1 Introduction

Modern aero-engines and heavy-duty gas turbines operate at a high turbine inlet temperature, which exceeds the melting temperature of the metal, to ensure a high specific thrust or power output and improved efficiency. Thus, efficient cooling technology should cool the turbine nozzle vanes and rotor blades. Film-cooling and impingement cooling have been proven as potential solutions to conquer the local high heat loading. Hence, film-cooling and impingement cooling are commonly applied in the leading-edge region suffering from the upstream hot gas. This composite technology is adopted for the pressure and suction sides of the advanced double-wall cooling design. The maximum cooling performance for the cooling technique is always the goal with equal or less coolant consumption. The high cooling performance of composite cooling depends on the improvements of film-cooling and impingement cooling separately and their proper combination.

The achievements in film cooling are mainly attributed to the in-depth understating of flow mechanisms and the innovation of hole shapes. Mayank et al. [1] employed the Large Eddy Simulation (LES) method to investigate the mechanism of vortex structures on film cooling effectiveness in a single inclined cylindrical film cooling hole and arrived at the following conclusions: when the blowing ratio M > 1.0, significant degradation in film cooling effectiveness occurs due to the adverse impact of the counter-rotating vortex pair (CRVP). Hence, numerous innovative film-cooling hole shapes [2,3,4,5,6,7] were proposed to weaken or eliminate the CRVP and thus improve the cooling effectiveness. However, the manufacturing difficulty is aggrandized for these geometrically complex film-cooling holes. In the past several decades, the limited application in practice has been the fan-shaped film cooling hole, considering the tradeoff between cooling performance and manufacturing cost.

Unlike the complex film-cooling hole, Bunker [8] proposed an alternative way to enhance the cooling effectiveness without modifying the simple cylindrical hole by spraying the thermal barrier coating (TBC). The successive trench, or discrete crater, can be formed depending on the cylindrical film-cooling hole's mask shape [9]. The comparative experiment from Dorrington et al. [10] revealed that the successive trenched hole gives a better cooling performance than the discrete cratered hole, and both are superior to the simple cylindrical hole. Tran al. [11] worried that the successive trench could inevitably affect the blade integrity and further pointed out that the discrete cratered hole should be more noteworthy. However, the slight improvement in the cooling performance for the cratered hole limits its application over the cylindrical hole, as shown in the experimental test work by Lu et al. [12]. The embedded crater shape affects the coolant flow trajectory and, thus, cooling effectiveness. Khalatov et al. proposed the hemispherical and triangular cratered film-cooling holes [13]. Their superiorities over the circular cratered hole were validated at M = 0.5–2.0 by performing the computational fluid dynamics (CFD) simulations. Subsequently, Khalatov et al. [14] investigated the influence of the triangular crater's leading edge (straight or concave curved). Their numerical results showed that the concave curved leading edge positively impacts the cooling performance improvement. Kalghatgi and Acharya [15, 16] invented a similar cratered film-cooling hole design with a V-shaped protrusion. They revealed the key role of the V-shaped protrusion is in cooling performance augmentation. The influence of the crater height was numerically studied by Zhang et al. [17]. They found that the largest crater height of 0.7D (D is the cylindrical hole diameter) gives the highest cooling effectiveness. An et al. [18] proposed a similar contoured cratered hole design but featured lateral and streamwise expansions. Their experimental data revealed that the contoured cratered hole yields a significantly improved cooling effectiveness compared to the cylindrical hole, especially at larger blowing ratios M > 1.0. The following numerical work in [19] showed that curved protrusion is crucial in generating the additional vortex pair with an opposite sense of rotation compared with the CRVP. Fu et al. [20] suggested the deep crater design with 1.0D height has the best cooling performance. Furthermore, Bai et al. [21] optimized the crater shape with five geometrical parameters at M = 0.5 and 1.5. The corresponding optimized cratered holes showed an increase of 17.21% and 101.96% in area-averaged cooling effectiveness compared to the benchmark cratered hole. Notably, the previous studies for the cratered film-cooling holes were carried out under adiabatic conditions, neglecting the coupling heat transfer between the solid and fluid domains. Recently, Zhang et al. [22] numerically evaluated the overall cooling effectiveness of several cratered cooling holes, considering the coupling of fluid convective heat transfer and solid heat conduction. Their conjugate heat transfer results showed that the countered cratered hole overwhelms the circular cratered holes.

