# Aerodynamic Drag Modeling of Alpine Skiers Performing Giant Slalom Turns

##### Abstract

Purpose: Aerodynamic drag plays an important role in performance for athletes practicing sports that involve high-velocity motions. In giant slalom, the skier is continuously changing his/her body posture, and this affects the energy dissipated in aerodynamic drag. It is therefore important to quantify this energy to understand the dynamic behavior of the skier. The aims of this study were to model the aerodynamic drag of alpine skiers in giant slalom simulated conditions and to apply these models in a field experiment to estimate energy dissipated through aerodynamic drag.

Methods: The aerodynamic characteristics of 15 recreational male and female skiers were measured in a wind tunnel while holding nine different skiing-specific postures. The drag and the frontal area were recorded simultaneously for each posture. Four generalized and two individualized models of the drag coefficient were built, using different sets of parameters. These models were subsequently applied in a field study designed to compare the aerodynamic energy losses between a dynamic and a compact skiing technique.

Results: The generalized models estimated aerodynamic drag with an accuracy of between 11.00% and 14.28%, and the individualized models estimated aerodynamic drag with an accuracy between 4.52% and 5.30%. The individualized model used for the field study showed that using a dynamic technique led to 10% more aerodynamic drag energy loss than using a compact technique.

Discussion: The individualized models were capable of discriminating different techniques performed by advanced skiers and seemed more accurate than the generalized models. The models presented here offer a simple yet accurate method to estimate the aerodynamic drag acting upon alpine skiers while rapidly moving through the range of positions typical to turning technique.

##### Author Information

^{1}Sport Science Institute, University of Lausanne, Lausanne, SWITZERLAND; ^{2}Department of Mechanical Engineering, Faculty of Engineering, The University of Auckland, Auckland, NEW ZEALAND; and ^{3}Department of Sport and Exercise Science, Faculty of Science, The University of Auckland, Auckland, NEW ZEALAND

Address for correspondence: Frédéric Meyer, Institut des Sciences du Sport de l’Université de Lausanne, Université de Lausanne, Batiment Vidy, 1015 Lausanne, Switzerland; E-mail: fmeyer.mail@gmail.com.

Submitted for publication June 2011.

Accepted for publication November 2011.

Athletes performing disciplines like running, speed skating, cycling, or cross-country skiing have always been interested in optimizing their aerodynamic drag to increase speed and achieve better performance (^{8,14,15,18}). In alpine skiing, the gravitational force is used to increase the skier’s kinetic energy, whereas the aerodynamic drag is one of the two nonconservative forces doing negative work on the skier. Quantifying this parameter is therefore important to understand skier performance.

Several studies have examined skier aerodynamic drag. Watanabe and Ohtsuki (^{19,20}) analyzed the aerodynamic drag of a variety of skiing postures in a wind tunnel study and skiing velocity in a field study. Later, Kaps et al. (^{7}) proposed a method to calculate snow friction and drag area (*CDA*) during straight downhill skiing using photocells. Theoretical drag analysis has been conducted by Savolainen (^{12}) to compare different skiers’ posture and determine factors limiting speed. Performance coefficients taking into account factors like mass (*M*), frontal area (*AF*), and drag coefficient (*CD*) have been developed with wind tunnel tests by Luethi and Denoth (^{9}). Thompson and Friess (^{17}) performed wind tunnel tests to improve the aerodynamic efficiency of speed skiers by optimizing their posture and equipment.

Although these studies have made valuable contributions toward our understanding of the aerodynamic properties of static skiing postures, they are limited in that alpine skiing is primarily a dynamic sport where the skier continually moves and changes positioning. To allow the drag analysis of skiers performing turns, Barelle et al. (^{2}) modeled the *CD* on the basis of athlete segment lengths and intersegmental angles, thereby allowing the determination of aerodynamic properties through a complete span of positions typically encountered in skiing. However, the segment lengths and intersegmental angles required to use this model are often difficult to obtain in field research settings.

Despite these previous studies looking at the aerodynamic properties of alpine skiers, the importance of air drag toward performance in this dynamic sport is poorly understood. Mechanical energy (i.e., the sum of kinetic and potential energy) was used by Supej (^{16}) to deduce dissipated energy during turns. Reid et al. (^{11}) used the same method in slalom, but the intrinsic factors influencing this energy dissipation have not yet been analyzed.

