CFD analysis of cyclist aerodynamics: Performance of ...

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CFD analysis of cyclist aerodynamics: Performance of different turbulence-1

modelling and boundary-layer modelling approaches 2

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Thijs Defraeye a, *

, Bert Blocken b, Erwin Koninckx

c, Peter Hespel

c and Jan Carmeliet

d,e 4

5 a Laboratory of Building Physics, Department of Civil Engineering, Katholieke Universiteit Leuven, Kasteelpark 6 Arenberg 40, 3001 Heverlee, Belgium 7 b Building Physics and Systems, Eindhoven University of Technology, P.O. Box 513, 5600 Eindhoven, The 8 Netherlands 9 c Research Centre for Exercise and Health, Department of Biomedical Kinesiology, Katholieke Universiteit 10 Leuven, Tervuursevest 101, 3001 Heverlee, Belgium 11 d Chair of Building Physics, Swiss Federal Institute of Technology Zurich (ETHZ), Wolfgang-Pauli-Strasse 15, 12 8093 Zürich, Switzerland 13 e Laboratory for Building Science and Technology, Swiss Federal Laboratories for Materials Testing and 14 Research (Empa), Überlandstrasse 129, 8600 Dübendorf, Switzerland 15 16 Keywords 17 18 Computational Fluid Dynamics; turbulence model; cyclist; aerodynamics; wind tunnel 19 20 Word count (Introduction to conclusions): 3494 words 21 22 23 Abstract 24 This study aims at assessing the accuracy of Computational Fluid Dynamics (CFD) for applications in sports 25 aerodynamics, for example for drag predictions of swimmers, cyclists or skiers, by evaluating the applied 26 numerical modelling techniques by means of detailed validation experiments. In this study, a wind-tunnel 27 experiment on a scale model of a cyclist (scale 1:2) is presented. Apart from three-component forces and 28 moments, also high-resolution surface pressure measurements on the scale model’s surface, i.e. at 115 locations, 29 are performed to provide detailed information on the flow field. These data are used to compare the performance 30 of different turbulence-modelling techniques, such as steady Reynolds-averaged Navier-Stokes (RANS), with 31 several k-ε and k-ω turbulence models, and unsteady Large-Eddy Simulation (LES), and also boundary-layer 32 modelling techniques, namely wall functions and low-Reynolds number modelling (LRNM). The commercial 33 CFD code Fluent 6.3 is used for the simulations. The RANS shear-stress transport (SST) k-ω model shows the 34 best overall performance, followed by the more computationally expensive LES. Furthermore, LRNM is clearly 35 preferred over wall functions to model the boundary layer. This study showed that there are more accurate 36 alternatives for evaluating flow around bluff bodies with CFD than the standard k-ε model combined with wall 37 functions, which is often used in CFD studies in sports. 38 39 1. Introduction 40 At racing speeds (± 50 km/h in time trails), the aerodynamic resistance experienced by a cyclist, also called drag, 41 is about 90% of his total resistance (Grappe et al., 1997; Kyle and Burke, 1984). The major part is caused by 42 form drag, related to the position of the cyclist on the bicycle. Many elite cyclists therefore try to optimise their 43 position for drag by means of field tests or wind-tunnel tests. With these techniques, the aerodynamic 44 improvements are usually assessed by trial and error, by evaluating the drag reduction. Rarely these 45 improvements are analysed more in detail by considering the resulting changes in the flow field since 46 measurements of the flow field are often time-consuming and can even be quite difficult to set up for field tests. 47 An alternative technique, which provides both drag and detailed flow-field information, is Computational Fluid 48 Dynamics (CFD), which has recently been used in cycling (Defraeye et al., 2010; Hanna, 2002; Lukes et al., 49 2004) but also in other sports disciplines like swimming (Bixler et al., 2007; Bixler and Riewald, 2002; Bixler 50 and Schloder, 1996; Gardano and Dabnichki, 2006; Lecrivain et al., 2008; Minetti et al., 2009; Rouboa et al., 51 2006; Zaïdi et al., 2008; Zaïdi et al., 2010), soccer (Barber et al., 2009), bobsleighing (Dabnichki and Avital, 52 2006) and ski jumping (Meile et al., 2006). All these studies consider flow around bluff bodies, i.e. mostly 53 humans, which have a quite streamlined shape (i.e. without sharp edges) and therefore no fixed boundary-layer 54 separation points, in contrast to other (sharp-edged) bluff bodies such as buildings. For these types of flows, the 55 CFD modelling approaches, applied in the aforementioned studies (see Table 1), have some limitations: (1) 56

* Corresponding author. Tel.: +32 (0)16321348; Fax: +32 (0)16321980.

E-mail address: [email protected]

Accepted for publication in Journal of Biomechanics, April 29, 2010

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Using steady Reynolds-averaged Navier-Stokes (RANS) modelling, the unsteady motions in the wake of a bluff 57 body are not captured. Only the mean flow is resolved and all scales of turbulence are modelled by a turbulence 58 model. There is however no universally valid turbulence model which is accurate for all classes of flows (Casey 59 and Wintergerste, 2000); (2) Using wall functions, the flow quantities in the boundary layer at the wall are 60 modelled instead of being resolved. Since wall functions are only valid under strict conditions, they can result in 61 inaccurate predictions of wall friction, and thus of the boundary-layer separation locations, in most complex 62 three-dimensional flows with separation and of boundary-layer transition from a laminar to a turbulent boundary 63 layer (Barber et al., 2009; Casey and Wintergerste, 2000). Instead, the boundary layer can be resolved explicitly 64 with low-Reynolds number modelling (LRNM), which should yield more accurate results but requires a much 65 higher grid resolution in the boundary-layer region. 66

