Computers & Fluids 84 (2013) 39–45

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Computers & Fluids

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Application and validation of a simplified numerical approach for theestimation of windage power losses in spur gears

0045-7930/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compfluid.2013.04.025

⇑ Corresponding author. Tel.: +33 04 72 77 06 63.E-mail addresses: sylvain.pallas@ecam.fr (S. Pallas), yann.marchesse@ecam.fr (Y.

Marchesse), christophe.changenet@ecam.fr (C. Changenet), Fabrice.ville@insa-lyon.fr (F. Ville), philippe.velex@insa-lyon.fr (P. Velex).

Sylvain Pallas a, Yann Marchesse a,⇑, Christophe Changenet a, Fabrice Ville b, Philippe Velex b

a Université de Lyon, ECAM Lyon, Laboratoire d’Energétique, Franceb Université de Lyon, CNRS, INSA-Lyon, LaMCoS UMR5259, France

a r t i c l e i n f o a b s t r a c t

Article history:Received 27 November 2012Received in revised form 26 April 2013Accepted 29 April 2013Available online 13 May 2013

Keywords:RANSWindage power lossSpur gears

In this paper a computational fluid dynamics (CFD) code is applied to three-dimensional simulations ofwindage power loss generated by spur gears rotating in air. Emphasis is placed on the simplificationwhich can be made in the numerical approach in order to gain cost and time needed for reaching a con-verged solution when using both fine mesh and important angular velocity. It appears that the similarityof the flows which are observed in the vicinity of both gear and disk sides is helpful. It is finally possible tocompute the flow pattern near the teeth without simulating the contribution of the sides when appropri-ate boundary conditions are applied at the inlet of a truncated computational domain. The predictions ofwindage power losses using this original method are satisfying.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

When considering high-speed gear transmission, the powerlosses generated by aerodynamic forces (pressure and viscous)may dominate the other losses [9]. In this case as mentioned byDawson [4] the windage power losses (WPLs) are defined as thepower loss due to the fluid drag generated by the gear rotationin air or air/oil mist. These losses may differ from the one observedwhen rotating gears run in an oil bath or dips into oil ‘‘slugs’’[3,14,15]. WPL can be the source of significant heating and mustbe therefore studied in order to mitigate its impacts. Numerousexperiments have been made in the past [4–6,16] essentially toproposed WPL models. However these models may be inaccuratefor gears whose characteristics are far from the ones used in exper-iments. Computational fluid dynamics (CFD) has the capability tosimulate airflows around rotating gears and can be then an exten-sion of experiments. In that way CFD method has been applied toanalyze windage losses for spur gears considering case or shroudeffect [1,8,11,12] or not [11,12] and for bevel gears [17]. Howeverthe quality of the numerical predictions greatly depends on boththe near wall cells size and the type of turbulent modeling. It hasbeen shown for example that using low-Reynolds number turbu-lent model like shear stress transport (SST) k–x gives satisfactionswhen the viscous sublayer of the boundary layer is resolved [12].However when tip velocity value becomes important the size of

the first cell near the wall dramatically decreases leading to anaugmentation of the numerical model. As a consequence the timeneeded for reaching a converged solution is very long and this canlimit the number of parameters to be varied in a parametric stud-ies. For example Hill et al. [10] built a numerical domain whichrepresents a volume around an entire gear with the plane of sym-metry and used a mesh comprising 8 � 106 cells for the study ofWPL generated by a gear (288 mm pitch diameter and 30 mmwidth) rotating at 700 rad/s. It has to be noticed moreover thatnear-wall grid spacing was defined in this reference only to accom-modate wall-function. The purpose of this article is thus to inves-tigate a method which makes it possible to estimate WPL using asimplified numerical approach to overcome this problem. As a firststep, a study considering a disk rotating in pure air is conducted.The aim of this part is to see the consequences of reducing themodel from the entire disk until a simplified domain. As a secondstep, using results which are observed on disk and gears leads toanother simplification.

2. Numerical approach

All the numerical predictions in this article which have been ob-tained using commercial software ANSYS–CFX 13.0 are based onthe following hypothesis and conditions:

(1) The airflows are simulated using Navier–Stokes equationsfor incompressible flows since the maximum tangentialvelocity is far below the sonic speed for all the computa-tions. The value of the maximum Mach number equals 0.30.

