Now, lubricated contacts—as they are present in internal combustion engines—might be assumed to operate in purely hydrodynamic lubrication. For this form of lubrication, a sufficiently thick oil film separates the two gliding surfaces from each other and the friction in this oil film generates the observed losses. Purely hydrodynamic lubrication is advantageous in principle as there is no actual contact between the surfaces—and consequently, no wear can occur. Of course, the reality is more complex as already the transient nature of operation for internal combustion engines prevents pure hydrodynamic lubrication (e.g., during starting).
Further, the Stribeck-curve shows that the minimum of the friction losses is located in the mixed lubrication regime where some surface contact is present. Consequently, the continued efforts to reduce the friction losses in internal combustion engines have led to the regular appearance of mixed lubrication. Excessive wear is typically countered with either oil-additive chemistry or surface coatings and is often combined with a very smooth surface topography.
2.2. Simulation Methodology
Accurately simulating the large range of operating conditions of journal bearings that extend from fully hydrodynamic lubrication to severe mixed lubrication involves many factors. As it is not possible to discuss these in deserved detail here, the authors refer to the original works [
16,
17], where the specific points are not only discussed in depth, but which also contain the relevant references to previous works. In the following, only a brief discussion of the key points and related references is given for this reason.
The basic simulation consists of a multibody system with elastically deformable finite element bodies, which are represented as condensed structures with reduced degrees of freedom [
18,
19]. The bodies are coupled with an oil film that is described using the Reynolds equation, with extensions developed by Patir and Cheng [
20,
21], to consider the effects of surface roughness on very small oil-film thicknesses:
where
x is the circumferential and
y the axial direction.
p and
h represent the local hydrodynamic pressure and oil-film thickness.
and
denote the sliding speeds of the facing surfaces. The pressure flow factors
,
and the shear flow factor
consider rough surfaces. The oil viscosity
is also a function of
x and
y and depends on local temperature, pressure, and shear rate. To consider cavitation in the low-pressure region, the mass-conserving cavitation model based on the Jakobsson–Floberg–Olsson (JFO) approach is used [
22].
represents the fill ratio for the mass-conserving cavitation model. In the cavitation region, the hydrodynamic pressure
p becomes the cavitation pressure and the fill ratio is below 1. In the lubricated region (high-pressure region), where
p is above cavitation pressure, the fill ratio becomes 1.
The complex lubricant rheology requires the consideration of the piezoviscous effect (increase of viscosity with increase of pressure), as well as taking into account the non-Newtonian behaviour (viscosity reduction due to high shear rates) of multigrade hydrocarbon-based lubricants. The method used in this work utilizes measured rheological data for these two effects and includes a correction to yield the correct high-temperature high-shear-rate viscosity (HTHS)-viscosity (The HTHS-viscosity is defined as the viscosity at high temperature (150
C) and high shear rate (10
1/s).) (see in particular, [
23,
24]). The lubricant rheology used in this work is also presented in detail in the next
Section 2.2.1.
To describe the occurrence of metal–metal contact in the simulation of an asperity contact model, a suitable boundary friction coefficient that considers the presence of friction-modifying additives [
25] and the surface roughnesses of the involved surfaces are required. The method used in this work employs the very well-known Greenwood–Tripp model [
26] in combination with surface roughness data from measured surfaces (for details, see in particular, [
23,
24,
27]). Accordingly, the asperity contact pressure
is calculated as follows:
where
is a dimensionless clearance parameter,
is the composite elastic modulus, and
K is the elastic factor which depends on surface roughness, asperity radius, and asperity density.
is a form factor defined by Greenwood and Tripp [
26]. The shear stress due to asperity contact is calculated by multiplying the asperity contact pressure with a boundary friction coefficient
:
where it was found in previous works [
23,
24] that a
of 0.02 gives a close agreement with experimental data.
In addition, it is crucial to use realistic surface shapes for the bearing edges (microgeometry) in the simulation as otherwise the metal–metal contact is largely overestimated [
24,
27]. Finally, the thermal processes within the oil film as well as the heat transfer to its surroundings need to be considered suitably. In [
28], the thermal processes of journal bearings under high dynamic loads are investigated in direct comparison to experimental data. It was shown that it is possible to reproduce the thermal properties of the journal bearings very accurately in comparison to experimental measurements. It was found that the stable thermal behaviour of the lubricant film in journal bearings can be very well described by an isothermal elastohydrodynamic (EHD) simulation method [
19] in combination with a suitable equivalent temperature that describes the temperature gradient in the journal bearing. From these results [
28], a very simple equivalent-bearing temperature relation was derived for the isothermal EHD-simulation that is consequently able to predict the journal bearing friction losses very accurately for a large range of different lubricants, journal speeds, and loads, as was shown in direct comparison to experimental data (see in particular, [
23,
24]). A complete and exhaustive discussion of these points is given in two open-access published book chapters [
16,
17] (the cited book chapters can be downloaded for free at [
1]).
Due to the inherent convergence issues with the piston-pin simulation, at least 9 working cycles (6480
crank angle) were simulated in total for every investigated case. Typically it was found—and an example is shown in
Figure 3—that the simulation needs 2–3 working cycles to reach a state which no longer changes completely in comparison to later cycles. While there exist smaller differences between later cycles and no ideal periodicity is achieved, the last 3 working cycles were averaged to ensure that the results are reliable. Especially, the investigated operating conditions involving part-load operation caused very slow resulting piston-pin rotations close to 0 rpm, which posed the biggest challenges numerically with the largest variations between cycles. Consequently, 18 working cycles have generally been simulated for part-load operation points to have enough data for a meaningful analysis of these operating points.
