CFD simulation of dip-lubricated single-stage gearboxes through …liu.diva-portal.org/smash/get/diva2:1268007/FULLTEXT01.pdf · 2018-12-04 · Scania CV AB, S odert alje Examiner:

CFD simulation of dip-lubricatedsingle-stage gearboxes throughcoupling of multiphase flow andmultiple body dynamics, aninitial investigation

Nasir Imtiaz

Linkoping University

Institutionen for ekonomisk och industriell utveckling

Div. of App. Thermodynamics and Fluid Mechanics

Examensarbete 2018 | LIU-IEI-TEK-A–18/03285-SE

Linkopings universitet

Institutionen for ekonomisk och industriell utveckling

Amnesomradet Mekanisk varmeteori och stromningslara

Examensarbete 2018 | LIU-IEI-TEK-A–18/03285–SE

CFD simulation of dip-lubricatedsingle-stage gearboxes throughcoupling of multiphase flow andmultiple body dynamics, aninitial investigation

Nasir Imtiaz

Academic supervisor: Hossein Nadali Najafabadi

IEI, Linkopings Universitet

Industrial supervisor: Samira Nikkar

Scania CV AB, Sodertalje

Examiner: Matts Karlsson

IEI, Linkopings Universitet

Linkopings universitet

SE-581 83 Linkoping, Sverige

013-28 10 00, www.liu.se

AbstractTransmissions are an essential part of a vehicle powertrain. An optimally designedpowertrain can result in energy savings, reduced environmental impact and increasedcomfort and reliability. Along with other components of the powertrain, efficiencyis also a major concern in the design of transmissions. The churning power lossesassociated with the motion of gears through the oil represent a significant portion ofthe total power losses in a transmission and therefore need to be estimated. A lack ofreliable empirical models for the prediction of these losses has led to the emergenceof CFD (Computational Fluid Dynamics) as a means to (i) predict these losses and(ii) promote a deeper understanding of the physical phenomena responsible for theselosses in order to improve existing models.

The commercial CFD solver STAR-CCM+ is used to investigate the oil distributionand the churning power losses inside two gearbox configurations namely an FZG(Technical Institute for the Study of Gears and Drive Mechanisms) gearbox anda planetary gearbox. A comparison of two motion handling techniques in STAR-CCM+ namely MRF (Moving Reference Frame) and RBM (Rigid Body Motion)models is made in terms of the accuracy of results and the computational require-ments using the FZG gearbox. A sensitivity analysis on how the size of gap betweenthe meshing gear teeth affects the flow and the computational requirements is alsodone using the FZG gearbox. Different modelling alternatives are investigated forthe planetary gearbox and the best choices have been determined. The numericalsimulations are solved in an unsteady framework where the VOF (Volume Of Fluid)multiphase model is used to track the interface between the immiscible phases. Theoverset meshing technique has been used to reconfigure the mesh at each time step.

The results from the CFD simulations are presented and discussed in terms of themodelling choices made and their effect on the accuracy of the results. The MRFmethod is a cheaper alternative compared to the RBM model however, the formermodel does not accurately simulate the transient start-up and instead provides justa regime solution of the unsteady problem. As expected, the accuracy of the resultssuffers from having a large gap between the meshing gear teeth. The use of com-pressible ideal gas model for the air phase with a pressure boundary condition givesthe optimum performance for the planetary gearbox. The outcomes can be used toeffectively study transmission flows using CFD and thereby improve the design offuture transmissions for improved efficiency.

Keywords: transmissions, churning losses, CFD, STAR-CCM+, FZG gearbox, plan-etary gearbox, RBM, MRF, unsteady framework, VOF, immiscible, overset meshingtechnique

i

PrefaceThis thesis is a part of the master’s degree at Linkoping University and has beendone in collaboration with NTAC (Transmission Development, Analysis and Testinggroup) at Scania CV AB during the spring of 2018.

AcknowledgementsI would like to express my sincere gratitude to my supervisors, Samira Nikkar (ScaniaCV AB) and Hossein Nadali Najafabadi (Linkoping University) for their discussionsand guidance throughout the thesis work as well as their help with the administrativeaspects of the thesis work. I also want to thank Hakan Settersjo (Scania CV AB)for his inputs during the presentations and his help in running simulations for theplanetary gearbox on Scania clusters, Paolo Fornaseri (Scania CV AB) for his inputsregarding the planetary model and Christian Windisch (Siemens PLM Software) forhis help in debugging the planetary simulations. Finally, a sincere thanks also goesto my examiner at Linkoping University, Matts Karlsson for his valuable inputs.

iii

Nomenclature

Abbreviations and Acronyms

Abbreviation Description

CFD Computational Fluid DynamicsFZG Technical Institute for the Study of Gears and Drive MechanismsVOF Volume Of FluidRBM Rigid Body MotionMRF Moving Reference FrameNTAC Transmission Development, Analysis and Testing groupPIV Particle Image VelocimetryPLM Product Lifecycle ManagementRANS Reynolds-Averaged Navier-StokesSST Shear Stress TransportE-E Eulerian-EulerianE-L Eulerian-LagrangianHRIC High-Resolution Interface CapturingCAD Computer-Aided DesignCAE Computer-Aided EngineeringLSQ Least Squares

Dimensionless numbers

Abbreviation Description

Re Reynolds numberCFL Courant-Friedrichs-Levyy+ Normalized wall distance

v

Symbols

Symbol Description

ρ DensityV Velocity vectorp Pressurefi Body force in ith directionU Velocity magnitudeL Characteristic lengthµ Dynamic viscosityui Time-averaged velocity component in ith directionui′ Oscillating velocity component in ith directionµt Turbulent viscosityk Turbulence kinetic energyτij Viscous stress tensorδij Kronecker deltaω Specific dissipation rate

V Mean velocityy Cell centroid distance from wallu∗ Wall friction velocityν Kinematic viscosityαi Volume fraction of phase iVi Volume of phase iVcell Volume of cellρi Density of phase iρcell Density of cellµi Viscosity of phase iµcell Viscosity of cellmqp Mass transfer from phase q to phase pSαq Source or sink of phase qucell Velocity in the cell∆t Time step∆x Local cell sizeφPi Interpolated property value at point Piαωk

Interpolation weight factorr Position vectorx Position in laboratory framexo Position of rotation axisvlab Velocity in laboratory framevr Velocity in moving frameωt Angular velocity of moving framevt Linear velocity of moving frame

vi

Contents

1 Introduction 1

1.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Literature study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Experimental and analytical studies . . . . . . . . . . . . . . 3

1.2.2 CFD studies . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Theory 6

2.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Turbulent flow and turbulence modelling . . . . . . . . . . . . . . . . 6

2.2.1 RANS equations . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.2 k-ω SST model . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Near wall modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 Multiphase flow modelling . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4.1 VOF model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5 Discretization scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5.1 Discretization scheme for time . . . . . . . . . . . . . . . . . 10

2.5.2 Discretization scheme for convective terms . . . . . . . . . . . 11

2.6 Overset meshing technique . . . . . . . . . . . . . . . . . . . . . . . . 11

2.7 Rotational motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.7.1 MRF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.7.2 RBM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.8 Modelling flows inside transmissions . . . . . . . . . . . . . . . . . . 13

2.8.1 FZG gearbox . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.8.2 Planetary gearbox . . . . . . . . . . . . . . . . . . . . . . . . 14

2.8.3 Churning losses in transmissions . . . . . . . . . . . . . . . . 15

3 Method 18

3.1 Softwares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.1.1 ANSA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.1.2 STAR-CCM+ . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 FZG gearbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.1 Geometry preparation . . . . . . . . . . . . . . . . . . . . . . 19

3.2.2 Setting up overset regions . . . . . . . . . . . . . . . . . . . . 20

3.2.3 Surface and volume meshing . . . . . . . . . . . . . . . . . . 20

3.2.4 Mesh sensitivity analysis . . . . . . . . . . . . . . . . . . . . . 22

3.2.5 Setting up motion . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.6 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 Planetary gearbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3.1 Geometry preparation . . . . . . . . . . . . . . . . . . . . . . 23

3.3.2 Setting up overset regions . . . . . . . . . . . . . . . . . . . . 24

3.3.3 Surface and volume meshing . . . . . . . . . . . . . . . . . . 24

3.3.4 Setting up motion . . . . . . . . . . . . . . . . . . . . . . . . 26

3.4 Solver settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.5 Solution assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.6 Additional investigations . . . . . . . . . . . . . . . . . . . . . . . . . 27

4 Results and Discussions 314.1 FZG gearbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2 Planetary gearbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5 Conclusions 46

6 Future work 48

Bibliography 51

Appendices 53

A FZG gear properties 53

B Solution assessment 54B.1 FZG gearbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54B.2 Planetary gearbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

C Single planet case 56

ix

List of Figures1 A schematic that illustrates the actual contact (No Gap) with the

simplified contact (Gap) where the gap has been obtained by scalingthe pinion (on the right) uniformly in three dimensions to 91 percentof its original size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 The laboratory and rotating frames of reference in MRF method . . 13

3 FZG back-to-back gear test rig . . . . . . . . . . . . . . . . . . . . . 14

4 Planetary gearbox where the main shaft has been removed for easyviewing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 Classification of load-independent power losses in a transmission . . 15

6 Schematic overview of the workflow for the two configurations . . . . 19

7 Symmetric geometry of FZG gearbox . . . . . . . . . . . . . . . . . . 19

8 Volume mesh at the symmetry plane in FZG gearbox . . . . . . . . 21

9 Boundary conditions in FZG gearbox . . . . . . . . . . . . . . . . . . 23

10 Prepared geometry of planetary gearbox . . . . . . . . . . . . . . . . 24

11 Volume mesh at the symmetry plane in planetary gearbox . . . . . . 25

12 Reduced geometry of planetary gearbox . . . . . . . . . . . . . . . . 29

13 Volume mesh at the symmetry plane in reduced planetary gearbox . 29

14 Development of volume fraction of oil at the symmetry plane usingthe MRF and the RBM models in STAR-CCM+ (All contours exceptat 2.25 revolutions of gear are instantaneous whereas the ones for 2.25revolutions represent the mean volume fraction). Volume fraction of 0indicates only air is present inside the discretized cell and 1 indicatesoil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

