Keywords: Texturation, lubrification hydrodynamique, contact segment-chemise, prédiction Cross-hatching, hydrodynamic lubrication, PRCL contact, prediction III

La microgéométrie de la surface de la chemise joue un rôle très important dans les pertes par frottement et dans la consommation de l'huile dans un moteur à combustion interne. Une des texturations classiques de cette surface est celle créée par pierrage. Elle se compose de plateaux plus ou moins lisses et de profondes stries croisées. L'épaisseur du film d'huile est influencée fortement par cette texturation. Un modèle simplifié du contact segment chemise a été créé en la présence de la microgéométrie. Puis, un code de calcul basé sur la méthode numérique multigrille a été développé. Ce code a été utilisé pour des études paramétriques avec des jeux de paramètres très variés. Les calculs quantifient l'influence de cette microgéométrie particulière sur la relation film d'huile -portance. Les résultats mettent en évidence deux mécanismes distincts de génération de portance selon le type du segment. Le segment parabolique porte par son convergent et les stries ne font que diminuer cette portance. Inversement, le segment plat ne porte pas et ce sont les stries qui génèrent la portance. Deux modèles de prédiction ont finalement été déduits, un pour les segments paraboliques et un pour les segments plats. Ces deux prédictions ont ensuite été validées par des calculs sur des surfaces mesurées.

Foreword

During my PhD study, I have had support from many people which I would like to thank.

First of all, I profoundly thank my supervisor Pr. Ton Lubrecht for offering me this PhD position and for his constant guidance, help and encouragement throughout this work. This work is mainly the result of the collaboration between LaMCoS (Contact and Structure Mechanics Laboratory -INSA de Lyon) and PSA Peugeot-Citroen. I would like to thank Pr. Combescure and Pr. Dureisseix for the welcome at LaMCoS. I also would like to thank PSA members: Dr. Cavallaro, Dr. Charles, Mr. El Fassi, Mr. Mermaz Rollet and Dr. Messé for their precious technical and financial aid.

I sincerely thank the jury members: Pr. Mazuyer (LTDS Ecole Centrale de Lyon), Pr. Monteil (Femto-st, ENS2M Besanon), Dr. Fillon (Institut Pprime, CNRS Universit de Poitier), Dr. Tian (Sloan Automotive Laboratory, MIT) and Dr. Charles (PSA) for accepting to evaluate my work.

There are many people who greatly contributed to the completion of this work, particularly Dr. Nans Biboulet who has supported me all along this work not only with valuable comments and suggestions but also with a lot of tips and tricks in coding. Dr. Philippe Sainsot and Mr. Jerôme Cavoret also helped me with measurements and surface analysis. Mr. Drouvin and Mr. Tanguy from Toyota Motorsport GmbH, greatly improved my understanding of PRCL contact with our fruitful discussions. Mrs. Sophie De Oliveira was always ready for assistance. I would like to express my gratitude to them. I would like to thank the members of the LaMCoS Laboratory, who not only created a pleasant work environment but also gave me lots of support in my work during the last four years: Lamia, Mathieu, Hugo, Florian, Ghizlane, Laure, Amina, Florence, Bachir, Mohamed, Hanfeng, Eliza and Hassen to name but a few. More generally speaking, I thank wholeheartedly all of my friends and relatives who were always there for me.

Finally, my wholehearted thanks are given to my wife Houda, my parents Souad and Moncef, my brother Hassen and my sister Fatma for their love. They have always supported and helped me in my study and arranged everything in my life. ness relation Distance between crossovers in y direction [

Introduction

According to "The International Organization of Motor Vehicle Manufacturers" 1 , over 80 millions motor vehicles were produced worldwide in 2012, including passenger cars, light commercial vehicles, heavy trucks, buses and coaches.These vehicles are almost equipped with internal combustion engines (IC engines). The produced engines are subject to particular requirements. In fact, the energy saving is of major concern. Customers and NGOs are more and more hard to please. The engines have to consume increasingly less fuel. As an example, the Global Fuel Economy Initiative 2 aims 4 l/100km by 2030 for light-duty-vehicle oil consumption (halving the 8 value of 2005). Almost 3 % of fuel consumption reduction are needed each year to reach this value. The thermal efficiency is scrutinized and the energy losses are deeply explored by manufacturers. The way to reach these ambitious goals is to increase the engine efficiency. In fact, the efficiency of a diesel engine is only about 29.5% tank to wheel and 22% for spark ignition engines typically gasoline engines [GOS 07]. Figure 1 shows the distribution of the energy losses in a diesel engine. It highlights the im- portance of the mechanical friction and then focuses the weight of the piston ring -cylinder 1 www.oica.net 2 www.globalfueleconomy.org liner (PRCL) friction in energy losses. According to this figure, the ring friction removes 0.5 % to 4 % of the engine efficiency. Consequently, reducing the PRCL friction is a straightforward way to increase the tank to wheel efficiency. There are several ways to reduce this friction but they are directly related to the PRCL clearance which means to the lubricant film thickness.

In addition to the energy saving issue, pollution and health are of major concern. The pollution standards in European legislation 3 , as an example, are getting more and more tough toward the car emissions. Table 1 shows clearly that not only the limit line has been approaching the zero emission limit, but also new requirements or criteria are added progressively such as the particle number per kilometre criterion added to the Euro5 stage. These emissions come directly from the combustion chamber exhaust which contains notably the combustion by-products, the unburned hydrocarbons and a certain amount of lubricant. A post-treatment does exist. The catalyst converts toxic pollutants to less toxic pollutants by means of redox reaction. However, this solution is expensive due, among others, material cost. Ideally, the pollutants have to be removed from the source which is the combustion chamber and notably the lubricant leakage. The amount of lubricant that is present during combustion has to be reduced. This quantity is controlled by the PRCL clearance. From above it is clear that, the lubricant film thickness acts directly on both the energetic and the environmental issues. It is important to understand the parameters that are inherent to the film forming (examples: ring shape, rheology, starvation, working conditions ...). The liner microgeometry is a predominant parameter that influences the film thickness and consequently the tribological performance. The liner surface features a special microgeometry. The scope of the thesis is to study the effect of this particular microgeometry on the load carrying capacity -film thickness relation. A classical texturing is the cross-hatching pattern obtained by the honing process. This texturing is highly directional with two major components: the plateaux and the crosshatched grooves. The classical parameters as R a , R q or valley/peak statistics may be unable to correctly describe this kind of microgeometry. Functional hydrodynamic analysis using statistical methods, notably Patir and Cheng's method, lumps the different components of the surface. There are methods decoupling these components (notably the homogenization method). Another alternative is to use a deterministic model. To correctly describe the roughness, the number of points has to be very large, impacting the computing time.

Solutions using direct method are not feasible and iterative methods will require a huge computing time. The use of the Multigrid method drastically reduces these calculational costs.

A code based on the multigrid method is developed. It is based on the deterministic model of the PRCL contact. This tool serves to study the influence of microgeometric parameters.

The resulting analysis permits one to quantify the importance of each parameter notably on the Load Carrying Capacity(LCC)-Film thickness relation. As a result, the prediction of the LCC is established. It gives an instantaneous approximation of the LCC for a given set of microgemetric parameters and for a range of film thicknesses.

Chapter 1 gives the technical context and enumerates the general considerations concerning the the piston-ring-cylinder-liner contact (PRCL). It introduces the lubrication and focuses particularly on the ring lubrication. The role of the ring pack is explained and the operating conditions are cited. The liner texture is presented and its influence on PRCL contact is shown via the literature. Chapter 2 deals with the PRCL deterministic modelling and the numerical method that is used (smooth, analytical and measured surfaces). It starts with the geometrical description then the different models that are used. The equations governing the problem are shown. Finally, the multigrid method is introduced. Chapter 3 studies three PRCL cases: the smooth case, the compression ring lubrication and the oil control ring lubrication. An analytical surface model is used to scrutinise the influence of the microgeometric parameters on lubrication. Predictions of the LCC are obtained. Chapter 4 links predictions to the real surface measurements.

Introduction

The contact between the piston ring and the cylinder liner is governed by complex laws. In fact, several phenomena occur in this contact impacting the whole behaviour of the engine and consequently its performance. These phenomena are various and are generally related to the geometry and to the operating conditions. Common engines have three rings that are geometrically different. They consequently have different behaviour and carry more or less load. In addition, the cylinder liner surface features a special micro-geometry. This texturing introduces fluctuations in generated pressure and consequently influences the load carrying capacity. This chapter precedes the detailed study of the PRCL contact, it introduces the technical context and enumerates the general considerations concerning the piston-ring-cylinder-liner (PRCL) contact. First, the lubrication phenomenon is explained, focusing on the role that it plays in an IC engine. In addition, the oil supply of the PRCL contact is shown. Thus, the hydrodynamic lubrication is briefly explained and the different regimes are clarified via the Stribeck curve.

The second section focuses on the different types of piston rings and their fundamental role in the IC engines. In the last part of this section, the operating conditions are enumerated. Some of the relevant literature that concerns these conditions is cited. The third section deals with the cylinder liner texture. It explains how the surface is obtained and describes the specificity of its geometry. Finally, it summarizes some of the existing literature on the influence of texturing on the PRCL contact.

Engine lubrication 1.2.1 Lubrication purpose

Engines contain components that move with respect to one another under a certain load. The friction between these two components generates a great amount of heat and causes damage to the solid surfaces leading to wear or in some cases to failure. The lubricant purpose is primary to minimize the friction between the moving parts, avoiding direct contact between these parts by interposing a substance between the opposing surfaces (Figure 1.1). The interposed substance is a liquid and is called lubricant. Well lubricated surfaces have low levels of friction, heat generation and wear. Lubrication can also be an easy solution to avoid corrosion. It separates the surface from its corrosive environment (through additive action). The lubricant permits a removal of small debris and dust, these contaminants could damage the surfaces. The fluid film lubrication provides the ability to carry loads, to absorb shocks and vibrations. The lubricant is also used as a coolant.

The previous abilities are more or less pronounced for each different contact. Four stroke IC engine oils must fulfil the previous functions and also meet a lot of requirements. Stepina and Veseley [STE 92] and also Mortier and Orszulik [MOR 97] detailed the requirements of such an oil (lubricating power, fluidity, high viscosity, thermal and oxidation stability, engine protection, lubricity...). These requirements are sometimes conflicting. The additives are of great importance for the lubricant. For example, the ZDDP additive is added to enhance the lubricity of the oil (the capacity to carry loads in the boundary regime). In fact, it avoids scuffing occurring generally during the running in process, when the surfaces may come into metal to metal contact and may locally weld.

Engine lubrication system

The moving parts in an engine (Figure 1.2) require a large amount of lubricant. Oil has to be sent to all of these zones to avoid failure (lubricating and cooling). The oil supply system differs from one engine to another but the principle is the same. Figure 1.3(a) illustrates a classical circuit containing a network of passages, in both the structure of the engine and its individual components. This circuit is the main method to lubricate the bearings, the crankshaft, the camshafts, the valves and sometimes the piston pin (Figure1.3(b)). The cylinder wall and the piston skirt are lubricated by the oil flung of the big end of the connecting rod using the centrifugal force (Figure1.3(c)). Some engines uses several jet spray points to enhance the oil distribution on the wall. The oil supply to the PRCL contact is not abundant, only a very small quantity reaches the top ring. Moreover this supply is not uniform circumferentially. In fact, the oil flung and the jet spray points do not distribute the lubricant uniformly due to their preferential directions. The rings operate under a starved regime and especially the top ring which is the farthest from the oil supply.

Hydrodynamic lubrication

Nowadays, lubrication is used in a multitude of domains notably in the machine elements like journal bearings, rolling elements, PRCL, cam tappets... The comprehension of the lubrication has permitted a great advance in engine design. To understand lubrication, one has to go back to 1883. In that year independent experimental work by Tower (U.K.) [TOW 83]

   ∂ ∂x ( ρh 3 12η ∂p ∂x ) + ∂ ∂y ( ρh 3 12η ∂p ∂y ) - ∂(u m ρh) ∂x - ∂(ρh) ∂t = 0 (x, y) ∈ ω 1 p = 0 (x, y) ∈ ω 2 (1.2)

ElastoHydrodynamic Lubrication (EHL)

The EHL is based substantially on three phenomena, the first is the hydrodynamic effect described in the previous paragraph. The second one is related to the deformation of the surface. The third is the piezoviscous effect relating the viscosity to pressure. The EHL with a neglected deformation and neglected piezoviscous effects, correspond to the hydrodynamic lubrication regime(HL) .

Stribeck curve

The Stribeck curve defines the lubrication regime of a given contact. It represents the friction coefficient as a function of the dimensionless number η u/w where η is the fluid viscosity, u is the velocity and w is the normal load (per unit length). Each contact has its own Stribeck curve. Figure 1.4 is an illustration of the curve. One can distinguish in this curve three zones corresponding to three different lubrication regimes: -Boundary regime At low speed, very high load or low viscosity the friction is at its highest value. The regime then is called boundary regime. It is characterised by a high proportion of solid to solid contact. The fluid does not carry any load. This load is entirely supported by the roughness contacts of both solids. The fluid acts as a reservoir for additives -Mixed regime Both the asperities and the film carry the load. In the mixed regime the surface roughness plays an essential part, it determines the slope of this part of the curve. In fact, when the surface is smooth the slope becomes high and the transition to the full film zone is fast.

-Full film regime The load is totally carried by the fluid. The lubrication can be either hydrodynamic or elastohydrodynamic, it depends on the pressure level generated. In fact, the order of magnitude of the maximum pressure is Mega Pascals for the HL against Giga Pascals for the EHL regime.

Piston rings 1.3.1 The ring pack role

The ring pack has to ensure two main functions: -the sealing function The combustion gas and the oil sump are the object of sealing. In fact, the unburned fuel and exhaust gas can contaminate the oil through entering the crankcase (blow-by phenomenon). The oil degrades chemically and it loses gradually its ability to lubricate. This sealing function is essentially ensured by the top ring. It avoids also the waste of power due to gas leakage. The oil should not enter the combustion chamber otherwise it burns, creating undesirable combustion results (the environmental constraint). The sealing avoids -the lubrication function The lubrication has to be maintained when sealing. A minimum oil quantity ensuring lubrication has to reach all the rings. There is a tradeoff between good lubrication and oil consumption (sealing and lubrication functions are contradictory). The oil flow is controlled by the ring pack and especially by the Oil Control Ring (OCR).

