High-speed spiral bevel gears are widely used in aero-engine accessory drives, helicopter transmissions and other compact systems where shafts intersect and high load capacity must be combined with smooth motion. At high pitch-line speed, however, the teeth also pump air. The resulting drag torque, known as windage, acts opposite rotation and converts mechanical power into heat. Unlike mesh loss, it does not vanish at zero load. Once pitch-line speed exceeds about 45–50 m/s, windage becomes a design constraint rather than a secondary correction. A well-fitted shroud can remove a large part of this loss; tests on a shrouded spiral bevel gear have measured reductions of about 70%, but the benefit depends strongly on shroud geometry and on the clearances to the tooth flank, heel and toe.
Where the loss comes from
Gearbox power loss is commonly divided into load-dependent mesh loss and load-independent viscous loss. The latter includes windage, churning and squeezing. Windage is the drag of air on the rotating teeth and faces; churning involves an oil–air mixture; squeezing arises as lubricant is displaced through the mesh. In a single-phase air environment, windage is the part that remains when the gear runs freely.
The flow around a gear is not a uniform shear layer. The tooth spaces act as passages of a centrifugal impeller. Air is drawn axially into the spaces near the root, accelerated by the flanks and discharged at the tip in the tangential and radial directions. On the pressure side, the relative flow stagnates and static pressure rises. On the suction side, the flow accelerates and pressure falls. The pressure difference integrated over the tooth produces a torque opposite rotation. Viscous shear on the flanks and end faces adds a smaller torque in the same direction.
For the spiral bevel gear considered here, pressure-driven torque accounts for 93–97% of the total windage loss, and its share rises with speed. The tooth flanks dominate: the sum of the heel-face and toe-face torques is about an order of magnitude lower than the tooth-flank torque. The maximum static pressure appears near the large end of the tooth. Flow separation at the tooth tip feeds a vortex inside each tooth space, so the loss is not simply skin friction on a smooth surface. The enclosed volume around the teeth, and therefore the amount of air that can enter and leave the spaces, has a direct effect on the loss.
Early work established the main features. A 1981 correlation by Anderson and Loewenthal for spur gears included geometry, operating conditions and fluid properties, but was limited to tip speeds of about 40 m/s. Dawson’s 1984 smoke-flow tests showed that a gear behaves like a centrifugal fan: air enters axially, is turned in the tooth spaces and leaves radially and tangentially. Those tests also showed that a shroud can reduce drag when its geometry and clearance are correctly chosen. Later experiments on a shrouded spiral bevel gear measured windage reductions of approximately 70%.
CFD model and operating envelope
The flow field is resolved with a finite-volume CFD model of a single spiral bevel gear in a closed, conical–cylindrical casing. The casing resembles the gearbox envelope but omits bearings and fixed supports. The shroud and the clearances to the tooth flank, heel and toe are parameterized so that their effects can be compared without rebuilding the complete gearbox.
The domain is divided into a rotating zone between the shroud and the gear or shaft, and a stationary zone between the shroud and the housing. An interface couples the two zones. The rotating zone uses the multiple reference frame, or MRF, approach. Gear teeth, heel and toe faces, and the shaft surface are no-slip rotating walls; the shroud and housing inner surfaces are no-slip stationary walls. All walls are treated as adiabatic and smooth. The absolute velocity formulation is used with a Coupled pressure–velocity scheme, PRESTO pressure discretization, second-order upwind momentum discretization and first-order upwind treatment of turbulence quantities.
The continuity and momentum equations in the absolute velocity form are
$$
\frac{\partial \rho}{\partial t}+\nabla\cdot(\rho\mathbf W)=0
$$
$$
\frac{\partial(\rho\mathbf V)}{\partial t}
+\nabla\cdot(\rho\mathbf W\mathbf V)
+\rho(\boldsymbol\omega\times\mathbf V)
=-\nabla p+\nabla\cdot\boldsymbol\tau+\mathbf S_m
$$
where the viscous stress is
$$
\boldsymbol\tau=\mu\left[\nabla\mathbf V+(\nabla\mathbf V)^{\mathsf T}-\frac{2}{3}(\nabla\cdot\mathbf V)\mathbf I\right]
$$
and the absolute and relative velocities are related by
$$
\mathbf V=\mathbf U+\mathbf W,\qquad \mathbf U=\boldsymbol\omega\times\mathbf r
$$
Here \(\rho\) is density, \(t\) is time, \(\mathbf W\) and \(\mathbf V\) are relative and absolute velocity vectors, \(\boldsymbol\omega\) is the rotational speed of the moving frame, \(\mathbf r\) is the radius vector, \(p\) is pressure, \(\mu\) is dynamic viscosity, \(\boldsymbol\tau\) is the viscous stress tensor and \(\mathbf S_m\) represents body forces.
