Matthew J. Hill and Robert F. Kunz

Pennsylvania State University, Philadelphia, Pennsylvania

A Computational Investigation of Gear Windage

NASA/CR—2012-217807

December 2012
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Matthew J. Hill and Robert F. Kunz

Pennsylvania State University, Philadelphia, Pennsylvania

A Computational Investigation of Gear Windage

NASA/CR—2012-217807

December 2012

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Prepared under Contract NNC10CA35C
Acknowledgments

I must fi rst thank Dr. Robert F. Kunz and Dr. Lyle N. Long for the opportunity to participate in this research and for all of their 

help and guidance. I also want to thank the other members of my dissertation committee members Dr. Philip Morris and Dr. Suren 

Rao for their help and insight. I need to extend my gratitude to both Ralph Noack and Rick Medvitz at the Applied Research 

Laboratory. Dr. Noack aided with SUGGAR and overset grid generation. Dr. Medvitz provided invaluable help with both grid 

generation and running OVER-REL for the initial Diab and NASA Glenn studies. I also need to thank Chuck Ritter and Bernie 

Berzonsky of the Applied Research Laboratory for all their help and with putting up with how often I broke their systems.

Dr. Robert Handschuh of NASA Glenn Research Center provided all of the experimental data from the NASA Glenn Gear Windage 

Test Facility. Ryan Ehinger from Bell Helicopter provided the gear geometry data for the helical gear study and was instrumental 

in its inclusion in this dissertation. I need to thank both the Penn State Aerospace Department staff and the ARL Garfi eld Thomas 

Water Tunnel staff for all their help. They made the process of completing this degree much easier. Finally, to my parents and 

family, I am grateful for all of their love and support. Their constant pestering that “ten years of college should be enough” did not 

hurt either. Funding for this research is provided by NASA under NASA Cooperative agreement NNX07AB34A. Computational 

support for this project was provided by the NASA Advanced Supercomputing (NAS) Division and is gratefully acknowledged.

The research presented in Section 4.5 was funded in part by the Vertical Lift Consortium, formerly the Center for Rotorcraft 

Innovation and the National Rotorcraft Technology Center (NRTC), U.S. Army Aviation and Missile Research, Development and 

Engineering Center (AMRDEC) under Technology Investment Agreement W911W6-06-2-0002, entitled “National Rotorcraft 

Technology Center Research Program”. The author would like to acknowledge that this research and development was accomplished 

with the support and guidance of the NRTC and VLC. The views and conclusions contained in this document are those of the

authors and should not be interpreted as representing the offi cial policies, either expressed or implied, of the AMRDEC or the U.S. 

Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding 

any copyright notation thereon.

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Available electronically at http://www.sti.nasa.gov

Trade names and trademarks are used in this report for identifi cation 

only. Their usage does not constitute an offi cial endorsement, 

either expressed or implied, by the National Aeronautics and 

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This work was sponsored by the Fundamental Aeronautics Program 

at the NASA Glenn Research Center.

Level of Review: This material has been technically reviewed by technical management. 
Abstract

A CFD method has been developed for application to gear windage aerodynamics.

The goals of this research are to develop and validate numerical and modeling approaches for these systems, to develop physical understanding of the aerodynamics

of gear windage loss, including the physics of loss mitigation strategies, and to

propose and evaluate new approaches for minimizing loss. Absolute and relative

frame CFD simulation, overset gridding, multiphase flow analysis, and sub-layer

resolved turbulence modeling were brought to bear in achieving these goals. Several spur gear geometries were studied for which experimental data are available.

Various shrouding configurations and free-spinning (no shroud) cases were studied. Comparisons are made with experimental data from the open literature, and

data recently obtained in the NASA Glenn Research Center Gear Windage Test

Facility. The results show good agreement with experiment. Interrogation of the

validative and exploratory CFD results have led, for the first time, to a detailed understanding of the physical mechanisms of gear windage loss, and have led to newly

proposed mitigation strategies whose effectiveness is computationally explored.

A Computational Investigation of Gear Windage

Matthew J. Hill and Robert F. Kunz 

Pennsylvania State University

Philadelphia, Pennsylvania 19104 

NASA/CR—2012-217807 iii
Table of Contents

List of Figures vi

List of Tables xi

Chapter 1

Introduction 1

1.1 A Brief Introduction to Gearing . . . . . . . . . . . . . . . . . . . . 2

1.2 Previous Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Modeling Gear Windage Loss . . . . . . . . . . . . . . . . . . . . . 14

1.4 Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.5 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.6 Objectives of Dissertation . . . . . . . . . . . . . . . . . . . . . . . 20

1.7 Organization of Dissertation . . . . . . . . . . . . . . . . . . . . . . 21

Chapter 2

Theoretical Formulation 22

2.1 NPHASE-PSU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.1.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . 23

2.1.2 Grid Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.1.3 Multiphase Flow Methods . . . . . . . . . . . . . . . . . . . 25

2.1.3.1 Two-Fluid Eulerian Formulation . . . . . . . . . . 26

2.1.3.2 Homogeneous Multiphase Formulation . . . . . . . 27

2.2 OVER-REL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.2.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . 28

2.3 Overset Grid Methods . . . . . . . . . . . . . . . . . . . . . . . . . 31

NASA/CR—2012-217807 iv
Chapter 3

Validation and Parametric Studies 33

3.1 NPHASE-PSU Results . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 OVER-REL RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3 Details of the 3-D Flow Field . . . . . . . . . . . . . . . . . . . . . 50

Chapter 4

Aerodynamic Studies of Windage Loss Mitigation Strategies 55

4.1 Shrouded Diab Gears . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2 Investigation of the Aerodynamics

of Shrouding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.3 NASA Glenn Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.3.1 NPHASE-PSU Results . . . . . . . . . . . . . . . . . . . . . 86

4.3.2 OVER-REL Results . . . . . . . . . . . . . . . . . . . . . . 93

4.4 Design Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.5 Bell Helical Gear Analysis . . . . . . . . . . . . . . . . . . . . . . . 103

4.5.1 Design Alternatives . . . . . . . . . . . . . . . . . . . . . . . 116

Chapter 5

Conclusions 121

Appendix A

Gear Windage Estimate Program 125

A.1 Program Description . . . . . . . . . . . . . . . . . . . . . . . . . . 125

A.2 GWLestimate.py . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Appendix B

Radial Viscous Power Loss Program 129

B.1 Program Description . . . . . . . . . . . . . . . . . . . . . . . . . . 129

B.2 Vprofiler.cpp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Bibliography 144

NASA/CR—2012-217807 v
List of Figures

1.1 Effect of lubrication failure on a gear. [1] . . . . . . . . . . . . . . . 3

1.2 Various gear types. [2] . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Schematic of typical gear tooth nomenclature. [3] . . . . . . . . . . 6

1.4 Dawson’s sketches of the airflow around (a) the spur gear and (b)

the helical gear. [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5 NASA High-Speed Helical Gear Train Facility. [5] . . . . . . . . . . 11

1.6 Diab Test Rig. [6] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.7 Gear windage loss estimates for a 13-inch pitch diameter spur sear 18

2.1 An example domain reduction of an axisymmetric system with periodic boundaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2 An example overset grid system. . . . . . . . . . . . . . . . . . . . 32

3.1 Grid topology of Diab Gear 4. . . . . . . . . . . . . . . . . . . . . . 34

3.2 Parallel efficiency of NPHASE-PSU on the NASA Ames Columbia

computer system for a spinning disk case (1.1x106cells). . . . . . . 35

3.3 Example NPHASE-PSU convergence history. . . . . . . . . . . . . . 36

3.4 Comparison between Diab experimental results and NPHASE-PSU

analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.5 Effect of turbulence model selection on viscous work prediction. . . 38

3.6 Breakdown of windage power losses for Diab Gear 1. . . . . . . . . 39

3.7 Breakdown of windage power losses for Diab Gear 2. . . . . . . . . 40

3.8 Secondary flows within the gear teeth. . . . . . . . . . . . . . . . . 41

3.9 Predicted surface pressure distribution for Diab Gears 1-4. Only

one side of symmetry plane is shown for each gear. The areas

shaded in red indicate high pressure areas. . . . . . . . . . . . . . . 41

3.10 Comparison of Diab Gear 1 and disk grids. . . . . . . . . . . . . . . 43

3.11 OVER-REL grid topology for Diab Gear 1. Periodic boundary is

highlighted in blue. Gear surface mesh is highlighted in green. . . . 44

3.12 Example OVER-REL convergence history of the axial torque component. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

NASA/CR—2012-217807 vi
3.13 Experimental, NPHASE-PSU, and OVER-REL results for Diab

Gears 1-4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.14 Comparison of results from experiment and OVER-REL for Diab

Gear 1, including viscous and pressure loss budgets. . . . . . . . . . 47

3.15 Comparison of Diab Gear 1 and disk measurements and OVERREL solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.16 Comparison of predicted torque per unit span contributed by viscous shear for the Diab disk (up to its outer radius) and Gear 1 (up

to its base radius). . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.17 Three-dimensional relative frame streamlines colored by static pressure for Diab Gear 1 at 850 rad/s. . . . . . . . . . . . . . . . . . . . 51

3.18 Axial projection of velocity vectors halfway between gear face and

gear centerline of Diab Gear 1 at 850 rad/s. Vector density of 0.5.

Background contours of local normalized projected relative velocity

magnitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.19 Axial projection of velocity vectors near gear centerline of Diab

Gear 1 at 850 rad/s. Vector density of 0.5. Background contours

of local normalized projected relative velocity magnitude. . . . . . 52

3.20 Static pressure coefficient contours on the gear leading tooth surface(left) and trailing tooth surface(right) of Diab Gear 1 at 850

rad/s. 1/2 of symmetrical gear shown. . . . . . . . . . . . . . . . . 53

3.21 ∆CP between gear leading and trailing tooth surfaces of Diab Gear

1 at 850 rad/s. 1/2 of symmetrical gear shown. . . . . . . . . . . . 53

3.22 Pressure torque per unit width vs. axial coordinate of Diab Gear 1

at 850 rad/s. 1/2 of symmetrical gear shown. . . . . . . . . . . . . 54

4.1 Four notional shroud configurations for Diab Gear 1 geometry. Figure is to scale. Black lines define the gear face, tip radius and base

radius, dark blue and red lines indicate the Large-Axial and SmallAxial shrouds, green and cyan lines indicate the Large-Radial and

Small-Radial shrouds. . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2 Grid topology of the Large-Axial-Large-Radial shroud configuration. 57

4.3 Cross-section of overset mesh topology of the Small-Radial shroud. . 58

4.4 Comparison of predicted windage losses between the unshrouded

and the four shrouded Diab Gear 1 configurations. . . . . . . . . . . 59

4.5 Comparison of predicted windage losses between the four shrouded

Diab Gear 1 configurations. . . . . . . . . . . . . . . . . . . . . . . 60

4.6 Static pressure coefficient contours on the leading tooth surface

(left) and the trailing tooth surface (right) of the Large-Axial-LargeRadial shroud case at 850 rad/s. 1/2 of symmetrical gear shown. . . 61

NASA/CR—2012-217807 vii
4.7 ∆CP between gear leading and trailing tooth surfaces of the LargeAxial-Large-Radial shroud case at 850 rad/s. 1/2 of symmetrical

gear shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.8 Contours of pressure and surface shear stress lines on the leading

surface for the four shrouded cases at 850 rad/s. . . . . . . . . . . . 63

4.9 Torque per unit width on the tooth face for the four shrouded cases. 64

4.10 Comparison of viscous losses per unit span for Diab Gear 1 at 850

rad/s for unshrouded and four shrouded configurations. . . . . . . . 65

4.11 Integrated radial mass flow through gear tooth. . . . . . . . . . . . 67

4.12 Integration surface for radial mass flow. . . . . . . . . . . . . . . . . 68

4.13 Contour plot of tangential velocity for unshrouded Diab Gear 1 at

700 rad/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.14 Contour plot of tangential velocity for the Large-Axial-Large-Radial

shroud case at 700 rad/s. . . . . . . . . . . . . . . . . . . . . . . . . 70

4.15 Comparison of tangential velocity profiles along R/Rtip = 0.95 between the unshrouded and the Large-Axial-Large-Radial shroud cases. 71

4.16 Comparison of tangential velocity profiles at different radial locations between the unshrouded and the Large-Axial-Large-Radial

shroud cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.17 Grid topology of the Large-Axial-Large-Radial shroud with corner

slots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.18 Grid topology of the Large-Axial-Large-Radial shroud with centerline slot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.19 Comparison of predicted windage losses between the unshrouded,

the fully enclosed Large-Axial-Large-Radial shroud, and slotted

Large-Axial-Large-Radial shroud configurations. . . . . . . . . . . . 75

4.20 Contour plot of tangential velocity for the Large-Axial-Large-Radial

shroud with corner slots at 700 rad/s. . . . . . . . . . . . . . . . . . 76

4.21 Three-dimensional absolute frame streamlines for the Large-AxialLarge-Radial shroud with slots at the corner at 700 rad/s. . . . . . 77

4.22 Contour plot of tangential velocity for the Large-Axial-Large-Radial

shroud with centerline slot at 700 rad/s. . . . . . . . . . . . . . . . 78

4.23 Three-dimensional absolute frame streamlines for the Large-AxialLarge-Radial shroud with centerline slot at 700 rad/s. . . . . . . . . 79

4.24 Comparison of tangential velocity profile along R/Rtip = 0.95 between the unshrouded, the fully enclosed Large-Axial-Large-Radial

shroud, and slotted Large-Axial-Large-Radial shroud configurations. 79

4.25 Comparison of tangential velocity profile along R/Rtip = 0.95 for

the unshrouded and the four fully enclosed shroud configurations. . 80

NASA/CR—2012-217807 viii
4.26 Contour plot of tangential velocity for the Large-Axial-Small-Radial

case at 700 rad/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.27 Contour plot of tangential velocity for the Small-Axial-Large-Radial

case at 700 rad/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.28 Contour plot of tangential velocity for the Small-Axial-Small-Radial

case at 700 rad/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.29 Overview sketch of the the NASA Glenn Research Center Gear

Windage Test Facility. [7] . . . . . . . . . . . . . . . . . . . . . . . 84

4.30 Photograph of the NASA Glenn Gear Windage Test Facility with

containment shielding in place. . . . . . . . . . . . . . . . . . . . . 85

4.31 Shroud assembly for test facility. [7] . . . . . . . . . . . . . . . . . . 86

4.32 NASA experimental gear windage power loss data for 13-inch spur

gear for the shrouded and unshrouded configurations. [7] . . . . . . 87

4.33 NASA experimental gear windage power loss data for 13-inch spur

gear for only the shrouded configurations. [7] . . . . . . . . . . . . . 88

4.34 Surface topology of the overset mesh used to describe the NASA

Glenn 13-inch spur gear. . . . . . . . . . . . . . . . . . . . . . . . 89

4.35 Overset mesh topology of the NASA Glenn 13-inch spur gear. . . . 90

4.36 NPHASE-PSU results for the unshrouded NASA Glenn 13-inch

spur gear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.37 Three-dimensional relative frame streamlines for the unshrouded

NASA Glenn 13-inch spur gear at 1000 rad/s. . . . . . . . . . . . . 92

4.38 Experimental and CFD results for the 13-inch pitch diameter NASA

spur gear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.39 Three dimensional relative frame streamlines colored by static pressure for the 13-inch pitch diameter NASA spur gear, Large-AxialLarge-Radial shroud, 700 rad/s. . . . . . . . . . . . . . . . . . . . . 95

4.40 Tooth geometry alternatives: leading surface rounding (top left),

leading+trailing surface rounding (top right), slotted (bottom left),

and trailing surface ramp (bottom right). . . . . . . . . . . . . . . . 96

4.41 Windage loss predictions for the baseline tooth geometry (Diab

Gear 1 with Large-Axial-Large-Radial shrouds) and four geometric alternatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.42 Torque per unit width predictions for the baseline tooth geometry (Diab Gear 1 with Large-Axial-Large-Radial shrouds) and four

geometric alternatives at 850 rad/s. . . . . . . . . . . . . . . . . . . 98

4.43 Predicted surface pressure coefficient and skin friction lines for baseline tooth geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.44 Predicted surface pressure coefficient and skin friction lines for tooth

geometry alternative: leading surface rounding. . . . . . . . . . . . 99

NASA/CR—2012-217807 ix
4.45 Predicted surface pressure coefficient and skin friction lines for tooth

geometry alternative: leading+trailing surface rounding. . . . . . . 100

4.46 Predicted surface pressure coefficient and skin friction lines for tooth

geometry alternative: tooth slot. . . . . . . . . . . . . . . . . . . . . 100

4.47 Predicted surface pressure coefficient and skin friction lines for tooth

geometry alternative: trailing surface ramp. . . . . . . . . . . . . . 101

4.48 Comparison of integrated radial mass flow from gear teeth between

the unshrouded, the baseline Large-Axial-Large-Radial shroud case,

and the Large-Axial-Large-Radial case with tooth trailing surface

ramps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.49 Bell 51-tooth helical gear and surface mesh. . . . . . . . . . . . . . 104

4.50 Shrouded Bell 51-tooth helical gear grid domain. . . . . . . . . . . . 105

4.51 Bell 139-tooth helical gear and surface mesh. . . . . . . . . . . . . . 106

4.52 Shrouded Bell 139-tooth helical gear grid domain. . . . . . . . . . . 107

4.53 CFD predictions of windage power loss vs. speed for Bell 51-tooth

helical gear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.54 CFD predictions of windage power loss vs. speed for Bell 139-tooth

helical gear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.55 Breakdown of pressure and viscous windage power losses for Bell

51-tooth helical gear. . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.56 Breakdown of pressure and viscous windage power losses for Bell

139-tooth helical gear. . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.57 CFD predictions of windage power loss vs. pitchline velocity for

Bell helical gears. . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.58 Predicted power loss ratio vs. pitchline velocity between the 139-

tooth and 51-tooth helical gears. . . . . . . . . . . . . . . . . . . . . 113

4.59 Three-dimensional relative frame streamlines colored entrance location for shrouded Bell 51-tooth gear at 1500 rad/s. . . . . . . . . 114

4.60 Three-dimensional relative frame streamlines colored entrance location for unshrouded Bell 51-tooth gear at 1500 rad/s. . . . . . . . 114

4.61 Three-dimensional relative frame streamlines colored entrance location for shrouded Bell 139-tooth gear at 550 rad/s. . . . . . . . . 115

4.62 Three-dimensional relative frame streamlines colored entrance location for unshrouded Bell 139-tooth gear at 550 rad/s. . . . . . . . 115

4.63 Bell 139-tooth helical gear with blocked leading edge. . . . . . . . . 116

4.64 Bell 139-tooth helical gear with leading edge spoiler. . . . . . . . . . 117

4.65 CFD predictions of windage power loss for modified Bell helical

gears. at 550 rad/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.66 Relative frame streamlines of the Bell 139-tooth helical gear with

blocked leading edge at 550 rad/s. . . . . . . . . . . . . . . . . . . . 119

4.67 Relative frame streamlines of the Bell 139-tooth helical gear with

leading edge spoiler at 550 rad/s. . . . . . . . . . . . . . . . . . . . 120

NASA/CR—2012-217807 x
List of Tables

1.1 Diab Gear Properties. . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.1 Diab Shroud Clearances . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2 Basic Gear Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.3 Shroud Wall Clearances Studied . . . . . . . . . . . . . . . . . . . . 84

4.4 Bell Helical Gear Data . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.5 Total Number of Cells (in millions) of Bell Helical Gear Grids . . . 103

NASA/CR—2012-217807 xi

Chapter 1

Introduction

Gears provide an extremely efficient method for the transmission of rotary motion

and power. It is possible to achieve efficiencies of 99% with a well designed gearbox,

but a number of power loss mechanisms have to be overcome to achieve such

efficiencies. These mechanisms include meshing (sliding and rolling) losses, bearing

losses, lubrication churning losses, and windage losses. The relative importance of

these mechanisms varies depending on the application. In high speed gearing

applications, windage effects become extremely important and can dominate the

other loss mechanisms. Unfortunately, windage losses are extremely difficult to

measure accurately and are the least understood of the other mechanisms.