The rings

A piston ring is an elastic ring which fits perfectly inside the cylinder by elastic deformation (Figure 1.5(b)). It is placed in a groove in the piston head. In order to enable mounting, a ring gap is provided. The resulting load is the tangential load or pretension and it is noted T . A ring pack contains between 2 and 5 rings but at least one compression ring. The most common car engines contain three rings (Figure 1.5(a)): -the top ring or the compression ring: its face is in general barrel-shaped or with a rectangular cross-section and it is chrome or molybdenum coated. It ensures the major part of sealing. It faces high temperatures and high pressures during the combustion phase. The pretension for this ring is in general between 5 and 20 N depending on the engine.

-the scraper ring (second compression ring): its profile is rectangular. It ensures more sealing and scrapes some oil at the same time (scraping during down ward motion). Its pretension is between 5 and 30 N.

-the oil control ring: Its geometry is more complex then the other rings. It performs the last task of sealing (or the first regarding the oil). It distributes the oil evenly onto the liner with its slots perforated in the peripheral direction. The oil control rings may have a spring inserted for additional tension to the pre-tension of the ring. The additional force on the oil control rings generates a total pretension of 10 to 60 N.

The operating conditions 1.3.3.1 The oil supply

The oil supply of the ring pack is extremely complex. Each ring is supplied differently. The sealing function forces the rings (notably the compression ring) to function with a limited amount of oil called starvation. The quantity of oil available depends particularly on the oil flows. The oil transport is influenced by multiple factors: the ring gap, the ring groove, the ring geometry, dynamics, liner surface, oil projection on cylinder wall, etc... This problem has been widely studied, one can name the following studies related to oil transport: -Casey [CAS 98] studied experimentally the oil flow. He concluded that oil transport pathways through gaps and grooves and inertial forces play a significant role.

-Experimental work has been carried out by Thirouard and Tian on oil transport in the ring pack [THI 01][THI 03a][THI 03b]. They used a multiple dye laser induced fluorescence to observe oil displacements in both axial and circumferential directions. The work showed global and detailed oil flows, via a region by region transport analysis.

-Organisciack [ORG 07] uses a starvation model to study the effect of the liner surface finish on oil redistribution. He suggests that the cross-hatched grooves and the discontinuous long grooves are the best microgeometries for good oil redistribution.

-Senzer [SEN 12] focused the oil transport via the OCR groove and the third land oil pattern (region between OCR and flutter ring). He highlighted the importance of the ring design and the operating conditions.

Kinematics and dynamics

To evaluate the dynamic effect on a ring, the moving parts have to be considered. Taylor and Dowson detailed the different forces acting on a piston ring [TAY 93]. The main forces acting are: -the gas pressures behind the ring, -the normal and tangential forces exerted by the PRCL contact, -the pretension of the ring, -the reaction of the piston groove flank (may be cancelled in some cases), -the inertia of the ring, -and the axial accelerations. The reaction of the piston groove flank is sensitive to the dynamic piston tilt (secondary motion), a theoretical model was developed by Tian for a twin land OCR which proves that the minimum oil thickness differ for the two lands especially in both thrust and anti-thrust sides [TIA 00]. Some transient phenomena can interfere on the equilibrium of the ring and on the oil distribution. Tian studied the ring flutter and the radial collapse [TIA 02a][TIA 02b]. He shows how the gas pressures can be redistributed and the stability of the ring changed by modifying small geometrical details. He also shows the importance of the coupling between dynamics and gas flow after a ring loses its axial stability. The honing process (Figure 1.7 (a)) is used in mass production for internal combustion engines. It is the final surface finish using an abrasive finishing process. Small abrasive stones rotate and oscillates creating plateaued surfaces (example in Figure 1.7 (b)). This process has been widely used for mass production in IC engines. Other processes, namely laser texturing [ETS 05] [ VIN 08], are used for more advanced engines. These processes have the advantage to be more flexible in defining the micro geometry.

Surface description

The honing process gives the cylinder liner a special surface texturing. It features a special micro geometry, with plateaux and cross-hatched grooves. Figure 1.8 (a) shows a new liner cut-up. The liner surface is totally honed and the cross hatching pattern is evenly distributed. On the run-in surface (1.8 (b), (c)) the plateaux are smoother. Measurements (Figure 1.8 (d), (e), (f )) show that the cross hatching parameters differ. In fact, the width, the depth of the grooves varies and discontinuities may occur. Reynolds equation remains valid for this kind of grooves, since the groove width is very large compared with the depth (cf. Organisciack [ORG 07] and Dobrica etal. [DOB 09]). The marking feature of this surface is the directionality. Figure 1.8 (g) shows two preferential directions.

The influence of the texture

The texturing acts as a micro-bearing, oil reservoir and debris trap [ETS 05]. These functions are crucial to avoid failure. Many experimental works on the influence of textured surfaced have been carried out over the previous years. They permit a better understanding but sometimes they are contradictory. Theorically, different approaches to deal with the influence of the texture are found in the literature. The first one is the stochastic approach. It is based on the random hypothesis of the roughness. Patir and Cheng [PAT 78] [PAT 79] have initiated this method which permits to derive an averaged Reynolds equation. In fact, the average influence of the roughness in a direction is reflected by a flow factor in that direction. The flow factors are determined by precalculations. This method has permitted to take into account the surface texturing with a fast approximation. In 1980, Rohde [ROH 80] presented a two-dimensions contact model of a rough liner and calculated for a whole cycle. He highlighted the effect of the roughness on the friction. Michail and Barber [MIC 95a][MIC 95b], using flow factors, assert that for the same operating conditions, angles below 45 degrees enhance the hydrodynamic effect by increasing the film thickness. Bolander on the secondary motion of the piston... This approach lumps the different components of the surface (grooves and plateaux) and does not consider the roughness directionality. Methods decoupling both components (grooves and plateaux), like the homogenization method [BAY 05] [ALM 11] are also used. Spencer [SPE 10] coupled the homogenization method with the flow factor method. Another alternative is to use a deterministic model of measured surfaces. This method is based on direct measurements of the liner. Caciu [CAC 06] filtered the micro-geometry and did not consider the curvature of the ring. Organisciak [ORG 07] studied the effect of starvation without considering the roughness on plateaux. He suggests to use narrower and denser cross-hatched grooves for better redistribution of the lubricant and to reduce friction. Pascovici et al. [PAS 09] investigated the partially textured surface effect. They highlighted the potential of the partial texturing in the lift off effect (contributing to the load carrying capacity). Dobrica et al. [DOB 10] demonstrated, using dimples, the importance of the grooves parameters and their aspect in pressure generation. They concluded that the full texturing can generate hydrodynamic lift for parallel surfaces while it has a negative effect for convergent plane-inclined surfaces. They also concluded that the texture effect seems proportional to the texturing density. Dobrica et al. [DOB 12] simulated scratched journal bearings. They performed a parametric study and concluded that the deeper / denser / more numerous the scratches, the poorer the bearing performance. Chen [CHE 08] [CHE 11] correlated experimental results with a deterministic model. He highlighted the importance of the roughness in the load carrying capacity for the flat ring. For the compression rings he found that the liner microgeometry is as important as the ring profile and the starvation degree.

Conclusion

This chapter presented the environment of the piston ring cylinder liner (PRCL) contact and enumerated the different phenomena that act on its tribological performance. These phenomena are various and are extremely coupled. The chapter started by presenting the engine lubrication, focusing on the role that it plays in an IC engine. The lubrication regimes were explained and the Stribeck curve shown. The second part of the chapter was dedicated to the rings. It highlighted the importance of these rings and the role that they play in lubrication and sealing at the same time, functions that are contradictory. Thus, some of the important operating conditions were enumerated such as the oil supply and the dynamics. The last part of the chapter showed the surface finish of the liner, second part of the PRCL contact. The literature was scrutinized to sum up the different models and results inherent to PRCL contact. These results are important in the understanding of the microgeometry effect on lubrication but only few studies deal with the texture effects in PRCL contact and especially the quantification and generalisation of the effect of the classical cross hatching parameters. The results in the next chapters contributes to the understanding of this effect whilst predicting the load carrying capacity -film thickness relation as a function of the cross hatching parameters.

Introduction

A full understanding of the phenomena taking part in the PRCL contact is necessary to enhance the contact performance. This understanding is established through modelling. Modelling is the art of transforming a real complex problem to "simple" mathematics. Modelling requires a precise knowledge of the real geometries and the real operating conditions, presented in Chapter 1. Chapter 2 links the previous chapter to the following ones. The first section presents the different models used in Chapters 3 and 4. Many simplified models are created corresponding to the main objective which is the understanding of the influence of the microgeometry. This first section contains the different assumptions involved in the PRCL contact model. It describes also how the rings and the liner are modelled, leading to equations governing the problem. Generally in lubricated contact problems three equations are obtained: -the Reynolds equation describing the lubricant flow, -the film thickness equation describing the geometry including the ring and the liner surface, -the force balance equation which is the fundamental equation of mechanics. The second section of the chapter is dedicated to the numerical tool, the key to solve the mathematical problem. First, the dimensionless equations are derived. Then the problem is discretized to obtain the numerical formulation. The numerical tool is based on the multigrid method. The multigrid method permits a fast and efficient transient calculation. The basics of this method are explained. Finally, the calculational algorithm is presented and an example of calculation is shown.

Model

General assumptions

Generally the PRCL contact is assimilated to a hydrodynamic lubricated contact (in contrast with an ElastoHydrodynamic lubricated contact). This fundamental assumption is based on the low pressures generated by low loads. In fact, these pressures generally do not exceed 100 MPa. Consequently, the lubricant density is considered to be constant all over the range of pressures. In addition the temperature is locally constant. The viscosity is then constant too (because of low pressure). As mentioned in paragraph 1.3.3, the ring is submitted to different loads. All is neglected but the action of normal forces (radial forces). The force balance is quasi static and the only forces acting are -the ring pretension and the gas pressure behind the ring from a side, -the load carrying capacity from the other side and the asperity contact resulting force (if considered). Locally, neglecting the acceleration, the velocity is constant and its direction is x.

Geometric model 2.2.2.1 The domain

The x direction

The contact occurs between the three ring carrying faces and the liner surface (Figure 2.1) . Each of the faces is independent of the others. This contact is time dependent due to the ring sliding motion (x direction). The pressurized zone is then moving. This zone corresponds to the calculational window and is noted [x a , x b ]. x a is the inlet meniscus position and x b is the outlet position. x a and x b are limited by geometrical considerations which is the ring axial thickness. They can also vary with the starvation degree (cf. paragraph 1.3.3.1). In fact, the amount of available oil can vary along the liner in both directions. It is assumed that x a corresponds to the starvation degree. The whole calculational domain is the domain where the window is moving for a given short period of time t f . The initial position is 0 (corresponding to 0 in time) and the last position is x f (corresponding to t f ). The calculational domain corresponds to a tiny zone of the liner surface. In fact, the period has to be short for two reasons: the first is purely numerical (huge number of points) and the second is for modelling purposes (parameters as the velocity, or the temperature are no longer constant). However, the short period remains sufficient for the main objective which is a local characterisation.

The y direction

The orthoradial direction can be flattened y (Figure2.2). In fact, the film thickness is very small compared to the ring nominal radius Φ/2 allowing such an assumption. Moreover, the conformability of the ring to the liner surface maintains the thin film assumption. The calculation window is [y a , y b ] in y direction. The values of y a and y b depend on the liner surface modelling and the boundary conditions.

Rings

The rings are perfectly circular, neglecting the end gaps.

Liner surface

Paragraph 1.4.2 described the liner surface. The cross-hatched surface obtained by honing is modelled in this paragraph. r(x, y) expresses the microgeometry of the surface. Three models are kept for use: the measured surface, the analytical model of the surface and the smooth surface. The measured surface A Sensof ar microscope with a confocal objective is used to measure the surface. The ×50 magnification allows 1mm 2 measurements with a high accuracy. In fact, beyond this area of 1mm 2 two main problems occur: -Stitching become difficult. It consists of automatically overlapping pieces of 200 × 200µm measured, obtaining the 1mm 2 surface.

-The number of points measured becomes very large (larger than 9 000 000). A magnification lower than ×50 would solve the previous problems. Unfortunately, the maximum slope that can be measured by a such lower magnification diminishes. Consequently the grooves, would be poorly measured. Considering the dimensions of the measurement and knowing that the ring conforms in the y direction, one can neglect the curvature in the y direction. Consequently the surfaces are flattened, the macro-curvature is not considered. Figure 2.4 shows two measured surfaces, the resulting useful surface and the removed macro-curvature. The macro-curvature in this figure is due to measurement (parallelism between the objective and the measured surface) and the initial y curvature. r(x, y) corresponds to the measurement values if (x, y) coincides with the measured point. Otherwise it corresponds to the interpolated value of the nearest four measured points.