15 Development of velocity field (m/s) of oil flow at the symmetry planewith the MRF and RBM models (All contours except at 2.25 revolu-tions of gear are instantaneous whereas the ones for 2.25 revolutionsrepresent the mean velocity). The difference between the two framesis due to the use of a volume fraction driven part. . . . . . . . . . . . 33

16 The mean velocity vectors (m/s) of oil flow at the symmetry planefrom the MRF and RBM models . . . . . . . . . . . . . . . . . . . . 34

17 The mean velocity vectors (m/s) at three planes in the transversedirection for the MRF and RBM models with the viewing directionfrom the left. The planes a, b and c represent the gear centerline,housing centerline and the pinion centerline locations respectively. . 35

18 Development of volume fraction of oil at the symmetry plane for threedifferent gap sizes (All contours except at 2.25 revolutions of gearare instantaneous whereas the ones for 2.25 revolutions represent themean volume fraction) . . . . . . . . . . . . . . . . . . . . . . . . . . 37

19 Development of velocity field (m/s) at the symmetry plane for threedifferent gap sizes (All contours except at 2.25 revolutions of gearare instantaneous whereas the ones for 2.25 revolutions represent themean velocity) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

20 Mean velocity plots at the gear and pinion centerlines shown as a andb respectively for the cases involving FZG gearbox after 2.25 revolutions 39

x

21 Development of volume fraction of oil and velocity field (m/s) at thesymmetry plane for the reduced planetary gearbox . . . . . . . . . . 42

22 Volume fraction and velocity (m/s) contours at the middle plane forthe planetary gearbox at 8 milliseconds . . . . . . . . . . . . . . . . 44

23 Monitor of total mass in the domain . . . . . . . . . . . . . . . . . . 5424 Monitor of CCN in the solution domain . . . . . . . . . . . . . . . . 5425 Monitor of total mass in the domain . . . . . . . . . . . . . . . . . . 5526 Monitor of CCN in the solution domain . . . . . . . . . . . . . . . . 5527 Mesh at the symmetry plane for the single planet case . . . . . . . . 5628 Development of volume fraction of oil and velocity (m/s) with the

revolutions of the sun at the symmetry plane of the single planet case 57

xi

List of Tables1 Size of gap for different scales of pinion . . . . . . . . . . . . . . . . 202 Surface remesher properties for FZG gearbox . . . . . . . . . . . . . 203 Surface controls properties for FZG gearbox . . . . . . . . . . . . . . 214 Volume mesh properties for FZG gearbox . . . . . . . . . . . . . . . 215 Churning torque and simulation time for two different meshes . . . . 226 Motion setup for FZG gearbox . . . . . . . . . . . . . . . . . . . . . 227 Surface remesher properties for planetary gearbox . . . . . . . . . . 248 Surface controls properties for planetary gearbox . . . . . . . . . . . 259 Volume mesh properties for planetary gearbox . . . . . . . . . . . . . 2510 Motion setup for planetary gearbox . . . . . . . . . . . . . . . . . . . 2611 Churning Torque measurements for the various cases involving FZG

gearbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3912 Cost analysis for the various cases involving FZG gearbox using 12

processors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4013 Evaluating the choice of MRF and RBM . . . . . . . . . . . . . . . . 4114 Churning torque measurements for the cases involving the planetary

gearbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4315 Cost analysis for the various cases involving the planetary gearbox

using 12 processors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4416 Data for FZG gears type C . . . . . . . . . . . . . . . . . . . . . . . 53

xii

1 IntroductionTransmission systems are used in vehicles to convert the power from internal com-bustion engine into an optimum drive force and engine speed, and to transport thisdrive force to the differential, through a drive shaft, which drives the wheels. Amongthe key mechanical components of a transmission are the gears, bearings and shaftswhich can become damaged or degraded due to friction during contacts. To main-tain the structural integrity of the system, these contact surfaces must be lubricatedand cooled through enough supply of oil throughout their operation.

The most common type of lubrication system used in an automotive transmissionis the dip-lubricated system where the gears are partially immersed in oil and asthe gears rotate they transport the oil to the meshing region. The cooling will beenhanced for a large amount of oil, however, this has a couple of negative aspects.Firstly, the damping effect of the oil increases which makes the torque transfer lessefficient and secondly, there is an increase in the drag on the moving/rotating com-ponents as they move through the oil [1]. The losses associated with the drag torquedue to rotation of gears submerged in oil are known as churning power losses andhave a negative impact on the torque transfer as well as the fuel efficiency partic-ularly in high-speed gearboxes and dip-lubricated transmission systems with highimmersion depths [2].

Traditionally, measurements on test rigs have been used as the principal meansfor prediction of churning power losses in transmissions. However, these measure-ments cannot be made at the design stages when no tests can be conducted. Thisgives rise to the need for appropriate models to predict these power losses in or-der to reduce them starting from the earliest stages of the design phase. Empiricalapproaches offer another alternative for investigation of these losses however theseare often limited to certain constraints and operating conditions, and considerableuncertainties exist when applying these empirical equations. Moreover, empiricalequations do not provide any information about the oil distribution [2].

CFD (Computational Fluid Dynamics) offers a very flexible means for investiga-tion of churning losses and also provides comprehensive information about the oildistribution. Various advantages when using CFD for prediction of oil flow in theinterior of transmissions in the industry include for example, shortening the develop-ment period, reducing the expenses for prototypes, and most importantly bypassinglimitations during actual tests for example, due to adherence of oil to the internalwalls of the housings [3].

1.1 Purpose

Different modelling approaches have been used in the past to simulate the oil flowinside simplified transmissions mostly consisting of a single gear or a gear pair [4, 2].Modelling gear motion is challenging in CFD simulations of gear systems. Simplesystems can be studied with a more accurate transient approach using the rigid mo-

1

tion of the mesh vertices to simulate the gear motion and the approach is known asRBM (Rigid Body Motion). However, for more complex systems consisting of sev-eral motions, this approach leads to a very high computational cost and therefore,a steady MRF (Moving Reference Frame) approach, where applicable, offers a goodcompromise between accuracy and computational cost. In this thesis, a thoroughcomparison between RBM and MRF is made using a single-stage FZG (TechnicalInstitute for the Study of Gears and Drive Mechanisms) gearbox.

A key aspect which determines the computational cost of gear simulations usingthe RBM approach is the size of the gap created between the two meshing gearsusually obtained by scaling the smaller of the two meshing gears. This is done tosufficiently resolve the flow in the meshing region at a reasonable computational cost.Figure 1 illustrates the actual contact and the gap contact. A large gap can reducethe computational costs for these simulations, however, it can lead to a less accurateprediction of critical parameters such as the churning losses. Therefore, a sensitivitystudy of the influence of gap size on the churning losses, and computational time isalso made using the FZG gearbox.

Figure 1: A schematic that illustrates the actual contact (No Gap) with the simplifiedcontact (Gap) where the gap has been obtained by scaling the pinion (on the right)uniformly in three dimensions to 91 percent of its original size.

In order to ascertain the applicability of CFD to industrial gearbox configurations,the flow inside a planetary gearbox from a Scania gearbox GRS905 is also simulatedtaking into account the observations from the investigations using the FZG gearbox.Different modelling choices are examined and the optimised simulation is ran whereimportant flow features including the churning losses have been computed.

1.2 Literature study

A summary of some of the literature searched during the thesis development is givenin this section pointing out the major aspects from each work that have been usedin the various stages of this thesis work. Three different techniques have been usedpreviously for studying flows inside transmissions and the churning power losses

2

namely test rig measurements, empirical equations and CFD simulations. Here, abrief overview of the previous work in each of these areas is given.

1.2.1 Experimental and analytical studies

Hartono et. al [1] used 2D-2C PIV (Particle Image Velocimetry) to study the flowfield inside an FZG gearbox for a variety of operating conditions including differ-ent pitch line velocities and oil levels. The experimental data on the velocity fieldin the gearbox has been provided as a means for validation of numerical calculations.

Kolekar et. al [5] investigated the effect of fluid properties on the churning losses fora simple test rig consisting of a spur gear rotating in a cylindrical housing through avariety of aqueous glycerol solutions and oils. The results demonstrated a significantinfluence of the oil disposition within the housing on the measured churning losseswhere the oil disposition is governed by effects of gravity, inertial forces, surfacetension and windage.

Changenet and Velex [6] proposed new formulas for evaluating churning losses forone pinion characteristic of automotive transmission geometry based on the resultsof a versatile test rig and on dimensional analysis. Moreover, the case of a pinion-gear pair in mesh was also investigated and it was concluded that, depending on thesense of rotation of the pair, the superposition of the individual losses of the pinionand of the gear leads to erroneous figures.

Boni et. al [7] experimentally investigated the churning power losses generatedin a dip-lubricated planetary gearbox under loaded conditions for different values ofoperating parameters i.e. rotational speed, oil sump level and temperature. It wasobserved that with the increase in speed, the oil sump in the planetary gearbox dis-appeared and the oil formed a ring towards the outer circumference of the housingdue to centrifugal force. Other important observations were that the component ofchurning losses associated with the planet carrier was not very significant and thatthe sun gear contributed to the churning losses only if the oil level is such that itreaches the region where the teeth of the sun and planet gears are meshed. Theauthors argue that the applicability of empirical equations, derived as a result ofchurning loss studies for single gear or gear pair investigations, to planetary gearsystems is limited because of the strong predominance of the centrifugal effects inplanetary gear sets as opposed to the free surface flows inside cylindrical gear setswhere the gravity forces have a larger influence.

1.2.2 CFD studies

The continuous advantages in computational capacity have brought CFD methodsinto the spotlight as a new way of investigating churning power losses [2]. Concli et.al [4] investigated the influence of some operating and geometrical parameters on thechurning power losses of a single gear by adopting an open-source code, OpenFOAM.The geometry was considered symmetric and the sliding mesh approach was usedwith the internal mesh rotated by a prescribed angle every time step. Moreover, the

3

MRF method was suggested as a useful means to reduce the computational effort ascompared to the sliding mesh approach. The results showed a significant influenceof gear parameters such as the tip diameter and the lubricant level on the churninglosses.

Li et. al [8] applied the VOF multiphase model and dynamic meshing to simu-late the fluid flow for a pair of mating gears using a two-dimensional solver. Theobtained results from the simulated flow were compared with the experimental vi-sualization results and found to be in good agreement.

