CFD simulation of dip-lubricated
single-stage gearboxes through
coupling of multiphase flow and
multiple body dynamics, an
initial investigation
Nasir Imtiaz
Link¨oping University Institutionen f¨or ekonomisk och industriell utveckling Div. of App. Thermodynamics and Fluid Mechanics Examensarbete 2018 | LIU-IEI-TEK-A–18/03285-SE
Link¨opings universitet Institutionen f¨or ekonomisk och industriell utveckling ¨
Amnesomr˚adet Mekanisk v¨armeteori och str¨omningsl¨ara Examensarbete 2018 | LIU-IEI-TEK-A–18/03285–SE
CFD simulation of dip-lubricated
single-stage gearboxes through
coupling of multiphase flow and
multiple body dynamics, an
initial investigation
Nasir Imtiaz
Academic supervisor: Hossein Nadali Najafabadi IEI, Link¨opings Universitet Industrial supervisor: Samira Nikkar
Scania CV AB, S¨odert¨alje
Examiner: Matts Karlsson
IEI, Link¨opings Universitet
Link¨opings universitet SE-581 83 Link¨oping, Sverige 013-28 10 00, www.liu.se
Abstract
Transmissions are an essential part of a vehicle powertrain. An optimally designed powertrain can result in energy savings, reduced environmental impact and increased comfort and reliability. Along with other components of the powertrain, efficiency is also a major concern in the design of transmissions. The churning power losses associated with the motion of gears through the oil represent a significant portion of the total power losses in a transmission and therefore need to be estimated. A lack of reliable empirical models for the prediction of these losses has led to the emergence of CFD (Computational Fluid Dynamics) as a means to (i) predict these losses and (ii) promote a deeper understanding of the physical phenomena responsible for these losses in order to improve existing models.
The commercial CFD solver STAR-CCM+ is used to investigate the oil distribution and the churning power losses inside two gearbox configurations namely an FZG (Technical Institute for the Study of Gears and Drive Mechanisms) gearbox and a 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 between the meshing gear teeth affects the flow and the computational requirements is also done using the FZG gearbox. Different modelling alternatives are investigated for the planetary gearbox and the best choices have been determined. The numerical simulations are solved in an unsteady framework where the VOF (Volume Of Fluid) multiphase model is used to track the interface between the immiscible phases. The overset 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 the modelling choices made and their effect on the accuracy of the results. The MRF method is a cheaper alternative compared to the RBM model however, the former model does not accurately simulate the transient start-up and instead provides just a regime solution of the unsteady problem. As expected, the accuracy of the results suffers 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 gives the optimum performance for the planetary gearbox. The outcomes can be used to effectively study transmission flows using CFD and thereby improve the design of future transmissions for improved efficiency.
Keywords: transmissions, churning losses, CFD, STAR-CCM+, FZG gearbox, plan-etary gearbox, RBM, MRF, unsteady framework, VOF, immiscible, overset meshing technique
Preface
This thesis is a part of the master’s degree at Link¨oping University and has been done in collaboration with NTAC (Transmission Development, Analysis and Testing group) at Scania CV AB during the spring of 2018.
Acknowledgements
I would like to express my sincere gratitude to my supervisors, Samira Nikkar (Scania CV AB) and Hossein Nadali Najafabadi (Link¨oping University) for their discussions and guidance throughout the thesis work as well as their help with the administrative aspects of the thesis work. I also want to thank H˚akan Settersj¨o (Scania CV AB) for his inputs during the presentations and his help in running simulations for the planetary gearbox on Scania clusters, Paolo Fornaseri (Scania CV AB) for his inputs regarding the planetary model and Christian Windisch (Siemens PLM Software) for his help in debugging the planetary simulations. Finally, a sincere thanks also goes to my examiner at Link¨oping University, Matts Karlsson for his valuable inputs.
