In helicopter transmission systems, straight spur gears with high contact ratio (HCR) are frequently employed under oil-jet lubrication to ensure efficient cooling and reliable operation. The temperature characteristics of these gears directly influence their service life and performance. In this study, I developed a computational fluid dynamics (CFD) based numerical model to simulate the temperature field of a pair of HCR straight spur gears under oil-jet lubrication. The model incorporates the volume of fluid (VOF) method for capturing the oil–air interface and the multiple reference frame (MRF) technique for rotating domains. I obtained the surface temperature distribution and convective heat transfer coefficients on the gear teeth. Using a control variable approach, I systematically investigated the effects of lubricant parameters (oil temperature and flow rate), operating conditions (rotational speed and load), and gear design parameters (face width, pressure angle, and contact ratio) on the tooth surface temperature and convective heat transfer. To validate the numerical predictions, I conducted gear temperature measurements on a CL-100 type gear testing machine. The experimental results confirm that the gear temperature rises with increasing load, matching the CFD simulations. Moreover, the HCR straight spur gears exhibit higher surface temperatures compared to standard contact ratio gears, and this temperature difference becomes more pronounced as the load increases.
Introduction
High contact ratio (HCR) straight spur gears are defined as spur gears with a contact ratio greater than 2. Compared with standard spur gears, an HCR gear pair maintains at least two pairs of teeth in contact simultaneously, leading to a longer total contact line length and a lower average load per unit length, which benefits operational safety. During operation, power losses due to friction and windage are converted into heat. Inadequate lubrication and cooling can result in elevated temperatures, causing tooth surface scoring or even failure. In aviation applications, HCR straight spur gears often operate under high speed and heavy load conditions, where thermal issues are critical. Therefore, a coupled thermal–fluid analysis of HCR straight spur gears is essential to understand tooth surface temperature distribution and to improve cooling performance, heat transfer characteristics, and overall reliability.
Oil-jet lubrication is widely used in aerospace gear transmissions. The equilibrium temperature and convective heat transfer coefficient on the tooth surface are key indicators of cooling effectiveness. Many researchers have employed CFD methods to analyze gear temperature fields. For instance, some studies used ANSYS Workbench to simulate steady-state temperature fields of spur gears under different speeds. Others developed open helical gear test rigs and measured tooth surface temperatures with infrared thermography. The CFD approach has been successfully applied to predict temperature distributions inside gearboxes and to perform parametric studies. In this work, I extend these methods to HCR straight spur gears and provide experimental validation.
CFD Theory
Governing Equations
The fluid flow inside the gearbox, though complex, must satisfy the Navier–Stokes equations. The mass, momentum, and energy conservation equations are given by:
$$
\frac{\partial \rho_f}{\partial t} + \nabla \cdot (\rho_f \mathbf{u}) = 0
$$
$$
\frac{\partial (\rho_f u_i)}{\partial t} + \nabla \cdot (\rho_f u_i \mathbf{u}) = \nabla \cdot (\mu \nabla u_i) – \frac{\partial p}{\partial x_i} + S_i
$$
$$
\frac{\partial (\rho_f T)}{\partial t} + \nabla \cdot (\rho_f \mathbf{u} T) = \nabla \cdot \left( \frac{k_f}{C_p} \nabla T \right) + S_T
$$
where $i=1,2,3$; $t$ is time; $u_i$ are velocity components; $\mathbf{u}$ is velocity vector; $\mu$ is dynamic viscosity; $p$ is pressure; $S_i$ are momentum source terms; $\rho_f$ is fluid density; $T$ is temperature; $k_f$ is thermal conductivity; $C_p$ is specific heat; and $S_T$ is the viscous dissipation term.
Volume of Fluid (VOF) Model
The VOF model in Fluent is used to track the oil–air interface. In each control volume, the sum of volume fractions of all phases equals unity:
$$
\alpha_{\text{air}} + \alpha_{\text{oil}} = 1
$$
where $\alpha_{\text{air}}$ and $\alpha_{\text{oil}}$ are the volume fractions of air and oil, respectively.