Windage losses are a result of the aerodynamic drag forces (pressure and viscous) that act on rotating gears. They are a function of the size, shape, and

rotation speed of the gears. Inside a gear transmission, windage manifests itself

as increased rotational resistance and added heating. These effects present several

problems for rotorcraft transmission systems. Rotorcraft transmissions are susceptible to windage effects because of the high rotation speeds introduced by their

gas turbine engines. Rotorcraft can require reduction ratios anywhere between

25:1 and 100:1 [5, 8] between engine and rotors. For example, the UH-60A helicopter transmission provides a reduction ratio of 81.042:1 from the engine input

(20,900 rpm) to the main rotor output (258 rpm) [9].

The increased rotational resistance consumes engine power which reduces range

and payload. Windage losses in a transmission can consume up to 3% of the

transmitted power from the engines. For a rotorcraft like the V-22 Osprey, whose

NASA/CR—2012-217807 1
engines produce 6,150 hp (4,590 kW), that can mean windage losses up to 180 hp

(137 kW). The added heating has design implications for the lubrication cooling

system which must be sized appropriately to absorb and dissipate all of the heat

generated within the gearbox. Proper functioning of the transmission lubrication

cooling system is critical on a rotorcraft because its unique weight and packaging

constraints require the gearing to be extremely lightweight. This means that the

gears have a relatively low heat carrying capacity and need to be actively cooled

during operation. If the cooling requirements (along with the size and complexity of

the cooling system) can be reduced, this could present significant benefits in range,

maintenance costs, and payload capacity. Furthermore, rotorcraft survivability

under transmission oil-out conditions is exacerbated by windage. In the oil-out

condition, the lubrication cooling system has ceased to function either due to

damage or component failure. Since windage manifests as added dissipative flow

heating, the loss of the cooling system can lead to a catastrophic failure of the drive

system. Figure 1.1 is from Bartz [1] and shows the effect of running high-speed

gears without lubrication. The gear teeth have deformed and have lost the ability

to transfer power.

Despite the significant relevance, design efforts throughout the gearing industry aimed at reducing windage losses have generally fallen into the trial and error

category. Nevertheless, it has been shown by some that modest geometric modifications to control the air flow path, such as shrouding, can very significantly

reduce both windage losses and lubricating oil consumption (80% and 40% reductions were observed, respectively, in Reference [10]). However, these hardware

specific approaches are empirical, time consuming, and expensive.

1.1 A Brief Introduction to Gearing

Gears are machine elements that transmit rotary motion and power by the successive engagements of teeth on their periphery. They provide an extremely efficient

method for such transmission, especially if power levels and accuracy requirements

are high. Gears come in a wide variety of types to serve different functions. Figure 1.2 provides sketches of some of the more common gear types. Spur gears are

used to transmit motion between parallel shafts. The teeth are radial, uniformly

NASA/CR—2012-217807 2
Figure 1.1. Effect of lubrication failure on a gear. [1]

spaced around the outer periphery and are parallel to the axis of rotation. Helical

gears can transmit motion between parallel or crossed shafts by meshing teeth that

lie along a helix at an angle to the axis of the shaft. Mating of the teeth occurs such

that two or more teeth of each gear are always in contact. This permits smoother

action than that of spur gears. However, helical gears also generate axial thrust

which causes slight power loss and requires thrust bearings.

Bevel gears are used to transmit rotary motion between two non-parallel shafts

and come in a variety of tooth designs. Straight bevel gears have straight teeth

that, if extended inward, would intersect at the axis of gear. Spiral bevel gears have

teeth that are curved and oblique. They provide a greater load carrying capacity

and transmit the load more smoothly than straight bevel gears. Zerol bevel gears

differ from spiral bevel gears in that their teeth are not oblique. Face gears have

teeth cut on the end face of a gear.

This dissertation focuses on spur and helical gears. Figure 1.3 is a schematic

NASA/CR—2012-217807 3
Figure 1.2. Various gear types. [2]

of the typical gear nomenclature and is from Reference [3]. The pitch circle is the

circumference of a gear measured at the point of contact with the mating gear and

the pitch diameter and pitch radius are the diameter and radius, respectively, of

the this circle. In the English system, the size of the teeth is described by the

diametral pitch (DP), which is the number of teeth per inch of pitch diameter. As

tooth size increases, the diametral pitch decreases. However, in the metric system,

the module is used in place of diametral pitch and it is simply the reciprocal of

NASA/CR—2012-217807 4
the diametral pitch. It is given in units of millimeters. Pressure angle denotes the

angle between a tooth profile and a radial line at its pitch point. The pressure angle

of an involute gear tooth is determined by the size ratio between the base circle

and the pitch circle. The standard pressure angles used in the gearing industry

are 14.5, 20, and 25◦, with 20◦ being the most common [11].

An involute gear tooth is a gear tooth whose profile is established by an involute

curve outward from the base circle. A involute curve is described by curve that is

drawn by the unwinding of a taut string from the base circle. The involute profile

can mathematically describe the thickness of the teeth as a function of radius:

t(R) = R



tS

RS

+ 2(inv(φS) − inv(φR))(1.1)

where tS denotes the thickness at the pitch radius, RS, and is determined by the

equation:

tS = π/(2 × DP) (1.2)

inv(φ) is the involute function and is given by Equation 1.3:

inv(φ) = tan(φ) − φ (1.3)

where φ is in radians. φS and φR are pressure angles calculated by the equation

φ = cos−1RB

R

(1.4)

where RB denotes the radius of the base circle.

NASA/CR—2012-217807 5
Figure 1.3. Schematic of typical gear tooth nomenclature. [3]

NASA/CR—2012-217807 6
1.2 Previous Research

The available, open literature on gear windage is somewhat sparse. Numerous

studies have been conducted on rotating disks or drums [12, 13, 14, 15, 16, 17],

but there have only been a handful of published studies on rotating gears. These

studies typically report on experimental evidence of the presence of windage and

the associated heating and power consumption [5].

The literature on rotating disk flows is rather extensive. Von Karman [12]

used an approximate method based on the momentum equation to investigate the

turbulent boundary layer on a rotating disk. Schlichting [13] provides an analytic

solution of the incompressible Navier-Stokes equations for an infinite spanning

rotating disk. Much of the literature has been in the area of gas turbine analysis [15,

16, 17]. These studies focus on the effects of shrouding, turbine disk placement, and

coolant flows in gas turbine engines. Daily and Nece [14] studied experimentally

and theoretically the rotation of a fully enclosed disk. The effect of axial clearance

was studied was studied while the radial tip clearance was kept small. Water

and four different lubricating oils were the test fluids. Daily and Nece identified

four possible flow modes within the disk enclosure: laminar flow with merged

boundary layers between the shroud wall and disk, laminar flow with separate

boundary layers, turbulent flow with merged boundary layers, and turbulent flow

with separate boundary layers. Maroti et al. [15] studied a rotating disk in air with

only axial shrouding (no outer radial shroud) and noticed an unsteady flow pattern

when the shrouds were radially larger than the disk. It was also observed that the

viscous torque on the disk was greater when the flow was unsteady. Bayley et

al. [16] studied air-cooled shrouded disk systems. However, the shrouds did not

fully enclose the disk as the shrouds shared the same radial dimensions. The

disks were cooled by air that was pumped from the axis of of the shroud system.

Bayley found that when there was no forced radial outflow of coolant, the shrouds

reduced the viscous moment on the disk by conserving the angular momentum

between the rotor and the stator shroud at high Reynolds numbers. When the

radial outflow was forced, the shroud caused the viscous moment to increase due

to increased shear stress at the outer radius as fluid passed through the shroud

clearance. Daniels et al. [17] conducted experiments on adjacent rotating disks

NASA/CR—2012-217807 7
enclosed in a shroud to determine the aerodynamic and torque characteristics. It

was found that co-rotating disks exhibited less viscous torque than counter-rotating

disks.

Gear theory texts, in fact, only qualitatively handle windage losses. For instance, Townsend [18] identifies a number of factors that influence windage losses.

The variables that affect windage include the size, shape, and speed of the rotating

elements, the oil-feed system, the overall casing design, the operating temperature

and viscosity of the oil, and the pressurization of the casing. Most importantly, he

noted that windage effects become important when pitch-line velocities begin to

exceed 10,000 ft/min (50 m/s). Townsend also provides a windage approximation

formula based on empirical studies of smooth rotating bodies in air, but leaves the

determination of windage largely to the experience of the designer.

Anderson and Loewenthal [19, 20] developed an analytic system to calculate

all of the losses in a gear train and provided a more sophisticated approximation

formula of gear windage based on turbine rotor wheel analysis. However, their

work on gear system power losses focused mainly on non-aerodynamic sources of

power loss (rolling, sliding, bearing, etc.) and their gear studies involved relatively

low-speed gearing. The pitch-line velocities did not exceed 8000 ft/min (40.6 m/s).

The first systematic study of gear windage was performed by Dawson [4] in

1984. He investigated how the geometric properties, speed, and the presence of

shrouding affected single isolated gears in air. Experimental measurements were

carried out on a set of rather large gears (the root diameters varied between 300 and

1160 mm) in his garage. The gears were constructed of hardboard and designed

to allow different tooth geometries to be fitted. Tests were primarily performed

on spur gears, but one helical gear was tested as well. The gears were operated

in a spin-down test ring where the test gears were accelerated to a maximum

rotation speed. The power supply was then disconnected and the gear was allowed

to freely decelerate. Windage power loss was calculated by measuring the rate of

deceleration. With his test stand, Dawson could achieve root-line velocities up to

around 90 m/s (17,900 ft/min) with the 1160 mm diameter gear.

Dawson concluded that the key source of gear windage losses was the teeth.

He observed with smoke that the teeth of a spur gear act similarly to those of a

centrifugal fan. Air is drawn in axially (from both ends of the teeth) and ejected

NASA/CR—2012-217807 8
out radially. Additionally, he noted that losses could be reduced by up to 41% by

simply inhibiting the axial inflow. This was done by placing paper “washers” at

the ends of the teeth. The helical gear exhibited quite different behavior, that of

an axial fan, when it was similarly observed. Air is drawn in axially and radially

from leading edge of the gear teeth and then expelled out axially, radially, and

tangentially from the trailing edge. When the edges of the teeth were blocked at

both ends or at just the leading edge, windage losses decreased. However, when

only the trailing edge was blocked, windage losses actually increased. Figure 1.4

provides sketches by Dawson of the flow patterns he observed.

Figure 1.4. Dawson’s sketches of the airflow around (a) the spur gear and (b) the

helical gear. [4]

Dawson also examined the effect of the gear case, since most gear trains are

typically enclosed. Various configurations of axial and radial shrouds of different

sizes and shapes were tested. All of the measurements were conducted on a spur

gear having the dimensions of 760 mm root diameter, a face width of 187 mm, and

a tooth module of 8 mm. Dawson published the power loss for these experiments

as a percentage of the power loss experienced by the same gear in free space.

The greatest reduction (66%) was seen with the gear completely enclosed in a

smooth cylindrical shroud. Dawson, however, did not make any observations of

how enclosures actually reduced windage losses but provided an empirical formula

for estimating spur gear windage losses that also accounted for shroud effects.

In a follow-up paper [21], Dawson performed further tests on helical gears.

In his previous paper, only one helical gear (30◦ helix angle) was studied, which

exhibited 30% less loss than an equivalent spur gear. The gear tests were performed

on a gear with a root diameter of 514 mm, a face width of 187.8 mm, and a 16 mm

NASA/CR—2012-217807 9
tooth module. The helix angle of a single gear was varied between 0◦ and 50◦

. It

was observed that power loss decreased as the helix angle was increased. However,

he only presented data for root-line velocities of about 20 m/s (4000 ft/min).

Dawson also provided an updated windage formula that accounted for the effects

of helix angle.

Following the work of Dawson, Winfree [10] conducted experiments on a single

high-speed bevel gear using various baffle configurations. Winfree’s focus was on

finding an optimal baffle configuration to minimize windage losses of a straight

bevel gear. The bevel gear had a diameter of 15 inches and was mounted on a

vertical shaft with its face pointed down at the sump. The test rig could achieve

pitch-line velocities up to 127 m/s (25,000 ft/min). The test stand was also enclosed

so that lubrication system effects could be studied. Windage was calculated by

measuring the power draw on the drive motor. With a bevel gear, Winfree observed

that there was a significant suction effect at the gear face. Fluid was pumped

radially from the inside to the outside through the teeth. Winfree determined that

if the pumping ability of the bevel gear could be restrained (e.g. with a shroud) a

gear system would become more efficient.

Handschuh and Kilmain [5, 8, 22] investigated the efficiency of a high speed

helical gear train and the effects of speed, load, and lubricant jet pressure on

operating performance and thermal behaviour. The test rig employed a closed-loop

torque regenerative system consisting of two identical gearboxes. One was the test

gearbox and the other was set as a slave. Both gearboxes consisted of a input

gear, three idlers and a bull gear. Figure 1.5 is an illustration of the test facility.

The facility could achieve speeds up 15,000 rpm with pitch-line velocities reaching

24000 ft/min (122 m/s) and loads up to 5000 hp (3730 kW). High rotational speeds

had a dramatic effect on efficiency and performance. Windage losses were found to

dominate meshing losses when light loads and high speeds were applied to a gear

meshing system. Shrouding was also found to be beneficial at the higher operating

speeds.

Houjoh et al. [23] instrumented meshing helical gears with pressure transducers

to determine the optimal method to supply lubrication to the gear teeth. The

transducers were placed in the space between teeth to trace the flow of air across

the width of the gears during meshing. Their results showed the axial travel of a

NASA/CR—2012-217807 10
Figure 1.5. NASA High-Speed Helical Gear Train Facility. [5]

strong low pressure wave (what the authors referred to as a negative pressure) as

meshing proceeds. It was also observed with helical gears that along with a strong

axial flow through the teeth from the leading edge to the trailing edge, there is

also an opposite flow from part of the trailing edge. This behavior was not seen

by Dawson [4].

In 2004, Diab et al. [6] presented a experimental and numerical study of single,

isolated gears rotating in air without an enclosure. Four different spur gears and

a disk were tested in a spin-down test rig. The test stand was similar to that of

Dawson [4], but it was designed to test industrial components at greater speeds in a

more controlled environment. Figure 1.6 is an illustration of the test rig. The gears

varied in diameter, width, and module and Table 1.1 lists the properties of the gears

and the disk that were studied. The experimental results were also used to evaluate

the windage estimation formulas of Anderson and Loewenthal [19], Dawson [4], and

those developed by Diab et al. The formulas developed by Diab showed improved

NASA/CR—2012-217807 11
results over the older formulas for the gears in the study. Dawson’s formula overpredicted the power losses for all cases, and Anderson and Loewenthal’s underpredicted windage for the smaller gears. Subsequent papers from these authors [24,

25] focused on modeling the pumping losses from the meshing of the gear teeth.

Figure 1.6. Diab Test Rig. [6]

Table 1.1. Diab Gear Properties.

Pitch Diameter Tooth Width Module Teeth

(mm) (mm) (mm)

Gear 1 288 30 4 72

Gear 2 144 30 4 36

Gear 3 144 60 4 36

Gear 4 144 60 6 24

Disk 300 30

Johnson et al. [26] studied the effects of shrouding on a spiral bevel gear mesh

test rig. Through transparent shrouding, Johnson observed that the air and oil

phases remained separate, indicating that analysis based on average fluid density

may not be physical. Lubrication was also found to reduce the effectiveness of

shrouding.

Seetharaman and Kahraman [27] proposed a physics-based model to predict

windage losses on meshed spur gears. In their model, windage losses were the sum

NASA/CR—2012-217807 12
of the aerodynamic drag losses associated with the individual gears and the losses

due to compression or trapping of the atmospheric medium between meshing gear

teeth. In this model, however, it was the compression losses in the meshing region

that were seen as the dominant source of windage losses, not the aerodynamic drag

forces. Their model was validated against windage data of two meshed spur gears

from Petry-Johnson et al. [28]. They did not validate their model against data

of isolated gears and the dominance of compression losses has not been seen by

others [25].

Due to the complexities of the physics and geometries involved, it is only recently that modern computational fluid dynamic (CFD) methodologies have been

employed in analysing gear windage. Farrall et al. [29] used the commercial unstructured CFD solver FLUENT to obtain a steady-state, single phase solution for

a shrouded bevel gear. Farrall attempted to computationally replicate the work of

Winfree, but calculated dramatically less windage reduction from shrouding than

did Winfree. This is likely due to the fact that Farrall did not study the same gear

or shroud design as Winfree.

Al-shibli et al. [30] performed a 2-D study of a spur gear using FLUENT.

The model domain consisted of two teeth and a side correlation factor adopted

from Townsend [18] was used to to account for 3-D effects (although these authors

state that work is under way to extend their simulations to 3-D [31]). Ref. [31] also

provides a fairly comprehensive review of the study of gear windage. Computations

were validated against data from experiments performed by Lord [32], but the side

correlation function actually predicted nearly 40% of the losses.

