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.
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.
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.
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.
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.
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.
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.