In modern mechanical transmission systems, helical gears are widely used due to their high load capacity and smooth operation. However, at high rotational speeds, windage power loss becomes a significant factor affecting transmission efficiency. This study focuses on analyzing the windage power loss of high-speed helical gears under various operating conditions and optimizing the design using computational fluid dynamics (CFD). The primary objective is to investigate the impact of rotational speed and shroud configurations on windage power loss, with an emphasis on helical gears in meshing conditions. We employ a simplified gearbox model and transient simulations to capture the complex fluid dynamics involved.
The fundamental equations governing the fluid domain include the conservation of mass, momentum, and energy. For incompressible flow, the continuity equation is given by:
$$ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0 $$
where \( \rho \) is the fluid density and \( \mathbf{v} \) is the velocity vector. The momentum conservation equation, or the Navier-Stokes equation, for a Newtonian fluid is:
$$ \frac{\partial (\rho \mathbf{v})}{\partial t} + \nabla \cdot (\rho \mathbf{v} \mathbf{v}) = -\nabla p + \nabla \cdot (\mu \nabla \mathbf{v}) + \mathbf{f} $$
Here, \( p \) represents pressure, \( \mu \) is the dynamic viscosity, and \( \mathbf{f} \) denotes body forces such as gravity. The energy equation is considered but simplified for isothermal conditions in this analysis. Given the high Reynolds numbers involved, the flow is turbulent, and we utilize the RNG k-ε turbulence model. The turbulent kinetic energy \( k \) and its dissipation rate \( \varepsilon \) are governed by:
$$ \frac{\partial (\rho k)}{\partial t} + \nabla \cdot (\rho k \mathbf{v}) = \nabla \cdot \left( \alpha_k \mu_{\text{eff}} \nabla k \right) + G_k – \rho \varepsilon $$
$$ \frac{\partial (\rho \varepsilon)}{\partial t} + \nabla \cdot (\rho \varepsilon \mathbf{v}) = \nabla \cdot \left( \alpha_\varepsilon \mu_{\text{eff}} \nabla \varepsilon \right) + C_{1\varepsilon} \frac{\varepsilon}{k} G_k – C_{2\varepsilon} \rho \frac{\varepsilon^2}{k} $$
where \( \mu_{\text{eff}} \) is the effective viscosity, \( G_k \) represents the generation of turbulent kinetic energy due to mean velocity gradients, and \( \alpha_k \), \( \alpha_\varepsilon \), \( C_{1\varepsilon} \), and \( C_{2\varepsilon} \) are model constants. The Reynolds number \( Re \) and Mach number \( Ma \) are critical parameters defined as:
$$ Re = \frac{\rho v l}{\mu} $$
$$ Ma = \frac{u}{a} $$
For helical gears operating at high speeds, \( Re \) often exceeds \( 3 \times 10^4 \), confirming turbulent flow, while \( Ma < 0.3 \) indicates incompressible flow conditions.
To model the helical gears, we developed a CFD-based approach using a simplified gearbox geometry. The helical gears have specific parameters, as summarized in Table 1. The model neglects minor components like shafts and bearings to focus on the fluid-structure interaction. The gear meshing region is handled using a tooth surface displacement method to ensure high-quality mesh generation in the narrow gaps.
| Parameter | Driving Gear | Driven Gear |
|---|---|---|
| Helix Direction | Left (L) | Right (R) |
| Number of Teeth | 380 | 23 |
| Normal Module (mm) | 1.75 | 1.75 |
| Pressure Angle (°) | 20 | 20 |
| Helix Angle (°) | 18.6 | 18.6 |
| Face Width (mm) | 50 | 50 |
The fluid domain is created by subtracting the gear and shroud volumes from the gearbox, resulting in a complex geometry for simulation. Mesh generation involves unstructured grids with local refinement near the gear teeth, totaling approximately 27 million elements. Dynamic meshing techniques, including spring-based smoothing and local remeshing, are employed to handle the gear rotation, controlled via user-defined functions (UDFs). The simulation settings include a pressure-based transient solver with a time step of \( 1 \times 10^{-6} \) s for gear pair analyses, ensuring accurate capture of fluid dynamics.
Windage power loss is calculated based on the torque exerted by the fluid on the gear surfaces. The total torque \( T_i \) for each gear (where \( i = 1 \) for driving gear and \( i = 2 \) for driven gear) is the sum of the torque on the tooth faces \( T_{fi} \) and the end faces \( T_{di} \):
$$ T_i = T_{fi} + T_{di} $$
The total windage power loss \( P_w \) for the gear pair is then:
$$ P_w = \sum_{i=1}^{2} \frac{n_i \pi}{30} T_i $$
where \( n_i \) is the rotational speed in rpm. This formula allows us to quantify the energy dissipation due to aerodynamic effects in helical gears.