Nomenclature
Abbreviations and Acronyms
Abbreviation DescriptionCFD Computational Fluid Dynamics
FZG Technical Institute for the Study of Gears and Drive Mechanisms
VOF Volume Of Fluid
RBM Rigid Body Motion
MRF Moving Reference Frame
NTAC Transmission Development, Analysis and Testing group PIV Particle Image Velocimetry
PLM Product Lifecycle Management RANS Reynolds-Averaged Navier-Stokes SST Shear Stress Transport
E-E Eulerian-Eulerian
E-L Eulerian-Lagrangian
HRIC High-Resolution Interface Capturing
CAD Computer-Aided Design
CAE Computer-Aided Engineering
LSQ Least Squares
Dimensionless numbers
Abbreviation DescriptionRe Reynolds number
CFL Courant-Friedrichs-Levy y+ Normalized wall distance
Symbols
Symbol Description
ρ Density
V Velocity vector
p Pressure
fi Body force in ith direction
U Velocity magnitude L Characteristic length µ Dynamic viscosity
ui Time-averaged velocity component in ith direction
ui0 Oscillating velocity component in ith direction
µt Turbulent viscosity
k Turbulence kinetic energy τij Viscous stress tensor
δij Kronecker delta
ω Specific dissipation rate
V Mean velocity
y Cell centroid distance from wall u∗ Wall friction velocity
ν Kinematic viscosity αi Volume fraction of phase i
Vi Volume of phase i
Vcell Volume of cell
ρi Density of phase i
ρcell Density of cell
µi Viscosity of phase i
µcell Viscosity of cell
˙
mqp Mass transfer from phase q to phase p
Sαq Source or sink of phase q
ucell Velocity in the cell
∆t Time step
∆x Local cell size
φPi Interpolated property value at point Pi
αωk Interpolation weight factor
r Position vector
x Position in laboratory frame xo Position of rotation axis
vlab Velocity in laboratory frame
vr Velocity in moving frame
ωt Angular velocity of moving frame
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
3.5 Solution assessment . . . 26 3.6 Additional investigations . . . 27
4 Results and Discussions 31
4.1 FZG gearbox . . . 31 4.2 Planetary gearbox . . . 41 5 Conclusions 46 6 Future work 48 Bibliography 51 Appendices 53 A FZG gear properties 53 B Solution assessment 54 B.1 FZG gearbox . . . 54 B.2 Planetary gearbox . . . 55
C Single planet case 56
List of Figures
1 A schematic that illustrates the actual contact (No Gap) with the simplified contact (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. . . 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 easy viewing . . . 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 using the MRF and the RBM models in STAR-CCM+ (All contours except at 2.25 revolutions of gear are instantaneous whereas the ones for 2.25 revolutions represent the mean volume fraction). Volume fraction of 0 indicates only air is present inside the discretized cell and 1 indicates oil. . . 32
15 Development of velocity field (m/s) of oil flow at the symmetry plane with the MRF and RBM models (All contours except at 2.25 revolu-tions of gear are instantaneous whereas the ones for 2.25 revolurevolu-tions represent the mean velocity). The difference between the two frames is due to the use of a volume fraction driven part. . . 33
16 The mean velocity vectors (m/s) of oil flow at the symmetry plane from the MRF and RBM models . . . 34
17 The mean velocity vectors (m/s) at three planes in the transverse direction for the MRF and RBM models with the viewing direction from 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 three different gap sizes (All contours except at 2.25 revolutions of gear are instantaneous whereas the ones for 2.25 revolutions represent the mean volume fraction) . . . 37
19 Development of velocity field (m/s) at the symmetry plane for three different gap sizes (All contours except at 2.25 revolutions of gear are instantaneous whereas the ones for 2.25 revolutions represent the mean velocity) . . . 38 20 Mean velocity plots at the gear and pinion centerlines shown as a and
21 Development of volume fraction of oil and velocity field (m/s) at the
symmetry plane for the reduced planetary gearbox . . . 42
22 Volume fraction and velocity (m/s) contours at the middle plane for the planetary gearbox at 8 milliseconds . . . 44
23 Monitor of total mass in the domain . . . 54
24 Monitor of CCN in the solution domain . . . 54
25 Monitor of total mass in the domain . . . 55
26 Monitor of CCN in the solution domain . . . 55
27 Mesh at the symmetry plane for the single planet case . . . 56 28 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
List of Tables
1 Size of gap for different scales of pinion . . . 20
2 Surface remesher properties for FZG gearbox . . . 20
3 Surface controls properties for FZG gearbox . . . 21
4 Volume mesh properties for FZG gearbox . . . 21
5 Churning torque and simulation time for two different meshes . . . . 22
6 Motion setup for FZG gearbox . . . 22
7 Surface remesher properties for planetary gearbox . . . 24
8 Surface controls properties for planetary gearbox . . . 25
9 Volume mesh properties for planetary gearbox . . . 25
10 Motion setup for planetary gearbox . . . 26
11 Churning Torque measurements for the various cases involving FZG gearbox . . . 39
12 Cost analysis for the various cases involving FZG gearbox using 12 processors . . . 40
13 Evaluating the choice of MRF and RBM . . . 41
14 Churning torque measurements for the cases involving the planetary gearbox . . . 43
15 Cost analysis for the various cases involving the planetary gearbox using 12 processors . . . 44
1
Introduction
Transmission 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 this drive force to the differential, through a drive shaft, which drives the wheels. Among the key mechanical components of a transmission are the gears, bearings and shafts which can become damaged or degraded due to friction during contacts. To main-tain the structural integrity of the system, these contact surfaces must be lubricated and cooled through enough supply of oil throughout their operation.
The most common type of lubrication system used in an automotive transmission is the dip-lubricated system where the gears are partially immersed in oil and as the gears rotate they transport the oil to the meshing region. The cooling will be enhanced 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 less efficient 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 torque due to rotation of gears submerged in oil are known as churning power losses and have 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 high immersion depths [2].
Traditionally, measurements on test rigs have been used as the principal means for prediction of churning power losses in transmissions. However, these measure-ments cannot be made at the design stages when no tests can be conducted. This gives 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. Empirical approaches offer another alternative for investigation of these losses however these are often limited to certain constraints and operating conditions, and considerable uncertainties exist when applying these empirical equations. Moreover, empirical equations 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 informainvestiga-tion about the oil distribution. Various advantages when using CFD for prediction of oil flow in the interior of transmissions in the industry include for example, shortening the develop-ment period, reducing the expenses for prototypes, and most importantly bypassing limitations during actual tests for example, due to adherence of oil to the internal walls of the housings [3].