The analytical surface

The measured surface contains smooth plateaux and relatively regular grooves (at least anglewise). The analytical model admits both assumptions: the perfectly smooth plateaux and the regular grooves. Only four parameters describe a full pattern: -α the half angle (called angle to simplify), -d 1 the distance between the cross-overs in x direction defining the density of the grooves. The grooves are sinusoidally shaped. Figure 2.5 shows the geometry of the cross-hatching pattern and the parameters. r(x, y) defines the surface topography.

               r (x, y, t) = -a 2 cos 2π λ |x cos α ± y sin α| cos α + 1 if |x cos α ± y sin α| cos α < λ 2 r (x, y, t) = 0 elsewhere (2.1)
Smooth surface r(x, y) is zero everywhere, this surface is simply taken as a reference, since the results of this surface are easy to obtain or already known. -the nominal film thickness h 0 which is constant unless in imposed load mode, -the ring profile for the x position, which is x 2 /(2R x ) for a parabolic shaped ring and zero for the flat one, -the r(x, y, t) term explained in subsection 2.2.2.3 . The film thickness equation reads:

Equations

Film thickness equation

h (x, y, t) = h 0 (t) + x 2 2R x + r (x, y, t) (2.2)
   1 12η ∂ ∂x (h 3 ∂p ∂x ) + 1 12η ∂ ∂y (h 3 ∂p ∂y ) -u m ∂h ∂x - ∂h ∂t = 0 (x, y) ∈ ω 1 p = 0 (x, y) ∈ ω 2 (2.3)

Boundary conditions

The boundary conditions in the x direction are Dirichlet conditions. This means that p(x a ) = p(x b ) = 0. In fact, the pressures on both sides of the ring are equal to the pressure of the lubricant saturated vapour. This means that these pressures are negligible compared to the contact pressure and considered nil. In the y direction either periodic conditions or Dirichlet conditions are imposed. In the first case p(y a ) = p(y b ) and for the second p(y a ) = p(y b ) = 0. Periodic conditions are used in case of periodic analytical surfaces. For measured surfaces the periodic conditions can not be applied due to the geometry. Imposing Dirichlet conditions can be justified by the fact that geometrical irregularities like very deep longitudinal grooves can occur at those boundary creating a zero pressure condition. Furthermore, if sufficiently distant these BC do not influence the central zone (further explanation will be provided in Paragraph 4.3.1).

Force balance equation

The force balance equation represents the load equilibrium. The applied load is balanced by the pressure which is generated on the surfaces.

ω=ω 1 ∪ω 2 p (x, y, t) dx dy = w(t) (2.4)
w is the resulting normal force and is expressed in N. The normal force per unit length in the y direction is w 1 which is the normal force divided by the width of the calculational window y b -y a (Figure 2.1) .

w 1 = w (y b -y a ) (2.5)
To have an order of magnitude and considering the action of the pretension force T only, w 1 reads [BRU 86]:

w 1 = T Φ/2 (2.6)

Dimensionless equations

The equations are solved using dimensionless variables for two main reasons:

-first, the dimensionless parameters enable one to obtain variables of the order of magnitude of 1 which is crucial for numerical precision, -second, they enable one to use the opportunity of similarity groups.

The dimensionless variables read

X = x x * (2.7) Y = y x * (2.8) H = h h * (2.9) P = p p * (2.10
)

W 1 = w 1 w * 1 (2.11)
The choice of the appropriate set of dimensionless parameters depends on the studied case. The table 2.1 resumes the different dimensionless parameters used in the four following cases:

-case 1: the measured surface and the ring is parabolic, -case 2: the analytical surface and the ring is parabolic, -case 3: the measured surface and the ring is flat, -case 4: the analytical surface and the ring is flat.

h * x * p * t * w * 1 Case 1 c √ R x c 12ηu m √ R x /c √ c u m / √ R x c 12ηu m R x /c Case 2 h 0 √ R x h 0 12ηu m √ R x /h 0 √ h 0 u m / √ R x h 0 12ηu m R x /h 0 Case 3 c λ 12ηu m λ/c 2 u m /λ 12ηu m λ 2 /c 2 Case 4 h 0 λ 12ηu m λ/h 2 0 u m /λ 12ηu m λ 2 /h 2 0 Table 2.1: Dimensionless parameters
The z-axis dimensionless parameter c make the film thickness and the groove depth dimensionless. The order of magnitude of these last physical parameters is 1µm. Thus, the value of c is chosen 1µm. In all the cases the Reynolds equation 2.3 becomes:

   ∂ ∂X (H 3 ∂P ∂X ) + ∂ ∂Y (H 3 ∂P ∂Y ) - ∂H ∂X - ∂H ∂T = 0 (X, Y ) ∈ Ω 1 P = 0 (X, Y ) ∈ Ω 2 (2.12)
The force balance equation is the same for the four cases. It reads

W 1 = Ω P (X, Y, T ) dX dY (Y b -Y a ) (2.13)
The film thickness is different depending on cases:

-case 1:

H = H 0 + X 2 2 -R (2.14)
H = 1 + X 2 2 -R (2.15)
-case 3:

H = H 0 -R (2.16)
-case 4:

H = 1 -R (2.17)

Numerical Method

Discretizing

The discretization phase transforms the continuous physical model to a discrete system. The discretization concerns the geometry then the parameters and consequently the equations. The space represented by the plane (X, Y ) is meshed from 0 to n i for the X direction and from 0 to n j the Y direction. So there are (n i + 1) * (n j + 1) points and h x × h y corresponds to the mesh size (Figure 2.7).

To each point (i, j) corresponds an approximation of P noted P h ij (h stands for the mesh size with the assumption h = h x = h y ) 1 . P h is the solution of Reynolds equation. A finite difference method is used to rewrite the Reynolds equation. The term H 3 reads

ξ i,j = H 3 i,j (2.18)
Since the coefficients ξ are strongly variable a harmonic averaging is used to evaluate intermediate points.

1 ξ h i±1/2,j = 1 2 1 ξ h i±1,j + 1 ξ h i,j (2.19)
1 In this section h and H may refer to mesh sizes, not to film thicknesses

1 ξ h i,j±1/2 = 1 2 1 ξ h i,j±1 + 1 ξ h i,j
(2.20)

Using central second order discretization, the stationary Reynolds Equation becomes (cf.

[VEN 00])

ξ h i+1/2,j P h i+1,j -(ξ h i+1/2,j + ξ h i-1/2,j )P h i,j + ξ h i-1/2,j P h i-1,j h 2 x + ξ h i,j+1/2 P h i,j+1 -(ξ h i,j+1/2 + ξ h i,j-1/2 )P h i,j + ξ h i,j-1/2 P h i,j-1 h 2 y - H h i+1,j -H h i-1,j 2h x = 0 (2.21)
Using the boundary conditions, Equation 2.21 can be written in matrix form as:

L.P h = r h (2.22)
L is the operator and r h is the right hand member (in this case the Couette term). An inversion of L permits to find the exact P h solution (direct method such as Gauss pivot). Unfortunately, this method has a huge numerical cost when n i × n j is large. Hence, the system is solved through the use of an iterative method (Figure 2.8).

The iterative method approximates the exact solution iteratively by subtracting a certain amount of the actual solution, proportionally to the residual. The operation of subtracting is called relaxation. The Gauss-Seidel relaxation is used. From an actual approximation P h i,j

, the new approximation P h i,j is derived

P h i,j = P h i,j + δ h i,j (2.23)
with

δ h i,j = -r h i,j (ξ h i+1/2,j + ξ h i-1/2,j )/h 2 x + (ξ h i,j+1/2 + ξ h i,j-1/2 )/h 2 y (2.24)
and the residual r h i,j is

r h i,j = f h i,j - ξ h i+1/2,j P h i+1,j -(ξ h i+1/2,j + ξ h i-1/2,j ) P h i,j + ξ h i-1/2,j P h i-1,j h 2 x - ξ h i,j+1/2 P h i,j+1 -(ξ h i,j+1/2 + ξ h i,j-1/2 ) P h i,j + ξ h i,j-1/2 P h i,j-1 h 2 y + H h i+1,j -H h i-1,j 2h x (2.25)
The current error is the difference between P h and P h . With every relaxation sweep this error is reduced. Venner and Lubrecht [VEN 00] indicate that this reduction is grid dependent.

The speed of convergence depends on the mesh size h (the larger the h, the faster the convergence). Hence, It is useful to use coarser grids than h to accelerate convergence. This is the purpose of the multigrid method. ]. The target is to solve the linear equation 2.22 on the target grid h using multigrid. The multigrid method could be introduced through the coarse grid correction cycle. This cycle contains five steps (Figure 2.9):

• Do ν 1 relaxation sweeps on the target grid h until smoothing the error (the convergence becomes slow). This step is called the pre-relaxation.

• Copy the problem to the coarse grid H = 2 × h via the transfer operator (restriction).

• Solve the coarse grid problem (eventually using ν 0 relaxations).

• Go back to the target grid, correcting the initial solution via the interpolation of the solved coarse grid problem.

• Do ν 2 relaxation sweeps on the target grid h . This is the post-relaxation.

Coarse grid correction cycle

Pre-relaxation

The problem reads: L h P h = f h . To simplify the illustration, the problem is assumed linear. Hence, a correction scheme is used. Actually, L is non linear and a full approximation scheme is used for calculations, cf. [VEN 00] for further explanations. An initial solution P h 0 is introduced and ν 1 relaxations sweeps are performed. The approximation P h of P h is obtained. The residual is then calculated

r h = f h -L h P h . But f h = L h P h and the error is v h = P h -P h . Thus, L h v h = r h . Restriction L h v h = r h is transferred to the coarse grid: L H v h = I H h r h . L H
is the approximation of L h on the coarse grid and I H h is the restriction operator. Generally, two types of restrictions are used : the injection (Figure 2.10)or the full weighting (Figure 2.11).

Coarse grid solving

After the restriction the problem reads: L H v H = r H . It can be solved using ν 0 sweeps of relaxations. v H is obtained.

Interpolation

The initial problem is corrected with the solution v H . Similar to the restriction, an interpolation operator I h H is used. For the coinciding points of both grids, the solution is directly injected from the coarse grid. The intermediate points are interpolated (Figure 2.12). The new target grid solution is

P h = P h + I h H v H .

Post-relaxation

Some errors are introduced by the interpolation process. Hence, ν 2 relaxations are performed to eliminate these errors. Finally, an improved approximation is obtained.

MLC(k, ν 1 , ν 2 , γ) If k > 1 -perform ν 1 relaxations on the grid k -restrict to k -1 -do γ times MLC(k -1, ν 1 , ν 2 , γ)
-interpolate to k -perform ν 2 relaxations on the grid k else -perform ν 0 relaxations For four levels, Figure 2.13 shows the V cycle for γ = 1 and the W cycle for γ = 2.

Calculation algorithm

The developed code is based on successive V-cycles for each time step. The average of the residual over the calculation window on the target grid indicates the convergence and is noted Res (Equation 2.26).

Res = Σ i,j |res i,j | n i × n j (2.26)
The stopping criterion to go to the next time step is Res < 10 -5 . This value ensures a sufficient convergence of the pressure. The algorithm for a measured surface can be written in a simplified way:

Measured surface HL (X a ,X b ,Y a ,Y b ,m s ,max lev ,ν 1 ,ν 2 )
-create max lev levels -load measured surface -interpolate the measured surface on target grid -Time=0 -end loop for -write output files Figure 2.14 shows a typical calculation scheme.

Numerical example

In this section, an example of a transient calculation is studied. A parabolic ring slides over a measured patch measuring 1.07 × 0.8 mm. Figure 2.15 shows the surface roughness and the sliding direction. The first case of Table 2.1 shows the dimensionless parameters that are used. Assuming that c = 1 µm and that R x = 1 cm, a length measuring 100 µm corresponds to a 1 dimensionless unit. Thus, the whole domain is 10.70 × 8.00.

The nominal minimum film thickness, simply called the film thickness, is h 0 = 0.5 µm corresponding to 0.5 dimensionless unit. This value is imposed in all time steps. A [-4, 2] × [-4, 4] window represents the zone under pressure (blue boxed in Figure 2.15). The chosen mesh size is 1/128 in both directions X and Y . Consequently, the nodes number of the finest grid is 769 × 1025 and the total time steps is T T = (10.7 -6)128/2 = 301. The code calculates the pressure all over the window for each time step. Thus the load carrying capacity (LCC) per unit length is calculated (other quantity can be calculated such as hydrodynamic force [BIB 10a]). Figure 2.16 shows the resulting LCC as a function of the time step T. It shows also the pressure for T = 1, 50, 100, 150, 200.

Assuming five levels for the multigrid V cycles solver, the coarsest grid is 49 × 65. For each level, 8 prerelaxations and 4 postrelaxations are performed, except for the coarsest grid where the problem is "completely" solved with relaxations reducing the residual in this level by a factor of 100 (number limited to 200 relaxations). Figure 2.17 illustrates the V-cycle used for the calculations.

31 hours were necessary for the multigrid code to ensure the desired convergence (Res < 10 -5 ) for all the time steps. This computing time is reasonable compared with the one using classical methods. To compare both methods, one has to introduce the work unit [VEN 00]. Assuming that the computing time is proportional to the number of the relaxed points (which is actually not, due to the different operators introduced for particular purposes, more details can be found in appendix B), the work unit is defined as a relaxation sweep. In one level grid corresponding to successive relaxations (Gauss Seidel) and more than one level permits one to evaluate the multigrid performance. Figure 2.18 shows the residual recorded at the end of each V-cycle as a function of the work units for different number of grids (7 to 1 from bottom to top). For 600 WU, the four level solver reduces the residual more than 10 6 while Gauss Seidel relaxation reduces it only 10 times. The convergence speed for Multigrid is largely faster. In general, it is clear that the higher the number of grids, the higher the convergence speed. It seems that one should increase the number of the grids all the time. Actually, the following calculations will be performed with only four grids because of the errors introduced by the coarse grid. In fact, the coarse grid, when it is sufficiently coarse, poorly represents the grooves. Thus, an error is transferred through interpolation and the convergence is not ensured. Not to interpolate locally is a solution for this problem.

Conclusion

This second chapter concerned the deterministic modelling and the numerical method that will be used in the next chapters. Many assumptions were made to simplify the problem and to focus on the microgeometry influence on hydrodynamic lubrication. Simplified models were created. Thus, the real problem was transformed into equations through the modelling of the geometry and the physical phenomena. Three equations were obtained:

-the Reynolds equation describing the lubricant flow, -the film thickness equation describing the geometry including the ring and the liner surface, ∇ ∇ Figure 2.18: The residual as a function of work units for different number of grids (7 to 1 from bottom to top)

-the force balance equation which is the fundamental equation of mechanics.

To solve these equations a code was developed based on Multigrid. The use of this method allows one to drastically reduce the computing time allowing a parametric study in the next chapter. The multigrid method was introduced in this chapter. Finally a simplified algorithm of the code is presented and an example is explained.