Liu et. al [2] investigated the oil distribution and churning power losses in a single-stage FZG gearbox by using the remeshing method for recomputing the mesh in themeshing zone between the pinion and the gear at each time step. The pinion andthe gear were both scaled to 99 percent of their actual size and it was expected thatthe results are not significantly influenced by this simplification. The results showeda good agreement both for the oil distribution and the churning loss torque withthe high-speed camera recordings and experimental measurements of the loss torque.

Concli and Gorla [9] identified the need for development of effective models forthe prediction of load-independent power losses and proposed a methodology forprediction of load-independent power losses in planetary gearboxes. The methodinvolves the usage of a customized domain and choice of mesh motion as well as thesolver setup for the evaluation of each component of the load-independent losses ofthe planetary gearbox namely the churning loss due to carrier, churning losses dueto gears and oil squeezing losses. The results showed that the most losses for boththe load-independent and load-dependent components came from the gears.

Cho et. al [10] used the overset meshing technique for the realistic simulation ofa planetary gearbox and a transaxle brake. Different oil levels have been inves-tigated for the planetary gearbox and the effect on the churning losses has beendocumented. The results showed an increase in churning losses with an increase inthe oil level.

1.3 Limitations

The actual contact between the gears is not considered here instead, a gap conditionis created due to the restriction of having at least 4-6 cells between the teeth andthe overset interface in the meshing region when using the overset mesh interfacein STAR-CCM+. This gap has been obtained by scaling the pinion and the planetgears in the FZG and planetary gearboxes respectively.

All the analysis for the two gearboxes has been done for oil level at the center-line of the housing and a single combination of rotational speeds.

The amount of analysis performed is limited by the computational resources avail-able at Linkoping University in the frame of Nasheim computers and indirect accessto clusters at Scania CV AB within the fixed amount of time.

4

2 Theory

2.1 Governing equations

The governing equations of the flow are the laws of conservation of mass (equation1) and momentum (equations 2 to 4). These laws can be described in differential orintegral form by applying the conservation laws to an infinitesimal fluid element ora finite region of the flow respectively. Here, the differential form of these governingequations is given. The fluid element is large enough to contain a huge number ofmolecules so that it can be viewed as a continuous medium.

The mass conservation for a compressible flow results from the fact that the rate ofincrease of mass inside a fluid element is equal to the net rate of flow of mass intothe fluid element across its faces

∂ρ

∂t+∇.(ρV ) = 0 (1)

where ρ is the density and V is the velocity vector. The first term represents therate of change of mass and the second term represents the net rate of flow of massout of the volume. The momentum equations result from equating the change inmomentum to the sum of forces acting on the fluid element

∂(ρu)

∂t+∇.(ρuV ) =

∂(−p+ τxx)

∂x+∂(τyx)

∂y+∂(τzx)

∂z+ ρfx (2)

∂(ρv)

∂t+∇.(ρvV ) =

∂(τxy)

∂x+∂(−p+ τyy)

∂y+∂(τzy)

∂z+ ρfy (3)

∂(ρw)

∂t+∇.(ρwV ) =

∂(τxz)

∂x+∂(τyz)

∂y+∂(−p+ τzz)

∂z+ ρfz (4)

where ρfx, ρfy and ρfz are the body forces in three directions. The terms on theleft hand side represent the rate of change in momentum and the net rate of flow ofmomentum out of the the volume, respectively. The terms on the right hand siderepresent the pressure and the shear stresses and the rate of change of momentumdue to sources [11].

In STAR-CCM+, the finite volume method is used to discretize these equationsin order to solve them numerically.

2.2 Turbulent flow and turbulence modelling

The Reynolds number is a measure of the relative importance of inertia forces andviscous forces. It is expressed as

Re =ρUL

µ(5)

where U is the velocity, L is the characteristic length and µ is the dynamic viscosity.Different flow regimes are characterized based on a so-called critical value of the

6

Reynolds number. Below this critical value, the flow is smooth and steady and theregime is called laminar flow. At values of Reynolds number above the same criticalvalue, the flow behavior changes drastically and the regime is called turbulent flow.Turbulent flows are characterized by three-dimensional randomness and apparentchaotic behavior. Most industrially relevant flows are turbulent in nature however, inorder to resolve every single whirl inside a turbulent flow a very high computationaltime is required and also most often the time-averaged behavior of the flow is ofmost significance. The governing equations of the steady mean flow are called theRANS (Reynolds-Averaged Navier-Stokes) equations.

2.2.1 RANS equations

In the current thesis work, RANS approach has been adopted to model the tur-bulence effect in the flow. The RANS equations are obtained by introducing theso-called Reynolds decomposition where the flow variables are decomposed into atime-averaged or steady and an oscillating component. The Reynolds decompositionis introduced for the velocity and the pressure according to

u(t) = u+ u′(t)v(t) = v + v′(t)

w(t) = w + w′(t)p(t) = p+ p′(t)

(6)

where¯and ′ denote the time-averaged and oscillating components respectively. TheRANS equations are then obtained by substituting these relations into the momen-tum equations and taking the time-average. These read

∂(ρu)

∂t+∇.(ρuV ) =

∂(−p+ τxx − ρu′2)∂x

+∂(τyx − ρu′v′)

∂y+

∂(τzx − ρu′w′)∂z

+ ρfx

(7)

∂(ρv)

∂t+∇.(ρvV ) =

∂(τyx − ρu′v′)∂x

+∂(−p+ τyy − ρv′2)

∂y+

∂(τyz − ρv′w′)∂z

+ ρfy

(8)

∂(ρw)

∂t+∇.(ρwV ) =

∂(τzx − ρu′w′)∂x

+∂(τzy − ρv′w′)

∂y+

∂(−p+ τzz − ρw′2)∂z

+ ρfz

(9)

where the new stress terms in the RANS equations ρu′iv′j are called the Reynoldsstresses. To close the equations, these terms must be estimated by additional tur-bulence models.

7

2.2.2 k-ω SST model

The Boussinesq approximation is the assumption that the Reynolds stresses areproportional to the mean velocity gradients according to

τij = −ρu′iv′j = µt

(∂ui∂xj

+∂uj∂xi

)− 2

3ρkδij (10)

where τij is the viscous stress tensor, µt is the turbulent viscosity, k is the turbulencekinetic energy and δij is the Kronecker delta. It follows that the RANS equationscan be written in closed form as long as the turbulent viscosity is determined. Dif-ferent turbulent viscosity models are available in STAR-CCM+ such as the k-ε, k-ω,Spalart-Allmaras models and the Reynolds stress transport models. For the currentthesis work the k-ω SST (Shear Stress Transport) turbulence model has been usedbecause of its superior performance in the near wall region.

The k-ω SST is a hybrid model which combines the k-ω formulation in the viscoussublayer and switches to a k-ε behavior in free stream, thus avoiding the sensitivityto inlet free-stream turbulence and extra damping functions. The turbulent viscosityis given as

νt = ρk

ω(11)

where ω is called the specific dissipation rate. The transport equations for k and ωread

∂(ρk)

∂t+∇.(ρkV ) = ∇.[(µ+ σkµt)∇k] + Pk − ρβ∗fβ∗(ωk − ωoko) + Sk (12)

∂(ρω)

∂t+∇.(ρωV ) = ∇.[(µ+ σωµt)∇ω] + Pω − ρβfβ(ω2 − ω2

o) + Sω (13)

where V is the mean velocity, σk, σω, Cε1 and Cε2 are model coefficients, fβ∗ and fβare free-shear and vortex-shedding modification factors respectively, Pk and Pω areproduction terms, Sk and Sω are user-specified source terms and ko and ωo are theambient turbulence values [12].

2.3 Near wall modelling

Turbulent flows are characterized by a more complicated boundary layer as comparedto the free shear flows which needs to be considered more specifically. It can bedivided into inner and outer regions. The inner region is further classified into threedifferent sub-layers namely, a viscous sub-layer (0 ≤ y+ ≤ 5), a buffer sub-layer(5 ≤ y+ ≤ 30) and a fully turbulent sub-layer (30 ≤ y+ ≤ 400). The dimensionlessvariable y+ is defined as

y+ =yu∗

ν(14)

where u∗ =√

τwρ is called the wall friction velocity, y is the distance of the centroid

of a particular cell from the wall and ν is the local kinematic viscosity.

8

The wall modeling strategy employed depends on the maximum value of the y+.Low y+ wall treatment is employed for y+ < 5 preferably if y+ = 1 or less. Thehigh y+ wall treatment is employed for y+ > 30. The All y+ wall treatment is acombination of the two above. In this case, the turbulence quantities such as k areevaluated by weighing the solutions coming from both the previous models accordingto

k = gklow + (1− g)khigh (15)

where g is based on the local Reynolds number for the cell. STAR-CCM+ uses theAll y+ wall treatment in combination with the k-ω SST model.

2.4 Multiphase flow modelling

Just as the most important flows for industrial application are turbulent, the mostcommon flows are multiphase flows. Some examples of multiphase flows includerainfall, air pollution, boiling, flotation, liquid-liquid extraction and spray drying[13]. The flow inside a transmission is a multiphase flow where two phases, oil andair, are present.

The methods to model multiphase flows can be classified into two different cat-egories, the E-L (Eulerian-Lagrangian) method and the E-E (Eulerian-Eulerian)method [14]. With the E-L method, the individual particles are tracked insidea continuous medium. The Eulerian framework is used for the continuous phasewhereas the Lagrangian framework is used for the discrete phase. On the contrary,with the E-E method, both phases are treated as continuous and are solved in theEulerian framework i.e. instead of tracking individual particles, the spatial distri-bution of the phases is described [15].

In STAR-CCM+, the E-E method is utilized in the Segregated multiphase flowmodel and the VOF (Volume Of Fluid), which has been explained in Section 2.4.1,model whereas the E-L method is employed in the Dispersed multiphase model, theDiscrete Element Model and the Lagrangian multiphase flow model. Because of itsability to track individual particles, the E-L method is more computationally de-manding and is suitable for flows with dilute mixtures. On the other hand, the E-Emethod is appropriate to simulate flows that can be considered continuous.

In the current thesis work, the E-E method is used as both air and oil form a con-siderable part of the volume of a transmission. From the two available E-E methodsin STAR-CCM+, the Segregated multiphase flow model solves two sets of governingequations, one for each continuous phase. In contrast, the VOF model utilizes theconcept of volume fraction hence solves only one set of governing equations. TheVOF model is suitable for free surface flows because of its ability to accurately trackinterfaces between two immiscible phases, it has therefore been chosen in this thesiswork.

