Spiral bevel gears carry high torque in a compact envelope, run smoothly and tolerate the lubrication regimes found in aero-engine accessory and fan-drive trains, which is why they dominate high-speed bevel stages in aircraft, marine and automotive transmissions. Their efficiency budget, however, is not set by meshing alone. Gearbox power loss separates into load-dependent mesh loss and load-independent viscous loss, the latter comprising windage, churning and squeeze effects. Windage and churning arise from the drag and transport of the air–oil mixture over the flanks and end faces, while squeeze loss comes from fluid displaced out of the meshing zone. Once pitch-line velocity passes roughly 45 m/s, the air dragged around by the gear becomes dynamically significant and the aerodynamic torque can no longer be neglected in the thermal and efficiency balance; above about 50 m/s a stationary shroud is normally required to keep that torque within acceptable limits.
What earlier work established
Spur-gear correlations developed by Anderson and Loewenthal in 1981 predicted windage from geometry, operating condition and fluid properties, but were validated only to about 40 m/s. Dawson’s 1984 smoke visualisation of a disc-type gear in free air showed the underlying kinematics: air is drawn axially into the gear, then thrown out radially and tangentially, so the toothed rim behaves like the impeller of a centrifugal fan. The same experiments showed that a shroud reduces windage, but that the benefit depends on shroud geometry and on the gap between shroud and gear. Test-rig work on a shrouded spiral bevel gear by Johnson and co-workers in 2007 measured reductions of up to about 70%, and Winfree’s 2013 campaign compared eight shroud configurations to derive design rules for a single gear.
Computational fluid dynamics has since become the main tool for resolving the flow field that experiments can only infer. Farrall and co-workers (2005) analysed a shrouded bevel gear and concluded that the gaps at the flanks, heel and toe each have an optimum, with windage rising on either side of it. Rapley and co-workers (2007–2008) quantified how shroud shape modifies the single-phase flow, Webb and co-workers (2010) linked dimensionless torque to tooth count and extended the modelling to oil-mist injection, and Chinese groups have applied cyclic-symmetry and rotating-reference-frame models to spur, face and planetary gears, including the fan-drive planetary train of a geared turbofan. For spiral bevel gears specifically, the influence of speed, rotation direction and shroud clearance has been examined, but the interaction between the three clearances — that is, the effect of shroud shape as a whole — has not been quantified. That interaction is the subject of the analysis below.
Flow structure and the origin of the resisting torque
Windage torque on a rotating toothed wheel has two physically distinct sources. The first is the pressure difference between the flank that faces the oncoming air (the pressure side) and the leeward flank (the suction side). Flow separates at the tooth tip; part of the stream passes over the tip, while the rest is drawn into the tooth space and forms a vortex there. The resulting pressure imbalance produces a moment that opposes rotation and must be overcome by additional shaft power. The second source is wall shear: viscous drag on the flanks, end faces and shaft surface produces a further resisting moment. Both mechanisms convert shaft work into heat, raising the oil and gearbox temperature and degrading transmission efficiency.
For the wheel studied here, the two contributions are far from equal. Over the whole simulated speed range the pressure-driven moment accounts for 93% to 97% of the total flank windage moment, with its share increasing as speed rises, because inertial effects grow faster than viscous ones. The distinction between surfaces matters as much as the distinction between mechanisms: the summed moment on the heel and toe faces is an order of magnitude smaller than the moment on the flanks, so the tooth surface dominates the loss and the maximum static pressure develops around the heel region of the flank.
Numerical model and boundary conditions
Because the flow around a single rotating gear in a housing is geometrically complex and turbulent, the simulations use a shear-stress-transport (SST) k–ω closure, which blends a k–ε formulation in the free shear layer with a k–ω formulation near the wall through a blending function, retaining the strengths of both. Rotation is handled with the multiple reference frame (MRF) approach: the region between shroud and gear rotates with the gear, the region between shroud and housing is stationary, and the two are coupled through a mixing interface. The rotating zone occupies the smaller part of the computational domain, so the absolute-velocity formulation is used, in which the momentum equation for the relative velocity W retains a Coriolis term written in terms of the absolute velocity V.
