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Tribology International 41 (2008) 914–919

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Numerical simulation of piston ring lubrication

Christian Lotz Felter

MAN B & W Diesel A/S, Technical University of Denmark, Nils Koppels Alle, Bygning 404, DK - 2800 Kgs. Lyngby, Denmark

Received 4 December 2006; received in revised form 20 November 2007; accepted 23 November 2007

Available online 22 January 2008

Abstract

This paper describes a numerical method that can be used to model the lubrication of piston rings. Classical lubrication theory is based

on the Reynolds equation which is applicable to confined geometries and open geometries where the flooding conditions are known.

Lubrication of piston rings, however, fall outside this category of problems since the piston rings might suffer from starved running

conditions. This means that the computational domain where the Reynolds equation is applicable (including a cavitation criteria) is

unknown. In order to overcome this problem the computational domain is extended to include also the oil film outside the piston rings.

The numerical model consists of a 2D free surface code that solves the time dependent compressible Navier–Stokes equations. The

equations are cast in Lagrangian form and discretized by a meshfree moving least squares method using the primitive variables u, v, r for

the velocity components and density, respectively. Time integration is performed by a third order Runge–Kutta method. The set of

equations is closed by the Dowson–Higginson equation for the relation between density and pressure. Boundary conditions are the non-

slip condition on solids and the equilibrium of stresses on the free surface. It is assumed that the surrounding gas phase has zero viscosity.

Surface tension can be included in the model if necessary. The contact point where the three phases solid, liquid, and gas intersect is

updated based on the velocity of the solid and the angle between the normals of the solid and the free surface.

The numerical model is compared with the results from an analytical solution of the Reynolds equation for a fixed incline slider

bearing. Then results from a more complicated simulation of piston ring lubrication are given and discussed.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Piston ring; Reynolds equation; Navier–Stokes equations; Free surface; Moving least squares

1. Introduction

The performance of piston rings in combustion engineshas been a topic of research for many years. Piston ringsact as sealing between the liner and the piston. The pistonrings are lubricated with oil and can thus be considered asslider bearings. This paper is concerned with theoreticalpredictions of the performance based on numericalsimulations.

Classical theory of lubrication is based on the Reynoldsequation, which can be derived from control volumeanalysis under certain simplifying assumptions [1]. Thisequation calculates the oil film pressure given the filmthickness, squeeze velocity, and the pressure at theboundaries (typically ambient pressure). It is well knownthat piston rings can have greatly changing runningconditions. At some points the piston ring might experience

ee front matter r 2007 Elsevier Ltd. All rights reserved.

iboint.2007.11.018

ess: clf@mek.dtu.dk

fully flooded conditions, at others it might suffer fromstarvation. This means that knowledge of the amount ofavailable oil becomes very important for successfulsimulations. However, the Reynolds equation does notapply outside the piston ring and therefore cannot modele.g. buildup of oil in front of the moving piston ring.Therefore it becomes difficult to relate the undisturbed oilfilm thickness on the liner and the immediate thickness infront of the piston ring.The literature on simulation of piston rings is vast. Since

the problem is very complex each publication typically doesnot treat all effects at once. However, the problem of theunknown inlet film thickness has been addressed by otherworkers. In the paper by Esfahanian et al. [2] a single ringis treated assuming fully flooded conditions at all times.Dowson [3] analyzes a single ring and a complete ring pack.In that paper the undisturbed oil film thickness is assumedto be one-half of the film thickness at the location underthe piston ring where qp=qx ¼ 0 (thus the effect of squeeze

ARTICLE IN PRESSC. Lotz Felter / Tribology International 41 (2008) 914–919 915

action on the oil flow is neglected). A more complicatedmodel is given in [4], where oil build up in front of thepiston ring is modeled by a trapezoidal geometry. The sameidea is followed in [5] expect the build up is assumed tohave a parabolic shape.

In this paper a 2D free surface model based on theNavier–Stokes equations is developed. The followingsection describes the governing equations and boundaryconditions. Then follows a section on the numericalmethod. Finally we compare analytical results from thefixed incline slider bearing with the numerical results, and amore general problem including a free surface is solved.

