Simulation and Modeling of the Hydrodynamic, Thermal, and Structural Behavior of Foil Thrust Bearings

by

Robert Jack Bruckner

Submitted in partial fulfillment of the requirements

For the degree of Doctor of Philosophy

Dissertation Advisor: Dr. Joseph M. Prahl

Department of Mechanical and Aerospace Engineering

CASE WESTERN RESERVE UNIVERSITY

August, 2004

Dedications

All of the work leading to and included in these pages is dedicated to the loving support of my family, Lisa, Eric, and Elisabeth Table of Contents

CHAPTER 1

Introduction to Foil Bearings ...... 17

1.1 Historical Context of Hydrodynamics...... 17

1.2 Foil Bearing State of the Art...... 19

1.3 Aviation Turbofan Engine Application...... 22

1.4 Typical Geometries and Characteristics of Foil Thrust Bearings...... 23

CHAPTER 2

Development of the Governing Equations...... 29

2.1 The Generalized Foil Bearing Problem...... 29

2.2 Reynolds Equation ...... 29

2.2.1 Development from Mass and Momentum Conservation...... 29

2.2.3 Cylindrical (Thrust Pad) Form of Reynolds Equation ...... 40

2.3 Density Models ...... 43

2.3.1 Power Law Models...... 43

2.3.2 The Energy Equation...... 44

CHAPTER 3

Solution Technique and Numerical Method ...... 54

3.0 Overview ...... 54

3.1 Derivative Expansions and Coefficient Functions...... 55

3.2 Finite Differencing Scheme...... 56

3.3 Boundary Conditions...... 57

3.4 Iteration Strategy...... 59

1 3.5 Mathematica Code ...... 60

CHAPTER 4

One-Dimensional Bearing Analysis ...... 65

4.1 Analytic Solutions...... 65

4.1.1 Rigid Straight Taper Bearing...... 66

4.1.2 Rayleigh Step Bearing...... 68

4.1.3 Double Taper Bearing ...... 72

4.1.4 Isobaric Channel Flow with Viscous Heat Generation ...... 77

4.2 Numerical Solutions...... 80

4.2.1 Parallel Channel Flow with Viscous Heat Generation...... 80

4.2.2 Power Law Density Models...... 82

4.2.3 Compliant Structure ...... 84

4.2.4 Viscous Heat Generation...... 85

4.3 General Observation...... 88

CHAPTER 5

Cylindrical (Thrust Pad) Bearing Analysis...... 90

5.1 Impacts of Side Leakage for Zero Preload ...... 90

5.2 Impacts of Side Leakage for Highly Loaded Bearings ...... 110

5.2.1 Incompressible, Constant Property Lubricant ...... 111

5.2.2 Impacts of Power Law Density Models ...... 115

5.2.3 Impacts of Compliant Foundation...... 119

5.5 Impacts of Viscous Heat Generation...... 124

5.6 General Observation...... 130

2 CHAPTER 6

Comparison of Simulations and Experiments...... 131

6.1. Low Load Wear ...... 131

6.2 Foil Thermocouple Test ...... 135

6.2 Torque versus Load Data...... 140

CHAPTER 7

Summary and Conclusion...... 142

Appendix A...... 146

Summary of Governing Equations ...... 146

Appendix B

Mathematica Iterator Loop ...... 150

Appendix C

Closed form solution to the Rayleigh step bearing ...... 151

Appendix D

Closed form solution to the double taper bearing ...... 155

References...... 159

3 List of Tables

Table 1. Generation I foil thrust bearing parameters...... 28

Table 2. Table of characteristic squeeze number and modified Reynolds number...... 33

4 List of Figures

Figure 1. Figure of Osborne Reynolds original hydrodynamic analysis...... 25

Figure 2. Typical foil journal bearing...... 26

Figure 3. Figure of foil thrust bearing having complex structure...... 26

Figure 4. Figure of the Generation I foil thrust bearing ...... 27

Figure 5. Typical top foil shape for generation I foil thrust bearings...... 27

Figure 6. Typical load deflection curve for an 8 pad generation I foil thrust bearing...... 28

Figure 7. Representation of a rectilinear foil bearing...... 52

Figure 8. Representation of a thrust foil bearing...... 52

Figure 9. Diagram of the energy equation control volume...... 53

Figure 10. Diagram of the division of the energy equation domain...... 53

Figure 11. Typical numerical grid in thrust pad geometry...... 60

Figure 12. Program flow chart...... 61

Figure 13. Sample of program convergence on load...... 62

Figure 14. Mathematica code to solve Reynolds equation in cylindrical coordinates. ....63

Figure 15. Mathematica code for solving the structural model in cylindrical coordinates.

...... 64

Figure 16. Load capacity versus runner drag and power loss for a rigid, straight taper

bearing with an incompressible, constant property lubricant...... 67

Figure 17. Operating characteristics of the straight taper bearing having contraction

ratios, k, of 10, 5, 2.2, and 1.5...... 68

Figure 18. Load capacity contours for the Rayleigh step bearing as a function of

contraction ratio, k, and change location, cl...... 70

5 Figure 19. Load capacity versus power loss for the Rayleigh step bearing...... 71

Figure 20. Operating characteristics of the Rayleigh step bearing having change

locations, cl, of 0.9, 0.718, 0.5, and 0.2...... 72

Figure 21. Load capacity plot for the double taper bearing as a function of inlet

contraction ratio, ka, and exit contraction ratio, kb, for a change location, cl, or 0.25.

...... 74

Figure 22.Load capacity plot for the double taper bearing as a function of inlet

contraction ratio, ka, and exit contraction ratio, kb, for a change location, cl, or 0.50.

...... 74

Figure 23. Load capacity plot for the double taper bearing as a function of inlet

contraction ratio, ka, and exit contraction ratio, kb, for a change location, cl, or 0.75.

...... 75

Figure 24. Contour plot of load capacity for the taper-parallel bearing as a function of

contraction ratio, ka, and change location, cl...... 76

Figure 25. Operating characteristics of the "optimized" taper-parallel bearing...... 77

Figure 26. Graph of required channel divergence for Isobaric flow with viscous heat

generation for inlet channel heights or 200, 500, and 1,000 micro-inches...... 79

Figure 27. Operating characteristics of parallel channel flow with viscous heat

generation...... 82

Figure 28. Operating characteristics of bearings using the power law density and

viscosity model...... 83

Figure 29. Operating characteristics of a bearing having a compliant foundation and top

foil...... 85

6 Figure 30. Operating characteristics of a bearing with the energy equation and power law

viscosity model...... 87

Figure 31. Operating characteristics of a bearing with the energy equation, power law

viscosity model, and compliant foundation...... 88

Figure 32. Photograph of a generation I foil thrust bearing showing wear in the center of

the pads after a low load test...... 91

Figure 33. Plot of the three film thickness profiles considered in the low load side

leakage calculations...... 92

Figure 34. Typical hydrodynamic performance for the incompressible, constant property

bearings...... 93

Figure 35. Centerline pressure ratios for the Cartesian bearing, α=1 ...... 94

Figure 36. Pressure ratio field for the Cartesian bearing, α=1, double-taper ...... 95

Figure 37. Pressure ratio field for the Cartesian bearing, α=1, parallel film...... 95

Figure 38. Centerline pressure ratios for the Cartesian bearing, parametric α, double -

taper profile ...... 96

Figure 39. Pressure ratio surface, α=1...... 97

Figure 40. Pressure ratio surface, α=2...... 97

Figure 41. Pressure ratio surface, α=3...... 98

Figure 42. Centerline pressure ratios for the Cartesian bearing, parametric α, ”smoothed”

profile...... 98

Figure 43. Pressure ratio surface, α=3...... 99

Figure 44. Pressure ratio surface, α=1...... 99

Figure 45. Pressure ratio surface, α=2...... 100

7 Figure 46. Pressure ratio distribution for the "smoothed" profile in radial plan form ...102

Figure 47 Thrust bearing pressure distributions for various geometry...... 105

Figure 48. Typical thrust bearing angular pressure distributions...... 106

Figure 49. Typical thrust bearing radial pressure distributions...... 106

Figure 50 Effect of varying bearing number for fixed geometry thrust bearing...... 107

Figure 51 Angular pressure distributions at the approximate maximum pressure radius.

...... 108

Figure 52. Example of a top foil shape when the compliant foundation is superimposed

on the static film thickness and pressure distribution...... 109

Figure 53. Close up photograph of the wear spot on the foil thrust bearing demonstrating

the similarity between calculations and experiment...... 109

Figure 54. Static top foil shape used for the side leakage effect calculations...... 111

Figure 55. Pressure ratio distribution for the incompressible, constant viscosity lubricant.

...... 112

Figure 56. Vector plot of mass flow in cylindrical coordinates for the incompressible,

constant viscosity lubricant...... 113

Figure 57. Vector plot of mass flow in rectilinear coordinates for the incompressible,

constant viscosity lubricant...... 114

Figure 58. Pressure ratio distribution for the power law model for density and viscosity,

n=5.0, a=0.5...... 116

Figure 59. Vector plot of mass flow in cylindrical coordinates for the power law model

for density and viscosity, n=5.0, a=0.5...... 117

8 Figure 60. Vector plot of mass flow in rectilinear coordinates for the power law model

for density and viscosity, n=5.0, a=0.5...... 118

Figure 61. Plot of static and deformed trailing edge film thickness for the compliant

foundation analysis...... 120

Figure 62. Plot of the deformed shape of the top foil...... 121

Figure 63. Pressure distribution corresponding to the film thickness created by the

deformed top foil...... 122

Figure 64. Vector plot of mass flow in cylindrical coordinates for the case of the

compliant foundation...... 123

Figure 65. Vector plot of mass flow in rectilinear coordinates for the case of the

compliant foundation...... 124

Figure 66. Pressure distribution corresponding to the viscous dissipation case...... 126

Figure 67. . Temperature distribution corresponding to the viscous dissipation case.....127

Figure 68. Vector plot of mass flow in cylindrical coordinates for the case of viscous

heat generation...... 128

Figure 69. Vector plot of mass flow in rectilinear coordinates for the case of viscous

heat generation...... 129

Figure 70. Inital film thickness for low speed simulations...... 132

Figure 71. Bump foil stiffness model...... 133

Figure 72. Typical low load hydrodynamic pressure field...... 134

Figure 73. Calculations showing typical low speed, low load foil shape. The white

regions show high spots where high-speed contact is most likely to occur...... 135

9 Figure 74. Experimental photograph showing witness marks after a low load, low speed

...... 135

Figure 75. Post-test photograph of the modified foil thrust bearing...... 136

Figure 76. Static Film Thickness...... 138

Figure 77. Bump Foil Stiffness Model ...... 138

Figure 78. Graphical Simulation Results...... 139

Figure 79. Close-up of "flat section" foil deflection ...... 139

Figure 80. Static and Deflected trailing edge film thickness...... 140

Figure 81. Calculated and Measured bearing temperatures...... 140

Figure 82. Bearing torque versus load for various lubricant models...... 141

10 Acknowledgements

The author would like to acknowledge with deepest gratitude the guidance, support, and motivation of Dr. Christopher DellaCorte of the NASA Glenn Research Center as well as the entire staff of the Oil-Free Turbomachinery team.

Professor Joseph M. Prahl, Chairman of the Department of Mechanical and Aerospace Engineering at Case Western Reserve University, the author’s dissertation advisor, is especially acknowledged for technical insight and interest in this research.

11 List of Abbreviations a temperature exponent in the viscosity power law relationship

CF coefficient function used in the Reynolds equation solution

CFr coefficient function used in the Reynolds equation solution

CFθ coefficient function used in the Reynolds equation solution

Cp specific heat capacity for the lubricant

D runner drag, width of a rectilinear bearing

E material modulus of the top foil

g static (no load) film thickness function

h film thickness function, static plus structural deflections

H characteristic film thickness

HPloss power loss due to runner shear stress

i finite differencing index variable for x or r

j finite differencing index variable for y or θ

Κc stiffness of the compliant foundation

Kb,f bending stiffness of the top foil

Kn Knudsen number at the bearing leading edge

L length of a rectilinear bearing

m mass flow rate

p hydrostatic pressure

Qs heat transfer between the lubricant and surroundings

ReL Reynolds number base on length scale L

ReM Modified Reynolds number

12 RR inner radius ratio for the cylindrical thrust bearing r radial direction coordinate in cylindrical thrust bearings t time coordinate, thickness of the top foil

T temperature ur lubricant velocity in the radial direction for cylindrical thrust bearings uθ lubricant velocity in the angular direction for cylindrical thrust bearings u lubricant velocity in the x-direction for rectilinear bearings

U characteristic velocity in the x-direction v lubricant velocity in the y-direction for rectilinear bearings

V characteristic velocity in the y-direction w lubricant velocity in the z-direction for rectilinear bearings

W characteristic velocity in the z-direction x direction of primary runner motion in a rectilinear bearing y direction of the film thickness z direction of the width of a rectilinear bearing

α width to length ratio for a rectilinear bearing

δ deflection of the top foil

Φ viscous dissipation function

γ ratio of specific heat of the lubricant

Γ mean free path of the lubricant

Λ bearing number for the rectilinear bearing

Λc bearing number for the cylindrical thrust bearing

13 µ viscosity of the lubricant

ν Poisson’s ratio of the top foil material

σ squeeze number

θ angular direction of the cylindrical thrust bearing

Subscripts

a ambient, bearing entrance conditions f conditions at / of the foil r conditions at / of the runner o non-dimensional parameters for the Reynolds equation development x,y,z quantities in the x, y, or z directions r,θ quantities in the r or θ directions

Overbars

(symbol) non-dimensionalized parameters

14 Simulation and Modeling of the Hydrodynamic-Thermal- Structural Interactions in Foil Thrust Bearings

Abstract

by

Robert Jack Bruckner

A simulation and modeling effort is conducted on foil gas thrust bearings. A foil bearing is a self acting hydrodynamic device capable of separating stationary and rotating components of rotating machinery by a film of air or other gaseous “lubricant”.

Although simple in appearance these bearings have proven to be complicated devices.

They are sensitive to fluid structure interaction, use a compressible gas as a lubricant, may not be in the fully continuum range, and operate in the range where viscous heat generation is significant. These factors provide a challenge to the simulation and modeling task.

The conservation equations of mass, momentum, and energy are applied to the problem.

The traditional Reynolds equation is developed with the addition of a Knudsen number effect due to thin film thicknesses. The energy equation is simplified by applying the thin layer assumptions such that fluid properties do not vary through the film. Heat transfer between the lubricant and the surroundings is also taken into consideration. The structural deformations of the bearing are modeled with a single partial differential equation. The equation models the top foil as a thin, bending dominated membrane whose deflections are governed by the biharmonic equation. A linear superposition of hydrodynamic load and compliant foundation reaction is included. The stiffness of the

15 compliant foundation is modeled after a set of discrete springs that support the topfoil.

The system of governing equations is solved numerically by a computer program written in the Mathematica computing environment. A generalized hydrodynamic analysis is conducted to systematically analyze each of the individual effects included in the development of the governing equations.

Previous analytic work on foil thrust bearings includes the modeling of the Reynolds equation with an isothermal density model and an additional model to predict top foil deflections. This work finds a substantial difference between bearing performance based on traditional lubricant models and that based on the energy equation model. Qualitative and quantitative comparisons are produced that demonstrate the utility of the current approach, which couples the Reynolds, energy, and structural equations.

16 CHAPTER 1 Introduction to Foil Bearings

1.1 Historical Context of Hydrodynamics

It could be considered that the field of analytic hydrodynamics can trace it genesis to

1842 in Belfast, Ireland with the birth of Osborne Reynolds. Even today, the governing equations of hydrodynamic lubrication, as well as many other important ideas in fluid flow and other natural phenomena, bear his name. Born as the son of the headmaster of the Dedham school in Essex, academics were ingrained into Osborne Reynolds as a very important part of life. However, prior to undertaking university studies he first gained real world experience through employment at the Edward Hayes Engineering Firm. This pattern of first gaining understanding through experience or observation then modeling the phenomenon with mathematics became the formula of success for Osborne Reynolds.

