## 1 Introduction  
Gas turbine engines are widely used in power generation, energy production and transport, distributed power supply, aviation propulsion, and other fields [1; 2]. The high temperature rises and low emissions are the development trend of combustors. Therefore, to develop more efficient gas turbines that better protect combustors and turbine blades from erosion in high-temperature environments while extending their service life, it is necessary to develop efficient cooling systems for gas turbines [3; 4]. Cooling holes play a crucial role in maintaining safe operating temperatures for engine components [5; 6]. [PERSON] [7] analyzed the interaction between vertical jets and cross flows by dividing the interaction between the jet and main flow into four regions: internal boundary layer flow, turbulent flow within the tube, boundary layer flow of main flow, and the intersection between main flow and jet. [PERSON] et al. [8] performed experiments on a vertical plate to measure the jet-mainstream interaction zone. They found complex mixing between the jet and mainstream flow in this region, distorting the vortex structures and creating vortex ring structures similar to Karman vortices. The mixing zone initially generated a pair of counter-rotating kidney-shaped vortices that block the main flow at the exits of the cooling vents, leading to two horseshoe-shaped vortices with weaker induced vortex streets. It can be seen that kidney-shaped vortices are one of the most important forms of film cooling, and their structure has a significant influence on the effectiveness of film cooling. The formation mechanism is mainly related to the entrainment of cold air through shear layers from both sides with the development of boundary layers within the hole walls [9], non-uniform velocity distribution at the inlet [10], and internal flows within the coolant cavities [11; 12]. Over time, much research has been performed to improve efficiency and effectiveness. One such advancement is shaped cooling holes, which have received considerable attention for their ability to improve heat transfer while minimizing pressure drop. Shaped cooling holes are non-circular or irregularly shaped perforations in engine components that allow coolant flow for thermal management due to various factors such as aerodynamics, material properties, and manufacturing constraints. Compared to conventional circular or square cooling holes, they offer several advantages, including the increased surface area for heat transfer, reduced pressure drop across the hole walls, and improved resistance to clogging by foreign objects. [PERSON] and [PERSON] [13] identified three main factors affecting film cooling: cold air/mainstream conditions, hole geometry/arrangement, and blade geometry. [PERSON] et al. [14] studied the effects of mainstream turbulence intensity on round/shaped holes. They found that increasing turbulence intensity at low blowing ratios weakened film cooling for round holes, while at high blowing ratios, it weakened film cooling performance for both types. [PERSON] et al. [15] conducted extensive experiments in which they investigated the effects of various parameters such as blowing ratio, density ratio, and turbulence intensity, along with geometric factors such as spacing and inclination angle on the effectiveness of film cooling by round holes. They also developed empirical correlations to predict the effectiveness based on the relevant coefficients. [PERSON] et al. [16] measured wall temperatures with thermocouples and studied the effect of density ratio on film cooling effectiveness. They found that decreasing the density ratio weakens the coverage and reduces the effectiveness. [PERSON] et al. [17; 18] used a hot-wire anemometer to measure the flow fields of circular coolant jets in cross-flowing main streams. Investigation of the effect of jet length and turbulence intensity on performance and measurement of energy spectra revealed that shorter lengths result in better lateral coverage due to the stronger interaction.