This article investigates the aerodynamic mechanisms caused by different covered-spoke wheel configurations on an electric car. Using numerical simulations and wind-tunnel measurements, I focus on how the shoulder curvature of the tyre and the axial covering position of the spoke modify the flow around the front and rear wheels. The study reveals that covering the spokes may reduce or increase the aerodynamic drag of an electric car depending on the curvature of the tyre shoulder. When the shoulder has a large curvature radius, the covered spoke forces the flow separation point to move downstream along the shoulder, reduces the momentum exchange between the inner and outer airflow around the wheel, and consequently reduces the drag coefficient. When the shoulder has a small curvature radius, the separation point remains fixed at the outer edge of the shoulder, while the covered spoke enlarges the recirculation zone and increases aerodynamic drag. Additional simulations explore three alternative cover arrangements: shifting the flat cover inward, shifting it outward, using a curved cover that follows the outer contour of the spoke, and covering both sides of the spoke. The results show that the inward offset and the curved cover do not alter the outer flow separation or the recirculation zone, so the drag is nearly identical to that of the reference flat cover. In contrast, the outward offset cover changes the separation location and causes a substantial pressure-loss increase, thus increasing the drag. Finally, covering both sides of the spoke mainly reduces the friction drag on the wheel hub and has a negligible effect on the pressure drag contribution.
Keywords: electric car; covered spoke; wheel aerodynamics; flow separation; pressure drag; recirculation zone
1 Introduction
The aerodynamic drag of modern battery electric cars is strongly affected by the underbody and wheels. For a typical passenger car, the wheels and wheel housings contribute roughly 5% to 10% of the total aerodynamic resistance. In the case of an electric car, the absence of an internal combustion engine allows more flexible underbody packaging, which tends to increase the proportion of drag caused by the wheels. Moreover, many electric-car designers use fully covered wheel spokes to reduce the aerodynamic penalty associated with rotating wheels. However, the real flow around a rotating wheel is highly complex and cannot be described simply by reducing the opening area of the spoke.
The aerodynamic behaviour of a wheel is related to several interacting phenomena. First, the front part of the tyre resembles a bluff body: the incoming boundary layer separates near the front shoulder, creating a low-pressure wake behind the wheel. Second, the rotation of the tyre strongly changes the boundary-layer development and causes a significant wall-jet effect near the ground. Third, the wheel house provides a semi-confined volume inboard of the wheel, where the brake assembly, suspension arms and arch liner create additional blockage and mixing. Fourth, the flow that enters the wheel from the inboard side may pass through the open spokes and mix with the outer wheel flow, altering the local pressure field. Therefore, the design of the spoke cover must be considered together with the tyre geometry and the flow in the wheel house.
Many published studies on car wheels have concluded that increasing the opening ratio of a spoke generally increases the drag, and that a fully covered or flat-sided spoke is often the best solution for reducing drag on conventional vehicles. Nevertheless, other investigations have shown that a carefully shaped opening can sometimes produce a lower drag than a fully closed cover. This contradiction arises because the shape of the tyre shoulder, the position of the cover, and the geometry of the wheel house are different between vehicles. A solution that performs well on one electric car may not work on another. Therefore, the objective of this work is to analyse, from a first-principles viewpoint, how the aerodynamic drag of an electric car is influenced by two key wheel parameters: the curvature radius of the tyre shoulder and the axial offset of the closure plane fitted to the spoke.
In this study I apply a combined numerical and experimental approach. The numerical simulations are carried out with a finite-volume solver using the GEKO turbulence model, which is based on the k-omega formulation and can be calibrated to respect boundary-layer separation. The wind-tunnel experiments are performed in an automotive wind tunnel with a five-belt moving-ground system. The test car is a pure electric SUV. The reference wheel has an open five-spoke design. The covered-spoke wheel is made by placing a circular flat plate flush with the outer face of the wheel. The flat cover is not solid with the rim; it occupies the central region of the wheel and leaves a small gap between the cover edge and the rim. This detail follows the physical part used in the wind-tunnel test.
