1 Introduction

Some teams aiming for victory in a mountain stage in cycling take control in the uphill sections of the stage. In this uphill drafting strategy, the supporting riders rotate through the front of the peloton and ride at a high pace that generally does not allow opponents to attack. There appear to be no systematic and extensive studies on the aerodynamics of uphill drafting in the literature. Most studies on cycling aerodynamics focused on the drag of a lone cyclist riding on a flat surface above 40 km/h (11.1 m/s) [1,2,3,4,5,6,7,8,9]. At these speeds, the aerodynamic resistance or drag is about 90% of the total resistance [1, 2, 10]. Therefore, reducing the drag of a cyclist is well known to be an important parameter for success on flat or descending sections. An efficient way to lower the aerodynamic resistance is the use of drafting. Studies on drafting were reported [11,12,13,14,15,16,17,18,19] in which maximum drag reductions of 48% for two cyclists [17], 57% for four cyclists [18], 60% for nine cyclists [19], and 95% for a complete peloton [20] were found.

During uphill sections, the speeds are lower, and depending on the slope and length are generally between 20 and 30 km/h (5.5–8.5 m/s) [21]. In this case, most of the resistive force of a cyclist is due to gravitational resistance [13]. Consequently, the benefit of drafting in terms of energy saving will be lower as well. Some studies suggested that for gradients higher than 7.2%, the speeds drop to 17 km/h and the air resistance can be neglected [22]. On the contrary, Ouvrard et al. [23] demonstrated by means of field testing that drafting did have an impact on uphill cycling performances. It was found that for an uphill time trial of 2.7 km length, and a mean slope of 7.4%, the time to complete the race could be reduced by 23 s (4.2%) while drafting behind a teammate. A mean reduction in resistive force of 2.3% was found, which was mainly attributed to reduced aerodynamic drag.

The main goal of a team when maintaining a high pace in the uphill sections of a race is to protect the team leader from opponent’s attacks. In this way, the team leader can maintain a constant power output, which is the most efficient strategy for long-duration cycling under constant environmental conditions [24,25,26,27,28]. Wilson [29] measured and estimated the power output from races and established a curve connecting the maximum recorded duration of a cyclist exerting a given power output. Taking into account that during uphill sections, the power output is already high, he found that even a small reduction in required power, such as the 2.3% found by Ouvrard et al. [23], could extend the duration one can maintain this power [29].

To provide further insights in the potential benefits of uphill drafting, this paper presents a systematic analysis of uphill drafting by means of drag reductions and power reductions obtained by computational fluid dynamics (CFD) simulations validated with wind tunnel (WT) measurements, for pacelines of up to eight cyclists.

2 Wind tunnel measurements

The WT experiments were performed in the closed circuit WT at Eindhoven University of Technology [30]. The cross section of the WT test section is (w × h) 3 × 2 m2. The cyclist models were placed on an elevated horizontal sharp-edged platform with integrated force transducer to limit the boundary layer development near the models (Fig. 1). The WT models originate from a previous study [20] and were cast from a synthetic resin (neolith) at quarter scale (frontal area = 0.03 m2) to yield a blockage ratio of 0.7% when considering the part of the cross section above the horizontal platform. This blockage ratio is below the recommended maximum of 5% [32]. The geometry of cyclist and bicycle was obtained via 3D scanning using the Artec Eva structured light 3D scanner [31], while the cyclist (height = 1.83 m, mass = 72 kg) was asked to hold an upright position with hands on the drops, as often seen during moderate uphill sections of the race. The procedure of processing the body geometry and reporting the results was approved by the ethical board committee of Eindhoven University of Technology with reference code (ERB2020BE_1859456_WT). While in CFD the actual 3D scans were used, reinforcements and minor simplifications were applied to the wheels for manufacturing of the WT models. The wheel-to-wheel separation distance between the cyclists was 0.0375 m (full scale = 0.15 m). A force transducer, designed especially for the expected forces of quarter-scale cyclist models [20], with an accuracy of 0.001 N was used, which corresponds to a systematic error of 3 × 10–6 m2 for CDA (coefficient of drag times frontal surface area).

Fig. 1
figure 1

Wind tunnel set-up. Dimensions in mm

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Tests were performed at quarter scale up to 25 m/s to ensure a close Reynolds number similarity with the full-scale speed of 6 m/s. During every measurement, the free stream wind speed and turbulence intensity were recorded near the cyclists by the use of a Cobra probe (Fig. 1c). The turbulence intensity at the position of the force transducer was 0.5%. Atmospheric pressure, air temperature and humidity were measured at the end of the test section. The drag force results were corrected to match 15 °C at 0 m (sea level) and standard atmospheric pressure 101,325 Pa. These conditions were also applied in the CFD simulations.