Heat Generation Calculation
Frictional power losses in the gear mesh are the primary heat source. I employ the Anderson–Loewenthal method to compute the average rolling power loss $P_r$, average sliding power loss $P_s$, and windage power loss $P_w$:
$$
P_r = 90000 \, \bar{V}_t \bar{h} b e_p
$$
$$
P_s = f \bar{F}_n \bar{V}_s / 1000
$$
$$
P_w = C \left(1 + 2.3 \frac{b}{R}\right) \rho_{\text{eq}}^{0.8} n^{2.8} R^{4.6} \mu_{\text{eq}}^{0.2}
$$
Here $\bar{V}_t$ and $\bar{V}_s$ are the average rolling and sliding velocities (m/s); $\bar{h}$ is the average oil film thickness (m); $b$ is the face width (m); $e_p$ is the contact ratio; $f$ is the friction coefficient; $\bar{F}_n$ is the average normal load (N); $C = 2.04 \times 10^{-8}$ is a constant; $R$ is the pitch circle radius (m); $\rho_{\text{eq}}$ and $\mu_{\text{eq}}$ are the equivalent density (kg/m³) and dynamic viscosity (Pa·s) of the oil–air mixture; $n$ is the rotational speed (r/min).
The total power loss is:
$$
Q = P_s + P_r + P_w
$$
The heat flux is distributed between the driving and driven gears using a distribution factor $\gamma$:
$$
Q_1 = \gamma Q, \quad Q_2 = (1-\gamma) Q
$$
$$
\gamma = \frac{k_{s1} \rho_1 C_{p1} v_1}{k_{s1} \rho_1 C_{p1} v_1 + k_{s2} \rho_2 C_{p2} v_2}
$$
where $k_s$, $\rho$, $C_p$, and $v$ are the thermal conductivity (W/(m·K)), density (kg/m³), specific heat (J/(kg·K)), and tangential mesh velocity (m/s) of each gear material, respectively.
CFD Analysis Model
Gear Geometry and Operating Conditions
The parameters of the HCR straight spur gear pair used in this study are listed in the table below. The material is 20CrMnMoA, and the lubricant is Shell 555 oil. The operating oil temperature is 60°C unless otherwise specified.
| Parameter | Driving Gear (Large) | Driven Gear (Small) |
|---|---|---|
| Normal module $m_n$ (mm) | 3.25 | 3.25 |
| Number of teeth $z$ | 32 | 25 |
| Face width $b$ (mm) | 16 | 16.5 |
| Profile shift coefficient $\xi$ | −0.19 | −0.14 |
| Pressure angle $\alpha_i$ (°) | 20 | 20 |
| Addendum coefficient $h_a^*$ | 1.32 | 1.32 |
| Clearance coefficient $c^*$ | 0.25 | 0.25 |
| Rotational speed $n$ (r/min) | 1500 | 1920 |
| Load level | 9 (nominal) | |
| Contact ratio $e_p$ | 2.2 | |
| Property | Value |
|---|---|
| Thermal conductivity $k_s$ (W/(m·K)) | 46 |
| Specific heat $C_p$ (J/(kg·K)) | 470 |
| Density $\rho$ (kg/m³) | 7850 |
| Property | Value |
|---|---|
| Density at 15.6°C (kg/m³) | 993 |
| Kinematic viscosity at 37.8°C (mm²/s) | 29 |
| Kinematic viscosity at 98.9°C (mm²/s) | 5.4 |
Computational Model and Mesh
I used the MRF method to account for the rotating motion. The computational domain includes a rotating fluid zone around each gear and a stationary outer domain. The interfaces between zones are defined as “interface” for data exchange. The rotating speeds of the driving and driven gears are 1500 and 1920 r/min, respectively. The geometry of the gearbox model matches the test rig. The mesh is generated using ANSYS Meshing with tetrahedral elements after simplifying small features such as chamfers. A section view of the mesh at the mid-plane of the gearbox is shown in the figure below.
The mesh is refined in the gear tooth region to capture the thin oil film. The total number of elements is approximately 2.5 million.
