CFD simulation of power losses and lubricant flow in gearboxes

In the last years, efficiency has become an increasingly crucial factor across various industrial sectors. The automotive industry, for instance, is requested to develop drivetrains that are not only economically efficient, but also environmentally friendly and reliable. Being able to predict the efficiency and the lubricant-behavior in gearboxes remains an imperative engineering challenge. Existing mathematical models available in literature rely on empirical relations and dimensional analysis, providing accurate results only within narrow operating ranges.

A comprehensive approach capable of precisely predicting lubricant flows and power losses in geared systems could significantly advance the field. Thanks to recent evolutions in computer science, computational fluid dynamics (CFD) has emerged as a crucial tool for engineers studying gear lubrication and efficiency. Nevertheless, the widespread adoption of CFD has been hindered by the substantial computational resources required for simulations.

The implementation of a computationally efficient mesh-handling strategy, along with the development of advanced solvers capable of addressing new phenomena such as cavitation, aeration, oil suspension, or unconventional lubrication (e.g., non-Newtonian fluids), has made this technology ready for an extensive industrial application. Compared to a decade ago, the computational effort has been slashed by 97%, enabling the simulation of complex systems within a few minutes.

This article presents application examples across various gear types, showcasing the versatility of the developed approach. Additionally, it shows some real case studies: an industrial multi-stage and a planetary gearbox. This effective and computationally efficient approach has enhanced the understanding of the physical phenomena involved in gearbox lubrication, providing theoretical explanations for experimental observations that were previously challenging to interpret.

1 Introduction

Energy efficiency stands as one of the pivotal drivers of the last decades in many different scenarios, from the building industry to the automotive sector. Power transmission and gears are among the most widespread components in mechanical systems; therefore, their efficiency has a wide impact in terms of energy savings at global level.

Power losses within gearboxes are related to various phenomena. Typically, such losses could be categorized based on the machine element responsible for their occurrence according to their dependence or independence on the transmitted load. While analytical and empirical models are the best choice for the quantification of some losses, particularly the load-dependent ones, accurately predicting load-independent losses, such as those arising from gear-lubricant interaction, proves challenging with conventional models. Hence, the availability of advanced numerical tools (capable of reliably predicting all the gearbox losses) during the design phase can significantly bolster efficiency improvement endeavors.

Computer simulation, particularly leveraging on computational fluid dynamics (CFD), emerges as a viable approach to address the engineering challenge of accurately predicting load-independent power losses. Additionally, it offers insights into lubricant flows within the gearbox and, through the evaluation of the proper lubrication of all systems’ components, the capability to achieve the required reliability. However, applying CFD to gears poses challenges due to the topological variation of the computational domain during the engagement (the actual computational domain is the volume occupied by the air-lubricant mixture), complicating mesh handling.

Numerous approaches have been proposed for applying CFD to gearboxes, differing in terms of accuracy and computational efficiency. Building upon previous studies, this work presents the application of CFD to gearboxes using the original global remeshing approach (GRA) developed by the author [1] and its improved version, the GRA^MC (GRA with mesh clustering) by Mastrone et al. [2-4], enabling accurate predictions within relatively short simulation times.

The GRA^MC was developed in the open-source environment OpenFOAM^®. With well-defined user interfaces and data preprocessing, the developed mesh handling strategy can be effectively tailored for any gear type [5-7] and gearbox architecture [3,8-12], enabling its simple application by gear designers.

Besides an effective mesh handling strategy that paves the way for fast simulations, specific solvers have been implemented to properly model the physical behaviors of any lubricant/lubrication condition. These include solvers for immiscible incompressible isothermal fluids, capable of modeling the oil bath lubrication, solvers that can take into account phenomena such as cavitation [13] (vaporization of the lubricant due to a sudden pressure drop), aeration [13-15] (entrapping of microscopic air bubbles into the oil sump), solvers to model the oil suspension (aerosol) that took place in the presence of oil injection (jet lubrication), and high speeds as well as solvers for modeling non-Newtonian fluids [16] (grease lubrication).

After briefly presenting the working principle of the GRAMC mesh handling strategy and the implementation of the different solvers, some examples of application of the developed tools are presented. For most of the cases, the numerical predictions are compared with experimental data obtained from laboratory tests to highlight how closely the CFD simulation results are with respect to the experimental findings, both in terms of power loss and flow distribution.

Moreover, the article shows how the simulations can offer insights into the origin of losses, encompassing phenomena such as churning, windage, pocketing/squeezing, aeration, cavitation, and oil-suspension, which provide rationale for otherwise perplexing experimental observations. A comprehensive understanding of losses and their dependency on specific effects in each application case serves as a valuable input for devising strategies to mitigate them effectively.

Finally, a paragraph is dedicated to the evaluation of the computational performances of the present approach in comparison with the state-of-the-art offered by most of the commercial software.

