.04 0.05

0.4

0.6

0.8

1

0 0.01 0.02 0.03 0.04 0.05

𝐶𝑝 𝐶𝑝

𝐶𝑝

𝐶𝑝

𝐶𝑝

𝐶𝑝

𝐶𝐷

103


The effect of surface roughness on the pressure distribution is shown in Fig. 79. The results of K

based on the pressure measurements are compared with those from Eq. 112 based on the equations

for K (Eq. 122 or Eq. 111 by distinguishing 𝐶𝑞𝑟 every 1 mm). The pressure drops faster towards the

shaft with the increase of 𝑘𝑠. With the increase of 𝐶𝐷, the values of 𝐶𝑝 decrease, which is caused

by the decrease of K (see Eq. 122 and Eq. 111).

𝐶𝐷 = 0 𝐶𝐷 = 5050

x x

𝑅𝑒

=0

.76

×1

06

x x

𝑅𝑒

=1

.9×

10

6

Solid points: By pressure measurements; Solid lines: Eq. 112.

Colors: Black: 𝑘𝑠 = 0.4 𝜇m; Blue: 𝑘𝑠 = 11.4 𝜇m; Green: 𝑘𝑠 = 26.5 𝜇m; Red: 𝑘𝑠 = 58.9 𝜇m.

Fig. 79: Distribution of 𝑪𝒑 along the radius for the rough disks

0.4

0.6

0.8

1

0 0.05 0.10.4

0.6

0.8

1

0 0.05 0.1

0.4

0.6

0.8

1

0 0.05 0.10.4

0.6

0.8

1

0 0.05 0.1

𝐶𝑝

𝐶𝑝

𝐶𝑝

𝐶𝑝

𝑘𝑠

104

5.2.3.3 Impact of Pre-Swirl on the Pressure Distribution with a Smooth Disk

To show the impact of centrifugal pre-swirl on the pressure distribution, the distributions of 𝐶𝑝

along x for 𝑅𝑒 = 0.76 × 106 and 𝑅𝑒 = 1.9 × 106 at G=0.072 with different 𝐶𝑎𝑚 at 𝐶𝐷 =

5050 are depicted in Fig. 80. With the increase of 𝛽, the pressure drops faster towards the shaft.

The values of 𝐶𝑝 are smaller for larger Re from both the simulation results and the pressure

measurements.

x 𝑅𝑒 = 0.76 × 106 x 𝑅𝑒 = 1.9 × 106

Solid Lines: Simulation results; Points: Pressure measurements;

Colors: Red: 𝛽 = 45°; Blue: 𝛽 = 15°; Green: no pre-swirl; Purple: 𝛽 = −15°; Orange: 𝛽 = −45°

Fig. 80: 𝑪𝒑 (x) curves for various 𝑪𝒂𝒎 at 𝑪𝑫 = 𝟓𝟎𝟓𝟎 and G=0.072

5.2.4 Thrust Coefficient

5.2.4.1 Impact of Centrifugal Through-Flow with a Smooth Disk

Based on the measurements, Hu et al. [18] correlate Eq. 113 for the thrust coefficient in a rotor-

stator cavity with centripetal through-flow ( 𝑘𝑠 = 0.4 𝜇m ). It is organized based on the

experimental results for centripetal through-flow. When compared with the experimental results, it

is similar to centrifugal through-flow, written in Eq. 125. In this thesis, 𝐶𝐷 is positive for centrifugal

through-flow. The comparisons of the results of 𝐶𝐹 for different G and 𝐶𝐷 are shown in Fig. 81.

The experimental results of 𝐶𝐹 are in good agreement with those based on the calculated pressure

along the disk with Eq. 115 and Eq. 125. The values of 𝐶𝐹 decrease with increasing 𝐶𝐷. In a rotor-

stator cavity with centripetal through-flow (𝐶𝐷 is negative) studied by Hu et al. [18], however, the

values of 𝐶𝐹 increase with increasing |𝐶𝐷|. The values of 𝐶𝐹 are smaller for a wider axial gap in

general.

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.023 0.046 0.069 0.092

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.011 0.022 0.033 0.044

𝐶𝑝 𝐶𝑝

105

𝐶𝐹 = [6.6 ∙ 10−3 ∙ ln(𝑅𝑒) − 0.113] ∙ 𝑒(−1.6∙10−4∙𝐶𝐷) ∙ [0.122 ∙ ln(𝐺) − 0.67] Eq. 125

𝐶𝐹 G=0.018 𝐶𝐹 G=0.036

𝐶𝐹 G=0.054 𝐶𝐹 G=0.072

Hollow points: Eq. 115; Solid points: Measurements; Dashed lines: Eq. 125.


Fig. 81: 𝑪𝑭 (𝑹𝒆) curves (𝒌𝒔 = 𝟎. 𝟒 𝝁𝐦)


For the rough disks, the impact of surface roughness on 𝐶𝐹 can be predicted with Eq. 126. A rougher

surface results in higher values of 𝐶𝐹 . This can be explained by the increase of 𝐶𝑝 . The

experimental results are also close to those based on the integral of pressure along the radius.

During the calculation of the pressure, the values of 𝐶𝑞𝑟 are calculated every 1 mm. Then the

values of K can be predicted based on the flow type. The results from Eq. 115 and Eq. 126 are in good

agreement with the experimental results, depicted in Fig. 82.