Detailed validation experiments are therefore required to quantify the accuracy of the applied CFD modelling 67 techniques for a specific flow problem. Most CFD validation studies in sports aerodynamics (see Table 1: Barber 68 et al., 2009; Bixler et al., 2007; Dabnichki and Avital, 2006; Gardano and Dabnichki, 2006; Meile et al., 2006) 69 looked at drag and lift forces but extensive comparison of flow quantities, i.e. velocities or surface pressures, was 70 generally not performed, except qualitatively, i.e. by means of flow visualisation. Such flow-field data provide 71 complementary information to the drag measurements: a good agreement with CFD for drag is not necessarily 72 the result of a correct flow-field calculation since the drag force is actually an integrated flow quantity (of 73 surface pressures). An attempt to provide more detailed flow-field evaluation data for sport applications was 74 presented by Defraeye et al. (2010). They performed wind-tunnel measurements on a cyclist, where, apart from 75 drag, also surface pressures were measured on 30 locations on the cyclist’s body. However, the use of a real 76 cyclist limited the amount of sensors and introduced some uncertainty on the pressure data, which was related to 77 the determination of the exact locations of the pressure plates on the cyclist’s body, the size of these plates and 78 the attachment of the plates onto the body. Improvements could be obtained by using a (scale) model of a cyclist. 79

In this study, wind-tunnel experiments on such a scale model of a cyclist are presented. Apart from three-80 component forces and moments, also high-resolution surface pressure measurements on the scale model’s 81 surface are performed, i.e. at 115 locations. These wind-tunnel data are used to compare the performance of 82 several commonly available turbulence-modelling approaches and boundary-layer modelling approaches of CFD. 83 A comparison with the measured surface pressures provides more insight in the accuracy and deficiencies of 84 each CFD modelling approach, than an evaluation only based on (drag) forces. The conclusions of this study can 85 also be relevant for CFD studies involving high-speed applications of similar bluff-body geometries, such as 86 swimmers, skiers, bobsleighers, etc. 87 88 2. Methods 89 2.1. Experimental setup 90 A digital model of a cyclist in the upright position was obtained from a real cyclist (see Defraeye et al., 2010), 91 using a high-resolution 3D laser scanning system (K-Scan, Nikon Metrology, Belgium) combined with post-92 processing software (Focus RE, Nikon Metrology, Belgium). This digital model was used as an input for the 93 manufacturing process of the scale model (see Figure 1) by means of rapid prototyping (scale 1:2). The bicycle 94 was not included in the scale model for CFD meshing purposes and to reduce the manufacturing costs and the 95 complexity of the scale model. Additional stiffening elements, with an aerodynamic shape, were included. The 96 surface of the scale model was given a smooth finishing by resin impregnation. A total of 115 pressure taps were 97 included in the scale model (see Figure 1), placed flush with the surface, and were connected by pressure tubes 98 (inside the hollow scale model) to the pressure transducer. The scale model was fixed on a stand (Figure 2), 99 where the upper part was given an airfoil-like profile to minimise drag. The half-sphere in Figure 2, which is also 100 a part of the stand, was required to house the pressure transducer since it had to be located close to the scale 101 model to limit the length of the pressure tubes. The scale model and stand were placed in the test section (2.25 m 102 high and 3 m wide) of a closed-circuit wind tunnel (Dutch-German Wind tunnels, Marknesse, The Netherlands) 103 on a six-component force balance, located below the test-section floor. 104

Measurements were carried out at a wind speed of 20 m/s and the turbulence intensity at the inlet of the test 105 section was 0.02℅. The wind direction was parallel to the (virtual) bicycle axis, representing head wind. The 106 frontal area of the scale model with stand and the stand separately are 0.23 m² and 0.11 m², respectively, 107 resulting in a blockage ratio of 3℅ for the scale model with stand. All three force components (Fx, Fy and Fz) and 108 moment components (Mx, My and Mz) were measured. The precision of the balance was 0.1% of the full-scale 109 range, namely 0.25 N for Fx (lateral) and Fy (drag), 0.5 N for Fz (lift), 0.09 Nm for Mx (pitch) and My (roll) and 110 0.05 Nm for Mz (yaw). The balance data were sampled at 10 Hz for 30 s. Surface pressures on the scale model’s 111 surface were measured with pressure taps with an accuracy of 7 Pa, i.e. 0.1% of the full-scale range. The 112 pressures were sampled at 512 Hz for 24 s. 113

Note that since a scale model has been used (scale 1:2 at a wind speed of 72 km/h, i.e. 20 m/s), the Reynolds 114 number of these experiments is lower than for real cyclists at racing speeds (real scale at wind speeds of ± 50 115 km/h). Reynolds number effects during wind-tunnel tests at higher wind speeds (25-35 m/s) were however 116