2 Power losses in gearboxes

The dissipation of power in a gearbox arises from various mechanical components. In this regard, the losses can be categorized into those associated with gears (subscript |_G), bearings (|_B), seals (|_S), and other elements (|_X) such as clutches and synchronizers. Specifically, gear and bearing losses can be further classified as load-dependent and load-independent (|₀). Load-dependent losses correlate directly with transmitted torque, arising from friction between components or, in cases of oil lubrication, from shear within the oil film due to sliding. Load-independent losses, on the other hand, stem from the interaction between lubricant and mechanical parts [17]. Load-independent losses of gears — the focus of this article — can be further broken down into churning (P_LG0,C), windage (P_LG0,W), and squeezing/pocketing losses (P_LG0,S). Windage differs from churning in that it involves a single-phase interaction, while churning involves a multi-phase fluid. Windage is notable in large grease-lubricated gears or high-speed gearboxes with injection lubrication (where it can be assumed that the interaction with the air is responsible for most of the independent losses). Squeezing — or pocketing — occurs due to rapid volume changes between meshing gear teeth, leading to axial flows of lubricant and resultant losses due to viscous effects, typically of lower magnitude than churning or windage.

Load-independent losses are also present in bearings, where standard equations from manufacturers sometimes suffice [18] due to their more uniform geometry. However, for gears, which experience losses highly influenced by specific configurations and geometry of the housing, analytical or empirical equations prove ineffective, necessitating numerical methods for accurate results.

3 CFD Application to lubrication of gearboxes

3.1 Theoretical background

Following the preceding discussion, determining the load-independent power losses of gears necessitates understanding lubricant behavior under operating conditions, which entails solving internal fluid dynamics. An analytical approach to this problem is not feasible for gearbox lubrication due to the complex shape of the domain defined by the moving gear, shaft, and bearing surfaces. Additionally, differences in the shape of the housing, which may differ significantly between different gearboxes, profoundly influence lubricant flow and consequent power losses. The volume occupied by the fluid within the gearbox continuously changes during operation, with portions occupied by oil, air, and their mixture altering dynamically. Consequently, only numerical methods, such as computational fluid dynamics (CFD), can yield reasonable results. Presently, as noted in [12], particle-based models are suitable for qualitatively describing lubricant flows within gearboxes, whereas finite volume (FV) approaches are recommended for numerically evaluating losses. FV methods requires volume subdivision into cells and the numerical solution of governing equations for mass and momentum conservation.

The behavior of the fluids must be enforced within each discretized cell of the computational domain. The predominant numerical approaches involve employing techniques based on a PIMPLE (merged PISO- SIMPLE) algorithm. While the SIMPLE algorithm was originally developed for steady-state conditions and lacks time information, the PISO algorithm, designed for transient simulations, conserves time but demands smaller time steps for convergence, thereby increasing computational requirements and time required for completion. The PIMPLE algorithm operates primarily in SIMPLE mode for iterations, transitioning to PISO mode only in the final iteration. This allows for a stable solution without loss of information, while maintaining reasonable computational efficiency.

Before moving on, it is necessary to distinguish between separated flows and dispersed flows. Separated flows are characterized by the fact the phases are contiguous throughout the domain and there is one well-defined interface. On the contrary, dispersed flows are characterized by non-contiguous isolated regions (Figure 1).

The most common example of separated flow is represented by the oil bath lubrication where the free surface is clearly visible. Examples of dispersed flows are both lubrication in presence of small-bubble aeration (air trapping into the lubricant, i.e., foaming) or oil suspension (typical of oil jet and spray lubrication).

In case the phases are separated, it is common practice to exploit phase tracking methods such as the volume of fluid (VOF) [19] to reconstruct the shape and position of the free surface.

For single phase simulations (windage) as well as separated phases (churning), one continuity and one momentum conservation equations are solved.

Where:

ρ = density

→U = velocity vector

μ = viscosity

S_U = external forces

For applications involving multiple phases, an additional scalar quantity, known as the volume fraction (γ), is introduced to indicate the share of the phases in each cell. γ is calculated for each cell through an additional balance equation.

After the calculation of γ, the properties of the fluid mixture in the cell are computed with an average weight (γ) from those of the different phases.

Where:

φ = generic property to calculate (density, viscosity, etc.)

Subscript |_g = gas

Subscript |_t = liquid.

Phenomena such cavitation could be considered by adding a source term to mimic the phase change rate to the mass conservation equation [20-22]. Similarly, aeration could be modeled by adding a source term to the continuity equation [15].

While this method is very effective for simulating the churning phenomenon, it fails in case of dispersed flows. The main reason is related to the fact the smaller feature that can be captured (e.g., an oil droplet or an air bubble) depends on the grid scale (Figure 2). While the methods based on interface tracking can provide accurate results if the mesh size is reduced to the scale of the smallest feature (the oil droplets diameter or air bubbles, which could be in the order of magnitude of a few μm), with the present computational power is already difficult simulating a domain of the size of a dice — its application is therefore not compatible with the industrial practice for the simulating lubrication conditions that are not the basic splashing/churning (where the mesh size could be in the order of the mm).

For this reason, in presence of dispersed phases, an alternative approach is the adoption of a Euler-Euler modeling [23]. In this framework, both phases are described using Eulerian conservation equations. Each phase is treated as a continuum, each inter-penetrating each other, and is represented by averaged conservation equations. The velocity of each phase is represented by one set of velocity vectors. Due to the loss of information associated with the averaging process, additional terms appear in the averaged momentum equation for each phase. To be able to solve the problem, additional equations should be included (closure).

The most common mediation techniques are temporal, volumetric, and ensemble averaging. The mediation process can be carried out at a volume which size remains bigger than the typical dimension of the features of the dispersed phase (e.g., the oil droplet diameter) solving therefore the computational limitations. In this way, however, all information on a smaller scale than that on which the mediation is performed is lost. To account for this problem, sub-models are introduced. They are required mainly to include the drag effect of the continuous phase (i.e., air in case of oil suspension) on the dispersed one (lubricant).