0

0.01

0.02

0.03

0.04

0.3 1.3 2.3 3.3

0

0.01

0.02

0.03

0.04

0.3 1.3 2.3 3.3

0

0.01

0.02

0.03

0.04

0.3 1.3 2.3 3.3

0

0.01

0.02

0.03

0.04

0.3 1.3 2.3 3.3

𝑅𝑒 (106) 𝑅𝑒 (106)

𝑅𝑒 (106) 𝑅𝑒 (106)

𝐶𝐷 𝐶𝐷

𝐶𝐷 𝐶𝐷

106

𝐶𝐹 = [6.6 ∙ 10−3 ∙ ln(𝑅𝑒) − 0.113] ∙ 𝑒(−1.6∙10−4∙𝐶𝐷) ∙ [0.122 ∙ ln(𝐺) − 0.67] ∙ 𝑒(880∙

𝑘𝑠𝑏

) Eq. 126

𝐺 = 0.018 𝐺 = 0.036 𝐺 = 0.072

𝑘𝑠

=3

.5 𝜇

m


Hollow points: Eq. 115; Solid points: Measurements; Dashed lines: Eq. 126.


Fig. 82: 𝑪𝑭 (𝑹𝒆) curves in dependence of 𝑪𝑫, 𝑮 and 𝒌𝒔

0

0.01

0.02

0.03

0.04

0.05

0.06

0.3 1.3 2.3 3.30

0.01

0.02

0.03

0.04

0.05

0.06

0.3 1.3 2.3 3.3

0

0.01

0.02

0.03

0.04

0.05

0.06

0.3 1.3 2.3 3.3

𝑘𝑠

=2

6.5

𝜇m


𝑘𝑠

=5

8.9

𝜇m


0

0.01

0.02

0.03

0.04

0.05

0.06

0.3 1.3 2.3 3.3

0

0.01

0.02

0.03

0.04

0.05

0.06

0.3 1.3 2.3 3.3

0

0.01

0.02

0.03

0.04

0.05

0.06

0.3 1.3 2.3 3.3

0

0.01

0.02

0.03

0.04

0.05

0.06

0.3 1.3 2.3 3.30

0.01

0.02

0.03

0.04

0.05

0.06

0.3 1.3 2.3 3.30

0.01

0.02

0.03

0.04

0.05

0.06

0.3 1.3 2.3 3.3

𝑅𝑒 (106)

𝑅𝑒 (106)

𝑅𝑒 (106)

𝑅𝑒 (106)

𝑅𝑒 (106)

𝑅𝑒 (106)

𝑅𝑒 (106)

𝑅𝑒 (106)

𝑅𝑒 (106)

𝐶𝐷

𝐶𝐷

𝐶𝐷 𝐶𝐷 𝐶𝐷

𝐶𝐷 𝐶𝐷

𝐶𝐷 𝐶𝐷

107

5.2.4.3 Impact of Pre-Swirl on the Thrust Coefficient with a Smooth Disk

The pre-swirl has a large impact on 𝐶𝐹 by changing the pressure distribution. The experimental

results of 𝐶𝐹 are compared for 𝐶𝐷 = 5050 and 𝑘𝑠 = 0.4 𝜇m in Fig. 83. With the increase of

𝛽, 𝐶𝐹 increases in general. The impact becomes lesser for wider axial gap width. The impact of

𝐶𝑎𝑚 on 𝐶𝐹 should be carefully considered, especially for small Re and small axial gaps. With the

increase of Re, the amounts of 𝐶𝐹 decline.

𝐶𝐹 𝐺 = 0.018 𝐶𝐹 𝐺 = 0.036

𝐶𝐹 𝐺 = 0.054 𝐶𝐹 𝐺 = 0.072

Points: Experiments.

Colors: Red: 𝛽 = 45°; Green: 𝛽 = 15°; Black: no pre-swirl; Yellow: 𝛽 = −15°; Blue: 𝛽 = −45°.

Fig. 83: Experimental results of 𝑪𝑭 versus 𝑪𝒂𝒎 for 𝑪𝑫 = 𝟓𝟎𝟓𝟎 at 𝐝𝐢𝐟𝐟𝐞𝐫𝐞𝐧𝐭 𝑹𝒆 and G

0

0.005

0.01

0.015

0.02

0.025

0.3 1.3 2.3 3.30

0.005

0.01

0.015

0.02

0.025

0.3 1.3 2.3 3.3

0

0.005

0.01

0.015

0.02

0.025

0.3 1.3 2.3 3.3

0

0.005

0.01

0.015

0.02

0.025

0.3 1.3 2.3 3.3

𝑅𝑒 (106)

𝑅𝑒 (106)

𝑅𝑒 (106)

𝑅𝑒 (106)

108

5.2.5 3D Diagram with Centrifugal Through-Flow

In this part, the 2D Daily&Nece diagram is extended with centrifugal through-flow by classifying

the tangential velocity profiles at x=0.945, x=0.79 and x=0.57 based on the results of numerical

simulations. Currently, five distinguishing lines can be found, depicted in Fig. 84 (a) (Hu et al.

[18]). Below and above the distinguishing lines are regime III and regime IV, respectively. The

distinguishing surface is drawn through the five distinguishing lines, shown in Fig. 84 (b). Near the

distinguishing surface, there is a mixing zone, where regime III and regime IV coexist in the cavity.

The distinguishing surface for centripetal through-flow (Hu et al. [16][17]) is also plotted to make

it a complete diagram.

G G

(a) Distinguishing line (b) Distinguishing surface

Fig. 84: 3D diagram distinguishing regime III and regime IV with centrifugal through-flow for

𝟎. 𝟑 × 𝟏𝟎𝟔 ≤ 𝑹𝒆 ≤ 𝟑. 𝟑 × 𝟏𝟎𝟔

5.2.6 Moment Coefficient


Comparing the torque measurements with the results from Daily and Nece [28] and Dorfman

[32][33], two correlations can be determined to predict the moment coefficient for centrifugal

through-flow (for a single surface of the disk, 𝑘𝑠 = 0.4 𝜇m), given in Eq. 127 and Eq. 128.