This mixture of experimental observations and theoretical analysis remains important in hydrodynamics especially in the current area of research, foil thrust bearings. After gaining experience with the engineering firm, Reynolds studied Mathematics at

Cambridge University where he was a classmate of John William Strutt, Lord Rayleigh, another great scientist and engineer of the time. Reynolds graduated in 1867 and became the first Professor of Engineering in Manchester where he held a position until his retirement in 1905.

The specific field of hydrodynamics was inspired to Osborne Reynolds by experimental research conducted by Beauchamp Tower of the Institute of Mechanical Engineers. The results of this research were first reported by that institute in 1883 and 1884 through a

17 report entitled “The Friction of Lubricated Journals”. In 1886 Osborne Reynolds

published the seminal work of hydrodynamics in the Philosophical Transactions of the

Royal Society, “On the Theory of Lubrication and its Application to Mr. Beauchamp

Tower’s Experiments, including an Experimental Determination of the Viscosity of Olive

Oil”. Figure 1 taken from Reynolds’ work shows the four cases of fluid flow considered

in this publication culminating in the straight tapered slider bearing analysis. In the

words of Reynolds a significant feature of lubrication is summarized below.

…The result of the whole research is to point to a conclusion

(important in Natural Philosophy) that not only in cases of intentional

lubrication, but wherever hard surfaces under pressure slide over each

other without abrasion, they are separated by a film of some foreign

matter, whether perceivable or not. And that the question as to whether

this action can be continuous or not, turns on whether the action tends to

preserve the matter between the surfaces at the points of pressure, as in

the apparently unique case of the revolving journal, or tends to sweep it to

one side , as is the result of all backwards and forwards rubbing with

continuous pressure…. [1]

This ability to maintain lubricating “matter” between hard surfaces and to use it efficiently is paramount to the operation of foil thrust bearings. This observation was made by Reynolds over a century before, and is shown in this work to be the single most important design consideration in modern highly loaded foil thrust bearings.

18

1.2 Foil Bearing State of the Art

A foil bearing is a self acting hydrodynamic device capable of separating rotating components of rotating machinery by a film of air or other gaseous “lubricant”. The benefits of such bearings to high speed turbomachinery can be quite significant. Among these benefits are reduced machine weight due to the elimination of the oil systems, removal of the traditional diameter – rotational speed limit imposed by rolling element bearings, and a synergistic use of working fluid as a lubricant enabling a contaminant free working fluid. However in order to realize these benefits, foil bearing must be able to support the required machine load and operate reliably and without burden to the machine. Excess burden to a turbomachine usually manifests itself in the detrimental use of working fluid as bleed flow to cool or pressurize specific components.

A typical foil journal bearing is shown in figure 2. It consists of a rigid, stationary shell that contains a compliant foundation. This compliant foundation can take different forms, but is usually portrayed as a thin layer of corrugated metal referred to as the bump foil. A smooth foil, known as the top foil, is then fastened to one edge of the bump foil. A rotating shaft fits inside the top foil. Foil bearings typically operate without static clearance between the stationary top foil and rotating shaft. In fact there is often a preload force between the two parts. At a certain speed know as the lift off speed, the hydrodynamic force overcomes the preload force and complete fluid film lubrication is established.

19 The foil journal bearing has received much research and development attention over the past two decades with the primary objective of increasing load capacity and understanding stiffness and damping characteristics of the bearings. Until recently the foil thrust bearing has received very little research and development attention. This circumstance is primarily due to the fact that until recently, the thrust loads in machines using foil bearings could be managed and reduced to acceptable levels through clever machine designs. However potential future applications include those in which the foil bearing supported turbomachinery is part of a larger system or a closed loop. For example, aviation turbofan engines and space nuclear Brayton-cycle alternators operate in such modes. In these applications the thrust loads are no longer merely a function of machine design, but are imposed on the machine by specific operating conditions. In order to extend the utility of foil bearings to these applications, the physics and details of the foil thrust bearing must be exposed through experiments and analysis.

One of the earliest published works on the idea of a foil bearing, in journal bearing form and using oil as the lubricant, was published in 1953 by H. Blok of the Technical

University at Delft in Holland [2]. In this work Blok describes an experimental apparatus in which a foil is wrapped around 180 degrees of a journal and loaded via a system of weights and pulleys. The results of friction and load tests are presented with theoretical equations representing the accommodation of the foil membrane to the hydrodynamic pressure. It is interesting to note here that the foil bearing seems to have been named due to the fact that the top foil had no appreciable bending stiffness. Historically, the term foil taken in technical context refers to a thin membrane with no appreciable bending

20 stiffness. One of the keys to highly loaded foil journal bearings has been to stiffen the top foils. Therefore it could be said that modern foil bearings no longer contain foils and should be renamed.

The foil gas bearing initially found application in the field of magnetic tape storage media where the tape behaved like a foil and it was also desirable that there be little or no dry sliding friction between the tape and other solid surfaces. The next major application of foil bearings occurred in the 1970’s with the air cycle machines on large commercial aircraft. These machines provide air conditioning and cabin pressurization for high altitude flights. Of paramount importance in the operation of the air cycle machines is to provide contaminant free air to the occupants of the vehicle. The foil air bearing enabled the removal of the oil system, the most serious threat to air contamination. The ability to operate efficiently for long periods of time with minimal maintenance was also enabled by the application of foil bearings. The 1980’s saw the application of foil bearings to turbocompressors. Again, the characteristic of no process fluid contamination was paramount. In addition, long life cryogenic capabilities were important for this application. In the late 1990’s the first commercially available oil-free turbogenerators became available and the first oil-free diesel engine turbocharger was demonstrated.

These machines also used foil bearings. A significant change in the nature of foil bearing occurred with this application. Until the 1990’s foil bearing applications have been limited to low temperature and low load capacity applications so that the bearing surfaces could be coated with soft, low friction materials such as Teflon to minimize the dry rubbing friction on startup, shutdown, and overload conditions. However, the

21 turbogenerators were the first high temperature, high load capacity application of foil bearings.

Applications that are on the horizon include the commercialization of the diesel turbocharger, space nuclear power generation, and aviation turbofan engines. Each of these applications requires the characteristics of high temperature, highly loaded foil bearings. Many significant challenges must be overcome to meet the demands of these future applications. One of the most significant challenges is the understanding of highly loaded foil thrust bearings. These applications include turbomachines which are either in a closed loop or part of a larger machine making the management of thrust loads through pressure balancing difficult. Therefore in order to extend the use of foil bearings to higher temperatures and higher loads a greater understanding of compressibility effects, viscous heat generations, and fluid structure interactions are required. The aim of this work is to provide insight into these phenomena to provide guidance to bearing and machine designers.

1.3 Aviation Turbofan Engine Application

The application of foil bearings to small and mid-range aviation turbofan engines can enable substantial savings in the design and operation of a vehicle designed with such engines. A propulsion system study [3] on a 50 passenger commercial regional jet and a notional 10 passenger supersonic business jet concludes that engine weight can be reduced by as much as 26%. These are a significant benefit to a machine as complex and mature as commercial turbofan engine. To place this benefit into perspective, the NASA

22 Ultra Efficient Engine Technology Program’s goal for reducing emissions from commercial aviation engines for carbon dioxide emissions reductions were 15% and 8% respectively for subsonic and supersonic flight. The application of foil bearings to these engines could achieve nearly a 3% reduction for both of these applications. Additionally, an oil-free turbofan engine is a much simpler machine that is less likely to experience unanticipated problems, simpler to maintain, and more affordable to operate.

1.4 Typical Geometries and Characteristics of Foil Thrust Bearings

A typical foil thrust bearing is shown in figure 3. The bearing consists of a solid backing plate with an annulus of discrete bearing pads or petals. Each of these pads is constructed from a smooth and continuous top foil that is fixed on the leading edge and free on the remaining three edges. The backside of the top foil is supported by a compliant foundation. Figure 3 shows a very complex foundation consisting of many layers of flat and corrugated foils. The hydrodynamic pressure rise is initially created by the physical contraction of the film thickness between the top foil and the thrust runner. The compliant foundation can take any number of styles and forms. However for the current work the specific form is that of a single corrugated bump foil supporting a single continuous top foil, such as that shown in figure 4 and is referred to as a generation I [4, 5] foil thrust bearing. Specific characteristics are summarized in Table 1.1. The nominal, undeflected top foil shape is shown in figure 5. This shape consists of two distinct sections, a straight taper of 0.002 inches in height followed by a flat section. The flat section represents a preload force holding the runner to the bearing. The second part of figure 5 also shows an approximation of the top foil shape under zero preload. In this

23 case both sections of the top foil form a straight taper with the angle of the second section being shallower than the first. Figure 6 contains a typical load deflection curve for an 8 pad generation I foil thrust bearing. This type of plot is used in the analysis to determine the spring constant of the compliant foundation. This curve consists of three distinct sections. The lower portion of the curve from a deflection of 1.5 to 2.5 mil is dominated by the seating of the top foil onto the bump foil. The central curved portion contains additional seating of the top foil as well as initial compression of the bump foil. The final portion is used to model the high load stiffness of the bump foil. Although figure 6 indicates some hysteresis due to the fact that the loading and unloading data points do not lie directly on one another, it is unclear whether this is due to test apparatus or the nature of the compliant structure itself. For the purposes of the current work, an “average” load deflection curve is considered to represent the compliant foundation of the foil thrust bearing.

24

Figure 1. Figure of Osborne Reynolds original hydrodynamic analysis. [1]

25

Figure 2. Typical foil journal bearing.

Figure 3. Figure of foil thrust bearing having complex structure.

26

Figure 4. Figure of the Generation I foil thrust bearing

Figure 5. Typical top foil shape for generation I foil thrust bearings.

27

Figure 6. Typical load deflection curve for an 8 pad generation I foil thrust bearing.

Typical Foil Thrust Bearing

Radius, inner 1.766 in. / 45 mm

Radius, outer 0.91 in / 23 mm

Radius, midspan 1.388 in. / 34 mm

Azimuthal extent 43 degree / 0.751 radians

Length, midspan 1.04 in. / 26 mm

Characteristic Speed 40,000 rpm

Midspan Speed 485 fps / 148 m/s

Lubricant air at sea level conditions

Top Foil material Inconnel X-750

Top Foil Thickness 0.005 in. / 0.13 mm

Compliant Foundation Corrugated bump foil

Table 1. Generation I foil thrust bearing parameters.

28 CHAPTER 2 Development of the Governing Equations

2.1 The Generalized Foil Bearing Problem

The generalized challenge of foil thrust bearing simulations entails the simultaneous solution of nine unknown variables. These unknowns include the three velocity components, pressure, temperature, density, viscosity, specific heat capacity, and structural deflection. Of course, nine equations or relationships are required in order to solve this simulation. These relations include the three scalar conservation of momentum equations, conservation of mass, conservation of energy, an equation of state, two property equations, and a structural model. The complete simulation of the structural behavior of the foil thrust bearing would introduce a host of additional unknowns and requisite equations. Therefore a simplified model is introduced in the hydrodynamic simulation. In order to reduce the size of this simulation even further a scaling analysis is performed that includes the traditional thin layer hydrodynamic assumptions. These assumptions allow for the elimination of the three velocity components and the related momentum equations. The simulation can then be reduced to three partial differential equation and three algebraic equations. The resulting simulation is then solved for the pressure, temperature, film thickness, density, viscosity, and heat capacity fields.

Summaries of the set of governing equations for rectilinear and cylindrical thrust plan forms are shown in Appendix A.

2.2 Reynolds Equation

2.2.1 Development from Mass and Momentum Conservation

29 The Reynolds equation is used for the hydrodynamic simulation of the foil thrust bearing.

It is developed from the basic conservation laws of mass and momentum of a continuum fluid element [6, 7]. The range of applicability for the equation can be extended into the sub-continuum range for Knudsen number greater than 0.01 if slip flow is accounted for when the velocity boundary condition is applied. Following the schematic and nomenclature of figure 7 the conservation equations can be written in the form of equations 2.1 –2.4.

Conservation of Mass

∂ρ ∂(ρu) ∂(ρυ) ∂(ρw) + + + = 0 (2.1) ∂t ∂x ∂y ∂z

Conservation of Momentum in the x-direction

 ∂u ∂u ∂u ∂u  ∂p ∂   ∂u  ∂   ∂u ∂υ  ρ + u + υ + w  = − + 2  µ  + µ +   ∂ ∂ ∂ ∂  ∂ ∂  ∂  ∂   ∂ ∂   t x y z  x x   x  y   y x  (2.2) ∂   ∂u ∂w  2 ∂   ∂u ∂υ ∂w  +  µ +  −  µ + +  ∂  ∂ ∂  ∂   ∂ ∂ ∂  z   z x  3 x   x y z 

Conservation of Momentum in the y-direction

 ∂u ∂υ ∂υ ∂υ  ∂p ∂   ∂u ∂υ  ∂   ∂υ  ρ + u +υ + w  = − + µ +  + 2  µ   ∂ ∂ ∂ ∂  ∂ ∂   ∂ ∂  ∂   ∂   t x y z  y x   y x  y   y  (2.3) ∂   ∂υ ∂w  2 ∂   ∂u ∂υ ∂w  +  µ +  −  µ + +  ∂   ∂ ∂  ∂   ∂ ∂ ∂  z   z y  3 y   x y z 

Conservation of Momentum in the z-direction

 ∂u ∂w ∂w ∂w  ∂p ∂   ∂u ∂w  ∂   ∂υ ∂w  ρ + u +υ + w  = − +  µ +  +  µ +   ∂ ∂ ∂ ∂  ∂ ∂  ∂ ∂  ∂   ∂ ∂   t x y z  z x   z x  y   z y  (2.4) ∂   ∂w  2 ∂   ∂u ∂υ ∂w  + 2  µ  −  µ + +  ∂  ∂  ∂   ∂ ∂ ∂  z   z  3 z   x y z 

30 A scaling analysis is then performed to determine the relative importance of each term in the conservation equations. The scaling and non-dimensionalization scheme in equation

2.5 is used to recast the conservation equations (2.6-2.9). The parameters used to simplify the equations are defined in 2.10.