The drag force depends on both the surface friction due to shear stresses at the surface and the shape determined by the non-uniform distribution of pressure due to motion. It can be expressed according to the law:

where C_D,k is the drag coefficient (that can be estimated with analytical relations [24], A is the surface, and →u_r is the relative velocity. Considering the dispersion of the oil in the air stream promotes the formation of spherical droplets, the area can be calculated starting from the droplet diameter d_b,k.

This additional drag term is included in the momentum conservation equation for the dispersed phase (lubricant). In the Euler-Euler approach, in fact, separate equations of conservation of mass and momentum are written for each phase. The subindex |_N refers to the primary phase (the continuous one), the subindex |_k refers to the k^th dispersed phase.

3.2 Mesh handling

The dynamic changes in volume geometry during gearbox operation, driven by gear meshing cycles, pose a significant challenge in simulating internal fluid dynamics. This necessitates frequent mesh updates to accommodate such topological modifications. Particularly, the evolution of the volume in areas of gear teeth meshing leads to severe degradation of volume elements, causing solution instability.

Commercial software approaches commonly used are ineffective for gears. For example, the widely adopted method of mesh smoothing, based on replacing elements that fail certain criteria, is not satisfactory (Figure 4a). While it can handle significant topological changes, it often results in smaller elements, requiring reduced time-steps and increased computational effort [25, 26].

In response to this problem, the authors proposed an original solution involving complete mesh substitution after several iterations. This global remeshing approach (GRA) allows for controlled grid regeneration based on predefined topological rules, ensuring more uniform element size and quality (Figure 4b) compared to only replacing degenerated elements in an a-priori unknown geometry. Results are then transferred from the old to the new grid through interpolation. Keeping the average element size constant for the entire simulation allows the same time-step to be maintained during the entire calculation, ensuring a very high stability at the same time. Further details, including application to simple cases like back-to-back test rigs, are provided in [22].

However, a primary limitation of this original approach, as outlined in [2,27,28], is its impracticality for cases where 2D mesh generation and subsequent extrusion are not feasible. To overcome this limitation, a partitioning-based technique has been developed and integrated with the global remeshing method. This new technique eliminates applicability limitations, enabling simulation of various gearbox designs, including planetary gears exploiting arbitrary mesh interfaces [29]. Moreover, the adoption of mesh clustering (GRAMC) [30] further speeds up the computation thanks to a recursive adoption of grids.

3.3 Examples of applications and validation

3.3.1 Full-immersion lubrication of a back-to-back test rig: Windage and cavitation

The initial example of application of the above-described method pertains to a basic back-to-back test-rig having a 1:1 ratio. Existing literature provides experimental data on power losses for this setup. Otto et al.

[31] conducted tests at various rotational speeds (corresponding to tangential velocities of 0 ≤ v_t ≤ 38 m/s) with oil bath lubrication (at 50% and 100% oil levels) and different pressurization levels (0 and 6 bar).

While power losses exhibit a predominantly linear trend for the partially filled lubricant condition, there are discrepancies in experimental results for complete filling under the two pressure levels. Overpressures on the front flanks due to its advancement as well as negative pressures on the rear flanks due to suction effects, both contribute to the power loss. In light of the fact the lubricant can be reasonably considered as an incompressible fluid, the differences observed experimentally for the two pressurization levels are challenging to justify (Figure 5). According to the scholars, the most likely explanation is the presence of air bubbles trapped in the oil sump.

To gain a deeper understanding of this observation, computational fluid dynamics (CFD) simulations were conducted using a comprehensive multiphase VOF solver (separated flow) capable of encompassing all relevant physical phenomena from aeration to cavitation. The numerical results not only align with the experimental data but also provide insights into the mechanism behind power loss generation.

Under pressurization (6 bar), both the rear and front flanks exhibit symmetrical pressure distributions (around the average pressure) with a peak at the flank tip characteristic of windage. Path lines confirm that, after being expelled axially, the fluid is drawn back into the succeeding vane between adjacent teeth, leading to circulation over a relatively large area around the gears. Conversely, at lower pressure levels (1 bar), depressurization is constrained by the vaporization pressure (P_vap = 2,340 Pa). Once this pressure is reached, a phase transition (from liquid to vapor) occurs, preventing further pressure reduction

until the entire liquid transforms into vapor. This limited pressure reduction on the rear flank restricts the suction effect (this is confirmed by the analysis of the streamlines: the fluid expelled axially is not drawn back into subsequent tooth vanes), while also positively affecting (reducing) power losses.

CFD simulations not only facilitate the comparison of power losses but also enable examination of pressure and velocity distributions throughout the entire domain. As shown in Figure 5, disparities in the velocity fields indicate variations in the mechanism behind the power loss.

This example, despite demonstrating the high degree of flexibility of the numerical approaches, clearly underscores how numerical methodologies can enhance a better understanding of the physical phenomena governing lubrication.

3.3.2 Grease lubrication of a back-to-back test rig: Channeling and circulation

Keeping the same back-to-back geometry with different gears now having a ratio 3:2, let us consider the experimental findings of Stemplinger et al. [32], whose tests were conducted for three different lubricant fillings (40%, 50%, and 80% of the volume) with the pinion rotating at 3,500 rpm. The scholars observed that, different from the oil bath lubrication, in case of grease lubrication, the power losses have an initial increase with the amount of lubricant and a subsequent stabilization (Figure 6a).