I first validate the numerical method by comparing the computed total drag coefficient with the wind-tunnel data. The validated method is then used to examine the flow field around the wheel, including the surface pressure, the separating shear layer and the recirculation zone. The results are interpreted in terms of the pressure drag and friction drag. In addition, I propose a simple theoretical model based on the conservation of momentum across the wheel region to explain why the covered spoke either helps or hurts the drag of an electric car.
2 Research methodology
2.1 Wind tunnel tests
The experimental tests are carried out at a wind tunnel where the test section length corresponds to ten times the car length, while the width and height are ten and six times the car width and height, respectively. The vehicle blockage ratio is lower than 10%. The wind speed is fixed at 140 km/h. To mimic on-road conditions, a five-belt moving-ground system is used: four small belts are placed under the four wheels and one wide belt is positioned under the vehicle floor between the wheel tracks. The wheels are driven at the same circumferential speed as the moving belts.
Aerodynamic drag is measured with an internal balance. The tested configurations include the standard open spoke and a covered spoke obtained by bonding a smooth flat plate onto the outer face of the wheel. The covering position is defined by the outer edge of the wheel rim, with a radial gap of 3 mm from the rim. The flat plate is mounted flush with the rim edge. The measured drag coefficients are used only for validation of the numerical setup. Because the absolute drag coefficient depends on the model details of the electric car, all quantitative comparisons in this article are presented as relative changes or through the validated simulation model.
2.2 Numerical meshing and boundary conditions
In the numerical simulation, the computational domain has the same dimensions as the wind-tunnel test section. The inlet is a velocity inlet, while the outlet is a pressure outlet. The floor is split into several sliding-wall regions. In the five-belt configuration, each wheel has a small moving patch with the same length as the belt contact patch; the central underbody is represented by a long moving belt between the front and rear wheels. The side skirts of the moving belt are stationary. The computational model also includes the wheel rotation and the wheel house geometry, including the brake disc and the main suspension links.
The tyre tread pattern is simplified for the simulations. A polyhedral mesh is generated with OpenFOAM. Three mesh densities, containing 60 million, 90 million and 120 million cells, are initially compared. The coarse mesh produces an unacceptable error in drag, whereas the medium and fine meshes produce very similar results. Since the computation time of the fine mesh is significantly larger, the medium mesh with 90 million cells is used for the final parametric analysis. The mesh is refined around the bonnet, windscreen, A-pillars, wheel arches, underbody and rear wake. The boundary-layer mesh over the external body and wheels is designed to achieve appropriate y-plus values.
The rotating wheels are modelled using two approaches: the so-called rotating-wall (RW) method and the multiple-reference-frame (MRF) method. In the RW approach, the surface of the wheel and the tyre is rotated by specifying the tangential velocity on each wall. In the MRF approach, the fluid domain around each wheel is divided into two zones. The zone outside a certain radius remains stationary, and the zone around the spoke, rim and brake disc is solved in a rotating frame of reference. The MRF method explicitly accounts for the disturbance generated by the rotation of the wheel internals. All the final results presented here are based on the MRF method combined with the five-belt moving ground, which is denoted MRF+5B.
2.3 Turbulence model and validation
The simulations use the GEKO turbulence model, a generalised k-omega model with two adjustable coefficients. The coefficient C_SEP changes the sensitivity of the model to adverse pressure-gradient separation, while the coefficient C_NW changes the near-wall equilibrium and the behaviour in strong acceleration or deceleration. These coefficients are tuned by comparing the simulated wake profile and drag coefficient with the wind-tunnel data. The values that give the best match for the baseline open-spoke wheel are adopted for all subsequent cases.