Continuity and momentum conservation in the rotating frame are written as
$$\frac{\partial \rho}{\partial t} + \nabla\cdot\left(\rho \mathbf{W}\right) = 0$$
$$\frac{\partial \left(\rho \mathbf{V}\right)}{\partial t} + \nabla\cdot\left(\rho \mathbf{W}\mathbf{V}\right) + \rho\left(\boldsymbol{\omega}\times\mathbf{V}\right) = -\nabla p + \nabla\cdot\boldsymbol{\tau} + \mathbf{S}_m$$
with the viscous stress evaluated from gradients of the absolute velocity,
$$\boldsymbol{\tau} = \mu\left[\nabla\mathbf{V} + \left(\nabla\mathbf{V}\right)^{\mathrm{T}} – \frac{2}{3}\left(\nabla\cdot\mathbf{V}\right)\mathbf{I}\right]$$
and the velocity triangle relating relative, absolute and peripheral velocity,
$$\mathbf{V} = \mathbf{U} + \mathbf{W}, \qquad \mathbf{U} = \boldsymbol{\omega}\times\mathbf{r}$$
where ρ is density, t time, W and V the relative and absolute velocity vectors, ω the rotational speed of the moving frame, r the radius, p pressure, μ dynamic viscosity and Sm a body-force source term.
Two dimensionless groups set the regime. The Reynolds number
$$Re = \frac{\rho v l}{\mu}$$
with density ρ, velocity v, characteristic length l and viscosity μ, compares inertia with viscous action; for a gear turning at tens of thousands of revolutions per minute in air, Re reaches the order of 106, so the flow is strongly turbulent. The Mach number
$$Ma = \frac{v}{a}$$
with local sound speed a, remains below 0.3 because the maximum air velocity is comparable to the pitch-line velocity, so the air is treated as a viscous, subsonic, incompressible medium.
A single gear is isolated from the housing so that neighbouring components do not contaminate the flow field, and the domain is bounded by a conical-cylindrical enclosure resembling the actual casing. Bearing and support detail is omitted, and the model is built parametrically so that the shroud-to-flank, shroud-to-heel and shroud-to-toe gaps can be varied without rebuilding the geometry. The fluid domain is discretised with unstructured tetrahedra in ANSYS ICEM CFD, with local refinement on the flanks, end faces and shaft surface. A mesh-independence check gave roughly 5 million cells and about 800 000 nodes, with converged solutions after some 300 iterations. Gear surfaces rotate as no-slip walls in the moving frame; shroud and housing surfaces are stationary no-slip walls; all walls are adiabatic and are treated as hydraulically smooth.
| Parameter | Pinion | Wheel |
|---|---|---|
| Number of teeth | 30 | 76 |
| Rotational speed (r/min) | 20 000 | 7 626 |
| Large-end module (mm) | 3.85 | 3.85 |
| Normal pressure angle (°) | 20 | 20 |
| Spiral angle (°) | 30 | 30 |
| Face width (mm) | 38.5 | 38.5 |
| Shaft angle (°) | 69.77 | |
| Spiral hand | Right | — |
Pressure–velocity coupling uses the Coupled scheme, pressure is discretised with the PRESTO format appropriate to high rotational speeds, momentum with second-order upwind, and the turbulence quantities with first-order upwind. Shaft speed is swept from 1 000 to 15 000 r/min with the gear viewed from the small end and turning clockwise in the baseline case.
Speed scaling of windage torque and power
Torque on each surface family is extracted in post-processing and combined into a single windage power,
$$P_w = P_{wf} + P_{wt} + P_{wh} + P_{ws} = \left(T_{wf} + T_{wt} + T_{wh} + T_{ws}\right)\frac{\pi n}{30}$$
where the subscripts refer to the flank, heel, toe and shaft, T denotes the corresponding moment and n the rotational speed in r/min. Both the pressure-driven and the viscous moment rise monotonically with speed, and so does the total. Beyond about 8 000 r/min — a large-end tangential velocity of roughly 120 m/s — the growth in total moment becomes steeper, and windage turns from a correction into a dominant term.
Fitting the simulated power values gives a power law that reproduces the data better than linear or quadratic alternatives:
$$P_w = 2.7252\times10^{-13}\,n^{3.1986}$$
with n in r/min. The exponent, close to 3.2 rather than the classical 3, is the transferable result; the prefactor is specific to this geometry, speed range and fluid. A cubic dependence on speed is the standard empirical expectation for turbulent windage, and its recovery here confirms that the model reproduces the accepted physical trend rather than an artefact of the discretisation.
Rotation direction and the impeller analogy
Because a toothed rim resembles a centrifugal impeller, the direction of rotation should matter, and it does. Four configurations were compared: shrouded and unshrouded, each run clockwise and counter-clockwise. Without a shroud, counter-clockwise rotation produces a windage moment about 22% higher than clockwise rotation. With a shroud, the two directions differ by less than about 5%, because the shroud suppresses the large-scale recirculation that the direction change otherwise modulates. All four curves show a local rise in dimensionless torque near 3 000 r/min, a feature that matches measurements made on high-speed bevel gears at the University of Nottingham.