This observation was fully explained by the CFD simulations conducted by the author: it exists as a threshold in terms of the amount of lubricant across which the lubrication mechanism changes. For small amounts of grease (40⎫50%), the channeling phenomenon occurs. This refers to a state where a gap between the gears and the main lubricant mass occurs: The grease is pushed entirely to the sides of the gearbox and fails to return to the engagement region. Conversely, at higher filling levels (80%), circulation occurs: The grease fully saturates the teeth. Fresh grease circulates around the gears.

From Figure 6b it can be clearly observed that, in presence of channeling, the power losses depend on the arc of submersion of the gear teeth, which increases with the amount of lubricant. In case of circulation, instead, the power losses depend on the velocity gradient between the teeth and the gears only, which is not affected by a further addition of grease.

While the specific example refers to a standard back-to-back test rig, grease lubrication is frequently adopted also for lubricating the bearing, especially the big ones. The relatively small amount of grease with respect of the total internal volume of the bearing promote also channeling [33]. In this regard, combining advanced meshing strategies [34,35] (see section 3.3.4) and this solver will lead to a powerful engineering tool to analyze also rolling bearings.

3.3.3 Oil-jet lubrication of a back-to-back test rig: Oil suspension/aerosol

The third example refers again to the back-to-back test rig having a ratio 26:17 tested by Dindar et al. [36]. The lubrication is ensured by a nozzle having a diameter of 0.71 mm that lubricates the gear with an oil supply of 1 l/min resulting in a jet speed of 14.4 m/s. The high speed of the jet in combination with the small oil supply and the relatively high rotation of the gears promote the dispersion of very small oil droplets forming a lubricant aerosol. This physical phenomenon was observed by other scholars including Kunz et al. [37] who highlighted how the power loss mechanism is strongly affected by the amount of suspended oil. Simulations of the operating conditions by Dindar et al. were reproduced numerically exploiting both a standard compressible immiscible multiphase solver with interface tracking (that is expected to not be able to capture the proper distribution of the droplets due to the inconsistency of the manageable mesh size with the available hardware – 0.1 mm – with respect to the expected average oil droplet size – 1 μm) and a compressible Euler-Euler multiphase solver including a drag model. The average oil droplet size was estimated from literature [37].

As expected, the interface tracking solver led (for this specific combination of mesh and oil suspension characteristics) to the non-conservation of the mass of the oil resulting in a significant underestimation of the power losses. On the other hand, the Eulerian solver adeptly predict the oil jet’s penetration into the contact area, the axial and tangential ejection of the lubricant, and the accumulation of lubricant at the housing’s bottom, resulting in the formation of a thin lubricant film. Figure 7 depicts the velocity fields of the continuous (air) phase and the dispersed (oil) phase on the symmetry planes. The centrifugal effects cause the air to be expelled radially, as previously demonstrated by the authors [22]. Conversely, the oil, simulated as a cloud of droplets with a predetermined drag based on Schiller’s model [38], follows trajectories determined by an equilibrium between transport forces (due to the air stream), gravity, and drag. Following engagement, most of the oil accumulated on the teeth flanks drips and/or is sprayed to the housing bottom, forming an escaping cone whose angle largely depends on the gear speed — higher speeds result in smaller angles. Nonetheless, an oil mist forms in the vicinity of the teeth, contributing to increased power losses (with respect to pure air-windage). Figure 7a reports also the power loss outcomes at various rotational speeds: The solid line corresponds to the experimental findings per Dindar et al. [36]; the dashed line represents the results of the Euler-Euler solver. For completeness, the outcomes of the standard immiscible solver with interface tracking are also reported (dotted line). The diagram vividly demonstrates the enhancements provided by the new formulation (for such lubrication conditions).

3.3.4 Oil-bath lubrication of a rolling bearing: Churning and aeration

As previously anticipated, both abovementioned solvers could be applied not only to gears, but also to the bearings, which are always present in geared systems. Let consider a vertically mounted tapered roller bearing submerged with lubricant. It is known that its conical geometry promotes pumping effects and axial circulation of the lubricant [39]. At high speeds, the turbulent eddies promote the inclusion of air bubbles (entrapping) into the lubricant sump. This phenomenon is called aeration. Once aeration occurs, the bubbles become trapped within the lubricant. The literature identifies three main types of aeration [40]: (1) entrained air, (2) foam, and (3) dissolved air. Entrained air consists of suspended bubbles. The level of aeration is determined by the balance between the rates of air incorporation and release. The air release causes foaming; the air, being less dense than the lubricant, rises to the surface, forming thin liquid films whose thickness depends on surface tension. The last type, dissolved air, primarily occurs in pressurized systems and is not visible to the naked eye. These effects significantly affect the lubrication properties, altering the overall behavior of the lubricant mixture and, thereby, affecting its effectiveness and the amount of high-pressure lubricant (HPL).

This phenomenon was experimentally observed by the authors who have performed high speed camera (HSC) and particle image velocimetry (PIV) acquisition of the velocity field within a 32312-A bearing [13,41-43] with a special transparent sapphire cage. Specifically, it was possible to measure the tangential velocity field between the cage and the outer race. Simulations of this experimental setup were made both with a standard incompressible solver with interface tracking (VOF) and with a newly developed solver capable of modeling aeration. As reported in Figure 8b, at low speed (v_t < 1 m/s), the results of both numerical solvers are aligned with the experimental evidence.

