Large Eddy Simulation of Film Cooling with Triple Holes: Injectant Behavior and Adiabatic Film-Cooling E ﬀ ectiveness

: This study investigated the e ﬀ ect of adding two sister holes placed downstream the main hole on ﬁlm cooling by employing large eddy simulation. Here, ﬁlm-cooling ﬂow ﬁelds from a triple-hole system inclined by 35 ◦ to a ﬂat plate at blowing ratios of M = 0.5 and unity were simulated. Each sister hole supplies a cooling ﬂuid at a ﬂow rate that is a quarter of that for the main hole. The simulations were conducted using the Smagorinsky–Lilly model as the subgrid-scale model, and the results were compared with those for a single-hole system for the same amount of total cooling air and same cross-sectional area of the holes. Relative to the single-hole system, the spanwise-averaged ﬁlm-cooling e ﬀ ectiveness in the triple-hole conﬁguration at M = 1.0 increased by as much as 345%. The subsequent proper orthogonal decomposition analysis showed that the kinetic energy of a counter-rotating vortex pair in the triple-hole system dropped by 30–40% relative to that of the single-hole system. This indicates that the additional sister holes promoted the suppression of the mixing of the coolant jet with the mainstream ﬂow, thus keeping the temperature of the latter low. Cross-sectional views of the root-mean-square temperature contours were also analyzed; with the results conﬁrming that the e ﬀ ect of the sister holes on the jet trajectory greatly contributes in promoting ﬁlm-cooling e ﬀ ectiveness as compared to the e ﬀ ect of the reduced mixing.


Introduction
The principle of idealized Brayton cycle states that gas turbine efficiency can be improved by increasing the inlet temperature of the turbine [1]. Here, to avoid the excessive thermal stress build-up on the turbine blades, the surface temperature of the blades must remain below a tolerable limit. Various cooling methods can be employed, among which the film-cooling technique has been widely used. In this technique, the cooling air which is bled from the compressor is injected through holes on the turbine blade surface-this reduces the surface temperature and protects the surface from the main flow. For the holes, a cylindrical shape is often appropriate because of its simplicity and ease of manufacture.
However, substantial investigations of the physics of film cooling in such a cylindrical hole system have revealed that a single cylindrical hole can be relatively vulnerable to the generation of the jet liftoff [2] and that a specific hole arrangement could produce film cooling that provides better thermal protection than single cylindrical holes do. Practically, specific hole arrangements make the intended vortex interactions between film cooling jets to increase film-cooling effectiveness.
Film cooling in turbine engines can be effectively studied at a reasonable cost using computational fluid dynamics (CFD). High turbulence is generated in the film-cooling flow field, after which a model is

Geometry and Boundary Conditions
The geometry of the single-hole system in the test case was taken from Seo et al. [13], whose experimental apparatus consisted of a row of five cooling holes. Herein, a single-hole configuration was adopted to simplify the computational domain, and a periodic boundary condition was applied on the mesh sides to reduce the computational cost. The computational domains of the single-and triple-hole systems are delineated by the orange dashed lines in Figure 1, whereas a schematic of the cooling holes in each case is shown in Figure 2. In the single-hole geometry (Figure 2a), the hole diameter D was 25 mm. The triple holes comprised two sister holes which are placed slightly downstream of the primary hole ( Figure 2b) as Processes 2020, 8,1443 3 of 17 described in Ely et al. [10]. In the triple-hole geometry, the hole diameters were calculated to match their total cross-sectional areas to the cross-sectional area of the single-hole. Thus, the primary hole diameter D was 20.4124 mm, and the diameter of each sister hole was D /2 = 10.2062 mm. The hole length-to-diameter ratio (L/D), injection angle (α), and pitch-to-hole diameter (P/D) were set to 1.6, 35 • , and 3.0, respectively. In addition, the holes were cylindrical, with no compound angles.
Processes 2020, 8, x FOR PEER REVIEW 3 of 18 cooling holes in each case is shown in Figure 2. In the single-hole geometry (Figure 2a), the hole diameter D was 25 mm. The triple holes comprised two sister holes which are placed slightly downstream of the primary hole ( Figure 2b) as described in Ely et al. [10]. In the triple-hole geometry, the hole diameters were calculated to match their total cross-sectional areas to the cross-sectional area of the single-hole. Thus, the primary hole diameter D′ was 20.4124 mm, and the diameter of each sister hole was D′/2 = 10.2062 mm. The hole length-to-diameter ratio (L/D), injection angle (α), and pitch-to-hole diameter (P/D) were set to 1.6, 35°, and 3.0, respectively. In addition, the holes were cylindrical, with no compound angles.