3 CFD simulations

3.1 Computational geometry, domain and grid

The simulations were performed at full scale and the models were placed in a domain with size according to CFD best practice guidelines [33, 34], based on the configuration with eight cyclists. The resulting domain dimensions were (l × w × h) 54.5 × 17.5 × 8.75 m3. Both the cyclist geometry and the hybrid hexahedral–tetrahedral computational grid topology were similar to those used before [9, 20]. Due to a lower cycling velocity in the current study, an additional grid-sensitivity analysis was performed. For the grid-sensitivity analysis, three grids were constructed for the single cyclist configuration in which surface and volume cell sizes were systematically refined (Fig. 2). This resulted in a coarse, medium and fine grid containing 26,952,545, 35,248,553 and 45,402,859 cells, respectively. Studies (e.g. Ref. [35]) highlight the importance of keeping the dimensionless wall unit y* below one for resolving the thin viscous sublayer to reproduce boundary layer flow and separation. The dimensionless wall unit is given by y* = u*yP/ν where u* is the friction velocity, yP is the normal distance of the cell center point P from the wall surface and ν is the local kinematic viscosity. Based on this definition and a free stream velocity of 6 m/s, yP = 0.026 mm would satisfy y* < 1. However, due to flow acceleration, the local friction velocity u* can increase, which would demand a smaller yP to satisfy y* < 1. Therefore, the choice was made to keep yP at 0.010 mm for all three grids.

Fig. 2
figure 2

Computational grid for the isolated cyclist: a coarse grid of 26,952,545 cells, b medium grid of 35,248,553 cells, and c fine grid of 45,402,859 cells

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The grid-sensitivity analysis was conducted with both the Transition SST (TSST) [38, 39] and Scale-Adaptive Simulation (SAS) approach [40, 41]. The results were compared in terms of the drag area for the combined cyclist-bicycle system. In general, the drag area did not change much for the finer grids. The use of the coarse grid resulted in a difference of 0.9 and 1.7% compared to the fine grid for, respectively, the TSST (0.4988 vs. 0.4945 m2) and SAS (0.4646 vs. 0.4725 m2) approach. For both the TSST (0.4938 vs. 0.4945 m2) and SAS (0.4729 vs. 0.4725 m2) approach, the drag area obtained on the medium grid differed by 0.1% from the results on the fine grid. Therefore, the medium grid was retained for the remainder of this study.

3.2 Boundary conditions

A uniform mean velocity of 6 m/s was imposed at the inlet with a turbulence intensity of 0.5%. These conditions ensure a similar Reynolds number to those of the quarter-scale models in the WT and represent the relative air movement due to cycling in calm conditions (zero wind speed). The bicycle surfaces were modeled as smooth no-slip walls. The cyclist body surfaces were modeled with an equivalent sand-grain roughness height of kS =  0.1 mm [20, 36]. Zero static gauge pressure was specified at the outlet. The bottom, top and lateral boundaries of the domain were treated as slip walls.

3.3 Approximate form of governing equations and solver settings

The CFD simulations were performed using the commercial CFD code ANSYS Fluent 19.2 [37]. Two approaches were adopted. The first, less computationally demanding approach, consisted of solving the 3D steady Reynolds-averaged Navier–Stokes (RANS) equations with the transition shear stress transport (TSST) turbulence model for closure [38, 39]. Curvature correction was applied. Pressure–velocity coupling was taken care of by the COUPLED algorithm. Pressure interpolation was second order and second-order discretization schemes were used for both the convection and the viscous terms of the governing equations. The Green–Gauss node-based scheme was used for gradient interpolation. To avoid possible divergence of the solution due to high-pressure gradients, the simulations were conducted under pseudo-transient conditions with a pseudo time step of 0.01 s for a minimum of 10,000 pseudo-transient time steps. Averaged results were obtained by averaging over the last 9000 time steps, after an oscillatory phase of the sampled drag values was reached.

