Numerical Calculation of the Oil Flow in a Truck Rear Axle Transmission and the Influence of Gear-Induced Air Flow

In gearboxes, a lubricant is used to reduce friction and to transfer heat. For gearboxes with circumferential speeds up to v_t = 60 m/s, dip lubrication is widely used [1,2]. The rotation of gears causes a flow of fluids, such as oil and air, in the gearbox. This fluidic interaction of rotating gears results in no-load gear power loss, which can be classified according to Niemann and Winter [3] as churning, squeezing, impulse, and windage. For dip-lubricated gearboxes, churning, squeezing, and windage are relevant. Churning refers to the displacement of fluid by teeth moving through a liquid fluid sump. A comprehensive study on churning was carried out by Polly et al. [4], Handschuh et al. [5], and Hildebrand et al. [6]. Squeezing describes the displacement of fluid from the tooth gaps of meshing teeth. Windage refers to the interaction of rotating gears with a homogenous fluid, which can be oil, air, or an oil–air mixture [7]. Gearbox fluid flow and no-load power losses are specifically important in the context of gearbox efficiency and heat balance [8,9].

Computational fluid dynamics (CFD) can be used to calculate the gearbox fluid flow and the no-load power loss. The mesh-based finite volume method (FVM) and the particle-based smoothed particle hydrodynamics (SPH) method are widely used here. The computational effort is high, especially in two-phase FVM calculations [10,11], which limits its applicability to practical gearboxes. The SPH has great potential for time-efficient calculation of gearbox fluid flow due to its meshless approach and good parallelization capability [12,13].

Maccioni and Concli [14] provide an overview of applied CFD methods and numerical investigations carried out in the context of the lubrication of machine elements and gearboxes. An overview of the application of SPH is given by Shadloo et al. [13]. CFD can be used to gain an in-depth understanding of the interaction mechanisms between rotating gears and fluid. Liu et al. [15] investigated the churning power loss of a dip-lubricated gearbox and determined the power loss contribution of each pinion and wheel. The work highlights the strong influence of the oil viscosity and the circumferential speed on the gearbox fluid flow. Detailed investigations on the mechanisms of the churning power loss of a gearbox with a guide plate have been carried out by Hildebrand et al. [16]. The work shows the influence of the fluidic interaction of rotating gears on the pressure and shear forces and, thus, on the no-load power loss. Hildebrand et al. [6] conducted a comprehensive parametric study on gearbox oil flow and its influence on the no-load loss torque, comparing numerical, analytical, and experimental results. The speed-dependent wetting of the gears was identified as a measure of interaction between gears and oil, which has a significant impact on the no-load loss torque. The fluid flow of a rear axle transmission was investigated by Qian et al. [11] with a focus on the influence of a guide plate on fluid flow. The results show that pressure forces inside and outside the guiding geometry strongly influence the fluid flow. Morhard et al. [17] used the CFD to support and improve the lubrication design of a multistage gearbox in a high-speed electromechanical powertrain. Quiban et al. [18] investigated the transition between churning and windage in relation to the immersion depth of gears. Shore et al. [19] investigated the influence of the oil viscosity and circumferential speed on churning and the effective immersion depth of gears. The results show that the churning loss torque increases with immersion depth, with a varying influence on the churning loss torque over the circumferential speed present for different viscosities. In addition to studies of gearbox fluid flow and no-load power loss, CFD is also used to investigate the thermal behavior of gearboxes. Hildebrand et al. [20] studied the heat dissipation of dry-lubricated gear geometries and highlighted the strong influence of the gearbox air flow on heat dissipation. Uerlich et al. [21] transferred numerical results on the oil distribution in a gearbox to a thermal network model. The reviewed literature highlights the significance of gearbox fluid flow in relation to its lubrication, power loss, and heat transfer. Using CFD has enhanced understanding of the interaction mechanisms between gears and fluid. This study investigates the application of SPH for modeling the fluid flow in a practical gearbox considering a truck rear axle transmission. First, the influence of gear-induced air flow in the context of a single-phase and two-phase modeling approach is studied in detail using a single-stage test gearbox. Second, the truck rear axle transmission is modeled and calculated based on that. The numerical results on the oil flow are compared with experimental results.