Therefore, the first aim of this study was to develop models of the aerodynamic *CD* of alpine skiers performing turns. The models should be able to take into account the skier’s postural changes, using parameters that can be measured in the field. Furthermore, the second purpose of this study was to use the developed models to analyze the energy dissipated by the aerodynamic drag of a skier using either a dynamic technique, where the skier exposes a relatively large *AF* to the wind, or a compact technique, where the skier maintains an aerodynamic position, while performing giant slalom turns on a ski slope.

## METHODS

##### Wind Tunnel Experiment

##### Participants

Fifteen recreational male and female skiers (mean ± SD: body *M* = 75.9 ± 9.7 kg, height (*H*) = 1.79 ± 0.07 m, age = 32.3 ± 6.7 yr) volunteered for the study. All participants were healthy, without any joint motion problems. Written informed consent was obtained from each participant before participation in the study, which was approved by the University of Auckland Human Participants Ethics Committee.

##### Wind tunnel setup

Testing was carried out at the University of Auckland in a wind tunnel with an open jet configuration, the jet having dimensions of 2.5 m (width (*W*)) by 3.5 m (*H*), and a maximum flow speed of 18 m·s^{−1}. Turbulence levels were approximately 0.5% in the flow direction, and the velocity profile was uniform.

Participants were positioned on a force platform capable of measuring drag in the longitudinal wind direction. The drag force (*D*) was calculated through measurement of the displacement of a distorting force block, by using a Linear Variable Displacement Transducer (RDP, Ltd., Heath Town, United Kingdom). A 16-bit analog/digital converter (NI-6034; National Instruments, Austin, TX) was used to acquire the signal on a PC at a frequency of 200 Hz. The Linear Variable Displacement Transducer was previously calibrated over a suitable range of loads. This transducer exhibits a high degree of linearity and repeatability, with an accuracy of approximately 1% of the measured reading and a repeatability of 0.5% (^{5}). In accordance with Sayers and Ball (^{13}), no flow corrections were required in the open-circuit tunnels because the blockage model (in our case, the person) was less than 1 m^{2} and the open area was 12.25 m^{2}.

The dynamic wind pressure was recorded with a Setra pressure transducer via a pitot-static probe (Airflow Developments, Ltd., High Wycombe, United Kingdom) positioned in the wind tunnel, upstream of the contracted section. Before the experiment, a second probe was positioned in the middle of the testing volume to determine the ratio of dynamic pressure between the two locations. The measured pressure at run time was then adjusted accordingly. The accuracy of the dynamic pressure is approximately 2% of the measured value with a repeatability of approximately 0.2%. Dynamic pressure, air temperature, and atmospheric pressure were recorded to enable the drag to be correctly nondimensionalized.

A limitation with many wind tunnel systems is the inability to measure the *AF* of an irregular moving object. This only allows the *CDA* to be calculated, which is of limited use in many subsequent calculations. To enable a true *CD* to be calculated, a real-time *AF* measurement system was developed. This consisted of a miniature camera (USB UI-1485LE; IDS Imaging, Obersulm, Germany) positioned in the wind tunnel upstream of the participant. The background was colored white, and the area covered by each pixel was calculated by measuring the size of a reference object positioned at the average plane of the participant. During the test, the participant was dressed in a black suit, which served both to provide a contrast for the photography and to provide clothing uniformity for the drag measurements. A 50% threshold was carried out on the grayscale image of the participant, generating a black-and-white picture. The total number of black pixels against the white background was then counted and converted into a true area in square meters, which was displayed to the subject every 0.5 s. The accuracy of this system was approximately 0.001 m^{2}. The black-and-white images were also used to determine the skiers’ *H* and *W* by counting the number of black pixels across the maximal horizontal and vertical distance between two anatomical reference points on the frontal plane.