Fig. 1. Sketch of the computational domain for the disk investigations (configura-tion 3) and presentation of the boundary conditions imposed at external surfaces.

Nomenclature

b tooth face width (mm)D disk diameter (mm)D1, D2 domain denominationh tooth height (mm)k turbulent kinetic energy (m2/s2)p static pressure (Pa)Pk turbulence production (kg3/m3 s3)R radius (mm)RD2 inlet radius of the truncated domain D2 (mm)t time (s)ui velocity component (m/s)ur radial velocity (m/s)uh tangential velocity (m/s)us wall friction velocity (m/s)xi Cartesian coordinate (m)y distance from the wall in y+ formulation (m)

y+ dimensionless distance from wall parameterV velocity (m/s)a coefficient used in inlet radial velocity formulationb coefficient used in inlet tangential velocity formulationdij Kronecker’s symbolf dimensionless parameterl dynamic viscosity (Pa s)lt eddy viscosity (Pa s)q fluid density (kg/m3)qu0iu

0j Reynolds stress (Pa)

sw wall shear stress (Pa)x specific dissipation rate (1/s)X angular velocity (rad/s)

40 S. Pallas et al. / Computers & Fluids 84 (2013) 39–45

(2) The fluid is supposed to be pure air and physical propertiesare evaluated at 25 �C (e.g. q = 1.185 kg/m3 andl = 1.831 � 10�5 Pa s).

(3) Convergence is reached when both the normalized residualsof the equations fall below 10�4 and the drag torque causedby windage keeps constant amplitude.

The averaged continuity and momentum equations for anincompressible viscous flow in the absence of body forces can beexpressed as [7]:

@ðquiÞ@xi

¼ 0 ð1Þ

@ðquiÞ@t

þ @

@xjðquiuj þ qu0iu

0jÞ ¼ �

@�p@xiþ @

@xjl @ui

@xjþ @uj

@xi

� �� �ð2Þ

where the Reynolds stresses qu0iu0j are expressed by using the eddy-

viscosity hypothesis as:

�qu0iu0j ¼ lt

@ui

@xjþ @uj

@xi

� �� 2

3qdijk ð3Þ

where k is the turbulent kinetic energy, and lt represents the eddyviscosity. The turbulent viscosity in (3) is expressed using the shearstress transport (SST) k–x model where the eddy viscosity reads:

lt ¼ qkx

ð4Þ

The values of k and x are derived from the transport equationsfor the turbulence kinetic energy and the specific rate expressed as[13]:

@

@tðqkÞ þ @

@xiðqkuiÞ ¼

@

@xjlþ lt

rk

� �@k@xj

� �þ Pk � b0qkx ð5Þ

and

@

@tðqxÞ þ @

@xiðqxuiÞ ¼

@

@xjlþ lt

rx

� �@x@xj

� �þ a

xk

Pk � bqx2 ð6Þ

In these equations a, b, b0, and rx are constants.The SST k–x turbulence model (low-Reynolds number model)

is used to solve the viscosity-affected region. Therefore the meshcriterion requirement y+ < 5 is necessary in order to use adequatelythe latter turbulent model. y+ = qyus/l is a dimensionless wallparameter where y is the distance from the center of the first cellsto the wall and us is the wall friction velocity given byus ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffijswj=q

pwhere sw is the shear stress at the wall. Since us is

linked to the free-stream velocity V, the size of the first cell de-pends on both y+ and V. When the SST k–x model is used withcoarser mesh, a low-Reynolds near-wall formulation is switchedto a wall-function method.

3. Modeling of a disk

The objective of this part is to investigate a simplified method toestimate the windage power losses generated by a rotating disk.Numerical predictions are compared with experimental data(D = 300 mm and b = 30 mm) obtained by Diab et al. [6]. Threenumerical configurations have been investigated:

1. An entire disk.2. A disk sector occupying angular space equal to 5�.3. Three half-width disk sectors occupying angular space equal to

5�, 10�, or 20� (Fig. 1).

