# Effects of Bulk Flow Pulsations on Film Cooling with Two Sister Holes
## 3 Numerical Methods  
Numerical solutions were obtained by using ANSYS Fluent v.19 [3; 24]. The CFD meshes were created in Pointwise v.18 [3; 25]. CFD simulations were performed in the LES, Smagorinsky-Lilly model. The LES approach helps resolve large-scale eddies in turbulent flow and facilitates the modeling of small-scale eddies with length scales smaller than the grid spacing. The time step was \(6.25\times 10^{-6}\) s, implying that the main flow convected the diameter of the single hole after 400 time steps [26; 27]. In each time step, approximately 10 sub-iterations were executed to obtain well-resolved data [28]. The fluid was assumed as Newtonian and incompressible. The mainstream velocity was 10 m/s and the mean  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Frequency (Hz)** & **0** & **2** & **16** & **32** \\ \hline \(Sr\) & 0 & 0.1005 & 0.8043 & 1.6085 \\ \(A\) & 0 & 1.82 & 0.57 & 0.44 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Pulsation amplitude \(A\) and frequency in Equation (1) at different Strouhal numbers (Sr).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Frequency (Hz)** & **0** & **2** & **16** & **32** \\ \hline \(Sr\) & 0 & 0.1005 & 0.8043 & 1.6085 \\ \(B\) & 0 & 0.04 & 0.05 & 0.16 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Pulsation amplitude \(B\) and frequency in Equation (2) at different Strouhal numbers (Sr).  
injectant velocity was 5 m/s, showing that both velocities were far below Mach = 0.3 [3,29]. The governing equations were as shown in [24]:  
\[\frac{\partial\rho}{\partial t}+\ \frac{\partial}{\partial x_{i}}(\rho\overline{u}_ {i})=0 \tag{3}\]  
\[\frac{\partial(\rho\overline{u})}{\partial t}+\frac{\partial}{\partial x_{j}} \left(\rho\overline{u}_{i}\overline{u}_{j}\right)=\frac{\partial}{\partial x _{j}}\left[\mu\left(\frac{\partial\overline{u}_{i}}{\partial x_{j}}+\frac{ \partial u_{j}}{\partial x_{i}}\right)-\frac{2}{3}\mu\frac{\partial\overline{u }_{l}}{\partial x_{l}}\delta_{ij}\right]-\frac{d\overline{p}}{dx_{i}}+\frac{ \partial\tau_{ij}}{\partial x_{j}} \tag{4}\]  
Here, \(\tau_{ij}\), which denotes the subgrid scale turbulent stress, required modeling by using the Boussinesq hypothesis, similar to RANS turbulence models, as follows:  
\[\tau_{ij}-\ \frac{1}{3}\tau_{kk}\delta_{ij}=\ -\mu_{t}\left(\frac{\partial \overline{u}_{i}}{\partial x_{j}}+\frac{\partial\overline{u}_{j}}{\partial x_ {i}}\right) \tag{5}\]  
where \(\mu_{t}\) represents the turbulent viscosity.  
Figures 3 and 4 illustrate the meshes in the \(xy\) and \(yz\) planes, respectively [3]. The computational domains of the single-hole and triple-hole systems were composed of 2.04 and 2.69 million hexahedron cells, respectively [3]. Figure 5 illustrates a close-up of the mesh near hole. \(P_{2}\) and \(P_{1}\) represent the static pressures at the inlet and outlet of the hole, respectively.  
Figure 4: Grids in the \(yz\) plane [21].  
Figure 3: Computational fluid dynamics grids in the \(xy\) plane [3].  
In [3], grid sensitivity tests were conducted in the steady case (Sr = 0) and at the blowing ratio, \(M\) of 0.5 for the single-hole and triple-hole systems.