Scaling Parameters

x y z pH 2 ρ xo = , yo = , zo = , po = , ρo = , L H D µ UL ρ a a (2.5) u υ w µ t uo = ,υo = , wo = , µo = ,to = U V W µa Θ

Conservation of Mass

1 ∂ρ U ∂(ρ u ) V ∂(ρ υ ) W ∂(ρ w ) o + o o + o o + o o = 0 (2.6) Θ ∂to L ∂xo H ∂yo D ∂zo

Conservation of Momentum in the x-direction

∂ 2 ∂ 2 ∂ 2 ∂ σρ uo +  H  ρ uo +  H  ρ υ uo +  H  ρ uo = o ReL  ouo ReL  o o ReL  owo ∂to  L  ∂xo  L  ∂yo  L  ∂zo 2  2  ∂p H ∂   ∂u  ∂  ∂u H ∂υ  − o +   µ  o  + µ  o +   o  2  o  o    (2.7) ∂x  L  ∂x   ∂x  ∂y   ∂y  L  ∂x  o o   o  o   o o  2   2  2  H  ∂   ∂uo  D ∂wo  2  H  ∂   ∂uo ∂υo ∂wo  +   µo +   −   µo + +   D  ∂z   ∂z  L  ∂x  3  L  ∂x   ∂x ∂y ∂z  o   o o  o   o o o 

31

Conservation of Momentum in the y-direction

∂υ 4 ∂υ 4 ∂υ 4 ∂υ σρ  H  o +  H  ρ o +  H  ρ υ o +  H  ρ o = o  ReL  ouo ReL  o o ReL  owo  L  ∂to  L  ∂xo  L  ∂yo  L  ∂zo 2  2  2 ∂p H ∂ ∂u H ∂υ  H ∂  ∂u  − o +   µ  o +   o  +   µ  o  (2.8)    o    2  o ∂y  L  ∂x   ∂y  L  ∂x   L  ∂y   ∂y  o o   o o  o   o  2  2  2 H ∂  H ∂υ ∂w  2 H ∂  ∂u ∂υ ∂w  +   µ   o + o  −   µ  o + o + o     o      o  L  ∂z   D  ∂z ∂x  3 L  ∂y   ∂x ∂y ∂z  o   o o  o   o o o 

Conservation of Momentum in the z-direction

∂ 2 2 ∂ 2 2 ∂ 2 2 ∂ σDρ wo + H D ρ wo + H D ρ υ wo + H D ρ wo =   o ReL    ouo ReL    o o ReL    owo  L ∂to  L  L ∂xo  L  L ∂yo  L  L ∂zo ∂ 2 ∂  ∂ 2 ∂  2 ∂   2 ∂υ ∂  − po +H µ  uo +D wo+D µ H o + wo (2.9)    o       o    ∂z  L ∂x  ∂z  L ∂x   L ∂y   L ∂z ∂y  o o   o o  o   o o  2 2 H ∂  ∂wo 2H ∂  ∂uo ∂υo ∂wo +2  µ  −   µ  + +  ∂  o∂  ∂  o∂ ∂ ∂   L zo   zo  3 L xo   xo yo zo 

Parameter Definitions

ρ H 2 SqueezeNumber = σ = a Θµa ρ = = aUL Re ynoldsNumber ReL (2.10) µa ρ 2 = = aUL  H  Modified Re ynoldsNumber ReM   µa  L 

The scaled equations (2.6-2.9) can be simplified by applying the assumptions of hydrodynamic lubrication. These assumptions specify that the squeeze number, modified

Reynolds number, and film thickness ratios, (H/L)2 and (H/D)2 are very small. These parameters can be evaluated numerically for the typical foil thrust bearing described in

32 chapter 1. Taking the midspan values from table 1.1, the parameters are calculated and shown in table 2.1.

 lbm  0.077 H 2 ρ H 2  ft3  SqueezeNumber = σ = a = f   Θµa  lbm  0.00001214   ft sec   lbm  σ max0.00001214  σ max µ  ft sec  Maximumfrequency = f max(Hz) = a =   2 ρ H  lbm  a 0.077 H 2  3   ft  ρ 2 = = aUL  H  Modified Re ynoldsNumber ReM   µa  L     lbm   ft  () 0.077 480 0.083 ft 2  3   sec  =  ft   H     lbm  1(in)  0.00001214   ft sec 

Table of maximum frequency (Hz)

H (in) 0.0002 0.0003 0.0005 σmax=0.1 56,758 25,226 9,081 σmax=0.05 28,379 12,613 4,540 σmax=0.01 5,675 2,526 908

Table of Modified Reynolds Number H (in) 0.0002 0.0003 0.0005 Rem 0.0101 0.0228 0.0634 Table 2. Table of characteristic squeeze number and modified Reynolds number.

For the highly loaded foil thrust bearing in the range of 200 to 500 micro-inch film thickness, the assumptions necessary to continue with the Reynolds equation

33 development are appropriate. The three momentum equations can then be written in the form of equations 2.11-2.13.

x-momentum

∂p ∂   ∂u  0= − o + µ  o  (2.11) ∂ ∂  o ∂  xo yo   yo 

y-momentum

∂p 0 = − o (2.12) ∂y o

z-momentum

∂p 2  ∂w  o D ∂  o  0=− +  µ   (2.13) ∂z  L ∂y  o∂y  o o   o 

The y-momentum equation then specifies that the pressure does not vary across the film thickness and is therefore only a function of x and z. This then allows the x and z momentum equations to be solved for the respective velocity components, u and w. The slip flow boundary conditions and closed-form velocity expressions in dimensional form are contained in equations 2.14-2.16. The parameter, Γ, which is found in the velocity boundary conditions, may contain both the mean free path of the gas molecules and a surface accommodation coefficient. However, for purely specular surfaces this parameter simply reduces to the mean free path, λ. This term is included in this analysis to extend the range of applicability of the resultant Reynolds equation into the sub-continuum range for Knudsen numbers greater than 0.01. This follows a similar form to that of Burgdorfer

[8] and become very important for highly loaded air bearings where the film thickness becomes very small and heat generation increases the mean free path of the lubricant as it

34 flows through the bearing. Furthermore, these velocity expression can be integrated across the film thickness to yield the volume flow rates in the x and z directions shown in equations 2.17 and 2.18. Lubricant shear, runner shear, runner drag, and power loss expressions are shown in equations 2.19-2.22.

Velocity boundary conditions

 ∂ u  u ( x , y = 0 , z ) = u r + Γ    ∂ x   ∂ u  u ( x , y = h , z ) = u − Γ f  ∂   x  (2.14)  ∂ w  w ( x , y = 0 , z ) = w r + Γ    ∂ z   ∂ w  w ( x , y = h , z ) = w − Γ   f  ∂ z 

x-direction velocity component

1  ∂p  2  h + Γ − y   y + Γ  u(x, y, z) = −  (hΓ + hy − y )+ ur   + u   (2.15) 2µ  ∂x   h + 2Γ  f  h + 2Γ 

z-direction velocity component

1  ∂p  2  h + Γ − y   y + Γ  w(x, y, z) = −  ()hΓ + hy − y + wr   + w   (2.16) 2µ  ∂z   h + 2Γ  f  h + 2Γ 

x-direction mass flow rate

y=h ρ ∂  3   ur + u   p  h 2   f  mx (x, z) = ∫ u(x, y, z)dy = −   + h Γ + h (2.17) = 2µ ∂x  6   2  y 0     

z-direction mass flow rate

y=h ρ ∂  3   wr + w   p  h 2   f  mz (x, z) = ∫ w(x, y, z)dy = −   + h Γ + h (2.18) = 2µ ∂z  6   2  y 0     

35

x-direction lubricant shear

∂ ∂ u − u u(x, y, z) = − 1  p ()− +  f r    h 2y   (2.19) ∂y 2µ  ∂x   h + 2Γ 

x-direction runner shear

 u − u  ∂u(x, y, z) h  ∂p   f r  µ = −   + µ (2.20) ∂y 2 ∂x  h + 2Γ  x = 0    

x-direction runner drag due to shear

LW ∂u(x, y, z)  =  µ  D ∫ ∫ dz dx 00 ∂y  (2.21) LW  ∂  u − ur   =  − h  p  + µ f   ∫  ∫    dz dx 00 2  ∂x   h + 2Γ       

x-direction runner power loss

  u − u   LW  h  ∂p   f r   HPloss = Du − u  = u − u  − + µ  dz dx (2.22)  f r   f r  ∫  ∫         00 2  ∂x   h + 2Γ       

The continuity equation can be integrated across the film thickness and simplified to the following form if it is assumed that the lubricant density does not vary across the film thickness. By application of Leibnitz’s rule for differentiation of integrals the rectilinear

Reynolds equation is determined by equation 2.24.

= = y h ρ ( x, z ) y h ρ ( x, z )u ( x, y, z) ∫ dy + ∫ dy y =0 ∂t y =0 ∂x (2.23) = = y h ρ ( x, z)v( x, y, z) y h ρ ( x, z )w( x, y, z) + ∫ dy + ∫ dy = 0 y =00∂y y = ∂z

36

Rectilinear Reynolds Equation

3 3 ∂  ρ h  6Γ  ∂p  ∂  ρ h  6Γ  ∂p   1 +   +  1 +   ∂x  12 µ h ∂x  ∂z  12 µ h ∂z          (2.24)     ρ h u + u  ρ h w + w  ∂   f r   ∂   f r   ∂ ()ρ =     +     + h ∂x  2  ∂ z  2  ∂t        

Non-Dimensionalization in Rectilinear Coordinates

The Reynolds, velocity components, runner drag, and mass flowrate equations for the rectilinear bearing can be written in non-dimensional forms shown below in equations

2.26 - 2.32. This scheme differs somewhat from the one which was previously used in the scaling analysis. The primary differences being that the pressures is scaled in a traditional gas dynamic sense, by the pressure ratio, while the bearing number, Λ, contains the hydrodynamic parameters of viscosity, speed, length, inlet pressure, and inlet film thickness. Also of note in this scaling is the relationship of the mean free path to the viscosity and pressure. This relation stems from current research in the field of micro- fluidics and the physics of fluids [9]. This model for the sub-continuum surface slip, while taking into account the mean free path effects, assumes both surfaces to be specular in molecular momentum reflections.

x z p ρ x = , z = , p = , ρ = , L D pa ρa u w µ t u = ,w = ,µ = ,t = (2.25) U W µa Θ Γ Γ µ D Kn = a ,Γ = = ,α = Γ hi a p L

37 Non-dimensional Reynolds Equation

∂  ρ h 3  6 K µ  ∂p  ∂  ρ h 3  6 K µ  ∂p   1 + n   + α 2   1 + n   ∂x  µ  p h  ∂x  ∂z  µ  p h  ∂ z          (2.26) ∂ ∂ ∂ ()ρ = Λ  ()ρ ()+ + α ()ρ ()+  + 1 h  h u f 1 h w f w r   ∂x ∂ z  Θ ∂ t

x-direction non-dimensional velocity

u u(x, y, z) = U µ µ  Kn   Kn  2 1+ − y   y+  (2.27) 3h  ∂ p  2 K µ  ph ph = −   y− y + n  +   + u   Λµ  ∂    2K µ  f  2K µ   x  ph  + n + n  1  1   ph   ph 

z-direction non-dimensional velocity

u w(x, y, z) = U  K µ   K µ  1 + n − y   y + n  (2.28) 2  ∂  K µ  3h p  2 n   ph   ph  = −   y − y + + wr   + w f   Λµ  ∂ z  ph  2K µ 2K µ     + n   + n   1  1   ph   ph 

Non-dimensional x-direction runner drag due to shear

∂ = D = µ u(x,y,z) Dc µ ∫∫ dxdz aLDU ∂ y h1      (2.30)      11 3h ∂ p  µ −1 µ 1  = −  +  + u   dxdz ∫∫ Λ  ∂   2K µ  f  2K µ  00  x  h + n h + n  1  1    ph   ph 

38

Non-dimensional x-direction mass flow rate

3 m ρ h  ∂ p  6K µ  m (x, z) = x = −  1+ n  + ()1+u ρ h (2.31) x ρ Λµ    f aUh1  ∂ x  ph  2

Non-dimensional z-direction mass flow rate

3 m ρ h  ∂ p  6K µ  m (x, z) = z = −  1+ n  + ()w +w ρ h (2.32) z ρ Λµ    r f aUh1  ∂ z  ph  2

2.2.2 Vector Form of Reynolds Equation

The Reynolds equation (2.24) can be written more generally, in vector notation, if it remains in dimensional terms. The vector form is convenient because it can readily be transformed into coordinates that are convenient for journal or thrust bearing simulations.

39 ∂(ρ ) ∇r • ()ζr = h ∂t where, 3 r ρh  6Γ  r ζ = MassFlowRateVector = 1+ ∇P − S 12µ  h  and,

ρhu + u  ρhw + w  r f r f r S =   iˆ,   ˆj 2 2 therefore,   3 ρhu + u   ρh  6Γ  ∂P  r f  − 1+  + iˆ,  µ ∂  (2.33)  12  h  x 2  ζr =     3 ρhw + w   ρh  6Γ  ∂P  r f  − 1+  + kˆ  µ ∂   12  h  z 2   

2.2.3 Cylindrical (Thrust Pad) Form of Reynolds Equation

This equation can then be cast into cylindrical coordinate where x maps into θ, z maps into r, and y remains the film-wise coordinate to represent the hydrodynamics of thrust bearings shown in figure 8.

3 3 1 ∂  ρ h  6 Γ  ∂p  ∂  ρ h  6 Γ  ∂p   1 +   +  r 1 +   ∂θ  µ ∂ θ  ∂  µ ∂  r  12  h   r  12  h  r  (2.34)

 ρ h  v + ω r    ρ h  v + v   ∂   rf r   ∂   rf rr   ∂ ()ρ h =     +  r    + ∂θ 2 ∂r 2 ∂ t        

This can be further simplified for stationary foils, concentric bearings, and time steadiness to yield equation 2.35.

3 3 1 ∂  ρ h  6 Γ  ∂p  ∂  ρ h  6 Γ  ∂p  ∂  ρ h ()ω r   1 +   +  r 1 +   =   (2.35) ∂θ  µ ∂θ  ∂  µ ∂  ∂ θ   r  12  h   r  12  h  r   2 

40 Non-Dimensional Thrust Pad Notation

This form of Reynolds equation for thrust bearings can then be non-dimensionalized by using the definitions of equation 2.36 to yield equation 2.37.

r p ρ u w µ r = , p = , ρ = ,u = ,w = ,µ = ro pa ρa U W µa (2.36) 2 6µaω  ro  Γa Λ =   ,Kn = pa  h1 h1

3 3 ∂  ρ µ ∂  ∂  ρ µ ∂  ∂ 1  h  + 6 K n  p  +  h  + 6 K n  p  = Λ ()ρ (2.37)  1    r 1   h r ∂θ  µ  p h  ∂ θ  ∂ r  µ  p h  ∂ r  ∂θ    

The traditional boundary conditions for the thrust bearing Reynolds equation is to set the static pressure around the perimeter of the thrust pad equal to a uniform pressure. In terms of the non-dimensionalization scheme this dictates a perimeter value of unity.

However to generalize the analysis, especially to analyze current foil thrust bearing thermal management designs, requires that the Dirichlet boundary condition to be written in functional form.

Pressure Boundary Conditions

p (θ , r = RR ) = pa (θ ) p (θ , r = 1 .0 ) = pb (θ ) (2.38) p (θ = 0 .0 , r ) = pc ( r ) p (θ = θ end , r ) = pd ( r )

The non-dimensional forms of the velocity components, volume flow rates, and mass flow rates can be written in the following forms.

41

θ-direction non-dimensional velocity

uθ uθ (r,θ , y) = ωro µ µ  Kn   Kn  2 1+ − y   y+  3h  ∂ p  2 K  ph ph = −   y− y + n  +   + uθ   (2.39) Λµ  ∂θ  ρ   2K µ  f  2K µ  r   h  + n + n  1  1   ph   ph  θ u f uθ f = ωro

r-direction non-dimensional velocity

ur ur(r,θ , y) = ωro µ µ  Kn   Kn  2 1+ − y   y+  3h  ∂ p  2 K µ  ph ph =−   y− y + n +ur  +ur   Λµ  ∂   r  2K µ  f  2K µ   r  ph  + n + n (2.40)  1  1   ph   ph  urr ur r = ωro ur f ur f = ωro

Non-dimensional θ-direction runner drag due to shear

 h1  ∂uθ (r,θ , y) 2 τ = τ   = µ r d rdθ = 4 ∫∫ ∂ ωro µa  y    (2.41) θ   2  end 1  3rh ∂ p   µ r ()ur(r,θ , y)−1  ∫∫−  + d rdθ Λ  ∂θ   2K µ  0 RR     h1+ n           ph  

42

Non-dimensional θ-direction mass flow rate

3 mθ ρ h  ∂ p  6K µ  θ θ = = −   + n  + ()+ ρ (2.42) m (r, ) ρ ()ω  1  1 u f h a ro h1 Λµ r  ∂θ  ph  2

Non-dimensional r-direction mass flow rate

3 m ρ h  ∂ p  6K µ  m (r,θ ) = r = −  1+ n  + ()ur +ur ρ h (2.43) r ρ ()ω Λµ    r f a ro h1  ∂r  ph  2

2.3 Density Models

Several density and viscosity models can be applied to the solution of the Reynolds equation. The first and simplest of these models would be the constant density, constant viscosity model in which both the non-dimensional density and viscosity are set to unity.

The next level of complexity can be modeled as an isothermal density model. If in this case it is also assumed that the viscosity of the lubricant is not a function of pressure but merely a function of temperature, then the model is applied mathematically by setting the non-dimensional density equal to the non-dimensional pressure and the non-dimensional viscosity equal to unity. This level of modeling is typically the most complex model used in the theory of hydrodynamic lubrication and there exist transformations and very efficient numerical methods to calculate solutions to the isothermal Reynolds equation.