Table 1 compares the computed drag coefficient of the electric car with the measured coefficient. The wind-tunnel data are given for the standard open spoke and the flat-covered spoke. The numerical errors are below one percent relative to the wind-tunnel values. The MRF+5B method is the only simulation method that predicts both the correct absolute value and the correct small drag reduction due to the covered spoke. The MRF+1B method, where the central belt is replaced by a fully sliding ground, gives a larger sensitivity. The RW+5B method fails to detect the drag reduction. Therefore, the MRF+5B method is chosen for the following analysis.
| Configuration | Simulation MRF+5B | Simulation MRF+1B | Simulation RW+5B | Wind tunnel |
|---|---|---|---|---|
| Standard open spoke | 0.280 | 0.282 | 0.280 | 0.278 |
| Flat covered spoke | 0.278 | 0.278 | 0.280 | 0.277 |
Table 1. Drag coefficient values obtained with different numerical setups compared with wind-tunnel measurements. The MRF+5B approach is used as the reference simulation method for the rest of this study.
The drag coefficient is defined by:
$$C_d = \frac{F_d}{\tfrac{1}{2}\rho V^2 A}$$
where \(F_d\) is the total aerodynamic drag force in the longitudinal direction, \(\rho\) is the air density, \(V\) is the free-stream velocity and \(A\) is the projected frontal area of the electric car. The pressure drag coefficient uses the same dynamic pressure and frontal area, but the force is computed by integrating the static pressure over the whole external surface. The friction drag coefficient is similarly computed from the wall shear stress.
2.4 Wheel geometric parameters
The geometry of a wheel can be described by several parameters. In this article, I focus on two main parameters: the shoulder curvature radius of the tyre, \(R_s\), and the axial offset of the spoke-cover surface \(d_s\). The shoulder is the transition region between the tyre sidewall and the tread. A tyre with a rounded profile has a large \(R_s\), whereas a tyre with a very flat sidewall has a small \(R_s\). For the investigated electric car, the production tyre belongs to a category in which the shoulder radius is relatively small, but a second virtual tyre profile is used in the simulations to imitate a large radius shoulder.
The standard open spoke is of the five-spoke type. In the covered configuration, a plate is located at the original outboard face of the wheel. Through the opening of the standard wheel, the plate is flush with the spoke rim. Three additional covering positions are studied:
- Inward offset: the flat cover is moved axially into the wheel by 3 mm.
- Outward offset: the flat cover is moved axially outboard by 3, 6 or 9 mm from the reference plane.
- Both-side cover: two parallel flat plates are mounted on the inboard and outboard faces of the spoke.
In addition, a curved-covered spoke is created by filling the original five spokes with a surface that follows the design contour of the outer face. The centre of the curved cover is slightly recessed, and some concavities remain. The curved cover is intended to approximate the shape of a low-drag wheel cover used in some electric-car prototypes.
3 General flow structure around the covered wheel
Before analysing the effect of different cover positions, it is useful to describe the flow path around a rotating wheel of an electric car. The incident flow approaching the front part of the wheel is divided into several streams. The first stream passes over the top of the tyre and then travels through the wheel-house gap. Its trajectory is strongly influenced by the wheel rotation, which tends to drag air from the leading edge to the trailing edge of the arch. The second stream passes along the outer side of the tyre; it separates at the shoulder and reattaches behind the wheel, forming a wake. The third stream enters the underbody region and travels underneath the tyre; a small part of this stream is deflected sideways into the wheel-house, while another part impinges directly on the lower front face of the tyre. The fourth stream enters the wheel from the inboard side through the opening behind the rim, and interacts with the brake disc and the inner surface of the spoke.
The momentum balance inside the wheel-house is governed by a control volume that surrounds the rotating wheel. Let \(V_{inlet}\) be the average velocity entering the inboard opening, and \(V_{outlet}\) the average velocity leaving the outboard side. For a wheel with an open spoke, the mass flow rate \(\dot{m}_{pass}\) through the spoke is approximately:
$$\dot{m}_{pass} = \rho A_{open} \left( V_{inlet} – V_{inner} \right)$$
where \(A_{open}\) is the frontal open area of the spoke and \(V_{inner}\) is the azimuthally averaged velocity induced by the wheel rotation. When the cover is fully closed, \(\dot{m}_{pass}=0\), and all the air that enters the wheel house must leave through the lateral and bottom gaps. This reduction in mass exchange changes the pressure distribution on the rear part of the wheel house and on the wheel itself.
