## Abstract

Optimizing bearing performance is based on effective lubrication, especially in high-speed machinery, where minimizing churning and drag losses is of significant importance. Over the past few decades, extensive research has been conducted into the better understanding of different aspects of bearing lubrication. These investigations have employed a combination of experimental methods and advanced computational fluid dynamics (CFD) models. This article provides a comprehensive overview of critical aspects of bearing lubrication, with a specific emphasis on recent advances in CFD models. Lubricant flow and distribution patterns are discussed while examining their impact on drag and churning losses. An extensive discussion is provided on the meshing strategies and modeling approaches used to simulate various flow phenomena within bearings. In addition, relevant trends and impacts of cage design on bearing lubrication and fluid friction have been explored, along with a discussion of prevailing limitations that can be addressed in future bearing CFD models.

## 1 Introduction

The operational efficiency and performance of a rolling element bearing (REB) hinges significantly on its lubrication condition. A substantial proportion of bearing failures can be attributed to suboptimal lubrication practices [1–3]. To prevent oil starvation and maintain efficiency, it is crucial to ensure proper lubrication. This is particularly important for machine components such as high-speed gearboxes, drivetrains, and wheel bearings, where minimizing churning and drag losses is essential for optimal performance [4–6]. Predicting these lubrication-related losses in REBs is vital to evaluate the influence of bearing design and operational parameters on system performance [6]. Recent decades have witnessed an increased focus on analytical modeling to understand bearing lubrication and its impact on friction and efficiency. Therefore, this article undertakes a review of computational fluid dynamics (CFD) models developed to study various types of lubricant flow and distribution patterns observed in bearing lubrication.

The lubrication regimes within REBs can be broadly classified into elastohydrodynamic lubrication (EHL) and hydrodynamic lubrication (HL). Figure 1 shows that the EHL regime occurs between the ball–raceway contact, while the HL regime exists primarily between ball-cage contact [7]. The EHL regime, characterized by elastic deformations of solid surfaces under significant pressures (typically between 0.5 and 3 GPa), requires a two-way coupling between the fluid flow and solid deformation solvers in the CFD-based models, termed fluid–structure interaction (FSI) [8,9]. Conversely, the HL regime exists at relatively low contact pressures (usually less than 5 MPa), which allows the assumption that surfaces remain rigid and nondeformable [7]. This significantly simplifies the modeling effort, as the bulk lubricant flow and its distribution within the bearing chamber, which falls under the HL regime, can be effectively simulated using only the fluid solver, excluding the region between the ball–race contact.

For the contact between the ball/roller and inner and outer races (EHL), researchers have extensively employed numerical solvers based on the Reynolds equations [10], incorporating various considerations such as thermal effects [11–15], improved pressure–viscosity relationships [16], non-Newtonian fluid rheology [17–21], transient effects [21–26], point contact modeling [27,28], and surface characteristics such as roughness [29–31], asperities [32,33], and dents [34,35]. With advances in computing power, sophisticated FSI models have been developed utilizing CFD-based methods to solve the full Navier–Stokes equations for the fluid and a space discretization method to evaluate solid deformation. Hajishafiee et al. [36], among others [37], developed a CFD-based partitioned FSI solver for line contact thermal EHL problems, including non-Newtonian fluid rheology. They described some deviations from the results obtained with a Reynolds-based solver under extreme conditions. However, their modeling suffered from high computational cost, with an average runtime of 20 days. Singh et al. [9,38] introduced an efficient coupled FSI model for line contacts and observed the effects of solid-body phenomena such as surface dents and cracks, subsurface inclusions, and plasticity, with an average computation runtime of 15 h. In a similar vein, Peterson et al. [8] provided a robust framework for detailed investigations of heavily loaded, lubricated point contacts, and depicted lubricant velocity streamlines near the contact. These comprehensive models enabled not only the assessment of the lubricant flow into the converging geometry of the contacting bodies but also their localized elastic deformations, with significant improvements in computation time.

The flow behavior within a bearing involves a complex interplay of various physical phenomena, including flow recirculation, oil starvation, cavitation, aeration, and capillary effects. The simplest CFD approach to analyze bulk lubricant flow in bearings is the single-phase method, where oil is assumed to be the only non-solid phase [39,40]. Since these models exclusively use oil, the simulated domain is representative of a fully flooded bearing, and these are commonly referred to as the full-film models. These models allow visualization of flow patterns around bearing cages and additionally estimate fluid frictional losses in various REBs [39–42]. Notably, these single-phase models are computationally efficient and can be further improved by leveraging symmetry and periodicity to simulate specific angular sections of the REB, involving single balls or rollers. It has also been shown that employing periodicity with three rolling elements in bearing lubrication studies [40,42] can adequately capture the drag and shielding effects from the preceding and subsequent rolling elements, respectively [43–45].