##### Experimental procedure

Before the test, each participant’s upright *H* and *M* were measured in meters and kilograms, respectively, and the corresponding body surface area (BodyS), which represents the total area of the skin, was calculated using the method of Boyd (^{4}):

To account for any weight-induced readings, the force transducer was zeroed with the participant on the balance under windless conditions at the start of each trial. The maximal *AF* (MaxA) of each participant (standing upright with the arms outstretched) was measured at the same time through the miniature camera. The wind tunnel was then run up to a speed of about 16 m·s^{−1}, corresponding to typical gate entry speeds in giant slalom conditions, and, therefore, the Reynolds number was approximately the same as it would be experienced by the participants in the field.

After a settling period, the participants assumed nine postures, with varying leg flexion and arm spacing (Fig. 1). A red/green light switch was set up in front of the participants to let them know when they had to change posture. Each posture was repeated three times and held for 15 s during which *D* and *AF* were measured and averaged.

The *CD* was calculated in the standard manner as follows:

where *ρ* is the air density and *V* is the wind speed. The dynamic pressure (ρ*V*^{2}/2) was measured with the Setra pressure transducer, *AF* was measured with the miniature camera, and *D* was measured with the force balance.

##### Model construction

When conducting field tests, a participant’s anthropometric data may not always be available (as in competition settings), whereas other information may be difficult to obtain (such as *AF*, which can only be obtained from full frontal pictures). Therefore, models based on different combinations of parameters (two to seven) were built to accommodate the information typically available in alpine field test conditions.

On the basis of the possible sets of available data, six models were built: four generalized models using all participants’ data, as well as two individualized models for each participant. Table 1 summarizes the parameters that were used in each of the six models.

Anthropomorphic parameters are inherent to an individual and do not vary with the position of the skier. They are therefore not relevant to build individualized models. However, the use of these parameters can improve the accuracy of the generalized models.

##### Field Experiment

The field experiment was carried out on an indoor ski slope. One of the 15 participants volunteered to perform two giant slalom runs in a white suit dotted with black hemispherical markers. He was asked to execute ample active movements on the first run and to remain more compact on the second run. A total of six gates were set up with a linear gate distance of 24 m and a lateral offset of 9 m. The first three gates were used to initiate the rhythm, and the last three were recorded. The slope angle was approximately 8 to 10 degrees. To record the skier’s position during the turn, six piA1000-48gm cameras with a 1004 × 1004-pixel resolution and running at 48 Hz (Basler AG, Ahrensburg, Germany) were placed around the slope, three on each side. The orientations of the two top and two bottom cameras were fixed. The two cameras in the middle were mounted onto specially built tripod heads that allowed operators to pan and tilt the cameras while maintaining camera sensor positions. Before the test, four calibration poles with three markers each were set up around the center of the capture volume and recorded with the cameras. Each calibration marker, reference point, and camera position was measured with a reflectorless total station (Sokkia Set530R; Sokkia Topcon, Kanagawa, Japan). Each camera was connected by Gigabit Ethernet to its own laptop, a battery pack, and a custom synchronization unit. The synchronization unit sent signals with the desired frequency to each camera, triggering the cameras to save images to the RAM memory of their associated laptops using a dedicated software (SwisTrack; Lausanne, Switzerland). When the synchronization unit was switched off or when the RAM memory was full, all the images were transferred to the hard drive.

Sequences provided by the multiple-camera system and three-dimensional positions of the points given by the total station were processed using the SIMI motion software (SIMI Reality Motion Systems GmbH, Unterschleissheim, Germany). The calibration markers were first used to determine the 11 standard parameters of the direct linear transformation calibration method (^{1}). Reference points were affixed to the side of the ski hall to allow for the cameras’ panning and tilting angles to be determined during the tests. The tridimensional reconstruction accuracy was controlled by comparing the gate position given by the total station with the position calculated with the software for the three visible gates on the two runs. The center of *M* (CoM) of the skier was calculated using the Hanavan method (^{6}). Position trajectories of the head, feet, and arms were exported to calculate the skier’s *H* and *W*. Because the anthropomorphic data were available but the *AF* was not, the GM3 and IM2 models were both used to determine the evolution of the aerodynamic *CD* over the turn cycle for the purpose of comparison. The energy losses due to aerodynamic friction (ΔEaero) were calculated at each step of the turn as follows:

where Δdist was the distance traveled by the skier’s CoM and *D* was determined by rearranging equation 2 to give the following:

where *V*_{Skier} is the speed of the skier’s CoM and *V*_{wind} is the component of the wind speed in the direction of the skier’s speed. GM3 and IM2 were used to give an estimate of the *CDA*, and combining equations 3 and 4 gives the dissipated aerodynamic energy during a Δ*t* interval (Δdist is replaced by *V*_{Skier} × Δ*t*):