𝐶𝑀3 = 0.011 ∙ 𝐺−

16 ∙ 𝑅𝑒−

14 ∙ [𝑒(10−4∙𝐶𝐷)] Eq. 127

𝐶𝑀4 = 0.014 ∙ 𝐺

110 ∙ 𝑅𝑒−

15 ∙ [𝑒(0.6∙10−4∙𝐶𝐷)] Eq. 128

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.1 1 10 𝐶𝐷

Centripetal

through-flow

Centrifugal

through-flow

𝐶𝐷 = 0

𝐶𝐷 = 2525

𝐶𝐷 = 5050

𝐶𝐷 = 3787

𝐶𝐷 = 1262

𝑅𝑒 (106)

𝐺 = 0.072

𝐺 = 0.054

𝐺 = 0.036

𝐺 = 0.018

𝑅𝑒 (106)

109

The experimental results of 𝐶𝑀 are compared with those from Eq. 127 and Eq. 128, depicted in Fig.

85. For 𝐺 = 0.018 and 𝐺 = 0.036, most of the flow regimes are regime III, shown in Fig. 85 (a)

and Fig. 85 (b). When G increases to 0.036, the flow regimes change from regime III to regime IV

with the increase of Re for 𝐶𝐷 = 0, 1262 and 2525, which is also indicated by the experimental

results of 𝐶𝑀 . For 𝐺 = 0.054 and 𝐺 = 0.072 , most of the flow regimes are regime IV. The

results of 𝐶𝑀 from experiments are in very good agreement with those from Eq. 128 in general. The

regime III may occur at small Re and large 𝐶𝐷. The amounts of 𝐶𝑀 increase with the increase of

𝐶𝐷. The values of 𝐶𝑀 drop faster with the increase of Re for smaller values of G. Compared with

measured 𝐶𝑀 for centripetal through-flow (Hu et al. [18]), the centrifugal through-flow will result

in larger values of 𝐶𝑀 at the same values of |𝐶𝐷|, which is in accordance with the conclusion of

Dibelius et al. [31].

𝐶𝑀 G=0.018 𝐶𝑀 G=0.036

𝐶𝑀 G=0.054 𝐶𝑀 G=0.072



Fig. 85: Curves for 𝑪𝑴 in dependence of 𝑹𝒆 for different values of 𝑪𝑫 and 𝑮 with centrifugal

through-flow

0

0.0005

0.001

0.0015

0.002

0.30 1.30 2.30 3.30

百万

0

0.0005

0.001

0.0015

0.002

0.30 1.30 2.30 3.30

百万

0

0.0005

0.001

0.0015

0.002

0.3 1.3 2.3 3.30

0.0005

0.001

0.0015

0.002

0.30 1.30 2.30 3.30

百万

𝑅𝑒 (106)

𝐶𝐷

𝑅𝑒 (106)

𝑅𝑒 (106)

𝑅𝑒 (106)

𝐶𝐷

𝐶𝐷 𝐶𝐷

110

On the distinguishing lines (see Fig. 84), the results from Eq. 127 should be equal to those from Eq.

128. The results of 𝐶𝑀3/𝐶𝑀4 for a non-dimensioned gap width G at the distinguishing lines are

presented in Fig. 86. The differences (may be attributed to the existence of the mixing zone) are

very small in general and cover an amount less than 5%.

𝐶𝑀3/𝐶𝑀4

𝐶𝐷 = 1262 𝐶𝐷 = 2525 𝐶𝐷 = 3787 𝐶𝐷 = 5050

Fig. 86: Results of 𝑪𝒎𝟑/𝑪𝒎𝟒 at the distinguishing lines for centrifugal through-flow

5.2.6.2 Impact of Surface Roughness of the disk

Comparing the torque measurements with the results from Daily and Nece [28] and Dorfman

[32][33], two correlations can be determined to predict the influence of surface roughness on the

moment coefficient (for a single surface), given in Eq. 129 and Eq. 130.

𝐶𝑀3 = 0.32 ∙ 𝐺−

16 ∙ 𝑅𝑒−

14 ∙ [𝑒(10−4∙𝐶𝐷)] ∙ (

𝑘𝑠

𝑏)

0.272

Eq. 129

𝐶𝑀4 = 0.41 ∙ 𝐺

110 ∙ 𝑅𝑒−

15 ∙ [𝑒(0.6∙10−4∙𝐶𝐷)] ∙ (

𝑘𝑠

𝑏)

0.272

Eq. 130

The results from Eq. 129 and Eq. 130 are compared with those from experiments in Fig. 87. For small

axial gaps (𝐺 = 0.018 and 𝐺 = 0.036), most of the experimental results of 𝐶𝑀 are in very good

agreement with those from Eq. 129. Regime IV may only occur at small 𝐶𝐷 and large Re for 𝐺 =

0.036 . For large axial gaps (𝐺 = 0.054 and 𝐺 = 0.072 ), the experimental results of 𝐶𝑀 are

close to those from Eq. 130. The experimental results of 𝐶𝑀 do not indicate the change of flow

regimes (see Fig. 84) with the increase of surface roughness, which is in accordance with the case

for centripetal through-flow. Kurokawa et al. [54] state that the impact of surface roughness of the

rotor on the thickness of the disk layer is weak in an enclosed rotor-stator cavity based on the

velocity measurements. Hence, the distinguishing lines are considered valid for the rotors when

𝑘𝑠 ≤ 81.3 𝜇m.

