Abstract
This paper investigates film cooling flow over a flat plate with five different cooling hole configurations. These configurations include a combined arrangement of cylindrical and fan-shaped holes. Numerical simulations are performed using the open-source computational fluid dynamics (CFD) platform OpenFOAM. The study is performed at two different mainstream Mach numbers (
and 
) and three different blowing ratios (
, 
, and 
), while maintaining a coolant density ratio of approximately 
. To enable accurate and physically consistent boundary conditions, a mathematical equation is presented to compute the total pressure of the coolant at the inlet as a function of blowing ratio, density ratio, and mainstream Mach number. Across all conditions, staggered fan-shaped configurations exhibited the highest cooling effectiveness. At the higher Mach number (
) and higher blowing ratio (
), an asymmetric coolant distribution was observed for the fan-shaped hole geometries. This asymmetry was attributed to flow separation within the fan-shaped coolant channel. This asymmetric coolant distribution leads to a significant drop in cooling performance, resulting in an approximate 60% reduction in the averaged film cooling effectiveness for staggered fan-shaped compared with 
at the same blowing ratio.
Subject terms: Engineering, Mathematics and computing, Physics
Introduction
The rapid advancement of aircraft propulsion systems and power plant efficiency demands exceptionally high turbine inlet temperatures. Under such extreme operating conditions, the temperature often exceeds the thermal resistance of turbine blade materials. To mitigate this issue, extensive research has been conducted on various cooling strategies. Generally, turbine cooling methods can be classified into internal cooling, where heat is dissipated through internal cooling channels, and external cooling, where cooling operations are externally incorporated to shield the blade surface1.
Among external cooling techniques, film cooling is one of the most widely employed methods for protecting turbine blades from excessive thermal loads2. This technique involves injecting coolant from discrete holes or pores in the turbine blade. Upon interaction with the high-temperature mainstream flow, the coolant forms a protective layer over the blade surface, reducing heat transfer between the main stream hot gases and blade surface thus enhancing the component’s operational lifespan. The effectiveness of film cooling is influenced by several parameters, including turbulence intensity, blowing ratio, surface roughness, hole geometry, injection angle, and coolant thermophysical properties. While various factors influence film cooling performance, the geometry of the cooling hole plays a decisive role in determining its effectiveness in mitigating thermal stresses. Consequently, extensive numerical and experimental studies have been conducted by many researchers to investigate the impact of the hole geometry on cooling performance.
Early research predominantly focused on cylindrical coolant holes3–8. Goldstein2 has presented a review of experimental research on flat plate film cooling prior to 1971. The influence of geometrical and flow parameters on film cooling characteristics for both incompressible and compressible regions was discussed. Sinha et al.3 experimentally investigated the downstream film cooling effectiveness of a single-row cylindrical hole with a fixed inclination angle while varying the density ratio from 
to 2.0. Their findings provided valuable insights into the influence of blowing ratio on cooling effectiveness. This study, along with other foundational research, laid the groundwork for subsequent investigations into alternative hole geometries aimed at improving cooling efficiency.
Further experimental work by Lutum and Johnson7 examined the influence of the length-to-diameter (L/D) ratio of cylindrical holes. The results indicated that effectiveness remained stable at higher L/D values but significantly decreased within the range 
, reaching its lowest effectiveness at 
. A key limitation of cylindrical holes is the counter-rotating vortex pairs (CRVPs) effect, which causes the coolant jet to lift off from the surface at high blowing ratios, reducing cooling efficiency. Walters and Leylek8 analyzed this phenomenon and found that at excessive blowing ratios, the coolant jet detaches from the surface, diminishing its protective effect. To address this challenge, researchers have explored modifications to cylindrical hole geometry. For example, Zhang et al.9 conducted numerical simulations comparing elliptical and cylindrical holes with a compound angle of 
. Their findings showed that both configurations improved film cooling effectiveness, with the elliptical hole performing better at lower blowing ratios, while the cylindrical hole with a 
compound angle exhibited superior effectiveness at higher blowing ratios.
Beyond cylindrical holes, researchers have extensively explored shaped-hole geometries to improve film cooling performance. Gritsch et al.10 evaluated the film cooling effectiveness of cylindrical hole and expanded-exit holes( i.e fan-shaped and a laid-back fan-shaped hole) at mainstream Mach number up to 1.2. Their key findings highlighted that expanded exit geometries provide superior performance to simple cylindrical hole. Bunker11 provided a comprehensive review of various shaped-hole designs, highlighting their substantial effectiveness improvements over conventional cylindrical holes. One of the notable advancements is the shallow surface trench design, introduced by Bunker12. This design improved cooling effectiveness by 50–75% compared to traditional cylindrical holes by optimizing coolant flow distribution. Cao et al.13 conducted both experimental and numerical investigations on four distinct hole types: cylindrical holes, fan-shaped holes, anti-vortex holes, and sister holes. Their analysis focused on the intensity of counter-rotating vortex pairs (CRVP). They found that sister holes enhanced cooling effectiveness by a factor of 0.3 to 1.5, while fan-shaped holes exhibited peak effectiveness at a blowing ratio of 2.0. Further optimization of fan-shaped holes was explored by Lee and Kim14, who investigated the influence of injection angle, lateral expansion angle, and L/D ratio. Their numerical study revealed that a converged inlet fan-shaped hole achieved approximately 46% higher cooling effectiveness than a conventional cylindrical hole.
The configuration of cooling holes significantly influences turbine performance. Ligrani et al.15 examined the flow through two row staggered cooling holes with a compound angle orientation, where the holes are spaced 3D apart in the spanwise direction. It was found that compound angle holes which inclined at 
in the streamwise direction and 
in the spanwise direction that provided significantly higher adiabatic film-cooling effectiveness compared to simple angle holes, especially in the near-hole region (
). The compound angle arrangement enhanced lateral spreading of the coolant and reduced jet lift-off, resulting in improved surface coverage. Moreover, decreasing the spanwise spacing from 3.9d to 3d further increased cooling effectiveness by 25–40% near the injection region and by 12–30% farther downstream. These findings underscore the benefits of compound angle, closely spaced film cooling holes for achieving superior thermal protection in gas turbine components. Saumweber16 conducted a comprehensive experiment on the interaction of cooling rows in a staggered arrangement. This study focused on the effects of hole geometry and row spacing on downstream cooling performance.
Brauckmann and von Wolfersdorf17 examined the influence of compound angle in a row of fanshaped holes (
).The findings indicate that increasing the compound angle enhances lateral coolant distribution, thereby improving spanwise cooling coverage. Kusterer et al.18 studied the compound angle configuration of double-jet film cooling (DJFC). In DJFC, two cylindrical holes are arranged side-by-side with a compound angle, allowing the jets to interact and form an anti-kidney vortex pair. This configuration was discovered to augment the effectiveness of film cooling. Recently, Taheri et al.19 conducted both experimental and numerical investigations on the impact of shaped multi-holes on the efficacy of film cooling. This study primarily focuses on arranging cylindrical holes into various configurations, which significantly enhances film cooling performance in both axial and lateral directions.
Naik et al.20 investigated the multi-row arrangement of fan-shaped holes on the turbine blades. Their findings highlighted the significance of variation of blowing ratios, which improves the film cooling effectiveness of a multi-row arrangement of fan-shaped holes for high-lift turbine vane and downstream blade. Li et al. 21 experimentally investigated the film cooling performance of multiple cylindrical hole rows on the suction surface of a rotating turbine blade. The study analyzed the influence of hole arrangement, blowing ratio, and rotational speed, showing that staggered rows enhance film coverage and that rotation slightly improves effectiveness at low blowing ratios while degrading it at higher ones. Meng et al. 22 conducted experimental and numerical investigations on the film cooling performance over the pressure side of a rotating turbine blade with 3, 4, and 5 rows of holes arranged in both in-line and staggered configurations. The results indicated that an increased number of hole rows enhances film coverage at higher coolant mass flow ratios due to improved jet superposition, whereas configurations with fewer rows are more effective at lower mass flow rates. Additionally, the hole arrangement significantly affects the film distribution and overall cooling effectiveness.