Although single-phase models offer useful insights into the overall flow patterns within a bearing, they are incapable of representing actual fill levels in REBs. For instance, in the case of vertically oriented bearings, the recommended fill level in a sump-lubricated bearing is usually 50% submersion for the lowest rolling element [46,47]. In such scenarios, the chamber contains both oil and air, necessitating multiphase CFD models. Multiphase CFD models also provide critical information on lubricant distribution patterns and help predict lubricant starvation within the bearing cages [48–51]. To accurately capture these starvation effects, the volume of fluid (VOF) approach is commonly used due to its accuracy in capturing the oil–air interface [52]. Models based on this approach have been able to accurately capture oil–air striations (thin bands of oil and air) observed within bearing cage pockets [48–51]. However, in this method, the definition of the oil–air interface relies on the user-selected value of the volume fraction in the postprocessing step. This value can be chosen arbitrarily, and often a value of 0.1 is selected to delineate the boundary between oil and air. To address this variability, the coupled level set (CLS)—VOF scheme maintains a consistent value of zero at the oil–air interface [53,54]. In addition, meshless approaches have emerged in recent years [55], such as the discrete phase model (DPM) or moving particle semi-implicit (MPS) method that discretizes the fluid instead of the spatial domain [56]. Despite their applicability to simulate lubricant behavior in tribological systems such as gearboxes [57,58] and REBs [59–61], these approaches have limited accuracy in predicting power losses for most engineering applications [55]. Figure 2 illustrates the findings from a comparative analysis highlighting the increasing complexity of different CFD models and their corresponding runtimes for a sample case. The single-phase CFD model efficiently captures flow around rotating components, while VOF and CLS-VOF models offer more realistic transient flow distributions. However, the precision of the CLS-VOF model comes with significantly higher computational time, making it impractical for REBs. A more accurate approach involves transitioning between the VOF and DPM [62] approaches to model mixed flows in REBs within realistic timeframes.

CFD models provide notable advantages over traditional hydrodynamic solvers. Most commercial CFD solvers solve the complete Navier–Stokes equations with turbulence models [63] instead of relying on the approximations of the Reynolds equation. These CFD solvers also excel at handling complex model geometries, including intricate cage pocket shapes [39,42,49,50], while allowing the application of periodicity and symmetry boundary conditions, leading to more efficient solutions in realistic bearing operations compared to numerical lubrication models, which are often constrained to a single contact [64]. Such features cannot be incorporated into the hydrodynamic solvers or simplifications like Petrov's law [7]. The discrepancies between the experimental friction data, CFD model, and Petrov's law are depicted in Fig. 3. Of particular note is the tendency for Petrov's law to overpredict frictional torque, notably within cage pockets, where starvation occurs. It is limited to analyzing full-film lubrication, a scenario that is not reflective of actual operating conditions. Multiphase CFD models can predict the lubricant distribution and consequently give a more accurate estimate of the friction inside bearings in such cases. Furthermore, CFD models also offer improved accuracy in capturing the dynamics of various components, such as using the moving reference frame [40].

Within the bearing chamber, maintaining an optimum lubricant fill level is crucial in striking a balance between the risks of starvation and excessive churning losses. Figure 4 illustrates how the lubricant level conceptually influences fluid churning losses [49,66,67], heat dissipation [68–70], and wear [71] in the bearing cage. Lubricant flow in and around the bearing cages also determines shear drag losses due to contact with the cage surface, as well as the churning losses incurred during motion through excess or viscous lubricant. These losses contribute to overall bearing friction [72], emphasizing the importance of analyzing lubrication in REBs. A key advantage of using CFD models is the ability to predict component-specific frictional loss due to different bearing geometries [40,73–75], providing detailed insights beyond experimental measurements that only estimate the overall frictional torque [40,75]. Therefore, once validated, CFD models become powerful tools, reducing reliance on costly and time-intensive testing [76–78] and providing additional insights into bearing lubrication.