Variable viscosity effects have been modeled in hydrodynamics by applying the Couette approximation [10, 11] to an uncoupled energy equation. However a self-similar solution of the temperature, density, viscosity, and pressure fields are not available.

2.3.1 Power Law Models

43 Temperature effects may begin to be modeled if a polytropic process is used to model the lubricant density as a function of pressure. The ideal gas law can then be used to model lubricant temperature and hence its viscosity. A Typical values of “a” for air at standard pressure and temperature is 0.5.

Polytropic Modeling p = C ρ n n p = ρ (2.44)  −   n 1  a a  n  µ = T = p   − − ρ 1 ( n 1) a = p n µ

2.3.2 The Energy Equation

In order to include the effects of viscous heat generation into the simulation of foil thrust bearings, the conservation of energy equation must be analyzed to account for thermal effects on the density and viscosity fields. As a starting point the energy equation is cast in terms of fluid enthalpy. A simplified energy control volume is shown in figure 9.

 ∂T ∂T ∂T  ρCpu + υ + w   ∂x ∂y ∂z    (2.45) ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = p + υ p + p +  T  +  T  +  T  + Φ u w  k   k  k  ∂x ∂y ∂z ∂x  ∂x  ∂y  ∂y  ∂z  ∂z 

The term Φ represents the dissipation function or viscous heat generation. By applying the previously imposed assumption of constant properties across the film thickness and that conduction within the lubricant is negligible compared to the convection and viscous heat generation, the energy equation can be simplified.

 ∂T ∂T  ∂p ∂p ρCpu + w  = u + w + Φ (2.46)  ∂x ∂z  ∂x ∂z

44 The appropriate form of the compressible dissipation function for rectilinear coordinates is shown below.

 ∂ 2 ∂ 2  Φ = µ u  +  w   (2.47)       ∂y   ∂y    

Integrating the energy equation across the film thickness in y, yields equations 2.48 and

2.49.

∂ ∂ ∂ ∂ h ρ  T + T  = p + p + Φ ∫0{ Cpu w  u w }dy (2.48)  ∂x ∂z  ∂x ∂z

2 2 h h  ∂u   ∂w  Φdy = {µ  +   }dy (2.49) ∫0 ∫0  ∂y   ∂y       

Simplifying,

∂T h ∂T h ∂p h ∂p h h ρCp udy + ρCp wdy = udy + wdy + {Φ}dy (2.50) ∂x ∫0 ∂z ∫0 ∂x ∫0 ∂z ∫0 ∫0

The term Φ represents the dissipation function or viscous heat generation. The velocity integrals of u and w can be re-written in terms of the volume flow rates or when multiplied by density the mass flow rates in the x and z directions respectively.

∂T ∂T m& x ∂p m& z ∂p h Cp m& x + Cp m& z = + + {Φ}dy (2.51) ∂x ∂z ρ ∂x ρ ∂z ∫0

The viscous dissipation function can be expanded in term of the velocity components.

2 2 3 2 2 2 h h  ∂u   ∂w   h  ∂p   ∂p   µU Φdy = {µ  +   }dy =   +    + (2.52) ∫0 ∫0     2  ∂y   ∂y   12µ  ∂x   ∂z    Γ      h1+ 2   h 

Although the thin layer of lubricant is assumed to be isothermal in the film-wise direction, the lubricant can still be thermally affected by surroundings, the foil and runner. This effect can be modeled by utilizing a thermal plug flow notion where the heat transfer and

45 generation are sufficiently fast so that uniform properties are maintained through the film thickness. An alternative to this line of reasoning would be to consider that heat transfer in the y-direction is conduction limited in the metallic boundaries not merely because of material properties but also due to the length scales along this coordinate. If the heat transfer at the foil and the runner are lumped together into a single heat transfer parameter, Qs(x,z), the dimensional form of the integrated energy equation in rectilinear coordinates can then be written in the following form.

  ∂ ∂ ∂ ∂  3  ∂ 2 ∂ 2  µ 2  T + T = m& x p + m& z p +  h  p  +  p  + U  + (2.53) Cpm& x Cpm& z        Qs ∂x ∂z ρ ∂x ρ ∂z 12µ ∂x ∂z Γ 2        +    h1 2     h  

The equation can be non-dimensionalized according to the same scheme used in the

Reynolds equation.

γ Cp  ∂T ∂T  m& x +αm& z  γ −1 ∂x ∂z    2 2   1  ∂p ∂p h3 ∂p   ∂p   Λ µ  = m& x +αm& z +   +α   + c +Q ρ ∂ ∂ Λ µ ∂ ∂  2  sc (2.54)  x z  c  x   z    2K µ     + n  3h1    ph  

Λ h Q (x, z) = c 1 Q sc µ 2 s 3 aU

The dimensional energy equation can also be represented in vector form according to the following equation.

 2 2  r 1 r h  ∂vθ   ∂v  Cp()ζ • ∇T = ()ζ • ∇p + µ  +  r  dy + Q (2.55) ρ ∫0  ∂y  ∂z s     

46 Finally this form of the energy equation can be written cylindrical coordinates to take on a form appropriate for thrust bearing modeling and non-dimensionalized to yield the following form.

γC ∂ ∂ p  T + m&θ T  m& r  γ −1 ∂r r ∂θ      3 2 2 2 1  ∂p m& θ ∂p  h ∂p   1 ∂p    Λµr  = m& + +   +   + +Q (2.56) ρ  r ∂ ∂θ  Λµ ∂ ∂θ  2  s  r r   r   r    2K µ     + n  3h1     ph  

Λ h Q (r,θ) = c 1 Q s µ ()ω 2 s 3 a ro

Two temperature boundary conditions are required for the energy equation; the first is

T(r,θ=θa) and the second T(r=ra, θ). At the leading edge of a thrust pad the lubricant is typically at a known temperature, which is represented by T(r,0)=ta(r). The second boundary condition can be reasoned by considering the implications of non-inertial flow.

∂p If a locus of points is defined by =0 for the angular extent of the thrust pad, this locus ∂r also defines the line of zero radial flow. In other words, flow that enters the pad at a smaller radius than the initial zero radial flow point either leaks out of the inner radius of the pad or exits the trailing edge. Conversely, flow that enters the pad at a larger radius than the initial zero radial flow point either leaks out of the outer radius of the pad or exits the trailing edge. Along this locus of point the energy equation becomes an ordinary differential equation 2.57, which may be solved for T(θ). The domain of the energy equation can then be divided into two parts as shown in figure 10. The inboard domain has a boundary condition along its outer radius and the leading edge while the

47 outboard domain has a boundary condition along its inner radius and the leading edge.

The energy equation can be solved in each of these two domains and combined for the complete solution. If the locus of points defining the maximum pressure along a constant

θ does not form a line of constant radius, then the energy equation domain can be divided into several subregions to approximate this scenario while maintaining this solution strategy.

         3  2 2  dT γ −1 1  dp  1 h  ∂p  r  Λµr  r  =    +    +   + Qs dθ γC ρ  dθ  m& Λµ  ∂θ   m&  2  m& p  θ   θ  2K µ  θ   3h1+ n           ph    dp T (rat = 0,θ = 0) = 1 (2.57) dr Λch Q (r,θ) = 1 Q s 2 s 3µa()ωro

2.4 Structural Model

A single equation is used to model the deflections of the compliant foundation / top foil combination. The models have their genesis in elementary beam theory and Stephen

Timoshenko’s theories for thin plates and shells [12, 13]. The major assumptions for the top foil model are that the foil is thin enough such that the entire membrane behaves like the neutral axis and that top foil stiffness is bending dominated such that the tensile stresses can be neglected. Additionally, the foil bending stiffness is assumed to be constant throughout the angular and radial extent of the bearing pad. This assumption allows the bending stiffness to be moved to the denominator of the right hand side of the equation. These assumptions limit the range of applicability to conditions of modest foil sag between discrete compliant foundation contacts. However it does provide a useful

48 and practical model for the investigation of fluid-structure interactions. The compliant

θ δ foundation is represented mathematically by the linear stiffness function,K c (r, , ) , which must be determined either experimentally or analytically outside of the current analysis. This formulation replaces the complex behavior of the bump foil with a virtual array of discrete springs located along the bump foil – top foil contact area. The structural model is summarized by the following equations.

P − K (r,θ,δ )δ ∇4δ = g c K b, f Et3 K = (2.58) b, f 12(1 −ν 2) Kc(r,θ,δ ) = bumpfoil model

K b, f is the bending stiffness of a thin plate according to Timoshenko

θ δ K c (r, , ) is the “pressure stiffness” of the compliant support material.

Two boundary conditions are applied along all the edges of the foil to satisfy the fourth order equation. Along the leading edge the structural condition is clamped where deflection and slope are zero. Along the remaining three edges the structural conditions are “free” where moment and shear are zero.

Clamped ∂δ δ (θ = 0) = 0, = 0 ∂θ θ = 0 (2.60) Free ∂δ 2 ∂δ 3 = 0, = 0 ∂θ 2 ∂θ 3 OnBoundary OnBoundary

Expanding the biharmonic operator for both rectilinear and cylindrical coordinate yields the following set of equations.

49 Dimensional Structure Equations

∂4δ ∂4δ ∂4δ P − K (x, z,δ )δ + + = g c (2.61) 4 2 2 4 ∂x ∂x ∂z ∂z Kb, f

∂2δ ∂4δ ∂δ ∂3δ ∂4δ ∂3δ ∂4δ ∂4δ 4 + + r − 2r + r4 + 2r3 − r2 + 2r2 ∂θ 2 ∂θ 4 ∂r ∂ ∂θ 2 ∂ 4 ∂ 3 ∂ 2 ∂ 2∂θ 2 r r r r r (2.62) P − K (r,θ,δ )δ = g c Kb, f

Rectilinear Non-dimensional Structure Equation

∂4δ ∂4δ ∂4δ 4 ()− − δ δ + α 2 + α 4 =  L  P 1 Kc(x, z, ) 4 2 2 4   ∂x ∂x ∂z ∂z  h1 Kb, f

K (x, z,δ )δ c h1 Kc (x, z,δ )δ = Pa h1 (2.63) Kb, f K = b, f 3 Pah1 3 = Et Kb, f 12()1 −ν 2 α = L D

50

Cylindrical Non-dimensional Structure Equation

∂2δ ∂4δ ∂δ ∂3δ ∂4δ ∂3δ ∂4δ ∂4δ 4 + + r − 2r + r 4 + 2r3 − r 2 + 2r 2 ∂θ 2 ∂θ 4 ∂r ∂r∂θ 2 ∂r 4 ∂r3 ∂r 2∂θ 2 ∂r 2∂θ 2 4  r  ()P −1 − K (r,θ,δ )δ = r 4 o  c  h1 Kb, f

K (r,θ,δ )δ (2.64) c h1 Kc (r,θ,δ )δ = Pa h1 Kb, f K = b, f 3 Pah1 3 = Et Kb, f 12(1 −ν 2)

51

Figure 7. Representation of a rectilinear foil bearing

Figure 8. Representation of a thrust foil bearing

52 Heat, +/-

Enthalpy, in Enthalpy, out

Work, in Heat, +/-

Figure 9. Diagram of the energy equation control volume.

Outboard region

Locus of maximum pressure

Inboard region

Figure 10. Diagram of the division of the energy equation domain.

53 CHAPTER 3 Solution Technique and Numerical Method

3.0 Overview

The equations developed in chapter 2 are solved in the Mathematica programming environment using finite differencing and the built-in Mathematica function NDSolve [14,

15, 16]. The Reynolds equation and the structural model are solved by the finite difference approach. The derivatives of pressure and deflection are discretized using central differencing. A sparse version of the numerical grid is show in figure 11. The number of nodes in the r and θ directions is independently specified, but the grid spacing is uniform. The current calculations are carried out using a 50 by 50 grid in the radial and angular directions. Numerical experiments have proven this to be a well balance point between computational time and resultant accuracy. The energy equation is more efficiently solved by using the NDSolve function of Mathematica. This numerical function is very effective at solving partial differential equations in which boundary conditions are applied in a way that “shooting” methods are appropriate. An iteration scheme is used to solve each equation in series for pressure, deflection, and temperature respectively. An equation of state is then used to obtain density and couple the energy equation back to the Reynolds equation. Relaxation factors are used on the deflection and temperature fields within the iteration loop to speed the convergence of a self-similar solution. The overall iteration and convergence strategy is outlined in the flowchart of figure 12. Verification of the numerical procedure is established by comparison with analytic results for cases of one dimensional incompressible flow, one dimensional wiper bearing flow, and two dimensional wiper bearing flow.

54 3.1 Derivative Expansions and Coefficient Functions

The solution technique used in this work for the Reynolds equation in cylindrical coordinates (2.37) begins with the expansion of the second derivative terms in both the radial and angular coordinates. The equation then takes on the following form.

∂ 2 p 2 ∂ 2 p ∂ p ∂ p Λ r ∂ + r + CF θ + r ()r ()CFr + 1 = ()ρ h 2 2 ∂θ ∂ ∂θ ∂θ ∂ r r CF where , 3 ρ µ (3.1) θ = h  + 6 K n  CF (r , ) 1  µ  p h  and ∂ CF θ (r ,θ ) = []Ln ()CF (r ,θ ) ∂ θ ∂ CFr (r ,θ ) = []Ln ()CF (r ,θ ) ∂ r

In a similar fashion the inhomogeneous biharmonic equation used to model the foil and compliant structure is expanded in cylindrical coordinates.

∂2δ ∂4δ ∂δ ∂3δ ∂4δ ∂3δ ∂4δ ∂4δ 4 + + r − 2r + r 4 + 2r3 − r 2 + 2r 2 ∂θ 2 ∂θ 4 ∂r ∂ ∂θ 2 ∂ 4 ∂ 3 ∂ 2∂θ 2 ∂ 2∂θ 2 r r r r r (3.2) r 4 ()P −1 −K (r,θ,δ )δ = 4 o  c r    h1 Kb, f

The energy equation is not expanded in the same manner as the Reynolds equation and the structural model. Being only first order in r and θ, the built-in Mathematica function

NDSolve can be efficiently used to solve this equation. The solution strategy for this equation has been outlined in chapter 2. Recapping, the thrust pad domain is divided into two region determined by the locus of maximum pressure points. The ordinary differential equation is solved along this locus to determine the boundary condition for

55 temperature as a function of θ. Each region can then be solved by utilizing this temperature function and the thrust pad leading edge temperature condition.

3.2 Finite Differencing Scheme

For the two equations that are solved using the finite differencing technique, the

Reynolds equation and the structural equation, conventional central differencing is used to determine the set of difference equations in the core of the domain. For the Reynolds equation, the derivatives in pressure are the only terms subjected to the finite differencing procedure. The coefficient functions, CFθ, CFr, and the right hand side terms are treated as known functions, although they are, strictly speaking, functionals of the pressure field.

For the purposed of iterating towards a solution the pressure field of the previous iteration is used to calculate these values for the current iteration. This technique allows for the solution of a linear problem and in effect trades numbers of iterations for reduced complexity compared to the solution of a non-linear problem. The finite difference form of the Reynolds equation is show as equation 3.3.

p (i, j + 1) − 2 p (i, j ) + p (i, j − 1) 2 p (i + 1, j ) − 2 p (i, j ) + p (i − 1, j ) + r h θ 2 hr 2 p (i, j + 1) − p (i, j − 1) p (i + 1, j ) − p (i − 1, j ) (3.3) + CF θ + r ()r ()CFr + 1 2 h θ 2 h θ Λ r ∂ = ()ρ h CF ∂θ

The structural equation is more naturally linear. With the exception of the pressure- displacement response of the compliant foundation, the equation is strictly linear if we again use the technique of using the pressure field of the previous iteration for the distributed load on the right hand side of the equation. If however a non-linear response of the compliant foundation must be modeled, the equation is linearized by including the higher order response within the stiffness function, Kc, and utilizing the structural

56 displacement, δ, from the previous iteration. The discretized structural equation is shown in equation 3.4.

