Numerical Visualization of Grease Flow in aGearboxHua Liu  ( liu@fzg.mw.tum.de )

Gear Research Centre https://orcid.org/0000-0001-5352-0999Florian Dangl 

Gear Research Centre (FZG)Thomas Lohner 

Gear Research Centre (FZG)Karsten Stahl 

Gear Research Centre (FZG)

Original Article

Keywords: lubrication, grease, geared transmission, numerical modeling

Posted Date: October 26th, 2020

DOI: https://doi.org/10.21203/rs.3.rs-95128/v1

License: This work is licensed under a Creative Commons Attribution 4.0 International License.  Read Full License

Numerical Visualization of Grease Flow in a Gearbox Liu, H.1)*; Dangl, F.1); Lohner,T.1); Stahl, K.1)

1) Gear Research Centre (FZG), Technical University of Munich (TUM), Boltzmannstrasse 15, 85748 Garching near Munich, Germany

*Corresponding author: liu@fzg.mw.tum.de

ORCID ID Liu, H.: 0000-0001-5352-0999

ORCID ID Lohner, T.: 0000-0002-6067-9399

ORCID ID Stahl, K.: 0000-0001-7177-5207

Abstract

Gearbox lubrication plays a decisive role in both load capacity and efficiency. It is necessary for the formation of a lubricant film in tribological contacts, and for heat dissipation. Grease lubrication is often applied to gearboxes at a low operating speed and means that, for example, less effort is required in sealing design compared to oil lubrication. Over the years, the com-putational fluid dynamics (CFD) method has frequently been applied to support the lubrication-related design of gearboxes. However, very few studies have focused on grease lubrication. Whereas oil can generally be considered as Newtonian fluid, grease displays complex non-Newtonian flow behavior. In this study, the grease lubrication of a single-stage test gearbox is investigated by means of a finite-volume based CFD model. The numerical results on the grease flow show good accordance with high-speed camera recordings, and clearly display the lubrication mechanisms of “channeling” and “circulating”. These first results on grease flow simulation in gearboxes offer a starting point for further numerical studies.

Keywords: lubrication, grease, geared transmission, numerical modeling

1. Introduction

The decisive aspects for the design of geared transmissions are noise-vibration-harshness (NVH) behavior, load capacity and efficiency. The last two aspects in particular greatly depend on the lubricant. Fluid lubrication is generally required for the formation of a lubricant film in tribological contacts, and for heat dissipation, which usually requires the majority of available lubricant. Fluid lubrication with liquids may refer to either oil or grease. Due to its efficient heat dissipation and film-formation capability, oil lubrication covers almost 97 % of all lubricant ap-plications in hydraulics, engines and transmissions [1].

Compared to oil lubrication, grease lubrication has a limited ability for heat dissipation, and is therefore often applied to applications at lower operating speeds and transmitted powers. It is advantageous in terms of sealing design compared to oil lubrication, as grease collars can assist with sealing. Moreover, due to its high consistency, grease tends to adhere to lubrication points, which can give effective protection against fretting and corrosion. Grease also can achieve a long service life without maintenance, and can be used for lifetime lubrication. Greases are distinguished according to their consistency, as described by the NLGI con-sistency number. According to Schultheiss [2], greases of low consistency (NLGI 00 and 0) are typically used in the lubrication of open gear drives, e. g. in rotary furnaces or ball mills, whereas greases of higher consistency (NLGI 1 and 2) are typically applied to smaller gear-boxes with difficult sealing conditions.

In the context of the lubrication of geared transmissions, the application of computational fluid dynamics (CFD) methods has developed into a helpful tool for assisting with the design of fluid flow in gearboxes. Most CFD studies have focused on oil lubrication, e. g. Liu et al. [3][4][5], Concli and Gorla [6–8], Kvist et al. [9] and Klier et al. [10]. Whereas oil can usually be consid-ered as Newtonian fluid at low shear rate and pressure, grease exhibits complex non-Newto-nian flow behavior.

There are only very few investigations of CFD modeling of grease flow. Westerberg et al. [11] modeled three lithium-soap greases of different consistency classes flowing through a straight channel by means of a 2D finite-volume-based CFD model, using a single-phase Herschel-Bulkley rheology model. The numerical results in terms of velocity profiles closely match the experimental and analytical results, capturing the yield- and shear-rate-dependent character-istics of grease flow. Based on [11], Westerberg et al. [12] modeled the grease flow in the grease pocket of a double-restriction seal geometry. Different rheological properties corre-sponding to NLGI 00, 1 and 2 greases, low and high rotational speeds as well as different temperatures were investigated. It was found that the flow and velocity distribution in the pocket is characterized by the rheological behavior of the grease. For high shear rates and tempera-tures, the grease approaches Newtonian behavior leading to reduced yield and shear-thinning characteristics. Chiranjit [13] conducted single-phase numerical simulations on the laminar flow of greases in channels with rectangular cross section. A Herschel-Bulkley rheology model was applied. The numerical results are compared to both analytical results and flow measure-ments with particle image velocimetry. By means of a 3D finite-volume-based CFD model, Gertzos et al. [13] modeled the grease flow in a hydrodynamic journal bearing based on a Bingham rheology model. The performance characteristics such as relative eccentricity and pressure distribution are derived and presented for several geometries and shear rates. The results are found to be in good accordance with experimental and analytical data from previous investigations on Bingham fluids. Yoo and Kim [14] conducted numerical investigations on thermal elastohydrodynamic lubrication (TEHL) in line contacts with grease using a Herschel-Bulkley model. The results show that thermal effects play an important role especially at high rolling speeds and that the flow index and viscosity parameter of the Herschel-Bulkley model essentially influence the TEHL contact. Nogi et al. [15] investigated the grease film thickness in EHL point contacts numerically at low speeds.