This article discusses the multifaceted aspects of lubricant flow within REBs, accompanied by a detailed discussion of recent advances in CFD modeling techniques that have facilitated the analysis of these intricate flow phenomena. Figure 5 depicts some noteworthy CFD studies that have analyzed bearing lubrication in recent years. Section 2 presents an overview of meshing strategies for developing efficient bearing CFD models. Section 3 investigates the effects of bulk lubricant flow and frictional losses associated with lubricant flow. Sections 4 and 5 discuss various studies concerning the distribution and starvation patterns of oil and models related to the dispersed flows in bearings, respectively. Finally, Sec. 6 highlights some common limitations in the existing CFD models and offers recommendations for addressing them in future models.

## 2 Meshing Strategies

Mesh generation is a pivotal step in any grid-based CFD model, as the element quality of the computational domain directly impacts the accuracy and efficiency of the simulation. For bearing CFD models, the optimal size and distribution of mesh elements are crucial to ensure model stability while minimizing computational effort. This is particularly important due to the significant variations in length scales encountered within a bearing's geometry. Rollers and raceways typically fall in the centimeter or sometimes in the meter ranges, while the clearances between these components and the cage are usually quite small and can be as small as microns. To address this challenge, simplifications to the geometry domain and novel meshing strategies can be employed to develop efficient meshes that capture relevant flow features without excessive computational effort.

Peterson et al. [39] adopted a well-established meshing technique for their deep groove ball bearing (DGBB) model. They reduced the ball diameter by 5%, which increased the minimum clearances, thus eliminating the need for a resource-intensive high-density mesh. Their mesh sensitivity analysis demonstrated an 80% reduction in computational effort while maintaining accurate drag calculations (within 10%). Figure 6 depicts a similar mesh convergence study performed by Aamer et al. [48] to assess the most efficient mesh by varying the number of mesh elements across the smallest gap. In addition, leveraging the cyclic symmetry of bearing geometries enables the use of a reduced domain [40,42,50,79], or if a bearing has a plane of symmetry perpendicular to the rotational axis (e.g., for a DGBB), symmetry simplification can also be implemented. Concli et al. [79] divided the computational domain into five regions for a cylindrical roller bearing (CRB), with independent mesh generation, allowing granular control over seeding of quadrilateral elements. This approach led to an 84% reduction in the total number of mesh elements, with negligible impact on computed power loss. Feldermann et al. [73] presented a novel mesh mapping method involving information transfer between a course and fine grid model. Initially, a coarsely meshed model was created neglecting the clearances between the rolling elements and the raceways. Using a rotating reference frame, the computed flow field was transferred to a densely meshed model representing the space between adjacent rollers, which facilitated an efficient and accurate simulation.

Selection of mesh element type is another important consideration for stable model development. Concli et al. [79] utilized a series of structured meshes with quadrilateral elements. Although a highly structured mesh is desirable, complex features within the bearing chamber, such as cage pocket shape (varying curvatures, protrusions, and grooves) and raceway flanges, can introduce challenges. Moreover, an inflation layer, which is a group of fine-structured mesh elements along the boundaries, can be utilized to enhance resolution in critical areas as depicted in Fig. 7. Tetrahedral mesh elements allow greater flexibility in meshing around geometric features while retaining sufficient mesh quality [39,40,48]. More recent advancements in meshing have enabled the use of polyhedral elements [42,49,50,80] that provide a good balance between an unstructured volume mesh and highly structured inflation layers along surfaces [81,82]. Figure 7 presents a comparison between the two mesh element types for a representative angular contact ball bearing (ACBB) geometry, highlighting the ability of the polyhedral mesh to increase mesh elements in critical clearances while minimizing excessive elements in less critical regions.

The aforementioned investigations exemplify the static meshing approach, assuming a fixed mesh topology throughout the model runtime, with imposed boundary conditions to simulate kinematic boundary conditions. A rotating reference frame can be incorporated to account for apparent forces such as Coriolis and centrifugal forces that alter fluid motion in rotating components. However, dynamically updating the mesh topology becomes crucial for scenarios that require computational domain modification over time [55]. Innovative meshing strategies like the sliding mesh and dynamic remeshing approaches offer more realistic representation of bearing component kinematics. Peterson et al. [39] introduced a sliding mesh method, employing distinct mesh zones for each rolling element with individual rotational velocities, enabling simulation of fluid churning in a bearing. Furthermore, the cage reference frame was assigned a rotational velocity with respect to the absolute reference frame, effectively capturing both the rolling movement of each element and their revolution around the bearing center. However, this approach faces limitations in solving problems that require geometric changes in the computational domain, such as simulating the motion between ball and cage [83] or solid-body deformations.