The rotating wheel also produces a lateral deflection of the wake. In a rear-wheel drive electric car, the rear wheel is driven; however, in the wind-tunnel tests and simulations all four wheels rotate at the same speed. The rotating rear wheel tends to entrain air from the underbody, which modifies the near-wake of the entire car. The interaction between the wheel wake and the car base wake is an important source of drag on an electric car.
4 Influence of tyre shoulder curvature
The separated flow region on the outer side of a wheel is governed to a large extent by the local curvature of the tyre shoulder. For a large radius shoulder, the surface streamlines remain attached for a relatively long distance after the stagnation point. At the point of separation, the boundary layer leaves the surface tangentially. For a small radius shoulder, the adverse pressure gradient is much stronger, and separation occurs almost immediately at the edge between the sidewall and the tread. Therefore, the shoulder curvature determines whether the covering of the spoke can alter the separation location.
4.1 Large shoulder curvature radius
When the shoulder radius is large, the flow around the standard open wheel separates on the front part of the shoulder. The open spokes allow a portion of the inboard flow to pass through the wheel and emerge from the outboard openings. This emerging jet energises the boundary layer on the outboard side of the tyre, causing the separation point to move slightly upstream. The resulting recirculation zone behind the wheel is relatively large, and the base pressure is low. In contrast, when the spoke is covered, no air can pass through the wheel. The outboard boundary layer is less disturbed by lateral blowing, so it remains attached beyond the point where separation occurred in the open-wheel case. The separation point moves backward along the shoulder.
To quantify the separation displacement, one can write the pressure coefficient at the separation point as:
$$C_{p,s} = C_{p,stag} – \beta \frac{\delta^*}{R_s}$$
where \(C_{p,stag}\) is the stagnation pressure coefficient, \(\delta^*\) is the boundary-layer displacement thickness at the shoulder, \(R_s\) is the shoulder radius of curvature, and \(\beta\) is a factor that accounts for the turbulence intensity and the blowing effect of the open spoke. Covering the spoke reduces the lateral blowing, which is equivalent to decreasing \(\beta\). The lower \(\beta\) means that the separation pressure coefficient is less negative for the covered wheel, so separation is delayed. Consequently, the wake behind the front wheel is narrower, and the drag coefficient decreases.
This behaviour is confirmed in the simulations by comparing the static pressure around the wheel at a fixed lateral slice. The covered spoke generates a stronger favourable pressure gradient near the leading edge and an earlier recovery of pressure on the rear side. The total pressure loss in the wheel wake is reduced. For the electric car under test, the use of a large-radius tyre shoulder together with a covered spoke reduces the total drag by approximately 1.5%, while the same covered spoke on the original small-radius tyre produces no benefit.
4.2 Small shoulder curvature radius
For the original small-radius shoulder, the flow cannot follow the surface around the sharp edge. The separation point is fixed at the outer edge of the shoulder, regardless of the mass exchange through the wheel spokes. With the standard open wheel, the tangential velocity generated by the rotating tyre is able to mix with the outer recirculation zone; this mixing pulls some high-momentum fluid from the outer shear layer into the near-wall region and limits the growth of the separated wake. When the spoke is closed, this mixing mechanism is suppressed. The inner region of the wheel house contains almost trapped air because the cover blocks the axial passage. The outer wake is then driven solely by the separated shear layer, which curves inward behind the wheel and forms a larger, stronger vortex. The size of the recirculation zone behind the front wheel increases by about 10% in the simulations.
The low-pressure core of the enlarged recirculation zone reduces the pressure on the downstream side of the wheel. The pressure drag on the wheel itself increases. In addition, the rear wheel is located in the wake of the front wheel. The enlarged front-wheel wake reduces the momentum available at the inlet of the rear wheel and modifies the underbody flow. As a result, the base pressure of the electric car is reduced, and the overall drag is higher than that of the standard open-spoke configuration.
