## 3 Pressure-Sensitive Paint Technique  
The PSP technique was used for the film cooling measurements in this study. PSP is a paint with luminescent molecules in an oxygen permeable polymer binder. If the luminescent molecules are illuminated by near-ultraviolet light, they emits light. The intensity of this light depends on the number of oxygen molecules, i.e., the pressure of the PSP coated surface. Due to oxygen quenching [28], the intensity of the emitted light is inversely proportional to the concentration of oxygen. Figure 3 shows the PSP calibration results. The PSP was calibrated with a separate vacuum chamber, and the same CCD camera and LED lamp that were used for the film cooling tests were also utilized for the calibration test. Because the intensity (\(I\)) of emitted light is also affected by the surrounding temperature, PSP calibration tests were conducted for different temperature conditions, and an appropriate calibration curve was selected for the film cooling test.  
In order to measure the film cooling effectiveness using PSP, two separate tests with different coolants were required. If air or foreign gases (nitrogen for the current study) are injected as a coolant and the intensity of light emitted from the PSP for each coolant case is compared, the film cooling effectiveness (\(\eta\)) can be calculated using Equation (1) [7; 29]:  
\[\eta=\frac{C_{\mathrm{mix}}-C_{\mathrm{air}}}{C_{\mathrm{fg}}-C_{\mathrm{air}} }=\frac{\left(P_{\mathrm{O_{2}}}\right)_{\mathrm{air}}-\left(P_{\mathrm{O_{2} }}\right)_{\mathrm{mix}}}{\left(P_{\mathrm{O_{2}}}\right)_{\mathrm{air}}}=1- \frac{1}{\left(1+\frac{P_{\mathrm{O_{2}}_{\mathrm{air}}}/P_{\mathrm{O_{2}}_{ \mathrm{ref}}}}{P_{\mathrm{O_{2}}_{\mathrm{2}}/\mathrm{O_{2}}_{\mathrm{ref}}} }1\right)\frac{W_{\mathrm{fg}}}{W_{\mathrm{air}}}}\right) \tag{1}\]  
In Equation (1), \(C\) and \(P_{\mathrm{O_{2}}}\) are the oxygen concentration and the partial pressure of oxygen, respectively. W is the molecular weight of the air or coolant. The subscripts air, mix, fg, and ref denote the mainstream, the mainstream-coolant mixture, the foreign gas, and the reference conditions, respectively.  
The partial pressure ratios in Equation (1) can be calculated from the PSP calibration results, as shown by Equation (2):  
Figure 2: Waveforms of the input voltage and current (6 kV input voltage case).  
\[\frac{\left(P_{\text{O}_{2}}\right)}{\left(P_{\text{O}_{2}}\right)_{\text{ref}}}= \text{f}\left(\frac{I_{\text{ref}}-I_{\text{blk}}}{I-I_{\text{blk}}}\right) \tag{2}\]  
For the film cooling measurements with PSP, four intensity images at difference conditions are required. Table 1 presents the conditions under which each image was acquired.  
The experimental uncertainty [30; 31] in the measured film cooling effectiveness was estimated using Equation (3). The estimated uncertainties were about 0.78% and 1.98% for the film cooling effectiveness values of 0.8 and 0.6, respectively. For the film cooling effectiveness value of 0.4, the uncertainty was estimated as 4.28%.  
\[\Delta\eta=\sqrt{\left(\frac{\partial\eta}{\partial I_{\text{air}}}\Delta I_{ \text{air}}\right)^{2}+\left(\frac{\partial\eta}{\partial I_{\text{N}_{2}}} \Delta I_{\text{N}_{2}}\right)^{2}+\left(\frac{\partial\eta}{\partial I_{ \text{ref}}}\Delta I_{\text{ref}}\right)^{2}+\left(\frac{\partial\eta}{ \partial I_{\text{blk}}}\Delta I_{\text{blk}}\right)^{2}} \tag{3}\]