(a) Single-hole configuration (b) Triple-holes configuration  The boundary conditions in each computational domain are specified in Table 1. The turbulence intensity of the mainstream flow was 0.2% at the main inlet (as reported experimentally [13]), and the main flow velocity was 10 m/s. The temperatures of the main flow and coolant at the inlets were 313 and 293 K, respectively. Moreover, the boundary layer thickness at the cooling hole center is about D.
As is well-known, injecting the film-cooling jet induces a counter-rotating vortex pair (CRVP) in the system (see Figure 3a). This vortex pair is sourced from the vortices generated in the boundary layer of the coolant tube wall and around the leading edge of the hole [12,14,15]. Interactions between the jet and main flow cause the injected coolant to split into two directions, whereas the vortices generated in the coolant are stretched and turned. As the CRVP vortices draw closer together, their mutual induction increases, promoting coolant jet liftoff. The consequent strong entrainment of the hot main flow under the jet increases the adiabatic temperature on the test plate and decreases filmcooling effectiveness. Moreover, when two sister holes are placed slightly downstream of the primary hole (Figure 2b), their jets generate an anti-CRVP, as illustrated in Figure 3b. Therefore, film-cooling effectiveness is increased by establishing vortex interactions between the jets and by reducing the intensity of the main CRVP [9][10][11][12].  cooling holes in each case is shown in Figure 2. In the single-hole geometry (Figure 2a), the hole diameter D was 25 mm. The triple holes comprised two sister holes which are placed slightly downstream of the primary hole (Figure 2b) as described in Ely et al. [10]. In the triple-hole geometry, the hole diameters were calculated to match their total cross-sectional areas to the cross-sectional area of the single-hole. Thus, the primary hole diameter D′ was 20.4124 mm, and the diameter of each sister hole was D′/2 = 10.2062 mm. The hole length-to-diameter ratio (L/D), injection angle (α), and pitch-to-hole diameter (P/D) were set to 1.6, 35°, and 3.0, respectively. In addition, the holes were cylindrical, with no compound angles.
(a) Single-hole configuration (b) Triple-holes configuration  The boundary conditions in each computational domain are specified in Table 1. The turbulence intensity of the mainstream flow was 0.2% at the main inlet (as reported experimentally [13]), and the main flow velocity was 10 m/s. The temperatures of the main flow and coolant at the inlets were 313 and 293 K, respectively. Moreover, the boundary layer thickness at the cooling hole center is about D.
As is well-known, injecting the film-cooling jet induces a counter-rotating vortex pair (CRVP) in the system (see Figure 3a). This vortex pair is sourced from the vortices generated in the boundary layer of the coolant tube wall and around the leading edge of the hole [12,14,15]. Interactions between the jet and main flow cause the injected coolant to split into two directions, whereas the vortices generated in the coolant are stretched and turned. As the CRVP vortices draw closer together, their mutual induction increases, promoting coolant jet liftoff. The consequent strong entrainment of the hot main flow under the jet increases the adiabatic temperature on the test plate and decreases filmcooling effectiveness. Moreover, when two sister holes are placed slightly downstream of the primary hole (Figure 2b), their jets generate an anti-CRVP, as illustrated in Figure 3b. Therefore, film-cooling effectiveness is increased by establishing vortex interactions between the jets and by reducing the intensity of the main CRVP [9][10][11][12]. The boundary conditions in each computational domain are specified in Table 1. The turbulence intensity of the mainstream flow was 0.2% at the main inlet (as reported experimentally [13]), and the main flow velocity was 10 m/s. The temperatures of the main flow and coolant at the inlets were 313 and 293 K, respectively. Moreover, the boundary layer thickness at the cooling hole center is about D. As is well-known, injecting the film-cooling jet induces a counter-rotating vortex pair (CRVP) in the system (see Figure 3a). This vortex pair is sourced from the vortices generated in the boundary layer of the coolant tube wall and around the leading edge of the hole [12,14,15]. Interactions between the jet and main flow cause the injected coolant to split into two directions, whereas the vortices generated in the coolant are stretched and turned. As the CRVP vortices draw closer together, their mutual induction increases, promoting coolant jet liftoff. The consequent strong entrainment of the hot main flow under the jet increases the adiabatic temperature on the test plate and decreases film-cooling effectiveness. Moreover, when two sister holes are placed slightly downstream of the primary hole (Figure 2b), their jets generate an anti-CRVP, as illustrated in Figure 3b. Therefore, film-cooling effectiveness is increased by establishing vortex interactions between the jets and by reducing the intensity of the main CRVP [9][10][11][12].