The second approach consisted of the use of Scale-Adaptive Simulation (SAS) [40, 41], coupled with the SST k-ω model [42]. The SAS model is an improved URANS formulation which employs URANS for wall-bounded flows and allows the resolution of turbulence structures in regions of high flow instabilities. SAS methods are most promising in flows with significant instability mechanisms present, such as in the near wake of bluff bodies [40, 43]. Pressure–velocity coupling was performed by the PISO algorithm. Pressure interpolation was second order. Second-order upwind discretization schemes were used for the turbulence model equations and bounded central differencing was used for the momentum equations. Gradient interpolation was taken care of by the Green–Gauss node-based scheme. To determine a suitable time step ∆t, a sensitivity analysis was performed for the single cyclist configuration. For all simulations, the Courant–Friedrichs–Lewy (CFL) number CFL = U∆t/∆x, with U = 6 m/s and ∆x = 0.03 m, was kept below one in the regions between the cyclists. Compared to the smallest tested time step size ∆t = 0.0005 s (CFL = 0.1), the use of larger time step sizes ∆t = 0.001 s (CFL = 0.2) and ∆t = 0.002 s (CFL = 0.4) resulted in respective differences of 0.11 and 1.47% in drag. Based on a trade-off between computation time and accuracy, the time step size of ∆t = 0.001 s was retained for all SAS simulations. Results were averaged after an oscillatory phase of the sampled drag values was reached. The number of time steps required to obtain a constant moving average of the drag values ranged from 5000 to 10,000.

4 Results

4.1 CFD validation: pacelines up to four cyclists

Figure 3 shows the results from the WT measurement and CFD simulations in terms of percentage drag of that of an isolated rider at equal speed. It is shown that all riders benefit from drafting. From the WT measurements, the rider in front (C1) experienced drag reductions of 3% (Fig. 3a) and 4% (Fig. 3b, c). The reduction in drag for the following riders was larger. A maximum reduction down to 44% was found for the fourth rider (Fig. 3c). The CFD simulations show a similar trend. The results obtained by the TSST turbulence model underestimated the drag of the trailing riders by 5.1% on average. The results obtained with the SAS model deviated from the WT measurements by 1.7% on average. For this reason, the SAS approach was used for the rest of the study.

Fig. 3
figure 3

Comparison of CFD and wind tunnel drag results for the a two, b three, and c four cyclists configuration, as a percentage of the drag of an isolated cyclist. Error bars represent the size of one standard deviation

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4.2 CFD simulations: pacelines up to eight cyclists

Figure 4 shows the results obtained by the SAS approach for pacelines of up to eight cyclists. The lowest drag percentage per configuration is marked in green. In line with studies on drafting cyclists [16, 19, 20, 44, 45] due to subsonic upstream disturbance, the drag on the leading cyclist decreases due to the other cyclists in its wake. The drag on the leading cyclist decreases with increasing number of cyclists up in the paceline to a maximum of 4% for configurations with five cyclists or more. In general, the drag decreases for positions further down the paceline. For configurations up to four cyclists, drag reaches a minimum for the last cyclist. For configurations of 5, 6, 7 and 8 cyclists, the second-to-last position is the position with the lowest drag. An explanation for this phenomenon is provided in [19] and confirmed by the contours of mean velocity ratio in Fig. 5b, d, as well as by Fig. 6 which shows the cross-sectional area of the wake (mean velocity ratio < 0.5) in a vertical plane situated at the most upstream point of the front wheel. The velocity ratio is defined as the magnitude of the local 3D velocity vector divided by the cycling speed of 6 m/s. The strong expansion of the wake originating from the first cyclist continues to about position five. From this point on, the expansion of the wake flattens out and all positions further downstream have a similar benefit from drafting in this wake. Because the last cyclist does not have the benefit of someone drafting behind them, they experience more drag.

Fig. 4
figure 4

Drag of every cyclist in configurations up to eight cyclists, as a percentage of the drag of the isolated cyclist. The cyclist with the lowest drag is colored in green (color figure online)

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Fig. 5
figure 5

Contours of a, c, e, g instantaneous and b, d, f, h mean a, b, c, d velocity ratio and e, f, g, h pressure coefficient in a, b, e, f vertical centerplane and in c, d, g, h horizontal plane at 1 m height

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Fig. 6
figure 6

Cross-sectional area of the wake (mean velocity ratio < 0.5) as a function of longitudinal distance for pacelines of up to eight cyclists

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The impact of the subsonic upstream disturbance is also visible in the instantaneous (Fig. 5e, g) and mean (Fig. 5f, h) contours of the pressure coefficient. The under-pressure (blue) area behind the leading rider is decreased in size and magnitude when one or more trailing cyclists are present. The largest area of overpressure (red) is visible in front of the leading cyclist and decreases in size for the positions further to the back of the formation. This is due the interaction with the under-pressure area behind the leading cyclist with the overpressure area in front of the trailing cyclist.