9

2.4.1 VOF model

In the multiphase phenomena, the kinematic and dynamic activity of the interfacebetween the phases plays an important role in most cases, so investigating interfacesis important in many multiphase flow analysis. The VOF model implemented inSTAR-CCM+ is able to resolve the sharp interfaces by tracing the volume of eachfluid instead of the motion of the particles. This is the so-called volume fraction andthe volume fraction of an arbitrary phase i is defined as

αi =Vi∑i Vi

=ViVcell

(16)

where Vi is the volume of phase i and Vcell is the volume of the cell.

In the VOF model, it is assumed that all phases share the same velocity, pres-sure and other fields which means that there is only one set of properties in eachcell even though more than one phase exists in it. As a result, only one set of massand momentum equations is solved for the fluid mixture and the properties of thefluid are calculated based on the physical properties of phases and the correspondingvolume fractions. These properties are defined as

ρcell =∑i

ρiαi (17)

µcell =∑i

µiαi (18)

where ρcell is the density of the cell, ρi is the density of phase i. Similarly, µcell is thedynamic viscosity of the cell and µi is the dynamic viscosity of phase i. In additionto the Navier-Stokes equations, the transport equation of the volume fraction issolved. This equation is given by

1

ρq

∂

∂t(αqρq) +∇.(αqρqvq) = Sαq +

n∑p=1

(mpq − mqp)

(19)

where mqp is the mass transfer from phase q to phase p and mpq is the oppositetransfer. Sαq is the source or sink of the phase q and vq is the velocity of thephase q. In the current case, the right hand side of equation 19 is zero since nomass transfer ocuurs between the two phases and the source term is defined as zero.Interested readers can refer to Hirt and Nichols [16] for more information about theVOF multiphase model.

2.5 Discretization scheme

The discretization schemes adopted in the simulations are given in this section.

2.5.1 Discretization scheme for time

The implicit unsteady scheme has been chosen as the time discretization scheme.When using the MRF method for the gear motion, a fixed time step of 8E-5 seconds

10

has been used since it satisfies the CFL (Courant-Friedrichs-Levy) criterion at thevolume fraction interface. For the cases involving the RBM model, the ConvectiveCFL Time-Step Control model has been used to ensure that the CFL criteria isalways met at the volume fraction interface. The details about the CFL number,this criteria and why it must be satisfied are presented in Section 2.5.2. Moreover,in order to ensure a reasonable simulation time, the first order implicit scheme hasbeen used.

2.5.2 Discretization scheme for convective terms

The Upwind difference scheme is recognized as a stable scheme which doesn’t intro-duce unphysical numerical oscillations. Hence, for the convective terms of the k−ωSST and momentum, the second order Upwind difference scheme has been adopted.

For the VOF model, the Upwind difference scheme is a robust and stable choice.However, it has a tendency to introduce a lot of numerical diffusion which leads tothe smearing of the volume fraction interface. On the contrary, the HRIC (High-Resolution Interface Capturing) scheme has less stability and robustness but it main-tains a sharp interface [12]. In STAR-CCM+, a hybrid method of a second orderupwind scheme and the HRIC scheme has been implemented as a discretizationscheme for the convective terms in VOF model. The blending between the twoschemes is governed by the local CFL number. It is given by

CFL =ucell∆t

∆x(20)

where ucell is the velocity in the cell, ∆t is the time step and ∆x is the local gridsize. Thus, in order to ensure that the HRIC scheme is used at the interface in orderto obtain a sharp interface, the CFL value at the interface should not exceed beyonda certain limit. By default, the HRIC scheme in STAR-CCM+ is optimized for freesurfaces operating at CFL< 0.5.

2.6 Overset meshing technique

The overset meshing technique, also known as chimera or overlapping grid technique,has been firstly developed by Steger et al. [17] in 1983. Initially, it was studied togenerate high quality local structured meshes by dividing the domain into smallersub-domains because of the inability to get a good mesh for particularly complexgeometries using an unstructured grid. Overset meshing technique in STAR-CCM+allows a more realistic simulation of gears engagement because of the inclusion ofthe meshing region between the teeth thus avoiding the need to, for example, scaledown one of the gears unphysically to get rid of the meshing region.

The implementation of the overset methodology in STAR-CCM+ can be dividedinto two steps. The first step is the decomposition of the domain into differentsub-domains and the second one is the choice of the coupling method between thesesub-domains in order to obtain an accurate, unique and efficient solution [12]. Abackground mesh is given for the entire computational domain and each sub-domain,

11

a gear or carrier in the current case, is enclosed by an overset mesh. These oversetmeshes can move freely in the computational domain as prescribed by the motion.Sufficient overlapping region must exist between all grids for the coupling to be pos-sible.

The cells in the overset meshing technique can be classified into active, passive,acceptor and donor cells. The classification of cells into these categories is donethrough the so-called hole-cutting process. The active cells are those in which dis-cretized governing equations are solved and passive cells are those in which theseequations are not solved beacuse these are overlapped by cells from an overset mesh.The acceptor and donor cells are present at the boundaries of each of the overlappinggrids. They transfer data between the overlapping meshes. The exact relationshipbetween the acceptor and donor cells depends on the interpolation scheme used [12].Here, a linear interpolation scheme has been used.

The interpolation of properties between the grids occurs according to the follow-ing equation

φPi =

ND∑k=1

αωkφDk

(21)

where φ is the interpolated property value at point Pi derived from the propertyvalues at points Di, balanced with the interpolation weight αωk

.

2.7 Rotational motion

A variety of engineering applications use rotating components. Different strategiesexist in commercial CFD solvers for modelling the rotational motion of these compo-nents. STAR-CCM+ allows its users to choose between the MRF and RBM modelsfor modelling rotating components.

2.7.1 MRF

The MRF model is a steady-state approximation in which individual cell zones moveat different rotational/translational speeds. It is a suitable choice when the flow atthe interface between the zones can be assumed as uniform.

In MRF implementation, the calculation domain is divided into sub-domains whichcan be assumed as translating and/or rotating according to the laboratory (inertial)frame. The governing equations in each sub-domain are obtained with respect tothe sub-domain’s reference frame. The position vector r of an arbitrary point ’P’ inthe computational domain, as shown in Figure 2 with respect to the origin of thezone rotation axis is given by

r = x + xo (22)

where x represents the position of the point ’P’ in the laboratory reference frame andxo represents the position of the origin of the zone rotation axis in the laboratoryreference frame.

12

Figure 2: The laboratory and rotating frames of reference in MRF method

[12]

The absolute velocity of the arbitrary point ’P’ in the laboratory reference frame isthen given by

vlab = vr + (ωt × r) + vt (23)

where vlab is the velocity in the laboratory frame, vr is the velocity in the movingreference frame, ωt is the angular velocity of the moving reference frame and vt isthe linear velocity of the moving reference frame which is zero in the current case.

Using the relative formulation with respect to the rotating reference frame givesthe governing equations for the frame as

∂ρ

∂t+∇.ρvr = 0 (24)

∂(ρvr)

∂t+∇.(ρvrvr) + ρ(2ωt × vr) + ρ(ωt × ωt × r) = −∇p+

∇.[µ(∇(vr) +∇(vr)T )] + ρg + F

(25)

where the third and the fourth terms on the left hand side in equation 25 representthe Coriolis and centripetal accelerations respectively [4]. In STAR-CCM+, theMRF model can handle both steady and transient simulations.

2.7.2 RBM

The RBM model in STAR-CCM+ involves the rigid motion of the mesh verticesaccording to user-specified rotations, translations or a combination of both. It isan unsteady approach which is more accurate than the MRF model especially whenthere is a strong interaction between the moving part and other stationary or movingparts in its vicinity. It is also, however, more computationally demanding than theMRF model.

2.8 Modelling flows inside transmissions

CFD modelling of transmission flows provides a useful means to optimize the lu-brication system so as to maximize cooling without increasing the losses too much.

13

The modelling of flow inside a transmission is complicated due to oil agitation bymultiple rotating components, mixing of oil and air to create unsteady and complexflows and existence of extremely fine spray and sudden changes in pressure in thegear meshing sections [18]. These factors together increase the computational timeof the gear simulations. In the current thesis work, flows inside an FZG gearboxand a planetary gearbox have been investigated.

2.8.1 FZG gearbox

The FZG gearbox is a single-stage gearbox based on the FZG back-to-back gear testrig (shown in Figure 3) which is the closest method to practice on predicting scuffingand wear properties of gear oils [1]. The inner dimensions of the gearbox are 270mm x 180 mm x 56 mm. The gears utilized are the FZG gears type C. More detailsabout the test rig and gear properties is given in Appendix A.

Figure 3: FZG back-to-back gear test rig

[19]

2.8.2 Planetary gearbox

A single-stage planetary gear train from a GRS905 gearbox, as shown in Figure 4,consists of an internally toothed ring gear (which can be fixed either to the sunwheel or the housing, depending on gear selection), a sun wheel (fixed to the mainshaft by splines), five planet gear wheels and a planet carrier (which forms the frontend of the output shaft). The main shaft which is fixed to the splines inside the sungear has been removed in Figure 4 for easy viewing. The outer radius of the ringgear is approximately 130 mm.

14

Figure 4: Planetary gearbox where the main shaft has been removed for easy viewing

2.8.3 Churning losses in transmissions

Power losses in transmissions are classified into load-dependent and load-independentpower losses. Load-dependent losses are related to mechanical power losses due tofriction at the contact of rolling parts. However, the load-independent losses arerelated to the interaction between the oil and the rotating/moving components andare independent of the transfer of torque between gears. These losses are furtherclassified based on the elements responsible for them. A classification of the load-independent losses is shown in Figure 5. Churning losses belong to the category ofload-independent power losses, and it is associated with the movement of gears inthe oil [20].

Figure 5: Classification of load-independent power losses in a transmission

The motion of the gears through the lubricant causes acceleration of the lubricantand consequently a loss of energy which is frequently referred to as the churning loss.Churning losses are mainly affected by the oil level, viscosity of the lubricant androtational speed of the gears. The drag torque responsible for these losses is knownas churning torque and originates from two different forces namely the pressure and

15

shear forces of the oil. The pressure and shear forces on an arbitrary face i areevaluated as

Fpres,i = (pi − pref ).a (26)

Fshear,i = −Tshear.a (27)

where pi is the pressure on face i, pref is the reference pressure, a is the face areavector and Tshear is the shear stress tensor [12].