Although a few studies on CFD modeling of grease flow are available, no numerical studies have been found on grease flow in gearboxes.

In this study, a FV-based CFD model of a single-stage FZG test gearbox was built to investi-gate the grease flow. The rheology of the grease is described by a modified Bingham model. Three different grease fill levels as well as two circumferential speeds are considered. The numerical results are compared with high-speed camera recordings. It is shown that the lubri-cation mechanisms of “channeling” and “circulating” can be reproduced. The work contributes to a better understanding of lubrication in grease-lubricated gearboxes.

2. Grease Lubrication in Gearboxes

Greases consist of base oil (65 – 95 %), thickener (5 – 35 %) and additives (0 – 5 %) [1]. According to DIN 51818 [16], the consistency of a grease is characterized by NLGI consistency numbers.

Circulating and channeling were identified as the two main lubrication mechanisms in grease-lubricated gearboxes (Fukunaga [17], Hochmann [18] and Stemplinger [19]) (cf. Figure 1). Cir-culating refers to certain operating conditions where grease is present at the tooth gaps, and is circulated. It provides a good lubricant supply to the gear mesh and heat dissipation. Chan-neling refers to certain operating conditions where the grease is barely present at the tooth gaps, and therefore only very little grease is supplied to the gear mesh for lubrication. Only an accumulation of separated oil supports the lubrication of gears. Channeling usually leads to poor lubrication and heat dissipation, and hence to a high risk of wear and scuffing. There is no general design criterion to predict the lubrication mechanism of grease-lubricated gear-boxes. It strongly depends on the temperature, grease fill level, load speed and grease com-position (Stemplinger [19]).

Siewerin et al. [20] showed that the grease flow can vary significantly with temperature. Greases with NLGI 1 show more availability on the tooth flanks at a sump temperature of 90 °C compared with 60 °C, thus supporting the circulating lubrication mechanism. Stemplinger [19], Fukuna [17] and Hochmann [18] proved a strong influence of the grease fill level on the lubrication mechanisms “circulating” and “channeling”. Stemplinger [19] performed experi-ments on a FZG back-to-back test rig, where a fill level of 54 % corresponds to the middle of the axes of pinion and wheel. He found that for grease fill levels of 40 and 50 %, channeling dominates, and the gears tend to dig a free space around themselves. The grease next to the gears is displaced and does not return due to a lack of replenishment, so that a gap forms between the rotating gears and the grease. For a grease fill level of 80 %, circulating predom-inates and the grease tends to stay at the tooth flanks and is circulated around. Stemplinger [19] suggests a grease fill level of 50 % as a good compromise between acceptable no-load gear losses and sufficient lubricant supply on the one hand, and heat dissipation on the other.

The thickener can also influence the lubrication mechanism. Fukunaga [17] states that the consistency is more important than the base oil viscosity. He shows that lithium- and aluminum-complex soap thickeners have the best phase separation, which is advantageous for both channeling and circulating. A calcium complex soap thickener tends to dig a free space and promotes channeling. Combined with limited phase separation capability, it leads to a poor lubricant supply of the gear mesh.

According to Stemplinger [19], the rotational direction has no remarkable influence on the lu-brication mechanism, whereas the speed has a strong impact on the grease flow behavior. At low circumferential speeds, circulating is the dominant lubrication mechanism. With higher cir-cumferential speeds, the predominant lubrication mechanism becomes channeling. This effect was also observed by experimental investigations with small module gears by Schultheiß [21].

Fischer et al. [22, 23] point out that the base oil type of a grease can strongly affect film for-mation in elastohydrodynamically lubricated contacts. The presence of a thickener layer can separate the contacting surfaces under starved lubrication.

Grease lubrication in gearboxes depends on a large number of influencing factors, which affect the efficiency, heat dissipation and load capacity. Due to the complex behavior of greases, experimental investigations are difficult to accomplish, particularly grease-flow visualization. The application of CFD can provide a detailed insight into the grease-flow behavior and lubri-cation in gearboxes.