4.3 Power calculations

The ascend of the Col du Tourmalet, the most climbed mountain pass in the history of the Tour de France [46], served as a case study for calculating the required total power PTOT for every cyclist in the paceline. The climb features a mean slope of 7.5%. A representative mean speed of 22 km/h (6.1 m/s) was selected based on elite cyclists’ performance [21, 47]. The power model of Martin et al. [13] was used to calculate PTOT required to overcome aerodynamic drag PAD, rolling resistance PRR, wheel bearing friction PWB, ascending and descending PPE, acceleration PKE, and friction in the drive chain EC:

$${P}_{\mathrm{TOT}}=\frac{{P}_{\mathrm{AD}}+{P}_{\mathrm{RR}}+{P}_{\mathrm{WB}}+{P}_{\mathrm{PE}}+{P}_{\mathrm{KE}}}{{E}_{\mathrm{C}}}$$

This study focused on riding at constant speed, so PKE was zero and hence was not considered here. In still air, the power to overcome the total aerodynamic drag PAD is defined by

$${P}_{\mathrm{AD}}=\frac{1}{2}\rho {(C}_{\mathrm{D}}A+{F}_{W}){U}^{3}$$

in which ρ is the air density (kg/m3), CD is the drag coefficient (-), A is the frontal area (m2), and U is the riding speed of the cyclist (m/s). The incremental drag area of the spokes is given by FW (m2). The power to overcome the rolling resistance PRR is described by

$${P}_{\mathrm{RR}}=U{C}_{\mathrm{RR}}mg$$

where CRR is the coefficient of rolling resistance (-), m is the total mass of bike and rider (kg), and g is the acceleration of gravity (= 9.81 m/s2). The total power lost to bearing friction torque PWB as a function of riding speed U is

$${P}_{\mathrm{WB}}=U\left(91+8.7U\right){10}^{-3}$$

The power associated with changes in potential energy PPE, due to ascending or descending for road grades of up to 10%, is related to the total mass of the bike and rider and to the road gradient G (rise/run) as

$${P}_{\mathrm{PE}}=UGmg$$

The total power PTOT is calculated from a number of input parameters depending on the bicycle, cyclists, road and environmental characteristics. The input parameters, including the computed drag area values for configurations up to eight cyclists, are provided in Table 1. It was assumed that all cyclists have equal morphological characteristics and that their CDA is invariant with speed. Standard atmospheric conditions were applied.

Table 1 Input parameters
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Figure 7a presents the required power for the isolated cyclist to overcome a slope of 7.5% as a function of cycling speed, with the different power components indicated. Figure 7b shows the cumulative power percentage. Changes in potential energy had the largest contribution to the total power. With a speed of 6 m/s, 80.2% of PTOT is required for PPE, 16.2% for PAD and 3.4 and 0.2% for PRR and PWB, respectively. Every required power component increases with increasing speed.

Fig. 7
figure 7

a Power as the sum of components PAD, PRR, PWB, and PPE and b cumulative power percentage required to overcome a slope of 7.5% as a function of speed for the isolated cyclist

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Figure 8 shows PReq per cyclist to overcome a slope of 7.5%, as a function of speed. PReq is defined as the required PTOT per cyclist, expressed as percentage of the required PTOT for an isolated cyclist. The leading (first) cyclist had a small benefit compared to the isolated cyclist. Larger benefits were found for the trailing cyclists. For two cyclists at 6 m/s, the second one needed 7.1% less power than the isolated cyclist. For pacelines of three to eight cyclists, this reduction was 7.6% (Fig. 8b–g). At 6 m/s, the reduction in PReq for the third cyclist was 8.7% for a paceline with three (Fig. 8b), 9.1% for a paceline with four (Fig. 8c), and grew to 9.3% for a paceline with eight cyclists (Fig. 8g). These drag reduction benefits increased with increasing speed. For a paceline with eight cyclists at 8 m/s, the benefit for the second and third cyclist was about 12 and 14%, respectively, while reductions in PReq of about 16 and 20% were found at 10 m/s. For the cyclists further down the paceline, this grew to values of about 22%, with a maximum reduction of 22.4% for the seventh cyclist.