1.1
Purpose
Different modelling approaches have been used in the past to simulate the oil flow inside 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. Simple systems can be studied with a more accurate transient approach using the rigid
tion of the mesh vertices to simulate the gear motion and the approach is known as RBM (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 good compromise between accuracy and computational cost. In this thesis, a thorough comparison between RBM and MRF is made using a single-stage FZG (Technical Institute for the Study of Gears and Drive Mechanisms) gearbox.
A key aspect which determines the computational cost of gear simulations using the RBM approach is the size of the gap created between the two meshing gears usually obtained by scaling the smaller of the two meshing gears. This is done to sufficiently 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 reduce the computational costs for these simulations, however, it can lead to a less accurate prediction of critical parameters such as the churning losses. Therefore, a sensitivity study of the influence of gap size on the churning losses, and computational time is also made using the FZG gearbox.
Figure 1: A schematic that illustrates the actual contact (No Gap) with the simplified contact (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 simulated taking into account the observations from the investigations using the FZG gearbox. Different modelling choices are examined and the optimised simulation is ran where important 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 given in this section pointing out the major aspects from each work that have been used in the various stages of this thesis work. Three different techniques have been used previously for studying flows inside transmissions and the churning power losses
namely test rig measurements, empirical equations and CFD simulations. Here, a brief 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 flow field inside an FZG gearbox for a variety of operating conditions including differ-ent pitch line velocities and oil levels. The experimdiffer-ental data on the velocity field in 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 for a simple test rig consisting of a spur gear rotating in a cylindrical housing through a variety of aqueous glycerol solutions and oils. The results demonstrated a significant influence of the oil disposition within the housing on the measured churning losses where the oil disposition is governed by effects of gravity, inertial forces, surface tension and windage.
Changenet and Velex [6] proposed new formulas for evaluating churning losses for one pinion characteristic of automotive transmission geometry based on the results of 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 the sense of rotation of the pair, the superposition of the individual losses of the pinion and of the gear leads to erroneous figures.
Boni et. al [7] experimentally investigated the churning power losses generated in a dip-lubricated planetary gearbox under loaded conditions for different values of operating parameters i.e. rotational speed, oil sump level and temperature. It was observed 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 housing due to centrifugal force. Other important observations were that the component of churning losses associated with the planet carrier was not very significant and that the sun gear contributed to the churning losses only if the oil level is such that it reaches the region where the teeth of the sun and planet gears are meshed. The authors argue that the applicability of empirical equations, derived as a result of churning loss studies for single gear or gear pair investigations, to planetary gear systems is limited because of the strong predominance of the centrifugal effects in planetary gear sets as opposed to the free surface flows inside cylindrical gear sets where the gravity forces have a larger influence.
1.2.2
CFD studies
The continuous advantages in computational capacity have brought CFD methods into 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 the churning power losses of a single gear by adopting an open-source code, OpenFOAM. The geometry was considered symmetric and the sliding mesh approach was used with the internal mesh rotated by a prescribed angle every time step. Moreover, the
MRF method was suggested as a useful means to reduce the computational effort as compared to the sliding mesh approach. The results showed a significant influence of gear parameters such as the tip diameter and the lubricant level on the churning losses.
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. The obtained 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 the meshing zone between the pinion and the gear at each time step. The pinion and the gear were both scaled to 99 percent of their actual size and it was expected that the results are not significantly influenced by this simplification. The results showed a good agreement both for the oil distribution and the churning loss torque with the high-speed camera recordings and experimental measurements of the loss torque. Concli and Gorla [9] identified the need for development of effective models for the prediction of load-independent power losses and proposed a methodology for prediction of load-independent power losses in planetary gearboxes. The method involves the usage of a customized domain and choice of mesh motion as well as the solver setup for the evaluation of each component of the load-independent losses of the planetary gearbox namely the churning loss due to carrier, churning losses due to gears and oil squeezing losses. The results showed that the most losses for both the load-independent and load-dependent components came from the gears.
Cho et. al [10] used the overset meshing technique for the realistic simulation of a 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 been documented. The results showed an increase in churning losses with an increase in the oil level.
1.3
Limitations
The actual contact between the gears is not considered here instead, a gap condition is created due to the restriction of having at least 4-6 cells between the teeth and the overset interface in the meshing region when using the overset mesh interface in STAR-CCM+. This gap has been obtained by scaling the pinion and the planet gears 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 Link¨oping University in the frame of Nasheim computers and indirect access to clusters at Scania CV AB within the fixed amount of time.
2
Theory
2.1
Governing equations
The governing equations of the flow are the laws of conservation of mass (equation 1) and momentum (equations 2 to 4). These laws can be described in differential or integral form by applying the conservation laws to an infinitesimal fluid element or a finite region of the flow respectively. Here, the differential form of these governing equations is given. The fluid element is large enough to contain a huge number of molecules so that it can be viewed as a continuous medium.