3. Rheology of Greases

The rheological behavior of fluids can be Newtonian or non-Newtonian (Chhabra and Richard-son [24]). Figure 2 gives a brief overview of the different rheological behaviors. The shear stress 𝜏 of Newtonian fluids (a) is proportional to the shear rate �̇�, with the dynamic viscosity 𝜂 as a constant of proportionality: 𝜏 = 𝜂 ∙ �̇�

(1)

The viscosity of non-Newtonian fluids depends on the shear rate. A fluid with increasing vis-cosity and increasing shear rate (shear thickening) is called dilatant (c). A fluid with decreasing viscosity and increasing shear rate (shear thinning) is called pseudoplastic (b). Bingham fluids (d, e) show a yield stress and fluid flow that only occurs when the external stresses are larger than the yield stress. The behavior can then be plastic (d) or pseudoplastic (e). Moreover, the fluid behavior may be time-dependent with increasing viscosity (rheopexic) and decreasing viscosity (thixotropic). Viscoelastic fluids have elastic and viscous properties.

Figure 1: Circulating (left) and channeling (right) as main lubrication mechanisms of grease-lubricated gearboxes (Stemplinger [19])

Oils show typically a pseudoplastic (b) non-Newtonian behavior at high pressure and shear rate. In the context of gearbox oil flow, it generally behaves in a Newtonian manner. Con-versely, greases display Bingham-like behavior (d, e). Due to the thickener, they behave like elastic solid materials as long as the external stress is below the yield stress. When the yield stress is exceeded, it starts to flow. Thereby, a plastic behavior with constant viscosity (d) or a pseudoplastic (e) behavior with decreasing viscosity and increasing shear rate can occur.

4. Object of Investigation and Operating Conditions

In this section, the con-sidered test gearbox and grease as well as the op-erating conditions are presented.

4.1 Test Gearbox

The object of investiga-tion is the test gearbox of a FZG back-to-back test rig according to ISO 14635-1 [25] (cf. Figure 3). It is based on the prin-ciple of power circula-tion.

The test gearbox fea-tures the same geometry as considered in previous studies (Liu et al. [3–5]). It contains a pair of spur gears with a gear ratio of 1.5. The front cover is transparent, which allows optical ac-cessibility and thus a validation of the lubricant distribution with high-speed camera recording. In this study, a high-speed camera of type Photron FASTCAM Mini AX200 is used.

4.2 Test Gears

Figure 2: Schematic representation of different rheological behavior of fluids (based on Chhabra and Richardson [24])

Figure 3: FZG back-to-back test rig (Siewerin [20])

Shear rate �̇�

She

arst

ress

𝜏

Shear rate �̇�V

isco

sity

𝜂a. Newtonian

d. Bingham fluid plastic

𝜏0

𝜂

𝜂

temperature sensor

load coupling

motor

torque measuring device

slave gear unit

test gear

test pinion

The considered gears are case-hardened test gears of FZG type C-PT as used in [3–5]. The main geometry data for both the pinion and wheel is shown in Table 1.

Table 1: Main geometry data of the considered test gears of FZG type C-PT a in mm z1|2 mn in mm αn in ° x1|2 in mm b1|2 in

mm da1|2 in

mm

Pinion (1) 91.5

16 4.5 20.0

0.182 14

82.5

Wheel (2) 24 0.171 118.4

4.3 Grease

A model grease with measured rheology data according to Table 2 is considered. The grease was sheared at 1,000 1/s and 25 °C. After that, its viscosity was measured at different shear rates. The arithmetic mean value was calculated from ten single measurements with the same grease.

The base oil was extracted from the grease and measured separately. The grease shows a clear dependency of the viscosity on the shear rate, whereas the viscosity of the base oil is independent of the shear rate.

Table 2: Measured shear rate and viscosity for the considered grease and its base oil at 25 °C Grease Base oil Shear rate �̇� in 1/s Viscosity η in mPas Viscosity η in mPas 0.1 2,012,170.00 49.74 1 176,862.50 49.74 10 47,838.20 49.74 100 6,864.04 49.74 1,000 1,163.52 49.74 10,000 262.06 49.74 15,000 220.27 49.74

4.4 Operating Conditions

In order to investigate the grease flow, the grease fill level and the circumferential speed are varied. Based on the literature, these two influencing factors have a strong influence on the two lubrication mechanisms: circulating and channeling (cf. section 2). Based on the investi-gations of Siewerin [20] and Stemplinger [19], the circumferential speeds of vt = {0.05; 0.38} m/s and the grease fill levels of {40; 54; 80} % are considered. The direction of rotation is chosen such that the wheel in Figure 3 rotates clockwise and the pinion counter-clockwise. The present secondary medium in the gearbox is air. After each measuring point, the gears were cleaned and the grease fill level was readjusted carefully.

Table 3: Considered operating conditions Rotational

speed of pinion n1 in rpm

Circumferential speed vt in m/s

Grease fill vol-ume in %

Grease temper-ature in °C

1 13 0.05 40 25 2 100 0.38 3 13 0.05 54 4 100 0.38 5 13 0.05 80 6 100 0.38

5. Numerical Model

The numerical model is implemented and solved using the commercial CFD software Ansys Fluent 19.0. The Eulerian Finite Volume Method is applied. By solving the conservation equa-tions, state variables like pressure, density and velocity of the medium can be determined at every position and every time within the domain.