Here, $Vcell,min$ is the volume of the smallest cell in the computational domain and *U* is the characteristic velocity of the problem. The choice of time-step size plays a critical role in ensuring computational stability and efficient runtimes, particularly under transient and high-speed conditions. In addition, adaptive time-stepping can be implemented to adjust the time-step size as the calculation progresses, rather than employing a fixed time-step for the entire computation. Different meshing strategies are needed to provide varying levels of geometrical complexity, and choosing an appropriate strategy is crucial for accurate and efficient solutions. This choice should align with the underlying physical principles to ensure reliable results.

## 3 Oil Flow and Frictional Losses

The flow of lubricant within the bearing chamber can result in parallel fluid layers without intermixing (laminar flow) or in disorderly or irregular flow patterns (turbulent flow) [86]. The specific flow pattern within the bearing depends on its operating speeds, lubricant properties, and, in particular, cage designs [39,42]. At higher speeds, the entry of oil into the contact area between the ball and the race can be impeded by the development of vortices near the ball as well as the airstreams at the side of the bearing [76,87]. This is why inlet and outlet regions of the bearing are commonly included in CFD models. Single-phase CFD models are powerful tools for analyzing such flow patterns and evaluating fluid frictional losses. These losses represent a significant portion of frictional losses in automotive applications [6,88–90]. Moreover, analyzing the oil flow field assists in identifying potential sources of windage losses within bearings [91].

Recent research trends have emphasized the importance of validating CFD models through experimental measurements such as using techniques like particle image velocimetry (PIV) [92,93]. Richardson et al. [94] used µ-PIV to capture lubricant recirculation vortices within a pocketed thrust bearing, similar to Wang et al. [95]. Recently, Maccioni et al. [96,97] developed a test rig to visualize lubricant flow inside a tapered roller bearing (TRB), mapping the lubricant velocity field in the cavity between the outer race and the cage at different speeds using PIV measurements. To understand the bulk flow patterns in a fully submerged bearing analytically, researchers have used the single-phase CFD approach to study bearing flow patterns for both horizontal [42] and vertical bearing orientations [40].

Early investigations used single-phase CFD models to analyze flow fields in journal [98–100] and thrust bearings [101]. Marchesse et al. [43,44] developed a single-phase model for a straight bearing configuration with three balls in tandem to analyze the flow patterns and drag losses for air as the only liquid phase. Gao et al. [102] and Wei et al. [103] also used a similar approach as previous researchers [43,44,104,105]. They illustrated the cooling effects caused by air flow on the bearing cage in the absence of any lubricant. While these models provided fundamental insights into the airflow patterns in REBs, it is important to recognize that the operational conditions of REBs typically do not entail completely dry environments. Therefore, employing CFD models with oil, rather than solely air, is more appropriate for effectively simulating bearing lubrication. Further research in bearing studies has continued for a better understanding of oil flow within the bearing chamber, as exemplified by the work of Peterson et al. [39]. They developed a single-phase CFD model to visualize the velocity streamlines and pressure contours over the cage surface for different oil properties and operating conditions, as shown in Fig. 8. They observed vortices near the inner and outer raceways, as well as vortex shedding near the ball surface, consistent with the findings of Yan et al. [87]. These phenomena also led to significant pressure concentrations in the converging region of the ball and outer race, as depicted in Fig. 8. Similarly, Lee et al. [106] used a single-phase CFD model to investigate the flow field for an aeroengine bearing chamber, which showed good agreement with their PIV measurements. Arya et al. [42] presented a comprehensive analysis for studying the bulk lubricant flow between the cage and raceways. They used bubble image velocimetry (BIV) and achieved both qualitative and quantitative validation of their single-phase CFD model. Their results demonstrated the significant impact of cage design on flow patterns, with smoother-walled cages showing laminar Λ-shaped flows, and cages with side protrusions inducing recirculating vortices, as illustrated in Fig. 9.