The objects of investigation, operating conditions, numerical models, and calculation procedure are described below.

Two objects of investigation are considered. Fundamental investigations on the influence of gear-induced air flow on the gearbox oil flow are carried out, focusing on the single-stage test gearbox of the FZG no-load power loss test rig. Focusing on a truck rear axle transmission transfers the knowledge gained to a practical application.

Previous investigations by the authors have also considered the single-stage test gearbox of the FZG no-load power loss test rig [6,16,20,22]. In this work, an adapted bearing is employed using deep groove ball bearings 61,806. Figure 1 shows the test gearbox with an acrylic front cover and mounted gears. The test gearbox has a width of 260 mm, a height of 171 mm, and a depth of 56 mm.

Figure 2 schematically illustrates the sectional view of the gearbox with the assignment of shafts, bearings, and sealings and with an indication of the engine. The gear considered is the FZG test gear of type C-PT [23]. Table 1 shows the main geometry data.

The rear axle transmission is of paramount importance for the robustness and efficiency of commercial vehicle drivetrains. The considered truck rear axle transmission is typically used in a 40-tonne truck. The rear axle transmission consists of a hypoid gear stage with the hypoid wheel connected to the differential. It is examined in a horizontal position, which corresponds to the truck driving on level ground. Figure 3 shows a rendering of the rear axle transmission with a view into the periphery with the assignment of the hypoid gear stage, the differential, the bearings, and the housing.

The axle flange bearing and axle differential are highly stressed parts of the transmission. Therefore, special consideration is taken in the design phase of the axle development to ensure a sufficient oil supply for both locations. The design of the rear axle transmission incorporates lubrication management functions. Oil thrown off by the gears during operation is collected in the housing and directed to the bearings via oil channels.

An oil collector area collects the oil and directs it to the tapered roller element bearings of the hypoid pinion. A geometric diverter divides the oil flow to lubricate the axle flange and pinion head side bearings. Each tapered roller element bearing produces an axial oil flow through the bearings corresponding to each contact angle. The head-sided bearing of the hypoid pinion returns the oil directly to the oil sump. At the flange-sided tapered roller element bearing of the hypoid pinion, a return channel is incorporated in the housing to return the bearing oil flow to the oil sump. Splash oil is collected at another oil collector area in the gearbox housing and directed through an oil channel to the tapered roller element bearings of the differential. Figure 4 shows the oil collection areas and the oil channels of the rear axle transmission.

It also schematically shows the oil flow from the oil collection areas to the bearings. Experimental investigations (cf. Sec. 2.4.2) were conducted on the rear axle transmission to measure the oil supply to the bearings. Oil was drained from the return flow channel of the pinion tapered roller element bearing (measurement channel I, cf. Fig. 4) and the differential tapered roller element bearing (measurement channel II, cf. Fig. 4).

In comparison to the single-stage test gearbox, the housing of the rear axle transmission is contoured to the gears, with the average clearance being approximately 20 mm.

The study focuses on the operating conditions of the truck rear axle transmission during regular operation. Specifically, the circumferential speeds v_t = {3.6; 4.2; 4.8} m/s and a static oil level in the horizontal mounting position with an immersion depth of e₂ = 30% of the outer diameter of the hypoid wheel are examined. An oil temperature of ϑ_oil = 60 °C is considered. The preliminary experimental investigations have identified a characteristic gearbox oil flow under these operating conditions.

The operating conditions of the FZG no-load power loss test rig are aligned with those of the rear axle transmission. To investigate the influence of gear-induced air flow on gearbox oil flow, a second static oil level with an immersion depth of the wheel of e₂ = 0% is considered in addition to e₂ = 30%. For the same reason, the circumferential speeds of v_t = {20; 30} m/s are investigated in addition to v_t = {3.6; 4.2; 4.8} m/s. A mineral oil is considered. Table 2 shows the considered oil properties.

The study employs the commercial software AVL preonlab 5.2.3. Isothermal simulations are being conducted.

