On the influence of viscosity formulation in CFD

simulation when predicting churning power losses

generated by partly immersed gears Yann MARCHESSE

1, Christophe CHANGENET

1, Fabrice VILLE

2

1 Université de Lyon, ECAM Lyon, LabECAM, Lyon, France.

2 Université de Lyon, INSA de Lyon, LaMCoS, Villeurbanne, France.

Session: Gears I — 2G Presentation type: Oral

Keywords: Gear transmission; churning power loss; Computational Fluid Dynamics.

1 Introduction

Problems may appear in gear transmission due to an inadequate adjustment of lubrication. This latter is not

simple to develop since while a small amount of oil is sufficient for lubricating bearings, like rolling-element

bearings, much more lubricant is needed for removing heat from the gears. The most common lubrication

method is splash lubrication in which a pinion runs partly immersed in an oil bath. The level of immersion is of

great importance because if the pinion is too much dipped churning power loss increases. On the contrary

when the immersion is too low both the heat removing and the ejected oil received by bearings are

insufficient. This level of immersion may be investigated using numerical method like Computational Fluid

Dynamics (CFD). Among all the methods that are able to handle interface between two fluids volume of fluid

(vof) method [1] has been employed in many studies (see among others references [2] and [3]). The purpose of

this method is to solve a single set of momentum equations for mixture properties. The volume fraction (αi) of

each of the fluids i throughout the domain is then tracked by solving a transport equation for one phase

volume fraction. The evaluation of the mixture properties (density and dynamic viscosity denoted respectively

ρ andµ) relies usually on a linear formulation between volume fractions and phase property parameters:

�� � �������� � 1 � ������ �� and �� � �������� � 1 � ������ ��

if the two fluids are air and oil. This seems consistent for density while it is questionable for viscosity that is

connected to transport effect. The purpose of this investigation is to see if the linear formulation leads to

numerical estimations that are consistent with experimental observations. For that experimental data have

been reached using a disk located in a specific churning test rig [4] comprising a shaft operated by an electric

motor via a notched belt. Churning losses are determined from direct torque measurements based on strain

gauges. In addition the volumetric flow rate expelled by the disk is evaluated by a system estimating the time

needed for filling one particular volume. The level of immersion and the angular velocity are both variable.

4000 rpm angular velocity value is only considered for experiments in this investigation.

Figure 1: cross sections of the

unstructured mesh and definition of

immersion (h).

2 Numerical approach

The numerical domain dimensions are identical to the experimental housing ones except that a plane of

symmetry has been employed here in computations and half-width of the real disk is simulated. A nil velocity

and an angular velocity are applied on nodes located on the housing and on the disk respectively. Symmetry

conditions are imposed in the rear surface. The properties of the two fluids (i.e. air and oil) are evaluated at

approximately 40°C (ρ = 1.185 kg/m3 and µ = 1.831×10

-5 Pa.s for the air, and ρ = 885 kg/m

3 and µ = 3.99×10

-2

Pa.s for the oil). Gravity acceleration is activated and oil immersion is applied on the mesh before increasing

gradually the angular velocity since 4000 rpm is reached. Simulations based on the Reynolds-Averaged Navier-

Stokes (RANS) theory have been carried out using SST k−ω low-Reynolds turbulent model [5] and solved on

unstructured mesh (Fig. 1). The convergence is reached when stable values are obtained both for the simulated

torque and for the volumetric flow rate that is received by a window located at the same place that the

experimental flowmeter.

3 Numerical and experimental results

One observes that the disturbance of the interface that is obtained numerically increases with the immersion

level as it is the case in the experiments (Fig. 2). However the numerical approach highly underestimates the

measured torque (60%) when the level of immersion is low, i.e. h/R = 0.2, 0.3 (Fig. 3.a). This is less the case

when h/R becomes greater or equal than 0.4 (Fig. 3.b) since relative error is less (40%). In the first case the

mixture volume fraction is weak due to the great presence of the air in the vicinity of the disk in the sump. As a

consequence the torque that acts on the disk is low. When immersion is higher this time both the waves

propagating in the housing and the ejection of oil lead to a greater torque value. As a consequence to that the

volumetric flow rate is quite nil when immersion is weak and is greater for higher immersion.

(a) (b)

Figure 2: flow pattern when h/R = 0.3 (a, experiment; b, numerical estimation).

(a) (b)

Figure 3: comparison between torque numerically obtained and measured (a, h/R = 0.2; b, h/R = 0.5) − T is the

time period.

3 Modification of the viscosity formulation

The linear formulation for estimating the mixture viscosity is changed to a nonlinear formulation:

�� � � �� �������� � 1

�� � 1���� � � ���

where β is a constant that can be adjusted. Depending both on the sign and the value of β the influence of the

air amount on the mixture viscosity differs. For example when β = 10 the air volume fraction must be equal to

0.77 in the cell for the mixture to reach 90 percent of the oil fraction. On the contrary, if β = −10 this quantity

becomes less and equals 1%. When using the nonlinear formulation in the computation approach one notices

that the torque values dramatically decrease when β = −10 in comparison with numerical results obtained

using linear formulation (Fig. 4). As it was mentioned above this is due to the great decrease of mixture even if

a small quantity of air is present in the cells near the disk. When β = 10 this time the torque values increase and

become more close to the experiments This must be explained by the mixture viscosity that remains nearly

constant while the air amount increases in the cells as it is expected by the nonlinear formulation.

Figure 4: influence of the mixture

viscosity formulation on the

numerical torque (h/R = 0.2).

CONCLUSIONS

The volume of fluid numerical method has been employed in the case of a dipping disk. The mixture viscosity

has been evaluated using at first the default linear formulation. In that case both the torque and the expelled

volumetric flow rate are underestimated in comparison with experimental data obtained in similar

configurations. When a modification of the latter formulation is done the torque values increase and become

more realistic.

REFERENCES

[1] Hirt C.W., Nichols B.D., Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries, , 1981, J. of Comp.

Phys., 39, pp. 201-225.

[2] Li L., Versteeg H., Hargrave G., Potter T., Halse C., Numerical Investigation on Fluid Flow of Gear Lubrication, 2009,

SAE Int. J. of Fuels and Lubricant, 1(1), pp. 1056-1062.

[3] Concli F., Gorla C., Della Torre A., Montenegro G., Churning power losses of ordinary gears: a new approach based

on the internal fluid dynamics simulations, 2014, Lubrication Science, DOI: 10.1002/ls.1280.

[4] Changenet C., Leprince G., Ville F., Velex P., A Note on Flow Regimes and Churning Loss Modeling, 2011, Journal

of Mechanical Design, 133, pp. 121009 1-5, DOI: 10.1115/1.4005330.

[5] F.R. Menter, Y. Egorov. A Scale-Adaptive simulation model using two-equation models. A.I.A.A. Journal, paper

2005-1095, Reno/NV (2005).