Extensive experimental and single-phase CFD studies have been conducted to measure friction in bearings [107–111]. Figure 10 illustrates the speed-dependent torque loss induced as a function of speed from various published works. The Schaeffler [112] and SKF [113] friction models are commonly used for estimating bearing friction for various bearing types and loading conditions [73,114,115]. These studies have broadly characterized friction in bearings as load dependent and speed dependent [74]. Load-dependent losses primarily arise at the roller-race contacts, which operate in the EHL regime. Numerous investigations have sought to characterize contact friction as a function of load and relative speeds [116–119]. On the other hand, speed-dependent losses mainly result from fluid drag generated by the orbital motion of rolling elements. These speed-dependent losses can be further categorized into (i) churning losses, caused by the shear stress of the lubricant, and (ii) drag losses, resulting from the displacement of the lubricant [73,120–122]. The empirical model introduced by Palmgren [110] among the earliest in the field and established the foundation for many recent studies [107,109,111] and modern standards [123]. However, these investigations were unable to reach an agreement on the contribution of drag losses to overall bearing friction. While some studies suggested that these losses were negligible [107,111], others predicted a more substantial contribution [113,124,125]. In addition, developing accurate empirical models often requires validation against experimental data across various bearing types, sizes, and operating conditions, limiting their generality.

Single-phase CFD models offer a valuable alternative to empirical models and several researchers have developed CFD models for analyzing frictional losses in various bearing types, such as DGBBs [40], TRBs [74,75], ACBBs [42], CRBs [73], and radial needle roller bearings (RNRBs) [40]. Feldermann et al. [73] characterized fluid drag losses for a CRB using a CFD model that aligned well with experimental data and the SKF model [113] up to speeds of 2000 rpm. Similarly, Liebrecht et al. [75] validated their CFD tool for TRBs, highlighting negligible drag contributions from the cage compared to raceways, similar to Ref. [73]. Gao et al. [102] investigated the impact of parameters such as rolling spacing and speed on churning losses using a simplified CRB geometry. Peterson et al. [40] employed an equivalent fluid model to investigate drag losses in DGBBs and RNRBs, achieving good agreement with their experiments at lower viscosity, but overestimating drag at higher speeds with more viscous oil. The development of these validated CFD models has provided a detailed estimate of fluid drag losses in REBs, exceeding the capabilities of empirical bearing friction models. However, it is essential to emphasize that single-phase CFD models cannot capture lubricant distribution in scenarios with lower fill levels, which are commonly encountered in actual bearing operations.

## 4 Oil Starvation

During operation, REBs are prone to oil starvation due to inadequate lubricant replenishment [76,126]. This critical deficiency can drastically alter the bearing operational characteristics [66,127], leading to excessive heat generation, vibrations, and accelerated wear [128]. Moreover, oil distribution within the cage pocket directly impacts the bearing cage friction, as well as dynamic motion of the bearing cage, due to its damping and stabilizing effects [66,129]. Inadequate lubrication increases cage friction and disrupts its dynamic motion, compromising its guiding functions and can lead to premature failure due to the excessive force exerted by the rolling elements [71]. Multiphase CFD modeling has enabled accurate prediction of oil distribution in bearings, offering insights to address these challenges.

Oil starvation in hydrodynamic rolling contacts often manifests as periodic “finger” patterns of air striations separated by narrow and continuous oil streams, as observed in various experimental investigations [130–136]. For a long time, ball-on-disc rigs have been the simplest tribological contacts for analyzing oil starvation in rolling contacts [137–140]. Figure 11 demonstrates the different oil film patterns between the ball and disc where the gap was maintained at 1 *µ*m. While these studies shed light on the underlying oil starvation phenomenon, the conclusions drawn from them may not directly apply to all REB contacts due to the geometric and dynamic constraints imposed by other components [141,142]. Damiens et al. [143] validated this distinction by observing a significant change in the oil distribution with the presence of a cage segment around the ball in a ball-on-disc rig.

To accurately simulate the rolling contact observed in a REB, Molina and Gohar [144] studied the lubrication in REB cage pockets using an isolated ball-cage segment. Their work primarily focused on experimental friction measurements and numerical modeling that specifically considered hydrodynamic friction at ball-pocket contact, while neglecting the crucial role of oil starvation. Houpert [145] explored roller-pocket contact and emphasized the substantial impact of incorporating lubricant starvation effects on cage pocket friction. Similarly, Gentle and Pasdari [146] integrated a “starvation factor” into their hydrodynamic friction model, which was considered essential for aligning with their experimental friction data [147]. Several noteworthy contributions have emerged from experimental studies [49,64,65,67], which observed a direct correlation between the quantity of oil within the cage pocket and the pocket friction torque. Furthermore, these studies identified the phenomenon of kinematic starvation [48,50,64,128,148], which is the speed-dependent loss of oil as depicted in Fig. 12. It should be noted that the phenomenon of “lubricant starvation,” characterized by the partial absence of lubricant in hydrodynamic contact [64,149,150], is different from “starved lubrication,” which is characterized by asperity contact [150,151] and usually occurs in mixed or boundary lubrication regimes, as shown in Fig. 1.