The SPH modeling approach of the test gearbox of the FZG no-load power loss test rig is based on a validated modeling approach in the context of the FVM that is documented in the author's previous work [16]. In accordance with this, a discretization of the fluid with the discretization length or particle size l = 1.0 mm is used. The bearings and sealings are neglected in the numerical model. In the single-phase SPH model, only the oil phase is discretized with particles. The influence of the air phase on the gearbox oil flow is modeled by a drag force model [25] that applies a force on the free-moving oil particles in the opposite direction of their motion and is dependent on their velocity. Figure 5 visualizes the single-phase SPH model of the test gearbox with the oil particles.

The numerical settings applied to the single-phase SPH model of the test gearbox are listed in Table 3. Here, CFL stands for the Courant–Friedrichs–Lewy number.

In the two-phase SPH model, the oil and air phases are discretized with particles. Figure 6 visualizes the two-phase SPH model of the test gearbox with the oil and air particles.

The numerical settings applied to the two-phase SPH model of the test gearbox are listed in Table 4.

The rear axle transmission is modeled using the single-phase SPH approach. In accordance with Sec. 2.3.1, the particle size l = 1.0 mm is used. This particle size is at least five times smaller than the diameter of the oil and measurement channels (see Sec. 2.1.2). The tapered roller element bearings are considered by setting the rotational speed of the roller element elements around the shaft axis at half the shaft speed. The self-rotation of the roller element elements is neglected. The gears and roller element bearings are not scaled in size. Figure 7 visualizes the SPH model of the rear axle transmission with the oil particles.

In addition to the measurement channels described in Fig. 4, the rear axle transmission has an additional measurement channel, which, however, is not considered within the scope of this study. Consequently, this channel is sealed off externally but filled with oil due to the initial static oil level. In Fig. 7, the inactive measurement channel can be seen behind measurement channel I.

The numerical settings of the SPH model of the rear axle transmission correspond to those of the single-phase SPH model of the test gearbox shown in Table 3.

The calculations were carried out on a workstation with 34 cores (AMD^® EPYC^® 75F3 @2.95 GHz) and 128 GB RAM. At the start of an initialized simulation, the gears are static. An acceleration time of t_a = 0.05 s is applied, during which the gears are linearly accelerated to their respective target circumferential speed. Following Refs. [16,20], a quasi-steady simulation is sought within the simulations. The quasi-steady-state is assessed based on the overall averaged energy of all particles. Figure 8 shows the averaged kinetic energy of all particles over time for e₂ = 30% and v_t = 4.8 m/s for the two-phase simulation of the test gearbox of the FZG no-load power loss test rig.

Figure 9 shows the averaged kinetic energy of all particles over time for the single-phase simulation of the truck rear axle transmission, and e₂ = 30% and v_t = 4.8 m/s.

The simulation for the test gearbox is for a physical time of t = 1 s, and for the truck rear axle transmission, t = 10 s is simulated. The curves exhibit an initial excitation at the beginning of the simulation due to the gears' acceleration. The curves then show a converging behavior, and after approximately t = 0.5 s, the simulations converge. The simulations for the test gearbox and e = 0% converge much earlier due to a more spatially confined fluid flow. The simulation of the truck rear axle transmission considers the oil volume flow at the measurement points (see Fig. 3) as a second criterion for the quasi-stationary state, in addition to the overall averaged kinetic energy of all particles. Table 5 summarizes the simulation cases in the study.

During numerical calculation, particles with unphysical behavior are deleted. For all simulation cases, the proportion of deleted particles compared to the total number of particles remained below 1%.

This section briefly outlines the experimental procedures for measurements at the test gearbox and the truck rear axle transmission.

The oil distribution in the test gearbox is recorded through an acrylic front cover using a high-speed camera Photron FASTAM MINI AX200 with 3000 frames per second. The considered oil temperature of ϑ = 60 °C is monitored by a temperature sensor located in the oil sump and another temperature sensor located on the outer ring of the engine-sided bearing of the wheel shaft.

The truck rear axle transmission is investigated on a test rig. Typically, the axle installation simulates conditions in a vehicle, e.g., the same inclination angle of the pinion shaft relative to the vehicle's longitudinal axis. An electric motor serves as a power supply, i.e., drives the axle flange (cf. Fig. 3). Temperature sensors are placed at the outer rings of the pinion bearings and in the oil sump.