Validation of Numerical Methods
The CFD calculations were performed in ANSYS Fluent v.19.1 [16], and the meshes were generated in Pointwise v.18.1 [17]. The numerical simulations were executed in the LES using the Smagorinsky-Lilly model as the subgrid-scale model. The time step was set to 6.25 × 10 −6 s. Assuming the mainstream convected the hole diameter after 400 time steps, the simulations were carried out with the computational time step ∆ = 0.0025 / [18,19]. Once the system had reached a statistical steady state, the statistics of the solution was collected in multiples of the period. In each time step, approximately 10 subiterations were executed to obtain well-resolved data [20]. The LES runtime using 18 cores of an Intel Xeon Gold 6148 processor was 1-2 months in each case.
The fluid was assumed as Newtonian and incompressible and had temperature-dependent variable properties. The compressibility effect was negligible because the main flow velocity was 10 m/s (Mach 0.029) and the jet injection velocities were 5 and 10 m/s at M = 0.5 and M = 1.0, respectively; that is, both velocities were far below Mach 0.3 [21]. The governing equations of the CFD simulations are the continuity and momentum equations. The filtered Navier-Stokes equations in the LES approach are given by [22] and where τij, the subgrid-scale turbulent stress, needs modeling using the Boussinesq hypothesis like RANS models as

Validation of Numerical Methods
The CFD calculations were performed in ANSYS Fluent v.19.1 [16], and the meshes were generated in Pointwise v.18.1 [17]. The numerical simulations were executed in the LES using the Smagorinsky-Lilly model as the subgrid-scale model. The time step was set to 6.25 × 10 −6 s. Assuming the mainstream convected the hole diameter after 400 time steps, the simulations were carried out with the computational time step ∆t = 0.0025D/U ∞ [18,19]. Once the system had reached a statistical steady state, the statistics of the solution was collected in multiples of the period. In each time step, approximately 10 subiterations were executed to obtain well-resolved data [20]. The LES runtime using 18 cores of an Intel Xeon Gold 6148 processor was 1-2 months in each case.
The fluid was assumed as Newtonian and incompressible and had temperature-dependent variable properties. The compressibility effect was negligible because the main flow velocity was 10 m/s (Mach 0.029) and the jet injection velocities were 5 and 10 m/s at M = 0.5 and M = 1.0, respectively; that is, both velocities were far below Mach 0.3 [21]. The governing equations of the CFD simulations are the continuity and momentum equations. The filtered Navier-Stokes equations in the LES approach are given by [22] ∂ρ ∂t and where τ ij , the subgrid-scale turbulent stress, needs modeling using the Boussinesq hypothesis like RANS models as where µ t is the turbulent viscosity on the subgrid scale. In the Smagorinsky-Lilly model, the turbulent viscosity is modeled as [18] µ t = ρL s 2 1 2 (triple-hole) million hexahedron cells. Figure 6 shows a close-up of the mesh near the hole, where P2 and P1 are the static pressures at the inlet and outlet of the hole, respectively, as mentioned earlier.
where μt is the turbulent viscosity on the subgrid scale. In the Smagorinsky-Lilly model, the turbulent viscosity is modeled as [18] Figures 4 and 5 show the meshes of the single-and triple-hole systems in the xy and yz planes, respectively. The computational domains of the systems were composed of 2.04 (single-hole) and 2.69 (triple-hole) million hexahedron cells. Figure 6 shows a close-up of the mesh near the hole, where P2 and P1 are the static pressures at the inlet and outlet of the hole, respectively, as mentioned earlier.    where μt is the turbulent viscosity on the subgrid scale. In the Smagorinsky-Lilly model, the turbulent viscosity is modeled as [18] Figures 4 and 5 show the meshes of the single-and triple-hole systems in the xy and yz planes, respectively. The computational domains of the systems were composed of 2.04 (single-hole) and 2.69 (triple-hole) million hexahedron cells. Figure 6 shows a close-up of the mesh near the hole, where P2 and P1 are the static pressures at the inlet and outlet of the hole, respectively, as mentioned earlier.     Tables 2 and 3. The results were obtained using the Smagorinsky-Lilly model in LES, and the tests were performed on five different grids. In the singlehole system, the effectiveness value along the centerline (z = 0, y = 0) on the third grid almost matched those of the finer fourth and fifth grids.   Tables 2 and 3. The results were obtained using the Smagorinsky-Lilly model in LES, and the tests were performed on five different grids. In the single-hole system, the    Tables 2 and 3. The results were obtained using the Smagorinsky-Lilly model in LES, and the tests were performed on five different grids. In the singlehole system, the effectiveness value along the centerline (z = 0, y = 0) on the third grid almost matched those of the finer fourth and fifth grids.     