Multiphase CFD modeling has emerged as a powerful tool for analyzing parameters that were challenging to grasp through experiments alone. A number of researchers have employed this technique to investigate how oil starvation manifests and impacts lubrication under various operating conditions [152–156]. Aamer et al. [48] observed the formation of oil–air striations inside an ACBB cage pocket, as shown in Fig. 13(a). These striations serve as visual indicators of oil starvation, and their width was found to vary with the bearing speed and pocket clearance. In another related study, Aamer et al. [49] also analyzed the effect of cage conformity on lubrication within the cage pocket of a CRB (Fig. 13(b)), highlighting the benefits of higher cage conformity for oil retention and reduced recirculation. These studies, while employing simplified isolated geometries, provided fundamental insights into the lubrication mechanisms within the cage pocket, laying the groundwork for further exploration of oil starvation in more complex REB simulations.

*k*is the curvature of the fluid interface, $\rho $ and $\mu $ are the equivalent density and viscosity of the oil–air mixture, respectively, given by:

The aforementioned studies employed a ball-on-disc configuration [137–143] or isolated cage pocket geometry [49,64,65, 67,144,147]. However, other researchers have used the actual bearing geometry without adopting these simplifications. This facilitated accurate consideration of the impact of the bearing raceway's rotation [158] and the effects of various cage designs [159]. A major factor that influences the oil distribution within a bearing chamber is its orientation, whether horizontal [42,50,96,97,160] or vertical [40,51,79,155,158,161,162]. In horizontally oriented bearings, where gravity's direction aligns with the rotational axis of the bearing, oil flow and distribution maintain symmetry around the circumference of the bearing [50], as depicted in Fig. 14(a). Periodic CFD models can be employed in such cases, e.g., Arya et al. [50], who presented an experimentally validated multiphase CFD model that depicted the oil starvation patterns within various cage pocket geometries of an ACBB. They emphasized the influence of the cage geometry on oil starvation inside the cage pocket and also observed that the lubricant was scraped from the edge of the cage, similar to Refs. [143,159], which resulted in spiraling striations on the ball surface.

Vertically oriented bearings, however, pose a different challenge due to gravity acting perpendicular to the rotational axis. This results in a nonuniform and nonrepeating distribution of oil around the bearing, necessitating a full bearing multiphase CFD model for accurate predictions, as shown in Fig. 14(b). Shan et al. [162] demonstrated such a model and observed a variation in oil distribution around the bearing circumference similar to Concli et al. [79]. Wu et al. [59,60] used the MPS method and observed a reduction in oil volume near the ball and cage at high speeds, with a higher concentration on the outer race, attributed to the centrifugal effect. Chen et al. [51,163] developed a periodic multiphase model to study the oil breakup in a vertically oriented ACBB. They observed a decrease in oil volume on the raceways with the capillary number, consistent with comparable findings reported in other published studies [139,153,154]. However, it is critical to note that utilizing periodicity in multiphase CFD modeling for a vertically oriented bearings is only practical at higher speeds $(dmN>1million)$, where centrifugal forces outweigh gravity. At lower speeds $(dmN<1million)$, the nonuniform and nonrepeating distribution of oil in the vertical orientation necessitates the utilization of full bearing geometry, as depicted in Fig. 14(b).

It is also noteworthy to mention that the dynamic interaction of the ball and cage during actual bearing operation also impacts oil flow and its distribution. These aspects were not considered in prior CFD studies. Deng et al. [76] and Liu et al. [83] addressed this gap by developing coupled models that integrate nonlinear dynamics of bearing components with CFD. These models predicted the fluid flow behavior for various loading scenarios, revealing dynamic changes in oil distribution in the cage pocket due to ball and cage collisions. Furthermore, some studies have addressed variations in the bearing chamber temperature with changing lubricant distribution [70,162]. This is prevalent in oil jet-lubricated bearings [76,77,105,156,162,164–166], where the maximum oil volume fraction is found near the jet inlet and decreases along the circumference, resulting in an elevated bearing temperature away from the jet. Despite the successful characterizations of multiphase oil–air lubrication in several studies, the agitation from bearing operations frequently introduces mixed flows and additional dispersed phases, particularly in high-speed conditions, requiring dedicated computational approaches.

