Cavitation Bubble Measurement in Tribological Contacts Using Digital Holographic Microscopy

Driven by the need for bearings with increased load capacity and reduced frictional losses and wear, lubricating flow has been the subject of extensive studies for well over a century [1]. Of particular current interest is the role that cavitation plays in the performance of piston rings, journal and sliding bearings [2]. Cavities are formed when the gas pressure drops below either the lubricant vaporisation pressure (vaporous cavitation) or the saturation pressure of dissolved air (gaseous cavitation). It is well known that cavitation can negate sub-cavity pressures in divergent contact regions. However, its detrimental effect on the wear of bearing surfaces through the process of erosion remains a topic of debate [3].

In this paper, the measurement of cavitation generated by a plane surface sliding over a cylindrical component is considered as shown in Fig. 1. This geometry has been well defined in both experimental and analytical studies as it is analogous to the geometry of a piston ring–cylinder liner conjunction [4]. The configuration was chosen for this work because it is easily fabricated out of precise yet inexpensive optical components facilitating the measurement of cavitation using interferometric methods.

Following the pioneering work of Reynolds in [5], there has been significant effort to produce suitable outlet boundary conditions for Reynolds-based lubrication models. Review papers by Dowson and Taylor [6] and Majumdar [7] and Braun [2] provide an excellent summary of the intervening investigations into cavitation in bearings.

If the inlet and outlet pressures at the edges of the bearing are assumed to be atmospheric (the full-Sommerfeld boundary condition), then the direct solution of Reynolds equation reveals that the fluid must support sub-ambient pressures and the bearing cannot support a load without cavitation. Gümbel [8] was the first to account for film rupture by considering the full film region to terminate at the minimum clearance, negating any sub-cavity pressures (half-Sommerfeld condition). Although mathematically straightforward, this assumption is not physical and does not account for flow continuity.

In order to satisfy flow continuity, there are essentially two possibilities: either the flow must pass around multiple cavities (in “streamers”) or it must flow over or under a single cavity within the diverging section of the contact. Both of these theories have been studied extensively. Multiple cavities between the bearing surfaces were proposed independently by Swift [9] and Stieber [10] who suggested that, for cavities that extend into the ambient region (i.e. open cavities), Reynolds equation could be used to describe the pressure with the additional boundary condition that both the pressure is equal to the outlet pressure and its gradient diminishes. The assumption of multiple cavities between the bearing surfaces was also progressed in a series of papers by Floberg [11], Jacobson and Floberg [12], Olsson [13] and Floberg [14] and together this work is collectively referred to as the JFO theory boundary conditions. For the first time, JFO theory presented the reformation boundary conditions necessary to describe closed cavities and a corresponding change in the predicted load-carrying capacity. Efficient algorithms to calculate the bearing characteristics iteratively were later proposed by Elrod [15, 16] and later by Vijayaraghavan and Keith [17].

The alternative notion, that the lubricant flows over or under a single cavity, has also been discussed and revisited several times. Birkhoff and Hays [18] recognised that according to the pressure distribution predicted by Reynolds equation, a flow reversal is predicted at a point in the diverging region. At this point, they propose that the flow separates to form a single cavity. This concept was extended considerably by Coyne and Elrod [19, 20] who introduced the influence of surface tension, gravity and inertia. Although the gravity and inertial effects are small, the surface tension causes a substantial pressure difference between the cavity and the surrounding fluid. This allows the cavity to “retreat” into the low-pressure region further from the point of minimum clearance and a corresponding decrease in load-carrying capability is predicted. A more recent analysis of gas cavity bubble shear and lubricant gas transfer conducted by Groper and Etsion [21] also used the single-cavity assumption.