Apart from the influence of the shape of the film cooling holes, the layout of the holes also plays a crucial role in determining the cooling effectiveness. Jiang et al. [23] were the first to numerically demonstrate that a double-row arrangement of film cooling holes provides better cooling performance than a single-row arrangement. Subsequently, Prakhar et al. [24] compared the cooling performance of four single-hole shapes: cylindrical, triangular, semi-cylindrical, and semi-elliptic, arranged in a staggered double-row configuration. Their findings revealed that the semi-elliptic hole shape offered the best cooling effectiveness. Furthermore, the impact of a mixed staggered layout combining semi-cylindrical and triangular film cooling holes on cooling performance was also discussed [25]. Prakhar [26] enhanced the film cooling performance downstream by introducing rectangular winglet vortex generators upstream of the film cooling holes. The research findings indicated that the cooling effect was optimized when the rectangular winglets were extended along the flow direction with an expansion angle of 15°.

In addition to the influence of the turbine's inherent structure on cooling performance, unburned fuel from the upstream combustion chamber may also interact with the oxygen-enriched film coolant jet, leading to secondary combustion and subsequently elevated local temperatures that can damage turbine blades. To address this issue, Anderson et al. [27] conducted a thorough investigation focusing on three commonly used configurations in modern turbines: normal holes, angled holes, and fan-shaped cooling holes. The results showed that the cooling performance of fan-shaped holes was most significantly affected by this phenomenon.

The heat transfer performance of the internal impingement cooling relies on the flow parameters and geometrical variables [28]. The flow parameters cover impingement Reynolds number, temperature ratio, turbulence intensity and scale, and crossflow effect. The geometrical variables include impingement hole diameter, spacing, impingement distance, shape, and row layout (inline or staggered). Haider et al. [29] revealed that the heat transfer coefficient decreases with increased impingement hole diameter at a fixed impingement Reynolds number. In their comparative study, Jordan et al. [30] concluded that the racetrack impingement hole has an improved heat transfer performance than the cylindrical hole. The recent developments in impingement cooling research and applications can be found in the review papers [31, 32]. The cooling performance of the composite cooling scheme depends on not only the individual cooling performance of film-cooling and impingement cooling but also the coupling effect between these two cooling technologies. Panda et al. [33] verified the superiority of composite cooling design over a single film-cooling scheme, featuring a high and uniform overall cooling effectiveness over the external wall. The experimental results in [34] draw a similar conclusion for composite cooling. Taslim et al. [35] focused on the influence of film-cooling extraction on impingement cooling. They found that the film-cooling extraction weakens the crossflow effect and improves the internal heat transfer performance by nearly 22%.

The experimental data in [36] proved that the impingement cooling results in an increased heat transfer performance over the internal wall and an improved overall cooling effectiveness. The effect of the impingement distance between the jet and film hole plate on the overall cooling effectiveness was investigated by Jung et al. [37]. They found that the effect is limited to within 2.7% when the impingement distance varied from 1 to 5D. Zhang [38] and Fu et al. [39] studied the influence of film-cooling holes on flat plate composite cooling. The overall cooling effectiveness for the cases with cylindrical, double-row, fan-shaped film-cooling holes at various blowing ratios M = 0.5–2.5 were compared. They concluded that the fan-shaped hole configuration precedes others. Liu et al. [40] established various decoupling models through numerical simulation methods, quantitatively analyzing the relationships between the overall cooling effectiveness and internal, bore, and film cooling. They also investigated the impact of blowing ratios, different sections, and cooling structures on these relationships. The results indicate that as the blowing ratio increases, the influence of internal impingement cooling on the overall cooling performance gradually becomes more significant.

From the above summary and discussion, it is evident that a cratered film cooling hole is a simple and effective method to enhance cooling performance. Considering the influence of solid heat transfer under gas-thermal coupling conditions, the study is brought closer to the practical application environment of combined cooling in engineering. However, most studies have focused on fan-shaped and cylindrical holes, with limited research exploring the combination of conjugate heat transfer in crater-shaped film-cooling holes and impingement cooling. In the case of film impingement composite cooling, the interaction between the flow field, internal heat transfer coefficient, and overall cooling effectiveness becomes particularly complex. To gain a more comprehensive understanding of this system, numerical simulations are necessary to investigate the influence of impingement hole diameter, impingement distance between the jet and the film cooling hole plate, and the coupling of impingement hole shape with cratered film cooling holes. Finally, comparing the conjugate heat transfer performance between cylindrical film cooling holes and two different contoured cratered film cooling holes is conducted to emphasize the superiority of cratered film cooling holes and provide a reference for practical engineering applications.