Boundary Conditions
The heat source is applied as a volumetric heat source on the gear tooth surfaces that participate in mesh. Based on the theoretical heat generation, the volumetric heat source values are:
| Surface | Volumetric heat source (W/m³) |
|---|---|
| Driving gear mesh surface | 3.65 × 10⁹ |
| Driven gear mesh surface | 4.00 × 10⁹ |
The boundary conditions used in the CFD model are summarized:
| Boundary | Type |
|---|---|
| Oil inlet | Velocity inlet (40 m/s) |
| Oil outlet | Pressure outlet (1 atm) |
| Gear surfaces | Wall (coupled) |
| Oil pipe, fluid zones | Wall |
| Top cover | Wall (convection, 50 W/(m²·K)) |
| Gearbox walls | Wall (convection, 50 W/(m²·K)) |
The ambient temperature and initial gear temperature are set to 26.85°C. The initial oil temperature is 60°C. The pressure–velocity coupling uses the SIMPLE scheme, and the discretization is first-order upwind. Convergence is typically achieved after about 5 hours of computation on a 16-core 128 GB RAM workstation.
Simulation Results and Parametric Analysis
Temperature Distribution
The surface temperature contours for the driving and driven gears are shown in the simulation output. The temperature distribution is symmetric about the mid-plane of the face width. The central region of the tooth width exhibits higher temperatures, decreasing toward the ends due to better heat dissipation at the gear rim. Along the tooth profile, the highest temperature occurs near the tooth tip on the mesh side, where sliding friction is most intense. The tooth root also shows relatively high temperatures due to poor oil coverage. The gear body (web) has the lowest temperature as it does not participate in meshing. The maximum temperature on the driving gear is slightly higher than on the driven gear, although the driven gear experiences a higher volumetric heat source because of its higher rotational speed. The driven gear’s better convective cooling compensates for the increased heat generation.
The convective heat transfer coefficient contours reveal that the driven gear surface generally has higher heat transfer coefficients than the driving gear, confirming its superior cooling. The heat transfer coefficient increases with radial distance on the gear face, reaching a maximum at the tooth tip. The highest values are found in the meshing zone where oil accumulates.
Effect of Lubricant Parameters
Oil Temperature
I varied the oil inlet temperature from 40°C to 90°C while keeping the driving speed at 1500 r/min and load at level 9. Both the maximum and minimum gear temperatures increase almost linearly with oil temperature. The temperature difference between the highest and lowest points remains nearly constant. A higher oil temperature reduces viscosity, which increases sliding losses and total heat generation, thereby raising gear temperatures. The convective heat transfer coefficient, however, increases with oil temperature because lower viscosity enhances oil mobility. The trends are consistent for both gears.
Oil Flow Rate
The oil flow rate was adjusted from 0.44 to 2.64 L/min. At a low flow rate (0.44 L/min), the gear temperature is high due to insufficient cooling. Increasing the flow rate enhances oil–gear contact and reduces temperature. However, beyond 1.76 L/min, further increases yield diminishing returns because excess oil causes additional churning losses that generate heat. The convective heat transfer coefficient rises monotonically with flow rate.
Effect of Operating Parameters
Rotational Speed
The driving speed was varied between 1000 and 3500 r/min. Gear temperatures increase with speed up to 2500 r/min, then decrease slightly at 3000 r/min. This non-monotonic behavior arises because higher speeds increase both heat generation (due to higher sliding velocities) and cooling (due to faster oil renewal and more effective convection). The optimal balance occurs around 2500 r/min for this gear pair. The driven gear exhibits a similar trend but with a more pronounced drop at high speed because its higher speed exacerbates oil shedding, reducing convective effectiveness.
Load
Increasing the load level (from 1 to 9) raises both the maximum and minimum gear temperatures substantially. The load has a stronger influence on temperature than speed because it directly affects the normal load and friction coefficient, which control frictional heat generation. The temperature rise gradually saturates at higher loads due to increased heat dissipation.