δ (i, j + 1) − 2δ (i, j) + δ (i, j −1) 4 hθ 2 δ + − δ + + δ − δ − + δ − + (i, j 2) 4 (i, j 1) 6 (i, j) 4 (i, j 1) (i, j 2) hθ 4 δ (i + 1, j) − δ (i −1, j) + r hr δ (i +1, j + 1) − 2δ (i + 1, j) − δ (i + 1, j −1) − δ (i −1, j +1) + 2δ (i −1, j) − δ (i −1, j −1) − r 2 hrhθ 2 δ (i + 2, j) − 4δ (i + 1, j) + 6δ (i, j) − 4δ (i −1, j) + δ (i − 2, j) + r 4 hr4 δ (i + 2, j) − 2δ (i + 1, j) + 2δ (i − 1, j) − δ (i − 2, j) + r3 hr3 δ (i +1, j) − 2δ (i, j) + δ (i − 1, j) − r 2 hr2 2 δ (i + 1, j + 1) − 2δ (i + 1, j) + δ (i + 1, j − 1) − 2δ (i, j + 1)  + 2r   2 2   hr hθ + 4δ (i, j) − 2δ (i, j −1) + δ (i −1, j +1) − 2δ (i − 1, j) + δ (i −1, j −1) 4  r  ()P −1 − K (r,θ,δ )δ = r 4 o  c  h1 Kb, f

(3.4)

3.3 Boundary Conditions

Traditional hydrodynamic boundary conditions are employed in the solution of the

Reynolds equation. These conditions dictate that the pressure along the boundary of the bearing pad remains constant and at ambient conditions. Within the non-dimensional scheme employed in this analysis this Dirichlet condition is satisfied by setting all numerical nodes on the boundary of the domain to unity. However, in order to provide flexibility in the Mathematica code written to solve this problem, the boundary condition

57 can be input as pressure as a function of the boundary coordinate. This type of boundary condition can be used to model the bearing when an external pressure in imposed for the purposes of cooling the bearing under highly loaded conditions. Sample equations of the

Reynolds equation boundary conditions are included in equation 3.5. c1 = Table[p[i,0] N[b1/.r→ rr[i]],{i,0,nr}]; c2 = Table[p[i,nθ] N[b2/.r→ rr[i]], {i,0,nr}]; c3 = Table[p[0,j] N[a1/.θ→ θθ[j]],{j,1,nθ-1}]; c4 = Table[p[nr,j] N[a2/.θ→ θθ[j]],{j,1,nθ-1}];

(3.5)

The boundary conditions for the structural equation are somewhat more cumbersome to apply. Being a fourth order equation, two conditions are applied at each boundary.

Along the leading edge a clamped condition is applied where the displacement and slope of the top foil are set equal to zero. Along the remaining 3 edges a “free” condition is applied in which the material moment and shear, second and third derivatives of displacement, are set equal to zero. In order to maintain a common size between physical domain and computational domain, one-directional finite differencing is used in the boundary conditions. . Sample equations of the structural model boundary conditions are included in equation 3.6. bc1 =Table[δ[i,1] 0,{i,3,nr-2}]; bc2=Table[δ[i,2] 0,{i,3,nr-2}]; bc3=Table[2δ[i,nθ]5δ[i,nθ-1]-4 δ[i,nθ-2]+δ[i,nθ-3],{i,3,nr-2}]; bc4=Table[5δ[i,nθ-1] 18δ[i,nθ-1]-24 δ[i,nθ-2]+14 δ[i,nθ-3]-3 δ[i,nθ-4],{i,3,nr-2}]; bc5=Table[2 δ[1,j] 5 δ[2,j]-4 δ[3,j]+δ[4,j],{j,1,nθ}]; bc6=Table[5 δ[2,j] 18 δ[2,j]-24 δ[3,j]+14 δ[4,j]-3 δ[5,j],{j,1,nθ}]; bc7=Table[2δ[nr,j] 5δ[nr-1,j]-4 δ[nr-2,j]+δ[nr-3,j],{j,1,nθ}]; bc8=Table[5 δ[nr,j] 18δ[nr-1,j]-24 δ[nr-2,j]+14 δ[nr-3,j]-3 δ[nr-4,j],{j,1,nθ}];

(3.6)

58 3.4 Iteration Strategy

No a prior solutions are utilized in the initialization of the iteration procedure for this simulation and modeling program. The standard set of inputs required to begin a calculation includes bearing number, fluid properties, undeflected film thickness, compliant foundation stiffness, foil bending stiffness, and density model. The solution procedure then commences by solving the Reynolds equation for an incompressible, rigid wall, bearing. Once the pressure field is known, the equations yielding structural deflections and temperature fields can be solved. The density field can then be calculated through the use of an equation of state. The updated density and hydrodynamic film thickness fields are then used to calculate new coefficient functions for the Reynolds equation and the second iteration progresses as the first with the exception of a relaxation factor added to the calculated deflections and temperatures in which the values from the two previous iterations are combined in a weighted average. This method damps the convergence criterion oscillations and arrives at a solution more quickly than an unrelaxed method. The convergence criterion is calculated after every iteration to ensure a self-consistent solution is reached at some user defined threshold of convergence criterion. Typically bearing hydrodynamic load is used as a convergence criterion.

Setting the desired bearing conditions in the initial iteration is herein referred to as direct convergence. At times direct convergence is not possible due to the desired conditions being far removed from the rigid wall, incompressible case. In such instances calculating bearing behavior along a speed line is the most useful approach to obtaining high speed and load solutions. In this approach the bearing performance is initially calculated at a low speed such that the incompressible, rigid wall assumptions are fairly accurate. When

59 a low speed solution is reached, the speed is incrementally increased until the desired condition is obtained. Figure 13 shows a typical convergence of the program as a plot of iteration number versus nondimensional load.

3.5 Mathematica Code

The numerical procedures discussed in the previous sections of chapter 3 have been coded into a Mathematica notebook. Figures 14 and 15 contain the source code functions that solve the Reynolds equation and structural model. These functions are called by an

“executive” notebook that contains the specific inputs, convergence criterion, and relaxation scheme for the problem at hand. A sample of an “executive” notebook is contained in Appendix B.

Figure 11. Typical numerical grid in thrust pad geometry.

60 Input bearing data: Geometry, lubricant properties, operating conditions, etc…

Calculate initial solution: Rigid, incompressible, Low speed

Move calculation to desired operating condition: Adjust speed and film thickness, Include density, viscosity, and structural model

Calculate final solution: Calculate temperature Calculate h=g+δ Calculate density Calculate Reynolds coeficients Calculate pressure Calculate load No Is convergence criterion met?

Yes Program Complete Successful solution

Figure 12. Program flow chart.

61

Convergence Example

4.5

4

3.5 d a o l

3

2.5

2

5 10 15 20 25 iteration

Figure 13. Sample of program convergence on load.

62

Figure 14. Mathematica code to solve Reynolds equation in cylindrical coordinates.

63

Figure 15. Mathematica code for solving the structural model in cylindrical coordinates.

64 CHAPTER 4 One-Dimensional Bearing Analysis

4.1 Analytic Solutions

For the case of zero side leakage (infinitely wide) bearings with rigid walls and constant temperature, density, and viscosity lubricant, the Reynolds equation can be solved analytically. Several of these solutions are presented in the first part of this chapter.

Specifically, solutions are presented for the straight taper bearing, the Rayleigh step bearing, and the double straight taper bearing. When defining the hydrodynamic film thickness many parameters can be used accomplish this task. In this analysis it is assumed that the minimum film thickness is determined through some means which may or may not include the hydrodynamics. Considerations which may determine this minimum film thickness include machine vibrations, surface finish (roughness), manufacturing tolerances, and symmetry. Once a minimum film thickness is determined the hydrodynamic solution can be cast into a set of parameters such as contraction ratios and location of slope change. These parameters are then optimized for maximum load carrying capacity and results are plotted. In addition to the load bearing film thicknesses analyzed, a purely Couette flow with viscous heat generation is included in this section.

This analysis begins to demonstrate the importance of viscous heat generation in the physical range of a highly load foil thrust bearing by calculating the required channel divergence to maintain isobaric flow. The speed and lubricant conditions used in these calculations were a surface speed of 480 fps, bearing length of 1.0 in., and standard sea level air. These conditions were chosen to approximate the midspan conditions for the generation I bearing described in Table 1.1.

65 4.1.1 Rigid Straight Taper Bearing

Perhaps the most analyzed of all hydrodynamic bearings is the rigid straight taper bearing.

The analysis here follows that of most classic solutions found in many fluid mechanics textbooks. The governing Reynolds equation and boundary conditions in dimensional form can be reduced to equation 4.1. If the solution is parameterized by the minimum film thickness and the physical contraction ratio, k, then the solution, hydrodynamic normal force (load), runner drag, and power loss are given by equations 4.2 – 4.5.

 ρ 3   ρ h (u ) d  h dp  = d  r  dx  12 µ dx  dx  2      p ( x = 0) = patm (4.1) p ( x = L ) = patm p ( x, k , h 2 ) − patm µ ()()− − = 6 UL L kh 2 h ( x, k , h 2 ) h ( x, k , h 2 ) h 2 (4.2) 12 Gch 2 2 k 2 − 1 h ( x, k , h 2 ) 2

2 6 µ UL  2()k − 1  l (k , h 2) =  Log ( k ) −  (4.3) 2 + 12 Gc ()k − 1 h 2 2  k 1 

6 µ UL  6 ()k − 1  d (k , h 2 ) =  4 Log (k ) −  (4.4) 12 Gc ()k − 1 h 2  k + 1 

6 µ U 2 L  6()k − 1  HPloss (k , h 2) =  4 Log (k ) −  (4.5) 12 Gc ()()k − 1 h 2 550  k + 1 

Plots of load versus runner drag and load versus power loss are shown in figure 16 for the cases of 500, 200, and 150 micro-inch minimum film thicknesses. The curves follow a similar pattern where they begin at the origin which corresponds to an infinite contraction ratio. As the contraction ratio is lowered the load, drag, and power loss all increase until a maximum load is reached at a contraction ratio of 2.2. The curves then

66 continue with decreasing load while drag and power loss continue to increase. The curves terminate at the point of zero load and maximum drag or power loss for a contraction ratio of 1.0. This corresponds to classic Couette flow for the incompressible, rigid wall, constant density, and constant viscosity case. The lubricant streamlines, entrance and exit velocity profiles, and pressure distributions are shown for the cases of

200 micro-inch minimum film thickness and contraction ratios of 10, 5, 2.2, and 1.5 are shown in figure 17. At large contraction ratios too much lubricant is swept into the bearing through momentum diffusion from the runner. As the pressure build, the unfavorable pressure gradient causes this fluid to be force out of the inlet on the stationary side of the bearing. At the point of maximum load the lubricant recirculation has nearly vanished and the velocity profiles at the inlet and exit exhibit a nearly zero wall shear at the stationary surface and runner surface respectively. As the contraction ratio is further reduced, the recirculation region completely disappears and the load begins to drop. These results indicate that there is some correlation between an efficient use of maximum through flow and maximum load for a given minimum film thickness.

Figure 16. Load capacity versus runner drag and power loss for a rigid, straight taper bearing with an incompressible, constant property lubricant.

67 k = 10. k = 5.

L = 1. in L = 1. in

h2 = 0.0002 in h2 = 0.0002 in

h1 = 0.002 in h1 = 0.001 in Load = 18.621 lbf in Load = 39.068 lbf in Drag = 0.036 lbf in Drag = 0.046 lbf in Power Loss = 0.031 HP in Power Loss = 0.040 HP in U = 480. ft s U = 480. ft s r- model =constant r- mo d e l =constant

m- model =constant m- mo d e l =constant

h- model = rigid h- mo d e l = rigid

k = 2.2 k= 1.5

L = 1. in L= 1. in

h2 = 0.0002 in h2 = 0.0002 in

h1 = 0.00044 in h1 = 0.0003 in Load = 60.462 lbf in Load = 49.491 lbf in Drag = 0.057 lbf in Drag = 0.064 lbf in Power Loss = 0.050 HP in Power Loss = 0.056 HP in U = 480. ft s U= 480. ft s r- model =constant r- model = constant

m- model =constant m- model = constant

h- model = rigid h- model = rigid

Figure 17. Operating characteristics of the straight taper bearing having contraction ratios, k, of 10, 5, 2.2, and 1.5.

4.1.2 Rayleigh Step Bearing

The Rayleigh step bearing is another film thickness profile that has been solved many times in the history of hydrodynamics. This bearing replaces a continuous physical contraction with two parallel flow segments joined into one bearing. This bearing allows for an additional parameter in the load optimization of the bearing. As in the straight taper bearing the minimum film thickness and overall contraction ratio are still present.

The location of the change in film thickness also enters the solution as a parameter.

Equation 4.1 remains as the governing equation. The expressions for the pressure distribution, load, drag, and power loss are contained in Appendix C.

68 A contour plot of load versus contraction ratio, k, and change location, cl, is shown in figure 18. This figure also shows the location of the maximum load for k = 1.866 and cl

= 0.718. This particular plot was generated for a minimum film thickness of 200 micro- inch. Although the magnitude of the maximum load will change with minimum film thickness, the qualitative shape of the contours and location of the extremum does not vary with h2. Since the contraction ratio effects were analyzed and discussed in the previous section, the focus in this section will be to fix the contraction ratio at its maximizing value of 1.866 and discuss the effects of the location of discontinuity, cl. A load versus power loss graph is included in figure 19 for minimum film thicknesses of

500, 200, and 150 micro-inches. The curves are represented for various change locations.

Beginning at the origin which corresponds to a cl of 1.0, the load and power loss increase to a maximum load at a cl of 0.718. As the cl is further decreased to 0.0 the load is decreased and the power loss is increased. Both endpoints of this curve correspond to purely Couette flow, the low power loss endpoint corresponds to parallel flow at the entrance film thickness while the high power loss endpoint corresponds to parallel flow at the minimum film thickness. Figure 20 shows the lubricant streamlines, entrance and exit velocity profiles, and pressure distributions are shown for the cases of 200 micro- inch minimum film thickness and contraction ratio of 1.866 and change locations of 0.9,

0.718, 0.5, and 0.2. The discontinuity in streamlines at the change location is due to the fact that the solution is stitched together by satisfying equality in pressure without regard to consideration to smooth, continuous, analytic functions of pressure. At small change location too much lubricant is swept into the bearing through momentum diffusion from the runner. As the pressure build, the unfavorable pressure gradient causes this fluid to

69 be force out the inlet on the stationary side of the bearing. At the point of maximum load the recirculated fluid has nearly vanished and the velocity profiles at the inlet and exit exhibit a nearly zero wall shear at the stationary surface and runner surface respectively.

As the change location is further increased, the recirculated flow completely disappears and the load begins to drop. These results again indicate that there is a correlation between an efficient use of maximum through flow and maximum load for a given minimum film thickness.

Figure 18. Load capacity contours for the Rayleigh step bearing as a function of contraction ratio, k, and change location, cl.

70

Figure 19. Load capacity versus power loss for the Rayleigh step bearing.

71 k= 1.866 k= 1.866

cl = 0.9 cl = 0.718

L= 1. in L= 1. in

h2= 0.0002 in h2= 0.0002 in

h1=0.0003732 in h1=0.0003732 in Load = 56.930 lbf in Load = 77.829 lbf in Drag = 0.066 lbf in Drag = 0.085 lbf in Power Loss = 0.058 HP in Power Loss = 0.075 HP in U= 480. ft s U= 480. ft s r-mo d e l = constant r-mo d e l = constant

m- mo d e l = constant m- mo d e l = constant

h- mo d e l = rigid h- mo d e l = rigid

k = 1.866 k = 1.866

cl = 0.5 cl = 0.2

L = 1. in L = 1. in

h2 = 0.0002 in h2 = 0.0002 in

h1 =0.0003732 in h1 =0.0003732 in Load = 65.376 lbf in Load = 29.057 lbf in Drag = 0.095 lbf in Drag = 0.103 lbf in Power Loss = 0.083 HP in Power Loss = 0.089 HP in U = 480. ft s U = 480. ft s r-mo d el = constant r-model = constant

m- mo d el = constant m- model = constant

h- mo d el = rigid h- model = rigid

Figure 20. Operating characteristics of the Rayleigh step bearing having change locations, cl, of 0.9, 0.718, 0.5, and 0.2.