At 15 °C the speed of sound in air is about 340 m/s. The maximum air speed near the gear is of the same order as the gear surface speed, so the Mach number remains below 0.3 and the flow can be treated as viscous, subsonic and incompressible. The Reynolds number,
$$
Re=\frac{\rho v l}{\mu}
$$
reaches the order of \(10^6\). Under these conditions the flow around the complex tooth geometry is strongly turbulent. The shear stress transport \(k\)-\(\omega\) model is used because it retains the near-wall behavior of the Wilcox \(k\)-\(\omega\) formulation while recovering the free-stream performance of the \(k\)-\(\varepsilon\) model.
The single-gear sweep covers 1,000–15,000 r/min. The gear pair itself has a rated operating point of 20,000 r/min for the 30-tooth driving gear and 7,626 r/min for the 76-tooth driven gear. The computational mesh has about 5 million cells and 0.8 million nodes, with local refinement on the tooth flanks, heel and toe faces, and shaft surface. The solution reaches the convergence criterion after roughly 300 iterations.
| Parameter | Value |
|---|---|
| Number of teeth | 30 driving, 76 driven |
| Rated speed | 20,000 r/min driving, 7,626 r/min driven |
| Large-end transverse module | 3.85 mm |
| Normal pressure angle | 20° |
| Spiral angle | 30° |
| Face width | 38.5 mm |
| Shaft angle | 69.77° |
| Hand | Right, driving gear |
Speed dependence
Both windage torque and windage power rise monotonically with speed. The increase is gradual at low speed and becomes steeper after about 8,000 r/min, where the heel tangential speed is approximately 120 m/s. At this point the air is strongly accelerated through the tooth spaces, the pressure difference across each tooth grows, and the loss becomes more sensitive to further speed increases.
The windage power extracted from the CFD results is obtained from the sum of the torques on the tooth flanks, heel face, toe face and shaft surface:
$$
P_w=(T_{\mathrm{tooth}}+T_{\mathrm{heel}}+T_{\mathrm{toe}}+T_{\mathrm{shaft}})\frac{\pi n}{30}
$$
where \(P_w\) is windage power in watts, the torques are in N·m and \(n\) is rotational speed in r/min. A least-squares fit over the full sweep gives
$$
P=2.7252\times10^{-13}n^{3.1986}
$$
with \(P\) in W and \(n\) in r/min. The exponent 3.1986 is close to a cubic law, consistent with classical windage correlations and with the view that the drag force scales with dynamic pressure and the torque with the square of speed times a characteristic radius. The pressure-driven share of the loss rises from about 93% to 97% over the same range.
Rotation direction and the impeller analogy
Viewed from the toe toward the heel, the gear can rotate clockwise or counter-clockwise. Four cases were compared: clockwise and counter-clockwise rotation, each with and without a shroud. Without a shroud, counter-clockwise rotation gives a windage torque about 22% higher than clockwise rotation at the same speed. With a shroud, the difference between the two directions falls within about 5%. A local rise in torque appears near 3,000 r/min in all four cases, consistent with earlier experiments at the University of Nottingham.
The direction effect can be read with the help of centrifugal impeller theory. The inlet blade angle \(\beta_1\) controls the direction at which fluid enters the passage, while the outlet blade angle \(\beta_2\) determines the head–flow characteristic. Backward-curved blades have \(\beta_2<90^\circ\), forward-curved blades have \(\beta_2>90^\circ\), and radial blades have \(\beta_2=90^\circ\). The helix of a spiral bevel tooth presents a similar choice of incidence and passage curvature. In one direction the tooth passage resembles backward-curved blading; in the other, the trailing edge tilts with rotation while the leading edge tilts against it. The shroud removes much of the axial inlet path and suppresses the external recirculation, so the flow becomes less sensitive to the sign of rotation.
Shroud clearance: orthogonal experiment
Three clearances define the fit between shroud and gear: the tooth-flank clearance \(A\), the large-end or heel clearance \(B\), and the small-end or toe clearance \(C\). Each was set at 1, 3, 5 and 7 mm. Because the three gaps are spatially adjacent, their interactions cannot be ignored. A four-level orthogonal array \(L_{16}(4^5)\) was used, with \(A\), \(B\), \(C\), \(A\times B\) and \(A\times C\) assigned to its columns. Sixteen steady CFD runs produced windage torques from 0.35111 to 0.65135 N·m.
An analysis of variance with the \(F\) test at a significance level of 0.1 gives a critical value of 3.29. The three main factors are significant, and the interactions are not. The heel clearance has the largest effect, followed by the toe clearance and then the tooth-flank clearance. The ranges of the level means are 0.158 for \(B\), 0.088 for \(C\), 0.067 for \(A\), 0.019 for \(A\times C\) and 0.010 for \(A\times B\).
| Source | Sum of squares | df | Mean square | F | Significance |
|---|---|---|---|---|---|
| A, tooth-flank clearance | 0.013 | 3 | 0.004 | 25.76 | Significant |
| B, heel clearance | 0.054 | 3 | 0.018 | 108.58 | Significant |
| C, toe clearance | 0.021 | 3 | 0.007 | 40.83 | Significant |
| Error, \(A\times B\) and \(A\times C\) pooled | 0.001 | 6 | 0.0002 | — | — |
The total sum of squares is 0.089 with 15 degrees of freedom. The best combination in the orthogonal array is \(A_1B_1C_1\), that is, 1 mm at all three locations. Its torque is 0.35111 N·m, the lowest of the sixteen runs. The result is consistent with the flow mechanism: a smaller enclosed volume admits less air into the tooth spaces, reduces turbulent kinetic energy and lowers the energy dissipated by the gear. In this sense, the shroud should follow the gear as closely as the mechanical design permits.