In configurations #2 and #3 periodic conditions are applied tothe radial edges at the limit of the computational domain. Normalcomponent of the velocity is set nil for the nodes in the plane ofsymmetry when present (i.e. configuration #3). An opening condi-tion is applied for the three surfaces surrounding the disk. Thiscondition allows the fluid to cross the boundary surface in eitherdirection and is specified by nil relative pressure value. State equa-tions are solved with respect to a rotating reference frame (RRF) forsteady state flows with Coriolis and Centrifugal effects being taken

S. Pallas et al. / Computers & Fluids 84 (2013) 39–45 41

into account. A nil velocity is therefore imposed at all wall nodes.When the entire disk is modeled (i.e. configuration #1) two typesof computation are made: using RRF method mentioned above orusing a transient computation associated with moving boundaryconditions for the nodes which are in the disk walls. A time stepof 1/100 of one passage duration (i.e. the time required for one diskrevolution) is then used. The chosen angular velocity correspondsto maximum one reached during experiments, i.e. 700 rad/s. Twovalues of dimensionless wall parameter are studied for configura-tion #3 (e.g. y+ equals 5 and 50) in order to evaluate the influenceof the near wall treatment on WPL predictions.

Fig. 2. Influence of near wall treatment on WPL predictions (configuration #3 –angular sector equals 20�).

Fig. 3. Sketch of the computational domain D1 and presentation of the boundary

3.1. Results

A good agreement is noticed between numerical predictionsand experimental data as the maximum relative error equals 4%(Table 1). Moreover for the entire disk, no difference arises whenconsidering a transient computation nor RRF approach as the rela-tive difference does not exceed 1%. These two approaches can thenbe considered as numerically similar. Furthermore both the valueof the disk sector or the use of a plane of symmetry does not influ-ence significantly the WPL predictions; the relative difference re-mains below 0.2% (in Table 1, WPS refers to calculationsperformed without using the plane of symmetry).

The influence of the near wall treatment is much more impor-tant than the numerical approaches mentioned previously. Indeedrelative error reaches 8% at maximum angular velocity when thecomputation is based on coarser mesh (i.e. y+ = 50) while thisquantity never exceeds 2% when y+ = 5 (Fig. 2).

The observations above are of great importance when simplifi-cations are expected in the numerical approach for the estimationof WPL for a rotating solid. As a consequence when a plane of sym-metry and/or rotationally periodic solutions is present the numer-ical approach may greatly be simplified. These physical aspects arepresent with spur gears rotating in a region far from walls. The pur-pose of the next section is then to take advantage of simplificationsstudied in this section to estimate WPL generated by spur gears.

conditions imposed at external surfaces.

Table 2Gear characteristics.

Gear#

Pitchdiameter(mm)

Toothwidth(mm)

Module(mm)

Tipdiameter(mm)

Pressureangle (�)

1 150 24 5 160 202 100 24 5 110 203 288 30 4 296 20

4. Modeling of a spur gear

Based on the simplified approach which has been previouslyhighlighted the computational domain (D1) comprises one half-width single tooth and the associated fraction of blank and theplane of symmetry (Fig. 3). The radial dimension of the domain istaken 15 times the tooth height while the axial dimension beyondthe gear side is set to one-half the real tooth width. The verticalsurface which is near the rotational axis (opening 1 in Fig. 3) is lo-cated at a radial distance equal to 10% of root radius. These valuesseem sufficient since increasing the size of the domain leads toinsignificant variation of WPL estimation (i.e. less than 1%). Thesame boundary conditions as the ones used for the disk are em-ployed here and RRF approach is still used.

The numerical predictions are compared with experimental re-sults from test rigs described by Diab et al. [6] and Changenet andVelex [2] in order to validate the use of the simplified configura-tion. The gear data are listed in Table 2.

Table 1Numerical predictions of WPL considering a disk (D = 300 mm; b = 30 mm) rotating at 700

Exp. Entire disk

Diab et al. [6] Transient RRF

WPL (W) 194.9 189.0 188.0

4.1. Results

The numerical results are close to the experimental measure-ments for angular velocities above 300 rad/s. The maximum differ-ence is observed for gear #3 at 700 rad/s and is equal to 80 Wwhich corresponds to a relative error of 11% (Fig. 4). More impor-tant relative errors are noticed however for lower angular veloci-ties but in a range where windage contribution is not significant.These results validate the use of a simplified numerical model

rad/s.