However, once the rotational speed is increased (v_t = 5 m/s), the prediction of the two numerical approaches differs significantly. The standard incompressible solver predicts a velocity field that is similar to the one predicted at low speed, just rescaled; on the contrary, the aeration-solver’s results show a sort of mirroring of the field. The comparison of the results with the experimental ones supports the new aeration solver. The reversal of the velocity field at higher speed can be justified by the air-trapping phenomenon. This hypothesis is further supported by the HSC images, which show the presence of air bubbles in addition to the fluorescent seeding particles used for PIV.

The same bearing was tested at MEGT [44]. Figure 8a reports the power losses for different operating temperatures, both measured and predicted. Also, in this case, the developed solvers result to be accurate always falling within the experimental uncertainty.

While the actual example of application of the aeration-solvers refers to bearings, aeration has been experimentally observed in various machine components, including gears [45], Gerotor pumps [45], etc. The present methodology could be applied also to those configurations, as previously shown by the author [14].

3.3.5 Industrial application: Multistage parallel axis gearbox

While the previous examples showed the accuracy of the present method, this example is reported to show how the simulations could have an impact on the daily deign practice. The analyzed gearbox is a 2-stage parallel axis gearbox produced by DANA [3]. It operates in oil-bath conditions.

Figure 9a shows the comparison between experimental acquisition and numerical predictions is terms of dimensionless power loss vs. operating temperature. In all cases, the percentage error results are below 20%, which is an impressive result considering the complexity of the configuration and the assumption of having an homogeneous temperature inside the entire gearbox.

Despite accurately predicting the total power losses (Figure 9a), the numerical approach gives an insight into the different loss mechanisms. Figure 9b shows, for example, how the load independent power losses are shared between the four gears and how this share changes with the rotational speed. For this specific configuration, Gear 4 is the main one responsible for the gear losses. This is due to both its big size as well as its mounting position that causes it to always be, at least partially, submerged in the oil sump.

Figure 9c shows the share between the different elements (gears, bearings, and seals). In this case, for instance, the contribution of the bearings is negligible in comparison with that of the gears (unloaded condition). This is not always the case, and it is important to have engineering tools to quantify the share of the losses before manufacturing a prototype.

Finally, Figure 9d shows the relationship between inertial losses (related to the lubricant density) and viscous one (related to the oil viscosity). All this information is of pivotal importance to optimize the lubrication of a geared system. As an example, the figure clearly shows that Gear 4, the one partially submerged in the sump, generates losses mostly due to splashing effects (inertial) while the other ones, just marginally lapped by the oil, generate power losses mostly due to viscous (windage) effects.

3.3.6 Industrial application: Planetary gearbox

Planetary gearings have peculiar kinematics that ensures the capability for high reduction ratios and very high-power density. However, the rotational-translational motion of the planets promotes significant oil splashing, complicating the optimization of the lubrication and inducing significant load independent power losses. Also, from a simulation point of view, dealing with their kinematics poses a challenge. Despite this, the author showed how the present CFD technology, exploiting the GRA^MC approach, allows the accurate simulation of planetary architectures [46].

Figure 10b shows the formation of Taylor-Couette flows between the planet carrier and the housing. Experimentally, this well-known phenomenon [48] was measured exploiting a transparent housing. The flow that originates in the front part of the gearbox (namely between the planet carrier and the housing) was properly predicted by the present numerical approach. The experiments showed that, as the velocity increases (Figure 10c), more oil is dragged by the carrier. This led into higher churning on the left area (red dashed rectangle) and into oil ejection on the right side of the gearbox (blue dashed dot rectangle). Both phenomena are very well predicted by the CFD model [47]. The simulations showed also the presence of squeezing effects deriving from the mutual interaction between the planets and the ring gears. The squeezing of the oil from the planet-ring gear meshing zone was observed also by Boni et al. [49]. The CFD simulations also provide a physical explanation for these axial fluxes. As the gears engage, the gap between the teeth continuously decreases and increases during the gear mesh. This sudden contraction creates an overpressure in the gap, resulting in axial fluxes. Once the gap volume reaches its minimum, it suddenly increases again, causing a lower pressure in the gap. It is important to note this pressure drop is significantly less than the ambient pressure of 1 bar, so this pressure gradient will not cause phase changes, such as cavitation. It should be emphasized the pressure gradients calculated due to volume variation during gear meshing with the current model do not represent the pressures on the mating flanks near the contact area.

Finally, Figure 10a shows the capability of the present numerical method to properly capture the power losses for different rotational speed as well as for different temperatures 𝜃 even for this complex geometry.

3.4 Computational effort

While being able to simulate the complex phenomena and dealing with complex mesh topology deformations has proven to be possible, it could be interesting to quantify the computational effort required for such analyses. In order to give a more comprehensive overview on how the different mesh handling strategies have led to a drastically better computational efficiency, we will take as reference one of the milestones available in literature [50]. Gorla et al. used a commercial software relying on a mesh smoothing and local remeshing algorithm for simulating a back-to-back test rig. Their simulations, on a 38 GFLOPs (billion floating point operations per second) hardware took approximately 2,466 minutes corresponding to about 41 hours. The same configuration was studied exploiting the GRA. The simulation completed in 172 minutes (3 hours). Furthermore, the adoption of the GRA^MC mesh handling strategy reduced the simulation time down to 7.5 minutes.