## 5 Dispersed Mixed Flows

Commonly used lubricating oils for REBs typically consist of nearly 8–12% dissolved air [167], depending on the type of oil, viscosity, pressure in the system, and type of additives. As the lubricant churns within the tight clearances of the bearing chamber, this air interacts with the oil, forming an aerated mixture. This dynamic interplay of oil and air leads to a frothy oil mixture characterized by distinct bubbles [168] and is significantly influenced by the agitation and splashing of lubricants during high-speed operations [42,169]. In addition, the lubricant can also fragment into dispersed droplets or mist, generating mixed flows. It is important to understand the behavior of these dispersed phases, as they not only affect lubricant performance [170] but also induce changes in their physical properties [171–173], load capacity [174,175], heat dissipation, and bearing operating temperature [68,176,177].

Recent experimental investigations have documented mixed flows in bearings [178–180], observing the emergence of lubricant droplets and ligaments resulting from the shedding of oil film. Maccioni et al. [96,97] used a sapphire outer race for a TRB to visualize the occurrence of lubricant aeration at inner race speeds above 900 rpm, which altered the velocity field. Arya et al. [42] used a high-speed camera and transparent ACBB cages to observe the motion of the aerated bubbles using BIV, as shown in Fig. 15. The authors also noted increased aeration at higher bearing speeds. Aamer et al. [49] also observed chaotic lubricant motion and aeration within CRB cage pockets. However, accurately simulating the physics of these dispersed oil and/or air particles with multiphase CFD models adds another layer of complexity.

Intricate bearing CFD models have emerged from the efforts of researchers to simulate mixed flow in bearings using a coupled DPM and VOF approach [184,190]. These investigations simulate oil–air interface features as well as the motion of dispersed particles. Some studies [61,191–195] have also examined the dynamics of oil droplet movement and the variation of droplet diameters within the bearing chamber using this approach. These investigations revealed key factors that influence the dispersion patterns of oil particles, including shaft speed, angular position of oil particles within the bearing, and different flow regimes. Furthermore, a dedicated mechanism allows transitions between VOF and DPM, particularly when the interfaces break down into distinct droplets [196–199]. The transition between these two approaches can occur as (i) a one-way transition, where only the VOF phase transitions to the DPM phase resembling the onset of aeration (where air escapes in the form of bubbles), or the breakup of oil into fine particles (or mist); or (ii) a two-way transition, where repeated transitions occur between them, allowing the prediction of the steady-state number of the dispersed phase. Previous studies [196–199] using this approach have effectively depicted complex flow structures characterized by dispersed phases resulting from the breakup of interfaces [184]. In particular, Dick et al. [62] applied this algorithm for the simulation of mixed flow in a ball bearing chamber. The authors observed finger-like oil patterns near the front wall of the bearing cavity, as shown in Fig. 16. However, there is a lack of dedicated multiphase models that study aeration and various other important characteristics of mixed flows in REBs, which is particularly important for high-speed conditions. Although multiphase CFD approaches accurately represent actual operating conditions for most practical REBs, the substantial computation time required to achieve a steady-state solution presents a major drawback (as illustrated in Fig. 2) and hampers their widespread adoption for various operating conditions.

## 6 Limitations and Future Directions

Current CFD models have successfully replicated experimentally observed lubrication phenomena, including oil flow around the bearing cage and its distribution near the cage pockets, along with the associated effects on bearing drag losses. Nevertheless, a few limitations persist with the current modeling techniques.

### 6.1 Simplifications and Assumptions.

With the available computational resources, certain simplifications are imperative to have efficient runtimes. Models with a reduction in ball diameter to 95% of its original size fall short in accurately predicting flow patterns, lubricant distribution, and friction in regions with enlarged gaps. Similarly, most published works fail to meet the recommended mesh resolution of at least ten cells in regions with boundary layer flows [63], observed in small gaps in REBs. Future advances in computing may provide a scope for higher mesh resolution, obviating the requirement for such simplifications.

Most CFD models discussed so far have assumed rigid walls between the ball and cage, as well as between the ball and raceways. This assumption neglects both the elastic deformations of balls and raceways (EHL) [8], as well as the viscoelastic deformations of cage material (soft EHL) [200]. Furthermore, the lubricant flow in the current bearing CFD models is also assumed to be independent of the material properties of the solid components. Different bearing components exhibit varying degrees of elasticity, and particularly bearing cages [65,201,202], which are not considered in existing CFD models. This oversight also overlooks the influence of cage material and inertia on its dynamic whirl behavior [201] and state [203], which in turn affects the flow within the bearing chamber [76,83]. This becomes particularly significant during high-speed operation, where there is a potential for cage expansion [203]. To properly account for the effect of deformable cage material, a fluid–solid coupling is needed. Therefore, integrating material effects into future bearing CFD models will offer a deeper understanding of these phenomena.