Li et al.23 conducted transient SAS simulations to explore the characteristics of turbine blade cutback trailing edges, which incorporated two rows of film holes (either cylindrical or fan-shaped) and pin fins. Their study demonstrated that upstream film holes, especially those with a fan-shaped configuration, significantly influenced vortex structures, thereby improving cooling effectiveness through strong interactions between the holes. Similarly, Veley and Thole24 illustrated that the geometry of the feed channel plays a crucial role in shaping secondary flows and, as a result, affects film cooling effectiveness in staggered, two-row film-cooling setups. Their experimental and CFD studies highlights the importance of considering hole–channel interactions in multi-row configurations. Furthermore, Hu and An25 discovered that vertically oriented slot holes provided more uniform coverage and greater multirow effectiveness compared to fan-shaped holes in a staggered, multirow film-cooling arrangement on a turbine vane suction surface, emphasizing the impact of hole geometry on multirow interactions.
Although extensive research has been conducted on the influence of cooling hole geometry, compound angle orientation, and shaped hole optimization, the effect of cooling hole arrangement remains relatively unexplored. Most existing studies primarily focus on individual holes or single-row configurations, resulting in a gap in understanding how different staggered arrangements of different shaped holes impact film cooling performance. Through numerical simulations, this research evaluates cooling effectiveness across different geometric configurations, offering new insights into optimizing film cooling strategies for enhanced turbine blade protection.
Numerical methodology
The present numerical investigation is conducted using OpenFOAM, an open-source computational fluid dynamics (CFD) platform. The ESI OpenFOAM-v2206 version is employed. A pressure-based, steady-state compressible solver namely, ’rhoSimpleFoam’ is utilized to perform the simulations. The governing equations are presented as follows:
Continuity equation:
 |
1 |
Momentum equation:
 |
2 |
Energy equation:
 |
3 |
Here, 
is the turbulent viscosity, 
is the thermal conductivity, K is the kinetic energy, h is the specific enthalpy, and 
denotes the viscous dissipation function.
The low Reynolds number formulation of the two-equation 
turbulence model was employed to account for turbulence. This formulation models the viscous sublayer without the use of any extra wall function treatment. This approach has been shown to provide good predictions of heat transfer in film cooling applications over a flat plate26–28. The details of the wall function modeling for the 
turbulence model in OpenFOAM are discussed in Liu et al29. The governing equation for turbulence kinetic energy 
and turbulence dissipation rate 
are given by:
 |
4 |
 |
5 |
 |
6 |
Where G represents the production of turbulent kinetic energy, and 
, 
, 
, 
, and 
are model constants set to their default values in OpenFOAM.
This solver uses SIMPLE-Algorithm (Semi-Implicit Method for Pressure-Linked Equation) by Patankar30 to iteratively solve for pressure-velocity coupling in fluid flows. Due to the complexity in numerical convergence in compressible film cooling simulations, a two-step approach was adopted. First, the simulation was initially conducted using first-order numerical schemes. The gradient terms were discretized using the ’cellLimited Gauss linear 1’ scheme, while the convective terms were treated using the ’Gauss upwind’ scheme. Under-relaxation factors were applied to all primary flow variables, including velocity, pressure, density, enthalpy, and turbulence quantities (k and 
). Specifically, a very low relaxation factor was applied to the density field to prevent sudden floating point errors that could arise from numerical instability associated with compressible flow during the early stages of the simulation.
After obtaining a stable and converged solution with the first-order schemes, the discretization was upgraded to enhance accuracy. The gradient terms were switched to the second-order ‘Gauss linear’ scheme, and the convective terms for velocity and enthalpy were switched to the ‘Gauss linearUpwind’, a second-order upwind scheme. However, to maintain numerical robustness and avoid oscillations in turbulence variables, the convective terms for turbulent kinetic energy (k) and dissipation rate (
) continued to use the ‘Gauss upwind’, a first-order scheme.
The pressure field was solved using the ‘Geometric-Algebraic MultiGrid (GAMG)’ solver with a ‘Gauss-Seidel’ smoother. All other variables including density, velocity, enthalpy, turbulent kinetic energy (k), and dissipation rate (
) were solved using the ‘Preconditioned Bi-Conjugate Gradient Stabilized (PBiCGStab)’ solver with the ‘Diagonal Incomplete LU factorization (DILU)’ preconditioner. Residual control criteria were defined for key variables to monitor convergence during the iterations: 
for pressure, 
for velocity, 
for enthalpy, and 
for both k and 
.
The thermophysical properties of air in the present simulation were defined using the ‘heRhoThermo’ model. The working fluid was treated as a ‘pureMixture’, with a constant specific heat capacity at constant pressure (
) set to 1007 J/kg
K. The gas behavior was modeled using the ‘perfectGas’ equation of state. Transport properties were calculated using Sutherland’s law by specifying the ‘sutherland’ transport model.
Non-Dimensional parameters in film cooling study
This section provides detailed definitions and explanations of key non-dimensional parameters, including the blowing ratio (
), density ratio (
), and film cooling effectiveness.
The blowing ratio (
) represents the relative momentum of the coolant flow in comparison to the main-stream flow. It is calculated as the ratio of the mass flux rate of the coolant at the cooling channel’s exit in to the mainstream flow to the mass flux rate of the main-stream flow, expressed as follows:
 |
7 |
The density ratio is defined as the ratio of the density of the coolant to the density of the main-stream flow and is expressed as follows:
 |
8 |
Film cooling effectiveness is defined as a non-dimensional temperature difference that assesses how effectively the coolant insulates the surface from the high-temperature main-stream flow. The general expression for film cooling effectiveness is expressed as follows:
 |
9 |
Where 
is the recovery temperature, 
is the adiabatic wall temperature at a specific streamwise (
) and spanwise (
) location, and 
is the total temperature of the coolant. The recovery temperature (
) is approximated as stagnation temperature, which is the temperature of the mainstream flow if it is brought to rest without any heat loss31. In the context of film cooling, 
is used as a reference temperature for the hot mainstream flow and is typically measured upstream of the coolant injection location, where the flow remains undisturbed by the cooling jets (e.g., at 
).
Figure 1 illustrates the reference locations for temperature measurements and the calculation of film cooling effectiveness. In this study, film cooling effectiveness is evaluated over the streamwise range 
and the spanwise range 
.
Fig. 1.
Flat plate surface with cooling arrangement and their spatial locations.
Film cooling effectiveness is quantified in different ways in this study. The first is centerline film cooling effectiveness (
), which measures the effectiveness along the centerline of the plate in x-direction i.e., from X/D=0 to X/D=24, and is expressed in Equation 
. The second is laterally averaged film cooling effectiveness (
), which provides an overall measure of cooling performance across the spanwise direction at a given 
location, and is given by Equation 
.
 |
10 |
 |
11 |
Also, the average film cooling effectiveness (
) over the plate surface is determined by performing a surface integral over the downstream region of the coolant holes bounded by 
and 
, as given in Eq. 12.
 |
12 |
Numerical model validation
For the validation of the present solver, the experimental research conducted by Gritsch et al.10 has been chosen. The dimensions and layout of the geometry are determined based on the experimental study and are presented in Fig. 2. The computational domain consists of a single cylindrical cooling hole, inclined at an angle of 
relative to the surface. The diameter (D) of the cylindrical hole and the length-to-diameter ratio (L/D) of the coolant channel are taken as 10.0 mm and 6.0, respectively.
Fig. 2.
Computational domain with the cylindrical coolant hole: (a) isometric view, (b) front view, (c) top view.
Experimental flow conditions for the cylindrical hole are outlined in Table 1.
Table 1.
Flow conditions for the cylindrical cooling hole10.
| Parameters | Value |
|---|---|
Mainstream inlet Mach number ( ) |
0.3 |
Total temperature at plenum coolant inlet ( ) |
290 K |
Turbulence intensity at mainstream inlet ( ) |
1.5% |
Turbulence intensity at plenum coolant inlet ( ) |
1.0% |
Density ratio ( ) |
1.85 |
Temperature ratio ( ) |
0.54 |
Blowing ratio ( ) |
2.0 |
A significant challenge in simulating compressible film cooling is determining the total pressure and total temperature boundary conditions from the given experimental non-dimensional parameters. The required computational boundary conditions include the total pressure at the mainstream inlet (
) and coolant plenum inlet (
), the total temperature at the mainstream inlet (
) and coolant plenum inlet (
), and the static pressure (
) at the outlet.