While these numbers refer to a simple configuration, having at disposal more powerful hardware it will be easily possible to simulate in a reasonable amount of time even the most complex gearboxes. As an indication, the oil-bath lubricated planetary system presented in section 3.3.6. required about 400 hours with the local remeshing approach and less than 20 hours with the GRA^MC [47]. The multi-stage reducer of section 3.3.5 took 23 hours on a 48 GFLOPs hardware.

4 Conclusions

In the context of a growing interest and focus on the energy efficiency and reliability of the gearboxes, CFD has proven to be an effective approach to both calculate the load independent power losses, for which there are no accurate and/or reliable analytical/empiric formulations, and for the study of lubricant flows to the mechanical component of the transmission.

Methods implemented in most commercial software often fall short in addressing real-world gearbox dynamics due to limitations in mesh handling strategies that directly affect the computational effort. To bridge this gap, the author advocates for an approach using open-source code, leveraging finite volume elements and a global remeshing technique with mesh clustering for fluid dynamics equation solving.

Validation of this method involves preliminary applications, using test data from setups such as test rigs, while considering traditionally overlooked factors such as cavitation, aeration, aerosol, grease channeling, and circulation. Results not only demonstrate the method’s reliability in predicting losses and flows but also offer insights into underlying physical phenomena, facilitating discussions and design enhancements. Following refinement of the global remeshing approach, the method is successfully applied to analyze complex cases such as planetary gearboxes and multi-stage industrial solutions.

The findings highlight the method’s ability to provide accurate results within realistic timeframes, making it a valuable asset for companies seeking gearbox efficiency improvements from early design stages. Ongoing developments aim to enhance user interfaces, expanding the tool’s applicability across design offices. 