Because tests were performed in a ski hall, wind speed can be neglected in this study. However, it is an important parameter and must be considered during outdoor experiments. For a whole turn, the total aerodynamic drag energy dissipated is obtained by summing the previous equation between *t*_{0}, the beginning of the turn, and *t*_{end}, the end of the turn:

##### Statistical Analysis

A backward stepwise linear regression was used to find the best predictive parameters of the models. The cutoff value for parameter acceptance was stated at *P* ≤ 0.1. The coefficient of determination (*R*^{2}) and the SD of the estimate of the models were calculated.

The validation of the generalized models was performed by removing one participant from the data set, recalculating the model coefficients with the remaining 14 participants, and then using the removed participant to compare the model prediction with an independent measure. A rotation through all 15 participants was performed, and the mean error was used to describe the model accuracy. The individualized models were validated in the same way, by removing the result of one posture from the data set, recalculating the model with the remaining eight postures, and applying the models to the removed posture.

Bland–Altman plots (^{3}) were used to compare the agreement between the generalized models and the experimental data. For the generalized and the individualized models, the 95% limit of agreement (±1.96 SD) was calculated. All the statistical analyses were performed with SPSS 16 software (SPSS, Inc., New York, NY), and significance was accepted at *P* < 0.05.

## RESULTS

##### Wind Tunnel Experiment

##### Developed models

Table 2 shows the multiplication coefficient of each parameter, as well as the *R*^{2}, the SD of the estimate, and the significance of each model (*P*) developed to estimate the *CDA*. The 0.1 cutoff of parameter acceptance discarded *M*, MaxA, and BodyS from GM1, as well as *M* and MaxA from GM3.

GM1 offered the best accuracy with *R*^{2} = 0.972, *P* < 0.001, and an SD of the estimate = 0.016 m^{2}. For GM2 and GM3, the *R*^{2} was 0.962 (SD = 0.019 m^{2}, *P* < 0.001) and 0.953 (SD = 0.021 m^{2}, *P* < 0.001), respectively. Finally, GM4 represented the worst model with an *R*^{2} equal to 0.933 (SD = 0.025 m^{2}, *P* < 0.001).

Bland–Altman plots between the generalized models and the experimental data are shown in Figure 2 for all the generalized models. The 95% limit of agreement is also reported for GM1 (±11.00), GM2 (±11.99), GM3 (±13.25), and GM4 (±14.18) in Figures 2A, B, C, and D, respectively.

For the individualized models, the backward linear regression did not remove any parameters. IM1 reached an average *R*^{2} = 0.995 and an SD of the estimate = 0.009 m^{2}. Validation between the models and the measures gave a 95% limit of agreement of ±4.52%. IM2 showed slightly worse results with *R*^{2} = 0.989, SD = 0.01 m^{2}, and a 95% limit of agreement of ±5.30%.

##### Field Experiment

To estimate his *CDA*, the following IM2 was individually developed for the skier who performed the field test:

Figure 3A compares the evolution of *CDA* over a turn cycle, using the IM2 defined in equation 7, for the active and compact techniques. The limit of agreement of 5.30% given for the IM2 is also plotted for each technique, showing a possible differentiation between the dynamic and the compact skiing technique for 56% of the turn. The darker gray area indicates an overlapping of the two techniques’ limit of agreement.

Figure 3B shows the comparison of the same data set, but using GM3. The limit of agreement of 13.25% is also plotted for each technique, showing an overlapping of the two techniques during the whole turn.

Equation 5 gives the total energy dissipated because of the aerodynamic drag and is illustrated in Figure 4 using either the IM2 (Fig. 4A) or the GM3 (Fig. 4B) for one turn performed with the two different techniques. The 95% limit of agreement is also reported, showing the disparity of energy dissipation. For the current giant slalom, an active technique gives around 3500 J of energy dissipated during one gate. This represents around 350 J more energy dissipation in one turn than a compact position, which means a loss of 10% more energy during the whole run.