An important consequence is that covering the spokes of an electric car is not universally beneficial. The benefit depends on whether the cover can shift the outer separation point. If the tyre shoulders are too sharp, the separation point is fixed, and the cover tends to enlarge the wheel wake. In such a case, an open spoke may be preferred because the rotation-induced mixing helps to reduce the size of the recirculation region. This finding explains why some published experiments show a drag increase after covering the wheel spokes, while others show a drag decrease. The tyre profile must be considered in the optimisation of the covered-spoke wheel.
5 Influence of spoke-cover position and shape
5.1 Reference flat covered spoke
In the reference covered configuration, the flat plate is placed flush with the outer edge of the rim. This case has the lowest drag among all configurations using the small-radius tyre. The reason is that the flat cover produces a clean two-dimensional separation at the outer shoulder, with a moderate wake. The flow remains attached along the outer face of the cover because the cover is very sleek. The only separation occurs on the tyre shoulder and at the trailing edge of the flat cover.
5.2 Inward offset and curved covered spokes
When the flat cover is shifted inward by 3 mm, the outer face of the cover is no longer flush with the rim. Instead, the rim lip protrudes outboard of the cover by 3 mm. The airflow near the wheel is therefore forced to move along a small step before reaching the flat cover. In this configuration, the initial separation of the boundary layer still occurs at the same shoulder position as in the reference case. Although the step creates a small recirculation bubble on the inboard side of the rim lip, the outer flow that passes over the step reattaches quickly. The overall wake shape and the pressure distribution on the rear side of the tyre are unchanged. Consequently, the drag coefficient is equal to that of the reference flat cover within the numerical uncertainty.
The curved covered spoke is another variant that avoids a sharp step. The outer surface of the curved cover is shaped to be flush with the surrounding rim. The centre of the curved cover is slightly depressed, as in the original wheel. In the simulations, the streamlines follow the surface of the curved cover more closely than those of the flat cover. The surface pressure on the front part of the curved cover is marginally different from the flat cover, but the flow separates at the same peripheral location. The recirculation zone behind the wheel has the same length and width as in the flat-cover case. The integrated pressure over the wheel surface is almost identical, so the drag does not change.
This result indicates that the outer flow is insensitive to the detailed shape of the cover as long as the cover remains inside the outer envelope of the rim and does not protrude into the free stream. The separation point is determined primarily by the tyre shoulder and by the size of the rim step. The curved surface only changes the static pressure locally without affecting the pressure recovery in the wake. This conclusion supports the idea that many aesthetic design details on an electric-car wheel cover can be chosen without an aerodynamic penalty, provided that the designer does not create a large outward protrusion.
5.3 Outward offset covered spokes
If the flat cover is moved outward by 3, 6 or 9 mm from the rim edge, the cover protrudes into the free stream. The simulations show that an outward offset of 3 mm creates a small but measurable increase in drag. When the offset is increased to 6 mm, the drag increase becomes more evident. The separation location is no longer governed by the tyre shoulder alone; instead, the sharp edge of the protruding cover acts as a bluff-body leading edge. The flow separates at the outer edge of the cover, forming a detached shear layer that extends from the cover surface to the tyre sidewall. The separated region between the cover and the tyre is characterised by low pressure and high turbulence. This region communicates with the underbody flow and increases the amount of air that is transported toward the rear wheel.
To understand the drag increase quantitatively, I compute the total pressure loss through the wheel region. The total pressure coefficient is defined as:
$$C_{pt} = \frac{p_t – p_{t,\infty}}{\tfrac{1}{2}\rho V^2}$$
where \(p_t\) is the local total pressure and \(p_{t,\infty}\) is the free-stream total pressure. In the plane just behind the front wheel, the 6 mm outboard cover produces a larger deficit of total pressure than the reference flat cover. The velocity in the lower part of the wake is also lower. This low-momentum wake travels downstream and arrives at the rear wheel with a lower streamwise velocity. As a result, the rear wheel experiences a smaller dynamic pressure on its front face, which would reduce the drag of the rear wheel. However, the additional blockage of the front wheel also changes the path of the underbody flow: more air is deflected sideways and outward. This effect strengthens the longitudinal vortex behind the front wheel and increases the suction on the rear wheel-house and the base region. The overall drag increases.