The mass conservation for a compressible flow results from the fact that the rate of increase of mass inside a fluid element is equal to the net rate of flow of mass into the 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 the rate of change of mass and the second term represents the net rate of flow of mass out of the volume. The momentum equations result from equating the change in momentum 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 the
left hand side represent the rate of change in momentum and the net rate of flow of momentum out of the the volume, respectively. The terms on the right hand side represent the pressure and the shear stresses and the rate of change of momentum due to sources [11].
In STAR-CCM+, the finite volume method is used to discretize these equations in 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 and viscous forces. It is expressed as
Re = ρU L
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
Reynolds number. Below this critical value, the flow is smooth and steady and the regime is called laminar flow. At values of Reynolds number above the same critical value, the flow behavior changes drastically and the regime is called turbulent flow. Turbulent flows are characterized by three-dimensional randomness and apparent chaotic behavior. Most industrially relevant flows are turbulent in nature however, in order to resolve every single whirl inside a turbulent flow a very high computational time is required and also most often the time-averaged behavior of the flow is of most significance. The governing equations of the steady mean flow are called the RANS (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 the so-called Reynolds decomposition where the flow variables are decomposed into a time-averaged or steady and an oscillating component. The Reynolds decomposition is introduced for the velocity and the pressure according to
u(t) = u + u0(t) v(t) = v + v0(t) w(t) = w + w0(t) p(t) = p + p0(t)
(6)
where¯and 0 denote the time-averaged and oscillating components respectively. The RANS 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− ρu02) ∂x + ∂(τyx− ρu0v0) ∂y + ∂(τzx− ρu0w0) ∂z + ρfx (7) ∂(ρv) ∂t + ∇.(ρvV ) = ∂(τyx− ρu0v0) ∂x + ∂(−p + τyy− ρv02) ∂y + ∂(τyz− ρv0w0) ∂z + ρfy (8) ∂(ρw) ∂t + ∇.(ρwV ) = ∂(τzx− ρu0w0) ∂x + ∂(τzy− ρv0w0) ∂y + ∂(−p + τzz − ρw02) ∂z + ρfz (9)
where the new stress terms in the RANS equations ρu0iv0j are called the Reynolds
stresses. To close the equations, these terms must be estimated by additional tur-bulence models.
2.2.2
k-ω SST model
The Boussinesq approximation is the assumption that the Reynolds stresses are proportional to the mean velocity gradients according to
τij = −ρu0iv0j = µt ∂ui ∂xj +∂uj ∂xi −2 3ρkδij (10)
where τij is the viscous stress tensor, µtis the turbulent viscosity, k is the turbulence
kinetic energy and δij is the Kronecker delta. It follows that the RANS equations
can 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-, k-ω, Spalart-Allmaras models and the Reynolds stress transport models. For the current thesis work the k-ω SST (Shear Stress Transport) turbulence model has been used because 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 viscous sublayer and switches to a k- behavior in free stream, thus avoiding the sensitivity to inlet free-stream turbulence and extra damping functions. The turbulent viscosity is given as
νt= ρ
k
ω (11)
where ω is called the specific dissipation rate. The transport equations for k and ω read ∂(ρk) ∂t + ∇.(ρkV ) = ∇.[(µ + σkµt)∇k] + Pk− ρβ ∗ fβ∗(ωk − ωoko) + Sk (12) ∂(ρω) ∂t + ∇.(ρωV ) = ∇.[(µ + σωµt)∇ω] + Pω− ρβfβ(ω 2− ω2 o) + Sω (13)
where V is the mean velocity, σk, σω, C1 and C2 are model coefficients, fβ∗ and fβ
are free-shear and vortex-shedding modification factors respectively, Pk and Pω are
production terms, Sk and Sω are user-specified source terms and ko and ωo are the
ambient turbulence values [12].
2.3
Near wall modelling
Turbulent flows are characterized by a more complicated boundary layer as compared to the free shear flows which needs to be considered more specifically. It can be divided into inner and outer regions. The inner region is further classified into three different 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 dimensionless
variable y+ is defined as
y+ =yu
∗
ν (14)
where u∗=qτw
ρ is called the wall friction velocity, y is the distance of the centroid
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. The high y+ wall treatment is employed for y+ > 30. The All y+ wall treatment is a combination of the two above. In this case, the turbulence quantities such as k are evaluated by weighing the solutions coming from both the previous models according to
k = gklow+ (1 − g)khigh (15)
where g is based on the local Reynolds number for the cell. STAR-CCM+ uses the All 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 most common flows are multiphase flows. Some examples of multiphase flows include rainfall, air pollution, boiling, flotation, liquid-liquid extraction and spray drying [13]. The flow inside a transmission is a multiphase flow where two phases, oil and air, 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 inside a continuous medium. The Eulerian framework is used for the continuous phase whereas 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 the Eulerian 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 flow model 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, the Discrete Element Model and the Lagrangian multiphase flow model. Because of its ability 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-E method 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 methods in STAR-CCM+, the Segregated multiphase flow model solves two sets of governing equations, one for each continuous phase. In contrast, the VOF model utilizes the concept of volume fraction hence solves only one set of governing equations. The VOF model is suitable for free surface flows because of its ability to accurately track interfaces between two immiscible phases, it has therefore been chosen in this thesis work.