We conducted simulations in three phases: first, to assess the effect of rotational speed on windage loss; second, to optimize shroud design for a single helical gear; and third, to evaluate shroud configurations for the entire gear pair. The rotational speed of the driving gear varied from 1,000 to 2,500 rpm, while the driven gear rotated at a proportional speed based on the gear ratio. Results indicate that windage power loss increases with speed, as shown in Table 2, due to higher fluid velocities and intensified turbulence.
| Driving Gear Speed (rpm) | Driving Gear Power Loss (W) | Driven Gear Power Loss (W) | Total Power Loss (W) |
|---|---|---|---|
| 1,000 | 450.2 | 1,200.5 | 1,650.7 |
| 1,500 | 680.9 | 1,650.8 | 2,331.7 |
| 2,000 | 987.4 | 2,001.2 | 2,988.6 |
| 2,500 | 1,350.6 | 2,500.3 | 3,850.9 |
For shroud optimization, we tested various configurations on the driving helical gear at 2,000 rpm. The shroud types included radial, axial, and combined designs with different clearances. The windage power loss results are summarized in Table 3. The combined radial and axial shroud with 5 mm gaps proved most effective, reducing power loss by 23.7% compared to the unshrouded case. This is attributed to restricted air entry into the tooth spaces, which minimizes turbulent kinetic energy generation.
| Shroud Type | Radial Clearance (mm) | Axial Clearance (mm) | Tooth Face Torque (N·m) | End Face Torque (N·m) | Power Loss (W) |
|---|---|---|---|---|---|
| None | N/A | N/A | 3.75 | 0.97 | 987.4 |
| Radial | 5 | N/A | 3.41 | 1.01 | 927.3 |
| Axial | N/A | 5 | 3.71 | 1.05 | 1,000.2 |
| Radial + Axial | 5 | 5 | 2.52 | 1.09 | 753.5 |
| Radial + 2 Axial | 5 | 5 | 2.44 | 1.12 | 744.0 |
Analysis of turbulent kinetic energy and velocity contours reveals that shrouds alter the flow field around helical gears. For instance, the radial shroud reduces air ingestion from the tooth sides, while axial shrouds limit entry from the ends. However, excessively small clearances can lead to localized high-velocity regions and increased turbulence in the meshing zone. In the case of helical gear pairs, we extended the study to include shrouds on the driven gear. Table 4 shows the windage power loss for different radial clearances on the driven helical gear, with fixed axial clearances of 5 mm. The optimal configuration achieved a 14.8% reduction in total power loss, but smaller clearances did not always yield better results due to complex flow interactions in the meshing region.
| Driven Gear Radial Clearance (mm) | Driving Gear Power Loss (W) | Driven Gear Power Loss (W) | Total Power Loss (W) |
|---|---|---|---|
| None | 1,088.8 | 1,899.8 | 2,988.6 |
| 2.5 | 952.9 | 1,631.5 | 2,584.4 |
| 5.0 | 976.8 | 1,591.1 | 2,567.9 |
| 7.5 | 985.0 | 1,572.3 | 2,557.3 |
| 10.0 | 984.0 | 1,832.6 | 2,816.6 |
The velocity distribution in the meshing area of helical gears is influenced by shroud placement. We observed that helical gears primarily draw air axially and expel it through the tooth faces. Shrouds can mitigate this by controlling flow paths, but if the clearance is too small, vortex interactions may increase power loss. The turbulent kinetic energy \( k \) is directly related to windage loss, as it represents the conversion of mechanical energy into fluid turbulence. For helical gears, the relationship can be expressed as:
$$ P_w \propto \int_V \rho \varepsilon \, dV $$
where \( \varepsilon \) is the dissipation rate and the integral is over the fluid volume. This highlights the importance of minimizing turbulence in critical regions.
In conclusion, our CFD-based analysis demonstrates that shrouds are effective in reducing windage power loss for high-speed helical gears. The optimal shroud design for a single helical gear involves combined radial and axial clearances of 5 mm, while for helical gear pairs, a balanced approach is necessary to avoid adverse flow effects. These findings provide valuable insights for designing efficient transmission systems involving helical gears, emphasizing the need to consider fluid dynamics in gear optimization. Future work could explore the impact of oil-air mixtures and different gear geometries on windage loss in helical gears.