Hill et al. [33] implemented a 3-D unstructured moving mesh method and applied it to isolated spur gears in air and validated their predictions against data

from Diab et al. [6] Marchesse et al. [34] followed the same approach as Al-shibli

et al. [30], but used a structured grid with the flow solver ANSYS CFX. However,

Marchesse et al. did not find the 2-D approach to be satisfying. The 2-D model

was then extended into three dimensions, and it was found that a 3-D CFD approach produced significantly improved calculations of gear windage losses. Their

CFD results produced flow patterns similar to those obtained by Hill et al.. Imai et

al. [35] investigated 3-D bevel gears in mesh in an air-oil atmosphere, but modeled

the gears as porous bodies.

NASA/CR—2012-217807 13
1.3 Modeling Gear Windage Loss

The simplest way to model the flow around a rotating gear is to neglect the effect

of the teeth and treat the gear as a simple disk. Schlichting [13] provides an

exact solution of the Navier-Stokes equations for the flow around a rotating disk.

Schlichting derives the frictional moment (M) of a disk in laminar flow and wetted

on both sides as:

2M = 0.616πρR4(νω3)

1/2

(1.5)

For turbulent flows, the viscous torque on the disk is written as:

2M = 0.073πω2R

5

(ν/ω R2)

1/5

(1.6)

where ρ is density, R is radius, ν is kinematic viscosity, and ω is the angular

velocity. These solutions only provide the minimum estimate of the windage drag

since both edge effects and pressure effects are neglected.

The windage estimation formula provided by Townsend [18] is based on empirical studies of smooth rotating bodies in air. The formula provides an approximation of windage power loss in horsepower:

P =

15

0.746

 n

1000

3



D

10004 

5L

100

+

D

100

(1.7)

where n is rotation speed in revolutions per minute, D is the diameter (inches) of

the rotating body, and L is the length (inches) of the body. Equation 1.7 completely

neglects the influence of the teeth and only provides a minimum estimate for

windage. This equation, though, has two parts that account for the losses from

the gear face and the losses from the periphery. Equation 1.8 represents the losses

from the periphery:

P =

15

0.746

 n

1000

3



D

10004 

5L

100

(1.8)

and it assumes that the outer surface is smooth. Townsend provides a modification

to account for the presence of gear teeth on the surface. This is shown in Equation 1.9 where the effect of the teeth are treated with a rough surface adjustment

NASA/CR—2012-217807 14
factor, Rf , that is a function of diametral pitch (coarser pitched gears will have a

higher roughness factor):

P =

15

0.746

 n

1000

3



D

10004 

5L

100  Rf

√

tanΨ



(1.9)

Ψ is the helix angle and adjusts the losses for differences in helix angle. Appropriate

values of Rf can be found in Table 12.5 of Reference [18].

The analytical formula provided by Anderson and Loewenthal [19] is based on

turbine rotor wheel analysis where windage power losses (Watts) are calculated

with Equation 1.10:

P = C1



1 + 2.3

t

R



ρ

0.8n2.8R4.6μ0.2

(1.10)

where C1 is a constant (2.04×10−8in S.I. units), t is the gear face width (meters),

R is the pitch radius (meters), n is rotation rate in revolutions per minute, and

ρ and μ are the density (kg/m3) and viscosity (centipoise), respectively, of the

atmosphere. To account for the oil atmosphere within the gearbox, the density

and viscosity of the atmosphere was corrected to reflect a 34.25 parts air to 1 part

oil combination as reported in Reference [36]. Their expression became:

PW = C2



1 + 2.3

t

R



n

2.8R4.6

(0.028μ + C3)

0.2

(1.11)

where C2 and C3 are 2.82 × 10−7 and 0.019, receptively. This equation, however,

does not account for the properties (diametral pitch, helix angle, etc.) of the gear

teeth.

Dawson [4] developed his own formula for gear windage power losses based on

his own empirical studies. He defined windage as a function of the rotation speed

N (RPM), the root diameter D of the gear (mm), the face width F (mm), and the

tooth module M (mm):

P = N

2.9

(0.16D

3.9 + D2.9F0.75M1.15) × 10−20Φλ (1.12)

where Φ represents an unknown function of the effective density of the air-oil

NASA/CR—2012-217807 15
atmosphere (Φ = 1 indicates an oil free atmosphere). The λ term represents the

effect of the gear case. A λ of unity is for a gear operating in free space, 0.6-0.7 is

for a gear in a large enclosure, and 0.5-0.6 is for tighter enclosures. He accounted

for the helix angle in his updated formula [21]:

P = 1.12 × 10−8C

0

ρN2.85D

4.7

ν

0.15λ (1.13)

where ν is the kinematic viscosity (m2/s) and C

0

represents a shape factor based

on the face width to diameter ratio and the number of teeth the gear possesses.

Figures 3 and 5 of Reference [21] provide the shape factors for spur gears and helical

gears, respectively. The root diameter, D, is in meters. Both Equations 1.12 1.13

provided windage power loss in kilowatts.

The formula presented by Diab et al. [6] calculated power loss in Watts as a

function of fluid density ρ (kg/m3), rotation rate ω (rad/s), pitch radius R (m)

and a total windage loss coefficient Ct:

P =

1

2

Ctρω3R

5

(1.14)

where two different formulas were provided to calculate Ct. The first formula was

based on a dimensional analysis:

Ct = αReβ



b

R

γ

Z

δ

(

h1

R

ψ

+



h2

R

ψ

)

(1.15)

where R, b, and Z are the geometrical gear parameters pitch radius, width, and

number of teeth, respectively. α, β, γ, δ, and ψ are constant coefficients that are

derived experimentally. The parameters h1,2 are used to account for the presence

of flanges near the teeth. When there is no obstacle on the side of the gear h1,2 =

0.5

1/ψR.

The second approach was based on a fluid flow analysis. The power loss coefficient (Ct = Cf + Cl) was divided between the losses from the gear face (Cf )and

from the gear teeth (Cl). The gear face contribution is given by Equation 1.16:

Cf =

2n1π

5 − 2m1

1

Re∗m1



R∗

R

5

+

2n2π

5 − 2m2

"

1

Rem2 −

1

Re∗m2



R∗

R

5

#

(1.16)

NASA/CR—2012-217807 16
where n1 and m1 are coefficients for laminar flows and have the values of 1.293

and 0.5, respectively. The coefficients n2 and m2 are for turbulent flows and have

values of 0.072 and 0.2, respectively. Re∗is the critical Reynolds number (∼= 3×105)

between laminar and turbulent flow. R is the pitch radius and R∗ =

p

μRe∗/ρω

is the critical radius separating the laminar and turbulent regions. The tooth

contribution coefficient is given by Equation 1.17:

Cl

∼= ξ

Z

4



b

R

 1 +

2(1 + XA)

Z

4

(1 − cosφ)(1 + cosφ)

3

(1.17)

XA represents the profile shift coefficient and φ is computed from the formula

φ = π/Z − 2(inv(αP ) − inv(αA)) (1.18)

where αP and αA are the pressure angles at the pitch point and tooth tip, respectively. The tooth analysis assumed that the fluid is expelled from an active tooth

area and that the pressure is uniform on the tooth. This has not been seen by the

author [33] or others [30, 31].

Figure 1.7 compares the windage approximation formulas for a simple 13 in.

pitch diamter spur gear. The gear is 1.12 inches wide and the teeth have 25◦

pressure angle. At the higher rotational speeds the differences between estimates

becomes quite pronounced. Figure 1.7 was produced by the Python program provided in Appendix A.

1.4 Physics

The flow physics inside a gearbox are quite complex. Gears are not designed to be

aerodynamically efficient. The shape and profile of the gear teeth has an important

function in transferring power between gears. Variations in angular velocity of the

driven gear will lead to vibrations in the gear train and will generally cause fatigue

cracks to form in the teeth resulting in early failure of the gear.

Unlike a smooth rotating disk, pressure torque on the teeth will play in important role in calculating the windage drag on gears. The teeth are also located at

the periphery of the gear where speeds are greatest (speed = ω ×r). It is very posNASA/CR—2012-217807 17
Figure 1.7. Gear windage loss estimates for a 13-inch pitch diameter spur sear

sible that depending on the rotation rate and the size of the gear, gear tip speeds

can enter the the range of compressible flow. Compressible flow effects typically

become noticeable above Mach numbers of 0.3. However, sonic speed is affected

by both the temperature and the composition (air-only versus air-oil) of the fluid

environment. Brennen [37] showed that the sonic velocity of a homogeneous mixture (e.g. air and only) can be much smaller than that of either of its constituents,

meaning that compressibility effects could be be a concern at lower speeds than

an air-only analysis would indicate. The teeth also introduce turbulence into the

flow. Gear teeth are not designed to pass smoothly through the air. Their faces

present a blunt surface in the direction of the direction of rotation.

The behavior of the air flow is also affected by the orientation of the teeth with

respect to the axis of rotation. The face of spur gear teeth are oriented parallel to

the axis of rotation while helical gear teeth are aligned along a helix at an angle to

the axis of the shaft. As was shown in Figure 1.4, it was observed that spur and

helical gears generate different flow patterns.

NASA/CR—2012-217807 18
Moving away from the effects of simple isolated gears, there is also the fact

that gears operate in mesh with or in close proximity to other gears. Diab et

al. [24] noted the possibility of sonic conditions existing within the tooth mesh

region. There is also the effect of the enclosure that needs to be considered. The

gear train studied by Handschuh and Kilmain [5, 8, 22] was enclosed in a relatively

tight (axially) enclosure, but additional radial shrouding was found to be necessary

to reduce losses.

Finally, there is the issue that meshing gears require lubrication. In high-speed

gear trains (pitch-line speeds above 10,000 ft/min) lubrication is used to both

reduce frictional losses between meshing teeth and to cool the gears. A significant

amount of heat is generated by frictional forces acting in the meshing region [5].

In high-speed gearing applications, lubrication is applied to the gears by an oil

spray system. As oil mist is sprayed on the gears, it builds up the oil film layer

on the teeth that is used to reduce meshing frictional losses. Immediately after

meshing, the gears are sprayed again to dissipate the heat generated by meshing.

Lubrication has to be continuously applied because centrifugal forces fling the oil

off the gear shortly after its application. The fluid mechanics of these systems thus

consists of highly turbulent air flow, disperse oil phase flow from the spray system

and oil fling-off, and continuous oil phase lubricating films on the gears.

1.5 Challenges

The complex physics involved in gear windage provide several challenges for CFD

analysis methods. The fluid mechanics involve complex separated air flow, disperse multiphase flow (oil droplets) and continuous multiphase films (lubricating

oil on gears), moving boundaries in contact, and all modes of heat transfer. The

complex internal geometries of the gearbox and the relative motion of the gears

also necessitate that a CFD solver support either adaptive grids or overset grid

systems.

With respect to the fluid mechanics, the CFD solver must be able to support

suitable turbulence modeling to accommodate the highly separated flows within the

gearbox and the cascade of energy through turbulence scales into viscous heating.

Viscous dissipation is critical in this case since it is often neglected in many CFD

NASA/CR—2012-217807 19
analyses. An oil-out condition exacerbates many of these challenges. In the oilout condition, all modes of heat transfer (conduction, convection, radiation) are

present and need to be accounted for in the CFD model. These flows are also highly

unsteady, non-aerodynamic high-Reynolds number flows (ρR2Ω/μ ∼= O(106

)). In

the most extreme cases, gears can have tip velocities close to a Mach number of

0.75, which means compressible flow effects will be present.

The solver also needs to contain a non-equilibrium multiphase capability that

can accommodate the disperse mist/droplet phase and the continuous film phases

to model the multiphase flow dynamics. Johnson et al. [26] indicated that simply

modeling the air-oil atmosphere as a homogeneous fluid may not be physical. The

relative importance of 1-phase and 2-phase physics in windage losses is still not

well known.

Finally, there is a paucity of validation data. There are only a handful of

experimental cases [10, 4, 6] in the open literature that provide enough useful data

for CFD validation. In the cases that are available, there is little data on the flow

itself. Typically, only power loss or temperature data is reported. Velocity field

data is nearly impossible to obtain because of the high velocities involved, and

the fact that lubrication tends to obscure optical measurement equipment. Also

fitting measuring equipment into a tightly enclosed gear space presents its own

issues. There are only two of cases that provide any kind of pressure data [7, 23].

1.6 Objectives of Dissertation

This research has a number of objectives with respect to gear windage. The first

is to develop numerical modeling approaches for the application of gear windage

aerodynamics. The second goal is to use these methods to develop physical understanding of gear windage aerodynamics, loss mechanisms, and loss mitigation

strategies. Additionally, it is expected that new approaches for minimizing windage

loss can be developed and evaluated through an understanding of the physics and

the numerical tools presented here. Finally, validation of the method is key to

relying on “computational discovery” elements of these objectives and date from

the open literature and the new NASA Glenn Gear Windage Test Facility will be

used to validate the approaches presented in this dissertation.

NASA/CR—2012-217807 20
1.7 Organization of Dissertation

The dissertation is organized as follows. Chapter 1 has introduced the problem of

gear windage and previous work on the subject. Chapter 2 reports the governing

equations and numerical procedures. Next, Chapter 3 describes the work done to

validate the numerical procedures. In Chapter 4, a series of numerical studies are

performed that provide insight into several of the physical mechanisms of windage

losses including the role of gear shrouding. This is followed by an overview of the

NASA Glenn Gear Windage Test Facility along with a selection of data obtained

to date. CFD simulation of the NASA Glenn windage loss tests are presented

and a discussion of these comparisons is provided. This is followed by the results

of a numerical experiment, guided by the findings described in the dissertation,

that demonstrate a benefit in windage performance beyond that observed with

shrouds alone. Finally, Chapter 5 presents a summary of the results, conclusions,

and suggestions for future work.

NASA/CR—2012-217807 21
Chapter 2

Theoretical Formulation

Two different CFD codes are used in this research. The first is NPHASE-PSU

which is a time accurate, unstructured, overset multiphase flow solver. In all

NPHASE-PSU simulations the entire gear is modeled, which allows for the study of

complex gearbox geometries and bodies in relative motion. The second CFD code

is OVER-REL. OVER-REL is a steady-state, incompressible, block-structured,

overset flow solver. Only one gear tooth/“blade row” is modeled using this solver

providing for a very efficient and numerically accurate physics analysis of gear

windage aerodynamics. The governing equations and the numerical methods used

in these two codes will be discussed further in this chapter. In this dissertation, all

verification and validation flow studies are either incompressible, or have maximum

local absolute Mach numbers of less than 0.35. The available data [4, 38, 10, 6]

shows that windage losses already become significant at low subsonic tip Mach

numbers. Accordingly, for all simulations presented here, an incompressible assumption is invoked and the energy equation is not solved.

2.1 NPHASE-PSU

NPHASE-PSU [39] is a parallel, face-based, arbitrary-element, unstructured multiphase flow solver that has been instrumented with overset mesh capability. The

baseline algorithm follows established segregated pressure based methodology. A

collocated variable arrangement is used and a lagged coefficient linearization is

applied [40]. Diagonal dominance preserving, finite volume spatial discretization

NASA/CR—2012-217807 22
schemes are used for the scalar transport equations. Continuity is introduced

through a pressure correction equation, based on the SIMPLE-C algorithm [41].

In constructing cell face fluxes, a momentum interpolation scheme [42] is used

which introduces damping in the continuity equation. Grid motion/deformation

terms are implemented in a Geometric-Conservation-Law (GCL) preserving fashion [43]. A dual-time formulation is usedwhere at each physical timestep, between

5 and 20 pseudo-timesteps of the SIMPLE-C algorithm are applied. Specifically, at

each pseudo-timestep, the discrete momentum equations are solved approximately

(using a simple point iterative scheme), followed by a more exact solution of the

pressure correction equation (using the PETSC [44] parallel LU preconditioning

and GMRES utilities). Turbulence scalar and energy equations are then solved in

succession. Parallelization is implemented in a standard fashion by invoking domain decomposition based on METIS [45] in a pre-processing step, and MPI-based

message passing in the CFD code. A high-Reynolds number k- turbulence model

and a sublayer resolved hybrid k-/k-ω turbulence model due to Menter [46] are

used in the studies that follow. No explicit transition model was employed as justified, for now, by the small contribution of near-axis viscous torques on windage

loss.

2.1.1 Governing Equations

The equations for the conservation of mass (Equation 2.1), momentum (Equation 2.2) and energy (Equation 2.3) can be written in integral conservation law

form for a unsteady compressible flow as:

∂

∂t Z

∀

ρd∀+

Z

S~

ρV~ · dS~ = 0 (2.1)

∂

∂t Z

∀

ρV d ~ ∀+

Z

S~

ρV~ V~ · dS~ = −

Z

S~

pdS~ +

Z

S~

τ · dS~ (2.2)

∂

∂t Z

∀

ρEd∀+

Z

S~

ρHV~ · dS~ =

Z

S~



τ · V~



· dS~ + Wf + qH (2.3)

NASA/CR—2012-217807 23
where V~ is the velocity vector, ρ is the fluid density, p is pressure, τ is the stress

tensor, qH is the heat added to the control volume, and Wf represents the work

done on the control volume. Since all verification and validation flow studies are

either incompressible, or have maximum local absolute Mach numbers of less than

0.35, an incompressible assumption is invoked for all simulations and the energy

equation is not solved. Future studies will require the use of the compressible flow

formulation of the governing equations.

A high-Reynolds number k- turbulence model and a sublayer resolved hybrid

k-/k-ω turbulence model due to Menter [46] are used. The model transport equations for k and  are:

∂

∂t Z

∀

ρkd∀ +

Z

S~

ρkV~ · dS~ =

Z

S~



μ +

μt

Prk



∇k · dS~ +

Z

∀

(P − ρ) d∀ (2.4)

∂

∂t Z

∀

ρd∀ +

Z

S~

ρV~ · dS~ =

Z

S~



μ +

μt

Pr



∇ · dS~ +

Z

∀

(C1P − C2ρ)



k

d∀ (2.5)

2.1.2 Grid Motion

In the case of moving bodies (e.g. rotating gears), the solution domain will change

with time due to the movement of the boundaries. Therefore, the grid also has to

move to accommodate the movement of the boundaries. Using Cartesian velocity

components in the absolute frame of reference (coordinate system remains fixed),

the only change in the equations of motion is the appearance of the relative velocity

in the convective terms. If we consider the 1-D continuity equation, for example:

∂ρ

∂t +

∂ρu

∂x = 0 (2.6)

By integrating Equation 2.6 over a control volume whose boundaries move with

time from x1(t) to x1(t) it becomes:

x2(t)

Z

x1(t)

∂ρ

∂t dx +

x2(t)

Z

x1(t)

∂ρu

∂x dx = 0 (2.7)

NASA/CR—2012-217807 24
After applying Leibniz’s rule to the first term and integrating the second, the

continuity equation becomes:

d

dt

x2(t)

Z

x1(t)

ρdx −



ρ2

dx2

dt − ρ1

dx1

dt 

+ ρ2u2 − ρ1u1 = 0 (2.8)

The dx/dt terms describe the velocities with which the grid moves. The terms in

the square brackets share a form similar to the last two terms involving the fluid

velocity which allows Equation 2.7 to be rewritten as:

d

dt

x2(t)

Z

x1(t)

ρdx +

x2(t)

Z

x1(t)

∂

∂x [ρ(u − ub)] dx = 0 (2.9)

To further simplify terms, dx/dt has been rewritten as ub. This analysis can be

extended into three dimensions by using the 3-D version of Leibnitz’s rule. Thus,

the conservation of mass, momentum and energy can be written for a compressible

flow through a moving mesh in integral conservation law form as:

∂

∂t Z

∀

ρd∀+

Z

S~

ρ



V~ − W~



· dS~ = 0 (2.10)

∂

∂t Z

∀

ρV d ~ ∀+

Z

S~

ρV~



V~ − W~



· dS~ = −

Z

S~

pdS~ +

Z

S~

τ · dS~ (2.11)

∂

∂t Z

∀

ρEd∀+

Z

S~

ρH V~ − W~



· dS~ =

Z

S~



τ · V~



· dS~ + Wf + qH (2.12)

where W~ is the velocity of the surface element, dS~, in the absolute frame of reference.