The blade-angle language of turbomachinery makes the asymmetry intelligible. The impeller inlet angle β1 fixes the direction in which fluid enters the blade passages, and the outlet angle β2 fixes the theoretical head; blading is classed as backward-curved for β2 < 90°, forward-curved for β2 > 90° and radial at 90°. Viewed from the small end, counter-clockwise rotation makes the leading and trailing edges of each spiral tooth lean away from the direction of motion, which is the backward-curved configuration; clockwise rotation tilts the trailing edge with the rotation, in the forward-curved sense, while the leading edge leans away to limit separation in the passage. The resulting velocity triangles at inlet and outlet, in which the absolute velocity resolves into a flow-rate-related radial component and a head-related tangential component, explain why the two directions load the air differently. The analogy is qualitative, but it predicts the observed ordering and is consistent with the shroud’s ability to flatten the difference.
Shroud clearance as a three-factor problem
Introducing a stationary shroud close to the rotating gear restricts the volume of air entrained into the tooth spaces and cuts the windage moment substantially. The remaining design question is how close the shroud should be at three locations: the gap to the tooth flank (factor A), the gap to the heel or large end (factor B) and the gap to the toe or small end (factor C). Because these gaps are spatially adjacent, their effects cannot be assumed additive, so the three factors were varied together in a four-level orthogonal array L16(45) with two interaction columns, A×B and A×C. Clearance levels were 1, 3, 5 and 7 mm for every factor, and the response was the windage torque on the gear.
| Level | A: flank gap (mm) | B: heel gap (mm) | C: toe gap (mm) |
|---|---|---|---|
| 1 | 1 | 1 | 1 |
| 2 | 3 | 3 | 3 |
| 3 | 5 | 5 | 5 |
| 4 | 7 | 7 | 7 |
| Run | A | B | A×B | C | A×C | Torque (N·m) |
|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 | 1 | 0.351 11 |
| 2 | 1 | 2 | 2 | 2 | 2 | 0.503 89 |
| 3 | 1 | 3 | 3 | 3 | 3 | 0.537 07 |
| 4 | 1 | 4 | 4 | 4 | 4 | 0.562 63 |
| 5 | 2 | 1 | 2 | 3 | 4 | 0.449 96 |
| 6 | 2 | 2 | 1 | 4 | 3 | 0.475 13 |
| 7 | 2 | 3 | 4 | 1 | 2 | 0.467 40 |
| 8 | 2 | 4 | 3 | 2 | 1 | 0.617 44 |
| 9 | 3 | 1 | 3 | 4 | 2 | 0.469 32 |
| 10 | 3 | 2 | 4 | 3 | 1 | 0.566 33 |
| 11 | 3 | 3 | 1 | 2 | 4 | 0.599 99 |
| 12 | 3 | 4 | 2 | 1 | 3 | 0.559 87 |
| 13 | 4 | 1 | 4 | 2 | 3 | 0.489 11 |
| 14 | 4 | 2 | 3 | 1 | 4 | 0.479 12 |
| 15 | 4 | 3 | 2 | 4 | 1 | 0.603 70 |
| 16 | 4 | 4 | 1 | 3 | 2 | 0.651 35 |
Range analysis of the level means ranks the factors unequivocally: heel gap (R = 0.158 N·m) dominates, toe gap (R = 0.088 N·m) follows, flank gap (R = 0.067 N·m) is third, and the interaction columns (R = 0.019 and 0.010 N·m) are an order of magnitude weaker. Level means for A and B increase monotonically with clearance, so the smallest gap is best in both cases; C behaves similarly but with a weaker, non-monotonic trend. The best combination is therefore A1B1C1, all gaps at 1 mm, which corresponds to the first array run and gives 0.351 11 N·m — 46% below the worst configuration at 0.651 35 N·m.
| Column | Level 1 | Level 2 | Level 3 | Level 4 | Range R |
|---|---|---|---|---|---|
| A | 0.489 | 0.502 | 0.549 | 0.556 | 0.067 |
| B | 0.440 | 0.506 | 0.552 | 0.560 | 0.158 |
| A×B | 0.519 | 0.529 | 0.526 | 0.521 | 0.010 |
| C | 0.464 | 0.553 | 0.551 | 0.528 | 0.088 |
| A×C | 0.535 | 0.523 | 0.515 | 0.523 | 0.019 |
Significance was tested by the F ratio at a level of α = 0.10, with the two interaction terms pooled into the error estimate because their ranges are small.
| Source | Sum of squares | Degrees of freedom | Mean square | F | F0.1(3, 6) |
|---|---|---|---|---|---|
| A | 0.013 | 3 | 0.004 | 25.76 | 3.29 |
| B | 0.054 | 3 | 0.018 | 108.58 | 3.29 |
| C | 0.021 | 3 | 0.007 | 40.83 | 3.29 |
| Error | 0.001 | 6 | 0.000 2 | — | — |
All three main effects are significant, with the heel gap the strongest by a factor of four in F. The physical reading is straightforward: the tighter the enclosure formed by shroud and gear, the smaller the mass of air dragged into the tooth spaces, the lower the turbulent kinetic energy in the passages and the smaller the dissipation. A point estimate built from the grand mean plus the significant main and interaction effects, ŷ = μ̂ + â1 + b̂1 + (ab)11 + ĉ1 + (ac)11, returns 0.351 11 N·m, identical to the measured best run, so the fitted main-effect structure is consistent with the array.