During the experimental investigation of film cooling over a flat plate, a zero streamwise pressure gradient was maintained. Therefore, in the numerical setup, a constant static pressure (
) was used to calculate the total pressure at the mainstream inlet (
) using the compressible flow Eq. 13. However, due to the interaction between the coolant and mainstream flow, slight variations in the static pressure were observed near the cooling hole region in the simulation results.
 |
13 |
The total pressure of the coolant (
) was estimated based on the specified blowing ratio and density ratio. By combining the definitions of blowing ratio (Eq. 7) and density ratio (Eq. 8), the relationship can be expressed as:
 |
14 |
The total pressure in the mainstream (
) and the total pressure at the coolant channel exit in the crossflow region (
) are expressed as:
 |
15 |
 |
16 |
Here, an incompressible assumption is adopted to simplify the relation between blowing ratio and stagnation pressures. Substituting Eqs. 15 and 16 into Eq. 14, the blowing ratio expression becomes:
 |
17 |
Here, 
represents the total pressure of the coolant at the channel exit into the crossflow. To determine the total pressure at the coolant plenum inlet (
), a pressure loss term (
) is introduced to account for the compressibility, viscous, and other dissipative effects between the coolant plenum and the crossflow region. Thus, the total coolant pressure (
) at the plenum inlet is:
 |
18 |
The total pressure of the coolant, 
, is determined by combining Eqs. 13, 17, and 18. This formulation expresses 
as a function of the blowing ratio (
), density ratio (
), and the total pressure of the mainstream flow (
), as shown in Eq. 19:
 |
19 |
The total temperature boundary condition for the validation case is determined based on the temperature ratio (
) provided in the experimental study. Table 2 provides a summary of the numerical boundary conditions applied in the validation case.
Table 2.
Numerical boundary conditions for the validation case study of the cylindrical cooling hole.
| Parameters | Value |
|---|---|
Mainstream total inlet pressure ( ) |
107853.39 Pa |
Total pressure at plenum coolant inlet ( ) |
120440.45 Pa |
Outlet static pressure ( ) |
101325 Pa |
Mainstream total inlet temperature ( ) |
537 K |
Pressure loss term (assumed) ( ) |
5000 Pa |
In this study, the total temperature boundary condition at main stream inlet is obtained from the compressible flow density ratio relation shown in Equation 
. This methodology is described by Bubb32.
 |
20 |
Next, the entire computational domain was meshed with hexahedral elements using ICEM CFD, as shown in Fig. 3. A fine mesh near the wall was achieved through boundary layer resolution with near-wall grading, and an O-grid was implemented around the cooling hole for better resolution. The final mesh for the cylindrical hole consisted of approximately 1.56 million cells.
Fig. 3.
Computational mesh view for the cylindrical cooling hole (a) Computational domain, (b) Top view of O-Grid mesh.
To verify the blowing ratio, the mass flux rate is evaluated across the computational domain using the ParaView ‘Calculator’ utility, as illustrated in Fig. 4. The mass flux rate (
) is mathematically defined as follows:
 |
21 |
Fig. 4.
Mass flux rate calculation at the mainstream and coolant inlet.
The mass flux rate at the mainstream inlet, represented as 
, varies as a function of 
at a particular 
-plane location. The average mainstream mass flux rate is computed using the following integral expression:
 |
22 |
Similarly, the mass flux rate at the coolant inlet, 
, is expressed as a function of 
at a particular 
-plane location within the cooling channel. The average coolant inlet mass flux rate is computed using the following integral expression:
 |
23 |
The integrals necessary for these calculations are conveniently executed using the ParaView ‘IntegrateVariables’ utility. According to the simulation results, the average mass flux rate at the mainstream inlet is 80.43 kg/m
s, while at the coolant inlet, it is 155.667 kg/m
s. The calculated blowing ratio is determined as the ratio of the averaged mass flux rate at the coolant inlet to that at the mainstream inlet. The computed blowing ratio, 
, closely matches the initially set blowing ratio of 
, which was established as a boundary condition. In case the computed blowing ratio is not sufficiently close to the required blowing ratio, the pressure loss term has to be re assumed appropriately.
Figure 5 compares the centerline film cooling effectiveness from the current simulation with the experimental data reported by10 for 
and 
. The simulation captures the characteristic rapid decline trend in effectiveness just after the injection point, followed by a gradual recovery downstream. While slight deviations are observed, particularly in the downstream region, the overall agreement is close. The root mean square error (RMSE) between the simulation and experimental data is found to be 3.28%, indicating good predictive capability of the numerical model. This validation confirms the reliability of the CFD model in predicting the film cooling effectiveness over the flat plate.
Fig. 5.
Centerline film cooling effectiveness (
) comparison with experimental data10 for the cylindrical hole case.
It is worthwhile to note that adjusting the computed blowing ratio to 2.0 does not cause any significant difference in the film-cooling effectiveness. The RMSE value between the 
and 
cases was found to be 0.85%. A potential source of the deviation could be other factors, such as numerical model errors, the adiabatic wall assumption in simulations, which may not hold strictly during the experimentation.
A validation study was also carried out for the fan-shaped cooling holes. The computational domain for the fan-shaped hole is similar to the cylindrical hole case, as discussed earlier in Fig. 2. The fan-shaped hole dimensions were adopted from the experimental study of Gritsch et al.10 The numerical boundary conditions for the fan-shaped case were derived from the experimental flow conditions presented in Table 3, following the same procedure described earlier for the cylindrical hole case. The resulting boundary conditions are summarized in Table 4. The entire computational domain was meshed using hexahedral elements in ICEM CFD, as shown in Fig. 6. The final mesh for the fan-shaped hole consisted of approximately 4.16 million cells.
Table 3.
Flow conditions for the fan-shaped cooling hole10.
| Parameters | Value |
|---|---|
Mainstream inlet Mach number ( ) |
0.6 |
Total temperature at plenum coolant inlet ( ) |
290 K |
Turbulence intensity at mainstream inlet ( ) |
1.5% |
Turbulence intensity at plenum coolant inlet ( ) |
1.0% |
Density ratio ( ) |
1.85 |
Temperature ratio ( ) |
0.54 |
Blowing ratio ( ) |
1.0 |
Table 4.
Numerical boundary conditions for the validation case study of the fan-shaped cooling hole.
| Parameters | Value |
|---|---|
Mainstream total inlet pressure ( ) |
129240.42 Pa |
Total pressure at plenum coolant inlet ( ) |
126664.41 Pa |
Outlet static pressure ( ) |
101325 Pa |
Mainstream total inlet temperature ( ) |
537 K |
Pressure loss term (assumed) ( ) |
10250 Pa |
Fig. 6.
Computational mesh view for the fan-shaped cooling hole (a) Computational domain, (b) Top view of O-Grid mesh.
Figure 7 presents a comparison of the centerline and laterally averaged film cooling effectiveness for the current simulation of the fan-shaped hole with the experimental data of Gritsch et al.10. For the fan-shaped hole validation case, the mainstream Mach number (
) and blowing ratio (
) were prescribed. The blowing ratio calculated from the simulation results was 
, which is close to the target value of 
used in the numerical boundary condition specification. The simulation successfully captures the trend of both centerline and laterally averaged film cooling effectiveness, as shown in Fig. 7a and 7b. The RMSE was computed for both centerline and laterally averaged film cooling effectiveness, yielding values of 3.48% and 1.19%, respectively, indicating that the numerical model accurately captures the experimental trends.
Fig. 7.
Comparison of centerline (
) and laterally averaged (
) film cooling effectiveness with experimental data10 for the fan-shaped hole case.
Flow problem description
To perform a detailed investigation of film cooling effectiveness, CFD simulations were conducted in the compressible regime using various hole configurations that incorporate a combination of cylindrical and fan-shaped cooling holes. This section outlines the complete numerical setup, including the geometric configurations, applied boundary conditions, meshing strategy, and the grid independence test (GIT) performed to ensure the accuracy and reliability of the simulation results. Each of these aspects is discussed in the subsequent subsections.