Response-surface fit and constrained optimum
For comparison across speeds and geometries, the torque is written as a dimensionless coefficient
$$
C_m=\frac{T}{0.5\,\rho\,\omega^2R^5}
$$
where \(T\) is windage torque, \(\rho\) is air density, \(\omega\) is angular velocity and \(R\) is a characteristic gear radius. With \(x_1\), \(x_2\) and \(x_3\) denoting the tooth-flank, heel and toe clearances in millimetres, a least-squares fit to the sixteen runs gives the polynomial form
$$
\begin{aligned}
C_m={}&1.103\times10^{-2}+3.010\times10^{-4}x_1+1.437\times10^{-3}x_2+2.133\times10^{-3}x_3\\
&+9.705\times10^{-6}x_1^2-1.919\times10^{-5}x_2^2-2.551\times10^{-4}x_3^2\\
&-3.394\times10^{-5}x_1x_2+7.896\times10^{-5}x_1x_3
\end{aligned}
$$
and the exponential form
$$
C_m=0.0698\,e^{\,0.0071x_1+0.0146x_2+0.0047x_3}-0.056
$$
Both fits decrease over the tested clearance range. Under the constraints \(0<x_i\le 7\)="" clearance.
| Model | \(x_1\) (mm) | \(x_2\) (mm) | \(x_3\) (mm) | \(C_{m,\min}\) |
|---|---|---|---|---|
| Polynomial | \(7.23\times10^{-5}\) | \(1.39\times10^{-5}\) | \(9.54\times10^{-6}\) | 0.01103 |
| Exponential | \(3.54\times10^{-5}\) | \(1.85\times10^{-5}\) | \(6.10\times10^{-5}\) | 0.01360 |
Zero clearance is not an operable design. At high speed, the low pressure in the tooth spaces can draw the shroud toward the gear. Shaft vibration, manufacturing and installation errors, thermal and centrifugal deformation, and the need to supply cooling and lubricating oil all require a margin. The practical lower bound is therefore 1 mm. Within the discrete test levels, 1 mm at all three locations gives the lowest measured torque; the continuous fits explain why the loss continues to fall as the clearance approaches zero, but the engineering optimum is set by the minimum safe gap rather than by the mathematical minimum alone.
Engineering implications and limits
The results give a clear order of priority for shroud design. Windage control becomes worthwhile when pitch-line speed exceeds about 45–50 m/s. The heel clearance should be controlled first, the toe clearance second and the tooth-flank clearance third. The shroud should be stiff enough to resist the pressure field and close enough to limit air exchange with the surroundings, but not so close that vibration or thermal growth causes contact. When no shroud is fitted, rotation direction can change windage by up to about 22%; with a shroud, the direction effect is much smaller, so the design should not rely on direction alone to reduce loss.
The numerical model treats air as a single-phase, incompressible fluid, uses a steady MRF approximation, and assumes smooth adiabatic walls. It does not include oil jets, oil mist, conjugate heat transfer, gear deflection or full gearbox interactions. Absolute power values therefore apply to the modeled geometry and operating range. The trends, especially the strong heel-clearance effect and the benefit of a close-fitting shroud, are the more transferable result. For oil-lubricated high-speed transmissions, transient multiphase CFD coupled with thermal and structural analysis would be required to capture churning and mist effects.
Conclusions
- Windage in a high-speed spiral bevel gear is dominated by pressure-driven torque on the tooth flanks. Pressure effects account for 93–97% of the loss and increase with speed. Heel and toe face torques are about an order of magnitude smaller, and the maximum pressure occurs near the large end of the tooth.
- Windage power rises approximately with the 3.2 power of rotational speed. The slope becomes steeper above about 8,000 r/min, where the heel tangential speed is roughly 120 m/s.
- Without a shroud, counter-clockwise rotation can produce about 22% more windage torque than clockwise rotation. With a shroud, the difference between directions falls within about 5%, because the shroud suppresses the axial inlet and external recirculation that make the tooth passages sensitive to incidence.
- Shroud clearance is a first-order design parameter. The heel clearance has the strongest influence, followed by the toe clearance and then the tooth-flank clearance. Among the tested levels, 1 mm at all three locations gives the lowest windage torque, 0.35111 N·m.
- The fitted response surfaces decrease toward zero clearance, but the practical minimum is 1 mm. Vibration, shaft displacement, manufacturing tolerance, thermal and elastic deformation, and oil supply set the lower limit. A close-fitting, stiff shroud with a controlled heel gap is the effective route to lower windage loss.