# Effects of Bulk Flow Pulsations on Film Cooling with Two Sister Holes
## 2 Geometry, Boundary Conditions  
The geometry of the single-hole system, which was considered as the baseline, was obtained from the investigation of [PERSON] et al. [9]. In their experimental apparatus, five cooling holes were positioned along a row. Here, the computational cost was reduced by applying periodic boundary conditions on the sidewalls [3]. The computational domains are displayed in Figure 1.  
The triple holes consisted of two small holes located downstream of main hole (Figure 2b), as stated by [PERSON] et al. [6]. In the triple-hole configuration, \(D\) of the primary hole and the sister hole were calculated to match the total cross-sectional areas of the triple holes to the cross-sectional area of the single hole [3]. When the two small holes are located downstream of the main hole, the coolant jets generate an anti-counter-rotating vortex pair (anti-CRVP), and \(\eta\) is increased through the generation of vortices interactions between the injectants and a reduction in the intensity of the main Counter Rotating Vortex Pair [3]. The hole-length-to-hole-diameter ratio (\(L/D\)) for the single-hole system was 1.6, the ratios for the primary hole and each sister hole in the triple hole were 1.8 and 3.6, respectively, the injection angle of the coolant (\(\alpha\)) was 35\({}^{\circ}\). The turbulence intensity was set as 0.2% at main inlet (as reported for the experiment in [9]), and the velocity of the main flow was 10 m/s. The temperatures of the mainstream and injectant at the inlets were 300 and 293 Kelvin, respectively. The boundary conditions were specified in Table 1. Figure 2 presents schematics of the cooling holes in each case. Figure 1: Computational domains. \begin{table}
\begin{tabular}{c c} \hline \hline
**Surface** & **Boundary Condition** \\ \hline Main inlet & Velocity inlet \\ Plenum inlet & Velocity inlet \\ Top & Symmetry \\ Test plate & Adiabatic wall \\ Outflow & Pressure outlet \\ Main sides & Periodic \\ Sides of plenum & Wall \\ Tube wall & Wall \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Boundary conditions. Figure 2: Schematics of hole systems [3]. The main flow velocity profile was uniform at the main inlet, similar to [9]. The main flow velocity at the inlet was:  
\[V_{main\ inlet}=A\ \sin(2\pi\cdot f\cdot t)+10\ \mathrm{m/s}. \tag{1}\]  
The values of the amplitude \(A\) of the mainstream velocity pulsation considered here were identical to those used by [PERSON] et al. [9]. The \(A\) and frequency values at the different Strouhal numbers (Sr) are presented in Table 2. The Sr values for the single-hole system and the main and sister holes in the triple-hole configuration were identical, even though their \(L/D\) values were different. The coolant velocity at the plenum inlet was calculated from the expression:  
\[V_{plenum\ inlet}=B\sin(2\pi\cdot f\cdot t)+0.164\ \mathrm{m/s}. \tag{2}\]  
While the values of the amplitude \(B\) of the pulsations in the plenum inlet flow velocity were not specified by [PERSON] et al. [9], they can be extracted from the \(P_{2}\)-\(P_{1}\) variation plots for different frequencies, which are presented in their paper. Here, \(P_{2}\) and \(P_{1}\) are the static pressures at the inlet and outlet of the hole. The \(B\) were acquired by matching \(P_{2}\)-\(P_{1}\) plots to numerical calculations through trial and error. In Equation (2), 0.164 corresponds to the plenum inlet velocity on the steady state. Table 3 shows the \(B\) values and frequencies at different Strouhal numbers. Increasing the pulsation frequency of the main flow increased the pressure difference between \(P_{2}\) and \(P_{1}\) since the phase shift increased [9], and this led to an increase in the amplitude of the coolant velocity pulsation, as is evident in Table 3.  
The Strouhal number is defined as \(\mathrm{St}=\frac{2\pi fL}{U_{C}}=\frac{L/IL}{1/2\pi f}\), and it denotes the ratio of the time needed for the coolant to pass through the hole to the duration of one pulsation cycle.