0.95

0.97

0.99

1.01

1.03

1.05

0.1 1 10

𝑅𝑒 (106)

111

𝐶𝑀 G=0.018 𝐶𝑀 G=0.036 𝑘

𝑠=

3.5

𝜇m

𝐶𝑀 G=0.054 𝐶𝑀 G=0.072

𝑘𝑠

=3

.5 𝜇

m

𝑘𝑠

=1

1.4

𝜇m

𝐶𝑀 G=0.018 𝐶𝑀 G=0.036

𝑘𝑠

=1

1.4

𝜇m

𝐶𝑀 G=0.054 𝐶𝑀 G=0.072

0.0003

0.0008

0.0013

0.0018

0.0023

0.0028

0.30 1.30 2.30 3.30

百万

0.0003

0.0008

0.0013

0.0018

0.0023

0.0028

0.30 1.30 2.30 3.30

百万

0.0003

0.0008

0.0013

0.0018

0.0023

0.0028

0.3 1.3 2.3 3.30.0003

0.0008

0.0013

0.0018

0.0023

0.0028

0.30 1.30 2.30 3.30

百万

0.0003

0.0008

0.0013

0.0018

0.0023

0.0028

0.0033

0.0038

0.30 1.30 2.30 3.30

百万

0.0003

0.0008

0.0013

0.0018

0.0023

0.0028

0.0033

0.0038

0.30 1.30 2.30 3.30

百万

0.0003

0.0008

0.0013

0.0018

0.0023

0.0028

0.0033

0.0038

0.3 1.3 2.3 3.3

0.0003

0.0008

0.0013

0.0018

0.0023

0.0028

0.0033

0.0038

0.30 1.30 2.30 3.30

百万

Re

(106)

Re

(106)

Re

(106)

Re

(106)

Re

(106)

Re

(106)

𝐶𝐷

Re

(106)

Re

(106)

𝐶𝐷

𝐶𝐷

𝐶𝐷 𝐶𝐷

𝐶𝐷

𝐶𝐷 𝐶𝐷

112

𝐶𝑀 G=0.018 𝐶𝑀 G=0.036 𝑘

𝑠=

26

.5 𝜇

m

𝐶𝑀 G=0.054 𝐶𝑀 G=0.072

𝑘𝑠

=2

6.5

𝜇m

𝑘𝑠

=5

8.9

𝜇m

𝐶𝑀 G=0.018 𝐶𝑀 G=0.036

𝑘𝑠

=5

8.9

𝜇m

𝐶𝑀 G=0.054 𝐶𝑀 G=0.072

0.0003

0.0008

0.0013

0.0018

0.0023

0.0028

0.0033

0.0038

0.0043

0.30 1.30 2.30 3.30

百万

0.0003

0.0008

0.0013

0.0018

0.0023

0.0028

0.0033

0.0038

0.0043

0.30 1.30 2.30 3.30

百万

0.0003

0.0008

0.0013

0.0018

0.0023

0.0028

0.0033

0.0038

0.0043

0.3 1.3 2.3 3.30.0003

0.0008

0.0013

0.0018

0.0023

0.0028

0.0033

0.0038

0.0043

0.30 1.30 2.30 3.30

百万

0.00030.00080.00130.00180.00230.00280.00330.00380.00430.00480.00530.0058

0.30 1.30 2.30 3.30

百万

0.0003

0.0008

0.0013

0.0018

0.0023

0.0028

0.0033

0.0038

0.0043

0.0048

0.0053

0.0058

0.30 1.30 2.30 3.30

百万

0.00030.00080.00130.00180.00230.00280.00330.00380.00430.00480.00530.0058

0.3 1.3 2.3 3.3

0.00030.00080.00130.00180.00230.00280.00330.00380.00430.00480.00530.0058

0.30 1.30 2.30 3.30

百万

Re

(106)

Re

(106)

Re

(106)

Re

(106)

Re

(106) Re

(106)

Re

(106)

Re

(106)

𝐶𝐷 𝐶𝐷

𝐶𝐷 𝐶𝐷

𝐶𝐷 𝐶𝐷

𝐶𝐷 𝐶𝐷

113

𝑘𝑠

=8

1.3

𝜇m

𝐶𝑀 G=0.018 𝐶𝑀 G=0.036

𝑘𝑠

=8

1.3

𝜇m

𝐶𝑀 G=0.054 𝐶𝑀 G=0.072



Fig. 87: Curves for 𝑪𝑴 in dependence of 𝑹𝒆 for different values of 𝑪𝑫 and 𝑮 with centrifugal

through-flow

5.2.6.3 Impact of Pre-Swirl on the Moment Coefficient with a Smooth Disk

The results of 𝐶𝑀 for centrifugal pre-swirl through-flow are depicted in Fig. 88 for 𝐶𝐷 = 5050

and 𝑘𝑠 = 0.4 𝜇m. The pre-swirl reduces the tangential velocity gradient in the disk layer with the

increase of 𝛽. Then, the wall shear stress on the disk surface will decrease correspondingly. 𝐶𝑀

therefore decreases with increasing 𝛽 (inlet angular momentum) for a set of given values of G and

Re. These trends are in accordance with theoretical analyses from Kurokawa et al. [52]. Currently,

the angular momentum at the inlet is not quantified, which deserves further investigations.