5.1 Governing Equations

The Navier-Stokes equations are a compilation of conservation equations with respect to mass, impulse and energy. A generalized form of the conservation equations is shown in equa-tion (2) (Schwarze [26]) with the specializations shown in

Table 4. Depending on the variables substituted in this equation, different forms of conservation result.

Table 4: Specializations of the generalized conservation equation (Schwarze [26])

Φ 𝐷Φ 𝑄Φ

Mass conservation 1 0 0

Impulse conservation �̅� ∇ ∙ 𝜏̅ −∇p + η∆�̅� + ρ�̅� + F

Energy conservation h −∇ ∙ �̅�´´ 𝜕𝑝 𝜕𝑡 + ∇ ∙ (𝜏̅ ∙ �̅�) By means of equation (2), an arbitrary state variable can be substituted for Φ. The first sum-mand on the left-hand side describes the partial derivation of the state variable. It describes the change in the state variable over time. The second summand on the left-hand side repre-sents the transport of flow information, which is known as the convective part of the state var-iable. The first summand on the right-hand side of the equation presents the transport of flow by random flow motion, which indicates the diffusive part of flow, whereas the second sum-mand on the right-hand side represents the different types of sources.

There are only very few simple flow configurations, such as the direct flow against the cylinder, which can be described by analytical equations. If the problem becomes more complex, nu-merical methods are required to solve these conservation equations. Thereby, the conserva-tion equations are transformed into a differential or integral form. The integral form known as the Finite Volume Method (FVM) is a common form for solving fluid mechanical problems due to its conservative character, which means that the state variables are conserved even after derivation. By applying the Gaussian Integral, the equation (2) reads as follows:

∫ 𝜕𝜕𝑡 (𝜌𝜙) 𝑑𝑉 + ∮(�⃑� 𝜌𝜙) ∙ �⃑� 𝑑𝐴𝑆𝐹 − ∮(Γ∇𝜙) ∙ �⃑� 𝑑𝐴𝑆𝐹 = ∫ 𝑄𝜙 𝑑𝑉𝐶𝑉𝐶𝑉 (3)

CV and SF indicate the control volume and its surface. By summing up the surface integral of the entire control volumes, the surface integrals of the inner control volumes cancel each other out, whereas the surface integrals of the outer control volumes remain in the system.

The conservation equations in integral form are solved by numerical approximation. First, the integrals are approximated by Gauss quadrature as function at one or a couple of points on the border of the control volumes. Second, the values on these points are determined by in-terpolation with known values in adjacent control volumes. Third, the derivations are replaced and solved by finite differences on the different integration points. The accuracy of the numer-ical approximation increases with the order of interpolation and the fineness of discretization.

For multiphase problems with immisible fluids, the Volume of Fluid (VoF) method is applied. Thereby, the fluid variables in each finite volume either originate from a single phase, or from

𝜕 𝜕𝑡 (𝜌𝛷) + ∆(𝜌�̅�𝛷) − 𝛻 ∙ (𝛤𝛻𝛷) = 𝑄𝛷 (2)

𝐹Φ 𝐷Φ

local tem-poral change within CV

convective flow across the border of CV

diffusive flow across the border of CV

sources

a mixture of phases, depending on the volume fraction 𝛼 in each finite volume (Hirt and Nichols [19], Zhang et al. [27]). For the density 𝜌, it reads: 𝜌 = 𝛼𝜌𝑞 + (1 − 𝛼)𝜌𝑝 (4)

The conservative form of the scalar advection equation for the volume fraction 𝛼 can be writ-ten as: 𝑑𝛼𝑑𝑡 = 𝜕𝛼𝜕𝑡 + 𝑢𝑖 𝜕𝛼𝜕𝑥𝑖 =

(5)

𝜕𝛼𝜕𝑡 + ∇(𝛼�⃑⃑� ) = (6)

Equation (6) is integrated over control volumes and a time interval 𝛿t with finite volume method. By applying the Gauss integral, equation (6) leads to:

∫ ( ∫ 𝜕𝛼𝜕𝑡 𝑑𝑉𝐾𝑉 ) 𝑑𝑡 + ∫ ( ∫ ∇(𝛼�⃑⃑� )𝑑𝑉𝐾𝑉 ) 𝑑𝑡 = 𝑡+𝛿t𝑡 𝑡+𝛿t

𝑡

(7)

By using the Gauss quadrature, equation (7) for the volume fraction 𝛼 becomes a bunch of linear equations, which can be solved numerically, whereas 𝑃 indicates the center of the con-trol volume and 𝑓 represents the centroid of its border: (𝛼𝑃𝑡+𝛿t − 𝛼𝑃t ) 𝑉𝑃 + +∑ [(1 − 𝜂)(𝛼𝑓𝑈𝑓⃑⃑⃑⃑ 𝐴𝑓⃑⃑⃑⃑ )𝑡 + 𝜂(𝛼𝑓𝑈𝑓⃑⃑⃑⃑ 𝐴𝑓⃑⃑⃑⃑ )𝑡+𝛿t] 𝛿t = 𝑛𝑓=1

(8)

According to the discretization scheme, 𝜂 can be set to 0 for Euler explicit, 1 for Euler implicit

or 12 for Crank-Nicholson scheme.