The purpose of the measurements is to evaluate the oil distribution in the transmission housing and to make qualitative statements about the lubrication of the bearings. Therefore, measurement channels I and II (cf. Fig. 4) are introduced that can be switched on and off. This modification allows oil to be guided out of the axle.

During quasi-stationary operation, the valve of either measurement channel I or II is opened manually. Beyond a minimum input speed, oil escapes the axle via the open measurement channel—this operating point is crucial as it relates oil supply to vehicle or input shaft speed, respectively. Oil draining via measurement channel I or II is collected by a metering cup. The time is recorded to calculate the corresponding volume flowrate. After the measurement, the oil from the metering cup is returned to the rear axle transmission. For each operating condition investigated, the measurements are repeated three times.

While this simple measurement approach proves to be very robust and efficient in detecting oil supply characteristics, it is acknowledged that at higher speeds, inaccuracy increases, e.g., due to significantly increasing oil volumetric flowrates that require a reduction of the measurement time. However, in practical axle development, little attention is paid to this since a lack of oil at the critical measurement spots is not likely to be at high axle speed.

This section presents the results of the experimental and numerical investigations. Section 3.1 focuses on the oil flow of the test gearbox, and Sec. 3.2 on the truck rear axle transmission.

In the first step, the influence of the air flow on the gearbox oil flow is investigated for the immersion depth e₂ = 0% and v_t = {4.8; 20; 30} m/s. In the second step, the immersion depth e₂ = 30% at v_t = {3.6; 4.2; 4.8} m/s is investigated to check the selected simulation setup for the truck rear axle transmission.

Figure 10(a) shows the measured quasi-stationary oil distributions for e₂ = 0% and v_t = {4.8; 20; 30} m/s. The results show that at v_t = 4.8 m/s, there is no interaction between the gears and the oil. The surface of the oil sump remains flat and smooth, indicating that there is no significant air flow. At v_t = 20 m/s, it can be seen that the air flow displaces the surface of the oil sump and causes waves on the surface of the oil sump. The continuous interaction of the wheel's tooth tips with the oil waves causes oil to be drawn out of the oil sump. At v_t = 30 m/s, the air flow opens the surface of the oil sump, allowing air to enter. There is a lot of oil splashing. Figure 10(b) shows the simulated oil distributions in the test gearbox for e₂ = 0% and v_t = {4.8; 20; 30} m/s using the two-phase SPH approach. Thereby, only the oil particles are rendered. The SPH approach models the interaction between the particles using a kernel function, with a weighting function determining the intensity of the interaction between the particles based on their distance. For e₂ = 0%, the SPH is expected to produce a numerical interaction between the passing tooth tip surfaces and oil particles in the oil sump below. At v_t = 4.8 m/s, it can be seen that the wheel drags little oil due to the numerical effect mentioned.

However, the oil sump has a flat surface, indicating that the influence of the air flow is negligible. At v_t = 20 m/s, the surface of the oil sump is strongly disturbed. Oil is displaced under the gears, and waves are formed, corresponding to the movement of the teeth of the pinion and wheel above the oil sump. The oil is distributed in the volume above the oil sump, and an interaction between the gears and oil is present. The interaction of the wheel is greater than that of the pinion. At v_t = 30 m/s, strong aeration of the oil in the oil sump occurs in addition to intense splashing.

The numerical results are in good agreement with the experimental results. The two-phase SPH predicts the influence of the circumferential speed on the oil flow and the gearbox oil flow well. Furthermore, the results confirm that the influence of the air flow on the gearbox oil flow is negligible at lower circumferential speeds.

Quiban et al. [18] noted a transition between churning and windage effects at low immersion depths around e₂ = 0%.

The results presented in Fig. 10 complement these findings by visualizing the influence of the air flow speed on the oil distribution and, thus, on the fluidic interaction with rotating gears. At higher speeds, the air flow causes the oil to be drawn out of the oil sump, subsequently distributing it throughout the gearbox housing. The two-phase SPH calculations took approx. 12 h for v_t = 4.8 m/s and 17 h for v_t = 30 m/s per simulated physical second.