Response surfaces and constrained optimisation
To make the results comparable across speeds, the torque is expressed as a dimensionless coefficient
$$C_m = \frac{T}{\tfrac{1}{2}\rho\omega^{2}R^{5}}$$
with T the windage moment, ω the angular velocity and R a reference radius. Least-squares regression of the sixteen runs, with x1, x2 and x3 the flank, heel and toe gaps in millimetres, gives a polynomial surface
$$\begin{aligned}
C_m = {} & 1.103\times10^{-2} + 3.010\times10^{-4}x_1 + 1.437\times10^{-3}x_2 + 2.133\times10^{-3}x_3 \\
& + 9.705\times10^{-6}x_1^{2} – 1.919\times10^{-5}x_2^{2} – 2.551\times10^{-4}x_3^{2} \\
& – 3.394\times10^{-5}x_1x_2 + 7.896\times10^{-5}x_1x_3
\end{aligned}$$
and an exponential alternative
$$C_m = 0.0698\,e^{\,0.0071x_1 + 0.0146x_2 + 0.0047x_3} – 0.056$$
Both surfaces are monotonic in the heel gap, which reproduces the dominance of factor B, and both carry a negative quadratic term in the toe gap, so the toe dependence is not monotonic within the sampled window.
Treating the shroud as a single-objective design problem under the casing envelope, the optimisation is
$$\min\ C_m\left(x_1, x_2, x_3\right) \quad \text{subject to} \quad 0 < x_1 \le 7,\; 0 < x_2 \le 7,\; 0 < x_3 \le 7\ \text{mm}$$
For the polynomial surface the optimum falls at x1 = 7.23×10−5 mm, x2 = 1.39×10−5 mm and x3 = 9.54×10−6 mm with Cm = 0.011 03; for the exponential surface it falls at x1 = 3.54×10−5 mm, x2 = 1.85×10−5 mm and x3 = 6.10×10−5 mm with Cm = 0.013 60. In both cases the mathematical minimum sits at the lower bound, effectively at zero clearance, which agrees with the orthogonal ranking but should not be read as a manufacturing target: neither fit was sampled below 1 mm, and the near-zero solutions are extrapolations of a quadratic model.
The lower bound on a practical clearance is set by mechanics, not aerodynamics. A shroud too close to a fast-running gear can be drawn toward the tooth tips by the pressure field; shaft whirl and vibration displace the axis; manufacturing and assembly tolerances, shroud deflection under differential pressure, and the oil flow required for cooling and lubrication all consume part of the available gap. Taking these constraints together with casing stiffness, transmitted power and available machining accuracy, 1 mm emerges as the recommended minimum, which coincides with the best orthogonal configuration.
Design implications and limits of validity
Three conclusions follow for high-speed spiral bevel gears. First, windage is a flank phenomenon driven by pressure: the pressure-side/suction-side imbalance supplies more than nine-tenths of the resisting moment and grows in relative importance with speed, so profiling or shroud measures aimed at the flanks are worth more than work on the end faces. Second, the loss follows an almost cubic speed law, which means that a gear moved from 8 000 to 15 000 r/min sees its windage power rise by roughly an order of magnitude, and thermal design must be sized for the top of the speed envelope rather than the cruise point. Third, a shroud is most effective when the enclosed volume is minimised, and the heel gap is the single most influential clearance, with the toe gap next; the interaction of the gaps is weak enough at the 0.1 significance level to be treated as error in a first-pass design.
The scope of these findings should be stated plainly. The model treats a single gear in air, isolated from the mesh and from neighbouring rotors, so it excludes the meshing-zone interaction, oil-mist two-phase effects and the housing flow that a full gearbox simulation would capture. The MRF approach is steady and frozen-rotor, so unsteady tooth-passing phenomena are averaged; walls are adiabatic and smooth, and no elastic deformation or thermal growth of either gear or shroud is represented. The power-law prefactor is tied to one geometry, and the response surfaces are valid only over the 1–7 mm range that was sampled. Within those limits, the results give a consistent picture: pressure-driven flow on the flanks sets the loss, speed amplifies it almost cubically, and the tightest shroud that mechanical and lubrication constraints allow is the best aerodynamic choice.