2.1.3 Multiphase Flow Methods

The issue of gear windage cannot be completely studied by only considering an

air-only environment. Gears operate in an environment of both air and oil. Gears

require lubrication to reduce friction during meshing and to provide cooling. A

NASA/CR—2012-217807 25
multiphase flow approach will thus needed to model the air and oil environment

and this is why NPHASE-PSU was selected.

Accurate predictions of multiphase flows can only be made if the overall CFD

model uses mechanistically-based closure laws for the various local phenomena.

These models describe the interfacial mass, momentum and energy transfer between the fields. There is a hierarchy of approaches to model such flows [37]. The

simplest method is to treat the air and oil atmosphere as a homogeneous mixture where all of the phases are in dynamic and thermodynamic equilibrium. The

next approach is the Eulerian two-fluid method where each phase is treated as a

continuum with separate velocity and temperature fields. Finally, there is the Lagrangian approach (or trajectory models) where the motion of the disperse phases

is assessed by following either the motion of the actual particles or the motion of

larger, representative particles. Both the homogeneous mixture method and the

Eulerian two-fluid method are available in NPHASE-PSU.

2.1.3.1 Two-Fluid Eulerian Formulation

The single-pressure ensemble averaged continuity and momentum equations are

cast in conservation law form as:

∂

∂t(α

k

ρ

k

) + ∂

∂xj

(α

k

ρ

kuk

j

) = X

k6=1

(Γlk − Γ

kl) (2.13)

∂

∂t(α

k

ρ

kuk

i

) + ∂

∂xj

(α

k

ρ

kuk

i u

k

j

) = −α

k

∂p

∂xi

+

∂

∂xj

"

α

kμk

t

 

∂uk

i

∂xj

+

∂uk

j

∂xi

!# (2.14)

+ α

k

ρ

k

gi + Mkl

i +

X

k6=1

(D

kl[ul

i − u

k

i

] + Γlku

l

i − Γ

kluk

i

)

Γ

kl is the mass transfer rate from field k to field l. Dkl is the drag force coefficient and Mkl

i

represents the non-drag interfacial force of field k arising from the

interface with field l. In general each field, k, will have a different density, volume

fraction, velocity, and viscosity. α

k

is the volume fraction for field k. Ensemble

averaging is achieved by repeating the measurement at a fixed-time and position

for a large number of systems with identical macroscopic properties (velocity, temNASA/CR—2012-217807 26
perature) and boundary conditions, and finding the mean values of the results. For

compressible flows and flows with heat transfer, it is necessary to solve an energy

equation. An enthalpy transport equation for each field is written as

∂

∂t(α

k

ρ

khk

) + ∂

∂xj

(α

k

ρ

kuk

j h

k

) = ∂

∂xj



α

k



μ

k +

μ

k

t

σ

k

h



∂hk

∂xj



+ S

k

h

(2.15)

In the context of multifield flows, separate turbulence transport scalars are

solved for each field. For example, the high Reynolds number k- model is written:

∂

∂t(α

k

ρ

k

k

k

) + ∂

∂xj

(α

k

ρ

kuk

j k

k

) = ∂

∂xj



α

k



μ

k +

μ

k

t

σ

k

k



∂kk

∂xj



+ P

k − αk

ρ

k



k + Sk

k

(2.16)

∂

∂t(α

k

ρ

k



k

) + ∂

∂xj

(α

k

ρ

kuk

j



k

) = ∂

∂xj



α

k



μ

k +

μ

k

t

σ

k





∂k

∂xj



+ C1



k

k

k

P

k − C2



k

k

k

α

k

ρ

k



k + Sk



(2.17)

where Sk and S are available source/sink terms to extract turbulence energy associated with breakup and and modify production due to interface dynamics and

mass transfer mechanisms [48].

2.1.3.2 Homogeneous Multiphase Formulation

For homogeneous multiphase flow, it is assumed that the fields are in dynamic and

thermodynamic equilibrium allowing equations 2.13 and 2.14 to reduce to:

∂

∂t(α

k

ρ

k

) + ∂

∂xj

(α

k

ρ

kuk

j

) = X

k6=1

(Γlk − Γ

kl) (2.18)

∂

∂t(ρ

mum

i

) + ∂

∂xj

(ρ

mum

i u

m

j

) = −

∂p

∂xi

+

∂

∂xj



μ

k

t



∂um

i

∂xj

+

∂um

j

∂xi

 + ρ

mgi (2.19)

where the set of momentum equations is reduced to a single equation for the

mixture. The superscript m represents the mixture quantities. In equations 2.18-

2.19, a high Reynolds number for the viscous term is assumed with dilation and

turbulence energy terms neglected.

NASA/CR—2012-217807 27
2.2 OVER-REL

OVER-REL is a parallel, cell-centered, incompressible, finite-volume code based on

structured overset multi-block grids and the time-marching, pseudo-compressibility

formulation of Chorin [49]. The inviscid fluxes are formulated from the Roe approximate Riemann solver [50] and extended to third-order accuracy through the

MUSCL scheme [51]. Second-order accurate central differences are used for the

viscous fluxes. Numerical derivatives are used to calculate the flux Jacobians.

A symmetric Gauss-Seidel method is applied to solve the resulting linear system

of equations. In the present application, the code’s turbomachinery analysis instrumentation is used; all simulations are carried out for a single gear tooth, with

periodic boundary conditions, in a non-inertial frame-of-reference rotating with the

gear. OVER-REL does not support compressible flow, multiphase flow, or unstructured meshes (necessary for complex gearbox geometries). However, it is better

suited than NPHASE-PSU for physics exploration analyses due to its efficiency

for cases where a single “blade-row”/gear tooth can be used (i.e., axisymmetric

shroud).

2.2.1 Governing Equations

As will be shown, much can be learned from systems that are either shroud-free

or have completely axisymmetric shrouds (i.e., fully enclosed radial shrouds). So

we can consider single-tooth domains that are periodic and steady in the frame

of reference of the rotating gear. The conservation of mass and momentum can

be written in integral conservation law form for a system rotating with constant

angular velocity ~ω as:

Z

S~

ρV~ · dS~ = 0 (2.20)

Z

S~

ρV~ V~ · dS~ = −

Z

S~

pdS~ +

Z

S~

τ · dS~

NASA/CR—2012-217807 28
−

Z

∀

ρ~ω × (~ω × ~r) d∀ − Z

∀

2ρ



~ω × V~



d∀ (2.21)

where V~ is the velocity vector in the relative frame of reference. The last two

terms on the right-hand side of Equation 2.21 represent the apparent centrifugal

and Coriolis forces, respectively.

It was shown in Hill et al. [33] that a sublayer-resolved two-equation turbulence

model performed better than a high-Reynolds number form with wall-functions in

predicting viscous losses on a spinning disk. Therefore, with this formulation of the

governing equations τ = (μ+μt)(∇V + (∇V )

T

) is adopted and a sublayer resolved

q-ω turbulence model due to Coakley [52] is employed. The q-ω model in OVERREL was selected because it has been validated for a number of turbomachinery

applications. Unlike NPHASE-PSU, OVER-REL does not have a k- turbulence

model. The dependent variables in this model are related to the turbulence kinetic

energy, k, and the turbulence dissipation rate, , through q =

√

k and ω = /k. In

this model, the eddy viscosity is obtained from:

μt = ρCμDq2/ω (2.22)

where Cμ = 0.09 and D is the near wall damping function:

D = 1 − e

−αρqdn/μ (2.23)

where α = 0.02 and dn is a measure of the normal distance to the nearest wall.

The modeled transport equations for q and ω are:

Z

S~

ρqV~ · dS~ =

Z

S~



μ +

μt

Prq



∇q · dS~ +

Z

∀

ρq

2



CμD

S

ω

− ω



d∀ (2.24)

Z

S~

ρωV~ · dS~ =

Z

S~



μ +

μt

Prω



∇ω · dS~ +

Z

∀

ρ



C1CμS − C2ω

2



d∀ (2.25)

where C1 = 0.405D + 0.045, C2 = 0.92, Prq = 1.0, and Prω = 1.3 and S represents

the strain rate invariant.

In a axisymmetric gear system, the overall size of the grid domain can signifiNASA/CR—2012-217807 29
cantly reduced by only gridding one tooth passage instead of many. This is done

by using periodic boundary conditions. Figure 2.1 illustrate the use of periodic

boundaries in a axisymmetric gear case. The area colored in green represents the

one tooth passage that is included in the computational domain. The blue region

represents the rest of the gear that is left out. The red lines indicate the periodic

boundaries of the domain. With periodic boundary conditions, the flux of all flow

variables at one periodic boundary is set to equal the flux at the opposite periodic

boundary.

Figure 2.1. An example domain reduction of an axisymmetric system with periodic

boundaries.

NASA/CR—2012-217807 30
2.3 Overset Grid Methods

The overset grid approach [54] uses a composite grid consisting of a set of overlapping component grids to discretize the domain. No point-to-point or face-toface matching is required between component grids. The solution on the component grids are linked by identifying appropriate intergrid boundary points (IGBPs)

where the solution is given by a specified boundary value obtain by interpolation

from another overlapping donor component grid. The overset domain connectivity

information (DCI), which consists of the identification of the intergrid boundary

points and corresponding interpolation sources, is obtained by an overset grid assembly step.

The Donor interpolation Receptor Transaction library [55] (DiRTlib) is a solverneutral library that encapsulates the functionality required by the solver to use

the overset domain connectivity information. It is independent of the solver grid

storage and topology, dependent variables, etc. and can be used with any solver.

The solver calls a few functions to initialize the library, load the DCI interpolation,

transfer the data to appropriate processors in a parallel execution environment, and

apply the interpolated data as boundary values at IGBPs. Solver functions must

be provided and are called by DiRTlib to get and put data in the correct solverdependent variable storage locations. When the solver executes in a distributed

memory parallel computational environment the solver must also inform DiRTlib of

the parallel decomposition enabling DiRTlib to get/put data from the appropriate

parallel process.

The current overset grid assembly process is performed using the SUGGAR

code [56]. SUGGAR stands for Structured, Unstructured, Generalized overset

Grid AssembleR. It is a general overset grid assembly code with the capability

to create the domain connectivity information at node and/or element centers for

most current grid topologies including any combination of structured Cartesian and

curvilinear, unstructured tetrahedral and mixed element, general polyhedral, and

octree based Cartesian grids. For static grid assemblies with no motion between

component grids the grid assembly is a pre-processing step. The case of solution

and time dependent motion, requires the solver to communicate the new body

and grid positions to the grid assembly process, wait for it to complete, and then

NASA/CR—2012-217807 31
load the new DCI. In the case of prescribed motion, such as used in the present

study, the DCI is computed a priori and saved in a file for each time step in the

simulation and the solver simply loads the file appropriate for each time step.

Figure 2.2 provides an example of an overset grid. The grids colored in red are

boundary layer resolved grids for the teeth. The blue gridlines represent the far

field domain.

Figure 2.2. An example overset grid system.

The donor interpolations produced by SUGGAR are a set of linear weights

that multiply the values at the donor members. For a cell centered flow solver the

interpolation stencil will use as members the cell in the donor grid that was found

to contain an IGBP and the neighboring cells that share a face with the donor

cell. The interpolation weights are computed using an unweighted least square

procedure.

NASA/CR—2012-217807 32
Chapter 3

Validation and Parametric Studies

The experimental data of Diab et al. [6] has been used to validate both NPHASEPSU and OVER-REL for the case of isolated unshrouded rotating spur gears. Diab

et al. tested four different spur gears and a disk isolated in free air on a spin-down

test rig. The properties of the gears were listed in Table 1.1. The effects of an

enclosure or lubrication were not part of the study. Diab et al. used a spin-down

test rig to measure windage losses. Here, a sequence of prescribed constant rotation

rate simulations are used to replicate the experiment.

3.1 NPHASE-PSU Results

Grids were generated for all four spur gears and the disk. For the gear studies

where the high Reynolds number k- turbulence model was used, near-wall grid

spacing was defined to accommodate wall-functions (e.g., y

+ ' 70 for gear 1,

ω = 1000 rad/s). The single plane of symmetry in the problem (at the gear

centerline) was exploited to reduce total cell count by a factor of two. Grid cell

counts for the different cases varied between 2.0 x 106(Gear 4) and 8.0 x 106(Gear

1). Grid generation was further simplified by employing a hybrid mesh topology, as

illustrated in Figure 3.1. Specifically, for the regions above the surface of the gear

teeth to the outer radial boundary, structured hexahedral cells were used. For

the region above the gear face surface, unstructured prism cells were employed.

Figure 3.1 shows the surface mesh of the gear in blue and the symmetry plane in

purple. The meshes were generated using the commercial grid generation software

NASA/CR—2012-217807 33
package GridgenV15.15 [57]. The computational domain of the isolated gear grids

was extended to approximately five times the gear radius from the gear surface in

all directions. It was found that this distance was adequate for defining a farfield

boundary condition with a specified pressure since the flow is nearly stagnant there.

Figure 3.1. Grid topology of Diab Gear 4.

An azimuthal step size of 1/40th of one tooth passage duration (the time it

takes one tooth to rotate to the position of the tooth adjacent to it) was used in all

CFD calculations. The timestep size ∆t can be calculated by using Equation 3.1:

∆t =

1

40 

2π

Nω

(3.1)

where N is the number of gear teeth and ω is angular velocity in radians per second.

This corresponds to 2880 timesteps per gear revolution for Gear 1, 1440 timesteps

for Gears 2 & 3, and 960 timesteps for Gear 4. All cases used 10 pseudo-time

iterations per physical timestep. All of the Diab gear NPHASE-PSU simulations

presented here were executed on the Columbia supercomputer at NASA Ames Research Center. The code scales very well on this system as illustrated in Figure 3.2.

NASA/CR—2012-217807 34
Figure 3.2 shows that by increasing the number of processors used to compute the

solution for a 1.1 x 106cell rotating disk case, NPHASE-PSU exhibits better than

ideal (linear) speedup in processing time when running on Columbia.

Figure 3.2. Parallel efficiency of NPHASE-PSU on the NASA Ames Columbia computer

system for a spinning disk case (1.1x106cells).

CFD runs were made for all four gears and the disk at a number of rotation

speeds. All cases were run for at least two complete revolutions to remove simulation startup transient behavior. Convergence histories show that transients leave

the solution after about one revolution as illustrated in Figure 3.3, where it is also

observed that the pseudo-time residual drops approximately two orders of magnitude in each physical timestep when 10 pseudo-timesteps are used per physical

timestep.

Figure 3.4 compares the experimental results of Diab et al. [6] with the computational NPHASE-PSU analysis of all four gears. The CFD analysis for all of

the gears exhibited very good agreement with experiment. The disk case, however, did not share this same level of agreement, as illustrated in Figure 3.5, where

NPHASE-PSU results are seen to underpredict the measured power loss. In order

NASA/CR—2012-217807 35
Figure 3.3. Example NPHASE-PSU convergence history.

to elucidate the reasons for the deterioration in solution accuracy observed for the

disk case, a number of observations and studies were made. First, it is observed

that the measured (and computed) windage loss power levels for the disk are much

smaller than the comparably sized spur gear (Gear 1, D ' 300 mm). This arises

due to the absence of any azimuthal pressure variation in the disc flow - torque

losses are due entirely to viscous effects, and these are clearly underpredicted. Indeed the absolute magnitudes of loss underprediction between the disk and Gear

1 are comparable (e.g. ' 50W @ 600s

−1

), so presumably this underprediction of

shear is present in all of the gear simulations, however its relative magnitude is

small for Gear 1.

To explore this further, the low-Reynolds number Menter k−/k−ω model [46]

was applied to the spinning disk case (using an appropriate sub-layer resolved

mesh). Figure 3.5 illustrates that improved turbulence modeling does benefit solution accuracy especially at higher rotation rates. This observation is not as important for the gear cases since the windage torques associated with azimuthally

varying pressure forces in the vicinity of the gear teeth dominate the viscous forces

as shown in Figures 3.6 and 3.7. It can be seen that, especially at lower rotation

rates, the contribution of viscous loss to total windage loss, is quite small. However, it is observed that at higher pitch-line velocities the relative magnitude of

the viscous torque increases with respect to the pressure torque. This can be seen

with Gear 1 (Figure 3.6) as well as with the smaller of Gear 2 (Figure 3.7). At a

NASA/CR—2012-217807 36
Figure 3.4. Comparison between Diab experimental results and NPHASE-PSU analysis.

rotation rate of 800 rad/s, Gear 1 has a pitchline velocity of 172.6 m/s. Gear 2

reaches a maximum pitch-line velocity of 86.4 m/s at 1200 rad/s.

A number of important physical features of the predicted flow field are available upon interrogation of the CFD simulations. Figure 3.8 shows a view of the

predicted secondary velocity vectors on the symmetry plane in the gear relative

frame of reference for one of the Diab cases. One can see a significant vortical

structure within the gear tooth region, and the tooth-to-tooth periodicity that has

been achieved in the transient simulation. Similar flow structures were seen by

Al-shibli et al. [30] and Marchesse et al. [34] in their CFD studies of gear windage.

Figure 3.9 displays the predicted surface pressure distributions for the four Diab

NASA/CR—2012-217807 37
Figure 3.5. Effect of turbulence model selection on viscous work prediction.

gears. Comparatively large pressure differences are observed between the leading

and trailing tooth faces - this difference being the source of the pressure component

of the spin-down torque. The highly three-dimensional nature of the flow in these

spurs gears is also clearly seen (the figure shows only 1/2 of each gear).