Computational domain and geometric configurations
The computational domain, illustrated in Fig. 8, represents a flat plate film cooling setup incorporating a combination of cylindrical and fan-shaped holes arranged in a staggered configuration. Dimensional details are annotated within the domain, with the diameter of the cooling holes (
) set to 
. The geometric parameters of both the cylindrical and fan-shaped holes are adopted from the experimental study conducted by Gritsch et al. 10. All cooling holes are inclined at an angle of 
with respect to the flat plate surface. The coolant is supplied through a plenum chamber situated beneath the flat plate, which acts as a high-pressure reservoir to ensure a uniform flow distribution at the coolant hole inlets.
Fig. 8.
Computational domain with the cylindrical and fan-shaped coolant holes: (a) isometric view, (b) front view, (c) top view (d) side view (e) cooling hole configurations.
Five different cooling hole configurations were developed to examine the influence of hole shape and arrangement. Four of these are staggered patterns, while one represents an inline configuration. The schematic layout of the hole arrangements is shown in Fig. 9. Each configuration features two coolant hole rows, with a uniform streamwise spacing and spanwise pitch of 
.
Fig. 9.
Different cooling hole configurations.
The configurations are described as follows:
Geom1: Cylindrical holes in both rows arranged in a staggered pattern (Fig. 9a),
Geom2: Cylindrical holes in the first row and fan-shaped holes in the second row, arranged in a staggered pattern (Fig. 9b),
Geom3: Fan-shaped holes in the first row and cylindrical holes in the second row, arranged in a staggered pattern (Fig. 9c),
Geom4: Fan-shaped holes in both rows arranged in a staggered pattern (Fig. 9d),
Geom5: Cylindrical holes in both rows arranged in an inline pattern (Fig. 9e).
These variations allow a systematic comparison of the effect of cooling hole geometry and configuration on the film cooling performance over a flat plate under close realistic compressible flow conditions.
Boundary conditions and flow parameters
A detailed summary of the boundary conditions employed for the parametric study is provided in Table 5. To model periodicity in the spanwise direction, the cyclicAMI (cyclic Arbitrary Mesh Interface) boundary condition is applied on the front and back boundaries of the computational domain. This condition facilitates the exchange of flow information across non-conformal mesh interfaces, enabling accurate treatment of periodic geometries without requiring mesh alignment on opposing boundaries.
Table 5.
Boundary condition details for the film cooling parametric study.
| Film cooling flow conditions | |||||||
|---|---|---|---|---|---|---|---|
| Mach number at main-stream flow inlet |  |
0.4, 0.6 | |||||
| Total temperature at main-stream flow inlet |  |
1500 K | |||||
| Turbulence intensity for main-stream inlet |  |
1.5% | |||||
| Turbulence intensity for coolant inlet |  |
1.0% | |||||
| Density ratio |  |
2.0 | |||||
| Blowing ratio |  |
1.0, 1.5, 2.0 | |||||
| Calculated boundary conditions | |||||||
|---|---|---|---|---|---|---|---|
Main-stream Mach number 
|
Main-stream Mach number 
|
||||||
 |
 |
 |
 |
 |
 |
||
 (Pa) |
113134.63 | 113134.63 | 113134.63 |
 (Pa) |
129240.42 | 129240.42 | 129240.42 |
 (Pa) |
111129.81 | 120060.83 | 131144.26 |
 (Pa) |
124232.71 | 145729.85 | 172455.84 |
 (K) |
746.17 | 762.84 | 782.33 |
 (K) |
741.57 | 776.17 | 814.43 |
| Calculated blowing ratio from simulation results | |||||||
|---|---|---|---|---|---|---|---|
 |
 |
 |
 |
 |
 |
||
To achieve the desired flow conditions in the simulation, the inlet mainstream gas is modeled to enter the domain at specified Mach numbers, namely 
and 
. By prescribing the total pressure along with the total temperature, the solver internally computes the corresponding static pressure and velocity field, thereby ensuring the target Mach number is achieved at the inlet. Similarly, for the coolant flow, total pressure and total temperature boundary conditions are applied at the coolant plenum inlet. These coolant conditions are carefully tuned to satisfy the desired blowing ratio (
) and density ratio (
), which are essential parameters in film cooling studies. These boundary conditions are determined as per the methodology presented in Section Numerical Model Validation.
To investigate the influence of blowing ratio on film cooling effectiveness across various cooling hole configurations, three blowing ratios (
, 
, and 
) were considered under two mainstream inlet Mach numbers (
and 
). A constant density ratio of approximately 
was maintained for all simulations.
A total of 30 simulations were conducted, accounting for the combinations of 3 blowing ratios, 2 mainstream Mach numbers, and 5 geometric configurations. The simulations were performed on the High-Performance Computing Cluster (HPCC) ’Magus’ at Shiv Nadar Institution of Eminence. Each simulation was executed in a parallel computing environment utilizing 128 processor cores, with an average computational time of approximately 30 minutes per case.
Meshing and grid independence test (GIT) study
A structured mesh was generated for all five geometric configurations using ICEM CFD, employing manual multi-blocking throughout the computational domain. Representative mesh topologies for Geom1 and Geom2 are illustrated in Figs. 10 and 11, respectively. An O-grid mesh structure was applied around both cylindrical and fan-shaped cooling holes to improve mesh quality and resolution near the cooling holes region. A highly refined mesh was utilized in the near-wall regions to accurately capture steep gradients in momentum and thermal boundary layers. Special attention was given to maintain the non-dimensional wall distance (
) close to 1, which is essential for resolving near-wall turbulence and ensuring reliable heat transfer predictions.
Fig. 10.
Geom1 mesh view (a) Computational domain, (b) Top view of O-Grid mesh.
Fig. 11.
Geom2 mesh view (a) Computational domain, (b) Side view, (c) Top view of O-Grid mesh.
The GIT was performed under high-speed flow conditions with a mainstream Mach number (
) and a blowing ratio (
). The RMSE was calculated by comparing the results obtained from coarser meshes against those from the finest mesh, serving as the reference. The RMSE is computed using the following expression:
 |
24 |
Here, 
and 
represent the film cooling effectiveness values obtained from the fine and coarse mesh solutions, respectively, at the 
streamwise location downstream of the cooling hole. The parameter 
represents the total number of streamwise data points considered along the centerline direction. The RMSE metric was computed separately for both the centerline film cooling effectiveness (
) and the laterally averaged effectiveness (
), providing a quantitative measure of mesh sensitivity.
Figure 12 presents the comparison of both centerline (
) and laterally averaged (
) film cooling effectiveness for different grid resolutions in the Geom2 configuration. As the mesh is refined from coarser to finer grids, a clear convergence trend is observed in both 
and 
plots, as shown in Fig. 12a and 12b, respectively.
Fig. 12.
Comparison of centerline (
) and laterally averaged (
) film cooling effectiveness for Geom2 with different grids.
Table 6 summarizes the RMSE values for both 
and 
, calculated with respect to the finest mesh (5.29 million cells) as the reference. The grid with 4.11 million cells exhibits very low RMSE values, 0.23% for 
and 0.52% for 
, confirming that it is sufficiently refined for accurate simulation results. Hence, the grid with 4.11 million cells was selected for all subsequent analyses to optimize computational efficiency without compromising accuracy.
Table 6.
RMSE values for Geom2.
| Grids | RMSE ( ) |
RMSE ( ) |
|---|---|---|
| 2.78 Million | 0.015352 | 0.023816 |
| 3.09 Million | 0.034511 | 0.012728 |
| 4.11 Million | 0.002324764 | 0.00522319 |
| 5.29 Million | – | – |
Based on the grid independence studies performed for Geom2, a consistent edge sizing strategy was adopted for the remaining geometries (Geom1, Geom3, Geom4, and Geom5) using multiblock structured meshing in ICEM CFD. This ensured comparable mesh resolution and solution accuracy across all configurations. The total number of mesh elements used for each geometric model is summarized in Table 7.
Table 7.