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limited, namely about 2% for drag force, lift force and pitching moment, and on average 15% for surface 117 pressures (see section 3.1). Also note that the approach-flow conditions in the wind-tunnel tests in this study and 118 in most other wind-tunnel experiments on cyclist aerodynamics (i.e. low turbulence intensity and a uniform 119 velocity profile) are representative for the case where only the cyclist is moving and where the wind speed of the 120 surrounding air is zero. This situation is typically found in indoor environments (e.g. a velodrome) or in the 121 outdoor environment if there is no or little wind. 122 123 2.2. Numerical simulations 124 2.2.1. Numerical model 125 A digital model of the scale model together with stand (as in Figure 2) was obtained using the 3D laser scanning 126 system and was used for computational modelling. This virtual scale model was placed in a computational 127 domain, representing the wind tunnel. The size of this domain and the imposed boundary conditions are 128 specified in Figure 3 (see Appendix 1 for additional information on the computational model). 129 130 2.2.2. Simulation parameters 131 The simulations are performed with the CFD software Fluent 6.3, which uses the control volume method. Steady 132 RANS is used in combination with different turbulence models: standard k-ε model (Launder and Spalding, 133 1972), realizable k-ε model (Shih et al., 1995), RNG k-ε model (Choudhury, 1993), standard k-ω model (Fluent, 134 2006; Wilcox, 1988; Wilcox, 1998) and the shear-stress transport (SST) k-ω model (Menter, 1994). All these 135 models are used with LRNM to take care of the viscosity-affected region, i.e. the boundary layer on the scale 136 model’s surface. Note that the k-ε models require low-Reynolds number modifications since they were primarily 137 developed for high-Reynolds number flows. Thereby, a two-layer approach is used where the turbulent core 138 region of the flow is resolved with the k-ε model and a low-Reynolds number model is used to resolve the 139 viscosity-affected region, for which the one-equation Wolfshtein model (Wolfshtein, 1969) is used in this study. 140 Note that this one-equation low-Reynolds number model is less complex than those used by the two-equation k-141 ω models since it only solves one transport equation for turbulence instead of two, which can lead to a reduced 142 performance for some flow regimes. Of these turbulence models, the realizable k-ε model is also used with wall 143 functions since k-ε models with wall functions are used in most aforementioned numerical studies in sports (see 144 Table 1), and since they are included in most commercial CFD codes. Note that for the use of wall functions, the 145 realizable k-ε model is preferred over the standard k-ε model for reasons of convergence stability. Two types of 146 wall functions are used: standard (Launder and Spalding, 1974) and non-equilibrium (Kim and Choudhury, 147 1995) wall functions. Note that standard wall functions are only valid under equilibrium boundary-layer 148 conditions (e.g. Casey and Wintergerste, 2000; Franke et al., 2007), which is not the case in regions of flow 149 separation, reattachment and strong pressure gradients. For the LES simulations, the dynamic Smagorinsky 150 subgrid-scale model is used (see Kim, 2004) with LRNM. An overview of the performed CFD simulations is 151 given in Table 2 (see Appendix 1 for additional information on the simulations). 152 153 3. Results 154 3.1. Forces and moments 155 Aerodynamic forces are usually quantified by dimensionless coefficients, e.g. drag or lift coefficients. These 156 force coefficients (CFx, CFy and CFz) and moment coefficients (CMx, CMy and CMz) relate the forces and moments 157 to the frontal area A (m

2) and the lever arm L (m): 158

2

i Fi

UF AC

2

ρ= (1) 159

2

i Mi

UM ALC

2

ρ= (2) 160

where ρ is the air density (kg/m3), U is the approach-flow wind speed (m/s) and i is an index (x, y or z). Often, 161

the force or moment area (ACFi and ALCMi) is reported, since this does not require an explicit determination of A 162 or L. Although ALCMi actually has the dimensions m³, it is referred to as moment area in this study. In Figure 4, 163 the dependency of the force and moment areas, from wind-tunnel experiments, with the approach-flow wind 164 speed is indicated. These areas become quasi-independent of the Reynolds number at wind speeds ≥ 20 m/s. The 165 force and moment areas of the scale model with stand, obtained by the wind-tunnel experiments, are compared 166 with the results from the various CFD simulations in Figure 5, by reporting their relative difference. Note that 167 ACFx, ALCMy and ALCMz are not compared since their value is very low due to symmetry, resulting in relatively 168 large measurement errors. The measurement errors on ACFy (drag area), ACFz (lift area) and ALCMx (pitch area) 169 are 0.8%, 4.0% and 0.5%, respectively. 170

Some general trends can be distinguished in the CFD predictions: an underprediction of the drag and pitch 171 areas by most turbulence models and an overprediction of the lift area. Apart from turbulence and boundary-172 layer model limitations, a possible reason for part of the differences with the wind-tunnel experiments could be a 173

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discrepancy in the boundary-layer thickness on the lower wall, which is a few centimetres in the wind tunnel. 174 Since no exact information was available however, it was assumed in the CFD simulations that the thickness was 175 zero at the inlet of the computational domain, resulting in a thickness of a few centimetres near the scale model. 176 The influence of this boundary-layer mismatch is however considered quite limited. Since the stand itself also 177 accounts for a significant part of the drag force, lift force and pitching moment, mainly due to its relatively large 178 frontal area (50% of that of the scale model with stand), discrepancies with the wind-tunnel data are not only 179 related to an erroneous prediction of flow around the scale model itself, but also around that of the stand, which 180 is quantified and discussed in detail in Appendix 1. 181