4.1.3 Double Taper Bearing

The double taper bearing allows for the addition of a fourth parameter in the solution to

Reynolds equation while approximating the undeflected shape of the generation I foil thrust bearings. The parameters that are now included in the solution are minimum film thickness (h2), change location (cl), contraction on the inlet portion (ka), and contraction on the exit portion (kb). Again, the Reynolds equation of 4.1 governs the hydrodynamics for this problem. Closed form analytic solutions are available and have been calculated using the symbolic computing routines of Mathematica. These solutions are contained in

Appendix D. Figures 21 - 23 show the behavior of load as a function of inlet and exit

72 contraction ratio, ka and kb, for various change location, cl, and a constant minimum film thickness of 200 micro-inches. These graphs indicate that the maximum load for this analysis occurs for a parallel exit portion, kb = 1. Therefore, the double taper bearing can be replaced with a “taper-parallel” bearing which greatly simplifies the analysis.

By holding constant the exit portion contraction, kb, the load optimization reduces to a three parameter problem similar to that of the Rayleigh step bearing. The minimum film thickness, h2, does not affect the qualitative nature of the optimization surface. Only the magnitude of the resultant load is affected by this parameter. A contour plot of the taper- parallel bearing is shown in figure 24. This figure shows the maximum load capacity occurs for an inlet contraction of 2.25 and a change location of 0.81. Figure 25 shows the pressure distribution, streamlines, and velocity profiles for this particular bearing. The load capacity for this film thickness profile under the assumptions of a rigid, constant density, constant property bearing is over 73 pounds per inch, which is the largest load capacity of the three cases analyzed in closed form. Similar to the previous cases, the maximum load capacity tends to occur just prior to the elimination of the recirculation on the inlet and at the point of zero wall shear at the runner exit.

73 Load Capacity Surface “Double Straight Taper Bearing” cl = 0.25

80 60 2 Load 40 1.8 capacity 20 0 1.6

2 1.4 Exit portion 3 contraction, kb 1.2 Inlet portion 4 contraction, ka 5

Figure 21. Load capacity plot for the double taper bearing as a function of inlet contraction ratio, ka, and exit contraction ratio, kb, for a change location, cl, or 0.25. Load Capacity Surface “Double Straight Taper Bearing” cl = 0.50

80 60 2 Load capacity 40 1.8 (lbf/in) 20 0 1.6

2 1.4 Exit portion 3 contraction, kb 1.2 Inlet portion 4 contraction, ka 5

Figure 22.Load capacity plot for the double taper bearing as a function of inlet contraction ratio, ka, and exit contraction ratio, kb, for a change location, cl, or 0.50.

74 Load Capacity Surface “Double Straight Taper Bearing” cl = 0.75

80 60 2 Load capacity 40 1.8 (lbf/in) 20 0 1.6

2 1.4 Exit portion 3 contraction, kb 1.2 Inlet portion 4 contraction, ka 5

Figure 23. Load capacity plot for the double taper bearing as a function of inlet contraction ratio, ka, and exit contraction ratio, kb, for a change location, cl, or 0.75.

75 1

0.8

0.6

0.4

0.2 ka=2.25, cl=0.81,kb=1.00

2 3 4 5

Figure 24. Contour plot of load capacity for the taper-parallel bearing as a function of contraction ratio, ka, and change location, cl.

76

k = 2.25

L= 1. in

h2 = 0.00020 in

h1 = 0.00045 in Load = 73.101 lbf in Drag = 0.064 lbf in Power Loss = 0.056 HP in U = 480 ft s r-model = constant

m- model = constant

h- model = rigid

slip- model = no

Figure 25. Operating characteristics of the "optimized" taper-parallel bearing.

4.1.4 Isobaric Channel Flow with Viscous Heat Generation

A final analytic result is presented which introduces the significance of viscous heat generation in the range of operation of highly loaded foil thrust bearings. The analysis of this problem looks to solve the Reynolds and Energy equation for isobaric flow, constant pressure, with the requisite film thickness as the solution. The governing equations are written in equations 4.6-4.7 where the dissipation function has been simplified due to the lack of pressure gradient effects.

77  ρ 3 ∂   ρ h (u ) d  h p  = d  r  dx  12 µ ∂x  dx  2      p ( x = 0) = patm (4.6) p ( x = L ) = patm

dT ρUh µU 2 Cp = dx 2 h or, (4.7) µ 2 dT = U 2 dx h CpρUh T (x = 0) = Ta

Under the conditions of isobaric flow, the left hand side of the Reynolds equation becomes zero which implies that the product of density and film thickness must remain constant. By utilizing the ideal gas equation of state the energy equation can then be written in the form of equation 4.8 and 4.9.

ρ µ 2 − pa d = U 2 Rgρ 2 dx h CpρUh ρ µ 2 ρ 2 d = − U 2 Rg dx h CpρUh pa ρah1 dh µU 2 2 Rgρ 2 − = − (4.8) h 2 dx h CpρUh p µ 2 ρ 2 2 µ ρ dh = U 2 Rg h = 2 URg h 1 dx h CpρUh pa ρah1 Cppaρah1 h µURgρh hdh = dx Cppaρah1 h(x = 0) = h1

4µURg 4µUx γ −1 h(x) = x + h12 = + h12 (4.9) paCp pa γ

78 Considering the results of experimentally observed load capacity, the analysis of the previous sections of chapter 4, and the experimental measurements on journal bearings of

[17], indicates that a highly loaded foil bearing operates approximately in the range of

1,000 to 200 micro-inches minimum film thickness. Figure 26 plots the requisite film thickness for isobaric flow at an initial 200, 500, and 1000 micro-inch gap. It stands to reason that any gap less than that shown in the figure would lead to a pressure rise due to what some references call the thermal or density wedge effect of hydrodynamics. This result begins to demonstrate the importance of viscous heat generation for the range of film thickness and runner speed appropriate to foil thrust bearing modeling.

Figure 26. Graph of required channel divergence for Isobaric flow with viscous heat generation for inlet channel heights or 200, 500, and 1,000 micro-inches.

79 4.2 Numerical Solutions

The numerical solutions contained in the following sections are computed using the methods and procedures that are discussed in chapter 3. An additional Mathematica notebook was written to solve the Reynolds, Energy, and Structural equation in one dimension. The properties and parameters used in these calculations are intended to simulate the midspan section of the generation I foil thrust bearing described in chapter 1.

The compliant foundation model used in the structural equation simulates the pitch and stiffness of the generation I bearing load-deflection curve. The first numerical analysis is that of a parallel channel flow which includes viscous heat generation. This analysis builds on the isobaric flow analysis of the previous section and is included to further emphasize the importance of viscous heat generation in the simulation and modeling of highly load foil gas bearings. The analyses continue with an examination of the effects of a power law density model and variable viscosity. In this case, the film thickness is rigid and selected to have the optimized “taper-parallel” profile that is calculated in the analytic section of this chapter. The effect of a compliant foundation and a top foil are also analyzed. In order to maintain a clear distinction between various effects, this calculation is performed with an incompressible, constant viscosity lubricant model.

Finally solutions are presented for the energy equation model with both a rigid and compliant bearing foundation.

4.2.1 Parallel Channel Flow with Viscous Heat Generation

The results of a parallel channel flow with viscous heat generation are presented in figure

27. The figure shows the pressure and temperature distributions along the length of the

80 bearing as well as the streamline and velocity profiles. The results of this calculation also indicate a load capacity of 13.1 pounds per inch for a relatively large gap of 900 micro- inches. The waviness of the streamlines near the trailing edge of the bearing occurs due to the combination of low lubricant density and high pressure gradients at the trailing edge, and reduced numerical precision used in the output graphics. The thermal or density wedge is demonstrated by this calculation. Due to the combined effects of compressive and viscous heating the temperature lubricant increases to the point of peak pressure, also know as the Couette point. Beyond the Couette point the temperature tends to decrease due to the recovery of the compressive heating. However unlike the traditional isothermal bearing analyses or the power law model, the viscous heating is irreversible and that energy remains in the lubricant. Since the density is coupled to the temperature and pressure through the ideal gas relationship, a variable density profile is induced in this bearing which causes a behavior analogous to a physical contraction.

81 T

k = 2.25

L = 1. in P h2 = 0.00090 in h1 = 0.00090 in Load = 13.064 lbf in Drag = 0.034 lbf in PowerLoss = 0.030 HP in U = 480 ft s r-model = energy equation

m- model =power law, a=0.5

h- model = rigid

slip- model = yes

Figure 27. Operating characteristics of parallel channel flow with viscous heat generation. 4.2.2 Power Law Density Models

The combined effects of the power law density model and variable viscosity are analyzed in this section for the taper-parallel film thickness profile. Figure 28 shows a collection of results in the format that is consistent with the previous analyses of this section. Each of the three results uses the identical film thickness profile and a square root relationship between viscosity and temperature. Sub-figure A is the result for a density exponent of 5 with no slip boundary conditions. Sub-figures B and C are results for a density exponent

82 of 3 with no slip and slip flow boundary conditions respectively. In sub-figure c the use of slip flow boundary conditions are indicated graphically by the gap between the bearing trailing edge and the exit velocity profile. A general observation that can be made from this figure is that the inclusion of variable density and viscosity reduce or eliminate the recirculation zone at the leading edge and increase the pressure gradient and hence the overdriven flow at the trailing.

k = 2.25 k = 2.25

L= 1. i n L= 1. in

h2 = 0.00020 i n h2 = 0.00020 in

h1 = 0.00045 in h1 = 0.00045 in Load = 117.030 lbf in Load = 44.119 lbf in Drag = 0.089 lbf in Drag = 0.067 lbf in Power Loss= 0.077 HP in Power Loss = 0.059 HP in U = 480 ft s U = 480 ft s r-mod el = power law, n=5 r- mod el = power law, n=3

m- mod el =power law, a=0.5 m- mod el =power law, a=0.5

h- mod el = rigid h- mod el = ri gi d

slip- mod el = no slip- mod el = no

Figure 28 A Figure 28 B

k= 2.25

L= 1. in

h2= 0.00020 in

h1= 0.00045 in Load = 47.469 lbf in Drag = 0.069 lbf in Power Loss= 0.060 HP in U= 480 ft s r-model = power law, n=3

m- model =power law, a=0.5

h- model = rigid

slip- model = yes

Figure 28 C

Figure 28. Operating characteristics of bearings using the power law density and viscosity model.

83

4.2.3 Compliant Structure

One of the primary purposes of the compliant structure in foil bearings is to achieve an

“optimum shape” under operating conditions. As such, a shape such as the taper-parallel shape which is optimized in section 4.1.3 should be the film thickness profile under loaded conditions. Therefore, the following analysis is somewhat pedagogical in nature, but is performed in order to shed insight into the behavior of the compliant foundation and top foil membrane effects. Figure 29 contains the results that include structural deformations of a compliant foundation with a top foil. These calculations are performed with an incompressible and constant viscosity lubricant model and a no slip velocity boundary condition. The wavy nature of the final bearing surface is primarily due to the corrugated nature of the compliant structure. The top foil membrane effects are indicated graphically by the increased film thickness at the trailing edge. More simplistic point deflection models that have been used in previous studies are unable to capture this effect.

84 k=0.00109335

L= 1. in

h2 = 0.00023 in

h1 = 0.0005 in Load = 41.075 lbf in Drag = 0.037 lbf in Power Loss = 0.032 HP in U = 480 ft s r- model = constant

m- model = constant

h- model = rigid

slip- model = no

Figure 29. Operating characteristics of a bearing having a compliant foundation and top foil.

4.2.4 Viscous Heat Generation

The final set of one dimensional result is presented in this section. Figures 30 and 31 contain the results of calculations using the Reynolds and Energy equations. Figure 30 shows results for the rigid foundation while figure 31includes a compliant foundation.

These figures contain the same information and graphics as have most of the figures in this chapter. The temperature profiles have also been included in these figures, similar to the manner in which the parallel channel flow information of section 4.2.1 is presented.

Again, the streamlines plotted near the trailing edge of the bearing become wavy due to

85 the combination of low lubricant density and high pressure gradients at the trailing edge, and reduced numerical precision used in the output graphics. The first important observation of these results is that the film thickness has been doubled compare to all previous in section 4.2 while the load capacity has remained in the same range and order of magnitude. This fact emphasizes the importance of heat generation for highly loaded foil bearings and demonstrates the thermal or density wedge. A second observation should be noted by the addition of the compliant structure. Initially the heat generation amplifies the pressure gradient at the trailing edge. This in turn causes a very large zone of over driven flow exiting the bearing. In the thrust bearing geometry this high temperature over driven flow is drawn into the next pad where the temperature is further increased, leading to a thermal run away. However, with the addition of a compliant structure, even a deflection of 10 micro-inches at the trailing edge, which is the difference in these two figures, can have a significant effect in reducing the trailing edge pressure gradient and, in turn, the over driven flow.

86 P k = 2.25

L= 1. in

h2 = 0.00040 in

h1 = 0.00090 in T Load = 61.245 lbf in Drag = 0.050 lbf in Power Loss = 0.043 HP in U = 480 ft s r- mo d e l = energy equation

m- mo d e l =power law, a=0.5

h- mo d e l = rigid

slip- mo d e l = yes

Figure 30. Operating characteristics of a bearing with the energy equation and power law viscosity model.

87 P k= 2.25

L= 1. in

h2 = 0.00041 in

h1 = 0.00090 in T Load = 43.289 lbf in Drag = 0.045 lbf in Power Loss = 0.039 HP in U = 480 ft s r- model = energy equation

m- model =power law, a=0.5

h- model = compliant

slip- model = yes

Figure 31. Operating characteristics of a bearing with the energy equation, power law viscosity model, and compliant foundation.

4.3 General Observation

The one dimensional analyses presented in this chapter provide insight into the effects of various aspects of the simulation and modeling of foil gas bearings. The analytic results of the first half of this chapter indicate that for maximum load capacity for a given minimum film thickness and efficient use of lubricant is important. The phrase “efficient use of lubricant” in this context implies that the recirculation zone at the leading edge and the over driven velocity at the trailing edge are small or do not exist. The importance of viscous heat generation and variable temperature effects are introduced with two simplistic examples. The first is an isobaric flow wall shape calculation and the second is

88 a parallel channel load capacity calculation. The effects of variable density, viscosity, and compliant structure are compared. The variable density and viscosity both in the power law models and the energy equation simulation steepen the pressure gradient causing both a more peaked pressure profile and an over driven exit velocity profile. The compliant structure, on the other hand, counteracts this tendency by deflecting more at the pressure peak which flattens out the pressure profile. In the case of the energy equation simulation with compliant structure a 10 micro-inch trailing edge deflection significantly reduces the pressure gradient and over driven exit velocity profile.

89 CHAPTER 5 Cylindrical (Thrust Pad) Bearing Analysis

5.1 Impacts of Side Leakage for Zero Preload

The impact of side leakage on the hydrodynamic pressure distribution of the cylindrical thrust pad is investigated numerically for the case of an incompressible lubricant and a rigid wall bearing. These simplifications are made in the analysis of this section to more clearly elucidate the effects of side leakage on bearing performance without the additional effects of compressibility, heat generation, and structural deformation. The film thickness shape that is used is defined in chapter 1 as the zero preload shape. This shape consists of a double taper. The effect of the discontinuity is also investigated by completing the analysis for both the straight wall case and a “smoothed” shape. This operating condition is of particular interest due to unexplained qualitative data obtained from laboratory experiments. The initiation of wear spot in the center of the pads is experienced under very light applied loads to the bearing. Figure 32 is a photograph of the wear spots on such a bearing after a short term, low load test.

90

Figure 32. Photograph of a generation I foil thrust bearing showing wear in the center of the pads after a low load test.

Numerical experiments have found local extrema in pressure in two dimensional rigid air bearing geometries where the film thickness undergoes a significant change in slope along the primary bearing direction. Although this phenomenon is present in both

Cartesian and cylindrical plan forms, it does not manifest itself in the zero-side leakage

(infinite-width) approximation. A mathematical and physical explanation is presented for this phenomenon which includes a qualitative example of this hydrodynamic interaction with the compliant foundation and top foil.