16

3 MethodIn this chapter, the softwares and the methodology used for the cases involving boththe FZG gearbox and the planetary gearbox have been presented. The sections in-clude the softwares section, FZG and planetary gearbox sections, solver settings, so-lution assessment and additional investigations sections. The solver settings and thesolution assessment sections provide the common settings and solution parametersfor the two gearboxes. The additional investigations section provides the strategiesused to overcome the instabilities initially encountered with the planetary gearboxsimulation.

3.1 Softwares

Mainly, two softwares have been used in this thesis and are being presented in thissection: ANSA for pre-processing and CAD-cleaning and STAR-CCM+ for meshingand simulations.

3.1.1 ANSA

The pre-processing software from BETA CAE Systems S.A. is a CAE pre-processingtool used for building and cleaning CAD as well as meshing. In this thesis, ANSAhas been used to prepare the two geometries prior to meshing in STAR-CCM+.This preparation included simplifying complex surfaces, removing holes and otherunwanted features and grouping of surfaces for easy selection in the CFD solver.

3.1.2 STAR-CCM+

STAR-CCM+, part of Siemens’ Simcenter portfolio, is the CFD solver used forthis thesis. The software is a commercial simulation software capable of solvingmulti-disciplinary problems in both fluid and solid continuum mechanics and hasan integrated mesher. The geometry already prepared in ANSA is imported intoSTAR-CCM+, where the surface and volume meshing is done and the solution tothe cases is calculated. STAR-CCM+ is capable of handling many different physicssettings including regions with RBM, MRF and multiphase flows.

3.2 FZG gearbox

This section provides a detailed description of the method followed for the casesinvolving the FZG gearbox. This includes all the stages from the preparation of thegeometry to the creation of volume mesh and definition of motion. Figure 6 showsthe overview of the common workflow for both gearbox configurations.

18

Figure 6: Schematic overview of the workflow for the two configurations

3.2.1 Geometry preparation

The geometry of the FZG gearbox is obtained by assembling the gear and the pinionwith a center-to-center distance of 91.5 mm. The shafts and the pockets are removedfrom the gears in ANSA and the housing is designed directly in STAR-CCM+. Tosave computational resources, a symmetric geometry of the FZG gearbox, as shownin Figure 7, is considered here because of symmetry about the transverse plane.

Figure 7: Symmetric geometry of FZG gearbox

As mentioned in Section 1.3 , the separation between gears is required when usingthe overset mesh interface between regions in STAR-CCM+. Hence, a gap is createdbetween the gears by scaling the pinion uniformly in three dimensions. For the gapsensitivity study, three different scales of the pinion have been used. The biggestscale used is 91 percent followed by 85 percent and 80 percent of the actual size ofthe pinion. The choice of the scales is made such that the gap size is between 25and 50 percent of the gear tooth height [12]. The gap sizes for the different scalesof the pinion are given in Table 1.

19

Table 1: Size of gap for different scales of pinion

Scale (%) Gap size (mm)

91 1.2

85 2

80 2.6

3.2.2 Setting up overset regions

Overset meshing technique involves the creation of overset regions surrounding eachof the moving components in the domain. Hence, cylindrical parts are created aroundthe gear and the pinion in the FZG gearbox. The gear and the pinion are then sub-tracted from these parts to obtain the respective overset regions. This gives a total ofthree regions, two overset regions and a background region consisting of the housing.

This is followed by the creation of interfaces between the three regions and theselection of method for the interpolation of solution across these interfaces which inthe present case has been chosen as linear since it is the most accurate method.

3.2.3 Surface and volume meshing

The next step is the generation of the surface mesh which covers the surface of thegeometry followed by the creation of the volume mesh which is the three-dimensionalmesh for resolving the flow. In STAR-CCM+, both the surface and volume meshescan be generated with the so-called automated mesh process. The quality of thevolume mesh is dependent on the quality of the surface mesh therefore, the surfacemesh is analyzed carefully. The properties of the surface remesher for the FZGgearbox in STAR-CCM+ are given in Table 2.

Table 2: Surface remesher properties for FZG gearbox

PartBase size

(mm)Target surface size

(mm)Minimum surface size

(mm)

Pinion overset 1 0.6 0.1

Gear overset 1 0.6 0.1

Gearbox 1 1 0.1

Additionally, two surface controls have been used in the mesh operations for thepinion and gear oversets. The first surface control is used to get a smooth transitionbetween the overset and background regions at the overset interface and the secondsurface control is used to sufficiently resolve the teeth profile of the pinion and thegear. It is also used to specify the prism layer settings for the teeth in order to getan acceptable wall y+ value as well as achieve a minimum of 4 cells between theteeth and the overset interface, which is a requirement of the overset hole-cuttingalgorithm. The settings of the surface controls are given in Table 3.

In addition to the surface controls, two volumetric controls have been used to haveboth a refinement in the meshing region and a smooth transition at the oversetinterfaces. The first volumetric control is common to the pinion overset and the

20

Table 3: Surface controls properties for FZG gearbox

Surfacecontrol

Targetsurface

size(mm)

Minimumsurface

size(mm)

Numberof prismlayers

Prism layertotal

thickness(mm)

1 0.5 0.5 1 0.5

2 0.6 0.1 5 0.6

background mesh where as the second one is common to the gear overset and thebackground mesh. This is done to have similar cell sizes at the overset interfaceswhich is an important factor to ensure mass conservation when using the oversetmeshing technique. The settings of the volume meshes for the three regions and thevolumetric controls are given in Table 4. The volume mesh at the symmetry planein the FZG gearbox is shown in Figure 8 where each color represents a differentregion.

Table 4: Volume mesh properties for FZG gearbox

Region/PartBase size(mm)

Pinionoverset

1

Background 1

Gearoverset

1

Volumetriccontrol 1

0.5

Volumetriccontrol 2

0.5

Figure 8: Volume mesh at the symmetry plane in FZG gearbox

21

3.2.4 Mesh sensitivity analysis

In order to verify that the results are independent of the mesh, two different mesheshave been investigated for the FZG gearbox. The approach is to start with a meshhaving a reasonably good resolution in areas of interest such as the meshing regionand at the interfaces between regions and then refine the mesh more in the areas ofinterest to see if the results vary too much.

Table 5 presents the measured churning torques for both the gear and the pin-ion for the two meshes and the corresponding simulation times. The cell count isvaried by changing the mesh size for the two volumetric controls. It can be seen thatless than one percent difference exists between the churning torques from the twomeshes. However, the finer mesh takes approximately eighty percent more time tosimulate 0.1 seconds of the motion. As a result, the coarser mesh has been chosenfor later analysis.

Table 5: Churning torque and simulation time for two different meshes

Rotationmodel

Pinion Scale(%)

Cell count(million)

Churning torque(Nm)

Simulation timefor 0.1 sec (hrs)

Gear Pinion

RBM 913.45 0.005023 0.001764 446.13 0.004975 0.00176 80

Percentage difference -0.96 -0.23 81.8

3.2.5 Setting up motion

At this stage, the motion is set up using either the MRF or the RBM model. TheFZG gearbox has two regions with prescribed motion, the gear and the pinion. Forthe FZG gearbox, both modelling options have been investigated and comparedwhile using the same scale of the pinion and the same volume mesh. This is doneto accurately judge the capabilities of both these methods. Additionally, the gapsensitivity study involves the usage of the RBM model with three different scales ofthe pinion as mentioned in Section 3.2.1. The motion setup for the FZG gearbox isgiven in Table 6. In order to achieve a steady state solution, two revolutions of thegear have been considered which corresponds to a physical time of approximately1.2 seconds.

Table 6: Motion setup for FZG gearbox

Pitch linevelocity(m/s)

Rotationalspeed(rpm)

Direction

Gear0.6

158.72 CCWPinion 105.86 CW

22

3.2.6 Boundary conditions

The surfaces in the plane where the gearbox has been cut have been given a symme-try boundary condition because of geometric symmetry about this plane. All othersurfaces of the domain including the surfaces of the gear and pinion are consideredas no-slip walls. The boundaries of the overset regions have been specified as oversetboundary condition in order to be able to create overset mesh interfaces betweenthe overlapping regions.

Figure 9: Boundary conditions in FZG gearbox

3.3 Planetary gearbox

Here, the method followed for the planetary gearbox has been described in detail.This section contains the same sub-sections as section 3.2.

3.3.1 Geometry preparation

ANSA is used to prepare the geometry for the planetary gearbox as well. Figure 10shows the planetary gearbox after preparation in ANSA where each color representsa separate part. In order to make the simulation computationally feasible, somedetails of the geometry are removed since they are unnecessary for the simulation.For example, the splines inside the sun gear have the removed and the pocket hasbeen filled with a protruding cylinder as can be seen in Figure 10.

23

Figure 10: Prepared geometry of planetary gearbox

Moreover, the surfaces have been grouped in ANSA to aid in the creation of a goodquality mesh and for easy application of appropriate boundary conditions in STAR-CCM+. Lastly, the planet gears have been scaled to 85 percent uniformly in threedimensions inside ANSA.

3.3.2 Setting up overset regions

The overset regions are created directly in STAR-CCM+ for the planetary gearboxas well. Seven overset regions are created for the planetary gearbox; five for theplanet gears, one for the sun gear and another for the planet carrier. This givesa total of eight regions for the planetary gearbox which can be seen in Figure 11.Each overset region is assigned a RBM motion. This is followed by the creation ofinterfaces between regions.

3.3.3 Surface and volume meshing

The planetary mesh has also been generated taking advantage of the automatedmesh operation in STAR-CCM+. The properties of the surface remesher for theplanetary gearbox in STAR-CCM+ are given in Table 7.

Table 7: Surface remesher properties for planetary gearbox

PartBase size

(mm)Target surface size

(mm)Minimum surface size

(mm)

Planet oversets 1 0.25 0.1

Sun overset 1 0.25 0.1

Carrier overset 1 0.6 0.1

Ring 1 0.25 0.1

Similar to the FZG gearbox, two surface controls are defined in the mesh operationsfor all overset regions in the planetary gearbox. Surface controls 1 and 2 perform the

24

same function here as they did in the FZG gearbox. The settings for these surfacecontrols are given in Table 8.