Fig. 8
figure 8

Required total power PReq as a function of speed to overcome a slope of 7.5%, expressed as a percentage of the power of an isolated cyclist, for the a two, b three, c four, d five, e six, f seven, and g 8 cyclists configuration

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A sensitivity analysis was performed concerning the impact of the main input values on the results (Fig. 9 and Table 1). Results were evaluated in terms of PReq. For the sake of brevity, only the configurations of up to four cyclists were included. With respect to PReq, positive relationships were found for m and G, while negative relationships were found for ρ, U and CDA. Changing the input values had little effect on PReq for the leading cyclist (Fig. 9a, c, f). Increasing or decreasing one of the input parameters by 20% yielded a maximum change of 0.1%. Larger changes were found for the trailing cyclists. If the combined mass of cyclists and bicycles increased by 20% for the two cyclists configuration, and the other parameters did not change, the reduction in PReq was about 1% (Fig. 9b). For the other cyclists in second (Fig. 9d, g), third (Fig. 9e, h) and fourth (Fig. 9i) position, values up to 1.2% were found. Comparable results existed for G. The negative relationships for ρ and CDA were of the same magnitude as those found for the positive relationships. Increasing one of these parameters by 20% yielded further reductions of PReq in the range of 1and 1.3% for the trailing cyclists. The sensitivity to speed was already highlighted in Fig. 8 and was demonstrated again here. With additional reductions up to 3.1% (Fig. 9h) for an increase of 20%, the sensitivity to speed was larger than for the other parameters.

Fig. 9
figure 9

Sensitivity analysis of the impact of several input parameters on the required power PReq for every cyclist. This for the a, c, f first, b, d, g second, e, h third and i fourth cyclist in a, b two, c, d, e three, and f, g, h, i four cyclists configuration. Initial input parameters are given in Table 1

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5 Discussion

Using the uphill drafting strategy a team can try to keep their opponents from attacking while making sure their team leader can maintain a constant power output and does not have to spend energy in countering attacks. Uphill drafting is favorable for cyclists who can maintain a high power output with respect to their weight (W/kg) for a long duration. As shown in Sect. 4.3, drafting at 6 m/s under the given circumstances can reduce the required power by about 7% for the second cyclist and up to 10% for the other cyclists in line. Considering the power output of elite cyclists during uphill sections is in the range of 300–400 W [21] and the relation between power output and duration by Wilson [29] (see Sect. 1), it is expected that power reductions in the range of 5–10% could extent the duration of this power output up to about 15 min. Future research could incorporate elite cyclists power output data during uphill performances, to better determine the actual time gain and adjust the uphill strategy accordingly.

Competitors can also use the wake drawn by other teams to benefit from drag and power reductions. However, when a team applies the uphill drafting strategy, they are generally convinced that their leader is the strongest cyclist in terms of power-to-weight ratio or that they can defend the time gains they can take in other stages (e.g. time trial stages). Other potential benefits are related to the psychophysiological part of cycling. Pleasure and positivity allow the cyclist to sustain higher exercise intensity [48,49,50]. It is suggested that the presence and leading roles of teammates increases positivity [23], increasing power output for the team leader.

The study was subjected to a number of limitations. Crosswind effects were not taken into account. The bicycle wheels did not rotate, the cyclists were not pedaling and all cyclists had equal body characteristics. In addition, in Sect. 4.3 it was assumed that the CDA remains constant with increasing cycling speed. While the effect of speed on CDA should be the subject of future work, it is expected that the impact of most of these limitations was well captured within the range of ± 20%, as applied in the sensitivity analysis (Sect. 4.3).

6 Conclusion

While uphill drafting is a common strategy in mountain stages in cycling, its effect is questioned and there are no systematic and extensive studies in the scientific literature on this topic. Therefore, this study presents an aerodynamic analysis on uphill drafting in cycling by means of WT measurements and CFD simulations. Drag reductions were computed and a power model was used to calculate potential power savings. For configurations of up to four cyclists, the last rider experienced the lowest drag. For configurations of five to eight cyclists, the second-to-last cyclist experienced the lowest drag. By drafting on a slope of 7.5% at a speed of 6 m/s, cyclists can save over 7% in terms of the required power. At higher speeds of 8 m/s, savings of over 12% can be achieved. Under the given conditions, the largest differences in required power are found between the cyclists in first, second and third positions. Comparable power reductions are found for all remaining cyclists in the paceline. The sensitivity analysis of input variables shows that uphill drafting allows power savings with different bicycle, cyclist, road and environmental characteristics.