2 Physical model and numerical setup

2.1 Case description

The current study considers a simplified flat-plate model with composite film-impingement cooling extracted from the experimental rig [38], except for the film cooling hole. A low-speed wind tunnel flat testbed was used for the experiment. During the experiment, the mainstream passes through the pipeline, heater, circular stabilizing section, and circular-to-rectangular transition section in sequence, driven by the centrifugal fan, before entering the test section. Subsequently, pressure-sensitive paint is applied to obtain the adiabatic cooling effectiveness of the film-cooling flat plate—the previous experimental work [38] employed cylindrical and fan-shaped film-cooling holes. However, the current study focuses on the contoured cratered film-cooling holes originally proposed by An et al. [18]. The cylindrical film-cooling hole is also employed to provide the benchmark. Figure 1 shows the geometries of the cylindrical and contoured cratered film-cooling holes. The cylindrical hole has a diameter of D = 4mm, an inclination angle of 30° relative to the streamwise direction, and a total hole axis length of 6D. The cylindrical part of the cratered hole has the same diameter and inclination angle as the cylindrical hole. The origin is located at the exit of the cylindrical hole. The crater depth, streamwise length, and width are denoted as H, Ls, and Ws, respectively. The curved protrusion of the crater has a width of Wp and a streamwise length of Lp. Two different craters are investigated in the present study, and the dimensions are listed in Tab. 1. The shape of crater 2 was optimized under a wide range of blowing ratios, 0.5–2.5, at the adiabatic condition in [41]. The target function was set as the average cooling effectiveness in the wall region (0 ≤ X/D ≤ 30, -3 ≤ Y/D ≤ 3). A certain range was defined for the parameters Ls/D, Ws/D, Lp/Ls, Wp/Ws, and H/D. Using the Neural Network Genetic Algorithm, an optimization search was conducted at blowing ratios of 0.5–2.5 to find the most suitable geometric parameters within this range. The specific optimization approach is consistent with the methods outlined in reference [21] (Table 1).

Fig. 1
figure 1

Film cooling holes

Full size image
Table 1 Dimensions of the craters
Full size table

Figure 2 illustrates the sketch of the flat-plate model with the composite film-impingement cooling, similar to the numerical work [39]. The coolant initially enters the coolant plenum at the bottom, then accelerates through four impingement cooling holes into the impingement cavity, strikes the target wall surface, and finally covers the external wall surface via one film-cooling hole. Two solid plates are included, i.e., the impingement plate and the target plate. These solid plates, the coolant and the impinging plenum, have the same streamwise length of 45D. The heights of the mainstream channel and the coolant plenum are both 10D. The thickness values for the impingement and target plates are 1D and 3D, respectively. The total streamwise length of the mainstream channel is 80D. The distance between the origin and left boundary of the target plate is 10D.

Fig. 2
figure 2

Sketch of the flat plate model with composite cooling

Full size image

Similarly, the distance between the nearest impingement hole's origin and the circle point is 10D. The spacing between two adjacent impingement holes is 7.5D. The lateral pitches of the film cooling hole and impingement hole are both 6D. Thus, the widths of the fluid and solid domains are 6D. The region ( 0 ≤ X/D ≤ 30, − 3 ≤ Y/D ≤ 3) downstream of the film-cooling hole exit is the region of interest to evaluate the overall cooling effectiveness.

The previous work [22] for the contoured cratered film-cooling with no impingement cooling showed that the overall cooling effectiveness altered indistinctively as the blowing ratio increased since M = 1.5. Hence, the current study focuses on the case of M = 1.5 to explore the impact of impingement cooling layout. The nominal composite cooling design has an impingement hole diameter of dj = 1D, an impingement distance of tj = 2D, and a cylindrical film-cooling hole. Table 2 lists the various types of impingement cooling layouts. Only one of the variables changes when the different impingement cooling layouts are studied. The impingement cooling hole varies from 0.5D to 1.5D. Two other impingement distances of 1D and 3D are considered. Four additional impingement hole shapes, including inlet expansion of 10 and 20°, and outlet expansion of 10° and 20°, are adopted to study the influence of hole contraction or expansion. When the diameter at the impingement hole inlet or outlet enlarges, the diameter at the other side is kept at 1D.