Meshing details.
| Geometrical models | Grid count (approx.) |
|---|---|
| Geom1 | 3.85 Million |
| Geom2 | 4.11 Million |
| Geom3 | 4.13 Million |
| Geom4 | 4.71 Million |
| Geom5 | 4.00 Million |
Results and discussion
This section presents a comprehensive analysis of the film cooling performance for all investigated geometrical models, highlighting the effects of mainstream Mach number and blowing ratio. The discussion is structured into three main subsections: (i) comparison of geometrical models, (ii) influence of mainstream inlet Mach number, and (iii) influence of blowing ratio. Each subsection includes both quantitative and qualitative assessments, based on centerline and laterally averaged film cooling effectiveness, as well as temperature contour visualizations.
Comparison of geometrical models
This subsection presents a comparative analysis of the five geometrical models under low Mach number (
) and low blowing ratio (
). The centerline effectiveness (
) and laterally averaged (
) effectiveness are evaluated downstream of the second row along this central axis, as described in Section Non-Dimensional Parameters in Film Cooling Study. Since the Geom5 model employs an inline cylindrical hole configuration and lacks a coolant hole at the centerline position, it is excluded from the centerline (
) effectiveness comparison. However, all five models are included in the comparison of laterally averaged (
) effectiveness.
Figure 13 illustrates the variation of centerline (
) effectiveness and laterally averaged (
) effectiveness along the streamwise direction for 
and 
. From the centerline effectiveness plot in Fig. 13a, it is evident that Geom1 and Geom2 yield higher 
compared to Geom3 and Geom4. This enhancement is primarily attributed to the presence of cylindrical holes at the centerline position in the first row of Geom1 and Geom2, which promotes a well-defined coolant jet in the central region.
Fig. 13.
Centerline (
) and laterally averaged (
) film cooling effectiveness comparison with different geometrical models at low Mach number (
) and low blowing ratio (
).
In contrast, Geom3 and Geom4 employ fan-shaped holes at the centerline in the first row (No hole at the center line in the second row). While fan-shaped holes offer improved lateral coolant dispersion, they tend to weaken the centerline streak by promoting wider jet spreading and forming two distinct side-by-side coolant streaks. As a result, both Geom3 and Geom4 exhibit lower 
compared to Geom1 and Geom2.
When comparing Geom1 and Geom2, only a slight variation in 
is observed, with Geom2 demonstrating marginally better performance. The lateral spreading of the coolant jets from the fan-shaped holes in the second row of Geom2 may contribute to the increased cooling effectiveness at the centerline. Similarly, a slight improvement in 
is observed for Geom4 compared to Geom3. This could be again due to the lateral spreading of jets by the two fan-shaped holes in the second row in Geom4.
Figure 13b compares the laterally averaged film cooling effectiveness (
) along the streamwise direction for all five geometrical models at 
and 
. It is evident that geometries incorporating fan-shaped holes (Geom2, Geom3, and Geom4) outperform those using cylindrical holes in terms of lateral coolant coverage and overall effectiveness. The highest 
is observed for Geom4, which benefits from enhanced lateral coolant spreading due to the presence of fan-shaped holes in both rows. Fan-shaped holes are known to produce a diffuser-like effect that enhances lateral dispersion of the coolant jet, resulting in broader surface coverage and improved thermal protection.
It can be observed from Fig. 13 that, for Geom4, the centerline effectiveness is considerably lower than the laterally averaged effectiveness. It should be noted that the centerline effectiveness is evaluated along the centerline (
) in the downstream region of the second-row holes. Since no coolant jet streak is concentrated along this centerline (unlike in the case of cylindrical hole geometries), the centerline effectiveness remains low. In contrast, the laterally averaged effectiveness is obtained by averaging across the entire span (at a given streamwise location), where the pronounced lateral dispersion of coolant jets from all the fan-shaped holes in Geom4 results in a higher laterally averaged effectiveness.
On the other hand, Geom5 exhibits the lowest 
among all configurations. The reduced performance is attributed to the inline arrangement of cylindrical holes, which limits lateral spreading. This behavior leads to non-uniform coolant coverage across the span and reduced film cooling effectiveness. This effect is clearly evident in the temperature contour and coolant jet isosurface shown in Fig. 14. The coolant jet isosurface is generated based on the average temperature of the mainstream and the coolant jet. The streamline plot in Fig. 14 also reveals the formation of counter-rotating vortex pairs (CRVPs), which is a characteristic feature of film cooling flow with cylindrical holes. These vortices form due to the interaction between the high-momentum mainstream flow and the shear layer emerging from the coolant jet hole. The CRVPs generate a strong upward flow that lifts the coolant layer away from the flat plate surface, a phenomenon commonly referred to as jet lift-off. This jet lift-off is clearly evident in the coolant isosurface shown in Fig. 14. Furthermore, the rotational motion of the CRVPs enhances the entrainment of hot mainstream gases into the core of the coolant jet, which limits lateral spreading, weakens surface coverage, and ultimately reduces the laterally averaged film cooling effectiveness.
Fig. 14.
Temperature contours, coolant isosurfaces, and surface streamlines at 
in the YZ plane for Geom5, at Mach number 
and blowing ratio 
.
The average film cooling effectiveness 
in the downstream region of the cooling holes for each geometry is presented in Table 8. Both Geom1 and Geom5 use cylindrical holes in both rows, but Geom1 adopts a staggered pattern while Geom5 uses an inline arrangement. Geom1 provides approximately 84% higher effectiveness compared to Geom5. The superior performance of Geom1 can be attributed to the staggered hole layout, which enhances the coverage of the coolant by forming four distinct coolant streaks downstream. In contrast, the inline configuration in Geom5 results in the formation of only two dominant coolant layers aligned along the hole axes, which limits the surface coverage and reduces overall effectiveness.
Table 8.
Average film cooling effectiveness (
) at 
and 
.
| Geometries | Average film cooling effectiveness ( ) |
|---|---|
| Geom1 | 0.2027 |
| Geom2 | 0.3852 |
| Geom3 | 0.3187 |
| Geom4 | 0.4840 |
| Geom5 | 0.1099 |
The remaining three configurations, Geom2, Geom3, and Geom4, all utilize staggered patterns but differ in hole types between the two rows. Geom4, which features fan-shaped holes in both rows, exhibits the highest average effectiveness of 0.4840, yielding a 139% increase compared to Geom1. The improvement in cooling performance for Geom2, Geom3, and Geom4 over Geom1 highlights the benefit of using fan-shaped holes in a staggered arrangement.
Since Geom5, which employs an inline arrangement of cylindrical holes, exhibits the lowest film cooling effectiveness among all configurations, it is excluded from the subsequent results and discussion. The following analysis, therefore, focuses only on the four staggered configurations, Geom1 to Geom4.
Variation of mainstream inlet mach number
This subsection presents a detailed analysis of the influence of mainstream inlet Mach number on the film cooling performance for all staggered configurations, evaluated separately at each blowing ratio.
For low blowing ratio (
)
Figure 15 illustrates the effect of mainstream Mach number variation on centerline effectiveness (
) and laterally averaged (
) effectiveness for all staggered cooling hole geometries at a low blowing ratio condition (
). In the centerline effectiveness plot shown in Fig. 15a, it is observed that for all configurations, increasing the Mach number from 
to 
leads to an improvement in 
, indicating enhanced cooling performance along the centerline. Similarly, in the 
plot presented in Fig. 15b, a slight increase in laterally averaged effectiveness is observed across all geometries with increasing Mach number, suggesting modest improvement in spanwise coolant coverage under high-speed flow conditions.
Fig. 15.
Centerline (
) and laterally averaged (
) film cooling effectiveness of geometries with variation in mainstream inlet Mach number at low blowing ratio (
).
To quantitatively assess the influence of Mach number on film cooling performance, the average film cooling effectiveness (
) for all four staggered geometries at a fixed low blowing ratio (
) is presented in Table 9. Across all configurations, a consistent improvement in 
is observed with increasing Mach number, indicating that higher mainstream momentum enhances the coolant jet’s ability to stay attached to the wall and spread downstream more effectively.
Table 9.