Significant efforts have been made to experimentally investigate the nature of cavitation in bearings [22‐26]. Pressure data are often measured experimentally to gain an understanding of the contact conditions. Dowson et al. [22] and Etsion and Ludwig [23] investigated the hydrodynamic film rupture in non-conformal contacts and pressure fluctuation in the region of cavitation in journal bearings, respectively. Both papers employed a combination of photography and pressure measurements techniques. Indeed, use of photography has been widely used to investigate cavitation bubbles in many bearings types. Dowson [24] observed cavitation bubbles in the lubricant between a fixed convex lens and a sliding steel plane surface. Braun and Hendricks [25] photographed cavitation bubbles in a journal bearing. Both studies viewed the bearing from above showing that cavitation occurs in the divergent region, while the fluid flows out of the contact through lubricant streamers between the fingers like bubbles. Using photographs taken from different angles, Heshmat [26] observations suggest that a continuous sub-layer is present on the moving surface in the cavitation region. Dellis and Arcoumanis [27] have studied the different bubble shapes formed in the lubricant film of a piston ring assembly also using photographic methods. More recently, Vladescu et al. [28] have used a laser-induced fluorescence to qualitatively visualise cavitation phenomena in surface-textured sliding bearings.

In this paper, the use of digital holographic microscopy is to realise high-resolution measurements of cavities within sliding bearings for the first time. A precision slide capable of bidirectional motion is used to drive glass-bearing surfaces fabricated from precision grade optical components. It is shown how the holographic recordings can be analysed to extract the bubble thickness with a lateral resolution of a few microns and a thickness resolution of a few tens of nanometres. Results from a series of experiments are discussed with reference to the theoretical models of lubricating flows.

In order to gain further understanding of the role of cavitation bubbles in lubricating flow, the cylinder–plane sliding contact configuration, shown in Fig. 2, was constructed from precision grade optical components with a prescribed surface form error (<50 nm) and surface roughness (R _a  < 1 nm).

The plane glass component is driven by a motorised translation stage capable of bidirectional motion with controlled speed (0–2.3 mm/s) and acceleration (up to 1.5 mm/s²). The cylindrical component (radius R = 13.25 mm, width D = 12.5 mm and length L = 25 mm) is fixed on a stationary metal surface. The lubricating oil used in this study is additive-free, high-viscosity classic green gear oil (SAE 140) with a viscosity η = 1.508 Pa s measured at room temperature (20 °C). The sliding plane glass is entirely supported by the hydrodynamic film entrained into the contact with the cylindrical lens, when a constant load of W = 2.65 N is applied to the cylindrical lens.

The digital holographic microscope that was used for this study is shown in Fig. 3. The plane wave from a Q-swithched, diode-pumped, frequency-doubled Nd:YLF laser of wavelength λ = 523 nm is divided into reference and object beams using a 50:50 beam splitter. The bearing is illuminated by the plane wave that constitutes the object beam and its image magnified by 80× using a microscope objective with NA = 0.55. The image is mixed with a diverging reference wave from the same source such that a near-image-plane hologram is recorded on a CCD sensor with 4,008 × 2,672, 9.0 μm square pixels. The camera is capable of 5.22 frames per second, however, by using commercially available CCD sensors with higher frame rates; it would be possible to investigate the initiation of cavitation and also cavitation in transient contacts.

Details of this microscope and the reconstruction process can be found elsewhere [29, 30]. However, it is worth considering some of the fundamental attributes of a coherent imaging system before proceeding.

The fundamental difference between incoherent and coherent imaging is illustrated by Fig. 4 which shows the intensity pattern recorded by the CCD. An enlargement of the black square reveals the fringes caused by the interference of the object beam and the reference beam. It is the position and contrast of these fringes that contain the information that is related to the amplitude and the phase of the recorded wavefront.

In order to reconstruct the hologram, the following demodulation process is used. If the complex field due to the object beam through the glass bearing is o(x, y) and the complex field due to the reference beam is r(x, y), then the intensity of Fig. 4, I(x, y), can be written as

In an off-axis hologram such as this, each of the terms has a different spatial bandwidth which means that the terms in Eq. (1) can be separated in the frequency domain.