2. Method

2.1. Governing equations

The fluid flow is governed by the 2D compressibleNavier–Stokes equations

du

dt¼

1

rm

q2uqx2þ

q2uqy2þ

1

3

q2uqx2þ

q2vqxqy

� �� ��

qp

qx

� �,

dv

dt¼

1

rm

q2v

qx2þ

q2vqy2þ

1

3

q2uqyqx

þq2vqy2

� �� ��

qp

qy

� �, (1)

where d=dt is the total derivative with respect to time t, u

the x-component of velocity, v the y-component ofvelocity, p the pressure, r the density and m the dynamicviscosity.

As can be seen from the expressions above viscosity isassumed to be constant, with a value given by an estimatedaverage. One possibility is to evaluate viscosity at the meantemperature along the liner (based on measurements) andneglect the dependency from pressure. The continuityequation is stated as

drdt¼ �r

qu

qxþ

qv

qy

� �. (2)

The system is closed using the Dowson–Higginsonequation of state [1, p. 71]

r ¼ r0 1þ0:6p

1þ 1:7p

� �, (3)

where r0 is the density when p ¼ 0, kg=m3 and p the gaugepressure, GPa.

Piston rings in general have a barrel shaped runningsurface, which gives a converging/diverging geometry inconnection with the liner. When the piston ring is movingpressure will build up on the converging part whilecavitation might appear on the diverging part, due to theassociated pressure drop. Currently the effect of cavitationis not included in the model, meaning that simulations areonly valid when cavitation is not present. A cavitationmodel will be added in a future work.

2.2. Boundary conditions

Three types of boundary conditions are considered inthis work. On solid walls the non-slip condition is used,which means that

ufluid ¼ uwall; vfluid ¼ vwall. (4)

In order to find the pressure on solid walls we write thefollowing equation:

nx

ny

!�

du=dt�RHSx

dv=dt�RHSy

!¼ 0, (5)

where nx is the x-component of solid wall normal, ny the y-component of solid wall normal, � denotes the scalarproduct of two vectors, RHSx the right-hand side of firstequation in (1) and RHSy the right-hand side of secondequation in (1) which after rearrangement gives aNeumann condition for pressure qp=qn ¼ � � �. Note thatthe velocity at the wall is known so one can evaluate du=dt

and dv=dt. Note also that p and r are not independent sincethey are coupled through the equation of state. Thus onecan either eliminate r using

r ¼ f ðpÞ

or eliminate p by writing

qp

qx¼

qp

qrqrqx

,

qp

qy¼

qp

qrqrqy

and then solve (5) for just one unknown quantity. We usethe second possibility solving for r. Because the equationof state for this application is nonlinear the solutionprocedure for r uses iteration.The boundary condition for a free surface is the balance

of normal and shear stresses on the interface separating thetwo fluids. This can be stated as

ðTijninjÞliquid ¼ ðTijninjÞgas,

ðTijnisjÞliquid ¼ ðTijnisjÞgas,

where T is the stress tensor, n the normal vector and s thetangent vector.It is assumed that the gas phase has constant pressure

and zero viscosity. The effect of surface tension isneglected. Substituting the expression for the stress tensor[6] and using the fact that the normal and tangent vectorsare perpendicular we get

pþ2

3m

qu

qxþ

qv

qy

� �� 2m

qu

qxn2

x

��

þqv

qxþ

qu

qy

� �nxny þ

qv

qyn2

y

��liquid

¼ ½p�gas,

qv

qxþ

qu

qy

� �ðn2

x � n2yÞ � 2

qu

qx�

qv

qy

� �nxny

� �liquid

¼ ½0�gas. ð6Þ

ARTICLE IN PRESSC. Lotz Felter / Tribology International 41 (2008) 914–919916

The pressure on the left-hand side (pressure in the liquidphase at the interface) can be calculated using (5) and (3).This gives three nonlinear equations for the threeunknowns u, v, and p.