## DISCUSSION

The most important finding of this study is the accuracy of the individualized models, which allow for very good estimation of skier aerodynamic properties while performing giant slalom turns. Indeed, these models, which explain 98.9% and 99.5% of the experimental data, have an accuracy better than 5.30% to determine the skier’s aerodynamic *CD*. The accuracy obtained is good enough for discrimination of different techniques performed by advanced skiers, as seen in Figure 3A.

The generalized models developed are a little less accurate, explaining between 93.3% and 97.2% of the experimental data, corresponding to 11.00% and 14.18% error for the 95% limit of agreement. Using anthropomorphic data to build the generalized models led to an improvement of only 2%. Therefore, the accuracy differences of about 8% between generalized and individualized models should be due to other factors not measured in this study such as differences in individual body posture held in the wind tunnel.

Similar to the model of Barelle et al. (^{2}), the generalized models developed in this investigation allow a global intraindividual comparison of a skier performing different techniques but not accurate differentiation between skiers. However, the parameters used in this study are less specific than the segment lengths and angles used by Barelle et al. (^{2}) and offer a wider and more generic use of the models. This allows the backward linear regression method to choose the relevant parameters and refuse parameters that are not necessary for the model. More flexibility is therefore possible for further parameter integration. Barelle et al. (^{2}) considered many more positions to allow the variation of the different parameters, but finally, the *CD* found in both studies corresponds very well for the different positions a skier can reach during a run.

The developed models help to understand intrinsic factors of energy dissipation as calculated by Supej (^{16}) and Reid et al. (^{11}). They both found high energy dissipation around the gate crossing and low energy dissipation during gate transition, which is inverted compared with the curves of Eaero in Figure 4. The energy dissipation due to snow friction, estimated by Meyer and Borrani (^{10}), indicates a higher importance of the ski–snow friction in giant slalom and curves corresponding to the results obtained by Supej (^{16}) and Reid et al. (^{11}).

The study undertaken here is a first approximation of the skier’s aerodynamic drag, which is correct in the field in the case of little or no ambient wind speed. In this case, the wind flow onto the skier will always be head-on, regardless of his/her direction of travel (no yaw angle). If there is a substantial wind speed, the aerodynamic drag experienced by the skier will change depending on his/her direction of travel (yaw angle different from zero). To model this scenario, further tests would have to be conducted at a range of a skier’s yaw angles in the wind tunnel. Then, a new dynamic model could calculate the skier’s aerodynamic drag considering wind speed and relative direction and the skier’s yaw angle at each point of the turn. In contrast, for small yaw angles, the current model serves as a good estimation of the aerodynamic drag.

One limitation of the current method is that the various postures in the wind tunnel used to develop the models (Fig. 1) are symmetric and differ from asymmetric skier positions achieved when turning, a fact that may jeopardize the model validity. Unfortunately, the repeatability of holding more turn-specific postures in the wind tunnel was poor because of the difficulty of holding unbalanced positions. A second limitation is that the models reported here use the wind tunnel measurements of a series of static positions to model skiers who change their position continuously while turning. It may be that the dynamic behavior of the aerodynamic drag of a skier in continuous movement may somehow differ from that of a set of static positions. However, wind tunnel measurements are currently limited to static positions as the ground force platform would record each CoM acceleration, making it difficult to isolate the aerodynamic *D*.

In conclusion, this article provides simple and functional models to calculate the aerodynamic drag of alpine skiers performing giant slalom turns. The developed models offer a mean accuracy between 4.52% and 14.18%, depending on the selected parameters. Using these models in skiing field studies may help to improve our understanding of the role of aerodynamic drag in skier performance. A functional model of ski–snow friction while performing turns still needs to be developed to have a full overview of where, how, and when athletes lose energy during turns.

This work was supported by a grant from the Swiss Federal Office for Sport.

The authors thank Dr. F. Formenti for his writing assistance.

The authors have no financial disclosure or conflict of interest to declare.

The results of the present study do not constitute endorsement by the American College of Sports Medicine.

## REFERENCES

**Keywords:**

WIND TUNNEL; ENERGY DISSIPATION; PERFORMANCE; AIR FRICTION; ALPINE SKI