Therefore, the adiabatic film-cooling effectiveness in the single-hole system was evaluated on the 2.04 million-cell grid. In the triple-hole system, the third grid with 2.69 million cells again showed almost the same centerline effectiveness as the fourth and fifth grids. Therefore, the 2.69 million-cell grid was selected for modeling the film-cooling effectiveness in the triple-hole system. Figure 9 shows the centerline and spanwise-averaged effectiveness values in the single-and triple-hole systems at M = 0.5 and 1.0. The centerline effectiveness decreased as the main flow traveled downstream. This degradation is attributable to the turbulence generation and rapid mixing between the main flow and jet, which increases the test plate temperature. For the same total cross-sectional hole area and same amount of total cooling air, the triple holes improved the centerline and spanwiseaveraged effectiveness values, respectively, by approximately 36% and 45% over the single-hole configuration at M = 0.5 and approximately 250% and 345% at M = 1.0. In Figure 9a-d, the LESpredicted centerline film-cooling effectiveness in the single-hole system showed a good match to the experimental data by Seo et al. [13], whereas the LES model under-predicted the triple-hole, spanwise-averaged effectiveness experimental data by Cao et al. [22] because the data were measured for L/D = 10. Thus, compared to lower L/D cases, higher L/D values of the hole increase the adiabatic   Therefore, the adiabatic film-cooling effectiveness in the single-hole system was evaluated on the 2.04 million-cell grid. In the triple-hole system, the third grid with 2.69 million cells again showed almost the same centerline effectiveness as the fourth and fifth grids. Therefore, the 2.69 million-cell grid was selected for modeling the film-cooling effectiveness in the triple-hole system. Figure 9 shows the centerline and spanwise-averaged effectiveness values in the single-and triple-hole systems at M = 0.5 and 1.0. The centerline effectiveness decreased as the main flow traveled downstream. This degradation is attributable to the turbulence generation and rapid mixing between the main flow and jet, which increases the test plate temperature. For the same total cross-sectional hole area and same amount of total cooling air, the triple holes improved the centerline and spanwise-averaged effectiveness values, respectively, by approximately 36% and 45% over the single-hole configuration at M = 0.5 and approximately 250% and 345% at M = 1.0. In Figure 9a-d, the LES-predicted centerline film-cooling effectiveness in the single-hole system showed a good match to the experimental data by Seo et al. [13], whereas the LES model under-predicted the triple-hole, spanwise-averaged effectiveness experimental data by Cao et al. [22] because the data were measured for L/D = 10. Thus, compared to lower L/D cases, higher L/D values of the hole increase the adiabatic film-cooling effectiveness under the same operating condition as the coolant velocity at the hole exit becomes more uniform [13]. The ratios of the spanwise-averaged effectiveness values in the triple-hole system to those in the single-hole system as functions of x/D for M = 0.5 and 1.0 are plotted in Figure 10. Note that increasing the blowing ratio improves the ratio of the spanwise-averaged effectiveness. This indicates that under high blowing ratios, configuring the main hole and two sister holes is more effective for film cooling than using an ordinary single hole mainly because the jet trajectory in the single-hole system is mostly governed by the jet liftoff at a high blowing ratio of M = 1.0. The result is very low adiabatic filmcooling effectiveness, as indicated in Figure 9c,d, and very high ratios of the spanwise-averaged effectiveness values, as shown in Figure 10. The maximum value of the ratio at M = 1.0 is observed that under high blowing ratios, configuring the main hole and two sister holes is more effective for film cooling than using an ordinary single hole mainly because the jet trajectory in the single-hole system is mostly governed by the jet liftoff at a high blowing ratio of M = 1.0. The result is very low adiabatic film-cooling effectiveness, as indicated in Figure 9c,d, and very high ratios of the spanwise-averaged effectiveness values, as shown in Figure 10. The maximum value of the ratio at M = 1.0 is observed around at x/D = 4 since the minimum value of the spanwise-averaged effectiveness in the single-hole system is shown around at x/D = 4 due to the jet lift off, as shown in Figure 9d. The ratios of the spanwise-averaged effectiveness values in the triple-hole system to those in the single-hole system as functions of x/D for M = 0.5 and 1.0 are plotted in Figure 10. Note that increasing the blowing ratio improves the ratio of the spanwise-averaged effectiveness. This indicates that under high blowing ratios, configuring the main hole and two sister holes is more effective for film cooling than using an ordinary single hole mainly because the jet trajectory in the single-hole system is mostly governed by the jet liftoff at a high blowing ratio of M = 1.0. The result is very low adiabatic filmcooling effectiveness, as indicated in Figure 9c,d, and very high ratios of the spanwise-averaged effectiveness values, as shown in Figure 10. The maximum value of the ratio at M = 1.0 is observed around at x/D = 4 since the minimum value of the spanwise-averaged effectiveness in the single-hole system is shown around at x/D = 4 due to the jet lift off, as shown in Figure 9d.   