2.4.1
VOF model
In the multiphase phenomena, the kinematic and dynamic activity of the interface between the phases plays an important role in most cases, so investigating interfaces is important in many multiphase flow analysis. The VOF model implemented in STAR-CCM+ is able to resolve the sharp interfaces by tracing the volume of each fluid instead of the motion of the particles. This is the so-called volume fraction and the volume fraction of an arbitrary phase i is defined as
αi = Vi P iVi = Vi Vcell (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 each cell even though more than one phase exists in it. As a result, only one set of mass and momentum equations is solved for the fluid mixture and the properties of the fluid are calculated based on the physical properties of phases and the corresponding volume fractions. These properties are defined as
ρcell = X i ρiαi (17) µcell = X i µiαi (18)
where ρcellis the density of the cell, ρi is the density of phase i. Similarly, µcell is the
dynamic viscosity of the cell and µi is the dynamic viscosity of phase i. In addition
to the Navier-Stokes equations, the transport equation of the volume fraction is solved. This equation is given by
1 ρq ∂ ∂t(αqρq) + ∇.(αqρqvq) = Sαq+ n X p=1 ( ˙mpq− ˙mqp) (19)
where ˙mqp is the mass transfer from phase q to phase p and ˙mpq is the opposite
transfer. Sαq is the source or sink of the phase q and vq is the velocity of the
phase q. In the current case, the right hand side of equation 19 is zero since no mass 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 the VOF 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
has been used since it satisfies the CFL (Courant-Friedrichs-Levy) criterion at the volume fraction interface. For the cases involving the RBM model, the Convective CFL Time-Step Control model has been used to ensure that the CFL criteria is always 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 has been 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 − ω SST 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 to the 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 order upwind scheme and the HRIC scheme has been implemented as a discretization scheme for the convective terms in VOF model. The blending between the two schemes is governed by the local CFL number. It is given by
CF L = ucell∆t
∆x (20)
where ucell is the velocity in the cell, ∆t is the time step and ∆x is the local grid
size. Thus, in order to ensure that the HRIC scheme is used at the interface in order to obtain a sharp interface, the CFL value at the interface should not exceed beyond a certain limit. By default, the HRIC scheme in STAR-CCM+ is optimized for free surfaces 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 to generate high quality local structured meshes by dividing the domain into smaller sub-domains because of the inability to get a good mesh for particularly complex geometries using an unstructured grid. Overset meshing technique in STAR-CCM+ allows a more realistic simulation of gears engagement because of the inclusion of the meshing region between the teeth thus avoiding the need to, for example, scale down one of the gears unphysically to get rid of the meshing region.
The implementation of the overset methodology in STAR-CCM+ can be divided into two steps. The first step is the decomposition of the domain into different sub-domains and the second one is the choice of the coupling method between these sub-domains in order to obtain an accurate, unique and efficient solution [12]. A background mesh is given for the entire computational domain and each sub-domain,
a gear or carrier in the current case, is enclosed by an overset mesh. These overset meshes 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 done through 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 these equations 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 overlapping grids. They transfer data between the overlapping meshes. The exact relationship between 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 X k=1 αωkφDk (21)
where φ is the interpolated property value at point Pi derived from the property
values at points Di, balanced with the interpolation weight αωk.
2.7
Rotational motion
A variety of engineering applications use rotating components. Different strategies exist 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 models for modelling rotating components.
2.7.1
MRF
The MRF model is a steady-state approximation in which individual cell zones move at different rotational/translational speeds. It is a suitable choice when the flow at the interface between the zones can be assumed as uniform.
In MRF implementation, the calculation domain is divided into sub-domains which can be assumed as translating and/or rotating according to the laboratory (inertial) frame. The governing equations in each sub-domain are obtained with respect to the sub-domain’s reference frame. The position vector r of an arbitrary point ’P’ in the computational domain, as shown in Figure 2 with respect to the origin of the zone rotation axis is given by
r = x + xo (22)
where x represents the position of the point ’P’ in the laboratory reference frame and xo represents the position of the origin of the zone rotation axis in the laboratory
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 is then given by
vlab= vr+ (ωt× r) + vt (23)
where vlab is the velocity in the laboratory frame, vr is the velocity in the moving
reference frame, ωt is the angular velocity of the moving reference frame and vt is
the 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 gives the 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 represent the Coriolis and centripetal accelerations respectively [4]. In STAR-CCM+, the MRF 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 vertices according to user-specified rotations, translations or a combination of both. It is an unsteady approach which is more accurate than the MRF model especially when there is a strong interaction between the moving part and other stationary or moving parts in its vicinity. It is also, however, more computationally demanding than the MRF 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.