Table 2 Variates of the impingement cooling layout
Full size table

As mentioned in the experimental work in [38], the air flows into the mainstream channel with a velocity of Vm = 25m/s and a turbulence intensity of 3.5%. The static pressure at the mainstream channel outlet is kept at 101.3kPa. The temperatures entering the mainstream channel and coolant plenum are Tm = 414K and Tc = 300K to mimic a density ratio of DR = 1.38. The coolant also flows into the coolant supply plenum with a uniform velocity Vc with adjustable values to ensure the blowing ratio. The turbulence intensity of the coolant at the plenum inlet is 1.0%. The blowing ratio M changes from 0.5 to 2.5 with an interval of 0.5. The adjustable value of Vc can be calculated with the following formula:

$$ V_{{\text{c}}} = \frac{{M \cdot V_{{\text{m}}} }}{DR} \cdot \frac{{A_{{\text{f}}} }}{{A_{{\text{c}}} }} $$
(1)

where Af is the sectional area of the film-cooling hole, and Ac is the area at the coolant plenum inlet.

The impingement and film-cooling plates are steel 4j36 with a thermal conductivity of 11W/(m·K). This results in a typical value of 0.22 for the Biot number in gas turbine engine condition, as shown in the numerical work [39].

The cooling performance is evaluated with the localηϕ, laterally-averaged ηϕl, and area-averaged overall cooling effectiveness ηϕave, expressed as follows:

$$ \eta_{\phi } = \frac{{T_{{\text{m}}} - T_{{{\text{ew}}}} }}{{T_{{\text{m}}} - T_{{\text{c}}} }} $$
(2)
$$ \eta_{{\phi {\text{l}}}} = \frac{1}{6D}\int_{ - 3D}^{3D} {\eta_{\phi } } dY $$
(3)
$$ \eta_{{\phi {\text{ave}}}} = \frac{1}{30D \times 6D}\int_{X = 0D}^{X = 30D} {\int_{Y = - 3D}^{Y = 3D} {\eta_{\phi } } dYdX} $$
(4)

where Tew is the wall surface temperature at the mainstream channel side.

The heat transfer performance of impingement cooling is evaluated with the internal heat transfer coefficient hc, which is defined as follows:

$$ h_{{\text{c}}} = \frac{q}{{T_{{{\text{iw}}}} - T_{{\text{c}}} }} $$
(5)

where q is the heat flux across the impingement plate, and Tiw is the temperature at the bottom wall of the impingement plate.

The vortex structure downstream of the film-cooling hole is evaluated quantificationally with the streamwise vortex intensity ωX, which is given by:

$$ \omega_{{\text{X}}} = \frac{{\partial V_{{\text{Z}}} }}{\partial Y} - \frac{{\partial V_{{\text{Y}}} }}{\partial Z} $$
(6)

VY and VZ are the velocity components along lateral and normal wall directions.

2.2 Numerical setup and turbulence method validation

In all the conjugate heat transfer simulations, the fluid domain consists of the coolant supply plenum, four impingement cooling holes, an impingement cavity, one film-cooling hole, and a mainstream channel. The solid domain covers the impingement plate and film-cooling plate. The interfaces between the fluid and solid domains are coupled with identical heat flux and temperature. The adiabatic and non-slip conditions are imposed at the wall surfaces sideways for the coolant supply plenum, impingement cavity, impingement plate, film-cooling plate, and the top surface for the mainstream channel. The planes Y/D =  − 3 and 3 in both fluid and domains are periodic to represent a lateral pitch of 6D.

The structured mesh is generated in the fluid domain, and the unstructured mesh in the solid domain using the ICEM CFD package, as shown in Fig. 3. The computational grids are refined with 25 layers to capture the velocity and temperature boundary layers near the wall surfaces in the fluid domains. The height of the first grid layer is 0.005D to ensure the corresponding y+ is near 1.0 and the stretching ratio is 1.2.