The local extrema that occur near the sudden change of the film thickness slope is due to leakage effects. The phenomenon has been noted for incompressible as well as compressible fluid case. However, it has not been found in the zero-side leakage cases.

The phenomenon also appears to be quite sensitive to the slope of the second half of the bearing.

91 The bearing film thickness profile used to analyze this phenomenon is a double taper shown in figure 33. The first half of the bearing consists of a steep taper while the second half is either held parallel to the runner or has a slight taper. In the case of the

Cartesian bearing the film thickness profile is held constant in the z-direction. In the cylindrical bearing the film thickness profile is held constant in the radial direction. The working fluid in this bearing is air at ambient pressure and temperature. The runner speed is 900 feet per second. In the case of the thrust bearing, the bearing number is held constant at 0.462.

h

x

Figure 33. Plot of the three film thickness profiles considered in the low load side leakage calculations. Typical hydrodynamic behavior is noted in figure 34 for the incompressible, zero-side leakage bearing having a parallel second half. The pressure increases through the first half of the bearing, reaching a peak near the change of slope. For this parallel gap region the governing equation takes the form of pxx = 0, which implies that the pressure must

92 decrease linearly to match the trailing edge boundary condition. If the second half of the bearing contains some physical contraction, then the appropriate form of the Reynolds equation still allows for pressure field curvature.

Double-Taper

p

parallel

h

x

Figure 34. Typical hydrodynamic performance for the incompressible, constant property bearings with zero side leakage.

The behavior of the pressure field for a square Cartesian bearing is shown in figures 35-

36. Although quite similar to the zero-side leakage case, one significant difference is the slight positive curvature of the pressure field for the parallel film case. Mathematically this can be attributed to the form of governing equation in the two-dimensional parallel film thickness case, pxx + α pzz = 0. The negative curvature in the transverse direction due to leakage must be offset by positive curvature in the bearing direction. Also of note in the case with the double taper bearing is the nature of the hydrodynamics to seek a maximum pressure in the second half of the bearing due to the continued physical contraction.

93

Figures 37 - 45 show how these effects combine and are enhanced by greater length to width ratios. This phenomenon ultimately leads to the case of local pressure extrema on the bearing. The last four of these figures is included to demonstrate that this phenomenon is not merely numeric due to the discontinuous change in film thickness.

“Smoothing” the sharp profile with a third order-interpolating polynomial generated the profile used in these calculations.

p

h

x

Figure 35. Centerline pressure ratios for the Cartesian bearing, α=1

94

Figure 36. Pressure ratio field for the Cartesian bearing, α=1, double-taper

Figure 37. Pressure ratio field for the Cartesian bearing, α=1, parallel film

95

p

h

x

Figure 38. Centerline pressure ratios for the Cartesian bearing, parametric

α, double -taper profile

96

Figure 39. Pressure ratio surface, α=1

Figure 40. Pressure ratio surface, α=2

97

Figure 41. Pressure ratio surface, α=3

p

h

x Figure 42. Centerline pressure ratios for the Cartesian bearing, parametric

α, ”smoothed” profile

98

Figure 43. Pressure ratio surface, α=3

Figure 44. Pressure ratio surface, α=1

99

Figure 45. Pressure ratio surface, α=2

Figure 46 is included to demonstrate that the additional radial terms of equation 2 enhance this phenomenon. In this example the pressure field is calculated for the same

“smoothed” film thickness profile used in the previous Cartesian case. The bearing number is held constant and equal to all previous examples. The radius ratio is roughly equal to 1/2, which is similar to the experimental bearings. However, the radial effects enhance the severity of the local extrema.

Figure 47 contains pressure ratio distributions for a variety of thrust bearing length to width ratios. In these nine cases the tip speed (outer radius runner velocity) is held constant at 900 feet per second while the length to width ratios and outer radii are varied.

In contrast to the Cartesian bearing, which begins seeing local pressure extrema at

α around 3, the thrust bearing demonstrates this phenomenon at αt of one. Figures 48

100 and 49 show the angular and radial distributions of pressure at various locations of a typical solution. In terms of comparing these results qualitatively to the bearings tested in the lab, the center figure with an outer radius of 2 inches and αt of 1 is the most similar.

Figures 50 and 51 expand on this geometry and show the effect of varying bearing number on the pressure distribution. The bearing number for a fixed geometry and inlet pressure can be varied by either a change in speed or inlet viscosity. In terms of speed these figures represent 18,049, 23,206, 28,363, and 33,159 revolutions per minutes respectively.

Figures 52 and 53 indicate similarities between this local pressure extrema phenomenon and the experimental bearing wear documented in the lab. The analytic foil shape shown in figure 52 is calculated by applying the incompressible, rigid thrust bearing pressure distribution to a simple “εP” compliant model where ε is stiff on the mid-span and soft on the edges which is typically considered “good” design practice for foil bearings.

101

Figure 46. Pressure ratio distribution for the "smoothed" profile in radial plan

form

102

Figure 47 Thrust bearing pressure distributions for various geometry.

103

Figure 47(continued) Thrust bearing pressure distributions for various geometry.

104

Figure 47(continued) Thrust bearing pressure distributions for various geometry.

105

Figure 48. Typical thrust bearing angular pressure distributions.

Figure 49. Typical thrust bearing radial pressure distributions.

106

Figure 50. Effect of varying bearing number for fixed geometry thrust bearing.

107

Figure 51 Angular pressure distributions at the approximate maximum pressure

radius.

108

Figure 52. Example of a top foil shape when the compliant foundation is superimposed on the static film thickness and pressure distribution.

Figure 53. Close up photograph of the wear spot on the foil thrust bearing demonstrating the similarity between calculations and experiment.

The phenomenon of local extema in a hydrodynamic bearing having a double-taper film thickness profile is due primarily to side leakage effects and, in the case of thrust bearings, is enhanced by the radial effects. The local pressure minimum predicted by this phenomenon appears to correspond to the wear patterns witnessed on the experimental bearings having zero preload and tested under very low applied loads.

109

5.2 Impacts of Side Leakage for Highly Loaded Bearings

The impact of side leakage for a highly loaded bearing is examined in this section.

Intuitively the effect is to reduce the mass flow rate through the bearing and therefore reduce the load carrying capacity of the bearing. The film thickness profile used in this analysis is the profile determined in section 4.1.3 and referred to as the taper parallel bearing. The rotational speed for these calculations is set to 40,000 revolutions per minute. For the case of the thrust bearing the x coordinate is replaced by the angular coordinate, θ, and the film thickness is constant along the radius. Most of the analysis performed in chapter 4 uses this profile, so the comparisons between one dimensional and two dimensional bearings are straightforward. The static foil shape is shown in figure 54.

The remainder of this chapter follows a similar format as that of chapter 4. The first analysis is conducted on a thrust bearing with an incompressible density and constant viscosity lubricant model. The power law density and viscosity model is then included in the analysis followed by the effects of a compliant foundation and top foil. Finally, the chapter concludes with the inclusion of the energy equation to examine the effects of viscous heat generation on the thrust foil bearing.

110 Top Foil - static

-0.0002

0.40.

-0.0003 0.6

-0.0004 0.8

0

0.2 0.4 1 0.6

Figure 54. Static top foil shape used for the side leakage effect calculations.

5.2.1 Incompressible, Constant Property Lubricant

The solution to the incompressible constant property lubricant thrust pad is shown in the figures 55 - 57. The pressure ratio and mass flow vectors are shown in these figures. As compared to the results of figure 25 the side leakage reduces the load capacity from 73.1 lbf / sq.inch to 31.1 lbf / sq inch with a torque of 0.39 inch lbf. The pressure distribution shows a peak near the change location of 81% of the pad angular extent which is similar to the one dimensional case. However, the pressure also demonstrates a strong radial variation which is required to satisfy the uniform pressure boundary conditions. Two

111 figures are included to show the mass flow vectors. The first vector plot is in cylindrical coordinates and is included to provide the physical intuition of the thrust bearing pad shape. The second vector plot is in rectilinear coordinates and is included to more readily see the side leakage since the bearing boundary is parallel to the axes.

Pressure Ratio

0.4

0.6

6

4

2 0.8

1

0

0.2

0.4

0.6

Figure 55. Pressure ratio distribution for the incompressible, constant viscosity lubricant.

112 0.8

0.6

0.4

0.2

0

0 0.2 0.4 0.6 0.8 1

Figure 56. Vector plot of mass flow in cylindrical coordinates for the incompressible, constant viscosity lubricant.

113 0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0.5 0.6 0.7 0.8 0.9

Figure 57. Vector plot of mass flow in rectilinear coordinates for the incompressible, constant viscosity lubricant.

114

5.2.2 Impacts of Power Law Density Models

The power law model for density and viscosity, n = 5. and a=0.5, is been added to the calculations in this section. Figures 58 - 60 show the pressure ratio and mass flow vectors. As compared to the results of chapter 4 the side leakage reduces the load capacity from 117. lbf / sq.inch to 37.7 lbf / sq inch with a torque of 0.58 inch lbf. The pressure distribution shows a peak near the change location of 81% or the pad angular extent which is similar to the one dimensional case. The pressure distribution in the angular direction shows more curvature than the incompressible, constant viscosity case, which is due to the nonlinearity introduced in the Reynolds equation. However, the pressure also demonstrates a strong radial variation which is required to satisfy the uniform pressure boundary conditions. Two figures are again included to show the mass flow vectors.

115 Pressure Ratio

0.4

0.6 10

7.5

5

2.5 0.8

1

0

0.2

0.4

0.6

Figure 58. Pressure ratio distribution for the power law model for density and viscosity, n=5.0, a=0.5.

116 0.8

0.6

0.4

0.2

0

0 0.2 0.4 0.6 0.8 1

Figure 59. Vector plot of mass flow in cylindrical coordinates for the power law model for density and viscosity, n=5.0, a=0.5.

117 0.6

0.4

0.2

0 0.5 0.6 0.7 0.8 0.9

Figure 60. Vector plot of mass flow in rectilinear coordinates for the power law model for density and viscosity, n=5.0, a=0.5.

118 5.2.3 Impacts of Compliant Foundation

The effect of a compliant foundation is analyzed in this section. The lubricant model is returned to the incompressible, constant property condition and the structural equation is introduced into the solution. The calculated load capacity and torque per pad are 12.0 pounds pre square inch and 0.067 inch pounds respectively. The reduction in these two parameters is primarily caused by the increase in film thickness due to the action of the hydrodynamic pressure on the compliant foundation and top foil. Figure 61 shows this graphically by plotting the film thickness along the trailing edge of the pad for the rigid and deflected case. The minimum film thickness increases to approximately 150% of its original thickness. The complete deflected foil shape is shown in figure 62. This plot shows both the effect of the corrugated compliant foundation structure as well as the top foil membrane effects. The pressure distribution and mass flow vector plots are also included in figures 63 - 65. An interesting subtly to note in the rectilinear mass flow vector plot is the behavior of the leakage on the inner radius. The angle of the vector indicates the relative amount of leakage versus through flow, which is a function of radial pressure gradient and local film thickness. The vectors near coordinate (0.5,0.3) and

(0.5,0.6) indicate less leakage and more through flow than the remainder of the inner radius. These locations correspond to the location of the bumps in the bump foil that are minimizing film thickness locally.

119

Figure 61. Plot of static and deformed trailing edge film thickness for the compliant foundation analysis.

120 FoilTop

0.40. -0.00025

-0.0003

0.6

-0.00035

-0.0004 0.8

-0.000455

0

0.2

0.4 1 0.6

Figure 62. Plot of the deformed shape of the top foil.

121 Pressure Ratio

0.4

0.6

3

2

0.8 1

1

0

0.2

0.4

0.6

Figure 63. Pressure distribution corresponding to the film thickness created by the deformed top foil.

122 0.8

0.6

0.4

0.2

0

0 0.2 0.4 0.6 0.8 1

Figure 64. Vector plot of mass flow in cylindrical coordinates for the case of the compliant foundation.

123 0.6

0.4

0.2

0

0.5 0.6 0.7 0.8 0.9

Figure 65. Vector plot of mass flow in rectilinear coordinates for the case of the compliant foundation.

5.5 Impacts of Viscous Heat Generation

The following section examines the effect of viscous heat generation on the cylindrical thrust pad geometry. The case is calculated with the same film thickness profile as the previous cases as shown in figure54. However, the rotational speed is set to 25,000 revolutions per minute. The heat transfer from the lubricant to the surroundings is set to

124 zero for these results. At these conditions the load capacity and torque of this configuration is 60.1 pounds per square inch and 0.41 inch pounds, respectively. The pressure ratio, temperature ratio, and mass flow vectors are plotted in figures 66 - 69. At these conditions the temperature effects become very significant as is seen in figure 67.

The temperature field qualitatively follows the pressure field with the exception of the additional temperature gained by the viscous dissipation function. This additional temperature is irreversible or non-recoverable heat which is indicated on the figure by the boundaries not returning to unity. In combination with the ideal gas relation this temperature effect creates a density wedge that enhances the pressure field and load carrying capacity of the bearing. The enhanced pressures also make an impact on the mass flow vector plots. The side leakage is much greater in this case than in previous cases due to the large radial pressure gradient that is created in this situation.

125 Pressure Ratio

0.4

0.6

15

10

5

0.8

1

0

0.2

0.4

0.6

Figure 66. Pressure distribution corresponding to the viscous dissipation case.

126 Temperature Ratio

0.4

0.6

10

5

0.8

1

0

0.2

0.4

0.6

Figure 67. Temperature distribution corresponding to the viscous dissipation case.

127 0.8

0.6

0.4

0.2

0

0 0.2 0.4 0.6 0.8 1

Figure 68. Vector plot of mass flow in cylindrical coordinates for the case of viscous heat generation.

128 0.6

0.4

0.2

0

0.5 0.6 0.7 0.8 0.9

Figure 69. Vector plot of mass flow in rectilinear coordinates for the case of viscous heat generation.

129

5.6 General Observation

The effect of side leakage on a cylindrical thrust bearing plan form is analyzed in this chapter for a variety of lubricant assumptions. Initially, zero preload and low applied load bearings indicate unique behavior and wear patterns in laboratory experiments. A possible hydrodynamic explanation for this behavior is the combined effect of side leakage with a double taper film thickness profile. Various density and viscosity models are investigated in conjunction with the side leakage effect. The cases of the incompressible lubricant and the power law lubricant models each follow a similar trend.

However, the case in which the viscous heat generation is included in the calculations shows different behavior. The primary reason for this difference in behavior is the large temperature rise which couples with the ideal gas relation to create a density wedge in the

Reynolds equation. This density wedge combines with the physical wedge to produce much a larger hydrostatic pressure rise than the traditional lubricant models.

130 CHAPTER 6 Comparison of Simulations and Experiments

6.1. Low Load Wear

Simulations of the combined effects due to foil thrust bearing hydrodynamics, heat generation, and structural compliance show realistic results that can be correlated to laboratory experience. The typical static film thickness profile for one pad of an 8-pad generation I thrust bearing is shown in figure 70. The first portion of the pad undergoes a large area contraction, on the order or 0.002 inches of height, while the second portion of the pad contains a slight contraction, on the order of a few hundred micro inches. The mathematical model of the bump foil sub-structure is shown graphically in figure 71.

The step-function nature used here more accurately models the initial flattening of the corrugated bumps under load, than would a more visually pleasing Sine wave type of function. The Reynolds and Energy equations are then coupled to these structural models to simulate the performance of the foil thrust bearing at low speed and load.

131

Figure 70. Inital film thickness for low speed simulations

132

Figure 71. Bump foil stiffness model.

Sample results for a low speed, 10000 rpm, and low load, 20 pounds, conditions are shown in figure 72 and 73. The effects of foil sag and non-radial corrugations in the bump foil combine to create zones of minimum film thickness (or zones of likely metal to metal high speed contact) within the domain of the pad rather than on the edges. The photograph of experimental results shown in figure 74 corroborates this result.