Introduction

To optimize the PRCL contact performance, one has to understand the phenomena occurring when the rings are in action. The film thickness is the main parameter, notably to evaluate friction. When looking over the Stribeck curve, one can easily see that the friction can be minimized by having the "right" film thickness. Applying a given load results in a corresponding equilibrium position of the ring. In fact, the film thickness generates enough pressure to sustain the applied load. Thus the load carrying capacity (LCC) -film thickness (F T ) relation has to be investigated. The grooves greatly affect the film generation. In fact, the groove parameter dimensions are comparable to the contact dimensions, especially the groove depth that has the same order of magnitude as the F T . The Reynolds equation, the equation governing the film generation, contains a h 3 term explaining this sensitivity to the microgeometry. Two questions are deduced: how do the groove parameters affect the film thickness and by how much?

To answer both questions the models established in Chapter 2 are used. First the smooth surface is investigated without any groove. This case was already studied in both the fully flooded and starved regime by respectively Moes [MOE 00] and Biboulet [BIB 13]. The results are exposed at the beginning of this chapter to serve as references and to validate the developed numerical code.

The second section of this chapter focuses on the top ring with a parabolic shaped profile. More than 2000 transient calculations are performed to explore the effect of the four groove parameters (depth, angle, width and distance between crossovers). At the beginning of this section the objective of these calculations is explained. Then the choice of the mesh size related directly to the calculation accuracy is explained. The results are then exposed, starting with the effect of the depth and then the density. A master parameter ∆ is then created, regrouping all the groove parameters. This leads to a single curve regrouping all the results. The master curve quantifies the load carrying capacity in a fixed starvation degree (X a = 4). Finally the effect of the oil starvation degree on the ∆ -LCC relation is investigated and quantified.

The last section of this chapter is dedicated to the oil control ring (OCR). A similar analysis as the top ring is performed. The use of the term 12ηu m R x /h 0 as the dimensionless parameter for the line load carrying capacity corresponding to the case 2 in Table 2.1 and leads to a simplified form.

W 1 = w 1 12ηu m R x /h 0 = 4.896 ηu m R x /h 0 12ηu m R x /h 0 = 0.

Interpretation

The dimensionless parameter for x a is defined as

X a = x a / √ R x h 0 . Thus writing in dimen- sional form w 1 becomes w 1 = 0.408 × 12ηu m R x h 0 + 4.809 R x h 2 0 /x 2 a + 5.745 R 2 x h 3 0 /x 4 a (3.4)
The load in the starved regime remains proportional to the mean velocity and to the viscosity but is no more inversely proportional to h 0 . It ranges between inversely proportional to h 0 and inversely proportional to h 3 0 . In fact, Equation 3.4 clearly shows that starvation decreases the load carrying capacity. From Equation 3.4, the fully flooded regime corresponds to x a = ∞ and in this case one can find the same expression as 3.1. For very starved conditions, the term with 1/x 4 a is predominant and the load becomes inversely proportional to h 3 0 . In this case, Equation 3.4 can be simplified to

w 1 = 0.852 ηu m x 4 a R x h 3 0 (3.5)
Figure 3.2 shows the load w 1 as a function of h 0 as in Figure 3.1 (the six dashed lines in Figure 3.2 are exactly the same lines of Figure 3.1 corresponding to the fully flooded conditions).

Top ring(red lines)

From top to bottom the two solid red lines correspond respectively to x a = 1.6 mm and to x a = 0.2 mm. The first is close to the dashed line that means that 1.6 mm are almost sufficient to obtain fully flooded conditions for a ring radius R x = 10mm. The second line correspond to the starved conditions.

OCR(black lines)

Both black solid lines, corresponding from top to bottom to x a = 0.2 mm and to x a = 0.1 mm for a ring radius R x = 5 m, are in very starved conditions. In fact the slope of these lines is approximatively -3 against -1 for the fully flooded regime. For the OCR, the generated LCC under starved conditions is very low compared with the fully flooded conditions. In fact, even if the OCR is well supplied with oil, its axial thickness is very thin considering a radius R x = 5 m. This LCC is also very low compared with the top ring LCC. To sustain the pretension, the film has to be extremely thin (< 0.1 µm close to TDC) which is, actually, not possible in hydrodynamic regime. The metal to metal contact will occur.

Code validation

The developed code, as described in Section 2.3, is a 2D code based on the multigrid method.

Given the difficulties of the code implementation, errors may occur. Fortunately, the Biboulet [BIB 13] Equation 3.3 can serve to validate the developed code. To obtain smooth conditions, the roughness is set to zero in the whole domain. The boundary conditions are periodic at Y a and Y b and nil for X a and X b . Calculations for various degrees of starvation are performed. The pressure at the center line for Y = 0 is shown in Figure 3.3 for three different starvation degrees: X a = -32 for a nearly fully flooded case, X a = -4 for an intermediate case and X a = -1 for a very starved case. To validate the developed code, a comparison is made between the results obtained by numerical calculations and the expression 3.3. The calculations shown in Figure 3.4 have a difference less than 0.5% with the Biboulet results which are based on a 1D code (solid line for the expression 3.3 and markers for numerical calculations). These calculations cover a large working domain going from severely starved conditions, corresponding to the left sloped part of the curve, to the fully flooded conditions corresponding to the right horizontal part. 3.3 Top ring: parabolic shape

Objectives

For a parabolic shaped profile, the pressure is build up by the wedge in the inlet zone. The grooves introduce fluctuations in the pressure profile (cf. Figure 3.5) causing pressure drops and consequently a global LCC reduction. The objective of this section is to quantify this decrease.

The parameters

The top ring model exposed in Chapter 2 and the analytical surface model are considered. The model is a seven parameters model. That means that these parameter, which are the four microgeometric parameters (cf. Figure 2.5), both boundaries of the calculation window X a and X b and the time period considered, perfectly describe the problem.

The parameters are dimensionless according to Case 2 of Table 2.1, for instance A = a/h 0 ,

D 1 = d 1 / √ R x h 0 , Λ = λ/ √ R x h 0 , X a = x a / √ R x h 0 and X b = x b / √ R x h 0 .

Time dependency

To reduce the calculation time, two observations are analysed in this paragraph: a perfect periodicity and a known period. Calculations are performed in the window [X a , X b ][Y a , Y b ] for a given period of time. Figure Variations are relatively small and perfectly periodic. This perfect periodicity is explained by the fact that, the transient term (H old i,j -H i,j )/∆T is well known since H is imposed. Each time step can be solved independently from the other. In other words, from a crossover to the next which will appear at the same position, the result is the same.

The previous observation suggests that the period is a pattern length (D 1 ), but actually it is the half of this length. Figure 3.7 shows the pressure profiles all over the period. This period, called T T corresponds exactly to a half pattern length path which means D 1 /2, in other words half the distance between two consecutive crossovers. In fact, Figure 3.8 clearly shows that the center line of each pressure profile mirrors the right part and the left part. It is the symmetry line of the geometry. That permits to reduce the computing time by two. Knowing that for each time step the ring is moving two meshes h x , the total number of time steps T T S reads

T T S = D 1 /(4 h x ) (3.6)
So one can consider a half pattern length problem where the window [X a , X b ] slides from a position 0 to a position D 1 /2. The result of each calculation is the pressure profile P (X, Y, T ).

Integrating the pressure, one obtains the value of the instantaneous line LCC which is W 1 (T ). Finally the mean value is calculated and is noted simply as W 1 . The relation X b = X a /2 is assumed. In fact, way before this position, cavitation has already occurred and zero pressure is established. Biboulet et al. [START_REF] Biboulet | Friction in starved hydrodynamically line lubricated contacts[END_REF] give the cavitation abscissa as a function of the starvation level for the smooth case.

W 1 = T T S T =1 jj j=0 ii i=0 P i,j,T h x h y /(Y b -Y a ) T T S (
X b = 0.6719 1 -1/(1 + 0.3418X 3/2 a + 0.2533X 3 a ) (3.8)
Figure 3.9 shows where the assumption is valid for the smooth case by comparing Equation 3.8 to X b = X a /2. The assumption is valid for X a ≥ 0.6 in the smooth case. In the grooved case, some pressure generation may occur after the cavitation position. For example, at the right of Figure 3.5, there is a small increase of the pressure after the cavitation position due to the presence of grooves. For deep and large grooves they may continue to generate pressure after the fixed X b position. However, the generated LCC can be neglected compared with the main LCC. In addition, physically, it is not possible to continue generating pressure after the cavitation position [ORG 07].

The problem becomes thus a five parameter problem. The four microgeometric parameters describing the grooves (α, A, Λ and D 1 ) and the parameter that determines the starvation degree (X a ).

Mesh-size

The numerical accuracy of the computed LCC depends on the mesh choice. The smaller the mesh, the better the approximation. Unfortunately, choosing a very small mesh will naturally increase the number of points. For example, reducing the mesh size by a factor of 2 will multiply the number of points by 4 and consequently the calculation time at least by 8. Thus an adequate mesh size is a mesh size that guarantees a sufficiently precise approximation in a reasonable calculation time. Table 3.2 compares several mesh sizes for two different cases.

A moderate case is chosen with Λ = 0.15, A = 4, α = 25 • , and D 1 = 1. The second case (extreme case) contains very deep grooves, the values are Λ = 0.2, A = 32,α = 25 • , and D 1 = 1. The LCC for each mesh is compared to the 1/384 mesh size taken as a reference (W 1ref ). A choice that guarantees a good accuracy to time ratio is 128 meshes per length unit. This enables a comparison between results even if in extreme cases (very narrow and/or very deep) the calculations will be less accurate. The calculation input is formed by five parameters: the four microgeometric parameters describing the grooves (α, A, Λ and D 1 ) and the parameter that determines the starvation degree (X a ). Figure 3.10 describes these parameters. The output of each calculation is W 1 as described by Equation 3.7. A first set of calculations is performed for a fixed X a . The choice of X a = -4 corresponds to a moderately starved condition (Figure 3.4). To give an order of magnitude, this value corresponds to x b -x a = 600µm for R x = 10mm and h 0 = 1µm. To cover a large domain, the depth range is from 2 -3 to 2 4 , doubling each time the value. This gives 8 values. The angle varies between 10 • and 40 • with a step of 5 • . The pair (Λ, D 1 ) has 36 different values. Figure 3.11 shows the distribution of these cross-hatching parameters. This leads to a total number of calculations of 2016.

Groove depth

The depth A is a major parameter for the pressure drop. Figure 3.12 shows that for small A values the pressure profile resembles a smooth line hydrodynamic pressure profile, the fluctuations are small with almost parallel horizontal lines (for isobar lines). For deeper grooves the straight horizontal lines start to disappear giving way to a diamond shape. The diamonds correspond to the groove footprint. For the rightmost profile, where the depth is 32, the pressure is only build up on the plateaux. The resulting relative LCC is analysed for various cases. Figure 3.13 shows the relative LCC W 1 /W 1s (s for smooth) as a function of 1/A for D 1 = 1 (blue) and D 1 = 2 (red), α = 25 • and for three values of Λ 0.025 (circles), 0.1 (squares) and 0.2 (triangles). The LCC is significantly affected by the change of the depth. The shallower the groove, the closer the relative LCC to the asymptotic value 1. All the shallow grooves merge with this asymptote. Otherwise, the slope is different for the five cases shown, which means that the other parameters are also important to quantify the relative LCC.

Groove density

D 1 , α and Λ are written into a single parameter ǫ defined by Equation 3.9. This parameter corresponds exactly to the groove surface ratio.

ǫ = Λ D 1 cos(α) 2 - Λ D 1 cos(α)
(3.9) Figure 3.14 shows the relative LCC for different angles (from 10 • to 40 • ). For a given angle, five values of the pair (Λ, D 1 ) are used: (0.1,0.5), (0.2,1), (0.3,1.5), (0.4,2) and (0.5,2.5). This leads to a constant ǫ for each angle. In fact, the ratio Λ/D 1 is constant for these pairs (cf. equation 3.9). Results in Figure 3.14 show that for the same groove density, with different (Λ, D 1 ) couples, the relative LCC is nearly the same. In other words, the LCC depends on the groove density, but not Λ and D 1 for this range of values. One can assume that ǫ is a major parameter, roughly independent of the value of its constituents.

Master curve

A master curve is a curve that permits an instantaneous approximation of the LCC starting from a given set of the cross-hatching parameters. A single parameter regrouping all the parameters is then elaborated. single parameter called ∆ (also in appendix C). Equation 3.10 gives its expression. This parameter is mainly composed of two terms: a term containing the groove density and a term containing the depth. This reinforces the idea that these two parameters are the two major parameters.

∆ = -log(ǫ) 1 + A + 3A 2 /8 (1 + A) 2.5 (3.10)
Figure 3.15 shows the relative LCC as a function of ∆ for all 2016 calculations. Each marker corresponds to a calculational result. The scattered plot obtained shows that above a value of ∆ about 0.7, all the grooves generate a LCC close to the smooth LCC. In other words, small values of ǫ and A give a relative LCC close to 1. The deep and dense grooves are on the contrary gradually approaching the zero value. A curve fit based on exponentials is carried out. The form proposed in Equation 3.11 correspond to the solid line drawn in Figure 3.15

W 1 /W 1s = 1 -e -9.
2∆ -e -5.4∆ -e -1.9∆ + e -13.5∆ + e -2.1∆ (3.11)

Starvation influence

The previous calculations were carried out with X a equals to 4. To extend relation 3.11 for other starved conditions, additional calculations are performed with X a ranging from 1 to 32. The value of 1 corresponds to relatively severe starvation conditions and the value 32 to close to fully flooded conditions. Figure 3.16 shows results for X a = 1 (△), X a = 4 (•) and X a = 32 (⊡). The curve fits proposed in Equation 3.12 are based on the Equation 3.11 with the introduction of a parameter f corresponding to the starvation degree X a according to the table 3.3. f is called starvation curve fit factor.