Fig. 3
figure 3

Computational meshes

Full size image

The computation mesh independence verification is carried out in advance to achieve the tradeoff between the calculation accuracy and computational cost. The composite cooling model with nominal impingement layout and crater 1 film-cooling hole at M = 1.5 is represented for computation mesh independence verification. The profiles of ηϕl at different computational mesh levels are shown in Fig. 4, It is seen that all predicted profiles of ηϕl have similar trends along the streamwise direction. However, higher ηϕl occurs for the coarsest mesh (3.8 million) than that for the medium mesh (4.3 million). There is little variation in ηϕl with a further increase of mesh number to the finest mesh (5.1 million). Hence, the computational mesh with a resolution of 4.3 million is determined for the following conjugate heat transfer simulations.

Fig. 4
figure 4

Computational mesh independence verification

Full size image

The current numerical study uses the Reynolds-Averaged Navier–Stokes (RANS) method to obtain the flow and temperature fields for the composite cooling using ANSYS CFX software. The total energy equation is adopted for the heat transfer model. The high-resolution scheme is used for the convective advection term. The convergence criterion is that the root square residuals of all solving variables are less than 10−6, and the temperature variation at three monitoring points along the film-cooling hole centerline is less than 0.1K.

While LES and Direct Numerical Simulation (DNS) can capture the intricate details of turbulence, RANS models often provide sufficient accuracy for most engineering applications despite their inability to resolve all the fine-scale features of turbulence. Furthermore, the error associated with RANS can be reduced by employing appropriate turbulence models. Consequently, validation of turbulence models is crucial to ensure their reliability and accuracy in practical engineering scenarios. The published research [13, 18,19,20,21, 33, 34] shows that the SST turbulence model has demonstrated high accuracy in predicting film cooling under adiabatic and conjugate conditions. The discrepancies between the model predictions and experimental results are within an acceptable range. Therefore, the SST turbulence model is a competitive option for the turbulence closure selection in this study. The experimental data of ηϕl for the composite cooling with a cylindrical film-cooling hole at M = 1.5 in [38] is verified using the k-ε and SST turbulence models. The prediction results are compared with the experimental data, as shown in Fig. 5. It is seen that the k-ε turbulence model generally underestimates the values of ηϕl relative to the experimental test, especially in the region X/D > 5. However, the SST turbulence model overstates in the downstream region X/D > 5. Comparatively, the predicted results using the SST turbulence model coincide well with the experimental data [38] and the CFD data using the same turbulence model [39]. The maximum deviation is − 18.53% at X/D = 15 for the k-ε turbulence model and 10.78% at X/D = 15 for the SST turbulence model. The impingement and film-cooling plate's side walls are set as adiabatic conditions in the current simulation. However, the experimental test cannot guarantee this ideal adiabatic condition. The mismatch of the adiabatic wall may contribute to the error. Besides, Fu et al. [39] pointed out that the inexact heat conductivity of steel 4j36 results in certain deviations in the downstream region X/D > 5. The numerical method with the SST turbulence model has been validated from the above analysis and further employed in the current study.

Fig. 5
figure 5

Validation of turbulence model

Full size image

3 Results and discussion

3.1 Effect of impingement cooling structure

This section compares the composite cooling scheme with the film-cooling hole (crater 1) at a typical blowing ratio M = 1.5, including the impingement hole diameter, distance, and hole shape.

Figure 6 shows the distributions of the external wall and hc over the internal wall for the composite cooling with three different impingement hole diameters (dj = 0.5D, 1D, and 1.5D). As shown in Fig. 6a, the impingement hole diameter has a remarkable influence on ηϕ in the region around and downstream of the film-cooling hole, there is little impact on the region upstream. Reducing the impingement hole size leads to an improvement, whereas increasing the impingement hole size results in a degenerative ηϕ. This change trend is consistent with that concluded by Fu et al. [39] for the composite cooling with a fan-shaped film-cooling hole. From the local internal heat transfer coefficient distribution over the film-cooling plate bottom surface in Fig. 6b, it is observed that the jet core with high hc increases gradually with the decrease of dj. Furthermore, the crossflow regulation alters the jet core shape more easily with a larger dj.