Disk sector

5� WPS 5� 10� 20�

198.0 198.2 198.0 197.8

Fig. 4. Windage power losses estimated experimentally and predicted fromnumerical approach which are based on domain D1.

Fig. 6. Distribution of radial (a) and circumferential (b) velocities at radial distancer = 140 mm for the disk and gear #3 (X = 700 rad/s).

42 S. Pallas et al. / Computers & Fluids 84 (2013) 39–45

which lies on a plane of symmetry and a computation made in arotating reference frame. This numerical approach is therefore use-ful for the predictions of windage losses which must be based onsufficiently fine mesh near the wall. This is the case when tip veloc-ity is important. The reduction of the cost (number of nodes) isthus important and the time required to reach a converged solu-tion is lower. In comparison with numerical approach which relieson entire spur gear the node number is approximately divided bytwice the number of teeth when similar wall treatments are em-ployed for the two configurations.

4.2. Disk side and gear side

The side and the tooth contributions are sometimes taken intoaccount separately in total WPL model [4,6]. The losses inducedby gear sides are thus taken similar to the ones generated by the

Fig. 5. Distribution of radial (a) and tangential (b) velocities at radial distancer = 100 mm for the disk and gear #3 (X = 700 rad/s).

sides of a disk which have similar dimensions. To make sure thatone can consider this hypothesis the velocity profiles which havebeen obtained numerically from the two solids can be compared.For that F and G functions are defined as:

FðfÞ ¼ urðfÞrX

GðfÞ ¼ uhðfÞrX

ð7Þ

where f ¼ z�ffiffiffiffiffiffiffiffiffiX=t

p(z is the axial component, X is the angular

velocity, and t is the kinematic viscosity), and ur and uh are respec-tively radial and tangential velocities. It appears that the numericalpredictions of these velocity profiles are similar for the disk andgear #3 which have similar outer diameter values (i.e. 300 mm)when radial distance is lower than nearly 90% of the radius andpitch radius values respectively for disk and gear #3 (Fig. 5). Thisis no more the case for higher radial distance where the influenceof the tooth is noticed for the gear (Fig. 6).

As a consequence the radial distribution of the friction coeffi-cient (cf = sw/0.5 � q(RX)2) which causes drag torque remainsidentical for the disk and the gear as long as the teeth do not sig-nificantly influence the flow in the vicinity of the side (Fig. 7). Ifone considers for example gear #3 this influence occurs at a radialdistance of 1 mm lower than the root radius value which corre-sponds to 10% of the tooth height. This low value validates theuse of WPL model for disk for the estimation of gear side losses.

5. Numerical prediction of WPL generated by spur gearswithout simulating their sides

5.1. Discussion about the simplification

The computations which have been made considering disk andspur gears highlighted the similarity of the flow pattern in the

Table 3Coefficient values for the model of radial and tangential velocities.

Radial velocity, ur

a1 a2 a3 a4 a5

0.1049 0.9085 51,660 15,756 1458.3

Tangential velocity, uh

Fig. 7. Radial distribution of friction coefficient for the disk and gear #3 rotating at700 rad/s (disk radius value has been used for the computation of gear frictioncoefficient).

S. Pallas et al. / Computers & Fluids 84 (2013) 39–45 43

vicinity of the sides. It is therefore interesting to investigate thecapacity of a numerical approach which does not compute the flowpattern near the sides of a spur gear to predict correctly the wind-age power loss of an entire spur gear. The new domain (D2) to beconsidered here has then to be truncated so that the entrance ofthe fluid would be located at a radial distance (which is writtenRD2) where the fluid flow is not influenced by the teeth. It has beenobserved in Section 4.2 that this region is just below the root ra-dius. In this new approach the radial value of the entrance is lo-cated at one tooth height lower than the root radius (Fig. 8).