In summary the results suggest that for the very high speed gears encountered in

rotorcraft (and other high performance aircraft) transmissions, viscous effects will

become more important. Secondly, it appears that the windage pressure torques

are predominantly located at the edges of the teeth of a spur gear.

NASA/CR—2012-217807 38
Figure 3.6. Breakdown of windage power losses for Diab Gear 1.

NASA/CR—2012-217807 39
Figure 3.7. Breakdown of windage power losses for Diab Gear 2.

NASA/CR—2012-217807 40
Figure 3.8. Secondary flows within the gear teeth.

Figure 3.9. Predicted surface pressure distribution for Diab Gears 1-4. Only one side of

symmetry plane is shown for each gear. The areas shaded in red indicate high pressure

areas.

NASA/CR—2012-217807 41
3.2 OVER-REL RESULTS

In the previous section, NPHASE-PSU was used to study all four gears and the disk

of Diab et al. [6]. The NPHASE-PSU simulations exhibited very good agreement

with experiment in modeling windage losses for all of the gear configurations. It

was also shown that a sub-layer resolved two-equation turbulence model performed

better than a high-Reynolds number form with wall functions in predicting viscous

losses on a disk. Unfortunately, using sub-layer resolved meshes to describe the

geometry of an entire gear and capture the unsteady flow physics is quite computationally expensive. The size of the grids easily exceeded 20 million cells. Although

this is not an impossible calculation with modern supercomputers (e.g. Columbia

at NASA Ames Research Center), it still requires significant CPU resources and

computation time.

Therefore, it was decided to change the overall grid topology to a single “bladerow”/gear tooth and exploit the axial symmetry of gear geometry instead of the

planar. This topology is better suited to the present physics exploration analyses

due to its efficiency. With this type of topology, the CPU resources that could been

used to run one sub-layer resolved NPHASE-PSU case, could be used to run at least

ten OVER-REL simulations. The solver OVER-REL is mature and validated for

these types of calculations so it was selected. In the present application, the OVERREL’s turbomachinery analysis instrumentation is employed. All simulations are

carried out for a single gear tooth with periodic boundary conditions in a noninertial frame-of-reference rotating with the gear.

First, free spinning simulations were carried out for Diab Gear 1 and disk using

OVER-REL. Non-overset multiblock structured meshes were employed for both as

shown in Figures 3.10 and 3.11. In Figure 3.11, one of the periodic boundaries

is highlighted by the dark blue mesh. The light blue lines represent the edges

of the various grid blocks that make up the domain, and the green and purple

surface represents the surface of the gear. Not shown in the figure is the outer

axial and radial boundaries, which were set as inviscid walls. Near-wall grids were

constructed to return wall cell y+ values < 1 and wall normal stretching ratios <

1.2 everywhere in order to adequately resolve the high Reynolds number boundary

layers that arise. The grid topologies, near-wall grid spacing, and grid stretching

NASA/CR—2012-217807 42
ratios were maintained as closely as possible between the gear and disk meshes.

In accordance with the periodic boundary conditions employed, one tooth passage

(2π/72) is modeled for both configurations. For a single tooth passage, the total

grid cell count did not exceed 1.6 x 106cells. In order to stably time-march the

OVER-REL solution, a very small inflow velocity and a pressure outflow boundary

were included adjacent to the maximum and minimum axial boundaries upstream

and downstream of the rotating elements. This artificial through-flow velocity was

successively reduced to where no perceptible changes in loss values were returned.

Figure 3.12 is an example convergence history of the axial torque component.

Figure 3.10. Comparison of Diab Gear 1 and disk grids.

Figure 3.13 includes the OVER-REL results for Gear 1 along with the previous

results from NPHASE-PSU. Very good agreement with experiment is seen here as

well. Figure 3.14 shows that for Gear 1, the pressure torque associated with the

integrated pressure difference between leading and trailing tooth surfaces, dominates the loss budget. As the rotation rate increases, viscous losses remain a nearly

constant fraction of total loss (10%). Figure 3.15 shows a comparison of loss results

between Gear 1 and the disk. The total losses are much smaller for the disk, which

are due to viscous shear alone. This large difference between the disk and Gear 1

NASA/CR—2012-217807 43
Figure 3.11. OVER-REL grid topology for Diab Gear 1. Periodic boundary is highlighted in blue. Gear surface mesh is highlighted in green.

is particularly striking if we compare their similar sizes as shown in Figure 3.10.

In Figure 3.16, the torque per unit span of radius contributed by viscous shear

is compared for the disk (up to its outer radius) and Gear 1 (up to its base radius).

The geometry of these systems requires that all of the pressure torque is due to the

pressure differences between the leading and trailing tooth surfaces. Figure 3.16

illustrates that the viscous losses are very similar between these configurations

(indeed the small differences in predicted viscous power values are due to very small

grid differences) indicating that 3-D effects (i.e. non-axisymmetric) associated with

pressure forces are almost completely responsible for the significant increase in loss

NASA/CR—2012-217807 44
Figure 3.12. Example OVER-REL convergence history of the axial torque component.

for the gear. Appendix B provides the C++ source code used to generate these

profiles.

NASA/CR—2012-217807 45
Figure 3.13. Experimental, NPHASE-PSU, and OVER-REL results for Diab Gears

1-4.

NASA/CR—2012-217807 46
Figure 3.14. Comparison of results from experiment and OVER-REL for Diab Gear 1,

including viscous and pressure loss budgets.

NASA/CR—2012-217807 47
Figure 3.15. Comparison of Diab Gear 1 and disk measurements and OVER-REL

solutions.

NASA/CR—2012-217807 48
Figure 3.16. Comparison of predicted torque per unit span contributed by viscous

shear for the Diab disk (up to its outer radius) and Gear 1 (up to its base radius).

NASA/CR—2012-217807 49
3.3 Details of the 3-D Flow Field

The physical mechanisms associated with the dominant pressure torque are studied by interrogating the CFD results for Diab Gear 1 with OVER-REL. In Figures 3.17 - 3.19, several 3-D visualizations are presented for the 850 rad/s case. In

Figure 3.17, a number of relative frame-of-reference streamlines, colored by local

static pressure, are plotted. These streamlines are seeded close to the gear face and

teeth and integrated in both directions. Some of the high speed (in the relative

frame) tangential flow near the gear face plane is diverted into the tooth passage,

where strong secondary flows are evident. Due to symmetry this axial transport

arises on both sides of the gear and therefore leads to impingement of oppositely

directed flows and radial ejection of momentum at the gear centerline. Figure 3.17

also shows an axial view of the same streamlines illustrating the complex secondary

motions and an indication of the radial ejection angle. Also evident is the radially

outward component of flow close to the face below the teeth. An axial projection

of relative velocity vectors is shown in Figure 3.18 at a plane halfway between the

gear face and gear centerline. A vector density of 0.5 (vector plotted for approximately every other grid point) is applied for clarity. Contours of local normalized

projected relative velocity magnitude are included (V

∗ =

p

V

2

y + V

2

z

/ωr). Two

counter-rotating passage vortices are present. Peak normalized secondary velocity magnitudes near 1

2

ωr are observed indicating the strength of these secondary

motions. The flow at this axial location is reminiscent of a rearward facing step

and/or cavity flow with attendant vortical recirculation regions. Figure 3.19 shows

the same plot but at the gear centerline. Here one sees the very significant radial

ejection quite clearly. The flow has a component directed upstream (against the

relative flow) near the leading surface at the tip radius. The magnitude of the

ejection flow induces significant blockage and we see values of V

∗ much less than

1 well beyond the tip radius.

In Figure 3.20, contours of static pressure coefficient, (CP = (p − p∞)/0.5ρV 2

tip)

are plotted for the leading gear tooth surface. As can be seen in the figure, some

of the high speed tangential flow near the gear face plane is diverted into the tooth

passage. There is a stagnation region where this flow impinges on the leading

surface, near the gear face. The leading surface pressure field can be compared

NASA/CR—2012-217807 50
Figure 3.17. Three-dimensional relative frame streamlines colored by static pressure

for Diab Gear 1 at 850 rad/s.

Figure 3.18. Axial projection of velocity vectors halfway between gear face and gear

centerline of Diab Gear 1 at 850 rad/s. Vector density of 0.5. Background contours of

local normalized projected relative velocity magnitude.

with the much lower surface pressures on the trailing surface. The net axial torque

due to pressure effects can be represented by the difference between the leading

and trailing surface pressure coefficients, which is shown in Figure 3.21. Clearly

the net torque is dominated by the impingement observed in Figures 3.17 and 3.20.

In Figure 3.22 the torque per unit width is plotted vs. distance from the gear face.

NASA/CR—2012-217807 51
Figure 3.19. Axial projection of velocity vectors near gear centerline of Diab Gear 1

at 850 rad/s. Vector density of 0.5. Background contours of local normalized projected

relative velocity magnitude.

Here, torque is nondimensionalized as:

T

∗

(x) =

R rtip

rinner

∆pdAθdr

dx "

1

1

2

ρV 2

refL

2

ref #

(3.2)

where ∆p is the pressure difference between the grid faces on the leading and

trailing surfaces (which have identical x-r vertex coordinates), dAθ is the tangential

projection of the area of the grid face, and r is the radial coordinate of the grid

face centroid, and the reference length, Lref , is the gear tip radius. We see that

indeed it is the near face region that dominates the pressure windage loss torque.

NASA/CR—2012-217807 52
Figure 3.20. Static pressure coefficient contours on the gear leading tooth surface(left)

and trailing tooth surface(right) of Diab Gear 1 at 850 rad/s. 1/2 of symmetrical gear

shown.

Figure 3.21. ∆CP between gear leading and trailing tooth surfaces of Diab Gear 1 at

850 rad/s. 1/2 of symmetrical gear shown.

NASA/CR—2012-217807 53
Figure 3.22. Pressure torque per unit width vs. axial coordinate of Diab Gear 1 at 850

rad/s. 1/2 of symmetrical gear shown.

NASA/CR—2012-217807 54
Chapter 4

Aerodynamic Studies of Windage

Loss Mitigation Strategies

4.1 Shrouded Diab Gears

As mentioned previously (Section 1.2), several researchers have measured improvements in gear windage loss performance when shrouds of various configurations

are employed [8, 10, 4, 38]. In this section the aerodynamics of shrouding are

explored in the context of geometrically simple configurations and several conclusions are drawn that may impact design decisions. The Diab Gear 1 configuration

studied in Section 3.2 is employed. Four notional shrouding arrangements are examined, nominally: Large-Axial-Large-Radial, Small-Axial-Large-Radial, LargeAxial-Small-Radial, and Small-Axial-Small-Radial. The shroud dimensions are

quantified and illustrated in Table 4.1 and Figure 4.1, respectively. These four

shroud configurations were chosen to be representative of the extrema of the fullshroud NASA Glenn tests detailed in Section 4.3.

Table 4.1. Diab Shroud Clearances

Minimum Maximum

Axial, (/Rtip) 0.0044 0.1733

Radial, (/Rtip) 0.0044 0.0970

NASA/CR—2012-217807 55
Figure 4.1. Four notional shroud configurations for Diab Gear 1 geometry. Figure is

to scale. Black lines define the gear face, tip radius and base radius, dark blue and red

lines indicate the Large-Axial and Small-Axial shrouds, green and cyan lines indicate the

Large-Radial and Small-Radial shrouds.

Figure 4.2 provides a cross-sectional view of the overall grid topology used in

the Large-Axial-Large-Radial configuration. In order to stably time-march the

OVER-REL solution and flush out start-up solution transients, a very small inflow

velocity and a pressure outflow boundary were included adjacent to the maximum

and minimum axial boundaries upstream and downstream of the rotating elements.

This artificial through-flow velocity was successively reduced to where no perceptible changes in loss values were returned. The smaller axial and radial shroud

NASA/CR—2012-217807 56
clearances gave rise to mesh topology constraints that lead to poor quality blockstructured meshes. Therefore, overset meshes were used for these cases. Figure 4.3

is a cross-sectional view of the overset mesh topology for a Small-Radial case. The

blue mesh is the boundary layer resolved grid of the shroud; the red mesh is the

boundary layer resolved grid of the gear.

Figure 4.2. Grid topology of the Large-Axial-Large-Radial shroud configuration.

In Figure 4.4, predicted windage loss vs. rotation rate is plotted for the unshrouded case, validated and studied previously (Section 3.2), and the four shroud

configurations. Each of the shrouds give rise to very significant improvements in

windage losses. The Large-Axial-Large-Radial shroud provides a 68% decrease in

loss at 850 rad/s; the Small-Axial-Small-Radial shroud provides approximately a

81% decrease at the same speed. Indeed, distinguishing the performance gains

NASA/CR—2012-217807 57
Figure 4.3. Cross-section of overset mesh topology of the Small-Radial shroud.

between the shrouded cases is helped by not including the unshrouded results

as in Figure 4.5. Examining this plot we see that reducing the axial and radial

clearances from large to small provide approximately the same level of additional

benefit over the Large-Axial-Large-Radial case, with the reduction of the radial

clearance providing somewhat more benefit in this particular case. Applying both

clearance reductions together provides the maximum benefit.

Figures 4.6 and 4.7 show the same pressure coefficient plots reported previously

(Figures 3.20 - 3.21), but here for the Large-Axial-Large-Radial shroud case at

850 rad/s. The load distribution on the leading surface shown in Figure 4.6 is

qualitatively similar to the unshrouded case (Figure 3.20), including the stagnation

region associated with near-face flow diverted into the tooth passage. However,

the range of CP values here are much smaller than for the unshrouded case. (Note

that the absolute values of CP should not be compared between cases here as this

level is determined in the incompressible flow solver using a domain “exit” value of

p∞ = 0). The trailing surface (at right in Figure 4.6) exhibits a much more axially

NASA/CR—2012-217807 58
Figure 4.4. Comparison of predicted windage losses between the unshrouded and the

four shrouded Diab Gear 1 configurations.

uniform pressure distribution than the unshrouded case (Figure 3.20), and again

the range of CP values here are much smaller. Most importantly, in Figure 4.7

a highly edge loaded ∆CP between leading and trailing surfaces can be seen, as

observed for the unshrouded case, but with a very reduced range of ∆CP which of

course leads to reduced torque. Figure 4.8 shows contours of pressure and surface

shear stress lines on the leading surface for the four shrouded cases at 850 rad/s.

It can be seen that the general features of impingement/stagnation near the edge

remain present for all of the configurations.

In Figure 4.9, the torque per unit width is plotted as a function of distance from

the gear face for all four shrouds at 850 rad/s. All four configurations exhibit edgeNASA/CR—2012-217807 59
Figure 4.5. Comparison of predicted windage losses between the four shrouded Diab

Gear 1 configurations.

loaded profiles, with their integrals consistent with the net loss trend reported in

Figure 4.5. Comparing Figure 4.9 with Figure 3.22 shows the dramatic reduction in

pressure torque that has been achieved for these configurations, but with retention

of the basic features of the pressure torque distribution.

Figure 3.16 plotted viscous losses per unit span for the unshrouded Diab Gear

1 case, illustrating the increase in shear torque with rotation rate and span. The

viscous losses per unit span are plotted for each of the four shrouded cases and the

unshrouded case at 850 rad/s in Figure 4.10. There is it observed that face shear

is smaller for all of the shrouded cases due to the “couette-flow-like” rotational

boundary layer that arises in the presence of an outer axial boundary. Also of

NASA/CR—2012-217807 60
Figure 4.6. Static pressure coefficient contours on the leading tooth surface (left) and

the trailing tooth surface (right) of the Large-Axial-Large-Radial shroud case at 850

rad/s. 1/2 of symmetrical gear shown.

interest in this figure is the increase in viscous loss for the Small-Axial shrouds

compared to the Large-Axial shrouds. These losses are smaller than for the unshrouded case but suggest that viscous losses can increase as a percentage of total

loss for very small axial shroud clearances.

NASA/CR—2012-217807 61
Figure 4.7. ∆CP between gear leading and trailing tooth surfaces of the Large-AxialLarge-Radial shroud case at 850 rad/s. 1/2 of symmetrical gear shown.

NASA/CR—2012-217807 62
Figure 4.8. Contours of pressure and surface shear stress lines on the leading surface

for the four shrouded cases at 850 rad/s.

NASA/CR—2012-217807 63
Figure 4.9. Torque per unit width on the tooth face for the four shrouded cases.

NASA/CR—2012-217807 64
Figure 4.10. Comparison of viscous losses per unit span for Diab Gear 1 at 850 rad/s

for unshrouded and four shrouded configurations.

NASA/CR—2012-217807 65
4.2 Investigation of the Aerodynamics

of Shrouding

It has been demonstrated both experimentally and computationally that enshrouding a gear reduces windage effects. However, up until now, very little has been

revealed about how shrouds actually accomplish this reduction. Winfree [10] observed that shrouding a bevel gear reduced the ability of the gear to pump fluid

from the gear axis and through its teeth. Essentially, that interpretation suggests

that the shroud acts as a restrictor to limit the amount of fluid that the gear can

can actually pump. This hypothesis can be tested for a spur gear with CFD using

the results to compute the mass flow rate of fluid that passes through the gear

teeth.

Figure 4.11 plots the integrated radial velocity entering/leaving the tooth region, which can be used to represent mass flow (  ̇m = ρAV ) through the teeth

(since density is constant) as a function of speed for unshrouded and shrouded

configurations of Diab Gear 1. For all cases, the radial velocity is integrated over

a surface that spans the tips of the gear teeth as illustrated in Figure 4.12. It

can be seen that the mass flow through teeth of the shrouded gears is less than

that of the unshrouded case. However, comparing just the unshrouded case with

the Large-Axial-Large-Radial case, the difference in the mass flow rates is not significant enough to explain for the large difference in windage losses between the

cases as seen previously in Figure 4.4. Thus, these results do support the Windfree hypothesis. Rather an alternative hypothesis has been developed, and this is

presented and supported here.

Figures 4.13 and 4.14 provide a cross-sectional view of the solutions for the

unshrouded and Large-Axial-Large-Radial cases, respectively, at 700 rad/s. The

cut-planes are located along the center of the tooth well and display contours of

tangential velocity (u

∗

t

). A zero tangential velocity (blue) indicates the fluid is close

to quiescent. Values of positive tangential velocity indicate the fluid is traveling

in the same direction as the rotation of gear. Figure 4.13 shows the radial ejection

of the fluid from the teeth of the unshrouded gear at the centerline of the gear. It

also shows that the ejected fluid has been given a tangential velocity component

by the rotating gear. (The asymmetry in the u

∗

t profile of the unshrouded case is

NASA/CR—2012-217807 66
Figure 4.11. Integrated radial mass flow through gear tooth.

due to the small inflow velocity that is required by the CFD solver. This artificial

through-flow was not found to effect the results.) However, for the shrouded case,

shown in Figure 4.14, the ejected fluid does not completely escape from the gear.