0.00030.00080.00130.00180.00230.00280.00330.00380.00430.00480.00530.00580.0063

0.30 1.30 2.30 3.30

百万

0.00030.00080.00130.00180.00230.00280.00330.00380.00430.00480.00530.00580.0063

0.30 1.30 2.30 3.30

百万

0.00030.00080.00130.00180.00230.00280.00330.00380.00430.00480.00530.00580.0063

0.3 1.3 2.3 3.30.00030.00080.00130.00180.00230.00280.00330.00380.00430.00480.00530.00580.0063

0.30 1.30 2.30 3.30

百万

Re

(106)

Re

(106)

Re

(106)

Re

(106)

𝐶𝐷 𝐶𝐷

𝐶𝐷 𝐶𝐷

114

𝐶𝑀 G=0.018 𝐶𝑀 G=0.036

𝐶𝑀 G=0.054 𝐶𝑀 G=0.072

Colors: Red: 𝛽 = 45°; Green: 𝛽 = 15°; Black: no pre-swirl; Yellow: 𝛽 = −15°; Blue: 𝛽 = −45°.

Fig. 88: Experimental results of 𝑪𝑴 𝐯𝐞𝐫𝐬𝐮𝐬 𝑹𝒆 for 𝒌𝒔 = 𝟎. 𝟒 𝝁𝐦 and 𝑪𝑫 = 𝟓𝟎𝟓𝟎

0.0005

0.001

0.0015

0.002

0.3 1.3 2.3 3.3

0.0005

0.001

0.0015

0.002

0.3 1.3 2.3 3.3

0.0005

0.001

0.0015

0.002

0.3 1.3 2.3 3.3

0.0005

0.001

0.0015

0.002

0.3 1.3 2.3 3.3

Re

(106)

𝛽

𝛽 𝛽

𝛽

Re

(106)

Re

(106)

Re

(106)

115

6. Applications of the Results in Radial Pumps

All the results presented in the previous chapters are obtained with the rotor-stator cavity model. It

is very important to apply the results to the design of the turbomachines. In this chapter, two

examples are given to show the applications of the results.

6.1 Flow in the Rear Chamber of a Submersible Multi-Stage Slurry Pump (SMSP)

6.1.1 Sand Discharge Groove Design

The complicated solid-liquid flow in a grooved rotor-stator cavity is strongly influenced by both

the axial and the radial gaps between the rotor and the wall. This part of the thesis presents the

results from numerical simulation on the influences of above two gaps on the solid-liquid flow

inside the grooved rear chamber, aiming to accomplish effective micro-sized sand exclusion. The

leakage flow in the rear chamber of a SMSP is selected as the research model, shown in Fig. 89 (a).

The flow pattern in the rear chamber of a SMSP is shown in Fig. 89 (b).

① : Shaft; ②: Shaft shoulder; ③:Disk; ④: Casing; ⑤: Wall layer; ⑥: Core; ⑦: Disk layer

Fig. 89: Leakage flow and flow pattern inside the rear chamber of a SMSP

Rear chamber

𝑧

Casing

∆r

Broken O-ring

Q

(a)

① ②

③

④

⑤ ⑥

⑦

(b)

0.8

b

b

Rotor

116

Since the micro-sized sand moves with the conveying fluid (Hu et al. [14][15]), the flow patterns

of solid-liquid flow with micro-sized sand may be similar to that of pure water. Due to mass

conservation, an axial convection of the fluid from the outer radius of the disk to that of the stator

takes place. A sand discharge groove, shown in Fig. 90 (a), is designed based on that axial

convection. The groove includes two parts: an annular groove at the outer radius of the stator and

a spiral groove on the inner surface of the stator. The flow in the groove is shown in Fig. 90 (b).

Since the sand moves from the outer radius of the disk to that of the stator, it may enter the annular

groove before it moves towards the disk. Due to the dominant tangential motion, the sand will be

discharged out of the rear chamber along the spiral groove. In this thesis, the ANSYS CFX 14.0

code is used to predict the solid-liquid flow in 6 grooved rear chambers. This part focuses on

selecting the optimum dimensions of both the axial and the radial gaps for accomplishing sand

exclusion.

(a) Design of the sand discharge groove (b) Flow in the sand discharge groove

Fig. 90: Sand discharge groove

6.1.2 Geometrical Parameters

The main geometries of the impeller are given in Table 16.

𝛽2 (o) 𝐷2 (mm) 𝐷1 (mm) 𝑏2 (mm) 휃1 (o) n (rpm) 𝑛𝑠

23 220 80 10 130 2900 690

Table 16: Geometrical parameters of the impeller

Four non-dimensional parameters, namely G, ∆x (non-dimensional radial gap width), Re and 𝐸𝑘

(Ekman number), are used to distinguish the different flow patterns. When 𝐺 << 1, 𝑅𝑒 >> 1

and 𝐸𝑘 << 1, the turbulent sheared flow is considered with dominant inertia effects. Sand is more

likely to move towards the shaft before entering the groove when the axial gap is too wide. At the

same time, the periphery flow will dramatically influence the cavity flow unless the radial gap

Flow in spiral groove

Annular groove

Spiral groove

Flow in annular groove

117

decreases to a certain value (Haudt et al. [35]). Hence, G and Δ𝑥 should be decreased. 6 cavities

are designed based on the above principle, noted from No.1 to No.6. The first three chambers are

designed to find the influence of the radial gap on the flow in the rear chambers. The radial gap of

the last three rear chambers will be designed based on that value, noted as 𝑐𝑠. The amounts of G

for No. 4 to No. 6 are minimized to increase the amount of sand which is transported into the groove.

According to the parameters of the 6 cavities, given in Table 17, the flow regimes in all the cavities

are flow regime IV. In this thesis, the diameter and the lead of the spiral groove are designed as 5

mm and 85 mm, respectively. The diameter of the annular groove is 5 mm.