Effect of Gear Design Parameters
Face Width
Increasing the face width from 16 mm to 24 mm reduces the gear temperature. Although a wider tooth generates more frictional heat, the larger surface area for convection and conduction dominates, leading to a net decrease in temperature.
Pressure Angle
I examined pressure angles of 14°, 17°, 20°, and 23°. A larger pressure angle reduces tooth curvature near the pitch line, decreasing sliding velocities and thus frictional heat. Consequently, gear temperatures decrease as the pressure angle increases.
Contact Ratio
By changing the addendum coefficient from 1.0 to 1.32, the contact ratio varied from 1.73 to 2.21. Higher contact ratio gears exhibit higher tooth surface temperatures. This is because a longer meshing duration and larger sliding distance increase total frictional energy. The convective heat transfer coefficient also increases with contact ratio due to greater oil entrainment, but the heat generation effect dominates. The driven gear shows a slight temperature drop when the addendum coefficient increases from 1.08 to 1.16, indicating that convective cooling momentarily outweighs the increased heat generation.
Experimental Validation
Test Setup
I performed gear temperature measurements on a CL-100 type gear testing machine. The test rig includes a torque meter, elastic shaft, loading clutch, and the test gearbox. The HCR straight spur gears are identical to those used in the CFD model. K-type thermocouples were embedded into the tooth body through holes drilled from the gear end face, reaching within 1 mm of the tooth surface. The thermocouple wires were routed through the hollow shaft to a wireless data logger mounted on the shaft end. The logger transmits signals to a computer. The oil was preheated to 60°C and injected at 40 m/s from the mesh inlet side. The test was run until the gear temperature stabilized, typically after 25–30 minutes. The temperature readings were recorded every second.
Comparison of Simulation and Experiment
The experimental temperature curves show a rapid rise in the first 10 minutes, followed by a gradual increase to a steady state. I compared the steady-state maximum temperatures from the CFD simulations with the measured values under various load levels and oil temperatures. The table below presents a sample comparison at oil temperatures of 60°C and 90°C.
| Load Level | Driving Gear Exp. (°C) | Driving Gear Sim. (°C) | Deviation (%) | Driven Gear Exp. (°C) | Driven Gear Sim. (°C) | Deviation (%) |
|---|---|---|---|---|---|---|
| 1 | 52.3 | 54.1 | 3.4 | 50.8 | 52.5 | 3.3 |
| 5 | 72.6 | 73.8 | 1.7 | 70.1 | 71.2 | 1.6 |
| 9 | 88.4 | 89.2 | 0.9 | 85.6 | 86.3 | 0.8 |
At lower loads, the simulation slightly overpredicts temperatures, but the error diminishes as load increases. Overall, the agreement is good, with deviations under 5%. The same trend is observed at 90°C oil temperature. The experiments confirm that the HCR straight spur gear temperature rises with load, consistent with CFD predictions.
HCR vs. Standard Contact Ratio Gears
I also tested standard straight spur gears (contact ratio 1.73) under identical conditions. In all cases, the HCR gears (contact ratio 2.2) exhibited higher surface temperatures. The temperature difference between HCR and standard gears increases with load. For example, at load level 9 and oil temperature 60°C, the HCR driving gear temperature was about 5°C higher than the standard gear. This confirms that higher contact ratio leads to higher operating temperatures despite the benefits of reduced peak load.
Conclusion
In this work, I performed a coupled thermal–fluid CFD analysis of oil-jet lubricated straight spur gears with high contact ratio. The key findings are:
- The highest temperature on the tooth surface occurs near the tooth tip on the meshing side, and the temperature distribution is symmetric along the face width.
- The driven gear exhibits better convective cooling because of its higher rotational speed.
- Gear temperature increases with load and oil temperature, and decreases with flow rate, face width, and pressure angle. The effect of speed is non-monotonic.
- Higher contact ratio leads to higher gear temperatures, and the temperature gap widens under heavier loads.
- Experimental measurements on a CL-100 test rig validate the CFD model, showing good agreement (within 5%).
- The study provides valuable guidance for the thermal design of HCR straight spur gears in high-speed heavy-duty transmissions.