# Aerodynamic Drag Reduction on Speed Skating Helmet by Surface Structures
## 2 Numerical Approach
### Slower Conditions and Mesh Convergence Research  
The CFD analysis was performed with the Ansys fluent 2020 R1. To catch the vortex around the helmet and in the wake region, a large eddy simulation (LES) turbulence model was adopted in our numerical analysis. LES is the widely accepted model in the industry to solve a variety of problems, in which turbulence models for the RANS equations are either inaccurate, or do not supply sufficient information. The governing equations employed for the LES are obtained by filtering the time-dependent Navier-Stokes equations. The filtering process effectively filters out the eddies whose scales are smaller than the grid spacing used in the computation. The filtered equations are written as follows. \[\frac{\partial\overline{u}_{i}}{\partial x_{i}}=0 \tag{1}\]  
\[\frac{\partial\overline{u}_{i}}{\partial t}+u_{j}\frac{\partial\overline{u}_{ i}}{\partial x_{j}}=-\frac{1}{\rho}\frac{\partial\overline{p}}{\partial x_{j}}+ \frac{\partial}{\partial x_{j}}\Bigg{[}v\Bigg{(}\frac{\partial\overline{u}_{ i}}{\partial x_{j}}+\frac{\partial\overline{u}_{j}}{\partial x_{i}} \Bigg{)}+\tau_{ij}\Bigg{]} \tag{2}\]  
Figure 3: The helmet model with surface structure used in the CFD simulations; (**a**) the riblet structured helmet model, (**b**) the bumped helmet model. Here, \(\
u\) is the kinematic viscosity of fluid, and \(\overline{u}\) and \(\overline{p}\) are the filtered velocity and pressure, respectively. In the spatially filtered Navier-Stokes equations, the stress term \(\tau_{ij}=\overline{u_{i}u_{j}}-\overline{u_{i}}\overline{u_{j}}\) is molded using the eddy-viscosity type SGS model of [PERSON], which is based on [PERSON]'s assumption,  
\[\tau_{ij}-\frac{2}{3}k_{t}\delta_{ij}=-2\
u_{t}\left(\overline{S}_{kk}-\frac{ 1}{3}\overline{S}_{kk}\delta_{ij}\right) \tag{3}\]  
\[\overline{S}_{ij}=\frac{1}{2}\left(\frac{\partial\overline{u_{i}}}{\partial x _{j}}+\frac{\partial\overline{u_{j}}}{\partial x_{i}}\right) \tag{4}\]  
where \(\overline{S}_{ij}\) is the rate of strain tensor computed from the resolved scales, \(\
u_{t}\) is the SGS viscosity and \(k_{t}\) is the turbulent kinetic energy. The unsteady pressure-based solver was used in our simulations and the SIMPLE scheme was chosen in pressure-velocity coupling. The second order scheme was set for the pressure and the bounded central differencing for the momentum with respect to the spatial discretization, while the gradients were evaluated as the least squared cell based gradients. The time step for the unsteady solver was set to \(10^{-5}\) s, and the total time was 2 s for every simulation. The residuals of continuity and velocity were monitored as the convergence criteria, which were the order of \(10^{-3}\) for continuity and the order of \(10^{-5}\) for velocity (x, y, z components) at every time step. The drag of every helmet model was analyzed to confirm whether the flow field had reached stability. The schematic of the mesh in the computational domain is shown in Figure 4. An unstructured mesh was established and refined near the helmet as well as in the wake region. The maximum element size, minimum element size, and growth rates were defined as 50 mm, 0.1 mm, and 1.2, respectively. A polygon surface mesh was defined on the surface of the helmet and the support, and a refinement was also applied to the surface structure (bumps and riblets). Regarding the boundary layer mesh, a prismatic inflation of 20 layers with the first layer height of 0.01 mm and the growth rate of 1.2 was applied. Figure 4: The schematic of the mesh in the computational domain. A mesh convergence study had been performed on the original helmet model with 4 sets of meshes, as shown in Table 1. The coefficient of aerodynamic drag (\(C_{\text{d}}\)) was calculated in this study as  
\[C_{\text{d}}=\frac{F_{\text{d}}}{\frac{1}{2}\rho AU^{2}} \tag{5}\]  
where \(F_{\text{d}}\) was the drag force, \(\rho\) was the density of air, \(A\) was the frontal area, and \(U\) was the free stream velocity.