Cavities Δx 𝐺 Re 𝐸𝑘

No.1 0.4×10-1

3.9×10-1

0.87×106 to 1.2×106

5.5×10-6 No.2 0.3×10-1

No.3 0.2×10-1

No.4

𝑐𝑠

3.1×10-1 8.6×10-6

No.5 2.3×10-1 1.5×10-5

No.6 1.6×10-1 3.4×10-5

Table 17: Parameters of the 6 rotor-stator cavities

6.1.3 Numerical Simulation

The analysis type is set as steady state. All the simulations are carried out using the Multiple Frames

of reference method. The value of inlet pressure is set according to the average pressure at the

entrance of the pump last stage from numerical simulation. The 3D numerical simulation is

conducted with the Ansys CFX 14.0 code. The simulation model includes five domains: inlet,

impeller, rotor-stator cavity, leakage gap and volute, shown in Fig. 91. The meshes of the rear

chamber with the spiral groove are generated using hybrid grids. All other domains are meshed

with structured grids. Since the volute has two outlets, the flows through the two outlet pipes are

considered as an equal value. The standard 𝑘 − 휀 model is used to predict the turbulent flow.

Schiller Naumann model is used to calculate the drag coefficient. Instead of resolving the boundary

layers, the scalable wall functions are used to model the viscous effects in the near wall regions.

The temperature of the water is set to 20℃. Based on the design requirements, the value of the

mass flow rate at the inlet, the pressure at the outlet and the solid volume fraction are set as 5.4

kg/s, 2.6 MPa and 10%, respectively. A mesh independent analysis is conducted to minimize the

error associated with a poor quality mesh [14][15]. The values of 𝑦+ range from 30.6 to 78, which

indicates that the scalable wall functions are valid.

118

Fig. 91: Domains and meshes of the simulation model

6.1.4 Results and Discussion

The contours of the non-dimensional radial velocity are shown in Fig. 92. The positive and the

negative values of 𝑉𝑟 indicate radial outward flow (centrifugal through-flow) and radial inward

flow (centripetal through-flow), respectively. Both the disk boundary layer (where 𝑉𝑟 ≥ 0.03) and

the wall boundary layer (where 𝑉𝑟 ≤ −0.03) can be predicted inside of the three cavities. The

values of 𝑉𝑟 in most of the cavities are close to zero, which indicates that there is a core in each

cavity. The simulation results show that when the non-dimensional radial gap ∆𝑥 decreases to

0.03, a small vortex near the shaft shoulder occurs. The vortex almost remains the same when the

non-dimensional radial gap ∆𝑥 decreases further to 0.02. It indicates that the periphery flow has

little influence on the cavity flow for ∆𝑥 ≤ 0.03. Considering the shaft deformation, ∆𝑥 = 0.02

is not selected as the optimal value. Hence, the value of 𝑐𝑠 for No.4, No.5 and No.6 is set as 0.03.

① : 𝑉𝑟 ≥ 0.03; ② : −0.03 < 𝑉𝑟 <0.03; ③ : 𝑉𝑟 ≤ −0.03

Fig. 92: Distribution of the radial velocity

The distribution of streamlines on the meridian plane is shown in Fig. 93. The movement of sand

is shown by the direction of the arrows. The flow near the spiral groove is turbulent. As shown in

Impeller

Volute with two outlets

Inlet

Cavity with sand discharge groove

Leakage gap

∆𝑥 ∆𝑥 ∆𝑥

119

Fig. 93 (a), most of the sand rotates in the chamber and moves towards the shaft before entering

the groove. Only a small portion of sand flows into the groove. From Fig. 93 (a to c)，it seems that

the amount of sand entering the groove increases with decreasing axial gap width. The cavity flow,

which is obviously not influenced by the spiral groove and the shoulder, is similar to Batchelor

type flow. In Fig. 93 (d), some particles move out of the groove towards the shaft.

(a) No. 2 (b) No. 3

(c) No. 5 (d) No. 6

Fig. 93: Movement of particles on the meridian plane

The distribution of the non-dimensional radial velocity 𝑉𝑟 is shown in Fig. 94 (a to d). The axial

distribution of the non-dimensional radial velocity reveals that there is a central core in each cavity

where the value is close to zero. Near both the disk and the wall, the values of the non-dimensional

radial velocity drop obviously with increasing radial coordinate after reaching their peaks.

Batchelor type flows can be predicted in all of the rear chambers. According to Fig. 94 (d), the

negative values of the non-dimensional radial velocity indicate that the flow moves towards the

shaft. One reason may be that the influence of the spiral groove and the shoulder on the fluid is

strong at that axial gap width.

120

r/(r+Δr) No.2 r/(r+Δr) No. 4

(a) (b)


(c) (d)

𝑧 = 0.2𝑠

𝑧 = 0.4𝑠

𝑧 = 0.6𝑠

𝑧 = 0.8𝑠

𝑧 = 0.9𝑠

𝑧 = 0.2𝑠: close to the disk; 𝑧 = 0.9𝑠: Close to the wall.

Fig. 94: Distribution of non-dimensional radial velocity

The simulation results of the non-dimensional tangential velocity are depicted in Fig. 95. The non-

dimensional tangential velocity increases with decreasing axial gap width. There is a balance

between the centrifugal force and the pressure gradient. Due to the no-slip wall condition, the non-

dimensional tangential velocities on the surface of the rotor and the stator are considered as 0.8

(the radius of the sealing is 0.8𝑏, see Fig. 89) and 0, respectively. With the increase of the non-

dimensional tangential velocity, the sand is likely to move away from the shaft. The simulation

results of the tangential velocity indicate that the axial gap should be minimized.