Finally a special boundary condition for the triple point

where the free surface touches the solid walls is needed.Clearly this point must be able to move and therefore thenon-slip condition does not apply. The physic of the triplepoint acts at the level of molecules and cannot be simulateddirectly by a continuum model. Instead we require no flowthrough the wall, and adjust the triple point velocity suchthat the contact angle tends to p=2 at all times. This can beachieved by

utriplepoint ¼ uwall þ nwally q,

vtriplepoint ¼ vwall � nwallx q, (7)

where

q ¼ ½cosðfssÞ � ðnwallx nfreesurface

x þ nwally nfreesurface

y Þ�C,

where fss is the prescribed steady state contact angle and C

the tuning parameter.See [7] for details and more advanced models. Pressure is

not evaluated at the triple points since the normal is notwell defined.

3. Numerical scheme

In this section we describe how the expressions from theprevious section are discretized. We employ the semi-discretization technique. For the spacial derivatives we usethe method of moving least squares. The method isoutlined below [8].

Consider the field f ¼ f ðxÞ ¼ f ðx; yÞ. If f is smoothwithin some distance from the point x� it can beapproximated by a polynomial surface. Define the func-tional

Jðx�Þ ¼

ZO

wðx� x�Þ½aðx�ÞTpðx� x�Þ � f ðxÞ�2 dx (8)

where

x� point of expansion;

w kernel function with compact support;

a coefficients to be determined;

p ¼ f1 x y x2 xy � � �g polynomial basis:

(9)

(We distinguish between pressure p and basis functions p.One element of p is denoted by the usual notation pi.) Weuse the following kernel expression:

wðh;xÞ ¼

1� 3=2q2 þ 3=4q3; qo1;

1=4ð2� qÞ3; 1pqo2;

0; 2pq;

8><>: (10)

where h is the smoothing length and q ¼ 2kxk=h. It is seenthat the kernel has compact support, which makes the

functional (8) local. Thus, in order to evaluate J we do notneed to sum over all points, but only those within thesmoothing length of x�. Furthermore the kernel has a bell-like shape, with qw=qqjq¼2 ¼ 0. This means that a point canenter or leave the support of J is a smooth manner.The coefficients a are found by minimizing J using the

stationarity condition qJ=qai ¼ 0. In a discrete setting thisbecomes

XI

wðxI � x�ÞXN

i¼1

piðxI � x�Þaiðx�Þ � f I

" #pkðxI � x�Þ ¼ 0,

(11)

where

N number of basis functions;

k ¼ 1; 2; . . . ;N;

I index of points within smoothing length.

(12)

After rearrangement the equations can be put in matrixform

Ma ¼

PI wðxI � x�Þðf I � f �ÞP

I wðxI � x�Þp1ðxI � x�Þðf I � f �Þ

..

.

PI wðxI � x�Þp5ðxI � x�Þðf I � f �Þ

8>>>>><>>>>>:

9>>>>>=>>>>>;, (13)

where M is called the matrix of moments. Note that theequation corresponding to the constant term in thepolynomial basis has been eliminated by moving f � ¼

f ðx�Þ to the right-hand side. This ensures that theapproximating surface passes through the field value atthe point of expansion (x� must coincide with one of thedata points xI ). This property is crucial for the method.After determining the coefficients the field f can beapproximated in the vicinity of x� by (with a1 ¼ f ðx�Þ)

f ðxÞ ¼XN

i¼1

piðx� x�Þai. (14)

It is seen that the units of the ai’s depend on the units of thefield function f. Similarly the partial derivatives at the pointof expansion x� are approximated by

qðjÞfqxðkÞqyðlÞ

¼XN

i¼1

qðjÞpið0Þ

qxðkÞqyðlÞai, (15)

where k þ l ¼ j. These expressions are substituted directlyinto the governing equations. The time integration isperformed by an explicit third order Runge–Kutta method.As indicated in (1) the computation is performed in themoving (Lagrangian) frame, which means that the positionof the computational nodes must be updated according to

dx

dt¼ u;

dy

dt¼ v. (16)

Due to the distortion of the flow field it is necessary toredistribute the computational nodes at some time interval.

ARTICLE IN PRESSC. Lotz Felter / Tribology International 41 (2008) 914–919 917

4. Results

First we compare three solutions for pressure from thefixed incline slider bearing. Consider the following situa-tion:

inlet hi ¼ 25 mm,

outlet ho ¼ 15mm,

width l ¼ 1mm,

speed u ¼ 1m=s,

viscosity m ¼ 0:05Pa s,

density r0 ¼ 900 kg=m3.