Another important aspect often overlooked in bearing CFD modeling is the thermal behavior of the lubricant during operation. While researchers aim for their experimental setups to operate under isothermal conditions by employing in situ temperature measurements, it is anticipated that shear heating of the oil between the cage and raceways [204] and the local heating surrounding bearing contact points [205,206], especially near ball–race contact, may lead to variations in the lubricant properties compared to the assumed values at the operating temperature. In addition, thinning of the lubricant at elevated temperatures can also lead to inadequate lubrication in these contacts. Therefore, it is critical that future CFD models consider appropriate thermal effects to account for these localized temperature variations.

### 6.2 Surface Properties.

The surface finish of a material has been identified as a significant factor influencing its adhesion and lubrication. Flow visualization studies often use transparent bearing components made from acrylic, which can cause differences in surface wetting as compared to actual bearing components made of steel [207]. For bearing applications, the surface finish can be altered by applying different types of coatings that can affect the lubrication and friction performance of the interacting surfaces [208–211]. Researchers have investigated the impact of incorporating oleophobic and oleophilic coatings on bearing components [159,208], revealing their ability to affect lubricant availability between ball–race contacts [212,213]. Similarly, studies have also shown that introducing surface features [214] or increasing surface energy can improve lubricant adhesion [215,216]. Consequently, modeling the effects of surface finish in future multiphase bearing CFD models holds significant potential in improving accuracy for simulating lubricant adhesion to bearing surfaces.

### 6.3 Grease Lubrication Modeling.

Despite the prevalent focus on oil lubrication in bearing CFD studies, nearly 90% of real-world applications rely on grease lubrication [217,218] due to its distinct advantages over oil lubrication [219]. Grease is a multicomponent lubricant that comprises base oil, thickener, and additives and exhibits bearing friction that is primarily dependent on the type of base oil [220,221]. During operation, grease undergoes churning and bleeding phases, with a significant portion being expelled from the bearing in the initial stages, ceasing active participation in lubrication [217,221]. The understanding of grease flow is complicated by a lack of consensus in the community and its nonlinear rheology [217,218]. Figure 17 presents the findings of the investigation by Noda et al. [222] who performed X-ray-based inspection of the grease flow behavior inside a ball bearing and highlighted the influence of cage shape on lubrication. Despite several experimental attempts to capture grease lubrication in REBs [222–227], analytical investigations through validated CFD modeling are notably absent [228,229]. Incorporating the nonlinear rheology of grease presents a potential avenue for future multiphase CFD studies.

## 7 Summary

This article reviews the experimental and analytical insights for various lubrication trends in REBs, emphasizing the significance of recent advancements in CFD modeling for studying these trends. It covers the advanced meshing techniques used by various researchers to develop efficient CFD models and discusses various models developed to evaluate fluid frictional losses arising from the lubricant flow within the bearing chamber. Furthermore, it investigates different intriguing trends for oil flow and starvation patterns, utilizing different single-phase and multiphase CFD approaches that have surfaced in recent years. The article also highlights several coupled approaches that simulate mixed flows in bearings, laying the groundwork for future lubrication models capable of simulating the impact of aeration and mixed flows.

Varying levels of flow information can be obtained with different CFD approaches, and through a comprehensive review of various CFD modeling techniques, the article suggests adopting an appropriate modeling approach depending on the complexity of the desired information about the lubricant flow. Single-phase CFD models are beneficial for evaluating different design parameters based on bulk flow characteristics and fluid drag losses, whereas multiphase CFD models are useful when simulating more realistic lubricant fill levels and mixed flows in the bearing chamber, albeit at the higher cost of computational efforts. Enhancements in meshing strategies can help mitigate the computational cost associated with these simulations. The article also discusses the common oversights in current CFD models, encouraging the development of more sophisticated methods that address these shortcomings. While current models capture most experimentally observed phenomena in REB lubrication, future models can address prevailing limitations, which will lead to the development of more robust and reliable CFD models, potentially minimizing the need for expensive test rigs to understand the intricacies of bearing lubrication.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

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