133

Figure 72. Typical low load hydrodynamic pressure field.

134

Figure 73. Calculations showing typical low speed, low load foil shape. The white regions show high spots where high-speed contact is most likely to occur.

Figure 74. Experimental photograph showing witness marks after a low load, low speed 6.2 Foil Thermocouple Test

135

A standard generation I 8-pad thrust bearing is modified by removing every other pad to create a 4-pad thrust bearing in which each pad had an angular extent of 43 degrees. The exposed area between the pads is further modified to include a cooling air feed hole near the leading edge mid-span of each remaining pad. A post-test photograph is shown in figure 75. Thermocouples are installed under the bump foils at approximately the 37.5% span – 32 degree location. The bearing is tested at nominal conditions of 50,000rpm, 15 pounds axial load, and cooling air inlet temperatures of 60-70 degrees F. These conditions result in bearing temperature measurements in the range of 120-137 degrees F.

No torque data was possible due to the rotational constraint of the cooling air line.

Figure 75. Post-test photograph of the modified foil thrust bearing.

136 The bearing performance is simulated numerically using the Reynolds equation, power law density and viscosity model (n=1.41, a=0.5), and the structural model. Initial (zero load) film thickness is shown in figure 76. It consists of a typical ramp-flat design in which the ramp height is 0.002 inches and the flat is separated from the runner by 650 micro-inches. The bump foils are modeled by the function shown in figure 77. The model captures the non-radial nature of the bump foils, the number and spacing of the contact lines, and the overall bearing stiffness. The initial “flattening” of the bumps is modeled by the use of a step function rather than a rounded function, which would look more like a bump foil arrangement but would not accurately model its behavior under load. Figure 78 shows the results of the numerical results of the simulation in terms of hydrodynamic pressure, temperature, and foil deflection. Figure 79 shows a magnified view of the initially flat section of the top foil. The minimum film thickness at the trailing edge, which shows an increase in the film thickness due to the membrane effects, is plotted in figure 80. The most significant result of these calculations is shown in figure

81. This figure plots the 37.5% span line for a 60 and 70 degree F inlet temperature as well as the thermocouple data from the test. There is agreement between the measured temperature and the hydrodynamic performance is very good. This example demonstrates the utility of applying the power law density model to the Reynolds equation. At low speeds and load where the viscous heating may still be considered insignificant, the power law model can correlate temperatures to pressures. This is significant because it is possible to obtain thermocouple data from foil bearing experiments, but it is nearly impossible to measure pressure directly without corrupting the bearing surfaces, lubricant flow field, and top foil stiffness.

137

Figure 76. Static Film Thickness.

Figure 77. Bump Foil Stiffness Model

138

Figure 78. Graphical Simulation Results

Figure 79. Close-up of "flat section" foil deflection

139

Figure 80. Static and Deflected trailing edge film thickness.

Temperature (F)

angular extent (radians)

Figure 81. Calculated and Measured bearing temperatures.

6.2 Torque versus Load Data

An important result for the comparison of analytic calculations with experimental data is the torque versus load characteristic. These are the two most significant parameters measured in a bearing test and any analytic result should be able produce such

140 information as well. A graph of bearing torque versus load is plotted in figure 82 for the various lubricant models that are used in chapter 5. The static film thickness of figure 54 is used for all of these calculations. It becomes clear from figure 82 that the significance of the density and viscosity is low. The bearing performance calculations that utilize the energy equation are unique in comparison to traditional models. Bearing data at high rotational speeds and high loads is not yet readily available in the open literature.

However, preliminary experimental results indicate that the energy equation simulation more accurately reproduces the laboratory results.

Figure 82. Bearing torque versus load for various lubricant models.

141 CHAPTER 7 Summary and Conclusion

A simulation and modeling effort is conducted on foil gas thrust bearings. Although simple in appearance these bearings have proven to be very complicated devices. They are sensitive to fluid structure interaction, use a compressible gas as a lubricant, may not be in the fully continuum range of classic fluid mechanics, and operate in the range where viscous heat generation is significant. These factors provided a challenge to the simulation and modeling effort.

The conservation equations of mass, momentum, and energy are applied to the problem.

For the case of high applied loads the modified Reynolds number and squeeze number make it appropriate to simplify the momentum equations to non-inertial flows. The fluid velocity components are then solved in closed form and substituted into the mass conservation equation. This procedure follows the traditional approach for the development of the Reynolds equation with the exception that a slip flow boundary condition is introduce to account for very thin film thicknesses. The energy equation is then simplified by applying the thin layer assumptions. The assumptions are justified and applied in the development of the Reynolds equation. The thin layer assumptions basically state that the film thickness is sufficiently thin such that fluid properties

(pressure, temperature, density, viscosity, and heat capacity) cannot vary through the film.

A term is introduced into the energy equation which may be used to account for heat transfer between the lubricant and the surroundings. Although this term is unused in the

142 present work, it will have utility in future analyses that examine cooling and thermal management issues in foil thrust bearings.

The structural deformations of bearings are modeled with a single partial differential equation. The equation models the top foil as a thin, bending dominated membrane whose deflections are governed by the biharmonic equation. A linear superposition of hydrodynamic load and compliant foundation reaction is included, making the equation inhomogeneous. The stiffness of the compliant foundation is determined through engineering judgment and analysis of the static load – deflection curve of the bearing.

A computer program written in the Mathematica computing environment is created to solve the set of equations and relations developed to analyze the foil thrust bearing. The

Reynolds and structural equations are solved by the finite difference method using central differencing techniques on all internal nodes. Boundary conditions are applied through the use of one-sided finite difference formula. The energy equation is solved by the application of the Mathematica function NDSolve. An iteration loop is then wrapped around the functions that continually updates the pressure, temperature, density, and deformation fields until a self-consistent solution is achieved.

A generalized hydrodynamic analysis is conducted to systematically analyze each of the individual effects included in the development of the governing equations. First analytic solutions are produced for one dimensional bearing flows with incompressible lubricants and rigid surfaces. These analytic solutions serve two purposes. The first is to provide

143 insight into the effects of film thickness on the performance of a bearing. The second purpose is to provide a known solution for the debugging phase of the programming that follows. Numerical solutions are then calculated for cases of variable density, variable viscosity, compliant foundations, and viscous heat generation. The current simulation and modeling approach that includes the viscous heat generation in the energy equation begin to show differences from the traditional isothermal lubricant models in these one dimensional examples.

The effect of side leakage on the cylindrical thrust bearing plan form is then numerically investigated. A similar format is used to that of the one dimensional calculations. Under high applied loads lower load capacities are noted and expected for the case of side leakage. However, for the case of zero preload and low applied loads a unique situation occurs that produces two local maxima of pressure. This effect combined with the compliant foundation may lead to the manifestation of a wear spot in the central portion of the thrust pads. The understanding of the combined effect of side leakage and subtle changes in film thickness is an important aspect in foil thrust bearings. This effect can manifest itself in the low applied load case as well as locations where foil sag between bump foils becomes significant.

Quantitative comparisons are produced that demonstrate the ability of this method to predict the temperature field of foil thrust bearings and the coupling of the temperature field to the hydrostatic pressure field. No previous work has demonstrated this ability.

Finally, bearing torque versus load curves are calculated that show the differences

144 between traditional bearing models and the current approach. The calculations show that the analyses which couple the Reynolds, energy, and structural equation will accurately model future bearing performance. An analytic model is developed that can provide insight into the hydrodynamic, thermal, and structural interactions of thrust foil bearings for future bearing designs. Results of this method will also provide system integration guidance for advanced foil bearing applications.

145 Appendix A Summary of Governing Equations

146 Rectilinear with No Side Leakage

Assumptions:

• Small modified Reynolds Number and Squeeze Number, non-inertial flow • Constant fluid properties through the film thickness • Bending Dominated top foil stiffness

Reynolds Equation and Boundary Conditions

 ρ 3  µ   d  h  6Kn  dp  d ()() 1+ = Λ ρh uf +1 dx  µ   dx  dx   ph  

p(x = 0) = pa, p(x =1.0) = pd

Energy Equation and Boundary Conditions

    γ 3  2   Cp  dT  1  dp h  dp   Λcµ m&x = m&x +   + +Q γ −1 dx ρ  dx  Λ µ  dx    2 sc     c     µ      +2Kn   3h1     ph 

T (x = 0) = Ta

Structural Model and Boundary Conditions

∂4δ 4 ( p − 1) − K (x,δ )δ =  L  c   ∂x4  h1  Kb, f

∂δ (x = 0) ∂2δ (x =1.0) ∂3δ (x = 1.0) δ (x = 0) = = 0, = = 0, ∂ x ∂x2 ∂x3

Ideal Gas ρ = p T Velocity

 Knµ  2  1+ ()1+u f + y()u f −1  u 3h  d p  2 K µ  ph = = −   − + n  +   w(x, y) = 0 u(x, y)   y y  µ U Λµ d x ph  + K n      1 2   ph 

Mass Flow

3 υ h  d p  6K µ  mf mf (x, y) = x = −  1+ n  + ()1+u h mf (x, y) = z = 0 x ρ Λµ    f z ρ aUh1 d x ph aUh1 2    2

Runner Drag

   1   ∂u( x , y = 0)  3h  d p  µ u f −1  f = µ dx = −   +   dx 1D ∫ ∫    µ  ∂ y Λ d x h  K n  0    1 + 2    ph 

147 Rectilinear

Assumptions:

• Small modified Reynolds Number and Squeeze Number, non-inertial flow • Constant fluid properties through the film thickness • Bending Dominated top foil stiffness

Reynolds Equation and Boundary Conditions

∂  ρ 3  6K µ  ∂  ∂  ρ 3  6K µ  ∂  ∂ ∂  h  + n  p  +α2  h  + n  p  = Λ ()ρ ()+ +α ()ρ ()+  1 1  h uf 1 h wf wr  ∂x  µ  ph  ∂x  ∂z  µ  ph  ∂z  ∂x ∂z          

p(x = 0, z) = pa(z), p(x = 1.0, z) = pd (z), p(x, z = 0) = pb(x), p(x,z = 1.0) = pc(x),

Energy Equation and Boundary Conditions

    γ 3  2 2 Cp  ∂T ∂T  1  ∂p ∂p h ∂p  ∂p  Λ µ  m&x +αm&z = m&x +αm&z +   +α  + c +Q γ −  ∂ ∂  ρ  ∂ ∂  Λ µ  ∂   ∂   2 sc 1 x z   x z  c  x  z    µ       +2Kn  3h1     ph 

T (x = 0, z) = Ta (z),T (x,z = zP max ) = TP max (x),

Structural Model and Boundary Conditions

4 ∂4δ ∂4δ ∂4δ  L  ( p −1) − K (x, z,δ )δ + α 2 + α 4 =   c 4 2 2 4   ∂x ∂x ∂z ∂z  h1 Kb, f

∂δ (x = 0, z) ∂2δ (x = 1.0, z) ∂3δ (x =1.0, z) ∂2δ (x, z = 0) ∂3δ (x, z = 0) ∂2δ (x, z =1.0) ∂3δ (x, z = 1.0) δ (x = 0, z) = = 0, = = 0, = = 0, = = 0, ∂ x ∂x2 ∂x3 ∂x2 ∂x3 ∂x2 ∂x3

Ideal Gas ρ = p T Velocity

 K nµ   Knµ  2  1+ ()1+u f + y()u f −1  2  1+ ()wr + w f + y()w f −wr  u 3h  ∂ p  K µ  ph w 3h  ∂ p  K µ  ph u(x, y, z) = = −   y − y 2 + n  +   w(x, y, z) = = −   y − y2 + n  +   Λµ    µ Λµ    µ U ∂ x ph  + Kn  U ∂z ph  + Kn      1 2      1 2   ph   ph 

Mass Flow

3 3 mf ρ h  ∂ p  6K µ  mf ρ h  ∂ p  6K µ  mf (x, z) = x = −  1+ n  + ()1+u ρ h mf (x, z) = z = −  1+ n  + ()w +w ρ h x ρ Λµ    f z ρ Λµ    r f aUh1 ∂ x ph aUh1 ∂ z ph 2    2   

Runner Drag

     ∂u( x, y, z ) 1 1  3h  ∂ p  µ u f − 1  f = ∫∫ µ dxdz = ∫∫−   +  dxdz ∂ Λ  ∂  K µ y 00  x  h     1 + 2 n    ph 

148 Cylindrical Thrust Pad

Assumptions:

• Small modified Reynolds Number and Squeeze Number, non-inertial flow • Constant fluid properties through the film thickness • Bending Dominated top foil stiffness

Reynolds Equation and Boundary Conditions

 3   3  1 ∂  ρ h  6 K n µ  ∂ p  ∂  ρ h  6 K n µ  ∂ p  ∂  1 +   +  r 1 +   = Λ r  ρ h  ∂θ   ∂θ   ∂θ   r  µ  p h   ∂ r  µ  p h  ∂ r     

p(r = 0,θ ) = pb (θ ), p(r =1.0,θ ) = pd (θ ), p(r,θ = 0) = pa (r), p(r,θ =θend) = pc (r),

Energy Equation and Boundary Conditions

    γ     3  2 2  2  Cp  ∂T m&θ ∂T  1 ∂p m&θ ∂p h  ∂p   1 ∂p   Λµr m& + = m& +  +   +  +   +Q γ −  r ∂ ∂θ  ρ  r ∂ ∂θ  Λµ  ∂   ∂θ     s 1  r r   r r   r   r  2         2K µ     3h1+ n        ph 

T (r = 0,θ ) = Ta (z),T (r,θ =θ P max ) = TP max (r),

Structural Model and Boundary Conditions

4 ∂2δ ∂4δ ∂δ ∂3δ ∂4δ ∂3δ ∂4δ ∂4δ  r  ()P −1 − K (r,θ,δ )δ 4 + + r − 2r + r 4 + 2r 3 − r 2 + 2r 2 = r 4 o  c 2 4 ∂ 2 4 3 2 2 2 2   ∂θ ∂θ r ∂r∂θ ∂r ∂r ∂r ∂θ ∂r ∂θ  h1  Kb, f

∂δ (r,θ = 0) ∂2δ (r = RR,θ) ∂3δ (r = RR,θ ) ∂2δ (r =1.0,θ) ∂3δ (r = 1.0,θ ) ∂2δ (r,θ =θend) ∂3δ (r,θ =θend) δ (r,θ = 0) = = 0, = = 0, = = 0, = = 0, ∂θ ∂r 2 ∂r 3 ∂r 2 ∂r 3 ∂θ 2 ∂θ 3

Ideal Gas ρ = p T Velocity µ µ  + Kn ()+ θ + ()θ −   + Kn ()+ + ()−  θ 2  ∂    1 1 u f y u f 1  2  ∂  µ   1 ur r ur f y ur f ur r  θ θ = u = − 3h  p  − 2 + Kn  +  ph  θ = ur = − 3h  p  − 2 + Kn  +  ph  u (r, , y) y y µ ur(r, , y) y y µ ωro Λµ  ∂θ  ρ   + Kn  ωr Λµ  ∂  ph   + Kn  r   h   1 2  o  r    1 2   ph   ph 

Mass Flow

3 3 mfθ h  ∂ p  6Knµ  mfr h  ∂ p  6Knµ  mf θ (r ,θ ) = = −  1 +  + 1 + u f h mf (r,θ ) = = −  1 +  + urr + ur f h ρ ()ωro h1 Λµ  ∂θ     r ρ ()ωro h1 Λµ  ∂     a r   ph  a  r  ph  2 2

Runner Torque

     2   h1  θend 1  3r h  ∂ p   µ r ()ur (r,θ , y) −1  τ = τ = ∫∫− + d rdθ  4       ω µ Λ  ∂θ  2K µ  ro a  0 RR    n    h1+           ph  

149 Appendix B Mathematica Iterator Loop

150 Appendix C Closed form solution to the Rayleigh step bearing

151

152 153

154 Appendix D Closed form solution to the double taper bearing

155

156

157

158 References

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159 13. Florin, G., Theory and Design of Surface Structures: Slabs and Plates, Trans Tech Publications, Rockport, MA, 1980.

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160