Figure 11 shows experimental results (a), single-phase SPH results (b), and two-phase SPH results (c) on the oil distribution for e₂ = 30% and v_t = {3.6; 4.2; 4.8} m/s. For the numerical results, only the oil particles are rendered. The flow field velocity of the oil particles in the x-direction is shown for each simulated oil distribution on an xy-plane with a vertical offset from the gear axis of the radius of the tooth tip circle of the pinion.

In the studied operating conditions, there is a pronounced gearbox oil flow. The oil flow is more intense on the wheel side, while there is a damming of the oil flow on the housing side walls. The comparison of the oil distributions at v_t = {3.6; 4.2; 4.8} m/s shows that higher circumferential speeds result in more oil being drawn out of the oil sump and distributed in the housing volume above it. The two-phase SPH approach predicts a more spatially extended distribution of oil than the single-phase SPH approach.

For the single-phase SPH, the analysis of the flow field velocities shows that there is no significant x-velocity on the pinion side, as the pinion side oil flow barely reaches the evaluation plane. On the wheel side, it can be seen that the oil flows from the teeth toward the right wall of the housing, as indicated by the positive x-velocity. In the corners of the housing, there is a weak backflow toward the left side wall, as indicated by the negative x-velocity. Focusing on the flow field velocities of the two-phase SPH shows a more pronounced flow field associated with a higher oil volume flowing through the evaluation plane. The flow field indicates that there is oil flow from the corners toward the gears along the front and rear housing walls on both the pinion and wheel sides. These backflows of oil make the oil distributions appear more pronounced in the front view compared to the single-phase SPH. In addition, these backflows show the influence of air flow on free oil flow in the simulations.

The results suggest that the immediate flow around the housing and the damming at the housing walls are well represented using single-phase SPH. A two-phase SPH is necessary to capture secondary flow influences such as those caused by air flow.

The single-phase SPH calculations took approx. 1 h, while the two-phase SPH calculations took about approx. 18 h per simulated physical second for v_t = 4.8 m/s.

The results demonstrate the impact of air flow on oil flow and distribution. With regard to the rear axle transmission, however, the influence is rated as low in the context of the flow characteristics due to the considered low circumferential speeds.

This section presents the results for the truck rear axle transmission using the single-phase SPH model. Based on Sec. 3.1.2, the rear axle transmission is simulated using the single-phase SPH due to its lower computational effort.

Figure 12 visualizes the simulated oil distribution in the rear axle transmission for e₂ = 30% and v_t = {3.6; 4.2; 4.8} m/s at the simulated physical times of t = {0.1; 0.5; 1.0; 2.0; 5.0; 10} s. The bearings are hidden in the images. Only the oil particles are rendered. The hypoid pinion moves the oil toward the housing wall, out of the image plane, as shown in the pictures. The hypoid wheel has a stronger interaction with the oil than the pinion, drawing the oil circumferentially and distributing it in the volume of the housing above the oil sump. The temporal evolution of the gearbox oil flow and the oil flow through the measurement channels are visible in the results, demonstrating the functionality of the oil channels in relation to the hypoid pinion bearings. The oil reaches the hypoid pinion bearings after t = 1 s for v_t = 4.8 m/s and t = 2 s for v_t = 3.6 m/s. Based on their contact angle, the bearings convey the oil axially. The oil flows out of measurement channel I after approximately t = 5 s at all circumferential speeds. In addition, at v_t = {4.2; 4.8} m/s, oil enters the measurement channel II after approximately t = 5 s. There is no oil in measurement channel II at the lowest circumferential speed of v_t = 3.6 m/s. Quasi-stationary oil flows are present after approximately t = 10 s.

Figure 13 illustrates the calculated oil flowrate $V˙oil$ at the measurement channels I and II. The higher the circumferential speed, the earlier the oil arrives at each measurement oil channel and the higher the flowrate.

The results show the influence of the circumferential speed on the gearbox oil flow. Specifically, the oil flow becomes more intense at higher circumferential speeds and flows faster toward the bearings and through the measurement channels. Notably, at v_t = 3.6 m/s, no oil is present; at v_t = 4.2 m/s, only a small amount of oil reaches the differential bearing.