In Fig. 1 the analytical solution of the Reynolds equation isplotted together with the pressure obtained from thesimulation program. The effect of compressibility isexamined by solving the Reynolds equation for anincompressible fluid and also using the Dowson–Higginsonequation of state. It is seen that compressibility leads to aslightly higher maximum pressure and that the location isshifted downstream. The solution of the Navier–Stokesequations gives results similar to the Reynolds equation. Itwas, however, not possible to achieve a fully convergedsolution of the Navier–Stokes equations because of an

0 0.2 0.4 0.6 0.8 1

x 10−3

0

1

2

3

4

5

6x 104

x [m]

gauge p

ressure

[P

a]

Reynolds (incompr)

Reynolds (compr)

Navier Stokes

Fig. 1. Pressure curves for the fixed incline slider bearing.

Fig. 2. Density plot for the fixed incline slider bearing. Red co

instability at the inlet boundary condition. It is expectedthat this explains the somewhat smaller maximum pressure.A better implementation of the inlet boundary condition isa subject for future work.Fig. 2 shows the density distribution in the fixed incline

slider bearing at the final time of the simulation, that is, atsteady state. It is seen that the density (from a graphicalview) is constant across the film. This feature is one of theassumptions under which the Reynolds equation is derived.The graph on Fig. 2 indicates that this assumption isperfectly valid for thin films at steady state.Second we simulate a more complicated situation

including a free surface. Initially the piston ring is at restand the oil film has uniform thickness outside the pistonring, see Fig. 3. The horizontal and vertical position of thepiston ring is prescribed by sine functions. In a realapplication the vertical position will be controlled by theforces acting from the surrounding gas and the oil film.Modeling of that effect is omitted in this presentation.Here are the simulation parameters:

ring shape parabolic,

ring height sh ¼ 1mm,

ring width l ¼ 2mm,

ring horizontal speed u ¼ 0:2p cosð200pt� p=2Þm=s,

ring vertical speed v ¼ 0:05p cosð400pt� p=2Þm=s,

ambient pressure pamb ¼ 0Pa,

initial oil film thickness hoil ¼ 850mm.

It should be noted that these settings are unrealistic for anyinternal combustion engine. However, we use these valuesto illustrate the performance of the free surface code. Itshould also be noted that cavitation is not included in themodel, so negative pressure can appear during thesimulation. Gravity forces are not included.The first plot shows the velocity field after the first 100

time steps. It is seen that the triple points are accelerated inorder to produce a contact angle of p=2 (velocity vectorsare scaled using the maximum speed). The following plotsshow the build up of a vortex that develops under thepiston ring. Also the deformation of the oil film is clearlyseen.

lor means high density and blue color means low density.

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Fig. 3. Snapshots from piston ring simulation. From top to bottom t ¼ 3� 10�5 s, t ¼ 8:1� 10�4 s, t ¼ 1:7� 10�3 s, t ¼ 2:8� 10�3 s, t ¼ 3:9� 10�3 s.

C. Lotz Felter / Tribology International 41 (2008) 914–919918

5. Discussion

Two simulation problems have been considered. For thefixed incline slider bearing the method agrees well whencompared with the results using the Reynolds equation.For the case of piston ring simulation the model is not fullydeveloped yet. From the modeling point of view the effectof cavitation needs to be considered. Currently, the modelallows the generation of negative pressure, which is notphysically admissible. The computational work required bythe method must be investigated further. The time requiredfor the calculation shown in this paper is approximately 1 hon a standard 3GHz desktop computer. However, whensolving real life problems the spacial resolution (andconsequently the time step) must be improved. Despitethese issues the presented method appears to be apromising alternative, when the Reynolds equation cannotbe applied.

6. Conclusion

A new approach for the numerical simulation of pistonring lubrication has been presented. The main idea is tosimulate also the free surface of the oil film outside thepiston rings. The method has been compared with resultsfor a confined geometry using the Reynolds equation withwell agreement. The current state of the model indicatesthat more work is needed in order to include the effect ofcavitation and also examine the computational costinduced by problems from real applications.

References

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