Figure 11 presents top views of the time-averaged film-cooling effectiveness contours on the test plate in each hole system. The additional sister holes largely increased the amount of coolant in contact with the test plate, which boosts film-cooling effectiveness. Moreover, the spread of the coolant in the spanwise direction in the triple-hole system was much larger than that in the single-hole configuration as the main CRVP weakens. Furthermore, lower film-cooling effectiveness was predicted at M of unity than at M = 0.5 because of strong jet liftoff.  Figure 11 presents top views of the time-averaged film-cooling effectiveness contours on the test plate in each hole system. The additional sister holes largely increased the amount of coolant in contact with the test plate, which boosts film-cooling effectiveness. Moreover, the spread of the coolant in the spanwise direction in the triple-hole system was much larger than that in the singlehole configuration as the main CRVP weakens. Furthermore, lower film-cooling effectiveness was predicted at M of unity than at M = 0.5 because of strong jet liftoff.

Contours of Film Cooling Effectiveness and Dimensionless Temperature
The cross-sectional views of the dimensionless mean temperature contours at x/D = 2, 5, 10, 15, and 20 in the two systems are illustrated in Figure 12. Initially, the dimensionless temperature near the test plate was higher in the triple-hole system than in the single-hole configuration, which increased film-cooling effectiveness on the test plate. As the main flow went downstream, the dimensionless temperatures near the test plate in the triple-hole system became similar to those in the single-hole system, leading to almost identical film-cooling effectiveness in the range of x/D > 15 (Figures 8 and 9) because of intensive mixing between the mainstream flow and coolant jet.  The cross-sectional views of the dimensionless mean temperature contours at x/D = 2, 5, 10, 15, and 20 in the two systems are illustrated in Figure 12. Initially, the dimensionless temperature near the test plate was higher in the triple-hole system than in the single-hole configuration, which increased film-cooling effectiveness on the test plate. As the main flow went downstream, the dimensionless temperatures near the test plate in the triple-hole system became similar to those in the single-hole system, leading to almost identical film-cooling effectiveness in the range of x/D > 15 (Figures 8 and 9) because of intensive mixing between the mainstream flow and coolant jet.  Figure 13 shows the central cross-sectional contours of the dimensionless mean temperatures in the z = 0 plane for both hole systems. The entrainment of the main flow under the cooling jet in the triple-hole system was noticeably weaker than that in the single-hole system. Many researchers have argued that the jet lifted off because of the high momentum of the coolant and reattached downstream to the plate at a high blowing ratio [2,22,23]. However, as illustrated in Figure 13b, the dispersion of the coolant jet increases without reattachment with the test plate as the flow goes downstream after the jet liftoff, followed by the entrainment of the main flow under the jet. In addition, the dimensionless temperature near the test plate in the near-hole region in the triple-hole system was much higher than that in the single-hole configuration as vortex interactions between the jets were established, thereby reducing the intensity of the main CRVP at M = 0.5 and 1.0 and increasing filmcooling effectiveness on the test plate.  Figure 13 shows the central cross-sectional contours of the dimensionless mean temperatures in the z = 0 plane for both hole systems. The entrainment of the main flow under the cooling jet in the triple-hole system was noticeably weaker than that in the single-hole system. Many researchers have argued that the jet lifted off because of the high momentum of the coolant and reattached downstream to the plate at a high blowing ratio [2,22,23]. However, as illustrated in Figure 13b, the dispersion of the coolant jet increases without reattachment with the test plate as the flow goes downstream after the jet liftoff, followed by the entrainment of the main flow under the jet. In addition, the dimensionless temperature near the test plate in the near-hole region in the triple-hole system was much higher than that in the single-hole configuration as vortex interactions between the jets were established, thereby reducing the intensity of the main CRVP at M = 0.5 and 1.0 and increasing film-cooling effectiveness on the test plate.