Fig. 6
figure 6

Contours of ηϕ and hc with different impingement hole diameters

Full size image

The variation of hc can be interpreted by the coolant streamline distribution along plane Y/D = 0 flowing through the impingement holes, as illustrated in Fig. 7. When M = 1.5, a smaller dj leads to a larger velocity across the impingement hole. The averaged velocities at the impingement hole exit are 3.2m/s, 7.1m/s, and 28.1m/s for dj = 1.5D, 1D, and 0.5D, respectively. The coolant flow strikes the target wall surface to generate the stagnation flow, providing a locally enhanced hc. On account of the entrainment effect by the stronger coolant strike, the impingement hole model with a smaller dj leads to an improved hc around the smaller stagnation region. There may be an optimal impingement hole diameter to balance the stagnation impact and entrainment effect. However, considering the manufacturing limitation in common, the smallest impingement hole diameter is restricted to 0.5D. Therefore, in the current study, the composite cooling model with dj = 0.5D yields the highest hc, shown in Fig. 6b. Besides, the side surfaces of the impingement cavity are blocked; thus, the coolant flows only through the film-cooling hole, resulting in a monodirectional crossflow effect. The crossflow more easily influences the coolant flow with lower velocity for the larger dj. The crossflow influence is especially severe for the leftmost impingement hole adjacent to the film-cooling hole, as shown in Fig. 7c.

Fig. 7
figure 7

Surface streamlines along sectional plane Y/D = 0 with different impingement hole diameters

Full size image

Figure 8 presents the profiles of ηϕl along the streamwise direction X/D and ηϕave for the composite cooling with different impingement hole diameters. As seen in Fig. 8a, the impingement holes with different diameters significantly alter the values of ηϕl in the downstream region X/D > 5 but slightly in the upstream region X/D < 0. Compared with the composite cooling model with nominal impingement holes dj = 1D, the model with dj = 0.5D provides higher ηϕl because of higher impingement cooling performance, but the model with dj = 1.5D has lower values. Because of the higher hc for the impingement hole with a smaller dj, the cooling capability of the coolants is expended, thus resulting in a slightly higher ηϕl at the region upstream of the film cooling hole. Figure 8b shows that the values of ηϕave decrease monotonically with the increasing dj. An improvement of 6.7% of ηϕave occurs for the model with dj = 0.5D than that with dj = 1D, whereas a decrease of 7.4% for the model with dj = 1.5D.

Fig. 8
figure 8

Distributions of ηϕl and ηϕave with different impingement cooling hole diameters

Full size image

As seen in Tab. 2, two additional composite cooling models are studied by changing the impingement distance as tj = 1D and 3D. Figure 9 illustrates the distributions of ηϕ and hc for the composite cooling with various impingement distances. Generally, the variation of tj yields a minor change in the ηϕ distribution in the region around X/D = 10 only, as seen in Fig. 9a. From the view of impingement heat transfer performance, it is seen in Fig. 9b that a smaller tj could slightly enhance the hc values in the region upstream of the film-cooling hole inlet.

Fig. 9
figure 9

Contours of ηϕ and hc with various impingement distances

Full size image

Figure 10 shows the profiles of ηϕl versus the non-dimensional streamwise length for the composite cooling with three different impingement distances and the corresponding ηϕave. It is observed that the composite cooling model with tj =  = 2D exhibits the lowest ηϕl among the studied three models, especially in the region X/D > 5. Besides, the impact of tj on ηϕave is faint, with a relative difference of less than 0.5%.

Fig. 10
figure 10

Values of ηϕl and ηϕave with various impingement distances

Full size image

Figure 11 provides the contours of ηϕ and hc for the composite cooling with different impingement hole shapes. There are five impingement hole expansion methods, i.e., inlet expansion 20°, 10°, no expansion, and outlet expansion 10°, 20°. As shown in Fig. 11a, the impingement hole expansion mode has some positive or negative impact on ηϕ, depending on the inlet or outlet expansion. The inlet expansion impingement hole positively affects the local ηϕ, whereas the out expansion one has a negative effect. The most affected region by the impingement hole expansion is located at X/D < 20 downstream of the film-cooling hole exit. However, it is observed that the distribution of ηϕis insensitive to the expansion angle. As shown in Fig. 11b, the impingement core of the no-expansion and outlet expansion holes is more obvious than that of the inlet expansion holes. This means that the coolant jet flow inside the impingement hole breaks away from the solid wall for the outlet expansion mode, similar to the free jet impingement flow to some extent. However, due to the flow acceleration inside the impingement hole, the inlet expansion model yields a larger region with a higher hc than the no expansion and outlet expansion models.