Table 8: Surface controls properties for planetary gearbox

Surfacecontrol

Targetsurface

size(mm)

Minimumsurface

size(mm)

Numberof prismlayers

Prism layertotal

thickness(mm)

1 2 2 1 1

2 0.6 0.1 4 0.6

Volumetric controls have been defined for the overset regions and the ring gear.Volumetric controls with half the base size are defined in the region close to theteeth for the planet, sun and ring gears. Volumetric controls with twice the basesize are defined for the regions in the carrier and the ring gear domains where thecells in the respective regions are cut by the hole-cutting process and therefore donot contribute to the solution process. The settings of the volume meshes for allregions and the two type of volumetric controls are given in Table 9. The volumemesh for the planetary gearbox is shown in Figure 11.

Table 9: Volume mesh properties for planetary gearbox

Region/PartBase size(mm)

Planet oversets 2

Sun overset 2

Carrier overset 2

Ring gear 2

Volumetric controls(fine)

1

Volumetric controls(coarse)

4

Figure 11: Volume mesh at the symmetry plane in planetary gearbox

25

3.3.4 Setting up motion

At this stage, the motion is set up using the RBM model. The planetary gearboxhas eight regions and the motion of these regions is given in Table 10.

Table 10: Motion setup for planetary gearbox

Sun gear 600 rpm CCW

Planet gearsRotation 528 rpm CWRevolution 160 rpm CCW

Carrier 160 rpm CCW

Ring gear stationary

3.4 Solver settings

The common settings for both the FZG and the planetary gearbox have been pre-sented in this section. The segregated flow solver has been used because it workswell for multiphase flows. As indicated before, the k− ω SST turbulence model hasbeen used since it captures the flow behavior well in the near wall region.

The simulations have been run for a temperature of 20◦C. Oil (Nytex 810) hasbeen considered as the primary phase with a density of 902 kg/m3 and dynamicviscosity of 5.77E-2 Pa.s, while air with a density of 1.18 kg/m3 and dynamic vis-cosity of 1.86E-5 Pa.s has been considered as the secondary phase. Since the surfacetension coefficent for oil is available only at 40◦C, this value has been used in spite ofthe simulations being run for a temperature of 20◦C. A contact angle of 15 degreeshas been given between the two phases and all simulations have been run for oillevel at the centerline of the housing.

The gradients have been specified using the Hybrid Gauss LSQ (Least Squares)method. The under-relaxation factors for velocity and pressure have been given as0.8 and 0.6 respectively. Convective CFL time step control with a target mean CFLnumber of 0.5 and max CFL number of 0.7 has been given. To find out more aboutwhat these settings represent, the interested reader can refer to an introductorybook on the subject (e.g. Versteeg and Malalasekera [11]).

3.5 Solution assessment

Un-normalized residuals are monitored within each time-step and are allowed toreach an asymptote. This corresponds to 15 inner iterations per time step. Differ-ent monitor points for velocity and volume fraction have been set up in regions ofinterest and are monitored within and between time steps. In addition, the CCNand total domain mass have also been monitored to ensure mass conservation sincethe overset meshing technique and the VOF model are not inherently conservative.The important monitor quantities are presented in Appendix B.

To calculate the churning losses, the pressure and the shear stress tensor across

26

the gears and carrier surfaces have been written to a report. These variables areused to calculate the pressure force and shear force as explained in Section 2.8.3which in turn gives the the churning torque for the respective gear or carrier.

3.6 Additional investigations

The common solver settings described in Section 3.4 have been initially adopted inthe planetary gearbox simulation. However, instabilities are encountered during thesimulation of the planetary gearbox with these settings. Since the solver settingswork well for the FZG gearbox, therefore two possible way forwards can be outlined:

1. Reduce the complexity of the current setup by simplifying the geometry ofplanetary gearbox.

2. Adopt some alternative modelling choices for the planetary gearbox simulationsuch that it leads to enhanced stability in the simulation behavior.

A useful first step when debugging gear simulations is to verify if the overset hole-cutting algorithm works well and whether the motion has been setup accurately.This is done by freezing all the solvers except the time, motion, load balancing andpartitioning solvers. After verification of these two aspects of the simulation, thetwo strategies outlined previously have been implemented and modified simulationsare executed.

In terms of simplifying the geometry, a key factor which has led to the decisionof starting with a very simple and less computationally expensive geometry is to beable to run the simulation locally during the debugging stage and thereby minimiz-ing both the time lost while the simulation is queued and unnecessary occupationof resources. Therefore, once it has been identified that the planetary geometryis not compatible with the solver settings used for the FZG gearbox, it is decidedthat a simpler version of the planetary geometry be used to investigate the effectof different solver settings and mesh resolution on the stability of the simulation.As a result, a simple geometry with the sun, a single planet and a circular housing,henceforth referred to as the single planet case, is used to establish the effects of thechanges adopted in the simulation. The details of this geometry and some of theresults are given in Appendix C.

Followed by the geometric simplifications, changes are introduced gradually in thephysics setup of the planetary gearbox simulation. Since the turbulence transportequation is the first one to diverge, the laminar model is considered for debugging.Later on, when the simulation has been stabilized, the Lag Elliptic Blending k − εmodel has been used instead of the previously used k − ω SST turbulence model.This has been done because it offers more stability to the simulation and has demon-strated a good predictive capability for flows subject to strong rotations, such as theflow inside a planetary gearbox, due to additional terms for modelling the effects ofanisotropy, curvatures and rotational effects [12].

Similarly, rotation speeds of the gears are defined using a conservative ramp over

27

1000 time steps as well as the final rotation speeds of the gears are reduced six timesas compared to the motion setup given in Table 10. The compressible ideal gasmodel is used for the air phase in contrast to the previously used constant densitymodel which means that the density is now determined by the local pressure andtemperature. This necessitates that the conservation of energy is added to the gov-erning equations of the flow in addition to the already present conservation of massand momentum. This on the one hand increases the computational expense but onthe other hand provides stability to the simulation and enables the use of biggertime steps.

Although STAR-CCM+ is capable of handling the pressure field inside closed do-mains through the definition of a so-called reference pressure, however, the stabilityof the solution can be enhanced through a manual specification of pressure in alocation using a pressure outlet, for example. This reduces the round-off error innumerical calculations involving pressure therefore, pressure outlet is specified ata face in the air phase [12]. Moreover, simulations are run in double-precision toprevent instabilities.

Once the single planet case has been running for a significant physical time withoutany instabilities, the geometry for the next stage is utilized. A big complexity withthe complete planetary geometry is the simulation of overlapping motions of theplanetary carrier with the sun and planet gears. Although, theoretically, the over-set meshing technique in STAR-CCM+ provides a good solution to simulate theseoverlapping motions, however, it has been decided to remove the carrier for the nextstage because of two major advantages:

1. It reduces the complexity of the simulation.

2. It significantly reduces the computational requirements on the simulation dueto both the symmetric nature of the geometry which reduces the mesh countin half as well as the reduction in the number of overset interfaces leading toa lesser time requirement for updating these interfaces at each time step.

Figures 12 and 13 respectively show the geometry and the mesh for this case hence-forth referred to as the reduced planetary case.

The final stage is the simulation of the complete planetary geometry as describedin Section 3.3. However, due to limitations with the memory allocation on clustersat Scania, this simulation could not be run on the clusters and has instead beenran locally on 12 processors and the simulation behavior is judged based on somepreliminary results.

28

Figure 12: Reduced geometry of planetary gearbox

Figure 13: Volume mesh at the symmetry plane in reduced planetary gearbox

29

4 Results and DiscussionsIn this section the results have been presented to validate their accuracy. Theseinclude the results for the different analysis performed on the FZG gearbox and theresults from the planetary gearbox configurations.

4.1 FZG gearbox

For FZG gearbox, the results from MRF and RBM models and the gap sensitivitystudy are presented. Figure 14 presents the development of the volume fraction ofoil at the symmetry plane of the FZG gearbox with time as obtained by using theMRF and the RBM models. The contours up till 2 revolutions of the gear showthe instantaneous volume fraction of oil whereas the contours for 2.25 revolutionsshow the mean volume fraction computed over the time between 2 and 2.25 revolu-tions since it is assumed that the flow has by then achieved a quasi-steady condition.

The first observation from this figure is that the MRF model does not give thetime-accurate behavior of the flow which is reasonable since the flow around thegear and pinion is modelled as a steady-state problem with respect to the MRFzones for the gear and the pinion as given in Section 2.7.1. As a result, it can beseen that the mean volume fraction of oil from the two strategies resembles eachother in the reservoir but not close to the gear and pinion teeth. This limitation ofthe MRF model is because it does not account for the relative motion of a movingzone with respect to adjacent zones. This is analogous to freezing the motion in aspecific position and observing the instantaneous flow field with the gears in thatposition [4].

The development of the velocity field of oil flow at the symmetry plane as obtainedby the MRF and RBM models is presented in Figure 15. It can be seen that thevelocity contours show a significant difference between the two models. Initially,both models give a similar velocity distribution but as the flow develops, the predic-tions from the two models are different. The RBM method predicts a more localizedregion of high velocity whereas the MRF method shows a wider distribution of ve-locity in the reservoir.

Upon looking closely at the velocity contours after 0.5 revolutions from the twomodels, it can be seen that there are velocity lines following the ’trail’ for gear mo-tion in RBM model where as they are directed in the opposite direction for the MRFmodel. This is essentially a representation of how the motion is being handled in thetwo methods. The RBM model moves the mesh thereby giving the correct velocity’trail’ whereas the MRF model introduces motion in the fluid to give a feeling of thegear rotation essentially rotating it in the opposite direction.

31

Figure 14: Development of volume fraction of oil at the symmetry plane using the MRFand the RBM models in STAR-CCM+ (All contours except at 2.25 revolutions of gearare instantaneous whereas the ones for 2.25 revolutions represent the mean volumefraction). Volume fraction of 0 indicates only air is present inside the discretized celland 1 indicates oil.

32

Figure 15: Development of velocity field (m/s) of oil flow at the symmetry planewith the MRF and RBM models (All contours except at 2.25 revolutions of gear areinstantaneous whereas the ones for 2.25 revolutions represent the mean velocity). Thedifference between the two frames is due to the use of a volume fraction driven part.