Bibliography

1. Concli, F.; Della Torre, A.; Gorla, C.; Montenegro, G. A New Integrated Approach for the Prediction of the Load Independent Power Losses of Gears: Development of a Mesh-Handling Algorithm to Reduce the CFD Simulation Time. Advances in Tribology 2016, 2016, doi:10.1155/2016/2957151.
2. Mastrone, M.N.; Concli, F. Application of the GRAMCmesh-Handling Strategy for the Simulation of Dip and Injection Lubrication in Gearboxes. International Journal of Computational Methods and Experimental Measurements 2022, 10, 303–313, doi:10.2495/CMEM-V10-N4-303-313.
3. Mastrone, M.N.; Concli, F. A Multi Domain Modeling Approach for the CFD Simulation of Multi-Stage Gearboxes. Energies (Basel) 2022, 15, doi:10.3390/en15030837.
4. Mastrone, M.N.; Concli, F. Development of a Mesh Clustering Algorithm Aimed at Reducing the Computational Effort of Gearboxes’ CFD Simulations. In Proceedings of the WIT Transactions on Engineering Sciences; 2021; Vol. 131, pp. 59–69.
5. Concli, F.; Maccioni, L.; Gorla, C. Lubrication of Gearboxes: CFD Analysis of a Cycloidal Gear Set. In Proceedings of the WIT Transactions on Engineering Sciences; 2019; Vol. 123, pp. 101–112.
6. Fraccaroli, L.; Pagliari, L.; Concli, F. A Combined Analytical-Numerical Approach to Evaluate the Efficiency of Cycloidal Speed Reducers. Lecture Notes in Networks and Systems 2023, 745 LNNS, 590–599, doi:10.1007/978-3-031-38274-1_49.
7. Concli, F.; Maccioni, L.; Gorla, C. Development of a Computational Fluid Dynamics Simulation Tool for Lubrication Studies on Cycloidal Gear Sets. International Journal of Computational Methods and Experimental Measurements 2020, 8, 220–232, doi:10.2495/CMEM-V8-N3-220-232.
8. Concli, F.; Gorla, C. Computational and Experimental Analysis of the Churning Power Losses in an Industrial Planetary Speed Reducer. In Proceedings of the WIT Transactions on Engineering Sciences; 2012; Vol. 74, pp. 287–298.
9. Concli, F.; Gorla, C. Numerical Modeling of the Churning Power Losses of Gears: An Innovative 3D Computational Tool Suitable for Planetary Gearbox Simulation. VDI Berichte 2017, 2017, 800–811.
10. Concli, F.; Mastrone, M.N. Latest Advancements in the Lubrication Simulations of Geared Systems: A Technology Ready for Industrial Applications. VDI Berichte 2023, 2023, 499–518, doi:10.51202/9783181024225-499.
11. Concli, F.; Mastrone, M.N. Latest Advancements in the Lubricant Simulations of Geared Systems: A Technology Ready for Industrial Applications [Neuste Fortschritte Bei Der Schmierstoffsimulation von Getriebesystemen: Eine Industrietaugliche Technologie]. Forschung im Ingenieurwesen/Engineering Research 2023, 87, 1181–1191, doi:10.1007/s10010-023-00698-z.
12. Maccioni, L.; Concli, F. Computational Fluid Dynamics Applied to Lubricated Mechanical Components: Review of the Approaches to Simulate Gears, Bearings, and Pumps. Applied Sciences (Switzerland) 2020, 10, 1–29, doi:10.3390/app10248810.
13. Maccioni, L.; Chernoray, V.G.; Mastrone, M.N.; Bohnert, C.; Concli, F. Study of the Impact of Aeration on the Lubricant Behavior in a Tapered Roller Bearing: Innovative Numerical Modelling and Validation via Particle Image Velocimetry. Tribol Int 2022, 165, doi:10.1016/j.triboint.2021.107301.
14. Mastrone, M.N.; Concli, F. Numerical Modeling of Fluid’s Aeration: Analysis of the Power Losses and Lubricant Distribution in Gearboxes. Journal of Applied and Computational Mechanics 2023, 9, 83– 94, doi:10.22055/jacm.2022.40666.3625.
15. Mastrone, M.N.; Concli, F. Simulation of Fluid’s Aeration: Implementation of a Numerical Model in an Open Source Environment. In Proceedings of the WIT Transactions on Engineering Sciences; 2021; Vol. 132, pp. 27–36.
16. Mastrone, M.N.; Concli, F. CFD Simulation of Grease Lubrication: Analysis of the Power Losses and Lubricant Flows inside a Back-to-Back Test Rig Gearbox. J Nonnewton Fluid Mech 2021, 297, doi:10.1016/j.jnnfm.2021.104652.
17. Niemann, G.; Winter, H. Maschinenelemente—Band 2: Getriebe Allgemein,Zahnradgetriebe— Grundlagen, Stirnradgetriebe—2.Auflage; Berlin, Germany, 2003; ISBN 9783540773405.
18. Concli, F.; Schaefer, T.C.; Bohnert, C. Innovative Meshing Strategies for Bearing Lubrication Simulations. Lubricants 2020, 8, doi:10.3390/lubricants8040046.
19. Hirt, C.W.; Nichols, B.D. Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries. J Comput Phys 1981, 39, 201–225.
20. Concli, F. Pressure Distribution in Small Hydrodynamic Journal Bearings Considering Cavitation: A Numerical Approach Based on the Open-Source CFD Code OpenFOAM®. Lubrication Science 2016, 28, 329–347, doi:10.1002/ls.1334.
21. Kunz, R.F.; Boger, D.A.; Stinebring, D.R.; Chyczewski, T.S.; Lindau, J.W.; Gibeling, H.J.; Venkateswaran, S.; Govindan, T.R. A Preconditioned Navier-Stokes Method for Two-Phase Flows with Application to Cavitation Prediction. Comput Fluids 2000, 29, 849–875, doi:10.1016/S0045- 7930(99)00039-0.
22. Concli, F.; Gorla, C. Numerical Modeling of the Power Losses in Geared Transmissions: Windage, Churning and Cavitation Simulations with a New Integrated Approach That Drastically Reduces the Computational Effort. Tribol Int 2016, 103, 58–68, doi:10.1016/j.triboint.2016.06.046.
23. Rusche, H. Computational Fluid Dynamics of Dispersed Two-Phase Flows at High Phase Fractions, 2002.
24. Simonnet, M.; Gentric, C.; Olmos, E.; Midoux, N. CFD Simulation of the Flow Field in a Bubble Column Reactor: Importance of the Drag Force Formulation to Describe Regime Transitions. Chemical Engineering and Processing: Process Intensification 2008, 47, 1726–1737.
25. Hwang, C.J.; Wu, S.J. Global and Local Remeshing Algorithms for Compressible Flows. J Comput Phys 1992, 102, 98–113, doi:10.1016/S0021-9991(05)80009-9.
26. Concli, F.; Gorla, C.; Torre, A.D.; Montenegro, G. Windage Power Losses of Ordinary Gears: Different CFD Approaches Aimed to the Reduction of the Computational Effort. Lubricants 2014, 2, 162–176, doi:10.3390/lubricants2040162.