The shroud keeps the accelerated fluid within the vicinity of the gear, most notably

near the gear teeth. The shroud confines the angular momentum imparted by the

gear to the fluid within the vicinity of the teeth.

To further elucidate how this effect impacts windage losses, Figure 4.15 plots the

nondimensionalized tangential velocity of the fluid along a line that runs parallel to

the gear axis and through the tooth space at a radial location of R/Rtip = 0.95. It

can be seen that as fluid approaches the gear face axially, the flow is immediately

accelerated to the local tangential velocity of the gear. In both shrouded and

NASA/CR—2012-217807 67
Figure 4.12. Integration surface for radial mass flow.

unshrouded cases, the flow is accelerated to the same tangential velocity upon

entering the tooth space, but with the shrouded case experiencing a significantly

smaller change in velocity. This difference reveals the dominant mechanism by

which shrouding reduces the windage losses. Above it was shown that for all cases

studied, shrouded and unshrouded, the near-face, near-tooth flow is drawn into

the tooth passage, by virtue of, lower pressures within the tooth passage. This

flow impinges on the leading tooth surface and the attendant stagnation pressure

rise was observed to give rise to dominant component of the windage torque. The

foregoing angular momentum results show conclusively that this deceleration and

attendant pressure rise is much smaller for the shrouded gears since the local

NASA/CR—2012-217807 68
Figure 4.13. Contour plot of tangential velocity for unshrouded Diab Gear 1 at 700

rad/s.

tangential velocity of the flow is much closer to the that of the gear. By conserving

the angular momentum imparted by the gear on the fluid, the shrouded gear uses

less force to pump the fluid. Examining the tangential velocity profiles at different

radial locations in Figure 4.16, it can be seen how viscous shear losses are also

affected by shrouding. By conserving the tangential velocity of the fluid near the

gear, the viscous drag experienced by the gear is reduced since less work is required

to accelerate the fluid to the speed of the gear.

NASA/CR—2012-217807 69
Figure 4.14. Contour plot of tangential velocity for the Large-Axial-Large-Radial

shroud case at 700 rad/s.

NASA/CR—2012-217807 70
Figure 4.15. Comparison of tangential velocity profiles along R/Rtip = 0.95 between

the unshrouded and the Large-Axial-Large-Radial shroud cases.

NASA/CR—2012-217807 71
Figure 4.16. Comparison of tangential velocity profiles at different radial locations

between the unshrouded and the Large-Axial-Large-Radial shroud cases.

NASA/CR—2012-217807 72
To further examine the physics of shrouding, two modifications of the LargeAxial-Large-Radial shroud configuration were tested. Openings were placed on

the outer radial boundary in each case, but at different locations. In the first

modification, slots were placed at the corners of the shroud as shown in Figure 4.17.

In the second, just a single slot was inserted directly above the center of the gear.

Figure 4.18 provides a view of this configuration. It can be seen in both cases

that a large outer domain was added to the original grid outside of the shrouded

domain. This was done because the flow conditions at the openings were unknown

and any outlet boundary condition could impose an arbitrary flow there.

Figure 4.17. Grid topology of the Large-Axial-Large-Radial shroud with corner slots.

Each case was run at a speed of 700 rad/s and in both cases the beneficial effect

of the shrouds was significantly reduced. The results are plotted in Figure 4.19.

NASA/CR—2012-217807 73
Figure 4.18. Grid topology of the Large-Axial-Large-Radial shroud with centerline

slot.

The corner slotted case provided only a 19% decrease in windage compared to

the unshrouded case at 700 rad/s and the centerline slotted case returned a 13%

decrease in windage loss at the same speed. In contrast, the fully shrouded configuration provided a 68% decrease at 700 rad/s. As can be seen in Figures 4.20 - 4.23,

the slots allow the fluid accelerated by the gear to escape away from the gear.

Figures 4.20 and 4.22 provide contour profiles of tangential velocity of the two

configurations. Figures 4.21 and 4.23 include three dimensional absolute frame

streamlines with the contour profiles of tangential velocity showing the leakage

of angular momentum from the shroud. This effect is further quantified by Figure 4.24 which plots the nondimensionalized tangential velocity in the same manner

NASA/CR—2012-217807 74
as Figure 4.15. Here we see that the tangential velocity of the fluid entering the

teeth is much closer in value to the unshrouded case than the fully shrouded case.

Figure 4.19. Comparison of predicted windage losses between the unshrouded, the fully

enclosed Large-Axial-Large-Radial shroud, and slotted Large-Axial-Large-Radial shroud

configurations.

Figure 4.25 provides the tangential velocity profiles along R/Rtip = 0.95 for

the unshrouded and fully shrouded gear configurations. Here it seen why the large

axial shroud cases experience less viscous torque than the small axial cases as first

observed in Section 4.1. The tighter clearances do not leave much room between

the walls of the shroud and the gear face to decrease the ∆ut between the surfaces.

Figures 4.26 - 4.28 provide contour profiles of tangential velocity of the Large-AxialSmall-Radial, Small-Axial-Large-Radial, and the Small-Axial-Small-Radial cases,

NASA/CR—2012-217807 75
Figure 4.20. Contour plot of tangential velocity for the Large-Axial-Large-Radial

shroud with corner slots at 700 rad/s.

respectively. Here, it is observed that the tighter shrouds keep the accelerated

flow much closer to vicinity of the gear teeth. Figure 4.25 also shows that the

∆utis greater for the tighter three shroud configurations, yet windage losses are

still smaller for these cases. This is explained by recalling Figure 4.12. The three

tighter shroud configurations experience less mass flow through the teeth and are

thus accelerating less fluid per unit time than the Large-Axial-Large-Radial case.

Thus these cases experience less windage power loss.

NASA/CR—2012-217807 76
Figure 4.21. Three-dimensional absolute frame streamlines for the Large-Axial-LargeRadial shroud with slots at the corner at 700 rad/s.

NASA/CR—2012-217807 77
Figure 4.22. Contour plot of tangential velocity for the Large-Axial-Large-Radial

shroud with centerline slot at 700 rad/s.

NASA/CR—2012-217807 78
Figure 4.23. Three-dimensional absolute frame streamlines for the Large-Axial-LargeRadial shroud with centerline slot at 700 rad/s.

Figure 4.24. Comparison of tangential velocity profile along R/Rtip = 0.95 between

the unshrouded, the fully enclosed Large-Axial-Large-Radial shroud, and slotted LargeAxial-Large-Radial shroud configurations.

NASA/CR—2012-217807 79
Figure 4.25. Comparison of tangential velocity profile along R/Rtip = 0.95 for the

unshrouded and the four fully enclosed shroud configurations.

Figure 4.26. Contour plot of tangential velocity for the Large-Axial-Small-Radial case

at 700 rad/s.

NASA/CR—2012-217807 80
Figure 4.27. Contour plot of tangential velocity for the Small-Axial-Large-Radial case

at 700 rad/s.

NASA/CR—2012-217807 81
Figure 4.28. Contour plot of tangential velocity for the Small-Axial-Small-Radial case

at 700 rad/s.

NASA/CR—2012-217807 82
4.3 NASA Glenn Tests

The NASA Glenn Research Center has recently installed a gear windage test facility [7]. The test facility is designed to parametrize the effects of gear geometry,

shroud geometries and sizes, different lubrication system configurations, system

pressures and temperatures, and gear meshing on windage loss. A sketch of the

test facility is shown in Figure 4.29 and Figure 4.30 is a photo of facility. The facility has a 150 hp (112 kW) DC drive motor that is connected to a 5.7:1 speed-up

gearbox. The output of the speed-up gearbox is then connected to a torque meter

prior to a coupling connection of the input shaft to the test gearbox. The input

and output shafts have hydraulically operated clutches that allow the facility (in

single or dual shaft mode) to be disconnected from the power source and/or magnetic brake attached to the output shaft. With the speed capability of the drive

motor and speed increasing gearbox and the dimensions of the test specimen, the

pitch line velocity can be as high as 280 m/s (55,000 ft/min).

The test gears can be run in the gearbox with or without shrouding. The

shrouding clearance can be adjusted radially and axially. The details of the gear

under study here is given in Table 4.2 and the maximum and minimum shroud

clearances are provided in Table 4.3. The gear has some modest geometric complexities compared to the idealized Diab gear studied previously. These include

chamfered teeth, filleted teeth roots, and a narrower body width between the teeth

and the hub. This dissertation reports the data for and analyses of the four extreme shroud configurations: Large-Axial-Large-Radial, Large-Axial-Small-Radial,

Small-Axial-Large-Radial, and Small-Axial-Small-Radial. A sketch of the shrouding arrangement is shown in Figure 4.31.

Table 4.2. Basic Gear Dimensions

Number of teeth 52

Module (Diametral Pitch), mm (1/in.) 6.35 (4)

Face Width, mm (in.) 28.4 (1.12)

Pitch Diameter, mm (in.) 330.2 (13.0)

Pressure Angle, deg. 25.0

Outside Diameter, mm (in.) 342.65 (13.49)

NASA/CR—2012-217807 83
Figure 4.29. Overview sketch of the the NASA Glenn Research Center Gear Windage

Test Facility. [7]

Table 4.3. Shroud Wall Clearances Studied

Minimum Maximum

Axial, mm (in.) 0.762 (0.030) 29.718 (1.17)

Radial, mm (in.) 0.762 (0.030) 16.65 (0.655)

Data from the NASA Windage Test Stand was measured in the following manner. Speed data is measured using inductive pickups that read a 60 tooth disc on

the end of each of the shafts. The output from the sensor (pulse / sine wave) is

sent to a frequency-to-voltage converter. The output from the converter is then

sent to a National Instruments card and read by Labview. Data was taken at 10

NASA/CR—2012-217807 84
Figure 4.30. Photograph of the NASA Glenn Gear Windage Test Facility with containment shielding in place.

Hz. The facility was operated at a series of increasing drive motor speeds. At each

of these conditions several different data were collected. The drive motor speed,

torque applied to rotate the test hardware, internal shroud (fling-off) temperature,

and internal shroud static pressure data were taken at steady drive motor speed

conditions. Data was taken and then the speed was incremented from a given

drive motor speed of 500 to 3125 RPM (or 2587 to 16168 RPM of the gear shaft).

The data was taken from motor controller speed, a commercially available torque

meter for torque, a thermocouple for inside shroud temperature (oil fling-off), and

a manometer. In order to determine the effects of the gear-only windage a separate

test was conducted with the entire system in place minus the gear [7].

Figure ?? was adapted from Ref. [7] and reports the air-only gear windage power

loss results for a number of tested configurations. This figure includes air-only data

from the unshrouded configuration, the fully enclosed shroud configurations, and

NASA/CR—2012-217807 85
Figure 4.31. Shroud assembly for test facility. [7]

the slotted shroud configurations. The slotted shroud configurations include a 1-

inch circumferential drain slot at the bottom of the shroud to allow for lubrication

testing. Figures 4.32 and 4.33 demonstrate the effect that was observed in the

CFD analysis performed in Section 4.2. By adding an opening at the periphery of

a shroud, the windage losses increase on a spur gear.

4.3.1 NPHASE-PSU Results

An overset grid system was developed for the 13-inch pitch diameter spur gear

configuration. The grid system was designed to facilitate comparisons between

different shroud configurations. One grid system describes the gear, and different

shroud designs are simply swapped out using the overset grid methodology provided by SUGGAR. The gear is described by a mesh consisting of 234 component

grids that add up to 15.7M cells. Figure 4.34 depicts the near-body grid topology

of the grid. Figure 4.35 illustrates the overset topology of the unshrouded configuration which has 23.4M total cells. Like the previous NPHASE-PSU simulations

(Section 3.1), the single plane of symmetry at the gear centerline was exploited in

NASA/CR—2012-217807 86
Figure 4.32. NASA experimental gear windage power loss data for 13-inch spur gear

for the shrouded and unshrouded configurations. [7]

the grid construction to reduce grid size.

Overset meshes were also constructed for the Large-Axial-Large-Radial and the

Large-Axial-Small-Radial configurations. The Large-Axial-Large-Radial configuration had a total cell count of 26M cells and the Large-Axial-Small-Radial case

had approximately 36.6M cells. Reducing the clearances between the gear and

shrouds required an increase in the number of cells in order to provide sufficient

overlap between the overset meshes. Unfortunately, these cases were never able to

achieve a converged solution.

Figure 4.36 shows NPHASE-PSU results against experimental data for the

unshrouded 13 inch NASA spur gear. The NPHASE-PSU results slightly overNASA/CR—2012-217807 87
Figure 4.33. NASA experimental gear windage power loss data for 13-inch spur gear

for only the shrouded configurations. [7]

predict the experimental results, which is not unexpected, but there is otherwise

good agreement with the experimental data. In the unshrouded experiment, the

shroud assembly and the top lid of the gearbox are removed. However, the rest

of the gearbox housing remains in place. In the CFD model, the outer walls of

the gearbox were not included. As was demonstrated in Section 4.1, even open

shrouds will provide some reduction in windage losses. Figure 4.37 shows the

predicted relative frame streamlines at 1000 rad/s along with iso-surfaces of high

pressure. This figure exhibits several of the features observed for the idealized

case including the diversion of the near face flow into the tooth passage and its

impingement on the leading surface, a strong axial secondary vorticity in the tooth

NASA/CR—2012-217807 88
Figure 4.34. Surface topology of the overset mesh used to describe the NASA Glenn

13-inch spur gear.

passage, a strong ejection of this flow near the tooth centerline and radial flow of

the near-face streamlines below the base of the teeth.

NASA/CR—2012-217807 89
Figure 4.35. Overset mesh topology of the NASA Glenn 13-inch spur gear.

NASA/CR—2012-217807 90
Figure 4.36. NPHASE-PSU results for the unshrouded NASA Glenn 13-inch spur gear.

NASA/CR—2012-217807 91
Figure 4.37. Three-dimensional relative frame streamlines for the unshrouded NASA

Glenn 13-inch spur gear at 1000 rad/s.

NASA/CR—2012-217807 92
4.3.2 OVER-REL Results

Grids were developed for the 13-inch pitch diameter spur gear configuration using

similar topologies as the Diab cases (Section 4.1). Grids were created for all four

of the shroud configurations listed in Table 4.3.

Figure 4.38 shows comparisons of the NASA Glenn experimental data and the

OVER-REL predictions for the four shroud configurations. The data shows the

same trends as the idealized Diab gear CFD studies reported in Section 4.1. Specifically, the Large-Axial-Large-Radial shrouding exhibits the highest loss levels, and

the Small-Axial-Small-Radial shrouding exhibits the lowest loss levels. The benefit realized by reducing both clearances is somewhat more substantial than for the

Diab case. The CFD results are seen here to provide fairly good agreement with

the measured values. An interesting observation in the CFD results is that the

Large-Axial-Small-Radial and Small-Axial-Large-Radial results are nearly identical along the entire speed line.

The qualitative correspondence between the Glenn and idealized Diab cases

presented earlier, suggests that the same physical loss mechanisms are acting on

both of them. Figure 4.39 shows a view of the predicted relative streamlines colored

by pressure for the Large-Axial-Large-Radial shroud case at 700 rad/s. This image

exhibits several of the features observed for the idealized case including diversion

of the near face flow into the tooth passage and impingement upon the leading

surface, strong axial secondary vorticity in the tooth passage, strong ejection of

this flow near the tooth centerline and radial flow of the near-face streamlines

below the base of the teeth.

4.4 Design Studies

It was demonstrated previously (Section 4.1) that axial and radial shrouding can

reduce windage losses. Some of the physics of these loss reduction schemes were

studied there. Despite the experimentally and computationally observed differences in loss magnitudes between unshrouded and various shrouded configurations,

in all cases a significant component of the torques associated with spin down arose

from impingement onto the leading surface of the high velocity relative-frame flow

NASA/CR—2012-217807 93
Figure 4.38. Experimental and CFD results for the 13-inch pitch diameter NASA spur

gear

drawn into the tooth passage. Accordingly, in this section we return to the geometrically idealized Diab Gear 1 configuration, and experiment numerically with

four proposed tooth geometry modifications aimed at mitigating this impingement

and attendant spin down torque. Figure 4.40 shows an oblique view of the four

alternative geometries considered: 1) leading surface tooth-edge rounding, 2) leading+trailing surface tooth-edge rounding, 3) double slots on the top of the teeth

and, 4) trailing surface ramp.

Figure 4.41 shows a comparison of these four simulations in the baseline LargeAxial-Large-Radial case. The leading surface rounding and double slot geometries

return nearly identical windage loss. The leading+trailing surface rounding returns

NASA/CR—2012-217807 94
Figure 4.39. Three dimensional relative frame streamlines colored by static pressure

for the 13-inch pitch diameter NASA spur gear, Large-Axial-Large-Radial shroud, 700

rad/s.

somewhat higher loss. However, the net loss obtained using the trailing surface

ramp is approximately 30% lower than the baseline configuration. The torque

per unit width for the five geometries is plotted in Figure 4.42. There it can be

seen that the ramp configuration exhibits much smaller torques within the tooth

channel and this clearly results in the reduced integrated loss for the entire gear.

Figures 4.43 - 4.47 are presented to further elucidate the physics involved in

these numerical studies. In each of these figures, predicted surface pressure coefficients contours are plotted along with selected surface skin friction lines. The

baseline and two rounded geometries exhibit largely the same qualitative flow

features, with the rounded cases “smearing” the leading face impingement and

trailing face detachment gradients. The tooth slots were conceived to “flush” the

peak axial vorticity/low pressure regions of incoming relative flow with higher velocity thereby reducing the axial pressure gradient and thereby diverting the high

relative velocity near-face flow into the passage. This appears here to not have

achieved the desired result. The trailing surface ramp geometry did have a signifNASA/CR—2012-217807 95
Figure 4.40. Tooth geometry alternatives: leading surface rounding (top left), leading+trailing surface rounding (top right), slotted (bottom left), and trailing surface ramp

(bottom right).

icant impact on the aerodynamics. Specifically, the relative flow near the face is

turned away from the gear. This turning induces a local pressure rise on the ramp

which contributes to spin-down torque. However, this flow has been diverted away

from the tooth enough that subsequent diversion of this flow into the tooth passage

has been virtually eliminated, resulting in almost no pressure rise on the leading

surface. This gives rise to the much smaller torques as shown in Figure 4.42. So

the improved net performance of the ramp configuration observed in Figure 4.41

is clearly due to the reduced integrated tooth passage torque more than offsetting

the increased torque associated with the ramp turning itself. Finally, Figure 4.48

shows the integrated radial velocity versus speed for the baseline shrouded and

unshrouded configurations and the ramp case.