0.8

0.85

0.9

0.95

1

-0.1 0 0.1

0.8

0.85

0.9

0.95

1

-0.1 0 0.1

0.8

0.85

0.9

0.95

1

-0.1 0 0.10.8

0.85

0.9

0.95

1

-0.1 0 0.1

𝑉𝑟 𝑉𝑟

𝑉𝑟 𝑉𝑟

121


(a) (b)


(c) (d)

𝑧 = 0.2𝑠

𝑧 = 0.4𝑠

𝑧 = 0.6𝑠

𝑧 = 0.8𝑠

𝑧= 0.9𝑠

𝑧 = 0.2𝑠: close to the disk; 𝑧 = 0.9𝑠: Close to the wall.

Fig. 95: Distribution of non-dimensional tangential velocity

The distribution of the solid volume fraction is shown in Fig. 96. The highest solid volume fractions

are shown in the spiral groove. A relative low solid volume fraction is predicted for case No.5. The

value of particle concentration in case No.6 is higher than that in case No.5 in general. Although

the tangential velocity is higher in case No.6, the concentration of sand in the groove is less than

that of case No.4 and case No.5. These results can be explained by analyzing the radial velocity

distributions in Fig. 94 (d). Near the outer radius of the disk, the flow moves towards the shaft

before it enters the groove. It seems that the geometries of the spiral groove and the shoulder have

a negative effect on the sand exclusion in case No. 6.

0.8

0.85

0.9

0.95

1

0.3 0.4 0.5 0.6 0.7 0.8

0.8

0.85

0.9

0.95

1

0.3 0.4 0.5 0.6 0.7 0.8

0.8

0.85

0.9

0.95

1

0.3 0.4 0.5 0.6 0.7 0.8

0.8

0.85

0.9

0.95

1

0.3 0.4 0.5 0.6 0.7 0.8

𝑉𝜑 𝑉𝜑

𝑉𝜑 𝑉𝜑

122

①: 𝜑𝑠 > 2%; ②: 1% ≤ 𝜑𝑠 < 2%; ③: 0.5%≤𝜑𝑠 < 1%; ④: 0.2% ≤ 𝜑𝑠 < 0.5%; ⑤: 𝜑𝑠 < 0.2%

Fig. 96: Distribution of solid volume fraction

In order to further analyze the accuracy of the results from the numerical simulations, the SMSP

with cavity No.5 is manufactured and tested. The test rig precedes the requirements of international

B-grade precision (ISO9906-1999). The uncertainties of the results are given in Table. 18. The

density and the kinematic viscosity of the working fluid are 1160 kg/m3 and 1.06 m2/s, respectively

at 20℃. The test rig has a circulating system at the bottom, which can reduce the settlement of sand.

The size of the sand is selected with screens. During the test, the concentration of the sand is

measured every 48 hours. An appropriate amount of sand is added to the slurry when the sand

concentration decreases more than 20%. The measured pump performance is given in Fig. 97.

Pressure Flow rate n Electric Voltage Electric current Power η

0.1% 0.2% 0.2% 0.1% 0.1% 0.14% 0.26%

Table 18: Uncertainties of results

0 5 10 15 20 25 30 35 400

50

100

150

200

250

300

350

400

450

Po

wer(

kW

)E

ffic

ien

cy

(%)

H

H (

m)

Q (m3/h)

0

10

20

30

40

50

ا¦

30

40

50

60

70

P

Fig. 97: Pump performance from experiments

① ① ① ① ② ②

②

②

③ ③

③

③

④ ④

④

④ ⑤ ⑤

⑤

⑤

H

η

P

No.2 No.4 No.5 No.6

Flow (m3/h)

Pre

ssu

re h

ead

(m

)

123

An abrasion test is also accomplished to verify the correctness of sand exclusion. The outlet

pressure is monitored with a pressure sensor during the experiments. When the mechanical seal is

broken, a high pressure difference contributes to a large leakage flow from the rear chamber to the

environment. It results in the plunge of the pressure head, shown in Fig. 98. The parameter 𝑝𝑟 is

defined in Eq. 131. The mechanical seal is considered broken when the values of 𝑝𝑟 are below 0.9.

According to the experimental results, the service life of the SMSP is significantly improved by

around 30% with cavity No. 5.

𝑝𝑟 =

𝑃𝑥

𝑃0

Eq. 131

Fig. 98: Pressure drop during the abrasion test

6.2 The Axial Thrust in a Deep Well Pump

6.2.1 Main Geometric Parameters

The bearing systems of pumps are fragile, especially when the large leakage flow occurs. Shi et al.

[84] describe an investigation of the axial thrust of a deep well pump based on both the numerical

simulation and the axial thrust measurements. The main geometric parameters of the pump are

listed in Table 19.

Geometric parameters Values Geometric parameters Values

Blade number 7 Impeller diameter at the back cover 132 mm

Blade inlet angle 30° Impeller inlet diameter 𝐷1 62 mm

Blade outlet angle 𝛽2 35° Impeller outlet width 𝑏2 12 mm

Wrapping angle 휃1 115° Impeller hub diameter 38 mm

Impeller diameter at the front cover 148 mm Shaft diameter 28 mm

Table 19: Main geometric parameters of the pump

0.6

0.7

0.8

0.9

1

0 200 400 600 800 1000

1

2

Prototype

Cavity No.5

𝑡ℎ (hour)

𝑝𝑟

Broken mechanical seal

124

The 3D model of the impeller and the guide vane are depicted in Fig. 99 (a) and Fig. 99 (b),

respectively [84].