to the plate at a high blowing ratio [2,22,23]. However, as illustrated in Figure 13b, the dispersion of the coolant jet increases without reattachment with the test plate as the flow goes downstream after the jet liftoff, followed by the entrainment of the main flow under the jet. In addition, the dimensionless temperature near the test plate in the near-hole region in the triple-hole system was much higher than that in the single-hole configuration as vortex interactions between the jets were established, thereby reducing the intensity of the main CRVP at M = 0.5 and 1.0 and increasing filmcooling effectiveness on the test plate.    Figure 14 illustrates the streamlines of cooling jets to a cross-sectional plane at x/D = 5 and film-cooling effectiveness on the test plate in the single-and triple-hole systems at M = 0.5 and unity. Unlike in a single hole, more streamlines of the coolant head to the region near the test plate in the x/D = 5 plane appeared in the triple-hole system, with these streamlines more dispersed, which resulted in higher adiabatic film-cooling effectiveness on the plate given that most streamlines from the sister holes have converged to the centerline. At M = 1.0, the strong jet liftoff caused the streamlines to cover smaller regions laterally in the x/D = 5 plane even in the triple-hole system. x/D = 5 plane appeared in the triple-hole system, with these streamlines more dispersed, which resulted in higher adiabatic film-cooling effectiveness on the plate given that most streamlines from the sister holes have converged to the centerline. At M = 1.0, the strong jet liftoff caused the streamlines to cover smaller regions laterally in the x/D = 5 plane even in the triple-hole system.  Figure 15 shows the cross-sectional contours of the mean x-vorticity at the x/D = 2 and x/D = 5 planes in the single-and triple-hole systems at M = 0.5 and unity. The mean velocity vectors are superimposed on the contours. As shown in Figure 15b,d of the triple-hole system, distinct anti-CRVPs were found from the sister holes around the x/D = ±0.5 and y/D = 0.2 regions in the x/D = 2 plane. The anti-CRVPs weakened the main CRVP and reduced the main jet liftoff, thereby reducing the entrainment of the hot mainstream gas under the coolant and increasing the dimensionless mean temperature and film-cooling effectiveness, as illustrated in Figures 11 and 13. However, the intensity of the anti-CRVPs became very small in the x/D = 5 plane because the anti-CRVP from the sister holes was not as strong as the main CRVP. In the triple-hole system, the upward velocity vectors around the line of z/D = 0 were less concentrated than those in the single-hole system. Thus, coolant liftoff was weakened by the generated anti-CRVPs. At M = 1.0, the positions of the core for the CRVPs were higher than those at M = 0.5 because of the strong jet liftoff generated.  Figure 15 shows the cross-sectional contours of the mean x-vorticity at the x/D = 2 and x/D = 5 planes in the single-and triple-hole systems at M = 0.5 and unity. The mean velocity vectors are superimposed on the contours. As shown in Figure 15b,d of the triple-hole system, distinct anti-CRVPs were found from the sister holes around the x/D = ±0.5 and y/D = 0.2 regions in the x/D = 2 plane. The anti-CRVPs weakened the main CRVP and reduced the main jet liftoff, thereby reducing the entrainment of the hot mainstream gas under the coolant and increasing the dimensionless mean temperature and film-cooling effectiveness, as illustrated in Figures 11 and 13. However, the intensity of the anti-CRVPs became very small in the x/D = 5 plane because the anti-CRVP from the sister holes was not as strong as the main CRVP. In the triple-hole system, the upward velocity vectors around the line of z/D = 0 were less concentrated than those in the single-hole system. Thus, coolant liftoff was weakened by the generated anti-CRVPs. At M = 1.0, the positions of the core for the CRVPs were higher than those at M = 0.5 because of the strong jet liftoff generated. temperature and film-cooling effectiveness, as illustrated in Figures 11 and 13. However, the intensity of the anti-CRVPs became very small in the x/D = 5 plane because the anti-CRVP from the sister holes was not as strong as the main CRVP. In the triple-hole system, the upward velocity vectors around the line of z/D = 0 were less concentrated than those in the single-hole system. Thus, coolant liftoff was weakened by the generated anti-CRVPs. At M = 1.0, the positions of the core for the CRVPs were higher than those at M = 0.5 because of the strong jet liftoff generated.   