NASA/CR—2012-217807 96
Figure 4.41. Windage loss predictions for the baseline tooth geometry (Diab Gear 1

with Large-Axial-Large-Radial shrouds) and four geometric alternatives.

NASA/CR—2012-217807 97
Figure 4.42. Torque per unit width predictions for the baseline tooth geometry (Diab

Gear 1 with Large-Axial-Large-Radial shrouds) and four geometric alternatives at 850

rad/s.

NASA/CR—2012-217807 98
Figure 4.43. Predicted surface pressure coefficient and skin friction lines for baseline

tooth geometry.

Figure 4.44. Predicted surface pressure coefficient and skin friction lines for tooth

geometry alternative: leading surface rounding.

NASA/CR—2012-217807 99
Figure 4.45. Predicted surface pressure coefficient and skin friction lines for tooth

geometry alternative: leading+trailing surface rounding.

Figure 4.46. Predicted surface pressure coefficient and skin friction lines for tooth

geometry alternative: tooth slot.

NASA/CR—2012-217807 100
Figure 4.47. Predicted surface pressure coefficient and skin friction lines for tooth

geometry alternative: trailing surface ramp.

NASA/CR—2012-217807 101
Figure 4.48. Comparison of integrated radial mass flow from gear teeth between the

unshrouded, the baseline Large-Axial-Large-Radial shroud case, and the Large-AxialLarge-Radial case with tooth trailing surface ramps.

NASA/CR—2012-217807 102
4.5 Bell Helical Gear Analysis

Bell Helicopter has recently developed their own gear windage test facility to test

shroud design configurations for a helical gear train. The facility consists of two

helical gears in mesh. Table 4.4 lists the properties of the gears under test. The

test facility is designed to parametrize the effects of gear geometry, shroud geometries and sizes, different lubrication system configurations, system pressures and

temperatures, and gear meshing on windage loss.

Table 4.4. Bell Helical Gear Data

Input Gear Bull Gear

Number of teeth 51 139

Pitch Diameter, mm (in.) 154.7 (6.09) 431 (16.97)

Helix Angle, degree 12

Module (Diametral Pitch), mm (1/in.) 3.033 (8.375)

Face Width, mm (in.) 67.2 (2.625)

Reduction Ratio 2.7255:1

Grids were developed for the helical gears using similar grid topologies as the

Diab (Section 4.1) and NASA Glenn (Section 4.3.2) studies. Figures 4.49 - 4.52

provide views of the surface grid topology at the gear teeth and the overall grid

topology. One shrouded and one unshrouded configuration was tested for each

gear. In the shrouded cases, both cases had an axial clearance of 2 mm at the

gear teeth. The smaller 51-tooth (51T) drive gear had a 1 mm radial clearance

and the 139-tooth (139T) gear had a 1.5 mm radial clearance. The geometry and

dimensions of the shroud were provided by Bell Helicopter. Table 4.5 lists the total

number of grid cells used in each case. Cases were run in the manner described in

Section 3.2.

Table 4.5. Total Number of Cells (in millions) of Bell Helical Gear Grids

Gear Shrouded No Shroud

51T 2.81 5.62

139T 3.85 6.20

NASA/CR—2012-217807 103
Figure 4.49. Bell 51-tooth helical gear and surface mesh.

Figures 4.53 and 4.54 show the predicted power loss as a function of speed for

the 51-tooth and 139-tooth gears, respectively. Both figures clearly show the effect

of shrouds in reducing windage losses. Figures 4.55 and 4.56 show the predicted

viscous and pressure loss budgets for the two cases. Shrouding has a profound

influence on the pressure losses. Pressure losses are greatly reduced and they are

no longer dominant over viscous losses. Figure 4.57 compares the total windage

losses of all four cases together. In this figure, pitch-line velocity replaces rotation

speed on the x-axis. Figure 4.58 plots power loss ratio between the large and small

helical gears as a function of speed. In the unshrouded configuration, the power

loss ratio remains fairly constant with speed. However, in the shrouded case, power

loss ratio decreases as speed increases. This is probably due to the fact that the

losses experienced by the small gear are quite small at low speed.

Figures 4.59 and 4.60 present a 3-D visualization of the flow through the gear

teeth of the smaller Bell gear. The streamlines colored in blue represent the flow

that enters the teeth from the leading edge of the gear. The red is for the flow

from the trailing edge. The surface of the gear is colored by the static pressure

coefficient. Figures 4.61 and 4.62 do the same for the larger gear. It is observed in

the unshrouded cases (Figures 4.60 and 4.62), that the flow from the leading edge

NASA/CR—2012-217807 104
Figure 4.50. Shrouded Bell 51-tooth helical gear grid domain.

dominates. Any flow that enters from the opposite end, is quickly expelled over

the teeth. A very large stagnation region on the leading edge side is also clearly

visible in both cases. With the addition of shrouding (Figures 4.59 and 4.61),

there is still a strong flow from the leading side, but the flow from the trailing

side is able to penetrate deeper into the tooth well before being turned away.

These different flow patterns also explain the difference of observations made by

Dawson [4] and Houjoh et al. [23] of the flow through helical gear teeth. The

helical gears in Houjoh et al. [23] were held in a moderately sized enclosure. Also

the large stagnation region at the leading edge has disappeared, replaced with

NASA/CR—2012-217807 105
Figure 4.51. Bell 139-tooth helical gear and surface mesh.

two small stagnation regions close the base of the rounded edges. It should be

noted that Figure 4.59 can be misleading. It appears that there is a high pressure

stagnation region on the leading tooth face along the entire width of the gear that

should yield higher torque values. However, this is countered by the high pressure

region on the opposite trailing tooth face, providing for a lower ∆Cp between

faces. It was explained previously in Section 3.3 that the net axial torque due

to pressure effects was due to the difference between leading and trailing surface

pressure coefficients.

NASA/CR—2012-217807 106
Figure 4.52. Shrouded Bell 139-tooth helical gear grid domain.

NASA/CR—2012-217807 107
Figure 4.53. CFD predictions of windage power loss vs. speed for Bell 51-tooth helical

gear.

NASA/CR—2012-217807 108
Figure 4.54. CFD predictions of windage power loss vs. speed for Bell 139-tooth helical

gear.

NASA/CR—2012-217807 109
Figure 4.55. Breakdown of pressure and viscous windage power losses for Bell 51-tooth

helical gear.

NASA/CR—2012-217807 110
Figure 4.56. Breakdown of pressure and viscous windage power losses for Bell 139-tooth

helical gear.

NASA/CR—2012-217807 111
Figure 4.57. CFD predictions of windage power loss vs. pitchline velocity for Bell

helical gears.

NASA/CR—2012-217807 112
Figure 4.58. Predicted power loss ratio vs. pitchline velocity between the 139-tooth

and 51-tooth helical gears.

NASA/CR—2012-217807 113
Figure 4.59. Three-dimensional relative frame streamlines colored entrance location

for shrouded Bell 51-tooth gear at 1500 rad/s.

Figure 4.60. Three-dimensional relative frame streamlines colored entrance location

for unshrouded Bell 51-tooth gear at 1500 rad/s.

NASA/CR—2012-217807 114
Figure 4.61. Three-dimensional relative frame streamlines colored entrance location

for shrouded Bell 139-tooth gear at 550 rad/s.

Figure 4.62. Three-dimensional relative frame streamlines colored entrance location

for unshrouded Bell 139-tooth gear at 550 rad/s.

NASA/CR—2012-217807 115
4.5.1 Design Alternatives

Like the design space exploration that was performed on the Large-Axial-LargeRadial shrouded Diab Gear 1 (Section 4.4), a similar exploration was performed

on the unshrouded 139-tooth Bell gear. Analysis of the unshrouded gears showed

a strong axial flow from the leading edge to the trailing edge of the gear teeth.

Therefore, two modifications were tested to reduce this axial flow. The first modification was to simply block the leading edge inlet with a wall, similar to that of

Dawson [4]. This was accomplished by taking advantage of grid block boundaries

located near the leading edge of the teeth. The original block-to-block boundaries

were simply changed to inviscid wall boundaries. A no-slip wall boundary condition would have required extensive modifications to the grid blocks to meet y+ wall

spacing requirements. The inviscid wall is shown in Figure 4.63 shaded in grey.

The second modification was based on the tooth ramp design study presented in

Section 4.4. However, unlike the spur gear study, the ramp was only added to the

leading edge side. Figure 4.64 illustrates the tooth ramp geometry.

Figure 4.63. Bell 139-tooth helical gear with blocked leading edge.

Both cases were only run at a speed of 550 rad/s, which is the maximum

operational speed for this gear. A 30% decrease in windage losses was achieved

with both configurations as shown in Figure 4.65. Figures 4.66 and 4.67 provide

NASA/CR—2012-217807 116
Figure 4.64. Bell 139-tooth helical gear with leading edge spoiler.

a 3-D view of the flow through the gear teeth. In both cases, the modifications

either block or divert the flow away from the inlet region. Most of the flow through

the teeth comes from the trailing edge side (streamlines colored in red) of the gear.

The streamlines colored in blue represent flow from the leading edge. It should

be noted that streamlines in Figure 4.66 that appear to go through the inviscid

wall are an artifact of the post-processing software. Comparing these results with

the those from the original unmodified case (Figure 4.62), it can be seen that the

stagnation region on edge of the gear tooth has also shifted from the leading edge

to the trailing edge and is significantly smaller.

NASA/CR—2012-217807 117
Figure 4.65. CFD predictions of windage power loss for modified Bell helical gears. at

550 rad/s.

NASA/CR—2012-217807 118
Figure 4.66. Relative frame streamlines of the Bell 139-tooth helical gear with blocked

leading edge at 550 rad/s.

NASA/CR—2012-217807 119
Figure 4.67. Relative frame streamlines of the Bell 139-tooth helical gear with leading

edge spoiler at 550 rad/s.

NASA/CR—2012-217807 120
Chapter 5

Conclusions

This dissertation has presented the development and application of a CFD method

for gear windage aerodynamics. The goals of this research have been to develop

and validate numerical and modeling approaches for these systems, to develop a

physical understanding of the aerodynamics of gear windage loss, including the

physics of loss mitigation strategies, and to propose and evaluate new approaches

for minimizing loss. Absolute and relative frame CFD simulations, overset gridding, multiphase flow analysis, and sub-layer resolved turbulence modeling were

brought to bear in achieving these goals. Several spur gear geometries were studied for which experimental data are available. Various shrouding configurations

and free-spinning (no shroud) cases were studied. Comparisons were made with

experimental data from the open literature, and data recently obtained in the

NASA Glenn Research Center Gear Windage Test Facility. The results showed

good agreement with experiment. Interrogation of the validative and exploratory

CFD results have led, for the first time, to detailed understanding of the physical mechanisms of gear windage loss, and have led to newly proposed mitigation

strategies whose effectiveness has been explored computationally.

The following conclusions have resulted from this research and have been presented:

1. Using the CFD method to interrogate the physics of these systems, it was

shown, for the first time, that for all configurations studied, the dominant

physical mechanism contributing to windage losses is the pressure field assoNASA/CR—2012-217807 121
ciated with diversion and impingement of the high speed relative flow on the

leading tooth surface.

2. Shrouding mitigates the magnitude of the pressure component of windage

losses, but not its dominance in the loss budget.

3. The interrogation of the validative and exploratory CFD results have shown,

for the first time, that shrouding reduces windage effects. Shrouds conserve

the angular momentum of the flow near the teeth so that the tangential

velocity of the flow that enters the gear teeth is much closer to that of the

gear. This reduces the work required of the gear to accelerate the flow.

4. The CFD studies have suggested a set of possible geometric tooth modifications to reduce windage loss further. The trailing edge ramp/spoiler appears

quite promising and the author hopes this is studied in the NASA Glenn

facility.

5. The CFD results show good agreement with open literature data and with

the NASA Glenn experiments. Experimental data from the NASA Glenn

test facility and and CFD analyses show that axial and radial gear shrouding

are effective in significantly reducing gear windage losses both independently

and when employed together. The NASA Glenn data also shows very similar

trends to the idealized shrouded gear configurations studied computationally.

6. The prediction of viscous losses and turbulence modeling was identified as a

shortcoming since power loss was consistently underpredicted for the viscousdrag-only spinning disk cases.

• Low Reynolds number turbulence modeling (with appropriately sublayer resolved meshes) exhibited improved performance in predicting

viscous losses.

• Due to the dominance of the pressure losses, high Reynolds number

two-equation modeling proved adequate for modeling the Diab spur

gear suite.

NASA/CR—2012-217807 122
• The budgets of viscous and pressure components of the windage torque

suggest that viscous effects will become much more important for gears

with pitch-line speeds approaching twice that investigated here.

• Viscous losses are reduced by shrouding as well, but are a small component of the total loss so this effect is not important.

• Small axial clearances were observed computationally to increase viscous

losses compared to larger shroud clearances, suggesting that this effect

could become important at higher speeds.

7. Overset meshing is and will continue to be a critical enabler in this effort.

The capability has been established and demonstrated here and will become

integral in further studies where the isolated gear assumption will not be

relevant.

8. CFD analysis of the helical gears showed qualitative agreement with previous studies. The flow through the teeth was mostly axial from the leading

edge to the trailing edge in the unshrouded configuration. When the gear

was enclosed, the dominance of the leading edge flow was reduced. The CFD

results also showed that shrouds are effective in significantly reducing gear

windage losses for helical gears. CFD studies also suggested two promising

geometric modifications that could be made to the leading edge of the gear

teeth to provide further reductions in windage loss. The author hopes experimental data will become available in the future to further validate these

results.

This dissertation has focused on the air-only effects of gear windage on isolated

gears. However, gears do not operate in either an air-only environment or alone.

The following items are areas of possible future work:

1. Apply multiphase CFD techniques to explore the physics of the air and oil

atmosphere on gear windage and the effectiveness of shrouding.

2. Explore the effects of meshing gears on windage effects and determine the

computational needs and requirements to provide design guidance.

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3. Evolve design guidance to mitigate windage losses through further CFD studies.

4. Further validate the CFD methodology as more data becomes available from

NASA Glenn.

NASA/CR—2012-217807 124
AppendixA

Gear Windage Estimate Program

A.1 Program Description

The python script GWLestimate.py generates the windage power loss predictions

ueing four different models: the Anderson and Loewenthal model[20], Dawson’s

1984 model[4], Dawson’s updated 1988 model[38], and the fluid flow model of

Diab et al.[6]. The code is currently configured for the NASA Glenn 13-inch gear

presented in Section 4.3.

A.2 GWLestimate.py

#!/usr/bin/env python

# GWLestimate.py

import re, sys, math, os

#Gear Variables

pitchradius = 0.1651 # meters

facewidth = 0.0284988 # meters

facewidth2 = 1000*facewidth # mm

baseradius = 149.6314 # mm

basediameter = 2*baseradius #mm

basediameter2 = basediameter/1000 # meters

NASA/CR—2012-217807 125
module = 6.35 # mm

teeth = 52

#Operating Conditions

muI = 1.72e-5 # kg/(m*s)

mu = muI*1000 # centipoise

rho = 1.225 # kg/m^3

nu = muI/rho # m^2/s

#mu = 0.015

speed = 50 # rad/s

maxspeed = 1200 # rad/s

file=open("glenn13.dat","w")

#Useful Constants

radpstorpm = (30/math.pi) # radians per second to rev. per minute

def computeDawson(rpm,basediamter,facewidth,module):

power = pow(rpm,2.9)*(0.16*pow(basediameter,3.9)+

pow(basediameter,2.9)*pow(facewidth,0.75)*

pow(module,1.15))*1e-20*1000

#print rpm, power

return power

def computeDawsonTwo(rpm,basediamter2, rho, nu):

cprime = 4 # approx. for Glenn 13

power = 1.12e-8*cprime*rho*pow(rpm, 2.85)*

pow(basediameter2,4.7)*pow(nu,0.15)*1000

#print rpm, power

return power

def computeAandL(rpm, pitchradius, mu, facewidth):

#Constants

NASA/CR—2012-217807 126
C1 = 2.82e-7 #S.I.

C2 = 0.019 #S.I.

C3 = 0.028*mu + C2

power = C1*(1+2.3*(facewidth/pitchradius))*pow(rpm,2.8)*

pow(pitchradius,4.6)*pow(C3,.2)*1000

#print speed, power

return power

def computeDiabFF(speed, teeth, pitchradius, facewidth, rho, muI):

# Constants

n1 = 1.293

m1 = 0.5

n2 = 0.074

m2 = 0.2

V = speed * pitchradius

L = 2*math.pi*pitchradius

Re = rho*V*L/muI

Restar = 300000.

Rprime = (muI*Restar)/(rho*speed)

Rstar = pow(Rprime,0.5)

# Face

Cf1 = (2*n1*math.pi/(5-2*m1))*(1/pow(Restar,m1))*

pow((Rstar/pitchradius),5)

Cf2 = (2*n2*math.pi/(5-2*m2))*( (1/(pow(Re,m2))) -

(1/pow(Restar,m2))*pow((Rstar/pitchradius),5) )

# Teeth

sigma = math.pi/teeth

Xa = 0 # profile shift coefficient, assume no shift for now

Cl = (teeth/4.)*(facewidth/pitchradius)*

pow((1.+(2.*(1+Xa)/teeth)),4)*

(1.-math.cos(sigma))*pow((1.+math.cos(sigma)),3)

# Total

Ct = Cf1 + Cf2 + Cl

NASA/CR—2012-217807 127
power = 0.5 * Ct * rho * pow(speed,3) * pow(pitchradius, 5)

return power

while speed < maxspeed+1:

rpm = speed*radpstorpm

power1 = computeAandL(rpm,pitchradius,mu,facewidth)

power2 = computeDawson(rpm,basediameter,facewidth2,module)

power3 = computeDawsonTwo(rpm,basediameter,rho,nu)

power4 = computeDiabFF(speed,teeth,pitchradius,facewidth,rho,muI)

file.write("%f \t %f \t %f \t %f \t %f\n"

%(speed,power1, power2, power3, power4))

speed = speed + 50

#END SCRIPT

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AppendixB

Radial Viscous Power Loss Program

B.1 Program Description

The C++ program vprofiler.cpp generates the radial viscous power loss profiles

from the solutions provided by OVER-REL. The program reads in a Tecplot formated ascii file called peeled-surfaces.dat. This file is generated by OVER-REL.

The file has multiple zones where data is stored at the node points. After successfully reading in the data file, the program performs the following tasks:

1. Compute cell center values of the velocity components u, v, and w, the face

area, and the skin friction coefficient Cf .