The power losses induced by the sides which are absent in thecomputation is predicted by the model proposed by Diab et al. [6].Moreover the boundary conditions to be imposed at the entrance

Fig. 8. (a) Scheme of the new domain D2 using truncated gear side for theestimation of WPL (h is the tooth height); (b) introduction to the parameters whichare used in Eq. (8).

must represent the conditions observed for a disk or a spur gearin this location. As a consequence the opening condition whichhas been used previously cannot be reemployed here and isreplaced by an inlet velocity boundary condition (Fig. 8). Basedon the numerical results obtained in Sections 3 and 4.1 a modelis proposed both for the radial and tangential velocities:

ur ¼ ða1XRD2 þ a2Þ � exp½�ða3R2D2 � a4RD2 þ a5Þ � z� ð8aÞ

uh ¼ ðb1XRD2 þ b2Þ � exp½�ðb3XffiffiffiffiffiffiffiffiRD2

pþ b4Þ � z� þ ðb5XRD2

þ b6Þ � exp½�ðb7RD2 þ b8Þ � z� ð8bÞ

where z corresponds to the normal distance outward the gear side(Fig. 8). Constants are given in Table 3. The axial component of thevelocity is weak and is not modeled here since it appears that spec-ifying or not this quantity at the entrance does not influence thenumerical prediction of the flow pattern. The radial velocities pre-dicted by model (8) compare well with the numerical predictionswhatever gear considered (Fig. 9a). However one notices that thisis less the case for tangential velocities since an overestimationfor the values from the model is observed for gears #1 and #2.

b1 b2 b3 b4 b5 b6 b7 b8

0.4964 4.1112 31.036 4394.4 0.3485 �2.8377 �1730.5 645.65

Fig. 9. Estimation of radial (F) and tangential (G) velocities from model (8) andnumerical predictions of these two components at a radial distance one toothheight lower than root radius for the three gears using domain D1 (X = 700 rad/s).

Fig. 10. Flow pattern predictions using D1 domain (Fig. 3) (a) and D2 domain (Fig. 8) (b).

Fig. 11. WPL predictions using D1 and D2 domains and by experiments.

44 S. Pallas et al. / Computers & Fluids 84 (2013) 39–45

5.2. Results obtained with domain D2

It appears that when using domain D2 with appropriate solutionsat the entrance of the domain (i.e. Eqs. (8a) and (8b)) the flow patternremains unchanged (Fig. 10). Furthermore predictions of the wind-age power losses are satisfying for gear #3 (Fig. 11). Relative error re-mains less than 8% when angular velocity is higher than 300 rad/sand finally are the same order as the ones observed when the contri-bution of the sides are completely simulated. More important valuesof relative errors are observed for the two other gears (i.e. 20% forgear #1 and 15% for gear #2). In these latter cases more importantWPL values are predicted when domain D2 is used than the valuesobtained with gear sides simulated. This observation may be dueto the overestimation of tangential velocity value from model (8)for these two gears as it was mentioned previously. However theseresults are still satisfying and this approach could be used in orderto investigate windage losses when high speed gears rotate.

6. Conclusion and discussion

In this paper an investigation were made in order to simplifythe numerical approach for the estimation of windage power lossinduced by moving spur gear. When considering a disk the use ofboth rotating reference frame where the airflow can be consideredas steady and a numerical domain which comprises a periodicgeometry leads to numerical results similar to the ones observedin experiments and to the data predicted when using no simplifiedapproach. This method has been therefore employed considering a

gear and gives results which are in accordance with experimentalmeasurements. Another simplification is moreover made lying onthe fact that the distribution of velocity components is similarfor the disk and the gears. A numerical domain which does notcomprise the side of the gears in the simulation is then investi-gated with appropriate values for radial and tangential velocitiesat entrance. These quantities are given by an analytical modelrather than a previous numerical computation. This new approachgives satisfying results. All of these simplifications are interestingas the time needed for reaching a convergence solution dramati-cally decreases. Furthermore the number of nodes is reduced evenwhen an important angular velocity is studied which normallyleads to important quantity of data to manage. For example whenconsidering gear #3 (288 mm pitch diameter and 30 mm width)rotating at 700 rad/s a mesh comprising less than 520,000 nodesand satisfying near wall criterion y+ < 5 is sufficient. A convergencemay thus be reached rapidly. It is clear than using a domain whichrepresents the air surrounding the entire gear would lead to bothgreater number of nodes and time needed for reaching a solution.Since the decreases of WPL generated by spur gears is susceptibleto be done from the variation of multiple parameters this novel ap-proach will be helpful in CFD investigations for the mitigation ofhigh speed transmissions.

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