33

Figure 16: The mean velocity vectors (m/s) of oil flow at the symmetry plane fromthe MRF and RBM models

The mean velocity vectors of oil flow at the symmetry plane for the MRF and RBMmodels after 2.25 revolutions of the gear are shown in Figure 16. A clear obser-vation from this figure is that for the MRF model, the flow inside the reservoir isdominated by the flow in the direction of the gear and the pinion on the side ofeach of these components respectively whereas for the RBM model there is a verystrong recirculation region on the pinion side of the reservoir and there is also asmall recirculation on the gear side as well. Both the models, however, make similarpredictions about the direction of velocity close to the left wall of the housing.

In Figure 17, the mean velocity vectors have been presented in three different planesa, b and c in the transverse direction corresponding to the gear, housing and pinioncenterline locations respectively for better understanding of the three dimensionalflow field close to the rotating components. Beginning from the gear centerline plane,both the models predict a flow dominated in the direction of the gear rotation anda change in the velocity direction as the flow reaches the bottom resulting in a re-circulation region.

For the housing centerline plane, the MRF model predicts a recirculation of thevelocity close to the back wall of the housing whereas the RBM model shows theflow in this plane to be largely driven by the large recirculation region on the pinionside of the gearbox. Similarly, the flow on the pinion centerline plane is shown to bedominated by the large recirculation region on the pinion side by the RBM modelwhereas the MRF model shows the velocity to be dominant in the direction of thepinion motion close to the symmetry plane and in the opposite direction near thebehind wall.

34

Figure 17: The mean velocity vectors (m/s) at three planes in the transverse directionfor the MRF and RBM models with the viewing direction from the left. The planesa, b and c represent the gear centerline, housing centerline and the pinion centerlinelocations respectively.

Figures 18 and 19 respectively represent the development of volume fraction and ve-locity on the symmetry plane for three different gap sizes obtained through differentscales of pinion. A few details of this analysis are quite visible from the figures. Thefirst of those details is the prediction of mixing of air in the reservoir which is quitevisible in Figure 18. It can be seen that with a large gap size, a lesser mixing of air ispredicted inside the reservoir and instead it predicts that more air is trapped insidethe gear teeth. This is obviously a drawback of having a large gap which means thatthe air has a chance to escape and thus a lesser mixing of air with the oil is predicted.

Similarly, a second observation from Figure 18 is that with a large gap size, thepinion transports more oil to the meshing region between the teeth as comparedto a smaller gap size. This is essentially because for a large gap, the pinion hasbeen scaled more which means that it has lesser distance to travel from the pointthat it leaves the oil surface to the point when it transports the oil to the meshingregion. As a result, it can be seen quite clearly in the mean volume fraction con-tours at 2.25 revolutions of the gear that the there is less oil on the tooth flangesof the pinion in case of a small gap as compared to the case of a large gap and theopposite is true for the gear where more oil is present on the flanges for smaller gaps.

Similarly, in Figure 19, it can be seen that the predictions for the development

35

of velocity on the gear side are almost the same for all gap sizes whereas on the pin-ion side the high velocity region is bigger for a smaller gap and smaller for a biggergap which is because a bigger gap means a smaller pinion and hence the behavior iswhat would be expected.

Figure 20 shows the mean velocities measured at the gear and pinion centerlinesfrom each of the cases involving the FZG gearbox where the centerlines have againbeen shown in the figure labeled as a and b respectively. It can be seen that the MRFmodel predicts a wider influence of both the gear and the pinion inside the reservoiras compared to the RBM model which is the same as what has been mentioned withregard to Figure 15 before.

Looking at the plots for the two gap sizes obtained using the RBM model, it isclear that the flow on the pinion side has a large recirculation region which is biggerfor the smaller gap because of a bigger pinion as indicated by the bulge in the plotfor both scales whereas on the gear side the flow does not have a counterflow region.However, the MRF model predicts a much smaller recirculation region on the pinionside indicated by a bulge closer to the bottom of the housing in the mean velocityplots in Figure 20 and can also be seen in Figure 16.

36

Figure 18: Development of volume fraction of oil at the symmetry plane for threedifferent gap sizes (All contours except at 2.25 revolutions of gear are instantaneouswhereas the ones for 2.25 revolutions represent the mean volume fraction)

37

Figure 19: Development of velocity field (m/s) at the symmetry plane for three differ-ent gap sizes (All contours except at 2.25 revolutions of gear are instantaneous whereasthe ones for 2.25 revolutions represent the mean velocity)

38

Figure 20: Mean velocity plots at the gear and pinion centerlines shown as a and brespectively for the cases involving FZG gearbox after 2.25 revolutions

Table 11 shows the churning torques measured for the different cases involving FZGgearbox. Comparing the torques evaluated for the MRF and the RBM cases, it canbe seen that the MRF model under predicts the churning torque as compared tothe RBM model. This is in turn due to the under prediction of the recirculationregions by the MRF model, as discussed with reference to Figure 20 for example,which contribute to the churning torque.

Table 11: Churning Torque measurements for the various cases involving FZG gearbox

Rotation modelGap size

(mm)Pinion scale

(percent)Churning Torque (Nm)Pinion Gear Total

MRF 1.2 91 0.00067 0.0031 0.00377

RBM1.2 91 0.0014 0.0051 0.00652 85 0.0012 0.0046 0.0058

2.6 80 0.0008 0.00425 0.00505

Similarly, when comparing the torques for different gap sizes, it can be seen thattorques for both the gear and the pinion decrease with the increase in gap size anddecrease in the pinion size which means that the change in size of one of the gearshas an effect on the churning torque for both the gears. Therefore, the additionof the individual losses for the gear and the pinion to get the overall losses canlead to erroneous results. This result is in line with the empirical model derived byChangenet and Velex [6]. Another interesting fact is that decreasing the pinion sizeby 11 percent, the churning torque from the pinion has reduced by half.

The computational cost for different cases involving FZG gearbox is presented inTable 12. Comparing the MRF and RBM cases with the same parameters, it can be

39

seen that the RBM model is more computationally intensive than the MRF model.This is specially true during the beginning of the motion where the RBM modeltakes twice as much time to simulate 0.1 seconds of the motion as compared to theMRF model. This is because the Convective CFL time step control uses smallertime steps in case of RBM model as opposed to the fixed time step used in case ofMRF model. Comparing two different mesh counts for the same gap size, it is clearthat increasing the mesh count two times causes an approximate two times increasein the computational time with 12 processors.

Table 12: Cost analysis for the various cases involving FZG gearbox using 12 processors

Rotationmodel

Gap size(mm)

PinionScale

(percent)

Inflationthickness

(mm)

Cell count(Million)

Simulation time (hrs)

0.1 sec1 rev

(0.567 sec)

Steadystate

(2.25 rev)

MRF 1.2 91 0.6 3.5 22 125 235

RBM

1.2 91 0.63.5 44 156 2966.13 80 - -

2 850.3

3.3542 175 -

1 50 - -2.6 80 0.6 3.35 32 134 192

Similarly, variation in the inflation size for the gap size of 2 mm also impacts thecomputational cost with a slightly higher computational cost for the larger inflation.This trend, however, is not observed for all cases since the optimum size of the in-flation layer depends on the size of the gap. For example, for the gap size of 2 mm,having an inflation layer with total thickness of 1 mm on both the gears means thatthe overlap between two overset zones will occur exactly at the end of the inflationlayers and as a result, smaller time steps are used in order to maintain the minimumamount of cells in the meshing region.

Comparing the different gap sizes in terms of the computational costs, it can beseen that increasing the gap lowers the computational costs considerably however,this is obtained at the cost of a loss in accuracy of critical parameters such as thechurning losses which are of interest.

The choice of the MRF and RBM models has been evaluated in Table 13. Since theprimary advantage with the MRF model is that it can be used in a steady frame-work therefore, a case with the MRF model in steady framework has been run whichdemonstrates the unsuitability of the steady framework for multiphase simulationsusing the VOF multiphase model. Due to this limitation of the VOF model, nextthe MRF model is used in an unsteady framework which leads to a physically accu-rate behavior of the VOF model in the reservoir but due to limitations of the MRFmodel in the vicinity of the teeth it is unable to predict the flow close to the teeth asdiscussed in Section 4.1. Nevertheless, it can still serve as a useful means for a roughestimate of the churning losses at the regime condition with a lower computationaleffort.

On the other hand, the RBM model is computationally intensive since it is involvesthe motion of the mesh but as demonstrated in Section 4.1, it gives a time-accuratedescription of the flow behavior throughout the domain and hence is capable of

40

Table 13: Evaluating the choice of MRF and RBM

Rotation Model Time Model Pros Cons

MRF SteadyLeast computationally

intensiveVOF model does not handle steady

simulationsCFL criteria does not apply Spatially inconsistent in time

MRF UnsteadyVOF model behaves physically

More computationally intensivethan steady MRF

Good alternative for evlauationof losses at the regime

Only provides a rough flowbehavior at steady state

RBM UnsteadyGives time-accurate description

of the interfaceMost computationally intensive

Accurate prediction of flowbehavior and critical

parameters

Restricted motion of oversetinterface to less than half of abackground cell per time step

predicting the churning losses more accurately.

4.2 Planetary gearbox

For the planetary gearbox, the results from different analysis are presented in thissection. As explained in Section 3.6, for the debugging phase, the single planet casehas been used. To see some results for this configuration, the reader can consultAppendix C. Here, the results for the reduced planetary gearbox and the completeplanetary gearbox have been presented and discussed. The churning torque measure-ments from different cases have been compared and a cost analysis is also included.

Figure 21 shows the development of volume fraction of oil and velocity field at thesymmetry plane for the reduced planetary gearbox. As a consequence of the highcomputational time of this simulation when using 12 processors, only one revolutionof the sun gear has been simulated and the interface deformation under the counterclockwise rotation of the sun gear and, clockwise rotation and counter clockwiserevolution of the planet gears can be observed in the volume fraction contours inFigure 21. This behavior of the simulation is encouraging after the initial instabili-ties especially in the pressure field which arose close to the beginning of the run andquickly resulted in the deterioration of the solution. It can be seen that the planeton the right drags a significant amount of oil as it moves through it and rises abovethe oil level in turn transporting the oil to the nearby planet as well as the ring gear.Similarly, the motion of the planet on left leads to a mixing of the air in the oilreservoir especially close to the sun. As noticed in the case of gap sensitivity studyinvolving the FZG gearbox, this can be a consequence of the size of gap between theplanet and the sun gear.