W 1 /W 1s = 1 -e -9.2 f ∆ -e -5.4 f ∆ -e -1.9 f ∆ + e -13.5 f ∆ + e -2.1 f ∆ (3.12)

Interpretation

In this subsection, the starting point is Figure 3.2 which shows the starvation influence in smooth case for three cylinder positions (TDC, MS, BDC). One can go further and analyse the influence of cross-hatched grooves in these cases. The objective here is to compare different groove influences and to evaluate the LCC decrease. The microgeometry is the same for the three positions, only the viscosity and the velocity change. Since the LCC is proportional to the product of the viscosity and the velocity, the LCC is shifted vertically. Figure 3.17 shows the load carrying capacity as a function of the central film thickness for the TDC, MS and BDC as in Figure 3.2. It compares the smooth case (dashed lines) with three grooved cases with the same density ǫ = 50% and different depths a = 0.25 µm (green), a = 1 µm (blue) and a = 4 µm (red), for each depth, results for two starvation degrees are shown (bottom to top respectively x a = 0.2 mm and x a = 1.6 mm).

To analyse this figure, two observations have to be kept in mind:

-The load is assumed to be around 250 N/m, since the pretension of the rings is of the same order of magnitude.

-Around and below 0.6µm, the central film thickness reaches the mixed regime, assuming that the surface roughness σ is about 0.2 µm and applying the well known law that says that above 3σ the lubrication regime is fully hydrodynamic. Both the TDC and BDC are in the mixed regime. In fact, the LCC generated by the oil is insufficient to bear the load in this example. This LCC is getting worse when the grooves are very deep (a = 4 µm). The proportion of the hydrodynamic LCC of the total load in the mixed regime is determined by the depth. The deeper the groove, the lower this proportion. Meanwhile for MS, the same load can easily be carried by the hydrodynamic LCC and the film thickness is between 1 and 10µm depending on the starvation degree. The influence of the grooves for these conditions is minimal even for deep grooves. Figure 3.18 compares the LCC for different groove densities. As for depth, the density greatly influences the LCC in TDC and BDC. Thus, it determines also the hydrodynamic proportion in mixed regime.

Oil control ring: flat shape 3.4.1 Objectives

For a smooth liner, the OCR's LCC is very low compared with the one generated by the top ring (Figure 3.2). Nevertheless, both carry almost the same load (order of magnitude 250N/m). That suggests that the mechanism that generates hydrodynamic pressure differs in both rings. While the top ring is characterised by its curvature and the pressure is build up by the wedge term (Equation 1.1), the OCR is considered flat and the pressure is generated by the grooves. The objective of this section is to analyse the LCC generated by the grooves through 2D hydrodynamic calculations.

The parameters

The OCR model exposed in chapter 2 and the analytical surface model are considered. The model is completely described by five parameters. The number of microgeometric parameters is reduced from 4, for the top ring, to 3 parameters for the OCR. Considering the fact that the groove width λ is the dimensionless parameter for the length scale, the dimensionless groove width Λ is equal to 1. The 3 groove parameters are then: the relative depth A, the distance D 1 and the groove angle α. Both boundaries of the calculation window X a and X b are also parameters to be considered.

As mentioned in the paragraph 3.3.2.1, the period in the pressure profile is the half of a cross-hatching pattern. It is more convenient to focus on what is happening in only the half of a cross-hatching pattern (D 1 /2) and then generalise to more than that. In fact, central patterns LCC's are not affected by the boundary conditions in sliding direction and then considered the same. Figure 3.19 shows pressure profiles for 1 to 6 patterns. The reference pattern is a half of a central pattern (black box). Figure 3.19 shows that except for the boundary halves, the LCC is nearly the same (more than 90%). This permits one to consider the case of a 4 pattern length domain as a reference case for the calculations. In other words X b -X a = 4D 1 . This length ensures that the central pattern LCC is not affected by the boundary conditions in sliding direction. The calculated result is the LCC generated in the central pattern (red rounded in Figure 3.19). Considering X b = -X a = 2D 1 , the input parameters become only the three groove parameters (A,D 1 and α). To analyse the results, the 1D analytical model results by Biboulet et al. [BIB 14b] are used. Biboulet found that the maximum pressure is twice the mean pressure. Figure 3.20 explains briefly the 1D model with a simplified geometry. The distance between two parallel grooves N is used in this model. The analytical solution found by Biboulet is:

P M = A A(N -1) + N -1/2 2 (1 + A) (1 + A)(N -1) + 1 2(N -1)(1 + A) 2 + A + 2 (3.13)
For 2D model, the parameter N is exactly the distance between two parallel grooves (cf. Figure 3.10). It can replace D 1 for the analysis with N = D 1 cos(α). The input parameters become A,N and α. Figure 3.21 shows examples of the pressure profile for different groove depths. The pressure is integrated over the central pattern to obtain the load carrying capacity. For convenience, the output W 1 of the 2D calculations is divided by the distance D 1 , corresponding to the mean pressure. Thus, one can obtain the double of this value. This value is incorrectly called the maximum pressure P M .

Mesh-size

As in paragraph 3.3.3, the accuracy of the LCC is studied as a function of the mesh size. Table 3.4 shows the relative error compared with the reference case of the LCC for a very fine mesh (first row in the table). The calculations were performed for α = 25 • and N = 2.5.The mesh size depends on the slope Λ/A with Λ = 1. That means that the deeper the groove, the finer the mesh has to be. The third row meshes are sufficient meshes to guarantee a good approximation except for A = 32 and a little A = 16. Unfortunately, taking a 1/192 mesh for A = 32 (instead of 1/96 for the third row) will impact the computing time and the memory used. In fact, for the case where α = 15 • and N = 10 the domain size is 4N/ cos(α) × N/ sin(α) ≃ 41 × 38 and thus the number of points is about 7872 × 7296 which is a huge number even for multigrid method! The calculations are performed with respect to the mesh sizes of the third row. That means that for a depth less than 4, the mesh size is 1/24, for A = 4 it is 1/48 and for more than 4 it is 1/96.

Influence of the groove parameters 3.4.4.1 Groove density

Knowing that D 1 = N/ cos(α) and that Λ = 1, the groove density or the groove surface ratio ǫ becomes:

ǫ = 1 N (2 - 1 N ) (3.14)
Equation 3.14 shows that for N = 1 the grooves cover all the surface and there is no plateaux.

For N ≫ 1 the grooves are very sparse. The calculations are performed for N = 2.5, 3, 3.5, 4, 5, 6, 8 and 10 corresponding respectively to ǫ = 64%, 55%, 49%, 43%, 36% 30%, 23%, and 19%. In Figure 3.22 P M is curvefitted using Equation 3.13. Only a few angles and depths are shown (α = 25 • , 45 • , 65 • and A = 1/8, 1, 8). The results show that there is a horizontal asymptote for N ≫ 1 for each case. The groove LCC is maximal and the maximum pressure is noted P M ∞ . The curve fit permits to extract each value of P M ∞ .

Groove angle

The calculations were performed for various angles (from 5 • to 65 • in 10 • steps). After curvefitting P M as a function of N, one obtain asymptotic values P M ∞ . Figure 3.23 shows with markers these P M ∞ as a function of the angle α and the depth A. When analysing these results, one can observe that the maximum pressure decreases when the angle becomes large. In fact, for 90 • the grooves are parallel to the sliding direction suggesting that the LCC is nil and P M = 0. A cosine sign correspond to this behaviour. Equation 3.15 curve fits the numerical results. P M ∞0 is the value at α = 0 • .

P M ∞ = P M ∞0 . cos 7/5 (α) (3.15)

Groove depth

The markers in Figure 3.24 present the values of P M ∞0 found by the angle curve-fitting. The depth is curvefitted using Equation 3.13 when N → ∞ (sparse grooves). The maximum pressure P M ∞0 reads:

P M ∞0 = 1 2 1.73 A (1 + 1.18 A) 2 (3.16)
The LCC has a maximum at A = 1/1.18 = 0.84. In other words, at this position the groove that carries the highest load is the groove corresponding to A = 0.84, which means ∆ Figure 3.23: Maximum pressure for very sparse grooves P M ∞ as a function of the angle α: from top to bottom A = 1, 1/2, 2, 1/4, 4, 1/8, 8, 1/16, 1/32 a/h 0 = 0.84. As a consequence, for a given film thickness h 0 , the groove that carries the highest load is the groove with a depth a = 0.84 h 0 .

Boundary effects

Thanks to the curve-fits, the value of P M can be predicted for a set of groove parameters.

The LCC in a single central pattern for this microgeometry can be deduced and is simply P M /2 × D 1 . As mentioned in section 3.4.2, the total LCC is the sum of the LCC of all patterns. While the central patterns are well approximated by P M /2 × D 1 (better than 90%), the patterns located near both boundaries, depending in which position they are, generate less LCC. To approximate the total LCC, various calculations are performed with the boundary patterns changing position. Table 3.5 shows, for three different cases, the ratio between a central LCC multiplied by the number of patterns and the average LCC calculated including the boundaries. For example for X b -X a = 4D 1 , the LCC reads:

W 1 = 0.88 × 4 × P M D 1 2 (3.17)
For less than one pattern, the load carrying capacity drops rapidly. For example for X b -X a = 0.5D 1 , the resulting LCC is about 30% of the curve-fitted value.

Interpretation

In this subsection the example of the microgeometry seen in subsection 3. and 4µm. Assuming that x a = 200µm, the domain length in the sliding direction is 4 × d 1 . Figure 3.25 shows the LCC generated by the same microgeometry for three different lubrication conditions: close to TDC (η = 0.004P a.s u m = 0.1m/s), MST (η = 0.012P a.su m = 4m/s) and close to BDC (η = 0.02P a.su m = 0.1m/s). The first observation is that the level of the LCC of a smooth surface (with a curvature) is very low compared with the one generated by the grooves. The flat surface model is confirmed.

The second observation concerns the importance of the groove depth in carrying the load. In fact, as seen from Equation 3.16, to each film thickness corresponds an optimum groove depth that carries the highest load. Figure 3.26 shows the LCC for three other sets of groove parameters, keeping α = 25 • , and a = 1µm and changing N to 2.2, 3.5 and 6.1 corresponding to ǫ = 70%, 50% and 30%. Even if there is a difference between the different cases, the LCC for sparse grooves and dense ones are close to each other. The groove density is important to determine the load level but for a given film thickness the depth will be more determinant.

Conclusion

Three PRCL models were investigated in this chapter via the load carrying capacity -film thickness relation. First, the smooth case was explored in the fully flooded regime (Moes [MOE 00]) and in the starved regime (Biboulet [BIB 13]). The importance of the starvation degree was highlighted for both compression and oil control ring. Second, the LCC generated by the top ring was investigated with the presence of cross-hatched grooves. A master curve was deduced from a single parameter regrouping all the microgeometric parameters. This prediction allows a fast approximation of the load carrying capacity for a given range of film thicknesses. Finally, the flat ring case was studied. The LCC is generated by the grooves in that case. The influence of the cross-hatching parameters has been studied. As for the top ) leading to a three regime curves (hydrodynamic, mixed and boundary regimes). Figure 3.27 illustrates such a curve. The red line represent the asperity contact and the blue one represents the hydrodynamic force.

Introduction

The ultimate objective of the liner hydrodynamic analysis is to optimise the microgeometry in order to enhance the engine performance. The predictions from Chapter 3 allow such an optimization. In fact, starting from microgeometry data, the load carrying capacity is predicted. As a consequence the tribological performance can be deduced (oil availability, friction...). The scope of the current chapter is to validate the chapter 3 prediction model. Measured surface calculations are used for this validation. Chapter 3 dealt with analytical liner surfaces. These surfaces are idealized models of the real surface. The real surfaces are rough and wavy (macroscopic curvatures). The real grooves are not uniform (neither constant width, depth or angle). This chapter links the analytical model to the real surface model through measured surface calculations. The prediction of the load carrying capacity starts from the liner microgeometry parameters. The model used in Chapter 3 has a unique set of parameters for each surface. However, measurements show that real surfaces are not that uniform. For validation purposes, three steps are necessary for each measured surface:

-extract the parameter distribution, -link the distribution to a unique set of parameters, -compare the prediction with the measured surface result.

The first part of this chapter compares both geometries, the analytical one and the measured one. The second part concerns the parabolic shaped rings (notably the top ring) and the validation of the corresponding prediction. The last part is dedicated to the validation of the flat ring (oil control ring) prediction model.

Geometry: Analytical vs Measured

This section starts by describing the measured surfaces samples. This description highlights the major differences between a measured surface and an idealised one. A second part is dedicated to the identification of the grooves from the measured surface samples. The extracted information will be used to create an intermediate model, linking the measured surface to the model predictions of Chapter 3.

Measured surface

A cylinder liner which was run for a considerable period under fired conditions is used for the measurements. It permits to evaluate the tribological performance under real conditions. Along the engine stroke the measured surfaces are worn to a different degree. In fact, near the top dead center (TDC) and near the bottom dead center (BDC) the wear is more pronounced than in the mid stroke (MST). This is due to the speed which is close to nil in both extremities and consequently due to the lubrication regime. The mixed and the boundary lubrication regimes can lead to important material removal (plastic deformation of asperity peaks could also occur). The reference position (REF) in Figure 4.1 clearly shows the difference between worn and unworn surfaces (the same height scale is used). Two qualitative observations are deduced:

-the plateau zones are smoother for worn surfaces, -the cross-hatched grooves are deeper, larger and denser for the unworn surfaces.

The measured surfaces are treated with the software Mountains to remove macrocur- In this chapter, two typical surfaces are analysed: a heavily worn and a normal worn surface (located respectively near BDC and near MST). Figure 4.2 shows both surfaces.

A thorough visual inspection of the surfaces permits to deduce important observations in addition to the previous ones: -There are some vertical grooves that are probably the result of a piston skirt contact (continuing to the REF zone). These vertical grooves can generate "Dirichlet Boundary Conditions". In fact, if they are sufficiently deep to create a tunnel under the ring, zero pressure is generated.

-Very deep and very small local grooves are measured (blue color). These grooves can interfere with the pressure generation.

-Straight grooves are not always continuous. Some discontinuities occur.

-Small bumps occur on MST plateaux (dark colors).