Of all LRNM k-ε models, the standard model seems to show the best overall performance, where only the lift 182 area is overpredicted significantly. The use of wall functions seems to decrease the accuracy to some extent, i.e. 183 with about 7%, but no significant differences were found between the two wall-function types. For the k-ω 184 models, the standard model shows large discrepancies with the wind-tunnel measurements, except for the lift 185 area. The SST k-ω model however shows a very good agreement for both force and moment areas (≤ 11% with 186 an average of 6%), and thereby it performs best of all evaluated turbulence-modelling approaches. Although LES 187 provides relatively accurate drag and pitch area predictions, the lift area is severely overpredicted, which is 188 mainly attributed to the stand (see Appendix 1). Note however that an accurate prediction of force and/or 189 moment areas does not necessarily imply that the flow field is resolved accurately, i.e. the prediction of over- 190 and underpressure zones, since these areas are actually integrated quantities. Since flow-field information could 191 provide additional insight for the comparison of CFD simulations, surfaces pressures on the scale model are 192 evaluated in the next section. 193 194 3.2. Pressure coefficients 195 Surface pressures are usually expressed by dimensionless pressure coefficients (CP): 196

surf inlP 2

(p p )C

U

2

−=

ρ (3) 197

where psurf is the pressure on the scale model’s surface and pinl is the static pressure at the inlet of the wind-tunnel 198 test section. For the CFD simulations, pinl is the average pressure at the inlet of the computational domain. Note 199 however that this inlet is not located at the same location as the inlet of the wind-tunnel test section, which is 200 done to limit the size of the upstream part of the computational domain. In Figure 6, the CP coefficients obtained 201 with the wind-tunnel experiments (CP,WT) are compared to the results from the CFD simulations (CP,CFD) for 202 different turbulence and boundary-layer modelling approaches. Due to the manufacturing process of the scale 203 model and the generation of the digital model for the CFD simulations, which led to some smoothing out of 204 surface details, there is some uncertainty on the locations of the pressure taps. To account for this uncertainty, 205 the reported CFD data are the averaged pressures within a circular zone (diameter 7 mm) on the surface of the 206 scale model. The uncertainty band for the CFD results in Figure 6 is the standard deviation from this averaged 207 value, and is quite small. For the uncertainty of the wind-tunnel data, the measurement error on the pressure taps 208 (7 Pa) is used. Note that a good agreement of CFD with wind-tunnel measurements implies that the data are 209 located near the solid line which is shown in the figures. The dotted lines represent 25% deviation from this solid 210 line. A more straightforward comparison between the results of the different CFD simulations can be done by 211 comparing the correlation coefficients (see Figure 6). Additional information on the flow field is given in 212 Appendix 1. 213

Roughly the same trends are found as in the previous section: (1) Of the LRNM k-ε models, the standard k-ε 214 model shows the best performance; (2) The use of wall functions leads to a decreased accuracy, compared to 215 LRNM; (3) Of the k-ω models, the standard model does not perform well, especially for the windward pressures, 216 while the SST k-ω model performs best of all evaluated turbulence-modelling approaches; (4) LES also performs 217 very well, i.e. comparable with the SST k-ω model. 218 219 4. Discussion 220 The results of both the force and moment areas combined with the surface pressures allow a detailed comparison 221 of the performance of the different turbulence and boundary-layer modelling approaches. Regarding turbulence 222 modelling, RANS combined with the SST k-ω model clearly shows the best overall performance, but also LES 223 performs very well, except for the lift area, which is mainly attributed to the stand (see Appendix 1). Although 224 LES outperforms all other RANS turbulence models and provides in addition information on the unsteady, i.e. 225 temporally-fluctuating, flow field around the scale model, it however imposes a much higher computational cost 226 (about 5-15 times more than RANS, depending on the RANS turbulence model used), which makes the RANS 227 SST k-ω model more attractive from a practical point of view. The reason for the good performance of the SST 228 k-ω model is probably because it uses a two-equation k-ω model formulation to solve the near-wall region, for 229 which the k-ω models were originally developed, while a k-ε model formulation, developed for high-Reynolds 230 number flows, is used to solve the turbulent core region of the flow. If a k-ω model is used to resolve the 231

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turbulent core region, which is done in the standard k-ω model, clearly larger discrepancies with the experiments 232 are found since this model was primarily not developed for resolving this region. 233

In this study, the standard k-ω model shows the worst performance of all LRNM RANS models which is in 234 contrast to the study of Zaïdi et al. (2010), in which this model clearly outperformed the standard k-ε model with 235 respect to drag predictions. These findings could be related to the fact that a highly aerodynamic shape was 236 considered by Zaïdi et al. (2010), namely a swimmer in diving position, combined with the use of a wall-237 function grid. For this flow problem, the wake zone is quite small and the skin-friction drag significantly 238 contributes to the total drag: about 20%, compared to about 5% in this study. Thereby, the prediction of 239 boundary-layer separation, which determines the size of the wake zone, and wall friction becomes critical for an 240 accurate drag prediction, but the wall functions, used by the k-ε models, generally cannot provide accurate 241 predictions here (Casey and Wintergerste, 2000). The k-ω models, which are in essence developed to deal with 242 near-wall, i.e. low-Reynolds number flows, could therefore provide more accurate results. The LRNM approach 243 is clearly identified as a more accurate boundary-layer modelling alternative than wall functions in this study 244 although LRNM requires a much higher grid resolution in the near-wall region, resulting in much more cells in 245 the computational model, especially at high wind speeds (see Appendix 1). 246

In many of the previous validation experiments (see Table 1) only drag and/or lift forces were quantified. 247 Since these are integrated flow quantities, they do not necessarily imply accurate flow-field predictions. 248 However, in this study, the detailed surface pressure measurements indicated that the accuracy of the different 249 modelling approaches could be compared well by considering force and moment measurements. It is however 250 important to note that a comparison based on only one parameter, e.g. the drag force, is not always sufficient. 251 Therefore it is recommended to compare multiple parameters, as in Figure 5, if possible together with flow-field 252 evaluation. 253