Average film cooling effectiveness (
) for 
and 
at 
.
| Geometry |
 ( ) |
 ( ) |
% Change |
|---|---|---|---|
| Geom1 | 0.2027 | 0.2141 | 5.62% 
|
| Geom2 | 0.3852 | 0.4072 | 5.71% 
|
| Geom3 | 0.3187 | 0.3366 | 5.62% 
|
| Geom4 | 0.4840 | 0.5189 | 7.21% 
|
This observed enhancement in film cooling effectiveness can be attributed to the fixed blowing ratio (
) and density ratio (
), where only the mainstream inlet Mach number is varied from 
to 
. To maintain a constant 
and 
while increasing the mainstream velocity (
) due to a higher Mach number, the coolant velocity (
) must also increase proportionally. This is achieved by increasing the coolant inlet total pressure. Consequently, both mainstream and coolant jet momentum increase at higher Mach numbers, despite the unchanged blowing ratio. This could be due to the domination of mainstream flow momentum over the coolant jet momentum (thinner boundary layer expected at higher 
), which promotes coolant jet bending and attachment to the wall surface.
.
For medium blowing ratio (
)
Figure 16 illustrates the effect of mainstream Mach number on all staggered geometries by presenting the centerline (
) and laterally averaged (
) film cooling effectiveness plots at a medium blowing ratio (
). As seen in Fig. 16a and 16b, both 
and 
exhibit a slight increase as the Mach number rises from 
to 
, consistent with the trend observed for the low blowing ratio (
) case discussed in Subsection 5.2.1. This marginal improvement in cooling performance is attributed to the increase in both mainstream and coolant jet momentum needed to maintain the constant blowing ratio (
) and density ratio (
) at higher freestream velocities.
Fig. 16.
Centerline (
) and laterally averaged (
) film cooling effectiveness comparison for different mainstream inlet Mach number at medium blowing ratio (
).
Figures 17 and 18 present the temperature contours, coolant isosurfaces, and streamlines plot in the YZ plane at 
for Geom4 at 
and 
, respectively, with a blowing ratio of 
. In both Mach number cases, a wide and uniform distribution of coolant across the downstream surface is observed, with higher coolant concentration near the injection region. As the coolant travels downstream, its footprint gradually diminishes due to diffusion and mixing with the mainstream, as evident in the temperature contour plots.
Fig. 17.
Temperature contours, coolant isosurfaces, and surface streamlines at 
in the YZ plane for Geom4, at Mach number 
and blowing ratio 
.
Fig. 18.
Temperature contours, coolant isosurfaces, and surface streamlines at 
in the YZ plane for Geom4, at Mach number 
and blowing ratio 
.
The coolant isosurfaces in both figures remain well attached to the plate surface, indicating effective film adherence. In the case of the fan-shaped hole, the formation of additional anti-counter-rotating vortex pairs (ACRVPs) is observed, as shown in the zoomed streamline views in the same Figs. 17 and 18. The presence of these ACRVPs between the CRVPs helps suppress coolant jet lift-off and enhances surface coverage. A slight improvement in cooling performance is observed at the higher Mach number case (
) compared to the lower Mach number case (
), which is reflected in the relatively lower surface temperatures shown in the temperature contours.
To quantitatively evaluate the effect of mainstream Mach number on film cooling performance, Table 10 summarizes the average film cooling effectiveness (
) for all four staggered hole configurations at a fixed medium blowing ratio of 
. The results are compared for two mainstream flow conditions: 
and 
.
Table 10.
Average film cooling effectiveness (
) for 
and 
at 
.
| Geometry |
 ( ) |
 ( ) |
% Change |
|---|---|---|---|
| Geom1 | 0.1214 | 0.1312 | 8.07% 
|
| Geom2 | 0.4125 | 0.4354 | 5.55% 
|
| Geom3 | 0.3537 | 0.3764 | 6.42% 
|
| Geom4 | 0.5711 | 0.6112 | 7.02% 
|
From the data, it is evident that increasing the Mach number from 
to 
leads to a consistent improvement in average film cooling effectiveness across all geometries. Geom4, featuring fan-shaped holes in both cooling rows, shows a 7.02% increase, achieving the highest absolute values of 
among all configurations.
For high blowing ratio (
)
Figure 19 illustrates the influence of mainstream Mach number on film cooling effectiveness for all staggered geometries at a high blowing ratio of 
. At this elevated blowing ratio, Geom1 with cylindrical holes in both cooling rows shows a noticeable improvement in both 
and 
with rising Mach number. Geom2, which features cylindrical holes in the first cooling row, shows a moderate increase in centerline effectiveness with increasing Mach number. This enhancement is attributed to the higher mainstream and coolant jet momentum, which slightly improves surface coverage.
Fig. 19.
Centerline (
) and laterally averaged (
) film cooling effectiveness comparison for different mainstream inlet Mach number at high blowing ratio (
).
In contrast, configurations featuring fan-shaped holes along the centerline (Geom3 and Geom4) exhibit a noticeable decline in centerline effectiveness (
) at higher Mach numbers. Likewise, the laterally averaged effectiveness (
) also decreases for all configurations that include at least one row of fan-shaped holes (Geom2, Geom3, and Geom4). This performance deterioration is primarily attributed to the separation of the coolant shear layer within the diffuser section of the fan-shaped holes under high Mach number and high blowing ratio conditions. Such separation causes the coolant jet to deflect, reducing its lateral spreading and resulting in a narrower and less uniform coolant film coverage on the surface. This deflection can be visualized in the temperature contours for Geom2, Geom3, and Geom4, shown in Figs. 20, 21, and 22, respectively. Additionally, to clarify the convergence behavior, the residual history is shown in Fig. 23 for 
and 
case for Geom3. It can be observed that for the second order accurate schemes, all the residual values are close to 
or below. Similar phenomena of coolant flow separation, asymmetric jet deflection and a decline in cooling efficiency under high blowing ratio conditions are reported in an experimental study by Jo et al.33 and in a large eddy simulation (LES) based study34. This phenomena is also observed in RANS-based simulation studies by Saumweber et al. 35 and Wei et al. 36.
Fig. 20.
Temperature contours, coolant isosurfaces, and surface streamlines at 
in the YZ plane for Geom2, at Mach number 
and blowing ratio 
.
Fig. 21.
Temperature contours, coolant isosurfaces, and surface streamlines at 
in the YZ plane for Geom3, at Mach number 
and blowing ratio 
.
Fig. 22.
Temperature contours, coolant isosurfaces, and surface streamlines at 
in the YZ plane for Geom4, at Mach number 
and blowing ratio 
.
Fig. 23.
Residual convergence history for Geom3 at Mach number 
and blowing ratio 
.
At a high blowing ratio (
), maintaining the required coolant mass flux rate necessitates an increase in the total coolant pressure in the plenum, leading to a higher coolant velocity in the coolant channel compared to the low blowing ratio case (
). This elevated velocity accompanied by the adverse pressure gradient causes flow separation within the diffuser section of the fan-shaped holes, which in turn contributes to lateral jet deflection and reduced film cooling effectiveness. The flow separation inside the fan-shaped hole can be visualized using Fig. 24. This figure compares the streamlines within the fan-shaped coolant holes of Geom4, colored by temperature contour, for 
and 
at 
. The distortion of the streamlines near the wall, the recirculation bubble and the surrounding flow separation region is clearly visible in Fig. 24(b). The temperature contours show high temperature in the separation region due to the recirculation of the hot mainstream gases.
Fig. 24.
Comparison of streamlines (colored by temperature contour), in the fan-shaped holes of Geom4 at 
(a) 
(b) 
.
It is worth noting that the flow may not attain a perfectly steady state under high Mach number and high blowing ratio conditions. To examine this behavior, a preliminary unsteady RANS (URANS) simulation was conducted for Geom 4 at 
and 
, using the first-order steady-state RANS solution as the initial field. The URANS results revealed that the coolant flow through the fan-shaped hole exhibits unsteady oscillations under these conditions, leading to time-dependent variations in the coolant jet trajectory. Representative instantaneous temperature contours at arbitrary time instants are shown in Fig. 25, which clearly demonstrate the coolant jet oscillations and flow separation within the fan-shaped holes. Nevertheless, both RANS and URANS simulations consistently predict flow separation inside the fan-shaped hole and asymmetric jet deflection, confirming the capability of RANS simulations to capture the essential flow features. However, for a more accurate characterization of film-cooling behavior at high Mach numbers and high blowing ratios, advanced unsteady techniques such as URANS or LES are more appropriate.