The modelling of flow inside a transmission is complicated due to oil agitation by multiple rotating components, mixing of oil and air to create unsteady and complex flows and existence of extremely fine spray and sudden changes in pressure in the gear meshing sections [18]. These factors together increase the computational time of the gear simulations. In the current thesis work, flows inside an FZG gearbox and 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 test rig (shown in Figure 3) which is the closest method to practice on predicting scuffing and wear properties of gear oils [1]. The inner dimensions of the gearbox are 270 mm x 180 mm x 56 mm. The gears utilized are the FZG gears type C. More details about 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 sun wheel or the housing, depending on gear selection), a sun wheel (fixed to the main shaft by splines), five planet gear wheels and a planet carrier (which forms the front end of the output shaft). The main shaft which is fixed to the splines inside the sun gear has been removed in Figure 4 for easy viewing. The outer radius of the ring gear is approximately 130 mm.
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-independent power losses. Load-dependent losses are related to mechanical power losses due to friction at the contact of rolling parts. However, the load-independent losses are related to the interaction between the oil and the rotating/moving components and are independent of the transfer of torque between gears. These losses are further classified 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 of load-independent power losses, and it is associated with the movement of gears in the 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 lubricant and 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 and rotational speed of the gears. The drag torque responsible for these losses is known as churning torque and originates from two different forces namely the pressure and
shear forces of the oil. The pressure and shear forces on an arbitrary face i are evaluated 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 area
3
Method
In this chapter, the softwares and the methodology used for the cases involving both the 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 the solution assessment sections provide the common settings and solution parameters for the two gearboxes. The additional investigations section provides the strategies used to overcome the instabilities initially encountered with the planetary gearbox simulation.
3.1
Softwares
Mainly, two softwares have been used in this thesis and are being presented in this section: ANSA for pre-processing and CAD-cleaning and STAR-CCM+ for meshing and simulations.
3.1.1
ANSA
The pre-processing software from BETA CAE Systems S.A. is a CAE pre-processing tool used for building and cleaning CAD as well as meshing. In this thesis, ANSA has been used to prepare the two geometries prior to meshing in STAR-CCM+. This preparation included simplifying complex surfaces, removing holes and other unwanted 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 for this thesis. The software is a commercial simulation software capable of solving multi-disciplinary problems in both fluid and solid continuum mechanics and has an integrated mesher. The geometry already prepared in ANSA is imported into STAR-CCM+, where the surface and volume meshing is done and the solution to the cases is calculated. STAR-CCM+ is capable of handling many different physics settings including regions with RBM, MRF and multiphase flows.
3.2
FZG gearbox
This section provides a detailed description of the method followed for the cases involving the FZG gearbox. This includes all the stages from the preparation of the geometry to the creation of volume mesh and definition of motion. Figure 6 shows the overview of the common workflow for both gearbox configurations.
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 pinion with a center-to-center distance of 91.5 mm. The shafts and the pockets are removed from the gears in ANSA and the housing is designed directly in STAR-CCM+. To save computational resources, a symmetric geometry of the FZG gearbox, as shown in 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 using the overset mesh interface between regions in STAR-CCM+. Hence, a gap is created between the gears by scaling the pinion uniformly in three dimensions. For the gap sensitivity study, three different scales of the pinion have been used. The biggest scale used is 91 percent followed by 85 percent and 80 percent of the actual size of the pinion. The choice of the scales is made such that the gap size is between 25 and 50 percent of the gear tooth height [12]. The gap sizes for the different scales of the pinion are given in Table 1.
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 each of the moving components in the domain. Hence, cylindrical parts are created around the 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 of three 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 the selection of method for the interpolation of solution across these interfaces which in the 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 the geometry followed by the creation of the volume mesh which is the three-dimensional mesh for resolving the flow. In STAR-CCM+, both the surface and volume meshes can be generated with the so-called automated mesh process. The quality of the volume mesh is dependent on the quality of the surface mesh therefore, the surface mesh is analyzed carefully. The properties of the surface remesher for the FZG gearbox in STAR-CCM+ are given in Table 2.
Table 2: Surface remesher properties for FZG gearbox
Part Base 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 the pinion and gear oversets. The first surface control is used to get a smooth transition between the overset and background regions at the overset interface and the second surface control is used to sufficiently resolve the teeth profile of the pinion and the gear. It is also used to specify the prism layer settings for the teeth in order to get an acceptable wall y+ value as well as achieve a minimum of 4 cells between the teeth and the overset interface, which is a requirement of the overset hole-cutting algorithm. The settings of the surface controls are given in Table 3.
In addition to the surface controls, two volumetric controls have been used to have both a refinement in the meshing region and a smooth transition at the overset interfaces. The first volumetric control is common to the pinion overset and the
Table 3: Surface controls properties for FZG gearbox Surface control Target surface size (mm) Minimum surface size (mm) Number of prism layers Prism layer total 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 the background mesh. This is done to have similar cell sizes at the overset interfaces which is an important factor to ensure mass conservation when using the overset meshing technique. The settings of the volume meshes for the three regions and the volumetric controls are given in Table 4. The volume mesh at the symmetry plane in the FZG gearbox is shown in Figure 8 where each color represents a different region.