254 5. Conclusions 255 In this study, a detailed experiment of flow around a scale model of a cyclist allowed an extensive comparison of 256 various CFD modelling approaches. It was found that the RANS SST k-ω model showed the best overall 257 performance, followed by the more computationally expensive LES, and that LRNM is clearly preferred over 258 wall functions to model the boundary layer. This study showed that there are more accurate alternatives for 259 evaluating flow around bluff bodies than the standard k-ε model combined with wall functions, which is often 260 used in CFD studies in sports. Although CFD did not provide the same accuracy as the wind-tunnel experiments 261 in this study, it has the significant advantage that detailed flow-field information is available, which can 262 contribute to the physical insight in the causes of the drag force. The results of this study can also be relevant for 263 CFD studies involving high-speed applications of similar bluff-body geometries, such as swimmers, skiers, 264 bobsleighers, etc. Note that for very aerodynamically-shaped bodies, e.g. a swimmer in diving position, the 265 influence of the boundary-layer modelling approach will probably become even more critical. 266 267 Conflict of interest statement 268 None 269 270 Acknowledgements 271 This study was funded by the Flemish Government and the Flemish Cycling Federation, which had no 272 involvement in: study design, collection, analysis and interpretation of data; writing of the manuscript; the 273 decision to submit the manuscript for publication. Special thanks go to Jos Smets (Belgian Cycling Federation), 274 Eddy Willemsen and the DNW wind-tunnel team, Harry Sools and the SIRRIS team. 275 276 277 References 278 Barber, S., Chin, S.B., Carre, M.J., 2009. Sports ball aerodynamics: A numerical study of the erratic motion of 279

soccer balls. Computers & Fluids 38 (6), 1091-1100. 280 Bixler, B., Pease, D., Fairhurst, F., 2007. The accuracy of computational fluid dynamics analysis of the passive 281

drag of a male swimmer. Sports Biomechanics 6 (1), 81-98. 282 Bixler, B., Riewald, S., 2002. Analysis of a swimmer's hand and arm in steady flow conditions using 283

computational fluid dynamics. Journal of Biomechanics 35 (5), 713-717. 284 Bixler, B., Schloder, M., 1996. Computational fluid dynamics. An analytical tool for the 21st century swimming 285

scientist. Journal of Swimming Research 11, 4-22. 286 Casey, M., Wintergerste, T., 2000. Best Practice Guidelines. ERCOFTAC Special Interest Group on “Quality 287

and Trust in Industrial CFD”, ERCOFTAC. 288 Choudhury, D., 1993. Introduction to the Renormalization Group Method and turbulence modeling. Fluent Inc. 289

Technical Memorandum TM-107. 290

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Dabnichki, P., Avital, E., 2006. Influence of the position of crew members on aerodynamics performance of a 291 two-man bobsleigh. Journal of Biomechanics 39 (15), 2733-2742. 292

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when using the CFD method. Journal of Biomechanics 43 (3), 405-411. 340

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Figure captions 341 342

343 Figure 1: Scale model of cyclist, manufactured by means of rapid prototyping. The locations of the 344 pressure taps are shown schematically by means of black dots. The stiffening elements between the elbows 345 and knees were not included in the actual scale model (see Figure 2). 346 347

348 Figure 2: Scale model with stand which is mounted on the wind-tunnel floor. The pressure transducer is 349 located in the half-sphere. 350

3 m

2.2

5 m 10.6 m

14.5 m

2.1 m

Ambient static

pressureUniform inlet velocity

20 m/s

Slip wall (symmetry)

zy

x3 m

2.2

5 m 10.6 m

14.5 m

2.1 m

Ambient static

pressureUniform inlet velocity

20 m/s

Slip wall (symmetry)

zy

x 351

Figure 3: Computational domain and boundary conditions. 352

8

0

0.02

0.04

0.06

0.08

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0.14

0 5 10 15 20 25 30 35

Wind speed (m/s)

AC

Fy

,WT, A

CF

z,W

T o

r A

LC

Mx

,WT

DRAG areaLIFT areaPITCH area

353 Figure 4: Drag, lift and pitch areas (of scale model with stand) of wind-tunnel tests (ACFy,WT, ACFz,WT, 354 ALCMx,WT) as a function of the approach-flow wind speed. 355 356

357 358

-4%

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sstk

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_S

WF

rk-ε

_N

WF

LE

S

Fy,CFD Fy,WT

Fy,WT

AC AC

AC

− Fz,CFD Fz,WT

Fz,WT

AC AC

AC

− Mx,CFD Mx,WT

Mx,WT

ALC ALC

ALC

−

DRAG area LIFT area PITCH area

sk-ε

rk-ε

rng

k-ε

sk-ω

sstk

-ω

rk-ε

_S

WF

rk-ε

_N

WF

LE

S-4%

29%

14%

-36%

47%

-48%

72%

4%

-3%-6%

50%

-20%

-10%

-20%

-33%

40%

-30%

-40%

-36%

-47%

48%

15%

11%

-50%

-40%

-30%

-20%

-10%

0%

10%

20%

30%

40%

50%

60%

70%

80%

sk-ε

rk-ε

rng

k-ε

sk-ω

sstk

-ω

rk-ε

_S

WF

rk-ε

_N

WF

LE

S

Fy,CFD Fy,WT

Fy,WT

AC AC

AC

− Fz,CFD Fz,WT

Fz,WT

AC AC

AC

− Mx,CFD Mx,WT

Mx,WT

ALC ALC

ALC

−

DRAG area LIFT area PITCH area

sk-ε

rk-ε

rng

k-ε

sk-ω

sstk

-ω

rk-ε

_S

WF

rk-ε

_N

WF

LE

S

359 Figure 5: Comparison of relative difference of drag, lift and pitch areas (of scale model with stand) of 360 wind-tunnel tests (ACFy,WT, ACFz,WT, ALCMx,WT) and various CFD simulations (ACFy,CFD, ACFz,CFD, 361 ALCMx,CFD). See Table 2 for the abbreviations of the CFD turbulence models. 362