Fig. 25.
Temperature contours at arbitrary time instants for 
and 
for Geom4: (a) 
, (b) 
, (c) 
, (d) 
.
The presence of asymmetric coolant ejection affects the vortex dynamics downstream. In particular, the asymmetry in the coolant ejection suppresses the formation of ACRVPs, which typically act to weaken the strength of the main CRVPs and help keep the coolant attached to the surface. In the absence or weakening of ACRVPs, the CRVPs become stronger and induce more upward motion, leading to increased coolant lift-off. This effect is clearly visible in the streamline plot shown in Fig. 20, 21, and 22. The enhanced lift-off results in partial detachment of the coolant film from the surface, thereby reducing its effectiveness and increasing the surface temperature.
The coolant jet emerging from the fan-shaped holes near the centerline exhibits a lateral deflection in the same direction as observed in Geom2 (Fig. 20), forming a narrow and skewed coolant streak. However, the fan-shaped holes located near the periodic boundary exhibit lateral deflection in the opposite direction, as observed in Geom3 and Geom4 (Figs. 21 and 22). This reversal in deflection is caused by the altered coolant shear layer separation within the fan-shaped hole adjacent to the periodic boundary. This altered shear layer development at the periodic face led to a mirrored coolant deflection compared to the centerline hole, resulting in an overall skewed and non-uniform coolant coverage.
The temperature contours on the plate surface in Fig. 20 shows very little variation in temperature in the immediate downstream of the cylindrical coolant holes. This is due to the coolant jet lift-off from cylindrical holes at high blowing ratio (
). This lift-off reduces the coolant footprint on the plate downstream surface. The ejection of the coolant jet from cylindrical holes can be visualized from the coolant isosurface plot shown in the same figure.
To quantitatively assess the influence of mainstream Mach number on film cooling effectiveness, Table 11 presents the average film cooling effectiveness (
) for all four staggered cooling hole configurations at a fixed high blowing ratio of 
, comparing two mainstream flow conditions: 
and 
. The results indicate that Geom1, which features cylindrical holes in both rows, exhibits a positive response to the increased Mach number, with 
improving by approximately 27.44%. This improvement can be attributed to enhanced coolant jet attachment and better spanwise coverage under high mainstream velocity conditions. However, it is important to note that despite this relative improvement, the absolute value of 
for Geom1 remains significantly lower than that of the other configurations, highlighting its limited cooling effectiveness overall.
Table 11.
Average film cooling effectiveness (
) for 
and 
at 
.
| Geometry |
 ( ) |
 ( ) |
% Change |
|---|---|---|---|
| Geom1 | 0.0718 | 0.0915 | 27.44% 
|
| Geom2 | 0.4296 | 0.2139 | 50.21% 
|
| Geom3 | 0.4031 | 0.1781 | 55.82% 
|
| Geom4 | 0.6168 | 0.2484 | 59.73% 
|
In contrast, all configurations incorporating at least one row of fan-shaped holes (Geom2, Geom3, and Geom4) experience a significant decline in average film cooling effectiveness (
) with increasing Mach number. Specifically, Geom2, Geom3, and Geom4 show reductions of 50.21%, 55.82%, and 59.73%, respectively. This quantitative trend clearly highlights the significant degradation in the performance of fan-shaped holes under such high Mach number and high blowing ratio conditions.
Variation of blowing ratios
This subsection investigates the influence of varying blowing ratios on all four staggered cooling hole configurations (Geom1, Geom2, Geom3, and Geom4) for both mainstream Mach numbers, 
and 
.
For Geom1 model
Figure 26 illustrates the impact of varying the blowing ratio on film cooling performance for the Geom1 configuration at 
and 
. The results clearly demonstrate a consistent decline in both 
and 
with increasing blowing ratio for both Mach numbers. This reduction in cooling effectiveness is attributed to the increased momentum of the coolant jet as the blowing ratio rises. At higher blowing ratios, the higher coolant momentum causes the coolant jet to lift off from the surface. This lift-off effect weakens the protective coolant film layer over the surface, thereby reducing both the 
and 
.
Fig. 26.
Centerline (
) and laterally averaged (
) film cooling effectiveness comparison with variation of blowing ratio for Geom1 model.
Table 12 reports the variation in average film cooling effectiveness (
) for blowing ratios 
, 
, and 
at both Mach numbers, 
and 
. As the blowing ratio is increased from 1.0 to 2.0 a reduction of 64.58% and 57.24% are observed for 
and 
cases, respectively. These reductions confirm a strong inverse relationship between blowing ratio and cooling effectiveness in the Geom1 configuration. While the general trend holds for both Mach numbers, the degradation in effectiveness is slightly more severe at the lower Mach number, highlighting the influence of mainstream velocity on coolant jet trajectory and surface attachment.
Table 12.
Average film cooling effectiveness (
) for Geom1 with varying blowing ratios at both 
and 
.
| Blowing ratios |
 ( ) |
 ( ) |
|---|---|---|
 |
0.2027 | 0.2141 |
 |
0.1214 | 0.1312 |
 |
0.0718 | 0.0915 |
For Geom2 model
Figure 27 illustrates the effect of varying blowing ratio on the film cooling performance of Geom2 configuration at Mach numbers 
and 
. In the 
plots, a clear decline in centerline effectiveness is observed as the blowing ratio increases for both Mach numbers. Geom2 employs a two-row staggered configuration with cylindrical holes in the first row and fan-shaped holes in the second row. Notably, the first row features a cylindrical hole located along the centerline, whereas the second row does not include any centerline holes. As the blowing ratio increases, the momentum of coolant issuing from the cylindrical holes rises, promoting jet lift-off and reducing wall attachment. This leads to a decrease in 
along the centerline.
Fig. 27.
Centerline (
) and laterally averaged (
) film cooling effectiveness comparison with variation of blowing ratio for Geom2 model.
In contrast, the 
plots show a different trend. For 
with increasing blowing ratio, 
slightly decreases in the immediate downstream region of the coolant holes but increases significantly in the far downstream region. This behavior is attributed to the fan-shaped holes in the second row, which enhance the lateral spread of the coolant, especially at higher blowing ratios. The initial drop in 
is due to the strong jet lift-off from the cylindrical holes, which weakens film coverage near the centerline. However, further downstream, the wide spreading of coolant from fan-shaped holes compensates for this, improving lateral coverage. For 
, it can be observed that increase in the blowing ratio from 1.5 to 2.0 has resulted in significant drop in 
, which is due to the deflection of the coolant jet as explained in Subsection 5.2.3.
Table 13 summarizes the variation in average film cooling effectiveness (
) for Geom2 for three blowing ratios (
, 
, and 
) at Mach numbers 
and 
. At 
, a steady increase in 
is observed with rising blowing ratio due to the enhanced coolant coverage provided by the fan-shaped holes used in Geom2.
Table 13.
Average film cooling effectiveness (
) for Geom2 with varying blowing ratios at both 
and 
.
| Blowing ratios |
 ( ) |
 ( ) |
|---|---|---|
 |
0.3852 | 0.4072 |
 |
0.4125 | 0.4354 |
 |
0.4296 | 0.2139 |
However, at 
, although the average film cooling effectiveness 
initially increases from 0.4072 at 
to 0.4354 at 
, it subsequently drops sharply to 0.2139 at 
. This significant decline in 
is expected because of the drop in 
.
For Geom3 model
Effect of blowing ratio on 
and 
at 
and 
for Geom3 is shown in Fig. 28. Unlike the Geom2 model, a moderate enhancement in 
can be observed with increasing blowing ratio, for 
in Fig. 28a. This improvement is primarily attributed to the presence of a fan-shaped hole located along the centerline in the first cooling row of the Geom3 configuration. At higher blowing ratios, the increased coolant momentum enhances lateral spreading without causing jet lift-off, unlike the behavior typically observed with cylindrical holes. This leads to improved surface coverage and sustained film cooling effectiveness in the downstream region.