In the simulation, the reference case was defined with a surface roughness of 0.06
m for the piston pin [
8] and 0.15
m for the small end and piston boss, respectively; in addition, the piston-pin clearance was defined with 6
m.
In addition, several variants have been investigated: different piston-pin clearances (4 and 9
m) as well as a reduced surface roughness (0.02
m), which are in agreement with the range of values found in literature [
8,
29,
30]. All variants are summarized in
Table 3.
2.2.1. Oil Rheology in the Simulation
To describe the physical properties of the lubricant realistically, an extensive rheological model is employed for the description of the 5W30 lubricant (as described by
Table 2) in the simulation [
16,
17].
Both the effects of pressure
p (piezoviscous effect) as well as the influence of local shear rate
(non-Newtonian effect) are considered in the simulation of the lubricant film by employing the well-known Barus and Cross equations; the viscosity (
) over temperature (
T) relation is described using the Vogel equation [
16,
17]:
The applied parameters are listed in
Table 4 and are calculated using results from previously published work [
23] and from lubricant characteristics obtained from laboratory oil analysis.
For the description of the pressure- and temperature-dependent density, the widely used Dowson/Higginson equation is used (see Equation (
5)), applying the parameters listed in
Table 5.
Figure 4 shows the resulting dynamic viscosity of the lubricant.
2.3. Thermoelastic Simulation of the Piston
In internal combustion engines, pistons made from aluminum are commonly used in combination with piston pins made from steel. These two different materials not only have a very different thermal expansion behaviour, but there also exists a very strong local temperature gradient between the piston top and the piston skirt. Both of these factors have a strong impact on the actual piston-pin clearance which does not only change strongly in magnitude but also becomes noncircular. This change in clearance can not only affect piston-pin friction [
32] and wear, but might also lead to a strongly increased noise emission [
33,
34].
The thermal simulation has been conducted using Simulia’s [
2]
Abaqus finite element solver package using a two step process:
The first step is to solve the heat-diffusion equation to obtain the temperature field for the piston. To realize this, suitable thermal boundary conditions have to be defined. As thermal diffusion processes are slow compared to the mechanical oscillation of the components [
35], a static heat analysis was performed. A heat power of 4.5 kW (corresponds to a heat flow of 9 × 10
W/m
) originating from combustion during full-load operation of the engine was defined to enter the piston top homogeneously. Further, so-called surface film temperatures and film coefficients were defined for different specific parts of the piston, which represent thermal boundary conditions describing the heat flow into the cylinder liner and to the oil mist inside the crank case. Surface-film temperature for the top ring, 2nd and oil ring, and piston skirt were defined as 160
C, 130
C, and 100
C, respectively. Inside the piston void, the piston is able to conduct heat to the crank case oil mist by defining the corresponding surface film as having a temperature of 80
C. These thermal boundary conditions are shown graphically in
Figure 5.
While the exact heat flows of this particular engine are not known and suitable assumptions had to be taken, it is of more importance for this work to ensure that a realistic temperature field for the piston is obtained from the performed thermoelastic simulation. In comparison to more detailed thermal simulations [
36], and in comparison to the temperature measurements available in literature [
8,
9,
10,
11,
12], the obtained piston temperatures are in good agreement and are shown in
Figure 6.
The second step of the thermal simulation is to use the obtained temperature field from the first step to calculate the resulting thermoelastic deformations of the piston. The resulting deformations are several orders of magnitude smaller than the geometrical dimensions of the piston, which justifies to treat these two simulation steps independently of each other. The obtained thermoelastic deformations are shown magnified by a factor of 100 in
Figure 7.
The central result of this thermal analysis are the thermoelastic deformations that yield a noncircular and much enlarged clearance between the piston pin and the piston bosses, see
Figure 8, where the thermal expansion of the piston pin has already been considered (With a piston-pin temperature of 160
C, the pins expand radially by about 9
m.).
For the piston pin to the con-rod small-end journal bearing, the clearance is not so strongly affected by the thermoelastic deformations as both parts (pin and con-rod) are made from steel. Therefore, a piston pin to small-end clearance of 6 m is used for the following investigations.
2.4. Simulation of the Piston Pin
For the simulation of the three oil films that support the piston pin, the following oil-film temperatures have been defined:
Piston boss: Due to the strong local temperature gradients, a pure isothermal oil-film simulation would be a strong simplification of the real situation. On the other hand, performing a full thermoelastohydrodynamic (TEHD) simulation considering the energy and heat equation—as was done by the authors in a previous work for a heavy loaded main bearing [
28]—is also not feasible. The simulation of a freely floating pin is already numerically intensive and plagued with poor convergence. Therefore, it is uncertain whether a combination with a full TEHD would—even with a huge numerical effort—lead at all to converged results. However, in contrast to the heavy loaded main and big end bearings where the actual friction in the oil film causes the temperature field, the situation is very different for the piston pin. For the piston pin, it is not the friction power loss in the oil film that generates the temperature field, but it is caused by the heat flow from the combustion (see also [
8]). Therefore, a suitable approach is to directly consider the temperature field from the piston thermal analysis in the EHD-simulation.
For the piston bosses, the locally varying temperature field from the thermal analysis is used in the EHD-simulation, see
Figure 9.
For the parts of the piston pin that are supported by the piston bosses, a temperature of 160
C is used in the simulation. This temperature represents roughly the average of the circumferentially varying piston boss temperature field [
8], which is justified due to the rotation of the pin. For the actual oil-film simulation, the average of both (local) temperatures is used for every circumferential point.
Small end: For the simulation of the small end journal bearing—which is not in direct contact to the piston—a constant 130 C oil-film temperature is defined.