Table 4: Volume mesh properties for FZG gearbox
Region/Part Base size (mm) Pinion overset 1 Background 1 Gear overset 1 Volumetric control 1 0.5 Volumetric control 2 0.5
Figure 8: Volume mesh at the symmetry plane in FZG gearbox
3.2.4
Mesh sensitivity analysis
In order to verify that the results are independent of the mesh, two different meshes have been investigated for the FZG gearbox. The approach is to start with a mesh having a reasonably good resolution in areas of interest such as the meshing region and at the interfaces between regions and then refine the mesh more in the areas of interest 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 simulatpin-ion times. The cell count is varied by changing the mesh size for the two volumetric controls. It can be seen that less than one percent difference exists between the churning torques from the two meshes. However, the finer mesh takes approximately eighty percent more time to simulate 0.1 seconds of the motion. As a result, the coarser mesh has been chosen for later analysis.
Table 5: Churning torque and simulation time for two different meshes
Rotation model Pinion Scale (%) Cell count (million) Churning torque (Nm) Simulation time for 0.1 sec (hrs) Gear Pinion RBM 91 3.45 0.005023 0.001764 44 6.13 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. The FZG gearbox has two regions with prescribed motion, the gear and the pinion. For the FZG gearbox, both modelling options have been investigated and compared while using the same scale of the pinion and the same volume mesh. This is done to accurately judge the capabilities of both these methods. Additionally, the gap sensitivity study involves the usage of the RBM model with three different scales of the pinion as mentioned in Section 3.2.1. The motion setup for the FZG gearbox is given in Table 6. In order to achieve a steady state solution, two revolutions of the gear have been considered which corresponds to a physical time of approximately 1.2 seconds.
Table 6: Motion setup for FZG gearbox
Pitch line velocity (m/s) Rotational speed (rpm) Direction Gear 0.6 158.72 CCW Pinion 105.86 CW
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 symmesymme-try about this plane. All other surfaces of the domain including the surfaces of the gear and pinion are considered as no-slip walls. The boundaries of the overset regions have been specified as overset boundary condition in order to be able to create overset mesh interfaces between the 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 10 shows the planetary gearbox after preparation in ANSA where each color represents a separate part. In order to make the simulation computationally feasible, some details 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 has been filled with a protruding cylinder as can be seen in Figure 10.
Figure 10: Prepared geometry of planetary gearbox
Moreover, the surfaces have been grouped in ANSA to aid in the creation of a good quality mesh and for easy application of appropriate boundary conditions in STAR-CCM+. Lastly, the planet gears have been scaled to 85 percent uniformly in three dimensions inside ANSA.
3.3.2
Setting up overset regions
The overset regions are created directly in STAR-CCM+ for the planetary gearbox as well. Seven overset regions are created for the planetary gearbox; five for the planet gears, one for the sun gear and another for the planet carrier. This gives a 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 of interfaces between regions.
3.3.3
Surface and volume meshing
The planetary mesh has also been generated taking advantage of the automated mesh operation in STAR-CCM+. The properties of the surface remesher for the planetary gearbox in STAR-CCM+ are given in Table 7.
Table 7: Surface remesher properties for planetary gearbox
Part Base 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 operations for all overset regions in the planetary gearbox. Surface controls 1 and 2 perform the
same function here as they did in the FZG gearbox. The settings for these surface controls are given in Table 8.
Table 8: Surface controls properties for planetary gearbox
Surface control Target surface size (mm) Minimum surface size (mm) Number of prism layers Prism layer total 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 the teeth for the planet, sun and ring gears. Volumetric controls with twice the base size are defined for the regions in the carrier and the ring gear domains where the cells in the respective regions are cut by the hole-cutting process and therefore do not contribute to the solution process. The settings of the volume meshes for all regions and the two type of volumetric controls are given in Table 9. The volume mesh for the planetary gearbox is shown in Figure 11.
Table 9: Volume mesh properties for planetary gearbox
Region/Part Base 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
3.3.4
Setting up motion
At this stage, the motion is set up using the RBM model. The planetary gearbox has 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 gears Rotation 528 rpm CW Revolution 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 works well for multiphase flows. As indicated before, the k − ω SST turbulence model has been 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) has been considered as the primary phase with a density of 902 kg/m3 and dynamic viscosity of 5.77E-2 P a.s, while air with a density of 1.18 kg/m3 and dynamic vis-cosity of 1.86E-5 P a.s has been considered as the secondary phase. Since the surface tension coefficent for oil is available only at 40◦C, this value has been used in spite of the simulations being run for a temperature of 20◦C. A contact angle of 15 degrees has been given between the two phases and all simulations have been run for oil level 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 as 0.8 and 0.6 respectively. Convective CFL time step control with a target mean CFL number of 0.5 and max CFL number of 0.7 has been given. To find out more about what these settings represent, the interested reader can refer to an introductory book 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 to reach 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 of interest and are monitored within and between time steps. In addition, the CCN and total domain mass have also been monitored to ensure mass conservation since the overset meshing technique and the VOF model are not inherently conservative. The important monitor quantities are presented in Appendix B.
the gears and carrier surfaces have been written to a report. These variables are used to calculate the pressure force and shear force as explained in Section 2.8.3 which 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 in the planetary gearbox simulation. However, instabilities are encountered during the simulation of the planetary gearbox with these settings. Since the solver settings work 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 of
planetary gearbox.