Groove identification 4.2.2.1 The topic

The optical comparison is not sufficient to quantify the surfaces. An identification of the grooves permits to quantify two major data inherent to each surface: the groove statistics (number of the grooves and their geometry) and the plateau roughness (cf. [GAD 02] for roughness parameters). The extracted data permits a better description of the surface. It is useful for various kind of analysis (wear analysis, hydrodynamic analysis, asperity contact analysis, etc,). This study focuses the hydrodynamic part related to the microgeometry effect on the load carrying capacity. Thanks to the predictions of Chapter 3, one can evaluate the load carrying capacity whatever the film thickness. The HL computing time for a given value of the film thickness and a given starvation degree can reach 24 hours even with the multigrid method, while the identification permits an instantaneous evaluation of the LCC (for any value of film thickness and starvation degree).

A code that automatically identifies the grooves was developed by Biboulet [BIB14a]. It aims to extract as much microgeometry data as possible. This identification assumes the same cosine form of the groove described in Chapter 3. A straight groove with the three associated parameters (the angle α, the depth a and the width λ) is fitted to the crosshatched grooves. The position of each groove is stored to generate the intermediate surfaces (

Plateau recognition

The plateaux constitute the rough surface between the different cross-hatched grooves. The groove identification permits one to isolate these plateaux. Assuming that all the plateaux are of the same level (heights), the mean plateau position is then defined as the mean value of the non identified zone, including all the imperfections. In this identification process it is assumed that all the cross-hatched grooves are identified. Unfortunately, the small grooves (narrow and shallow) are hard to detect. An undetected groove will tend to lower the plateau height. Figure 4.5 illustrates the displacement of the mean height caused by unidentified grooves. Other phenomena contributing to an incorrect location of the plateaux are: -the very deep and very small local grooves, -the macrocurvature which can not be completely eliminated. Table 4.1 shows the mean height value R and the value of the roughness parameters R a and R q for both the entire measured surface and the plateau (the measured surface without identified grooves). The groove identification process moved the mean line by 0.19 µm in MST and 0.10 µm in BDC. It also acts on the values of R a and R q . These parameters are corrected, and can indicate in a better way the surface finish (eventually for a peak analysis in the mixed regime...). -The drop of the number of the shallow grooves and also the number of narrow grooves is not physical. In fact, the groove identification is hard when its dimensions are close to the plateau roughness.

R in µm

-There is a small shift to the left in width and depth values from the MST to BDC. In fact, the material removal, caused by the boundary lubrication regime in BDC, makes the grooves smaller (shallower and narrower). -The angle distribution contains no particular information, unless the fact that the distribution is relatively wide (15 • to 25 • ). This is due to the honing process. There are several ways to combine parameters. The simplest way is to average the distributions. More complicated models can be considered including a combined interaction between different depth ranges, etc. In this study, the simplest model is chosen: an averaging of the microgeometry distribution. To associate with each surface a single value of each of the parameters (zero sub script), one has to take into account the density of the grooves. In fact, to each groove i a density ǫ i is associated ( Σǫ i = ǫ total due to cross-overs). The parameters are averaged using ǫ i , for example a 0 = Σ(ǫ i a i /Σǫ i ). The solid triangle marks in -the analytical surface with a single set of parameters defined in Table 4.2 and noted AS (as in Chapter 3 model), -the prediction which corresponds to the Chapter 3 prediction (Equation 3.12), -the smooth line contact corresponding to Equation 3.3.

Transient hydrodynamic calculations are performed on the surfaces with x a = 400 µm for four values of h 0 (h 0 = 0.5, 1, 2, 4 µm for MST and h 0 = 0.25, 0.5, 1, 1.5 µm for BDC). Knowing that the maximum roughness values are 0.8 µm for MST and 0.34 µm for BDC, only three values of h 0 are calculated on the measured surfaces. Actually, to avoid extremely thin films or asperity contact, the lowest value of h 0 calculated for the measured surfaces is 1 µm for MST and 0.5 µm for BDC. The mesh size that gives a sufficient accuracy is ∼ 0.78 µm.

Figure 4.8 shows the load carrying capacity as a function of the film thickness. The averaged values obtained by HL calculations are shown in this figure. The square is for MS, the circle for IS and the triangle for AS. The curves are, from bottom to top, the prediction and the smooth surface. Some observations are deduced: -The three markers corresponds perfectly for large h 0 while for 1 µm at MST and for 0.5 µm at BDC the measured surface (MS) has slightly higher LCC. In fact when approaching the plateaux roughness, the peaks influence the pressure profile in a non linear manner. Figure 4.9 shows an example of these peak effects. Since the final result is averaged over time, this effect vanishes or at least is greatly reduced. error relative to prediction.

-The pressures for the measured surface (⊡) are higher than those of the intermediate analytical surfaces (•). This is mainly due to the bad height location of the plateaux in certain areas. Even with this error the approximation is still good enough to correctly model the performance of the measured surface.

-Comparing the analytical surfaces (△) and the intermediate analytical surfaces (•) permits to validate the associated set of parameters (Table 4.2). Table 4.4 reports the values of LCC. It is concluded that the averaging done to obtain these parameters is sufficient to approximate the surface. It is worth noting that for a surface that contains a sparser distribution or with one very deep groove the same approach as previous will be less accurate. In fact, one should give more weight to deep grooves. -The difference between the calculated analytical surfaces (△) and the prediction (bottom line) is less than 15% (in absolute value). This difference is due to the curve-fitting and is considered acceptable.

Oil control ring 4.4.1 Boundary conditions

As for the parabolic shaped ring, the flat ring is analysed. The measured surface is also compared to predicted results. These calculations are performed for the same values of h 0 as the parabolic ring (for MST h 0 = 0.5, 1, 2, 4 µm and for BDC h 0 = 0.25, 0.5, 1, 1.5 µm). The same mesh size as top ring is used for calculations (∼ 0.78 µm). The surfaces are the same as in the parabolic shaped ring case. The calculations are stationary and the whole surface is calculated at once. The whole domain is 1070µm long. The resulting LCC is calculated proportionally for x b -x a = 400µm (×400/1070). The boundary conditions in x = x a and x = x b can alter this proportionality but this is simply neglected. The results are shown in Figure 4.10 for a 400µm land, corresponding to a single land length of a twin land OCR. In this figure the square is for MS (⊡),the circle for IS (•), triangle for AS (△),plus sign for Analytical and periodic BC (+) and the prediction is a solid line. The triangle markers and the cross markers fit perfectly. They correspond to the analytical surface with respectively Dirichlet and periodic BC. In fact, with flat rings the effect of the zero pressure boundaries vanishes rapidly, as for a given film height the pressure generation is much less than for the parabolic ring.

Film thickness-LCC relation

The comparison between different surfaces is done gradually from the prediction to the measured surface.

-Prediction (solid line) vs Analytical Surface AS(△) Using the set of parameters defined in Table 4.2 for MST and for BDC, and then respectively equation 3.16, equation 3.15 and also an interpolation function of the N number, one can obtain the w 1 -h 0 relation. This prediction is shown in solid lines in Figure 4.10. The prediction line is about 40% higher than the analytical values for the MST and 20% for BDC. This is due to the combined action of the approximation of the boundary conditions in sliding direction and the curve-fitting.

-Analytical Surface AS(△) vs Intermediate Surface IS (•)

The set of parameters defined in Table 4.2 remained the same for the analytical surface despite the fact that the pressure generation for flat ring is more complicated than for parabolic rings. In fact, when the range of the film thickness variation is relatively large, the grooves that are carrying most of the load are different in each region of this range which contradicts the simplification to use a single set of parameters. MST in Figure 4.10 illustrates this fact, for h 0 = 0.5 µm and h 0 = 1 µm AS and IS load carrying capacities correspond, while for h 0 = 2 µm and h 0 = 4 µm an error starts to form. The averaged value of the depth is 0.848 µm while there are two grooves of more than 2 µm deep. These deep grooves generate more pressure than those of 0.848 µm depth around the film thickness of 2 µm. The same fact can be observed in BDC, the error is greater when the film thickness is greater.

-Intermediate Surface IS(•) vs Measured Surface MS (⊡) Two observations concerning the origin of the difference between circles (IS) and squares (MS) in Figure 4.10: -for MST, Figure 4.11 shows that the shape is approximately the same but there is a global shift of the LCC. Figure 4.12 shows that the pressure is everywhere higher by a certain factor. The shift becomes bigger when approaching the surface as if the plateaux in measured surface are higher. In fact, this can be explained by the incorrect location of the plateau globally (explained previously in subsection 4.2.2.2). Visualizing Figure 4.3 suggests a +0.4µm shift of the plateau. This shift, if applied to the LCC-h 0 curve gives the red squares in Figure 4.13. With this correction, the squares (MS) perfectly follow the slope of the circles (IS).

-for BDC, Figure 4.11 shows that in addition to a very small global shift there is a bump on the left of the curve that is exceeding the mean value. Figure 4.12 shows at the top left corner that the pressure generated is relatively high. In fact, exactly at this location, there is a very deep groove with a 8µm measured depth (cf. Figure 4.4). The flat ring LCC is very sensitive to this kind of grooves, it generates over-pressure. This perturbation can be eliminated with considering only the right half of the surface. The corrected values are in red squares in Figure 4.13. This correction permits to fit the squares (MS) to the circles (IS).

Conclusion

The main objective of this chapter is to validate predictions of chapter 3 via measured surface calculations. The major difference between the analytical model (used for prediction) and the measured model is the microgeometry. The first one has smooth plateaux and the second has rough and wavy plateaux. The first one also has a unique set of groove parameters while the second has a distribution of grooves that are not uniform. For the comparison of the surfaces, a procedure was used to extract microgeometrical parameters from the measured surfaces. The parameters distribution has been studied. Thus, the extracted data was linked to a unique set of parameters. This single set was used for prediction, according to Chapter 3 predictions, permitting then a comparison with the measured surface results. The performance of the parabolic ring and the flat ring in term of load carrying capacity (LCC) in a measured surface was compared to the corresponding prediction for two kind of surfaces (located at midstroke and at bottom dead center). This comparison led to the prediction validation. The prediction is an instantaneous way to approximate the LCC for a large range of film thickness and starvation degree, while a transient code will run for 24 hours even with the efficient multigrid method for only a single value of film thickness and a single starvation condition.

Conclusion

The understanding of the phenomena that are acting in piston ring cylinder liner (PRCL) contact permits the enhancement of the whole engine performance, reducing the oil consumption and respecting the environment constraints. In fact, the PRCL contact contributes to more than 30% of the energy loss by friction and contributes also to the lubricant consumption. The liner texturing is of great importance in the PRCL contact and consequently to the tribological performance. It influences the film thickness, acting on ring behaviour. The liner surface features a special microgeometry. The scope of the thesis is to study the effect of this particular microgeometry on the load carrying capacity-film thickness relation. The texturing that is the most common is the cross-hatching pattern obtained by the honing process. This texturing is highly directional with two major components: the plateaux and the cross-hatched grooves. Statistical methods such as the Patir and Cheng method are not able to distinguish both components. The deterministic method is a straightforward method to analyse the effect of such a microgeometry. The correct description of the surface needs a large number of points. Classical solution methods such as Jacobi or Gauss Seidel relaxation are too slow to permit a reasonable computing time. An efficient and robust method does exist and is the Multigrid method. The use of the Multigrid method drastically reduces the computing time. The PRCL model starts by distinguishing the flat ring (oil control ring precisely) and the parabolic shaped rings (top ring and scrapper ring: compression rings). The liner surface is modelled by three different models: the first is the classical smooth surface which has been widely studied, the second is the analytical surface which is a simplified model that assumes periodicity of the geometry with smooth plateaux, the third model is the measured surface. A code based on multigrid methods is developed. It permits a fast 2D transient hydrodynamic calculations, taking into account the microgeometry. This code permits, via an analytical surface model, to perform more than 2000 transient calculations for the top ring and more than 1000 calculations for the flat ring. This parametric study, takes into account the different parameters defining the microgeometry notably the groove depth, the groove width, the groove angle and the groove density. It also considers the starvation degree. These calculations serve to predict via analytical expression the load carrying capacity as a function of the different parameters and the starvation degree. Predictions for compression ring and the oil control ring load carrying capacity are build. These predictions reinforce the idea that the pressure generation is very different for both cases. In fact, for the top ring the grooves contribute only in the load carrying capacity decrease and for the OCR the grooves generate the load carrying capacity. The measured surface calculations permit the validation of these predictions. To link both models, a procedure that identifies cross-hatched grooves is used. The obtained information is analysed and a single set of parameters is deduced. This set is used for prediction. Thus, both results can be compared. This comparison led to the prediction validation. Some perspectives and future work can be considered:

• Short term perspectives:

-Experiments should be held to validate the predictions with a test campaign including sufficient surface measurements for various liners and various positions. The experiments should start by differentiating the flat case and the parabolic case since the mechanism that generates the pressure is not the same.

-The study of the case of a huge radius R x will link between both models.

• Mid-term perspectives:

-The starvation remains an important problem that is not well understood or quantified in textured surfaces. It should be detailed more and more.

-One of the problems that are related to multigrid method is the coarse grid introduced error. The microgeometry is not well described by this grid. More suitable description should be found (cf.

) + V 1 + z h (V 2 -V 1 ) (A.5)
The different volume flows read The film thickness h is written according to the fixed frame related to the liner (motionless replaced by the block relaxation (cf. line relaxation [START_REF] Venner | [END_REF]). The principle of the block relaxation method is simple: instead of relaxing point by point, one relaxes block by block. The whole block is solved at once by a direct method. In this work the LU decomposition is used as a direct solver. The block relaxation is more efficient than the point relaxation but it uses costly computing time and memory. This block relation is used for pre-relaxing. Blocks should be small (5×3 in this work) to ensure a reasonable computing time but sufficient to solve the problem. shows that the residual is relatively well distributed over the surface. This means that the local problems were overcome. The block pre-relaxation permit fast convergence regarding the number of cycles and regarding the work units, as defined in Chapter 2. The definition does not take into account the pre-relaxation which is reasonable when using the point relaxation. With the block relaxation, it has a cost and is unpredictable and is a function of the size of a single block, of the largest value of the residual, of the number of neighbouring points to the cavitation zone, of the number of points inside the grooves (or the highest slopes),... All this means that the computing time can, in some cases, be better for point pre-relaxation than for the block pre-relaxation (not in the case studied in this appendix), but for robustness purposes the block relaxation is always used. Assuming that f (A) = R * /R * smooth , g(ǫ) (cf. Equation C.2) is determined by looking for a function that decreases when the density increases. In fact, knowing that very dense grooves and very deep grooves have almost the same effect on load carrying capacity, the value of ∆ must be nearly the same for both cases (dense and deep). Equation C.8 shows that when A is increasing ∆ decreases. This explains why g has to be a decreasing function. In addition to the decreasing requirement, g has to ensure that the result (LCC as a function of ∆) is not sparse nor scattered. In other words, two given calculated LCC's, of approximatively the same ∆, have to be as close as possible one to each other. The function -log fits these requirements. The expression of ∆ is ∆ = -log(ǫ) 1 + A + 3A 2 /8 (1 + A) 5/2 (C.9)

q x = h 0 u dydz = -h

Reynold equation for the parabolic ring

.2 shows ∆ as a function of ǫ and A. For deep grooves (A > 10) ∆ is small, for dense grooves (ǫ > 0.8), ∆ is also small. The deep grooves and the dense grooves give the same order of magnitude of ∆. ∆ increases when the groove depth decreases and when the density decreases. For very sparse grooves and for very shallow grooves, ∆ is large. This meets the idea that those types of grooves have the same behaviour and does not influence much the LCC.

4a

  Measured surface calculations: Prediction and validation 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Geometry: Analytical vs Measured . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Measured surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Groove identification . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Top ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Film thickness-LCC relation . . . . . . . . . . . . . . . . . . . . . . . 4.4 Oil control ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Film thickness-LCC relation . . . . . . . . . . . . . . . . . . . . . . . 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion A Reynolds equation derivation B Numerical solutions and pre-relaxation operator C The single parameter ∆ VI Notation Groove depth at the centre[m] c z axis dimensionless parameter[m] d 1

Figure 1 :

 1 Figure 1: Energy distribution in a diesel engine:(a) Total energy (b) Mechanical friction (c) Piston, rings and rod friction (d) Rings friction [RIC 00]

Figure 1 . 1 :

 11 Figure 1.1: Two bodies A and B in contact (a) Dry contact (b) Lubricated contact.

Figure 1 . 2 :

 12 Figure 1.2: IC Engine: the moving parts [HEY 88]

Figure 1 . 4 :

 14 Figure 1.4: A Stribeck curve with its 3 zones from the left to the right: the boundary regime, the mixed regime and the full film.

Figure 1 . 5 :

 15 Figure 1.5: The ring pack: (a) Piston with three rings (b) A ring

Figure 1 . 6 :

 16 Figure 1.6: Typical viscosity along the stroke.

Figure 1

 1 Figure 1.7: (a) Schematic representation of the honing process (b) Typical measured honed surface

Figure 2 .

 2 Figure 2.1: Calculational domain

Figure 2 . 2 :

 22 Figure 2.2: Flattened surface in y direction

Figure 2 . 3 :

 23 Figure 2.3: Piston ring shapes

Figure 2 .

 2 3 roughly shows the piston ring shapes. The top ring and the second ring are quite similar. Their carrying surface profile is parabolic with a radius of curvature R x . It ranges between 5 and 25 cm. The twin land OCR has two carrying surfaces. Both of them are considered flat. In fact their radius are about 10 m. The three rings have smooth surfaces, their surface roughness is neglected.

Figure 2 . 4 :

 24 Figure 2.4: Flattened measured rough surfaces

Figure 2 . 5 :

 25 Figure 2.5: Cross hatching pattern

Figure 2 .

 2 Figure 2.6 regroups the ring model and the surface model. The film thickness at each point (x, y) is the sum of three terms:-the nominal film thickness h 0 which is constant unless in imposed load mode, -the ring profile for the x position, which is x 2 /(2R x ) for a parabolic shaped ring and zero for the flat one, -the r(x, y, t) term explained in subsection 2.2.2.3 . The film thickness equation reads:

Figure 2 . 6 : 2 . 2 . 3 . 2

 262232 Figure 2.6: The film thickness (a) Three ring contact (b) OCR film (c) Top ring film (d) Liner surfaces

Figure 2 . 7 :

 27 Figure 2.7: The discretized domain

Figure 2 .

 2 Figure 2.8: Iterative method

Figure 2 . 9 :Figure 2 . 10 :

 29210 Figure 2.9: Coarse grid correction cycle

Figure 2 .

 2 Figure 2.11: Full weighting restriction

Figure 2 . 12 :

 212 Figure 2.12: 1D linear interpolation stencil

Figure 2 .

 2 Figure 2.13: Multigrid schemes: V-cycle and W-cycle

Figure 2 .

 2 Figure 2.14: Calculation scheme

Figure 2 .

 2 Figure 2.15: Calculation example: Sliding parabolic ring over a measured surface patch

Figure 2 . 17 :

 217 Figure 2.17: A five levels V-cycle and the corresponding Work units for each grid

3. 2

 2 Figure 3.1: Load carrying capacity per unit length as a function of the film thickness (fully flooded) in three positions: close to Top Dead Center (TDC), MidStroke (MST) and close to Bottom Dead Center (BDC): bottom to top, top ring (R x = 10mm) and OCR (R x = 5m)

Figure 3 . 2 :

 32 Figure 3.2: Load carrying capacity per unit length as a function of the film thickness: solid lines (smooth starved) dashed lines (smooth fully flooded), for the top ring in red (bottom to top: x a = 0.2, 1.6 mm) and for the OCR in black (bottom to top: x a = 0.1, 0.2 mm).

Figure 3 . 3 :

 33 Figure 3.3: Pressure at the center line Y=0 for a smooth surface with different starvation degree (nearly fully flooded to very starved: X a = -32, X a = -4 and X a = -1)

Figure 3 . 4 :

 34 Figure 3.4: Code validation: Load carrying capacity as a function of the starvation degree: calculation (markers) vs analytical expression (solid line)

Figure 3 . 5 :Figure 3 . 6 :

 3536 Figure 3.5: Pressure profile on the center line Y=0: smooth (red) vs grooved (black)

Figure 3 . 7 :

 37 Figure 3.7: Pressure profiles for D 1 = 1, Λ = 0.2, α = 25 • and A = 0.25, 1, 4, 32 for different time steps

Figure 3 . 8 :Figure 3 . 9 :

 3839 Figure 3.8: Symmetry in pressure distribution

Figure 3 . 10 :Figure 3 .∞Figure 3 . 12 :

 3103312 Figure 3.10: The dimensionless microgeometric parameters and the dimensionless starvation degree parameter

Figure 3 .Figure 3 .Figure 3 .

 333 Figure 3.13: Relative LCC as a function of 1/A for D 1 = 1 (blue) and D 1 = 2 (red), α = 25 • and Λ = 0.025, 0.1, 0.2 (respectively circles, squares and triangles)

Figure 3 .

 3 Figure 3.16: Fitted curves for X a = 1 (△), X a = 4 (•) and X a = 32 (⊡)

Figure 3 . 17 :

 317 Figure 3.17: Load carrying capacity per unit length as a function of the film thickness (solid lines: grooved, dashed: smooth), a = 0.25µm, ǫ = 50% (green solid), a = 1µm, ǫ = 50% (blue solid) and a = 4µm, ǫ = 50% (red solid)

Figure 3 .Figure 3 . 19 :

 3319 Figure 3.18: Load carrying capacity per unit length as a function of the film thickness (solid lines: grooved, dashed lines: smooth), a = 1µm, ǫ = 30% (green solid), a = 1µm, ǫ = 50% (blue solid) and a = 1µm, ǫ = 70% (red solid)

Figure 3 .

 3 Figure 3.20: 1D model and the pressure profile [BIB 14b]

Figure 3 .

 3 Figure 3.21: Pressure profile for different groove depths

Figure 3 .

 3 Figure 3.22: Maximum pressure as a function of the groove density N for α = 25 • , 45 • and 65 • (top to bottom) and for A = 1/8 (•), A = 1 (△), and A = 8 (⊡)

Figure 3 . 24 :Table 3 . 5 :Figure 3 . 25 :

 32435325 Figure 3.24: Maximum pressure for sparse grooves and zero angle P M ∞0 as a function of A (solid line for curve fit and dashed line for Biboulet formula [BIB 14b])

Figure 3 .

 3 Figure 3.26: Load carrying capacity per unit length as a function of the film thickness for four microgeometries: a = 1µm, N = 2.2 and α = 25 • , a = 1µm, N = 3.5 and α = 25 • , a = 1µm, N = 6.1 and α = 25 • (solid lines bottom to top) and the smooth case (dashed line).

Figure 3 .

 3 Figure 3.27: Load carrying capacity per unit length as a function of the film thickness: Example of the force generation in the mixed regime

Figure 4 . 1 :

 41 Figure 4.1: Measured surface positions

Figure 4 . 2 :

 42 Figure 4.2: Measured surfaces: (a) Mid-stroke (b) Bottom dead center

Figure 4 . 3 :Figure 4 . 4 :

 4344 Figure 4.3: Groove identification at MST: (a) Measured surface (b) Intermediate analytical surface (c) 1D comparison between measured (blue) and intermediate analytical (red)

Figure 4 .

 4 3(b) andFigure 4.4(b)). These generated surfaces are the analytical equivalent of the real measured surface where the plateaux are smooth and the grooves are sinusoidal. The height scale (color) is the same, simplifying an optical comparison with Figure 4.3(a) and Figure 4.4(a).

Figure 4 .

 4 3(c) andFigure 4.4(c) show the superposition of the measured surface (blue) and the intermediate surface (red). It can be concluded that all the major grooves are identified in both position and that the depth and the width approximations are correct.

Figure 4 . 5 :

 45 Figure 4.5: Mean line displacement

Figure 4

 4 .6 and Figure4.7 show the distributions of the depth, the width and the angle.

Figure 4 .

 4 6(a) and Figure4.7(a) show the number of identified grooves for different depth ranges, where the depth correspond to the curve-fitted value (cf.

Figure 4 .

 4 3(c) andFigure 4.4(c)). The deepest identified grooves are 2.7µm and 2.2µm deep for receptively MST and BDC and the mean values are respectively 0.93µm and 0.66µm.

Figure 4 . 6 :

 46 Figure 4.6: MST grooves parameters: (a) Depth distribution (b) Width distribution (c) Width over depth slope distribution (d) Angle distribution

Figure 4 .

 4 6(c) and Figure 4.7(c) permit a better comparison between MST and BDC. They show the distribution of the width over depth (the inverse of the slope).

Figure 4 .

 4 6(d) andFigure 4.7(d) concern the angle distribution. The average value is about 22 • for both surfaces. Comparing the depth and the width distributions, other observations can be made:

Figure 4 . 7 :

 47 Figure 4.7: BDC grooves parameters: (a) Depth distribution (b) Width distribution (c) Width over depth slope distribution (d) Angle distribution

  Figure 4.6 (c) and Figure 4.7 (c) represent this associated set of parameters.

4. 3 . 2 Figure 4 . 8 :

 3248 Figure 4.8: The LCC for a parabolic ring (w 1 ) as a function of the film thickness (h 0 ) for MST and BDC conditions: (⊡) for MS,(•) for IS, (△) for AS, (solid line) for prediction and (dashed line) for 1D smooth.

Figure 4 . 9 :

 49 Figure 4.9: Pressure distribution in MST with h 0 = 1µm at t = ∆t (surfaces left to right: MS, IS and AS).

  d

Figure 4 . 10 :

 410 Figure 4.10: The LCC of a flat ring as a function of the film thickness h 0 in MST and BDC conditions: (⊡) for MS,(•) for IS, (△) for AS, (+) for Analytical and periodic BC, (solid line) for Prediction.

Figure 4 .

 4 Figure 4.11: The LCC as a function of y for MST (top to bottom h 0 = 1, 2, 4) and BDC (top to bottom h 0 = 0.5, 1, 1.5) and for both measured surfaces and intermediate surfaces (top to bottom).

Figure 4 . 12 :Figure 4 .

 4124 Figure 4.12: Pressure profiles in MST (top, h 0 = 1µm) and BDC (bottom, h 0 = 0.5µm) for Intermediate Surface (left) and Measured Surface (right)

2

 2 Figure A.1 (b) shows the parabolic ring and its sliding direction x (it means thatW 1 = W 2 = V 1 = V 2 = 0). The liner is motionless, and the ring is moving in the decreasing x direction. The velocity of the ring is noted U. One obtains U 2 = -U. Thus U

  Figure B.1: Dimensionless pressure profile on a grooved analytical surface (colormap) for a flat ring with N = 4, α = 25 and A = 4.

Figure B. 2 :

 2 Figure B.2: The mean residual as a function of the number of cycles, from bottom to top: block pre-relaxation + partial interpolation, block pre-relaxation + full interpolation , point pre-relaxation + partial interpolation and point pre-relaxation + full interpolation.

  Figure B.2 shows that the block pre-relaxation diminishes the residual level and converges within a few cycles when using the partial interpolation.

  Figure B.3 (d)

Figure B. 3 :FinallyR

 3 Figure B.3: Residual profile at the end of the calculations (a) point pre-relaxation + partial interpolation (b) point pre-relaxation + full interpolation (c) block pre-relaxation + partial interpolation (d) block pre-relaxation + full interpolation

Figure C. 2 :B. 2

 22 Figure C.2: ∆ as a function of ǫ and A

  .2.1 Fully flooded regime . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Starved regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Code validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Top ring: parabolic shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 The parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Mesh-size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Influence of the groove parameters . . . . . . . . . . . . . . . . . . . 3.3.5 Starvation influence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.6 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Oil control ring: flat shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 The parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Mesh-size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Influence of the groove parameters . . . . . . . . . . . . . . . . . . . 3.4.5 Boundary effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.6 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Smooth surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

Table 1 :

 1 EU Emission Standards for Passenger Cars in mg/km.

		NOx		Tot.HydroCarb. THC+NOx	PM		PN in ♯/km
	STAGE & DATE Diesel	P etrol	Diesel	P etrol	Diesel	P etrol	Diesel	P etrol	Diesel	P etrol
	Euro 1 07-1992	-	-	-	-	970	970	140	-	-	-
	Euro 2 01-1996	-	-	-	-	700/900 500	80/100	-	-	-
	Euro 3 01-2000	500	150	-	200	560	-	50	-	-	-
	Euro 4 01-2005	250	80	-	100	300	-	25	-	-	-
	Euro 5a 09-2009	180	60	-	100	230	-	5	5	-	-
	Euro 5b 09-2011	180	60	-	100	230	-	5	5	6e11	-
	Euro 6 09-2014	80	60	-	100	170	-	5	5	6e11	-

Table 2 .