The development of the velocity field at the symmetry plane can also be seen inFigure 21. It can be seen that a high velocity region exists in the vicinity of thegears initially as would be expected. As the flow develops, the distinct high veloc-ity regions around the planet and sun gears start to disappear and instead a singlehigh velocity region appears towards the center of the housing with high velocitiesconcentrated around the gear teeth in the meshing region. This is in line with theobservations made by Cho et. al [10].

41

Figure 21: Development of volume fraction of oil and velocity field (m/s) at the sym-metry plane for the reduced planetary gearbox

42

The churning torques have been measured for the single planet and the reducedplanetary cases at 0.015 seconds and are presented in Table 14. It can be seen thatthe magnitude of churning torque predicted by both the cases is different for boththe revolution component of the planet gear and the rotation component of the sungear whereas the predictions for the rotation component of the planet gear are quiteclose. The biggest deviation exists in the churning torques associated with the re-volving action of the planet gear from the two cases. This makes sense because inthe single planet case, the housing is farther because of the absence of the ring gearand secondly because of the larger quantity of oil in contact with the planet in thiscase because it is the only planet in this configuration. This means that during therevolving action it has to independently drag the oil in the housing hence resultingin a larger churning torque contribution from the revolving action of the planet.

A cost analysis is made for the three cases involving the planetary gearbox in Table15. This table clearly demonstrates the computationally intensive nature of gearsimulations using the overset meshing technique. Where it takes around 5 hours tosimulate 1 millisecond of motion for the single planet case, it takes almost six timesas much when it comes to simulating the same amount of time in the reduced plan-etary gearbox. This on the one hand is due to the increased mesh count associatedwith more regions in the simulation and on the other hand, it is due to the increasein the number of interfaces between the components and therefore leads to a highertime requirement for updating these interfaces at each time step. The same trendis observed when comparing the simulation time for the planetary gearbox with thereduced planetary gearbox except that in this case, another contributing factor tothe increase in cost is the doubling of the domain size because of the inclusion ofthe carrier in the simulation which excludes the usage of a symmetric geometry.

Some very preliminary results for the complete geometry of the planetary gear-box have been given in Figure 22. Both the volume fraction and velocity contourshave been obtained at the middle plane and at 8 milliseconds. On 12 processors, ithas taken around 700 hours to simulation this degree of motion.

Table 14: Churning torque measurements for the cases involving the planetary gearbox

GearChurning Torque (Nm)

Single planetcase

Reducedplanetary case

Planetary case

PlanetRotation 0.035 0.034 -

Revolution 0.054 0.02 -

Sun 0.03 0.038 -

43

Table 15: Cost analysis for the various cases involving the planetary gearbox using 12processors

ConfigurationSimulation time (hrs)

0.001 sec 0.25 rev (0.029 sec)

Single planet case 5 139

Reduced planetary case 28 780

Planetary case 112 -

Figure 22: Volume fraction and velocity (m/s) contours at the middle plane for theplanetary gearbox at 8 milliseconds

44

5 ConclusionsA common experimental setup used for the prediction of scuffing and wear propertiesof gears i.e. FZG gearbox and a planetary gearbox have been used to successfullystudy the effect of different modelling choices on the prediction of oil distributionand the churning power losses which are of interest for Scania and can be estimatedthrough CFD models. Moreover, the computational requirements for the choiceshave been compared. Among the key modelling choices, the VOF model has beenused to resolve the interface between the two immiscible phases and overset meshingmethod is used in combination with the MRF or the RBM model for modelling gearrotation.

Two major investigations are conducted using the FZG gearbox. These includea comparison of two motion modelling alternatives in STAR-CCM+, namely theMRF and RBM models, and a sensitivity analysis on how the size of gap betweenthe gear teeth affects the flow and the computational requirements. These investiga-tions have revealed the inadequacy of the MRF method to study the time-accuratephenomena inside a gearbox although it can provide a rough estimate of the churn-ing losses at mean flow with lesser resources and can be of benefit for the industry.Moreover, it has been demonstrated that an increase in the gap size can reduce thecomputational costs but it can cause the loss of important flow features such as themixing of air and oil in the reservoir.

Through a careful analysis of different modelling choices for the planetary gear-box, the best physical setup for simulation of oil flow inside complex configurationslike the planetary gearbox has been determined. It has been found that using thecompressible ideal gas model for the air phase as well as specification of a pressureboundary in the air domain enhances the stability of the simulation. The resultsfrom planetary gearbox point to the effectiveness of CFD as a tool both for sim-ulating oil flow as well as evaluation of churning losses inside industrially relevanttransmission configurations such as the planetary gearbox used in this thesis work.

46

6 Future workIn terms of the prospective extensions for this master’s thesis, the cluster configu-ration at Scania needs to be adopted with the help of STAR-CCM+ support suchthat sufficient physical memory is made available for the simulation of the completeplanetary gearbox. This will add to the accuracy of the churning torque measure-ments for the gears and can lead to further recognition of CFD as the state of theart method for oil flow simulations inside transmissions.

Since the oil in a gearbox also fulfills important requirements concerning the coolingsystem of a gearbox, a potential extension of this thesis work can include simulationof oil temperature and heat dissipation in the gearbox. Moreover, a study of heatconduction at the gearbox walls and gear teeth can also be done. A CFD frame-work for simultaneous analysis of the load-independent power losses and the coolingaction of the oil for an operating condition will be a huge advantage towards thedevelopment of future transmission systems.

48

References[1] Hartono EA, Golubev M, Chernoray V. ’PIV Study of Fluid Flow Inside a

Gearbox’. 10th International Symposium On Particle Image Velocimetry. 2013;.

[2] Liu H, Jurkschat T, Lohner T, Stahl K. ’Determination of oil distribution andchurning power loss of gearboxes by finite volume CFD method’. TribologyInternational. 2017;.

[3] Saegusa D. ’CFD analysis of Lubricant Fluid Flow in Automotive Transmis-sion’. SAE Technical Paper. 2014;.

[4] Concli F, Gorla C, Torre AD, Montenegro G. ’Churning power losses of ordi-nary gears: a new approach based on the internal fluid dynamics simulations’.Lubrication Science. 2014;.

[5] Kolekar AS, Olver AV, Sworski AE, Lockwood FE. ’Windage and ChurningEffects in Dipped Lubrication’. Journal of Tribology. 2014;.

[6] Changenet C, Velex P. ’A Model for the Prediction of Churning Losses in GearedTransmissions-Preliminary Results’. Journal of Mechanical Design. 2007;.

[7] Boni JB, Neurouth A, Changenet C, Ville F. ’Experimental investigations onchurning power losses generated in a planetary gear set’. The Japan Society ofMechanical Engineers. 2017;.

[8] Li L, Versteeg HK, Hargrave GK, Potter T, Halse C. ’Numerical Investigationon Fluid Flow of Gear Lubrication’. SAE International. 2008;.

[9] Concli F, Gorla C. ’A new methodology for the prediction of the no-loadlosses of gears: CFD and experimental investigation of a planetary gearbox’.International conference on gears, At Munich, Germany, Volume 2. 2013;.

[10] Cho J, Hur N, Choi J, Yoon J. ’Numerical simulation of oil and air two-phaseflow in a planetary gear system using the overset mesh technique’. ISROMAC.2016;.

[11] Versteeg HK, Malalasekera W. ’An Introduction to Computational Fluid Dy-namics: The Finite Volume Method’. Pearson Education Limited. 2207;.

[12] Siemens PLM Software. ’STAR-CCM+ User guide v.13.04’. 2018;.

[13] Clift R, Grace JR, Weber ME, Weber MF. ’Bubbles, Drops and Particles’.London: Academic Press INC LTD. 1978;.

[14] Worner M. ’A Compact Introduction to the Numerical Modeling of MultiphaseFlows’. Forschungszentrum karlsruhe GmbH, Karlsruhe. 2003;.

[15] Sjostrand M. ’CFD Simulations of Two-Phases Flows Passing Through a Dis-tributor’. Department of Applied Mechanics, CHALMERS. 2008;.

50

[16] Hirt CW, Nichols BD. ’Volume of Fluid (VOF) Method for the Dynamics ofFree Boundaries’. Journal of computational physics (1981) vol 39 page 201-225.1979;.

[17] Steger JL, Dougherty FC, Benek JA. ’A Chimera grid scheme’. AmericanSociety of Mechanical Engineers, Fluid Engineering Division. 1983;.

[18] Saegusa D, Kawai S. ’CFD Analysis of Lubricant Fluid Flow in AutomotiveTransmission’. SAE Technical Paper. 2014;.

[19] FZG gear test rig; 2018. https://www.strama-mps.de/en/products/

test-rigs/fzg-gear-test-rig/.

[20] Concli F. ’Oil Squeezing Power Losses in Gears : A CFD Analysis’. AFM120041.2012;.

51

https://www.strama-mps.de/en/products/test-rigs/fzg-gear-test-rig/

https://www.strama-mps.de/en/products/test-rigs/fzg-gear-test-rig/

A FZG gear propertiesThe important properties of FZG gears of type C are given in Table 16.

Table 16: Data for FZG gears type C

Parameter Unit Magnitude

Center distance mm 91.5

Number of teethPinion - 16Gear - 24

Module mm 4.5

Pressure angle degree 20

Face width mm 14

Pitch diameterPinion mm 73.2Gear mm 109.8

Tip diameterPinion mm 82.5Gear mm 118.4

53

B Solution assessmentThis appendix presents the important monitor quantities for the different configu-rations.

B.1 FZG gearbox

Figure 23: Monitor of total mass in the domain

Figure 24: Monitor of CCN in the solution domain

54

B.2 Planetary gearbox

Figure 25: Monitor of total mass in the domain

Figure 26: Monitor of CCN in the solution domain

55

C Single planet caseHere, the mesh and the results for the single planet case used for debugging theplanetary gearbox simulation have been given. These results have been crucial interms of establishing the impact of the changes in the simulation settings.

Figure 27: Mesh at the symmetry plane for the single planet case

56

Figure 28: Development of volume fraction of oil and velocity (m/s) with the revolu-tions of the sun at the symmetry plane of the single planet case

57

Documents

CFD simulation of dip-lubricated single-stage gearboxes through …liu.diva-portal.org/smash/get/diva2:1268007/FULLTEXT01.pdf · 2018-12-04 · Scania CV AB, S odert alje Examiner: