At a pitch-line speed of about 114 m/s, the 30-tooth pinion and 76-tooth wheel of an aircraft spiral bevel pair no longer move through still air. The teeth drive a rotating vortex, generate a centrifugal field, and leave low-pressure wakes. The resulting pressure and viscous shear on tooth flanks, end faces, and shafts is the windage part of load-independent spin loss. Once tooth speed exceeds about 50 m/s, windage becomes large enough to affect efficiency and heat rejection.
Transient CFD of this pair, with and without a shroud on the driven wheel, shows how clearance and mesh opening change torque. The best tested shroud reduced driven-wheel windage torque by 55.3% and total pair windage power by 38.4%. That difference matters: the wheel dominates the loss, but the pinion and the open mesh region prevent a single-gear percentage from carrying over directly to the pair.
Why Windage Matters in a High-Speed Bevel Pair
Load-independent spin loss includes bearing viscous dissipation, oil churning or windage, and compression/expansion of the lubricant or oil-air mixture in the mesh. In an air-dominated case, gear windage is set by the wetted area moving through the air, the air density, and the local velocity. For a single gear, the loss rises roughly with the cube of speed and the fifth power of diameter. In this pair, the 284.9 mm wheel turns at 7,626 r/min and the 103.95 mm pinion at 20,900 r/min. The speed ratio partly offsets the diameter ratio, but the wheel still carries most of the loss: about 1,781 W of the 2,704 W total, versus 924 W on the pinion. Its windage torque without a shroud is 2.23 N·m, compared with 0.422 N·m on the pinion. This is why the shroud is applied to the driven wheel.
Gear Pair and Transient CFD Model
| Parameter | Pinion | Wheel |
|---|---|---|
| Teeth z | 30 | 76 |
| Rotational speed n (r/min) | 20,900 | 7,626 |
| Large-end module mt (mm) | 3.85 | 3.85 |
| Pressure angle αn (°) | 20 | 20 |
| Spiral angle β (°) | 30 | 30 |
| Face width B (mm) | 38.5 | 38.5 |
| Shaft angle Σ (°) | 69.77 | |
| Hand | Right | Right |
| Pitch diameter D (mm) | 103.95 | 284.9 |
Tooth surfaces were generated by local synthesis and fitted in UG. The assembly was simplified to solid shafts and gear bodies; bearings, connectors, fillets, and chamfers were removed. A small tooth thinning left 2–3 boundary-layer cells between meshing flanks. Because the contact is point contact, this gap has little effect on windage, while it avoids mesh friction in the model.
The flow domain is the air between the gear surfaces and the housing. With a shroud, the domain is split into the gear-shroud region and the shroud-housing region. An ICEM unstructured mesh used a 4 mm maximum cell size and six boundary layers up to 1 mm thick on tooth surfaces, giving about 13 million cells.
Fluent solved the transient, pressure-based coupled flow. The RNG k-ε model was used because standard k-ε can distort strongly swirling and curved flows; RNG corrects the turbulent viscosity and handles high strain and streamline curvature better. A UDF set the wall rotation at 20,900 r/min for the pinion and 7,626 r/min for the wheel. Dynamic remeshing and spring smoothing updated the domain as the teeth moved. The time step was 2 × 10−6 s for 6,000 steps; momentum used second-order upwind and turbulence used first-order upwind. Air was treated as incompressible, and the Reynolds number was about 9 × 105.
Shroud Configurations and Loss Accounting
The shroud is a grade 45 steel shell bolted to the bottom of the main housing. It wraps the driven wheel flanks and has a sector cutout at the mesh. The test variables were the shroud-to-wheel clearance and the mesh-opening angle. In each run, the heel, toe, and flank clearances were set to the same value: 7, 5, or 1 mm. The opening angle was 45° or 60°. The unshrouded case is the baseline.
Torque was integrated over three surface groups on each gear: tooth flanks Tfi, end faces Tdi, and shaft surfaces Tsi. The total windage power is
$$P_w = \sum_{i=1}^{2} \frac{n_i \pi}{30} T_i, \qquad T_i = T_{fi}+T_{di}+T_{si}$$
where ni is in r/min and Ti is in N·m.
| Configuration | Clearance (mm) | Opening (°) | Pinion torque (N·m) | Wheel torque (N·m) | Total windage power (W) |
|---|---|---|---|---|---|
| No shroud | — | — | 0.422 | 2.23 | 2704.4 |
| Shroud | 7 | 45 | 0.405 | 1.60 | 2164.1 |
| Shroud | 7 | 60 | 0.370 | 1.50 | 2019.5 |
| Shroud | 5 | 45 | 0.408 | 1.40 | 2010.9 |
| Shroud | 5 | 60 | 0.406 | 1.45 | 2045.8 |
| Shroud | 1 | 45 | 0.397 | 0.997 | 1665.1 |
| Shroud | 1 | 60 | 0.408 | 1.13 | 1795.3 |
Clearance dominates. The 1 mm/45° case gives the lowest total, 1665.1 W, and the lowest wheel torque, 0.997 N·m. The pinion torque stays between 0.370 and 0.422 N·m across all runs, while the wheel torque falls from 2.23 to 0.997 N·m. The shroud therefore acts mainly on the wheel.
Opening angle is secondary and not monotonic. At 7 mm clearance, the 60° opening gives a lower total than 45° (2019.5 W versus 2164.1 W); at 5 mm and 1 mm, 45° gives the lower total. The differences are small compared with the effect of reducing clearance from 7 mm to 1 mm.
| Configuration | Tooth flanks (W) | End faces (W) | Shafts (W) | Total (W) |
|---|---|---|---|---|
| No shroud | 1892.81 | 757.12 | 54.47 | 2704.39 |
| 1 mm, 45° | 1182.15 | 382.95 | 99.95 | 1665.05 |
Tooth flanks account for about 70% of the total in both cases, end faces for 23–28%, and shafts for 2–6%.
What the Flow Field Shows
Pressure contours on the driven wheel show a consistent pattern. The maximum appears on teeth about to enter mesh; the minimum appears on teeth about to leave. As a tooth approaches contact, air in the tooth space is squeezed; as it leaves, the space expands and pressure drops. The shroud changes the surrounding pressure level but does not remove this mesh-induced pattern. The same behavior appears on the pinion.
In the X–Y section, the shroud lowers air speed in the region between shroud and housing. With a smaller clearance, the peak speed in the shroud-wheel gap rises, but the volume of high-speed fluid shrinks. This is the intended isolation effect: the shroud keeps the bulk of the housing air from being spun by the wheel, while the close gap produces a local high-shear layer. The 1 mm case therefore has a more confined high-velocity region than the 5 or 7 mm cases.
Turbulent kinetic energy contours show where turbulent dissipation is concentrated. At 7 mm, parts of the field had higher turbulent kinetic energy than the unshrouded case, so a large clearance is not simply a scaled-down version of a close shroud; it can leave a strong unsteady gap flow without effectively isolating the wheel. As clearance falls to 1 mm, the average turbulent kinetic energy near the tooth surfaces decreases. The opening angle has a weaker effect on turbulent kinetic energy than clearance. In every configuration, the mesh region has the highest turbulent kinetic energy, because the moving boundaries and repeated squeeze/expansion exchange energy there.
Best Configuration and Practical Limits
The 1 mm/45° shroud is the best of the tested configurations: 0.397 N·m on the pinion, 0.997 N·m on the wheel, and 1665.1 W total. Relative to the unshrouded case, wheel torque falls by 55.3% and total pair windage power by 38.4%; pinion torque falls by 5.9%. The 55.3% figure is a driven-wheel reduction, not a pair-total reduction. The 1 mm clearance is also a practical limit, because smaller gaps would tighten manufacturing and assembly tolerances.
Reported single-gear shrouds have reached 70–75% reduction. The lower pair-level value is consistent with the open mesh sector, the unshrouded pinion, and cross-flow between the two sides of the shroud. Closing the mesh region completely would interfere with tooth contact.
The absolute values are for single-phase air. In a real gearbox the medium is an oil-air mixture with higher density and viscosity, so the loss and the optimum clearance may shift. The model also neglects air compressibility and the compression/expansion work in the mesh, both of which add loss mechanisms. An unsealed housing exchanges mass and energy with the surroundings, increasing the participating fluid and windage. The flow volume below the driven wheel could be reduced further, but that is a housing-level change rather than a shroud parameter.
Design Implications
- Treat windage as a pair-level transient problem when mesh opening and squeeze flow matter; a steady single-gear model can rank shroud concepts but does not capture the mesh interaction.
- Place the shroud on the wheel, where diameter-driven loss is largest, and keep clearance as small as assembly allows; 1 mm gave the best result here.
- Use a 45° mesh opening as a starting point; opening angle is secondary to clearance and did not follow a monotonic trend across all clearances.
- Report whether a percentage reduction refers to the wheel or to the pair total; the two differ substantially (55.3% versus 38.4% in this case).
- Validate absolute power loss with oil-air mixture and sealed or unsealed housing conditions before using it for thermal sizing.
For this 30/76 pair at about 114 m/s pitch-line speed, a close-fitting shroud on the driven wheel is an effective windage control. The gain is concentrated on the wheel, and the pair total improves less than the wheel alone. The strongest benefit came at the smallest practical clearance: 1 mm with a 45° mesh opening. A 7 mm clearance still reduced loss, but it left a larger high-speed region and gave a smaller benefit.