After solving the variable for the volume fraction 𝛼, the continuity equation for each phase in each control volume can be described as: 𝜕(𝛼𝑞𝜌𝑞)𝜕𝑡 + ∇(𝛼𝑞𝜌𝑞𝑢𝑞) = 𝑆𝛼𝑞 + ∑(�̇�𝑝𝑞 − �̇�𝑞𝑝)𝑛

𝑝=1 (9)

As described before, after the equation (9) is transformed into integral form, the domain is discretized and the integrals are decoupled by Gaussian quadrature in its implicit scheme: (𝛼𝑞𝑛+1𝜌𝑞𝑛+1 − 𝛼𝑞𝑛𝜌𝑞𝑛)∆𝑡 𝑉 + ∑(𝜌𝑞𝑛+1𝑈𝑓𝑛+1𝛼𝑞,𝑓𝑛+1)𝑓 = [𝑆𝛼𝑞 + ∑(�̇�𝑝𝑞 − �̇�𝑞𝑝)𝑛+1

𝑝=1 ] (10)

After determining the volume fraction in each control volume, the conservation equations for momentum are solved with the averaged fluid variables in a joint momentum equation through all control volumes in the domain.

The fluids are considered as compressible. As thermal effects are expected to be small at the considered no-load conditions with low speeds, thermal influences are neglected and therefore the energy conservation equation is not considered.

A pressure-velocity coupled scheme based on algorithm SIMPLE is used, whereas the mo-mentum equations are discretized by second order upwind. The VoF equations are interpo-lated by a downwind compressive scheme, which is able to preserve a good resolution for the interfaces between the phases even with coarse meshes or high Courant numbers. Turbulence models are not applied in this study due to small Reynolds numbers (cf. section 5.3).

5.2 Rheological Model

The flow of a grease can be described as exhibiting, Bingham-like behavior (cf. section 3), for which several models are available to describe the relationship between shear stress, shear rate and viscosity. In this study, a modified Bingham model is used.

The classical Bingham plastic model can be seen as a special form of the Herschel-Bulkley model (Ehrentraut [28]). As long as the stress does not exceed the yield stress 𝜏 < 𝜏0, the grease behaves like an elastic solid body.

When the yield stress is exceeded 𝜏 > 𝜏0, it shows a linear behavior and reads (Chhabra and Richardson [24]): 𝜏 = 𝜏0 + 𝜂𝐵𝐻�̇� (11)

Thereby, 𝜏0 denotes the yield stress and 𝜂𝐵𝐻 the Bingham viscosity. Eq. (11) can also be written in terms of viscosity:

𝜂 = 𝜂𝐵𝐻 + 𝜏0�̇� (12)

A modified Bingham model is obtained by introducing the exponent 𝑛 and interpreting the Bingham viscosity as the base oil viscosity of the grease 𝜂𝑂𝑖𝑙: Based on the measured relationship of shear rate and viscosity of the considered grease (cf. Table 2), the modified Bingham model can be parameterized by regression analysis. Figure 4 illustrates the measured and predicted shear stress and viscosity over shear rate. Based on the fitted parameters τ0 = 250 Pa and n = 0.76, very good correlation is found.

The modified Bingham model is implemented in the simulation model by compiling a user-defined material property function. Depending on the kinematics of the test gears and the shear rate �̇�, the non-linear grease viscosity η will be calculated for each time point and position in the flow domain. It is used as an input variable for the conservation equations. Compared to oil flow simulations based on Newtonian fluids, this implicit calculation of viscosity requires more computational effort.

5.3 Model Description

The conservation equations are solved iteratively for both phases within the finite volumes in the flow domain (cf. section 5.1). Therefore, a discretization of the flow domain is required.