The single-phase SPH calculations of the rear axle transmission took approx. 3 h for v_t = 3.6 m/s and 4 h for v_t = 4.8 m/s per simulated physical second.

The experiments involved measuring the oil volume flow in measurement channels I and II at the circumferential speeds v_t = {3.6; 4.2; 4.8} m/s (cf. Sec. 2.4.2). An oil volume flow was observed in measurement channel I for all circumferential speeds. The oil volume flow in measurement channel I increased over the speed. No oil volume flow was present in measurement channel II at v_t = 3.6 m/s. However, a small oil volume flow was detected in measurement channel II from v_t = 4.2 m/s onwards, which increased with speed.

Table 6 compares the oil volume flows in the oil channels of the truck rear axle transmission as determined numerically and experimentally. The numerical results were obtained by time-averaging the transient curves shown in Fig. 13 over the time interval t = [8 10] s. The flowrate changes are shown relative to the first operating point at which each measurement channel carries oil. The magnitude and trends over the speed of the oil volume flow in the measurement channels detected in the experiment are in accordance with the simulation results (cf. Fig. 13).

The oil flows in the measurement channels result from the dragging of oil from the oil sump by the gears, the inflow of oil to the collector areas (cf. Fig. 4) as well as the conveying effect of the tapered roller element bearings (see Sec. 2.1.2). It can be stated that the single-phase SPH method accurately captures these interactions. The deviations observed at higher speeds can be attributed to the fact that bearing oil flow is not adequately represented by the selected modeling approach.

This study investigated the application of SPH for modeling the fluid flow in a practical gearbox considering a truck rear axle transmission. First, the influence of gear-induced air flow in the context of a single-phase and two-phase modeling approach was studied using a single-stage test gearbox. Second, the truck rear axle transmission is modeled and calculated based on that. The numerical results on the oil flow were compared with experimental results. The following conclusions can be drawn:

• Rotating gears create an air flow that can encounter the oil sump. At higher circumferential speed, this air flow can impact oil displacement, even if the gears do not penetrate the oil sump in the static state. The two-phase SPH can simulate the influence of air flow on gearbox oil flow in detail.

• When the influence of the air flow on the gearbox oil flow is comparably small, the single-phase SPH yields more efficient results than the two-phase SPH without compromising the quality of the results too much.

• The oil flow characteristics of a truck rear axle transmission, including the influence of the tapered roller element bearings' conveying effect, are accurately predicted by the single-phase SPH.

• The single-phase SPH method enables a calculation time of approximately 4 h per simulated physical second for the considered truck rear axle transmission with a mid-range workstation.

Future work may include a comprehensive confrontation of single-phase and two-phase SPH results in the context of the fluidic interaction of gears, e.g., with a focus on the no-load loss torque. A GPU-based calculation expects a further increase in calculation efficiency. In addition, future work may address the calculation of the bearing oil flow in more detail by local particle refinement or by the use of bearing sub-models.

• The presented results are based on the research project FVA no. 857/II; self-financed by the Research Association for Drive Technology e.V. (FVA). The authors would like to thank for the sponsorship and support received from the FVA and the members of the project committee.

There are no conflicts of interest.

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

a =

center distance in mm

b =

face width in mm

l =

element size in mm

t =

time in s

v =

circumferential speed in 1/min

x =

coordinate axis in m

y =

coordinate axis in m

z =

coordinate axis in m

V =

volume in m³

$V˙$ =

volume flowrate in ml/min

d_a1 =

tip diameter of the pinion in mm

d_a2 =

tip diameter of the wheel in mm

e₂ =

immersion depth of the wheel in %

m_n =

normal module in mm

t_a =

acceleration time in s

t_sim =

simulated time in s

z₁ =

tooth number of the pinion

z₂ =

tooth number of the wheel

β =

helix angle in deg

θ =

contact angle in deg

ν =

kinematic viscosity in mm²/s

ρ =

density in kg/m³

σ =

surface tension in N/m

α_n =

pressure angle in deg

ϑ_oil =

oil temperature in °C