2. Compute the velocity components in cylindrical coordinate frame where the

x-axis is the axis of rotation.

3. Compute the local shear stress and velocity magnitudes

4. Compute tangential shear stress.

5. Compute local tangential viscous force.

6. Compute local viscous torque.

7. Sum all local viscous torques for validation against previously computed

OVER-REL results.

NASA/CR—2012-217807 129
8. After validation, output to a text file of area averaged viscous loss versus

radius.

B.2 Vprofiler.cpp

/*

* vprofiler.cpp

*

* Created on: Apr 20, 2010

* Author: Matthew Hill

*

* HOW TO RUN

* ==========

*

* This program is run from the command line in the following

* fashion:

* vprofiler filename speed zones

*

* filename: name of the file to be read

* speed: angular velocity (rad/s) of body

* zones: number of zones that are to be read from file

*

* The program outputs two files:

* vprofile.dat - radial dist. of visc losses along

* one face of the gear

* vprofile_total.dat - radial dist. of visc loss over all faces.

*

*/

#include <iostream>

using std::cerr;

using std::cout;

using std::endl;

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using std::ios;

#include <fstream>

using std::ifstream; //input file stream

using std::ofstream; //output file stream

#include <cstdlib>

using std::exit; //exit function prototype

#include <cmath>

int main(int argc, char *argv[]){

//argv[1] = filename and location;

//argv[2] = angular speed;

if (argc != 4) {

cerr << " Invalid input options" << endl;

}

cout << "PROGRAM START" << endl;

// Open input, make sure it exists before running

//ifstream inClientFile("peeled-surfaces.dat", ios::in);

ifstream inClientFile(argv[1], ios::in);

if ( !inClientFile){

cerr << "File could not be opened" <<endl;

exit(1);

}

inClientFile.ignore(80,’\n’); //ignore the first line

// allocate variables and memory

//int zones = 11; // if diab ogear

int zones = atoi(argv[3]); // = 3; // if disk

cout << "File has " << zones << " zones."<< endl;

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int I,J,points,ncells;

// REFERENCE VALUES

double Dref = .15;

double rho = 1.0;

double Vref = 81.4;

//

double sum_areas = 0.0;

double total_vpower = 0.0;

int stations = 40;

double* vpower_radial = new double[stations];

double* radial_station = new double[stations+1];

double r_min = .04; //

double rmax;

if (zones == 3){ rmax = 0.15;}//diab disk radius in meters

else {rmax= 0.144;} //diab gear 1 radius in meters

double delr = (rmax-r_min)/double(stations);

double rad = r_min;

double omega = atof(argv[2]);// radians per second,

for (int i = 0 ; i < stations; i++) {

radial_station[i] = rad;

rad += delr ;

vpower_radial[i] = 0.0; // initialize

}

radial_station[stations] = rmax;

ofstream outClientFile2("vprofile.dat", ios::out);

outClientFile2.precision(7);

for (int zone = 1; zone <= zones; zone++){

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// read header

cout << "Zone = " << zone << endl;

inClientFile.ignore(17,’,’); //ignore: zone T = " gear ",

inClientFile.ignore(2,’=’); //ignore: ,I=

inClientFile >> I;

inClientFile.ignore(4,’=’); //ignore: ,J=

inClientFile >> J;

inClientFile.ignore(80,’\n’); //ignore rest of line

cout << "I = " << I << endl;

cout << "J = " << J << endl;

points = I*J;

// read data

double* x = new double[points];

double* y = new double[points];

double* z = new double[points];

double* cp= new double[points];

double* u = new double[points];

double* v = new double[points];

double* w = new double[points];

double* cf = new double[points];

for (int i = 0; i < points; i++){

inClientFile >> x[i];

inClientFile >> y[i];

inClientFile >> z[i];

inClientFile >> cp[i];

inClientFile >> u[i];

inClientFile >> v[i];

inClientFile >> w[i];

inClientFile >> cf[i];

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}

//ignore rest of line and move on to next:

inClientFile.ignore(80,’\n’);

// compute #cells

ncells = (I-1)*(J-1);

cout << "ncells = " << ncells << endl;

//compute cell areas, centroids, centroid radius and theta

int cell_n, point_n;

double* area = new double[ncells];

double* radius = new double[ncells];

double* theta = new double[ncells];

double* xc = new double[ncells]; // x location of area centroid

double* yc = new double[ncells]; // y location of area centroid

double* zc = new double[ncells]; // z location of area centroid

double x1,x2,x3,x4;

double y1,y2,y3,y4;

double z1,z2,z3,z4;

double A1, A2, s1, s2, s3, s4, s5, s6;

double area_zone = 0.0; // start at zero

for (int j = 0; j < J-1; j++){

for (int i = 0; i < I-1; i++){

cell_n = i + (j)*(I-1) ;

point_n = i + j*I;

// multiply by Dref to convert to dimensional units

// from non-dimensional:

x1=(x[point_n])*Dref;

y1=(y[point_n])*Dref;

z1=(z[point_n])*Dref;

x2=(x[point_n+1])*Dref;

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y2=(y[point_n+1])*Dref;

z2=(z[point_n+1])*Dref;

x3=(x[point_n+I])*Dref;

y3=(y[point_n+I])*Dref;

z3=(z[point_n+I])*Dref;

x4=(x[point_n+I+1])*Dref;

y4=(y[point_n+I+1])*Dref;

z4=(z[point_n+I+1])*Dref;

// area centroid location

xc[cell_n] = 0.25*(x1+x2+x3+x4);

yc[cell_n] = 0.25*(y1+y2+y3+y4);

zc[cell_n] = 0.25*(z1+z2+z3+z4);

// Area is computed by considering each face as two triangles

// then taking pythagorian sum of the areas of the respective

// projections on the three principal planes

s1 = (x1*y2+x2*y3+x3*y1-x1*y3-x2*y1-x3*y2);

s2 = (y1*z2+y2*z3+y3*z1-y1*z3-y2*z1-y3*z2);

s3 = (z1*x2+z2*x3+z3*x1-z1*x3-z2*x1-z3*x2);

s4 = (x2*y4+x4*y3+x3*y2-x2*y3-x4*y2-x3*y4);

s5 = (y2*z4+y4*z3+y3*z2-y2*z3-y4*z2-y3*z4);

s6 = (z2*x4+z4*x3+z3*x2-z2*x3-z4*x2-z3*x4);

A1 = 0.5*sqrt((s1*s1) + (s2*s2) + (s3*s3));

A2 = 0.5*sqrt((s4*s4) + (s5*s5) + (s6*s6));

area[cell_n] = A1+A2;

radius[cell_n] = sqrt(zc[cell_n]*zc[cell_n]+yc[cell_n]*yc[cell_n]);

theta[cell_n] = atan2(zc[cell_n],yc[cell_n]);

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area_zone += area[cell_n];

}

}

cout << "Area of zone is = " << area_zone << endl;

sum_areas += area_zone;

// compute v_r, v_th, radius, theta

double vr1, vr2, vr3,vr4;

double vth1, vth2, vth3,vth4;

double* vr = new double[ncells];

double* vth = new double[ncells];

for (int j = 0; j < J-1; j++){

for (int i = 0; i < I-1; i++){

cell_n = i + j*(I-1);

point_n = i + j*I;

//v_radial

vr1 = v[point_n]*cos( atan2(z[point_n],y[point_n]))

+ w[point_n]*sin(atan2( z[point_n],y[point_n] ));

vr2 = v[point_n+1]*cos( atan2( z[point_n+1],y[point_n+1] ))

+ w[point_n+1]*sin( atan2( z[point_n+1],y[point_n+1] ));

vr3 = v[point_n+I]*cos( atan2( z[point_n+I],y[point_n+I] ))

+ w[point_n+I]*sin( atan2( z[point_n+I],y[point_n+I] ));

vr4 = v[point_n+I+1]*cos( atan2( z[point_n+I+1],y[point_n+I+1] ))

+ w[point_n+I+1]*sin( atan2( z[point_n+I+1],y[point_n+I+1] ));

// multiply by ref. velocity

vr1 = Vref*vr1;

vr2 = Vref*vr2;

vr3 = Vref*vr3;

vr4 = Vref*vr4;

//v_theta

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vth1 = -v[point_n]*sin( atan2( z[point_n],y[point_n] ))

+ w[point_n]*cos(atan2( z[point_n],y[point_n] ));

vth2 = -v[point_n+1]*sin( atan2( z[point_n+1],y[point_n+1] ))

+ w[point_n+1]*cos( atan2( z[point_n+1],y[point_n+1] ));

vth3 = -v[point_n+I]*sin( atan2( z[point_n+I],y[point_n+I] ))

+ w[point_n+I]*cos( atan2( z[point_n+I],y[point_n+I] ));

vth4 = -v[point_n+I+1]*sin( atan2( z[point_n+I+1],y[point_n+I+1]) )

+ w[point_n+I+1]*cos( atan2( z[point_n+I+1],y[point_n+I+1] ));

// multiply by ref. velocity

vth1 = Vref*vth1;

vth2 = Vref*vth2;

vth3 = Vref*vth3;

vth4 = Vref*vth4;

vr[cell_n]=0.25*(vr1+vr2+vr3+vr4);

vth[cell_n]=0.25*(vth1+vth2+vth3+vth4);

}

}

// compute Tau_mag, Vel_mag, Tau_th

double* Tau_w = new double[ncells];

double* Tau_th = new double[ncells];

double* Cf = new double[ncells];

double cf1, cf2,cf3,cf4;

double Vmag;

for (int j = 0; j < J-1; j++){

for (int i = 0; i < I-1; i++){

cell_n = i+ j*(I-1);

point_n = i + j*I;

cf1 = cf[point_n];

cf2 = cf[point_n+1];

cf3 = cf[point_n+I];

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cf4 = cf[point_n+I+1];

Cf[cell_n]=0.25*(cf1+cf2+cf3+cf4);

Vmag = sqrt(vr[cell_n]*vr[cell_n]+vth[cell_n]*vth[cell_n]);

Tau_w[cell_n] = 0.5*Vref*Vref*rho*Cf[cell_n];

Tau_th[cell_n] = vth[cell_n]*Tau_w[cell_n]/Vmag;

}

}

// viscous forces/torques/power at each cell face

double* F_th = new double[ncells];

double* Torque = new double[ncells];

double* Power = new double[ncells];

for (int i = 0; i < ncells; i++){

F_th[i] = Tau_th[i]*area[i];

Torque[i] = F_th[i]*radius[i];

Power[i] = Torque[i]*omega;

}

// sum of all forces/torques/power of each cell face

double vTorque = 0.0;

double vPower = 0.0;

for (int i = 0; i < ncells; i++){

vTorque += Torque[i];

vPower += Power[i];

}

total_vpower += vPower;

// Get the total radial distribution of visc. forces

for (int i = 0; i < ncells; i++){

for (int j = 0 ; j < stations; j++){

NASA/CR—2012-217807 138
if ( (radius[i] > radial_station[j])

&& ((radius[i] < radial_station[j+1])) ){

vpower_radial[j] += Power[i];

}

}

}

cout << "Viscous torque from zone "<< zone << " is = "

<< vTorque << endl;

cout << "Viscous power from zone " << zone << " is = "

<< vPower << endl;

// get the radial distribution of forces on 1 side of the gear

double vradial, delr, rlower, rupper;

if (zones == 3 && zone == 1){ // diab disk, original grid config

for (int i = 0; i < I-1; i++){

vradial = 0.0;

for (int j = 0; j < J-1; j++){

cell_n = i+ j*(I-1);

point_n = i + j*I;

vradial += Power[cell_n];

}

y1=(y[point_n])*Dref;

z1=(z[point_n])*Dref;

y2=(y[point_n+1])*Dref;

z2=(z[point_n+1])*Dref;

rlower = sqrt(y1*y1+z1*z1);

rupper = sqrt(y2*y2+z2*z2);

delr= rupper-rlower ;

outClientFile2 << radius[cell_n] << " "

<< fabs(vradial)/delr << endl;

}

}

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//inviscid wall w/axial hole diab gear grid

else if (zones == 12 && zone == 10){

for (int i = 0; i < I-1; i++){

vradial = 0.0;

for (int j = 0; j < J-1; j++){

cell_n = i+ j*(I-1) ;

point_n = i + j*I;

vradial += Power[cell_n];

}

y1=(y[point_n])*Dref;

z1=(z[point_n])*Dref;

y2=(y[point_n+1])*Dref;

z2=(z[point_n+1])*Dref;

rlower = sqrt(y1*y1+z1*z1);

rupper = sqrt(y2*y2+z2*z2);

delr= rupper-rlower ;

outClientFile2 << radius[cell_n] << " "

<< fabs(vradial)/delr << endl;

}

}

//inviscid wall w/axial hole diab disk grid

else if (zones == 6 && (zone == 3 || zone == 6)){

for (int i = 0; i < I-1; i++){

vradial = 0.0;

for (int j = 0; j < J-1; j++){

cell_n = i+ j*(I-1) ;

point_n = i + j*I;

vradial += Power[cell_n];

}

y1=(y[point_n])*Dref;

z1=(z[point_n])*Dref;

y2=(y[point_n+1])*Dref;

z2=(z[point_n+1])*Dref;

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//this case is unique since i index start

// at outer radius, not inner

rlower = sqrt(y1*y1+z1*z1);

rupper = sqrt(y2*y2+z2*z2);

delr= rlower - rupper; //-rlower ;

outClientFile2 << radius[cell_n] << " "

<< fabs(vradial)/delr << endl;

}

}

else if (zones == 11 && zone == 5){ //ogear diab grid

for (int i = 0; i < I-1; i++){

vradial = 0.0;

for (int j = 0; j < J-1; j++){

cell_n = i+ j*(I-1) ;

point_n = i + j*I;

vradial += Power[cell_n];

}

y1=(y[point_n])*Dref;

z1=(z[point_n])*Dref;

y2=(y[point_n+1])*Dref;

z2=(z[point_n+1])*Dref;

rlower = sqrt(y1*y1+z1*z1);

rupper = sqrt(y2*y2+z2*z2);

delr= rupper-rlower ;

outClientFile2 << radius[cell_n] << " "

<< fabs(vradial)/delr << endl;

}

}

// diab gear w/ baffles

else if (zones == 13 && (zone == 10 || zone == 12)){

//zone number is really 14 in file but we

// are ignoring that face completely when the program gets called

for (int i = 0; i < I-1; i++){

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vradial = 0.0;

for (int j = 0; j < J-1; j++){

cell_n = i+ j*(I-1);

point_n = i + j*I;

vradial += Power[cell_n];

}

y1=(y[point_n])*Dref;

z1=(z[point_n])*Dref;

y2=(y[point_n+1])*Dref;

z2=(z[point_n+1])*Dref;

rlower = sqrt(y1*y1+z1*z1);

rupper = sqrt(y2*y2+z2*z2);

delr= rupper-rlower ;

outClientFile2 << radius[cell_n] << " " <<

fabs(vradial)/delr << endl;

}

}

// De-allocate arrays

delete[] x;

delete[] y;

delete[] z;

delete[] cp;

delete[] u;

delete[] v;

delete[] w;

delete[] cf;

delete[] area;

delete[] radius;

delete[] theta;

delete[] xc;

delete[] yc;

delete[] zc;

NASA/CR—2012-217807 142
delete[] vr;

delete[] vth;

delete[] Tau_w;;

delete[] Tau_th;

delete[] Cf;

delete[] F_th;

delete[] Torque;

delete[] Power;

} // end zone loop

inClientFile.close();

cout << "Total area of all grid zones = " << sum_areas << endl;

cout << "Total power of all grid zones = " << total_vpower << endl;

cout << "Total power of whole gear = " <<

72.0*total_vpower << "W" <<endl;

// output vpower radial profile

ofstream outClientFile("vprofile_total.dat", ios::out);

outClientFile.precision(7);

for (int i = 0 ; i < stations ; i++){

outClientFile << radial_station[i] << " "

<< fabs(vpower_radial[i]) << endl;

}

outClientFile.close();

outClientFile2.close();

delete[] vpower_radial;

delete[] radial_station;

cout << "PROGRAM END" << endl;

return 0;

}

NASA/CR—2012-217807 143
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NASA/CR—2012-217807 148
Vita

Matthew J. Hill

Matthew John Hill was born in Upper Darby, PA on December 20, 1981. In

1989, he moved to Hong Kong and attended the Hong Kong International School

until 1992. In that year, he moved backed to Pennsylvania and attended St.

Pius X and then Devon Preparatory School, where he graduated in 2000. After

graduation, he moved to New York City to pursue his Bachelor of Engineering in

Mechanical Engineering at The Cooper Union for the Advancement of Science and

Art. After graduating in 2004, he returned to Pennsylvania once again to continue

his studies at The Pennsylvania State University. Working under Dr. Lyle Long,

he earned his M.S. in Aerospace Engineering in 2006. It was at Penn State where

he entered the world of in computational fluid dynamics and rotorcraft engineering

working on coupled fluid-structure simulations of helicopter rotors. Afterwards, he

continued on at Penn State as a Ph.D. student in Aerospace Engineering working

for Dr. Rob Kunz studying gear windage. Following graduation, Matthew plans

to work somewhere...

NASA/CR—2012-217807 149
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4. TITLE AND SUBTITLE

A Computational Investigation of Gear Windage 

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Hill, Matthew, J.; Kunz, Robert, F. 

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This report was submitted as a dissertation in Aerospace Engineering in partial fulfillment of the requirements for the degree of Doctor of 

Philosophy to the graduate faculty of the Pennsylvania State University, Philadelphia, Pennsylvania, December 2010. 

14. ABSTRACT

A CFD method has been developed for application to gear windage aerodynamics. The goals of this research are to develop and validate 

numerical and modeling approaches for these systems, to develop physical understanding of the aerodynamics of gear windage loss, 

including the physics of loss mitigation strategies, and to propose and evaluate new approaches for minimizing loss. Absolute and relative 

frame CFD simulation, overset gridding, multiphase flow analysis, and sub-layer resolved turbulence modeling were brought to bear in 

achieving these goals. Several spur gear geometries were studied for which experimental data are available. Various shrouding 

configurations and free-spinning (no shroud) cases were studied. Comparisons are made with experimental data from the open literature, and 

data recently obtained in the NASA Glenn Research Center Gear Windage Test Facility. The results show good agreement with experiment. 

Interrogation of the validative and exploratory CFD results have led, for the first time, to a detailed understanding of the physical 

mechanisms of gear windage loss, and have led to newly proposed mitigation strategies whose effectiveness is computationally explored. 

15. SUBJECT TERMS

Gears; Drive systems; Gear windage; Transmission; Computational fluid dynamics 

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