According to Figure 5 (top), the geometry of the considered test gearbox (cf. Figure 3) is pre-pared by focusing on the geometry relevant for fluid flow. Components like screws and dowel

𝜂 = 𝜂𝑜𝑖𝑙 + 𝜏0�̇�𝑛 (13)

Figure 4: Measured and predicted relationship between shear stress and shear rate (left) and between viscosity and shear rate (right) of the considered grease

1

10

100

1 000

10 000

100 000

1 000 000

10 000 000

100 000 000

0 10 1 000 100 000

She

ar s

tres

s in

mP

a

Shear rate in 1/s

Grease FVA785I/F1

Base oil

ModifiedBinghammodel

1

10

100

1 000

10 000

100 000

1 000 000

10 000 000

0 10 1 000 100 000

She

ar v

isco

sity

in

mP

as

Shear rate in 1/s

Vis

cosi

tyη

in m

Pas

Shear rate in 1/s

She

arst

ress

in

mP

a

Shear rate in 1/s

Fitted parameters: = 250 Pa; n = 0.76 Base Oil

ModifiedBingham

Grease Fitted parameters: = 250 Pa; n = 0.76 Base Oil

ModifiedBingham

Grease

100 101 103 105 100 101 103 105

pins, as well as small housing curvatures, are neglected. The whole geometry is then sub-tracted from a cuboid, so that a negative raw model of the test gearbox is created. Due to the symmetry of the test gearbox, only one half is modeled, while the middle plane of the test gearbox is set to symmetry boundary condition.

Figure 5 (bottom) shows the mesh model of the considered test gearbox. Although it has been already considered and generally explained by the authors in Liu et al. [5], some descriptions on the mesh model are repeated to improve readability. The mesh model is divided into two domains, which are connected with each other sharing the same grid points: the outer domain and the remeshing domain. The outer domain is mainly discretized by hexahedral elements and remains static, whereas the contours of pinion and wheel of the remeshing domain rotate at a predefined speed. The remeshing domain is used to balance out the mesh deformation caused by the gear kinematics. It consists of a deformable meshing structure with tetrahedral elements, which are stretched or rebuilt with every time step. When the mesh quality, evalu-ated by e.g. skewness and orthogonal quality, falls below a minimum mesh quality, the affected elements are reconstructed. Due to the very small gap between the tooth flanks, i.e. gear backlash, bad element quality and numerical singularities would occur during numerical anal-ysis. Therefore, the pinion and the wheel are scaled to 98 % of their actual size. The scaling is not expected to significantly influence the predicted grease distribution. The element grid size in the outer domain is between 2.0 and 3.0 mm, whereas in the remeshing domain it is set to 1.25 mm, resulting in a total element number of 0.7 M.

The time steps are set to 1.28 ms and 0.67 ms, which equals 0.1° rotation of the pinion at n1 = 13 rpm and n1 = 100 rpm (cf. Table 3). The CFL number at these time steps is 0.05, which is much lower than 1 and assures numerical stability [29]. A local convergence criterion of 10-

5 is used for all equations. A global convergence criterion of 10−3 complies for every time step, which is considered as sufficient for the grease distribution from a global point of view.

Adhesion and surface tension usually becomes important when the gravitation or exterior force is near-zero. In this study, the adhesion angle between both phases is set to 90° at walls, from which the adhesion force is derived.

The global Reynolds number can be estimated by 𝑅𝑒 = 𝜔𝑟𝑃2𝜈 and is <<1 for the considered

operating conditions. The local cell Reynolds number as the ratio of local inertia and viscous

Figure 5: Derivation of CFD simulation model from CAD model

Outer Domain Remeshing Domain Simulation Model

pinion contourwheel contour

symmetry plane

Mesh

Mo

del

Mo

del

Pre

para

tio

n

CFD Raw Model

wall

CAD Model CAD Simplification

force 𝑅𝑒 = 𝜌𝑢𝐿𝜂 is mostly <1. The maximum local cell Reynolds numbers appear in air phase

around the gears and is <<100. This indicates that the considered mesh size is suitable. Fur-ther information in terms of modeling can be found in previous studies (Liu et al. [4, 5, 30]).

6. Results and Discussion

In this section, simulated grease flow distributions are investigated and compared with high-speed camera recordings. First, different rotational positions during a start-up case are con-sidered. Second, quasi-stationary states of the considered operating conditions are shown.

6.1 Grease Distribution during Start-Up

First, CFD simulations were conducted for the start-up of the grease-lubricated gear stage. Figure 6 compares the simulated grease distribution with high-speed camera recordings for different grease fill volumes of 40 %, 54 % and 80 % at one-quarter, one-half, and three-quarter wheel rotation. The circumferential speed of vt = 0.05 m/s (cf. Table 3) is considered. The start-up phase for these gears, from 0 m/s to operating speeds (cf. Table 3), was achieved via a constant acceleration ramp within 0.125 s. This acceleration time is a reasonable value for this asynchronous motor.

Figure 6: Simulated and recorded grease distributions for grease fill volumes of 40 %, 54 % and 80 % at one-quarter, one-half, and three-quarter wheel rotation

The area of pinion and wheel initially covered with grease in the sump before start-up is larger at higher grease fill volumes. When the gears start to rotate, the grease sticks in the tooth gaps and at the gear face, and is dragged out of the grease sump (cf. one-quarter of wheel rotation). The amount of dragged grease increases with increasing grease fill volume from 40 % to 80 %.