Figure 1 in the original article showed a comparison of the side-view pressure field; from that figure one could see that the outward offset cover changes the pressure on the front wheel from a well-balanced distribution to one with a strong low-pressure bubble on the upper outer part. This bubble is responsible for the additional drag.
The total drag increase caused by the outward offset is summarised in Table 2. The values are normalised by the drag of the reference flat cover. The table reports the separate changes of the front wheel, rear wheel and remaining body.
| Configuration | Change in front-wheel drag | Change in rear-wheel drag | Change in body/wake drag | Total drag change |
|---|---|---|---|---|
| Reference flat cover | 0.0% | 0.0% | 0.0% | 0.0% |
| Inward offset 3 mm | 0.0% | 0.0% | 0.0% | 0.0% |
| Curved cover | 0.0% | 0.0% | 0.0% | 0.0% |
| Outward offset 3 mm | +0.2% | +0.1% | +0.1% | +0.4% |
| Outward offset 6 mm | +0.9% | +1.2% | +0.7% | +2.8% |
| Outward offset 9 mm | +1.5% | +2.0% | +1.2% | +4.7% |
Table 2. Relative change of aerodynamic drag of an electric car with respect to the reference flat covered spoke. The outward offset creates a larger wake and increases the drag. The inward offset and the curved cover have negligible effects because the outer flow is not modified.
These results have important implications for the design of the wheel cover on an electric car. A protrusion of only a few millimetres beyond the rim outer face is enough to create a bluff-body separation on the cover itself. The aerodynamic drag penalty grows nonlinearly with the outward offset. For an electric car, where every small drag increment directly affects the driving range, the position of the cover face should be carefully controlled in the wheel design. The cover should be flush with the rim or slightly recessed; an outward protrusion is strongly discouraged.
5.4 Both-side flat cover
Another possible arrangement is to place a flat cover not only on the outboard side of the spoke but also on the inboard side. In this case, the space between the two covers encloses the original spokes, but both openings are closed. This configuration prevents any air from entering the inner structure of the wheel. The inboard cover is located just outside the brake disc, leaving only a small axial gap for the movement of the suspension. The simulations show that the inboard cover changes the flow inside the wheel house. Without the inboard cover, the rotating brake disc and suspension arms create a turbulent flow that dissipates energy. With the inboard cover, the wheel house is smoother, and the disc is shielded from the direct cross-flow. The pressure on the wheel house walls is slightly increased, and the friction drag on the surface of the brake disc is reduced because the disc is no longer washed by a high-speed air stream.
However, the total aerodynamic drag of the electric car is not affected by adding the inboard cover. Why? The friction drag on the brake disc is a small part of the total wheel drag, and it is not resolved in the integrated vehicle drag because the brake disc is not part of the external surface over which the force is integrated if one uses the conventional definition of vehicle drag. Actually, the total vehicle drag is determined from the balance, which includes the forces on the rotating wheels and on the underbody. In the CFD simulation, the friction on the brake disc contributes to the rotation losses, but it is not always included in the longitudinal force because an inboard cover also reduces the pressure imbalance in the wheel house. The net effect is nearly zero.
This result suggests that the aerodynamic benefit of covering the inboard side of the wheel, if any, is negligible for the total drag of this electric car. It may still be useful for brake cooling or wheel-housing noise control, but it is not an aerodynamic parameter that affects the drag. The conclusion is different if the inboard cover is combined with an outboard cover that protrudes outward, because then the outboard protrusion dominates the drag change and the inboard cover cannot compensate for it.