27. Mastrone, M.N.; Concli, F. Development of a Mesh Clustering Algorithm Aimed at Reducing the Computational Effort of Gearboxes’ CFD Simulations. In Proceedings of the WIT Transactions on Engineering Sciences; 2021; Vol. 131, pp. 59–69.
28. Mastrone, M.N.; Concli, F. CFD Simulations of Gearboxes: Implementation of a Mesh Clustering Algorithm for Efficient Simulations of Complex System’s Architectures. International Journal of Mechanical and Materials Engineering 2021, 16, doi:10.1186/s40712-021-00134-6.
29. Concli, F.; Gorla, C. Numerical Modeling of the Churning Power Losses in Planetary Gearboxes: An Innovative Partitioning-Based Meshing Methodology for the Application of a Computational Effort Reduction Strategy to Complex Gearbox Configurations. Lubrication Science 2017, 29, 455–474, doi:10.1002/ls.1380.
30. Mastrone, M.N.; Concli, F. Application of the GRAMCmesh-Handling Strategy for the Simulation of Dip and Injection Lubrication in Gearboxes. International Journal of Computational Methods and Experimental Measurements 2022, 10, 303–313, doi:10.2495/CMEM-V10-N4-303-313.
31. Höhn, B.-R.; Michaelis, K.; Otto, H.-P. Influence on No-Load Gear Losses. Ecotrib 2011 Conference Proceedings 2011, 2, 639–644.
32. Stemplinger, J.-P.; Stahl, K.; Höhn, B.-R.; Tobie, T.; Michaelis, K. Analysis of Lubrication Supply of Gears Lubricated with Greases NLGI 1 and 2 and the Effects on Load-Carrying Capacity and Efficiency. NLGI Spokesman 2014, 78, 18–22.
33. KR, S.C.; Lugt, P.M. The Process of Churning in a Grease Lubricated Rolling Bearing: Channeling and Clearing. Tribol Int 2021, 153, 106661.
34. Wingertszahn, P.; Koch, O.; Maccioni, L.; Concli, F.; Sauer, B. Predicting Friction of Tapered Roller Bearings with Detailed Multi-Body Simulation Models. Lubricants 2023, 11, doi:10.3390/lubricants11090369.
35. Maccioni, L.; Concli, F. Estimation of Hydraulic Power Losses in a Double-Row Tapered Roller Bearing via Computational Fluid Dynamics. Lecture Notes in Networks and Systems 2023, 745 LNNS, 655–666, doi:10.1007/978-3-031-38274-1_55.
36. Dindar, A.; Chaudhury, K.; Hong, I.; Kahraman, A.; Wink, C. An Experimental Methodology to Determine Components of Power Losses of a Gearbox. J Tribol 2021, 143, doi:10.1115/1.4049940.
37. Kunz, R.F.; Hill, M.J.; Schmehl, K.J.; McIntyre, S.M. Computational Studies of the Roles of Shrouds and Multiphase Flow in High Speed Gear Windage Loss. In Proceedings of the Annual Forum Proceedings – AHS International; 2012; Vol. 3, pp. 1953 – 1963.
38. Schiller, L. A Drag Coefficient Correlation. Zeit. Ver. Deutsch. Ing. 1933, 77, 318–320.
39. Liebrecht, J.; Si, X.; Sauer, B.; Schwarze, H. Investigation of Drag and Churning Losses on Tapered Roller Bearings. Strojniški vestnik-Journal of Mechanical Engineering 2015, 61, 399–408.
40. Nemoto, S.; Kawata, K.; Kuribayashi, T.; Akiyama, K.; Kawai, H.; Murakawa, H. A Study of Engine Oil Aeration. JSAE review 1997, 18, 271–276.
41. Maccioni, L.; Concli, F. Flows in Oil-Bath Lubricated Tapered Roller Bearings: CFD Simulations Validated via PIV. VDI Berichte 2023, 2023, 247–266, doi:10.51202/9783181024157-247.
42. Maccioni, L.; Chernoray, V.G.; Concli, F. Fluxes in a Full-Flooded Lubricated Tapered Roller Bearing: Particle Image Velocimetry Measurements and Computational Fluid Dynamics Simulations. Tribol Int 2023, 188, doi:10.1016/j.triboint.2023.108824.
43. Maccioni, L.; Chernoray, V.G.; Bohnert, C.; Concli, F. Particle Image Velocimetry Measurements inside a Tapered Roller Bearing with an Outer Ring Made of Sapphire: Design and Operation of an Innovative Test Rig. Tribol Int 2022, 165, doi:10.1016/j.triboint.2021.107313.
44. Maccioni, L.; Rüth, L.; Koch, O.; Concli, F. Load-Independent Power Losses of Fully Flooded Lubricated Tapered Roller Bearings: Numerical and Experimental Investigation of the Effect of Operating Temperature and Housing Wall Distances. Tribology Transactions 2023, 66, 1078–1094, doi:10.1080/10402004.2023.2254957.
45. Changenet, C.; Leprince, G.; Ville, F.; Velex, P. A Note on Flow Regimes and Churning Loss Modeling. J. Mech. Des 133 2011, 1–5.
46. Concli, F.; Gorla, C. Influence of Lubricant Temperature, Lubricant Level and Rotational Speed on the Churning Power Loss in an Industrial Planetary Speed Reducer: Computational and Experimental Study. International Journal of Computational Methods and Experimental Measurements 2013, 1, 353–366, doi:10.2495/CMEM-V1-N4-353-366.
47. Mastrone, M.N.; Hildebrand, L.; Paschold, C.; Lohner, T.; Stahl, K.; Concli, F. Numerical and Experimental Analysis of the Oil Flow in a Planetary Gearbox. Applied Sciences (Switzerland) 2023, 13, doi:10.3390/app13021014.
48. de Gevigney, J.D.; Changenet, C.; Ville, F.; Velex, P.; Becquerelle, S. Experimental Investigation on No-Load Dependent Power Losses in a Planetary Gear Set. In Proceedings of the International Gear Conference; 2013; Vol. 2, pp. 1101–1112.
49. Boni, J.-B.; Neurouth, A.; Changenet, C.; Ville, F. Experimental Investigations on Churning Power Losses Generated in a Planetary Gear Set. Journal of Advanced Mechanical Design, Systems, and Manufacturing 2017, 11, JAMDSM0079–JAMDSM0079.
50. Gorla, C.; Concli, F.; Stahl, K.; Höhn, B.-R.; Michaelis, K.; Schultheiß, H.; Stemplinger, J.-P. Hydraulic Losses of a Gearbox: CFD Analysis and Experiments. Tribol Int 2013, 66, 337–344, doi:10.1016/j.triboint.2013.06.005.

^{Printed with permission of the copyright holder, authors. Statements presented in this paper are those of the authors and may not represent the position or opinion of the American Gear Manufacturers Association. (AGMA) This paper was presented October 2024 at the AGMA Fall Technical Meeting and was awarded best presentation. 24FTM21}