For a grease fill volume of 40 %, the absent back flow of grease to the gear areas in the sump can be clearly recognized after half a wheel rotation. The initially grease-free gear areas rotate through the grease sump without picking up further grease. This relates to the yield stress of

t = 1.7 s t = 3.5 s t = 5.2 s

t = 1.7 s t = 3.5 s t = 5.2 s

t = 1.7 s

t = 1.7 s

t = 1.7 s t = 3.5 s t = 5.2 s

t = 1.7 s t = 3.5 s t = 5.2 s

t = 3.5 s t = 5.2 s

t = 3.5 s t = 5.2 s

54 (

mid

dle

axis

)80

40

Gre

ase f

ill vo

lum

e i

n %

Circumferential speed vt = 0.05 m/s

¼ wheel rotation ½ wheel rotation ¾ wheel rotation

the considered grease. When there is no stress acting on the grease in the sump larger than the yield stress, it behaves like an elastic solid body. After three-quarter of wheel rotation, the dragged grease by the wheel interacts with the pinion in the gear mesh. By that, grease is transferred from the wheel to the pinion as well as from the pinion to the wheel after half a wheel rotation.

Compared to the grease fill volume of 40 %, the observed behavior at a grease fill volume of 54 % is very similar, except for the fact that there is more grease clinging on the side area and teeth on both the pinion and wheel, so that there is a larger amount of grease exchange be-tween the pinion and wheel in the gear meshing area.

At a grease fill volume of 80 %, only a small part of the wheel protrudes from the grease sump, whereas the pinion is fully covered with grease. Even the part of the wheel that was initially uncovered picks up grease due to grease exchange between pinion and wheel, so that a pro-nounced circulation of grease takes place.

The degree of grease coverage of gears correlates qualitatively well between simulation re-sults and high-speed camera recordings. For a grease fill volume of 80 %, the bead above the gear meshing zone predicted by the simulation is not found experimentally.

The results show that, due to a very limited backflow of the considered grease to the gear areas in the sump, the interaction of areas initially with grease covered of both the pinion and the wheel in the gear mesh is important for grease lubrication. Thus, the grease distributions are investigated at a later time, once grease exchange in the gear mesh has taken place for several pinion and wheel rotations. These results are presented in the following section.

6.2 Grease Distribution after Four Wheel Rotations

Figure 6 and Figure 7 show simulated grease distributions after four wheel rotations for the considered operating conditions in Table 3. Whereas the simulation results in Figure 6 are illustrated in a slightly angled view, the high-speed camera recordings are still shown from the front view on the gearbox. highlights the simulated grease fraction from the front and side view in the middle plane of the gearbox.

The three columns of Figure 6 correspond to the grease fill volume of 40 %, 54 % and 80 %, whereas the two rows relate to the circumferential speeds vt of 0.05 m/s and 0.38 m/s. It can be observed that the influence of circumferential speed vt is subordinate compared to the in-fluence of grease fill volume. A quasi-stationary grease distribution tends to occur when a grease exchange between initially with grease covered and uncovered areas has taken place, so that the entire circumference of the pinion and wheel are covered with grease.

The circumference of both pinion and wheel are uniformly covered with grease after four wheel rotations for all fill volumes. The amount of grease around the circumference of the gears, as well as the part squeezed out of the gear mesh, clearly increases with increasing grease fill volume. The simulated degree of grease coverage of gears correlates qualitatively well with the high-speed camera recordings. Only for a grease fill volume of 80 % is the bead above the gear meshing zone (as predicted by the simulation) not observed experimentally.

By means of Figure 8, it can clearly be observed that the pinion and wheel dig a surrounding volume with low grease fraction into the grease sump for fill volumes of 40 % and 54 %. A grease exchange between the initially covered and uncovered areas of pinion and wheel has already taken place. The amount of grease around the circumference as well as at the gear mesh is higher for a grease fill volume of 54 % compared to 40 %. At 54 %, a chamfered grease fill level of about 30° on the left-hand side of the wheel is formed. However, the grease dragged along the circumference is not enough to form a “grease belt” over the wheel, which would result in a good grease supply of the gear mesh. Thus, channeling (cf. section 2) is the dominating lubrication mechanism at grease fill volumes of 40 % and 54 %. Circulating is lim-ited to exchange of grease in the gear mesh and not pronounced. This observation can be correlated with the lubrication mechanism “channeling” found in experimental investigations in literature (section 2).

For the grease fill volume of 80 %, it is shown that the circumferences of both gears are almost fully filled with grease. As a “grease belt” is formed over the wheel, the initially uncovered area picks up grease after each encounter with the pinion. Due to the resulting good grease supply to the gear mesh, circulating becomes the dominating lubrication mechanism (cf. section 2). The large cohesion force of grease leads to notable grease transport, which can be observed by the lowered grease fill level to the left side of the wheel and the right side of the pinion. The exaggerated bead above the gear mesh zone is ascribed to the deviation between the meas-ured and predicted values for the Bingham model (cf. Figure 4). At very low shear rates, this can have a strong influence on the results.

t = 3.6 s

Grease fill volume in %

40 54 (middle axis) 800.

050.