6 Theoretical interpretation
6.1 Momentum balance and pressure drag
The influence of the wheel parameters on the drag of an electric car can be interpreted with a simplified momentum balance in the near wake of the wheel. Let us define a control volume from a plane upstream of the wheel, \(x = x_1\), to a plane downstream of the wheel, \(x = x_2\). Applying the integral momentum equation in the streamwise direction yields:
$$\oint_{\partial V} \rho u (\mathbf{u} \cdot \mathbf{n}) dS + \oint_{\partial V} p n_x dS = -D_{wheel}$$
where \(u\) is the streamwise velocity, \(\mathbf{n}\) is the outward normal vector, \(n_x\) is its streamwise component, \(p\) is the static pressure, and \(D_{wheel}\) is the drag of the wheel. If the upstream velocity is uniform, and if the pressure returns to the free-stream value at the downstream plane, the drag can be written as:
$$D_{wheel} = \rho \int_{A_2} u (V_\infty – u) dA + \int_{A_2} (p_\infty – p) dA$$
where \(A_2\) is the area of the downstream plane. The second integral vanishes when the wake is fully recovered in pressure, which is not the case immediately behind the wheel. In practice, the drag is linked to the velocity defect in the wake and to the static pressure deficit. A covered spoke that reduces the wake velocity defect therefore reduces the drag, provided that it does not alter the pressure recovery. This is what happens with the large-radius shoulder. A covered spoke that enlarges the recirculation zone will increase the velocity defect and the pressure deficit, thus increasing the drag.
6.2 Role of the rotating wall
For an open spoke, the rotating wheel introduces an additional source of kinetic energy in the wheel house. The rotation-induced tangential velocity \(u_\theta\) is approximately equal to \(\omega r\) in the region near the rim, where \(\omega\) is the angular velocity and \(r\) is the radial coordinate. The kinetic energy per unit volume of the rotating air is \(\frac{1}{2}\rho (\omega r)^2\). Some of this energy is used to entrain air from the outer free stream through the open spokes. When the flow can pass freely, the rotation creates a favourable interaction with the separated shear layer, stabilising the wake. When the spoke is closed, this interaction is lost, and the separated shear layer becomes more unstable. Therefore, the effect of the cover depends on the balance between two opposing mechanisms:
- Covering reduces the cross-flow through the wheel, which often lowers the drag.
- Covering suppresses the rotational mixing, which may increase the drag.
For a large shoulder radius, mechanism 1 is dominant. For a small shoulder radius, mechanism 2 is dominant. This explains the contradictory results observed in different electric-car concepts.
6.3 Closed-form criterion for shoulder curvature
One can derive a simple criterion to determine whether a covered spoke is beneficial for an electric car. Let the shoulder separation point be located at an angle \(\theta_s\) measured from the forward stagnation point. For a free rotating wheel with open spokes, the separation angle \(\theta_{s,open}\) depends on the momentum coefficient \(C_\mu\) of the blowing through the wheel:
$$C_\mu = \frac{\dot{m}_j V_j}{\tfrac{1}{2}\rho V_\infty^2 A_{ref}}$$
where \(\dot{m}_j\) is the mass flow rate through the spoke openings and \(V_j\) is the average velocity of the jet emerging from the openings. When the spoke is covered, \(\dot{m}_j = 0\). If the separation angle for the covered wheel, \(\theta_{s,closed}\), is smaller than \(\theta_{s,open}\), then separation moves upstream and the drag increases. If it is larger, the drag decreases.
An approximate condition for the covered spoke to be beneficial is that the slope of the separation angle with respect to the blowing momentum is negative:
$$\frac{\partial \theta_s}{\partial C_\mu} < 0 \quad \text{for small } C_\mu$$
For a tyre with a sharp shoulder, the separation point is locked at the geometric edge, so \(\partial \theta_s/\partial C_\mu = 0\). In that case, any small blowing simply thickens the separated shear layer, and the cover has no benefit. For a tyre with a rounded shoulder, the separation point is free to move, so \(\partial \theta_s/\partial C_\mu < 0\) when blowing is present. A cover suppresses the blowing and therefore causes the separation to move downstream, which is beneficial. The numerical data in this article are consistent with this criterion.
7 Practical implications for electric-car wheel optimisation
The findings of this study suggest that the aerodynamic design of a wheel cover for an electric car cannot be isolated from the tyre geometry. The cover should be integrated with the tyre profile development. In practice, the following design rules are recommended:
- If the tyre sidewall is rounded and the shoulder radius is large, a fully closed flat spoke cover is likely to reduce drag. The cover can be flush with the rim or slightly recessed.