 2 2 shows the work units. For a given time step, the comparison between

	M ax lev	1	2	3	4	5	6	7
	W U	200	112	68	46	35	29.5	26.7

Table 2 .

 2

2: Work units for different number of grids

  Influence of the groove parameters . . . . . . . . . . . . . . . . . 3.3.4.1 The parameters choice . . . . . . . . . . . . . . . . . . 3.3.4.2 Groove depth . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4.3 Groove density . . . . . . . . . . . . . . . . . . . . . . . 3.3.4.4 Master curve . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Starvation influence . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.6 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Oil control ring: flat shape . . . . . . . . . . . . . . . . . . . . . . . . . . Groove density . . . . . . . . . . . . . . . . . . . . . . . 3.4.4.2 Groove angle . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4.3 Groove depth . . . . . . . . . . . . . . . . . . . . . . . . 3.4.5 Boundary effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.6 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	3.4.4.1
	Chapter 3
	Influence of cross-hatched grooves:
	Load carrying capacity -Film
	thickness relation

Contents 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Smooth surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Fully flooded regime . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Starved regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Code validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Top ring: parabolic shape . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 The parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2.1 Time dependency . . . . . . . . . . . . . . . . . . . . . 3.3.2.2 Eliminating X b . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Mesh-size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 3.4.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 The parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Mesh-size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Influence of the groove parameters . . . . . . . . . . . . . . . . .

  Table 3.1: Viscosity and velocity at TDC, MST and BDC 3.2.2 Starved regime Biboulet et al. formula The fully flooded regime for ring lubrication assumes that the ring is endlessly supplied with oil and that the ring axial thickness is infinitely large (infinite parabola). Actually neither the first assumption nor the second is valid. The starvation is introduced and it acts on relation 3.2 and W 1 is no longer constant. In fact in a completely starved regime, when no oil is available, W 1 is nil. Biboulet et al. [BIB 13] extended relation 3.2 to the starved line contact. The dimensionless relation reads:

	Position	T DC	M ST	BDC
	η in Pa.s	0.004	0.012	0.02
	u m in m/s 0.1		4	0.1
		W 1 =	0.408 1 + 4.809/X 2 a + 5.745/X 4 a	(3.3)
					408	(3.2)

Table 3 . 2

 32

	Mesh size		1/24	1/48	1/96	1/128	1/192	1/384
	Moderate case	W 1	0.248	0.199	0.184	0.182	0.180	0.179
		error	38%	11%	2.8%	1.6%	0.6%	-
	Extreme case	W 1	3.02e-1	7.71e-2	2.93e-2	2.30e-2	1.95e-2	1.68e-2
		error	680%	358%	75%	36%	16%	-

: Accuracy of W 1 for different mesh sizes 3.3.4 Influence of the groove parameters 3.3.4.1 The parameters choice

Table 3 . 3 :

 33 Starvation curve fit factor f

	-Xa	1	2	4	8	16	32
	f	0.85	0.85	1.03	1.15	1.20	1.20

Table 3 . 4 :

 34 Accuracy of W 1 for different mesh sizes

	Depth A M esh ref	2 -5 1/96	1 1/96	2 1/96	4 1/192 1/384 1/384 1/384 8 16 32
	W 1ref	0.0162 0.142	0.121	0.0833 0.0509 0.0276 0.0141
	M esh	1/48	1/48	1/48	1/96	1/192 1/192 1/192
	0.0163 0.141 0.8% (W 1 -W 1ref )/W 1ref 0.2% W 1 M esh 1/24 1/24	0.119 1.5% 1/24	0.0811 0.0496 0.0277 0.0157 2.6% 2.6% 0.4% 11% 1/48 1/96 1/96 1/96
	W 1	0.0165 0.1431 0.122	0.0829 0.0489 0.0299 0.0209
	(W 1 -W 1ref )/W 1ref 1.7% M esh 1/12	0.4% 1/12	1.1% 1/12	0.5% 1/24	3.8% 1/48	8% 1/48	47% -
	0.0172 0.159 12% (W 1 -W 1ref )/W 1ref 6% W 1 M esh 1/6 1/6	0.150 24% 1/6	0.104 25% 1/12	0.0551 0.0420 -8% 52 -1/24 --
	W 1	0.0193 0.212	0.237	0.187	0.0890 -	-
	(W 1 -W 1ref )/W 1ref 18%	48%	95%	125%	74%	-	-

  The groove identification process generates a set of microgeometrical parameters that can now be analysed. The objective is to relate the measured surface and the intermediate surface to the Chapter 3 model which has a single value of each parameter.

				Ra in µm	Rq in µm
	MST	only plateau	0	0.280	0.370
		all the surf.	-0.190	0.417	0.570
	BDC	only plateau	0	0.132	0.371
		all the surf.	-0.105	0.258	0.482
	Table 4.1: Mean height and roughness parameters on plateaux and complete surface
	4.2.2.3 Groove parameters

Table 4 .

 4 2 resumes the set of parameters.

		MST	BDC
	a 0	0.848µm	0.590µm
	α 0 21.98 •	18.98 •
	λ 0	34.09µm	27.63µm
	ǫ tot 54.10%	40.15%

Table 4 . 2 :

 42 Associated set of paramters

		MST (u m = 4m/s, η = 0.012P a.s)	BDC (u m = 0.1m/s, η = 0.02P a.s)
	Surface		Smooth		Grooved		Smooth		Grooved
	BC	Periodic Dirichlet	Periodic Dirichlet	Periodic Dirichlet	Periodic Dirichlet
	h 0 = 0.25µm -	-	-	-	363	353 (3%) 266	254 (5%)
	h 0 = 0.5µm	4066	3863(5%) 2803	2588(8%) 169	161 (5%) 139	130 (7%)
	h 0 = 1µm	1776	1629(9%) 1405	1262(11%) 74	67 (10%) 65	59 (10%)
	h 0 = 1.5µm	-	-	-	-	43	38 (12%) 39	35 (11%)
	h 0 = 2µm	695	606(13%) 599	516(14%) -	-	-	-
	h 0 = 4µm	229	188(18%) 209	170(19%) -	-	-	-

Table 4 . 3 :

 43 Comparison between the LCC generated in MST and BDC with Dirichlet BC and periodic BC for smooth surfaces and grooved ones (values of the LCC are in N/m and errors between parenthesis)The Dirichlet BC for the lateral sides alters the pressure generation compared to an infinitely large 1D contact. It is obvious that for sufficiently distant boundaries, the Dirichlet BC have no influence on the central pressure in the contact (to avoid this problemDalissier [DAL 12] duplicates the measured grooves to use periodic boundary conditions). The objective of this paragraph is to estimate the influence of Dirichlet BC. The smooth case is taken as a reference. Table4.3 shows the load carrying capacity of a contact with Dirichlet BC compared with the infinitely large 1D contact for both smooth and grooved surface in MST and BDC. The grooves parameters correspond to table 4.2. The calculated LCC for Dirichlet BC cases is a time and y-direction averaged value. Only the center of the contact, 3/4 th of the domain width, is taken into account for comparison. The error shown between parenthesis in Table4.3 is less than 20% for the worst case. One can take less than 3/4 th of the domain width to ameliorate this value, but this will give less surface covered and fewer grooves analysed. The choice is to keep this value of 3/4 th for the next analysis and it is considered as sufficiently correct to evaluate the averaged LCC.

	4.3 Top ring
	4.3.1 Boundary conditions

Table 4 .

 4 4 details the calculated values and the MST (u m = 4m/s, η = 0.012P a.s) BDC (u m = 0.1m/s, η = 0.02P a.s)

	Surfaces	Sm.	MS	IA	AS	Pred.	Sm.	MS	IA	AS	Pred.
	h 0 = 0.25µm -	-	-	-	-	363	-	259	245	250
	error	-	-	-	-	-	+45% -	+3%	+1%	-
	h 0 = 0.5µm	4066	-	2785	2588	2294	169	145	132	130	137
	error h 0 = 1µm	+77% -1776 1446	+21% +12% -1312 1262 1256	+23% +5% 74 63	-4% 60	-6% 59	-66
	error h 0 = 1.5µm	+41% +15% +5% ---	+0% -	--	+12% -5% 43 36	-10% -11% -35 35 39
	error h 0 = 2µm	-695	-557	-531	-516	-574	+10% -8% --	-11% -11% ----
	error h 0 = 4µm	+27% +1% 232 179	-3% 174	-6% 170	-189	--	--	--	--	--
	error	+17% -10% -13% -15% -	-	-	-	-	-

Table 4 . 4

 44

: Comparison of LCC for MST and BDC conditions, for MS (measured surface), IS(intermediate surface), AS (analytical surface), prediction (Pred.)and 1D smooth(Sm.).

  Alcouffe et al. [ALC 81] and Noutary et al. [NOU 12]).

	Thus	u =	-1 2η	∂p ∂x	(hz -z 2 ) + U 1 +	z h	(U 2 -U 1 )	(A.4)
	By the same way
		v =	-1 2η	∂p ∂y	(hz -z 2

  Equalling the entering flows with the exiting ones, one obtains the generalized form of the Reynolds equation

											3 12η	∂p ∂x	dy +	U 2 + U 1 2	hdy	(A.6)
				q y =	0	h	v dxdz =	-h 3 12η	∂p ∂y	dx +	V 2 + V 1 2	hdx	(A.7)
											q 1 = W 1 dxdy	(A.8)
			q 2 = W 2 dxdy -U 2	∂h ∂x	dxdy -V 2	∂h ∂y	dxdy -	∂h ∂t	dxdy	(A.9)
	W 1 -W 2 = -	1 12η	∂ ∂x	h 3 ∂p ∂x	-	1 12η	∂ ∂y	h 3 ∂p ∂y
						+	U 2 + U 1 2	∂h ∂x	+	V 2 + V 1 2	∂h ∂y
						+	h 2	∂ ∂y	(V 2 + V 1 ) +	h 2	∂ ∂x	(U 2 + U 1 ) -	∂h ∂t
											(A.10)

  One can re-write Equation A.16 according to a frame that is fixed to the moving ring. The variable x is changed to x 1 according to the following relationx = x 1 -Ut (A.17)Knowing that ∂x = ∂x 1 , Equation A.16 becomes

	The Couette term becomes
				-U m	∂h ∂x	= -U m	x R x	-U m	Ut R x	-U m	∂R ∂x	(A.14)
	And the transient term becomes
										∂h ∂t	=		∂h 0 ∂t	+	U 2 t R x	+	xU R x	(A.15)
	Adding these two terms leads to
		-U m	∂h ∂x	+	∂h ∂t	= U m	x R x	+ U m	Ut R x	-U m	∂R ∂x	+	∂h 0 ∂t	(A.16)
			-U m	∂h ∂x		+	∂h ∂t		= U m	x 1 R x	+	∂h 0 ∂t	-U m	∂R ∂x	(A.18)
	The velocity can be expressed									U = -	∂x ∂t	(A.19)
	Thus												U	∂R ∂x	+	∂R ∂t	= 0	(A.20)
	Consequently							U m	∂R ∂x	+	∂R ∂t	= -U m	∂R ∂x	(A.21)
	Equation A.18 becomes
		-U m	∂h ∂x	+	∂h ∂t		= U m	x 1 R x	+	∂h 0 ∂t	+ U m	∂R ∂x	+	∂R ∂t	(A.22)
	The film thickness reads				h(x 1 , t) = h 0 (t) +	x 2 1 2R x	+ R(x 1 , t)	(A.23)
	and	U m	∂h ∂x	= U m	x 1 R x	+ U m	∂R ∂x	and	∂h ∂t	=	∂h 0 ∂t	+	∂R ∂t	(A.24)
	Finally the Reynolds equation reads
		1 12η	∂ ∂x	h 3 ∂p ∂x		+	1 12η	∂ ∂y	h 3 ∂p ∂y	= U m	∂h ∂x	+	∂h ∂t	(A.25)
	Reynolds equation for the flat ring
															x 2 2R x	+	U 2 t 2 2R x	+	xUt R x	+ R(x)	(A.13)

). It reads

h(x, t) = h 0 (t) + (x + Ut) 2 2R x + R(x) = h 0 (t) +

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Appendix A

Reynolds equation derivation

This appendix details how the Reynolds equation is derived for both the parabolic and the flat ring.

Generalised Reynold equation

The Reynolds equation is based on the thin film assumption between two moving surfaces 1 and 2. An elementary volume between these surfaces is considered. Figure A.1 (a) shows the coordinate system x, y, z and the velocity for both surfaces 1 and 2, respectively

Using the definition of the viscosity η, the pressure gradient for x reads

The integration with respect to z gives

where C 1 and C 2 are determined using boundary conditions

Appendix C

The single parameter ∆

This appendix details how the single parameter ∆ is derived. The objective of the parameter ∆ is to regroup both the groove density ǫ and the groove depth A in one single parameter with

The effect of the groove depth is analysed in this paragraph, independently of its density (which regroups three parameters that are D 1 , Λ and α). For this purpose, a 1D groove is considered with a unit width (Figure C.1). The expression of this groove is

The effect of this groove is compared with the smooth case. The film thickness for the

and for the smooth case

The terms in H 3 in the Reynolds equation indicate the conductivity of the contact in term of Poiseuille flow. Thus, the inverse of this value is the resistance that the geometry opposes