38

Cir

cu

mfe

ren

tial

sp

eed

vtin

m/s

t = 27.7 s

t = 27.7 s

t = 27.7 s

t = 27.7 s

t = 27.7 s

t = 27.7 s

t = 3.6 s

t = 3.6 s

t = 3.6 s

t = 3.6 s

t = 3.6 s

Figure 7: Simulated grease distribution for the considered operating points after four wheel rotations

Figure 8: Simulated grease volume fraction from the front and side view in the middle planes of the gearbox for the considered operating points after four wheel rotations

7. Conclusion

In this study, a Finite Volume based CFD model of a single-stage gearbox was built to inves-tigate the grease flow of a high-consistency grease under isothermal conditions. The rheolog-ical behavior of grease was described by a modified Bingham model, which was parametrized by regression analysis of measurements at different shear rates. The simulated grease distri-butions showed good accordance with high-speed camera recordings for different grease fill volumes and circumferential speeds. The considered circumferential speeds had no significant influence on the grease flow, whereas the grease fill volume played a decisive role. The lubri-cation mechanisms “channeling” and “circulating” shown experimentally were discerned by the simulation results. The study showed first qualitative insights on the numerical calculation of grease flow in gearboxes.

Acknowledgements

The presented results are based on the research project STA1198/14-1 supported by the Ger-man Research Foundation e.V. (DFG). The authors would like to thank Dr. Dornhöfer from

Robert Bosch GmbH for providing the rheological measurement and model results, as well as Mr. Pfadt from Klüber Lubrication for supplying the grease for the experimental investigations.

Declarations

The presented results are based on the research project STA1198/14-1 supported by the Ger-man Research Foundation e.V. (DFG). The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. The applicants in this proposal declare on his honor that the information provided is accurate and complete.

Author´s Contribution

H. Liu: Conceptualization, Consulting, Prestudy, Experimental investigation, Evalua-tion of results, Writing - Original draft preparation

F. Dangl: State of the art, Experimental investigation, CFD – Simulation, Model imple-mentation, Evaluation of results, Writing - Original draft preparation

T. Lohner: Writing - Original draft preparation, Consulting K. Stahl: Writing - Original draft preparation, Consulting

0.05

0.38

Cir

cu

mfe

ren

tial

sp

eed

vtin

m/s

Grease fill volume in %

40 54 (middle axis) 80

t = 27.7 s

t = 3.6 s

t = 27.7 s

t = 3.6 s

t = 27.7 s

t = 3.6 s

Oil fraction0.500.25 0.75 1.000.00

Nomenclature

a Center distance in mm 𝛼 Volume fraction

BH Bingham αn Pressure angle in °

b Tooth width in mm 𝛤 Diffusion coefficient

CV Control volume �̇� Shear rate in 1/s 𝐷𝛷 Diffusive flow ∆ Laplace operator

da Tip diameter in mm 𝛿t Time interval in s

F Exterior force in N η Dynamic viscosity in mPas 𝐹𝛷 Convective flow ρ Density in kg/m³

f Centroid of border of control volume 𝜏 Shear stress in Pa

g Gravity in m/s² 𝜏0 Yield stress in Pa

h Specific enthalpy in J/kg 𝛷 Arbitrary state variable

i12 Gear ratio 𝛻 Nabla operator

L Characteristic length in mm 𝜈 Kinematic viscosity in mm²/s

m Consistency index 𝜔 Angular speed in rad/s

mn Normal module in mm 𝜕 𝜕𝑡 Partial derivation of time

�̇�𝑝𝑞 Mass flow rate phase 1 to phase 2 �̇�𝑞𝑝 Mass flow rate phase 2 to phase 1

n Flow index │Rotational speed in rpm│time step

P Center of control volume

p Phase 1

rP Pitch radius in mm

𝑄𝛷 Sources

q Phase 2

q´´ Specific heat in J/kgK

Re Reynolds number 𝑆𝛼𝑞 Source term of phase 2 in kg/s

SF Surface of control volume

𝑈𝑓 Volume flow rate in m³/s

V Volume of control volume in m³

vt Circumferential speed in m/s

x Addendum modification coefficient

z number of teeth

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Figures

Figure 1

Circulating (left) and channeling (right) as main lubrication mechanisms of grease-lubricated gearbox-es(Stemplinger [19])

Figure 2

Schematic representation of different rheological behavior of �uids (based on Chhabra and Richard-son[24])

Figure 3

FZG back-to-back test rig (Siewerin [20])

Figure 4

Measured and predicted relationship between shear stress and shear rate (left) and between viscosityand shear rate (right) of the considered grease

Figure 5

Derivation of CFD simulation model from CAD model

Figure 6

Simulated and recorded grease distributions for grease �ll volumes of 40 %, 54 % and 80 % at one-quarter,one-half, and three-quarter wheel rotation

Figure 7

Simulated grease distribution for the considered operating points after four wheel rotations

Figure 8

Simulated grease volume fraction from the front and side view in the middle planes of the gearbox for theconsidered operating points after four wheel rotations