Figure 5 is the absolute value of the Fourier transform of Fig. 4 which illustrates the frequency bands corresponding to the different terms in Eq. (1). The centre circle represents the band of first two terms in Eq. (1), while the two bright regions in the corners of the spectrum represent the holographic image of the third and fourth terms, respectively. Holographic reconstruction can be achieved using the Fourier transform of either the third or fourth term. Since the reference field is known, the phase and amplitude of object field can be calculated. Hence, the reconstruction is simply a filtered version of the recorded intensity pattern.

A major advantage of coherent imaging using digital holographic microscopy rather than its incoherent counterpart is that both the recorded fields are available for further analysis. That is, the recovered object field, o(x, y), can be written as,where ϕ(x, y) is the phase and a(x, y) the amplitude distribution. In this study, the phase is the most important parameter as illustrated in Fig. 6 which shows the phase of the demodulated hologram shown in Fig. 4. It can be seen that the cavitation bubbles are now revealed, but there are underlying fringes that are principally due to wavefront aberration in the illuminating beam. In order to remove this contribution, it is useful to subtract the phase distribution recovered from a holographic recording of the apparatus when there is no cavitation (i.e. when the slide is stationary).

An example of the subtracted phase is shown in Fig. 7. The information revealed in this and other similar images will now be discussed with reference to the theoretical models of lubricating flows.

In the experiment, regular shape bubbles such as those shown in Fig. 7 were observed after a short time after start-up when the slide is moving at constant velocity towards the top of its speed range, in this case after approximately 2 s when it is travelling at 2.3 mm/s. The multiple regularly spaced cavitation bubbles have closed cavities as suggested by the JFO theory. The bubbles shown in Fig. 7 are generated as result of the motorised translation stage moving from top to bottom of the image. The distances on the vertical axis are taken from the point of minimum clearance. Therefore in the region, where the bubbles are formed, there is an increasing clearance between the bearing surfaces. The gap between the bubbles (the streamer) becomes smaller as the distance from the centre increases. This is in accord with the continuity of mass flow. The bubble thickness measurements are now considered in detail.

The paper has presented, for the first time, precise measurement of cavitation bubbles using digital holographic microscopy. The measurements are carried out for a high-viscosity lubricant intervening between cylindrical and sliding plane glass components. The benefit of using digital holography instead of incoherent imaging methods is that digital holography records both the phase and amplitude of the transmitted wavefront. Using this information, it is possible to make high-resolution measurements of bubble thickness and position.

The data from the phase images show that lubricant flows between the cavitation bubbles as predicted by the “multiple-cavity” assumption models, expounded by Swift–Stieber result and the JFO assumptions. Analysis of the high-resolution thickness measurements shows that the bubble thickness profile follows the curvature of the cylindrical bearing surface. Closer inspection, however, suggests that a thin film of lubricant flows over (or under) the cavitation bubbles as suggested by the “single-cavity” assumption of Birkhoff and Hays and Coyne and Elrod. Considering the pressure equilibrium between the inside and outside the bubble, it is clear that the bubble must be in contact with at least one of bearing surfaces. In this case, the results suggest that the bubbles are in contact with the cylindrical bearing surface. In conclusion, there is evidence for aspects of both the single- and multiple-cavity theories.

The experimental configuration allows for the control of speed and direction of the lubricated bearing surfaces. Using an argument based on symmetry, it was possible to measure the bubble formation position with respect to the minimum clearance position. A comparison with a selection of widely used cavitation models has been made, the results showing that the multiple-cavity theory (JFO) is the best prediction for the low load, low speed conditions as in the current study. It is noted, however, that the results are taken at steady-state conditions, the constant film thicknesses produced by these conditions vary considerably amongst the models presented. It is, therefore, concluded that the position of bubble formation should not be the only discriminant and cannot be solely used to validate the predictive models reliably. For this reason, further work is planned to make simultaneous measurement of bubble and lubricant film thickness.