Figure 17 illustrates the portion of the total kinetic energy at x/D = 2 for Modes 1, 2, 3, and 4 in the single-and triple-hole systems at M = 0.5 and M = 1.0, which showed Mode 1 as the most dominant. At M = 0.5, POD Mode 1 contributed to about 60% and 36% of the total kinetic energy in the singleand triple-hole systems, respectively, which means that CRVP was dominant for both, although   [24,25]. On the basis of the figure, CRVP was most distinct in the velocity vector field at x/D = 2. Figure 17 illustrates the portion of the total kinetic energy at x/D = 2 for Modes 1, 2, 3, and 4 in the single-and triple-hole systems at M = 0.5 and M = 1.0, which showed Mode 1 as the most dominant. At M = 0.5, POD Mode 1 contributed to about 60% and 36% of the total kinetic energy in the single-and triple-hole systems, respectively, which means that CRVP was dominant for both, although relative to the former, the portion of the kinetic energy in the latter system was largely decreased by about 40% because of a weakened main CRVP by anti-CRVPs from the two sister holes. Similarly, at M = 1.0, POD Mode 1 contributed to about 71% and 49% of the total kinetic energy in the single-and triple-hole systems, respectively, indicating that the portion of the kinetic energy was decreased by nearly 30%. Indeed, it could be inferred that compared to that in the single-hole system, the kinetic energy of the CRVP in the triple-hole system was reduced by about 30-40%. nearly 30%. Indeed, it could be inferred that compared to that in the single-hole system, the kinetic energy of the CRVP in the triple-hole system was reduced by about 30-40%.   Figure 18 illustrates the cross-sectional views of the root-mean-square (RMS) temperature contours at x/D = 2, 5, 10, 15, and 20 for the mixing region between the mainstream flow and coolant in the single-and triple-hole systems. The mixing area was more expanded in the triple-hole system in the spanwise direction, especially in the near-hole region as x/D = 2, because the weakened main CRVP allowed wider spreading and closer contact of the coolant jet with the test plate. Although no significant difference in the mixing intensity was observed in both hole systems, the Trms in the x/D = nearly 30%. Indeed, it could be inferred that compared to that in the single-hole system, the kinetic energy of the CRVP in the triple-hole system was reduced by about 30-40%.   Figure 18 illustrates the cross-sectional views of the root-mean-square (RMS) temperature contours at x/D = 2, 5, 10, 15, and 20 for the mixing region between the mainstream flow and coolant in the single-and triple-hole systems. The mixing area was more expanded in the triple-hole system in the spanwise direction, especially in the near-hole region as x/D = 2, because the weakened main CRVP allowed wider spreading and closer contact of the coolant jet with the test plate. Although no  Figure 18 illustrates the cross-sectional views of the root-mean-square (RMS) temperature contours at x/D = 2, 5, 10, 15, and 20 for the mixing region between the mainstream flow and coolant in the single-and triple-hole systems. The mixing area was more expanded in the triple-hole system in the spanwise direction, especially in the near-hole region as x/D = 2, because the weakened main CRVP allowed wider spreading and closer contact of the coolant jet with the test plate. Although no significant difference in the mixing intensity was observed in both hole systems, the T rms in the x/D = 2 plane reached maximum values of 6.25, 6.24, 6.08, and 5.19, as shown in Figure 18a-d, respectively, which indicates a similar but slightly decreased mixing intensity in the triple-hole system relative to the single-hole system. This reduced mixing intensity is beneficial to film cooling because increased mixing between the coolant and mainstream flow consequently decreases film-cooling effectiveness [22]. Nevertheless, it could be stated that in the triple-hole system, the effect of the sister holes on the jet trajectory contributes greatly to improve film-cooling effectiveness as compared to the effect of less mixing between the main flow and coolant. mixing between the coolant and mainstream flow consequently decreases film-cooling effectiveness [22]. Nevertheless, it could be stated that in the triple-hole system, the effect of the sister holes on the jet trajectory contributes greatly to improve film-cooling effectiveness as compared to the effect of less mixing between the main flow and coolant.