9

-2

-1.5

-1

-0.5

0

0.5

1

1.5

-2 -1 0 1

CP,WT (-)

CP

,CF

D (

-)

-2

-1.5

-1

-0.5

0

0.5

1

1.5

-2 -1 0 1

CP,WT (-)

CP

,CF

D (

-)

-2

-1.5

-1

-0.5

0

0.5

1

1.5

-2 -1 0 1

CP,WT (-)

CP

,CF

D (

-)

-2

-1.5

-1

-0.5

0

0.5

1

1.5

-2 -1 0 1

CP,WT (-)

CP

,CF

D (

-)

-2

-1

0

1

2

3

-2 -1 0 1

CP,WT (-)

CP

,CF

D (

-)

-2

-1.5

-1

-0.5

0

0.5

1

1.5

-2 -1 0 1

CP,WT (-)

CP

,CF

D (

-)

-2

-1.5

-1

-0.5

0

0.5

1

1.5

-2 -1 0 1

CP,WT (-)

CP

,CF

D (

-)

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

-2 -1 0 1

CP,WT (-)

CP

,CF

D (

-)

sk-ε

rk-ε

rngk-ε

sk-ω

sstk-ω

rk-ε_SWF

rk-ε_NWF

LES

0.92

0.90

0.87

0.95

0.90

0.87

0.86

0.95

-2

-1.5

-1

-0.5

0

0.5

1

1.5

-2 -1 0 1

CP,WT (-)

CP

,CF

D (

-)

-2

-1.5

-1

-0.5

0

0.5

1

1.5

-2 -1 0 1

CP,WT (-)

CP

,CF

D (

-)

-2

-1.5

-1

-0.5

0

0.5

1

1.5

-2 -1 0 1

CP,WT (-)

CP

,CF

D (

-)

-2

-1.5

-1

-0.5

0

0.5

1

1.5

-2 -1 0 1

CP,WT (-)

CP

,CF

D (

-)

-2

-1

0

1

2

3

-2 -1 0 1

CP,WT (-)

CP

,CF

D (

-)

-2

-1.5

-1

-0.5

0

0.5

1

1.5

-2 -1 0 1

CP,WT (-)

CP

,CF

D (

-)

-2

-1.5

-1

-0.5

0

0.5

1

1.5

-2 -1 0 1

CP,WT (-)

CP

,CF

D (

-)

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

-2 -1 0 1

CP,WT (-)

CP

,CF

D (

-)

sk-ε

rk-ε

rngk-ε

sk-ω

sstk-ω

rk-ε_SWF

rk-ε_NWF

LES

0.92

0.90

0.87

0.95

0.90

0.87

0.86

0.95

363 Figure 6: Comparison between pressure coefficients of wind-tunnel tests (CP,WT) and various CFD 364 simulations (CP,CFD) (with uncertainty bands). The correlation coefficients are also indicated. Note that the 365 vertical axes of sk-ω and LES have a different scale. The uncertainty band for CP,CFD is the standard 366 deviation from the averaged value within a circular zone (diameter 7 mm) on the surface of the scale 367 model. For the uncertainty of CP,WT, the measurement error on the pressure taps (7 Pa) is used. The 368 dotted lines represent 25% deviation from the solid line. See Table 2 for the abbreviations of the CFD 369 turbulence models. 370

10

Table 1: Overview of turbulence and boundary-layer modelling approaches of previous CFD studies in 371 sports. 372

373 Table 2: Overview of modelling specifications of the performed CFD simulations. 374 Abbreviation Turbulence-modelling

approach

Turbulence

model

Boundary-layer modelling approach

sk-ε Steady RANS standard k-ε LRNM (Wolfshtein model)

rk-ε Steady RANS realizable k-ε LRNM (Wolfshtein model)

rngk-ε Steady RANS RNG k-ε LRNM (Wolfshtein model)

sk-ω Steady RANS standard k-ω LRNM

sstk-ω Steady RANS SST k-ω LRNM

rk-ε_SWF Steady RANS realizable k-ε Standard wall functions

rk-ε_NWF Steady RANS realizable k-ε Non-equilibrium wall functions

LES LES Smargorinsky LRNM

375

Author Application 2D/

3D

Steady

/ Unsteady

Turbulence &

BL modelling

Validation

Bixler and Schloder

(1996)

Swimming (hand) 2D Steady/unsteady (b) sk-ε, rngk-ε,

RSM

Drag force (a)

Bixler and Riewald

(2002)

Swimming (arm) 3D Steady sk-ε (NWF) Drag & lift force (a)

Dabnichki and Avital

(2006)

Bobsleighing (bob

& riders)

3D Steady sk-ω (WF) Drag & lift force + FV

Gardano and

Dabnichki (2006)