2. Adopt some alternative modelling choices for the planetary gearbox simulation such 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 and partitioning solvers. After verification of these two aspects of the simulation, the two strategies outlined previously have been implemented and modified simulations are executed.
In terms of simplifying the geometry, a key factor which has led to the decision of starting with a very simple and less computationally expensive geometry is to be able to run the simulation locally during the debugging stage and thereby minimiz-ing both the time lost while the simulation is queued and unnecessary occupation of resources. Therefore, once it has been identified that the planetary geometry is not compatible with the solver settings used for the FZG gearbox, it is decided that a simpler version of the planetary geometry be used to investigate the effect of 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 the changes adopted in the simulation. The details of this geometry and some of the results are given in Appendix C.
Followed by the geometric simplifications, changes are introduced gradually in the physics setup of the planetary gearbox simulation. Since the turbulence transport equation 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 − model 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 the flow inside a planetary gearbox, due to additional terms for modelling the effects of anisotropy, curvatures and rotational effects [12].
Similarly, rotation speeds of the gears are defined using a conservative ramp over
1000 time steps as well as the final rotation speeds of the gears are reduced six times as compared to the motion setup given in Table 10. The compressible ideal gas model is used for the air phase in contrast to the previously used constant density model which means that the density is now determined by the local pressure and temperature. 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 mass and momentum. This on the one hand increases the computational expense but on the other hand provides stability to the simulation and enables the use of bigger time 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 stability of the solution can be enhanced through a manual specification of pressure in a location using a pressure outlet, for example. This reduces the round-off error in numerical calculations involving pressure therefore, pressure outlet is specified at a face in the air phase [12]. Moreover, simulations are run in double-precision to prevent instabilities.
Once the single planet case has been running for a significant physical time without any instabilities, the geometry for the next stage is utilized. A big complexity with the complete planetary geometry is the simulation of overlapping motions of the planetary carrier with the sun and planet gears. Although, theoretically, the over-set meshing technique in STAR-CCM+ provides a good solution to simulate these overlapping motions, however, it has been decided to remove the carrier for the next stage because of two major advantages:
1. It reduces the complexity of the simulation.
2. It significantly reduces the computational requirements on the simulation due to both the symmetric nature of the geometry which reduces the mesh count in half as well as the reduction in the number of overset interfaces leading to a 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 described in Section 3.3. However, due to limitations with the memory allocation on clusters at Scania, this simulation could not be run on the clusters and has instead been ran locally on 12 processors and the simulation behavior is judged based on some preliminary results.
Figure 12: Reduced geometry of planetary gearbox
Figure 13: Volume mesh at the symmetry plane in reduced planetary gearbox
4
Results and Discussions
In this section the results have been presented to validate their accuracy. These include the results for the different analysis performed on the FZG gearbox and the results from the planetary gearbox configurations.
4.1
FZG gearbox
For FZG gearbox, the results from MRF and RBM models and the gap sensitivity study are presented. Figure 14 presents the development of the volume fraction of oil at the symmetry plane of the FZG gearbox with time as obtained by using the MRF and the RBM models. The contours up till 2 revolutions of the gear show the instantaneous volume fraction of oil whereas the contours for 2.25 revolutions show 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 the time-accurate behavior of the flow which is reasonable since the flow around the gear and pinion is modelled as a steady-state problem with respect to the MRF zones for the gear and the pinion as given in Section 2.7.1. As a result, it can be seen that the mean volume fraction of oil from the two strategies resembles each other in the reservoir but not close to the gear and pinion teeth. This limitation of the MRF model is because it does not account for the relative motion of a moving zone with respect to adjacent zones. This is analogous to freezing the motion in a specific position and observing the instantaneous flow field with the gears in that position [4].
The development of the velocity field of oil flow at the symmetry plane as obtained by the MRF and RBM models is presented in Figure 15. It can be seen that the velocity 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 localized region 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 two models, 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 direcmo-tion for the MRF model. This is essentially a representation of how the motion is being handled in the two 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 the gear rotation essentially rotating it in the opposite direction.
Figure 14: Development of volume fraction of oil at the symmetry plane using the MRF and the RBM models in STAR-CCM+ (All contours except at 2.25 revolutions of gear are instantaneous whereas the ones for 2.25 revolutions represent the mean volume fraction). Volume fraction of 0 indicates only air is present inside the discretized cell and 1 indicates oil.
Figure 15: Development of velocity field (m/s) of oil flow at the symmetry plane with the MRF and RBM models (All contours except at 2.25 revolutions of gear are instantaneous whereas the ones for 2.25 revolutions represent the mean velocity). The difference between the two frames is due to the use of a volume fraction driven part.