Fig. 99: Geometry of the impeller (left) and the guide vane (right)

6.2.2 Simulation Set-Up

The fluid domains are modeled in Creo 2.0. To predict the inner flow, the axial thrust and the

frictional torque, numerical simulations are carried out using the ANSYS CFX 14.0 code by Shi et

al. [84]. The CFX set-up by Shi et al. [84] is described in this part. The simulation type is set as

steady state. The standard 𝑘 − 휀 turbulence model in combination with the scalable wall functions

is selected. The chosen boundary conditions are total pressure at the inlet and mass flow at the

outlet, respectively. The convergence criteria for all the numerical simulations are set as 10−4 in

maximum type. To minimize the error associated with the quality of the mesh, a mesh sensitivity

analysis is accomplished by comparing the simulation results of the axial thrust.

6.2.3 Results for Axial Thrust Coefficient

The pressure acting on the impeller of a centrifugal pump is shown in Fig. 100 (a). Based on Eq.

124, the thrust coefficient at surface 1 and surface 2 can be calculated when the leakage flow is

estimated (centripetal through-flow). The maximum difference between the simulation results [84]

and the measurements [84] of the axial thrust is 5.9%. The values of 𝐹𝑎𝑏 − 𝐹𝑎𝑓 are plotted versus

𝐶𝐷 in Fig. 100 (b). The experimental results of 𝐹𝑎𝑏 − 𝐹𝑎𝑓 are obtained by subtracting the sum of

the impulse force at the impeller eye 𝐹𝑎𝑠 and the force on the impeller passage 𝐹𝑎𝑝 (from

numerical simulations [84]). The trend of the results from Eq. 124 is in better agreement with the

125

experimental results than that from the equation by Kurokawa et al. [53] when |𝐶𝐷| ranges from

1150 to 4630.

(a) (b)

Fig. 100: Axial thrust in a centrifugal single stage well pump [84]: (a) Pressure distribution and

(b) Comparison of 𝑭𝒂𝒃 − 𝑭𝒂𝒇

-50

0

50

100

150

200

1000 2000 3000 4000 5000

𝐹𝑎

𝑏−

𝐹 𝑎𝑓

Suction

pressure

|𝐶𝐷|

Pre

ssu

re

in

the

bac

k c

ham

ber

Pressu

re in th

e

front ch

amb

er

Surface 3

Su

rfac

e 2

Su

rface 1

𝐹𝑎

126

7. Summary of the Results

The key parameters studied within the scope of this thesis are listed as follows:

a. Through-flow coefficient 𝐶𝐷:

① Centripetal through-flow;

② Centrifugal through-flow;

b. Core swirl ratio K;

c. Non-dimensional axial gap width G;

d. Global Reynolds number Re;

e. Local circumferential Reynolds number 𝑅𝑒𝜑;

f. Equivalent surface roughness 𝑘𝑠;

g. Pre-swirl;

h. Axial thrust coefficient 𝐶𝐹;

i. Moment coefficient 𝐶𝑀.

According to the results from both the numerical simulations and the experiments, some achieved

results and determined correlations are as follows:

a. Two correlations to evaluate the core swirl ratio K with a smooth disk for both centripetal

and centrifugal through-flow;

b. Two correlations for K by introducing the impact of surface roughness of the disk for both

centripetal and centrifugal through-flow;

c. Some experimental results of K in a rotor-stator cavity with centripetal or centrifugal pre-

swirl through-flow with a smooth disk;

d. Two equations for the axial thrust coefficient 𝐶𝐹 for the first time in the field of

turbomachinery to introduce the impacts of 𝐶𝐷, Re, G and 𝑘𝑠 for centripetal or centrifugal

through-flow;

e. A 3D diagram distinguishing regime III and regime IV for both centripetal and centrifugal

through-flow;

f. Two sets of correlations to predict the influences of 𝐶𝐷, Re, G and 𝑘𝑠 on 𝐶𝑀 for regime

III and regime IV.

The applications of the results in the future are offered:

a. An extensive data base of 𝐶𝐹 and 𝐶𝑀, which is of a huge worth for the designers of radial

pumps and turbines;

b. New correlations for 𝐶𝐹 and 𝐶𝑀 with good accuracy, which can be easily implemeted

into the design tools;

127

c. Effective sand exclusion in the rear chamber of radial slurry pumps, which can improve

the service life of sealings;

d. Reduction of the axial thrust, which can improve the service life of bearing systems in

radial pumps or turbines;

e. Reduction of the frictional losses of the disks, which can reduce the energy consumption

of a radial turbomachine by modifying the chamber geometry.

128

8. Outlook

Nevertheless a large amount of work has been accomplished, there are still some limitations in this

work.

1. The distinguishing lines for regime III and regime IV are obtained by evaluating the

tangential flow component based on numerical simulations. This has to be put on an

experimental level by measuring the velocity components in both the tangential and the

radial directions with a two-component LDV system in the future.

2. To quantify the impact of pre-swirl on the cavity flow, more investigations should be

accomplished in the future with more variations of the pre-swirl angle.

3. Currently, it is not possible to obtain rougher surfaces than those investigated in this thesis.

More complicated, the surface roughness is a 3D parameter, while the measured 𝑅𝑧 is the

average of the five highest peaks and the five deepest valleys (2D parameter). How to

expand the 2D surface roughness into 3D more correctly also deserves further investigation.

4. The impacts of boundary conditions at the inlet on K, 𝐶𝐹 and 𝐶𝑀 deserve further

investigation.

5. To better understand the scientific meaning of the correlations, massive fundamental

researches are still required in the future work.

6. Experiments in real machines have to be conducted to verify the results obtained in this

thesis and to show the applicability of the deduced correlations.

129

References

1. Altmann, D., 1972, Beitrag zur Berechnung der turbulenten Strömung im Axialspalt zwischen

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