# Aerodynamics of Cycling Skinuits Focused on the Surface Shape of the Arms
## 2 Methods
### CFD Analysis  
The three-dimensional model shown in Figure 4 was created based on the full-scale mannequin used in the wind tunnel experiment. The analysis of this model was performed with CFD software (PowerFLOW5.5, Dassault Systemes, Tokyo, Japan) based on the lattice Boltzmann method [18,19]. The lattice Boltzmann method is a simulation technique to model the thermal flow field of a fluid in which the fluid is approximated by a collection of virtual particles and the collisions and propagation of the particles and their moments are successively calculated using the velocity distribution function of the particles. More specifically, in order to compare how the airflow around the cyclist would change according to different wind speeds, the flow around the model was analyzed with two types of speed intervals (case 1: wind speed = 16 m/s, case 2: wind speed = 20 m/s) in this experiment. The objects of analysis were a racing bicycle and a cyclist wearing a helmet, gloves, skin suit and bike shoes. The posture of an actual cyclist was scanned with a 3D scanner and used for analysis. The crank angle was fixed at an angle parallel to the ground similar to the mannequin in the wind tunnel. In the calculations, the tire and wheel were treated as stationary with no rotation and the ground was treated as a stationary floor. The analysis was performed with the spatial grid resolution (VR9) at an offset of 10 mm from the actual shape. The analysis space (x, y, z) consists of areas of respectively 4.2 m, 0.8 m and 1.8 m and approximately 200 million cells with a standard size of 5 mm. Additionally, the standard cell size of the cyclist model surface is 1.25 mm. The physical conditions were obtained at a temperature of 20 \({}^{\circ}\)C and a representative density of 1.204 kg/m\({}^{3}\) under atmospheric pressure (Table 1). The behavior of the multi-particle motion system can be represented by the fundamental laws of mechanics that govern single-particle motions at the molecular scale. The Boltzmann equation can formulate a problem in terms of the distribution function \(f\) (\(x\), \(v\), \(t\)), which indicates the molecular number density at position \(x\), time \(t\) and velocity \(v\). This equation (in the absence of external forces) can be expressed as follows:  
\[\frac{Df}{Dt}=\ \frac{\partial f(x,\ v,\ t)}{\partial t}+v\cdot\
abla f(x,\ v,\ t)=C(x,\ v,\ t) \tag{1}\]  
In this equation, the total differential on the left-hand side represents the connective motion of the particles, while the right-hand side represents the complex intermolecular interactions (collisions). By integrating the distribution function, it is possible to obtain macroscopic variables such as the fluid density, velocity and pressure. \begin{table}
\begin{tabular}{c c} \hline
**Item** & **Values** \\ \hline Turbulence model & VLES (analyzed with Powerflow 5.5) \\ Representative pressure [Pa] & 101,325 \\ Representative temperature [\({}^{\circ}\)C] & 20.0 \\ Representative density [kg/m\({}^{3}\)] & 1.204 \\ Representative kinematic viscosity [m\({}^{2}\) /s] & 1.49 \(\times\) 10\({}^{-5}\) \\ Calculated physical time [sec] & 1.27 (case 1; 16 m/s) \\ Minimum voxel size [mm] & 1.25 \\ Total voxels & 191.20 million \\ Fine Equivalent Voxels (FEV) & 100.61 million \\ Total surfels & 7.09 million \\ FES (fine equivalent surfels) & 6.85 million \\ \hline \end{tabular}
\end{table} 
Table 1: Analysis conditions. The main purpose of the collision operator is to orient the velocity distribution function to the equilibrium distribution. The Bhatnagar, Gross and Krook collision operator [20] was used here and can be defined as follows:  
\[C(x,\ v,\ t)\ =\ -\frac{1}{\tau}\ [\ f(x,v,t)-f^{eq}(x,\ v,\ t)\ ] \tag{2}\]  
Assume that \(\tau\) is the relaxation time of the fluid and \(f^{eq}(x,\ v,\ t)\) is the equilibrium distribution function. To solve these equations more efficiently, they were discretized on a three-dimensional cubic lattice using a D3Q19 model [20; 21]. This model discretizes the given velocity space into 19 discrete velocities.