## 2 Numerical and Experimental Methodologies
### Large Eddy Simulation Based on Smagorinsky-Lilly Model  
In order to overcome the limitations of unsteady RANS equations that are time-averaged equations of motion for fluid flow, LES was performed based on the Smagorinsky-Lilly model first proposed by [PERSON] [25]. In LES, large eddies are resolved directly, while small eddies are modeled. LES, therefore, falls between direct numerical simulation (DNS) and RANS in terms of the fraction of the resolved scale. The governing equations in LES are obtained by filtering the original N-S equations. The filtered continuum and N-S equations are given as follows:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x_{\rm i}}(\rho \overline{u_{\rm i}})=0 \tag{1}\]  
\[\frac{\partial}{\partial t}(\rho\overline{u_{\rm i}})+\frac{\partial}{ \partial x_{\rm j}}\Big{(}\rho\overline{u_{\rm i}u_{\rm j}}\Big{)}=\frac{ \partial}{\partial x_{\rm j}}\Big{(}\sigma_{\rm ij_{\rm j}}\Big{)}-\frac{ \partial\overline{p}}{\partial x_{\rm i}}-\frac{\partial\tau_{\rm ij_{\rm j}} }{\partial x_{\rm j}} \tag{2}\]  
where \(\overline{u_{\rm i}}\) is the resolved velocity in i-direction (i = 1, 2, and 3 correspond to the x, y and z directions), \(\sigma_{\rm ij}\) is the stress tensor due to molecular viscosity and \(\tau_{\rm ij_{\rm j}}\) is the subgrid-scale stress and are defined by the following equations:  
\[\sigma_{\rm ij_{\rm j}}\equiv\left[\mu\left(\frac{\partial\overline{u_{\rm i }}}{\partial x_{\rm j}}+\frac{\partial\overline{u_{\rm i}}}{\partial x_{\rm i }}\right)\right]-\frac{2}{3}\mu\frac{\partial\overline{u_{\rm i}}}{\partial x _{\rm l}}\delta_{\rm ij_{\rm j}} \tag{3}\]  
\[\tau_{\rm ij_{\rm j}}\equiv\rho\overline{u_{\rm i}u_{\rm j}}-\rho\overline{u_ {\rm i}u_{\rm j}} \tag{4}\]  
The subgrid-scale stresses originating from the filtering operation are unknown, and require modelling. The subgrid-scale turbulence models employ the Boussinesq hypothesis [26] as in the RANS models because they refer to the representation of important small-scale physical processes that  
Figure 4: y+ value distribution on suction and pressure sides of the axial fan.  
occur at length-scales that cannot be adequately resolved on a computational mesh and therefore the subgrid-scale turbulent stresses can be computed from the following:  
\[\tau_{ij}-\frac{1}{3}\tau_{k,k}\;\delta_{ij}\;=\;-2\;\mu_{t}\;\overline{S_{ij}} \tag{5}\]  
where \(\mu_{t}\) is the subgrid-scale turbulent viscosity. The isotropic part of the subgrid-scale stresses \(\tau_{k,k}\) is not modelled, but added to the filtered static pressure term. The rate of strain tensor for the resolved scale is represented by \(\overline{S_{ij}}\) and is defined by the following expression:  
\[\overline{S_{ij}}\;\equiv\;\frac{1}{2}\left(\frac{\partial\overline{u_{ij}}}{ \partial\chi_{j}}+\frac{\partial\overline{u_{ij}}}{\partial\chi_{i}}\right) \tag{6}\]  
This simple model was first proposed by [25]. In the Smagorinsky-Lilly model, the eddy-viscosity is modelled by  
\[\mu_{t}=\;\rho\;L_{5}{}^{2}\big{|}\overline{S}\big{|} \tag{7}\]  
where \(L_{5}\) is the mixing length for subgrid scales and is computed using \(\big{|}\overline{S}\big{|}\;\equiv\;\sqrt{2\;\overline{S_{ij}}\;\overline{S_{ ij}}}\)  
\[L_{5}=\min\;(\kappa d,C_{5}V^{1/3}) \tag{8}\]  
where \(\kappa\), \(d\), \(C_{5}\) is the von Karman constant, the distance to the closest wall and the Smagorinsky constant, respectively and \(\Delta\) is computed according to the volume of the computational cell using \(\Delta=V^{1/3}\). Here, \(V\) is the volume of the computational cell. In the Smagorinsky-Lilly Model [25; 27], the \(C_{5}\) value of 0.1 is recommended to yield the best results for a wide range of flows, and is the default value in ANSYS Fluent.