Swimming (arm) 3D Steady - Drag & lift force

Meile et al. (2006) Ski jumping (skier) 3D Steady sk-ε Drag & lift force

Rouboa et al. (2006) Swimming (arm) 2D Steady/unsteady (b) sk-ε Drag & lift force (

a)

Bixler et al. (2007) Swimming

(swimmer)

3D Steady sk-ε (NWF) Drag force

Barber et al (2009) Soccer (balls) 3D Steady rk-ε (LRNM) Drag force + FV

Lecrivain et al. (2008) Swimming

(swimmer)

3D Unsteady (c) - Drag force (

a)

Zaïdi et al. (2008) Swimming

(swimmer)

2D Steady sk-ε (NWF) -

Zaïdi et al. (2010) Swimming

(swimmer)

3D Steady sk-ε (NWF), sk-

ω (WF)

Drag force (a) + FV

Defraeye et al. (2010) Cycling (cyclist) 3D Steady/unsteady (d) sk-ε (LRNM),

LES (LRNM)

Drag force & surface

pressures

sk-ε: standard k-ε model; sk-ω: standard k-ω model; rngk-ε: RNG k-ε model; RSM: Reynolds stress model; rk-ε: realizable

k-ε model; WF: wall functions; NWF: non-equilibrium wall functions; FV: flow visualisation; BL: boundary-layer; (a)

validation was performed by comparison with data of previous experimental studies of other researchers; (b) accelerated

flow; (c) movement of arm during simulation; (

d) steady approach flow but unsteady wake flow.

11

Appendix 1: CFD simulations 376 377 Boundary conditions 378 At the inlet, a low-turbulent and uniform inlet velocity was imposed, namely 20 m/s with a turbulence intensity 379 of 0.02℅. Note that the inflow conditions for turbulence for LES were specified by taking into account 380 fluctuations (vortex method; Mathey et al., 2006). Due to the low turbulence level however, these fluctuations 381 died out quite rapidly, i.e. before reaching the cyclist. The surfaces of the scale model, the stand and the lower 382 wind-tunnel wall were modelled as no-slip boundaries (i.e. walls) with zero roughness. For the remainder of the 383 wind-tunnel walls, a slip-wall boundary (symmetry) was used in order to avoid resolving the boundary layer here, 384 which would have required a high grid resolution close to these walls. Slip walls assume that the normal velocity 385 component and the normal gradients at the boundary are zero, resulting in flow parallel to the boundary. At the 386 outlet of the computational domain, the ambient static pressure was imposed. The reader is referred to Casey and 387 Wintergerste (2000) for best practice guidelines for CFD simulations. 388 389 Spatial and temporal discretisation 390 The grid is a hybrid grid, consisting of prismatic cells in the boundary-layer region on the scale model’s surface, 391 tetrahedral elements in the vicinity of the scale model and hexahedral elements further away. The two different 392 boundary-layer modelling approaches that are compared, namely wall functions and LRNM, have different grid 393 requirements in the near-wall region: y

+ values of the wall-adjacent cell of about 1 and below 5 are required for 394

LRNM while wall functions require the y+ value to be in the range of 30 to 500 (Casey and Wintergerste, 2000). 395

Therefore two slightly different grids were built, according to the specified grid requirements, which only 396 differed by the grid resolution in the boundary-layer region. The resulting grids contain 5.5 x 10

6 and 7.7 x 10

6 397

cells for wall functions and LRNM respectively, where the resulting distance of the first computational cell 398 centre to the wall is about 15 µm for LRNM. The average cell size in the wake region is 0.03 m. These grids 399 were built based on a grid sensitivity analysis according to best practice guidelines in CFD. The grid 400 discretisation error was estimated by means of Richardson extrapolation and was about 3% for the drag force. 401

For unsteady LES simulations, the temporal discretisation is dependent on the spatial discretisation. Both are 402 related by the CFL (Courant-Friedrichs-Lewy) number: 403

u tCFL

d

∆= (4) 404

where u is the characteristic velocity in the cell, ∆t is the time step and d is the characteristic cell dimension. 405 Time steps resulting in CFL numbers of 1 are suggested in the wake region (Spalart, 2001). For the simulations, 406 the choice of the time step and averaging period was also based on a sensitivity analysis. A time step of 4.3x10

-4 407

s was chosen, resulting in CFL numbers below about 5 in the majority of the domain, with maximal values that 408 do not exceed 10, and values of about 0.5 in the wake. A dimensionless simulation time of about 1.4 flow-409 through-times was found to be sufficient to obtain stationary, i.e. stable averaged, values for drag and surface 410 pressures, whereas the flow-through-time (tFT) is defined as: 411

FTD

UTt

L= (5) 412

where U is the free-stream (approach flow) wind speed (20 m/s), T is the averaging period (1 s) and LD is the 413 length of the computational domain (14.5 m). 414 415 Simulation parameters 416 Second-order discretisation schemes are used throughout, except for momentum in LES simulations, for which a 417 central differencing scheme is used. The SIMPLE algorithm is used for pressure-velocity coupling. Pressure 418 interpolation is second order. For LES simulations, second-order implicit time stepping is used. For the RANS 419 simulations, convergence was assessed by monitoring the velocity and turbulent kinetic energy on specific 420 locations in the flow field, surface friction on the surface of the scale model and the resulting drag force on the 421 scale model. For the LES simulations, 20 iterations per time step were found to be sufficient to have 422 convergence within a certain time step whereas the convergence behaviour was assessed in a similar way as 423 mentioned for RANS. 424 425 Flow-field evaluation 426 In Figure A, the flow fields (colour contours of mean wind speed, 0-25 m/s) of the different CFD modelling 427 approaches are shown in the vertical centreplane. Note that for LES, the time-averaged values are presented. The 428 RNG k-ε model shows a slightly different flow pattern, compared to the other k-ε models, which show a similar 429 flow field as LES. The standard k-ω model shows a much larger wake zone, compared to all the other turbulence 430 models. The influence of both types of wall functions on the flow field is limited, when they are compared to the 431 flow field of the LRNM realizable k-ε model. 432