Q-Criterion Iso-Surfaces
In the Q-criterion, Q is defined as the difference between the magnitudes of the rotation rate and strain rate, representing the dominance of the former over the latter [26][27][28]. The instantaneous contours of a Q-criterion with iso-surfaces of level 80,000 at M = 0.5 and unity are displayed in Figure  19. Q-criterion contours in the single-hole system clearly showed various vortex structures, such as

Q-Criterion Iso-Surfaces
In the Q-criterion, Q is defined as the difference between the magnitudes of the rotation rate and strain rate, representing the dominance of the former over the latter [26][27][28]. The instantaneous contours of a Q-criterion with iso-surfaces of level 80,000 at M = 0.5 and unity are displayed in Figure 19. Q-criterion contours in the single-hole system clearly showed various vortex structures, such as the CRVP, hairpin and jet shear layer vortices, and horseshoe vortex, whereas those in the triple-hole configuration had smaller hairpin and jet shear layer vortices and were not very distinguishable. These smaller and indistinct hairpin vortices represent the weakened CRVP as the hairpin vortex is closely related to the generation of the CRVP [22]. A weakened CRVP positively affects the cooling performance on the plate because it increases film-cooling effectiveness on the test plate.  Figure 20 shows the cross-sectional contours of the mean velocity magnitude in the hole measured around the hole exit (y/D = −0.2). As mentioned earlier, the ratio of the delivery tube length to the hole diameter in the single-hole system was 1.6, whereas it was 1.8 and 3.6 for the main hole and sister holes in the triple-hole system, respectively, because the triple holes had the same total cross-sectional hole area as the single hole. Therefore, the mean velocity magnitude in the main hole is not uniform but has high-and low-momentum regions, whereas that in the sister holes shows more developed coolant flow. The cooling jet having a uniform velocity profile is advantageous to filmcooling performance because of less jet liftoff. Thus, it could be stated that relative to the single-hole system, the sister holes improve film-cooling effectiveness by the generation of the anti-CRVP, as well as higher L/D ratios.  Figure 20 shows the cross-sectional contours of the mean velocity magnitude in the hole measured around the hole exit (y/D = −0.2). As mentioned earlier, the ratio of the delivery tube length to the hole diameter in the single-hole system was 1.6, whereas it was 1.8 and 3.6 for the main hole and sister holes in the triple-hole system, respectively, because the triple holes had the same total cross-sectional hole area as the single hole. Therefore, the mean velocity magnitude in the main hole is not uniform but has high-and low-momentum regions, whereas that in the sister holes shows more developed coolant flow. The cooling jet having a uniform velocity profile is advantageous to film-cooling performance because of less jet liftoff. Thus, it could be stated that relative to the single-hole system, the sister holes improve film-cooling effectiveness by the generation of the anti-CRVP, as well as higher L/D ratios.

Cross-Sectional Contours of the Mean Velocity Magnitude in the Hole
cross-sectional hole area as the single hole. Therefore, the mean velocity magnitude in the main hole is not uniform but has high-and low-momentum regions, whereas that in the sister holes shows more developed coolant flow. The cooling jet having a uniform velocity profile is advantageous to filmcooling performance because of less jet liftoff. Thus, it could be stated that relative to the single-hole system, the sister holes improve film-cooling effectiveness by the generation of the anti-CRVP, as well as higher L/D ratios.

Conclusions
Film-cooling simulations of the single-and triple-hole systems on a flat plate were carried out in this paper at blowing ratios of M = 0.5 and unity by employing LES. On the basis of these simulations, when two sister holes which were placed slightly downstream the main hole, the coolant jets from the former generated an anti-CRVP, which increased film-cooling effectiveness. Thus, for the same total cross-sectional hole area and same amount of total cooling air, the triple-hole system increased the centerline and spanwise-averaged effectiveness values by approximately 36% and 45% at M = 0.5 and by approximately 250% and 345% at M = 1.0 as compared to that done by the single-hole configuration. Moreover, the ratios of the spanwise-averaged effectiveness values in the triple-hole system indicate that it is the more effective system for film cooling under a high blowing ratio.
Meanwhile, the time-averaged film-cooling effectiveness contours on the test plate and the cross-sectional views of the dimensionless mean temperature contours in the two hole systems showed that additional sister holes can increase the amount of coolant contacting with the test plate and boost film-cooling effectiveness. Specifically, the mean dimensionless temperature contours in the z/D = 0 plane showed that in the single-hole system at M of unity, as the cooling jet goes downstream, the jet gets dispersed without reattachment with the test plate, and the entrainment of the main flow under the jet is generated. Furthermore, the cross-sectional mean x-vorticity contours in the triple-hole system showed that the generated anti-CRVPs weaken the main CRVP. The results of the POD analysis confirmed that the kinetic energy of the CRVP in the triple-hole system was reduced by about 30-40% relative to that in the single-hole system. Streamlines of the cooling jets to the cross-sectional plane at x/D = 5 also showed that most streamlines from the sister holes converge to the centerline. As with the Q-criterion contours in the triple-hole system, the hairpin and jet shear layer vortices were smaller and indistinguishable, unlike in the single-hole system.
Lastly, on the basis of the cross-sectional views of the RMS temperature contours, the effect of the sister holes on the cooling jet trajectory contributes significantly to the increase in film-cooling effectiveness, as compared to the effect of less mixing. Accordingly, the sister holes improve film-cooling effectiveness using higher L/D ratios.
Author Contributions: S.I.B. carried out the simulations, analyzed the CFD data and wrote the paper. J.A. supervised the research, analyzed the CFD data and wrote the paper. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.

Conflicts of Interest:
The authors declared no conflict of interest.