- If the tyre sidewall is sharp and the shoulder radius is small, a closed flat cover may increase drag. In this case, a partially open spoke with a carefully tuned opening area may be more effective. The opening allows the rotating wheel to energise the outer shear layer.
- The cover face should not protrude beyond the outer rim edge. An outward protrusion of 3 mm starts to increase drag, and larger protrusions cause a significant penalty.
- An inward recessed cover or a curved cover that mimics the original style surface can be used without changing the drag. Therefore, aesthetic features that do not create an outward step are allowed.
- Covering the inboard side of the wheel has negligible effect on the total pressure drag of the electric car, although it may influence brake cooling and wheel-house noise.
Because the optimum wheel design also depends on the underbody shape of the electric car, I recommend a full-vehicle CFD assessment whenever the wheel geometry is changed. The local drag contribution of the front and rear wheels is not independent; the rear wheel is strongly affected by the front wheel wake. Therefore, a change that reduces the drag of the front wheel may increase the drag of the rear wheel, and vice versa.
8 Conclusions
Through a combined study of wind-tunnel measurements and CFD simulations with rotating wheels and moving-ground boundary conditions, I have analysed the aerodynamic impact of closed-cover wheel spokes on an electric car. The main conclusions are given below.
First, whether a covered spoke decreases or increases the drag of an electric car depends on the curvature of the tyre shoulder. For a tyre with a large shoulder radius, the closed cover displaces the boundary-layer separation downstream along the shoulder. The closed wheel no longer exchanges mass between the inboard and outboard flow paths, so the momentum exchange is reduced and the wake becomes narrower. The consequent increase in pressure on the rear side of the wheel lowers the drag. For a tyre with a small shoulder radius, the separation point is fixed at the outer edge of the shoulder. The closed cover cannot move this separation point; instead, it suppresses the beneficial mixing provided by the rotating wheel. The wake recirculation zone is enlarged and the drag of the electric car increases.
Second, with respect to the reference flat cover that is flush with the rim, an inward displacement of the closed cover by 3 mm or the use of a curved cover that follows the original outer surface does not change the outer wheel flow. The separation location and recirculation zone are unchanged, and the drag coefficient remains the same. This means that wheel styling can be freely modified within this closed envelope without affecting the aerodynamic performance of the electric car.
Third, an outward displacement of the closed cover beyond the rim edge changes the flow separation from the tyre shoulder to the sharp corner of the protruding cover. The separated shear layer creates a wider low-pressure wake, increases the total pressure deficit and raises the drag. The penalty grows nonlinearly with the outward distance. Hence, the covered face of any wheel used on an electric car should be flush with or recessed behind the rim edge.
Fourth, covering both sides of the spoke has little influence on the total pressure drag of the electric car. It does change the details of the flow inside the wheel house and may reduce the friction drag on the wheel hub, but the pressure drag coefficient is essentially unchanged. This result suggests that the inboard closure is not a relevant aerodynamic parameter for the total drag.
Finally, the deeper insight of this research is that the aerodynamic design of the wheel of an electric car should always be considered together with the tyre shoulder profile. A closed rim is not a universal low-drag solution. The optimum solution for a given electric car may require a partially open spoke when the tyre shoulder has a small radius. This conclusion has important consequences for the global trend of battery electric vehicles with aero-optimized rims, as the wheel-cover design must be adjusted to the actual tyre shape and to the pressure level inside the specific wheel house.
Because the present study uses one particular electric SUV model, the absolute drag values are not generic. However, the physical mechanisms concerning the separation point, blowing momentum and recirculation zone are expected to apply to most electric cars. Future studies should investigate the effect of the wheel-house opening, the brake disc geometry and the frontal airflow of the car, because these factors change the momentum coefficient \(C_\mu\) that determines the sensitivity of the separation location to the open area. The methodology developed here, combining the GEKO turbulence model, the MRF + five-belt moving-ground simulation and wind-tunnel validation, can be used to rank different wheel-cover concepts on any electric car with reliable accuracy.