12

sk-ε

rk-ε

rngk-εsk-ω

sstk-ω

rk-ε_SWF

rk-ε_NWF

LES

25 m/s

0 m/s

20 m/s

15 m/s

10 m/s

5 m/s

sk-ε

rk-ε

rngk-εsk-ω

sstk-ω

rk-ε_SWF

rk-ε_NWF

LES

25 m/s

0 m/s

20 m/s

15 m/s

10 m/s

5 m/s

25 m/s

0 m/s

20 m/s

15 m/s

10 m/s

5 m/s

433 Figure A: Flow field (contours of mean wind speed) in a vertical centreplane for different CFD modelling 434 approaches. See Table 2 for the abbreviations of the CFD turbulence models. 435 436 Force and moment areas of the stand: Comparison between wind-tunnel experiments and CFD 437 To quantify the discrepancies related to the stand itself, the force and moment areas of the stand (i.e. without the 438 scale model), obtained by separate wind-tunnel experiments, are compared with the results from CFD 439 simulations of the stand in Figure B. Note that the reported differences are normalised by the force and moment 440 areas of the scale model with the stand (of the wind-tunnel experiments), which is done to allow a comparison 441 with the relative differences reported in Figure 5. Although the contribution of the stand to the discrepancies in 442 drag and pitch areas is rather limited (in general < 10%), it is responsible for the largest part of the discrepancies 443 in the lift area, which is mainly related to the flow-field prediction around the half-sphere (see Figure 2). Note 444 that roughly the same trends can be noticed as in Figure 5, regarding the performance of the different CFD 445 modelling approaches. 446

13

3%

48%

29%

48%

-8%

54%

-9%

41%

29%

1%

18%

-8%

2%

50%

-9%

-1%-5%

-9%-8% -9%-9% -9%

59%

31%

-50%

-40%

-30%

-20%

-10%

0%

10%

20%

30%

40%

50%

60%

70%

80%

rk-ε

_S

WF

rk-ε

_N

WF

LE

S

sk-ε

rk-ε

rng

k-ε

sk-ω

sstk

-ω

rk-ε

_S

WF

rk-ε

_N

WF

LE

S

sk-ε

rk-ε

rng

k-ε

sk-ω

sstk

-ω

Fy,CFD,STAND Fy,WT,STAND

Fy,WT

AC AC

AC

− Fz,CFD,STAND Fz,WT,STAND

Fz,WT

AC AC

AC

− Mx,CFD,STAND Mx,WT,STAND

Mx,WT

ALC ALC

ALC

−

DRAG area LIFT area PITCH area

sk-ε

rk-ε

rng

k-ε

sk-ω

sstk

-ω

rk-ε

_S

WF

rk-ε

_N

WF

LE

S

3%

48%

29%

48%

-8%

54%

-9%

41%

29%

1%

18%

-8%

2%

50%

-9%

-1%-5%

-9%-8% -9%-9% -9%

59%

31%

-50%

-40%

-30%

-20%

-10%

0%

10%

20%

30%

40%

50%

60%

70%

80%

rk-ε

_S

WF

rk-ε

_N

WF

LE

S

sk-ε

rk-ε

rng

k-ε

sk-ω

sstk

-ω

rk-ε

_S

WF

rk-ε

_N

WF

LE

S

sk-ε

rk-ε

rng

k-ε

sk-ω

sstk

-ω

Fy,CFD,STAND Fy,WT,STAND

Fy,WT

AC AC

AC

− Fz,CFD,STAND Fz,WT,STAND

Fz,WT

AC AC

AC

− Mx,CFD,STAND Mx,WT,STAND

Mx,WT

ALC ALC

ALC

−

DRAG area LIFT area PITCH area

sk-ε

rk-ε

rng

k-ε

sk-ω

sstk

-ω

rk-ε

_S

WF

rk-ε

_N

WF

LE

S

447 Figure B: Comparison of relative difference of drag, lift and pitch areas (of stand) of wind-tunnel tests 448 (ACFy,WT,STAND, ACFz,WT,STAND, ALCMx,WT,STAND) and various CFD simulations (ACFy,CFD,STAND, 449 ACFz,CFD,STAND, ALCMx,CFD,STAND). Note that the differences are normalised with the force and moment 450 areas of the scale model with the stand from wind-tunnel tests (ACFy,WT, ACFz,WT, ALCMx,WT). See Table 2 451 for the abbreviations of the CFD turbulence models. 452 453 References 454 Mathey, F., Cokljat, D., Bertoglio, J.P., Sergent, E., 2006. Assessment of the vortex method for Large Eddy 455 Simulation inlet conditions. Progress in Computational Fluid Dynamics 6 (1-3), 58-67. 456 Spalart, P.R., 2001. Young person’s guide to Detached-Eddy Simulation grids. NASA Contractor Report CR-457 2001-211032. NASA, Virginia. 458 459