Fig. 28.
Centerline (
) and laterally averaged (
) film cooling effectiveness comparison with variation of blowing ratio for Geom3 model.
For 
, a similar trend is observed as the blowing ratio increases from 
to 
, where an increase in 
is evident in Fig. 28a. However, at 
, a substantial decline in 
occurred due to coolant jet asymmetric deflection. Regarding the laterally averaged effectiveness (
), a consistent enhancement is observed with increasing blowing ratios for both Mach numbers, except for the case of 
and 
. In this particular operating condition, a notable reduction in 
is recorded.
Table 14 presents the average film cooling effectiveness (
) for blowing ratios 
, 
, and 
at Mach numbers 
and 
for Geom3. At 
, 
increases with blowing ratio, from 0.3187 at 
to 0.3537 at 
, and further to 0.4031 at 
. This progressive rise indicates that higher coolant momentum at low mainstream Mach numbers improves lateral coolant spreading and surface coverage, enhancing film cooling effectiveness for Geom3.
Table 14.
Average film cooling effectiveness (
) for Geom3 with varying blowing ratios at both 
and 
.
| Blowing ratios |
 ( ) |
 ( ) |
|---|---|---|
 |
0.3187 | 0.3366 |
 |
0.3537 | 0.3764 |
 |
0.4031 | 0.1781 |
Whereas, for 
, a moderate increase in 
is observed from 
to 
, and a significant drop in 
from 
to 
. Overall, it can be observed that 
values of Geom3 are lesser than Geom2 for all the cases, indicating that Geom2 configuration is preferable compared to Geom3.
For Geom4 model
Variation of 
and 
along the streamwise centerline for different blowing ratios at 
and 
for Geom4 model are shown in Fig. 29. For 
, an increase in blowing ratio consistently enhances both 
and 
, indicating improved coolant coverage and attachment. A similar trend is observed for 
when the blowing ratio increases from 
to 
, resulting in a noticeable improvement in both metrics. However, a further increase to 
at 
led to a sharp decline in both 
and 
as expected due to asymmetric coolant jet behavior.
Fig. 29.
Centerline (
) and laterally averaged (
) film cooling effectiveness comparison with variation of blowing ratio for Geom4 model.
Table 15 presents the average film cooling effectiveness (
) for blowing ratios 
, 
, and 
at Mach numbers 
and 
for Geom4. At 
, 
increases steadily with rising blowing ratio, indicating enhanced cooling performance due to improved lateral coolant spread and surface coverage.
Table 15.
Average film cooling effectiveness (
) for Geom4 with varying blowing ratios at both 
and 
.
| Blowing ratios |
 ( ) |
 ( ) |
|---|---|---|
 |
0.4840 | 0.5189 |
 |
0.5711 | 0.6112 |
 |
0.6168 | 0.2484 |
At 
, a similar enhancement is observed initially, with 
rising from 0.5189 at 
to 0.6112 at 
. However, at 
, 
drops sharply by nearly 60%, highlighting the detrimental impact of excessive coolant jet momentum at high mainstream Mach number, which likely causes coolant jet deflection and asymmetric flow behavior downstream.
These results highlight the importance of optimizing the blowing ratio. A moderate increase in 
can improve the film cooling performance, while excessive values, especially at high Mach numbers, may lead to coolant jet lift-off and asymmetry in the coolant layer formation for fan-shaped holes, thereby reducing surface coverage and overall cooling effectiveness.
Conclusion
This study comprehensively investigated the film cooling effectiveness of five multi-row cooling-hole configurations (four staggered: Geom1 - Geom4, and one inline: Geom5) under two mainstream Mach numbers (
and 
) and three blowing ratios (
, 
, and 
), with a constant density ratio of approximately 
. Performance was evaluated using centerline (
), laterally averaged (
), and area-averaged (
) film cooling effectiveness.
Across the operating conditions, configurations incorporating fan-shaped holes in staggered layouts, particularly Geom4 ( fan-shaped holes in both rows) provided the highest 
and 
owing to stronger lateral dispersion, reduced jet momentum at the exit, and the presence of ACRVPs that weakened CRVPs induced lift-off. Conversely, purely cylindrical-hole configurations (Geom1, Geom5) suffered from narrow coolant footprints, stronger CRVPs, and pronounced lift-off, yielding the lowest 
and 
. Overall, the film cooling effectiveness across geometries followed the order: Geom4 > Geom2 > Geom3 > Geom1 > Geom5, under most operating conditions.
For low and medium blowing ratios (
), increasing the mainstream Mach number from 
to 
produced a modest, but consistent rise in 
, 
, and 
for all staggered configurations. However, at high blowing ratio (
), fan-shaped configurations (Geom2–Geom4) experienced large drops in 
when Mach number increased to 
. This is driven by shear-layer separation inside the diffuser, asymmetric jet deflection, suppression of ACRVPs, and consequent loss of lateral coverage. Cylindrical Geom1, although still poor in absolute terms, showed a relative gain with Mach number at 
.
For Geom1 (cylindrical-holes), 
, 
, and 
monotonically decreased with increasing 
at both the mainstream Mach numbers due to momentum-driven lift-off. For Geom2 and Geom3 (mixed cylindrical and fan-shaped rows), and Geom4 (both rows fan-shaped), increasing 
from 1.0 to 1.5 generally improved effectiveness at both Mach numbers.
Overall, the results demonstrate that excessive blowing ratio at high Mach number severely penalizes fan-shaped design cooling efficiency. These findings underscore that optimal film cooling performance in compressible regimes requires the joint optimization of hole geometry (favoring fan-shaped exits), row arrangement (staggered over inline), and operating conditions (avoiding excessively high 
at high 
). The proposed total-pressure specification for the coolant, tied to 
, 
, and 
, provides a practical pathway to accurately impose boundary conditions in compressible film cooling simulations. The present study is limited to the flat plate geometry. Extending the study to real gas turbine blade under rotating conditions can give further insights.
Acknowledgements
The authors gratefully acknowledge FOSSEE, IIT Bombay, for providing manpower support to this work. The High Performance Computing Cluster (HPCC) ‘Magus’ at Shiv Nadar Institution of Eminence significantly contributed to the research results presented in this paper.
List of symbols
- Ma
Mainstream inlet Mach number
- DR
Density ratio,
Recovery temperature
Total coolant inlet temperature
Turbulence intensity at mainstream inlet, %
Total pressure at coolant inlet
- P
Static pressure
Density
Pressure loss
Centerline film cooling effectiveness
Average film cooling effectiveness
- D
Cooling hole diameter
Mass flux rate at mainstream inlet
Mass flux rate at coolant inlet in crossflow
- c
Subscript denoting coolant flow
- CFD
Computational fluid dynamics
- RMSE
Root mean square error
- CRVPs
Counter-rotating vortex pairs
Mean flow velocity
- RANS
Reynolds-Averaged Navier-Stokes
- BR
Blowing ratio,
- TR
Temperature ratio
Adiabatic wall temperature
Total mainstream inlet temperature
Turbulence intensity at coolant inlet, %
Total pressure at mainstream inlet
Heat capacity ratio
Total pressure at coolant crossflow
Film cooling effectiveness
Laterally averaged film cooling effectiveness
- L
Cooling hole length
Mass flux rate
Average mass flux rate at mainstream inlet
Average mass flux rate at coolant inlet in crossflow
- m
Subscript denoting mainstream flow
- GIT
Grid independence test
Number of data points
- ACRVPs
Anti-counter-rotating vortex pairs
- SAS
Scale adaptive simulation
- URANS
Unsteady Reynolds-Averaged Navier-Stokes
Author contributions
Devendra Pratap Singh and Veasna Mom equally contributed to the conception, simulation work, analysis and writing. Sathi Rajesh Reddy contributed to the conception, reviewing and approving the final manuscript.
Funding
Open access funding provided by Shiv Nadar University.
Data availability
The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.
Declarations
Competing interests
The authors declare no competing interests.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Devendra Pratap Singh and Veasna Mom contributed equally to this work.
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Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Data Availability Statement
The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.