Fig. 11
figure 11

Contours ofηϕ and hc with different impingement hole shapes

Full size image

Figure 12 illustrates the influence of impingement hole shape on profiles of ηϕl and ηϕave. It is seen that the inlet expansion models give the best overall cooling performance, regardless of ηϕl andηϕave, and the outlet expansion models have the worst overall cooling performance. Nevertheless, the specific expansion angle has almost negligible influence on the overall cooling performance. The maximum variation of ηϕave is 1.5% in all five expansion models in the current study. This lesser impact is attributed to the inadequate expansion or convergence inside the impingement hole, resulting from the small thickness of the impingement plate with only 1D.

Fig. 12
figure 12

Profiles of ηϕl and ηϕave with different impingement hole shapes

Full size image

3.2 Effect of film-cooling hole shape

In this part, the composite cooling structures with different film-cooling holes and nominal impingement cooling layouts (dj = 1D, tj = 2D, no expansion hole) are investigated at M = 1.5 to explore the impact of the film-cooling hole. Three film cooling holes are included, i.e., the simple cylindrical hole, the original contoured cratered hole (crater 1), and the optimal contoured cratered hole (crater 2). Qualitative and quantitative assessments are conducted to determine which film cooling hole structure performs better. A mechanical analysis explains the reasons behind these observations, focusing on fluid dynamics and heat transfer mechanisms within the cooling system. The aim is to identify the optimal film cooling hole configuration that maximizes cooling efficiency and understand the underlying fluid dynamic principles that govern its performance.

The distributions of ηϕ and hc with different film-cooling holes are shown in Fig. 13. Because of the same impingement cooling layout and flow parameters, the distributions of hc are almost identical regardless of the specific film-cooling hole shape, as illustrated in Fig. 13b. However, the film-cooling holes significantly alter the distribution of ηϕ over the external wall. For the cylindrical film-cooling hole case, overall cooling effectiveness along the lateral direction has a minor change due to the poor coolant coverage at M = 1.5. Furthermore, impingement cooling dominates the composite cooling model with a cylindrical film-cooling hole. For the cratered film-cooling cases, the single-peak profiles occur for the ηϕ distribution. This is attributed to the coolant accumulation near the centerline and better coolant coverage, as explained in [41] for the adiabatic film-cooling condition.

Fig. 13
figure 13

Contours of ηϕ and hc with different film cooling holes

Full size image

Figure 14 illustrates the contours of ωX and surface streamlines over plane X/D = 10 for the composite cooling models with different film-cooling holes. It is known that when the CRVP is dominated in the downstream flow field, the coolant departs from the wall surface, resulting in a deteriorative coolant coverage and poor film-cooling performance. The cylindrical film-cooling hole model exhibits a large-scale CRVP structure. The cratered 1 film-cooling hole alters the downstream vortex structure remarkably. The original CRVP is broken into two sections by an additional Anti-CRVP and dragged toward the wall surface with a shallow penetration into the mainstream flow. This extra Anti-CRVP yields a wider coolant coverage and higher film-cooling performance, as shown in Fig. 13. For the cratered 2 film-cooling model, due to the wider exit and deeper crater, the CRVP is eliminated, and the large-scale Anti-CRVP dominates the downstream flow field, resulting in the much-improved overall cooling effectiveness.

Fig. 14
figure 14

Contours of ωX and surface streamlines over plane X/D = 10 for the composite cooling with various film-cooling holes

Full size image

The comparisons of ηϕl and ηϕave among the composite cooling configurations with these three film-cooling holes are given in Fig. 15. The cratered film-cooling hole configurations demonstrate a higher laterally averaged overall cooling effectiveness over the whole studied streamwise region than the cylindrical film-cooling hole case. The improvement in the downstream region benefits from the better coolant attachment to the external wall, higher adiabatic cooling effectiveness, and heat conduction in the streamwise direction upstream of the film-cooling hole. The crater 2 film-cooling hole provides the best coolant coverage near the film-cooling hole exit, which is attributed to its larger lateral width at the hole exit. The value of ηϕl for the crater 1 film-cooling hole case monotonously increases with the increasing streamwise length. However, the cylindrical film-cooling hole case has a minimum ηϕl at X/D = 2, resulting from the coolant departure downstream of the film-cooling hole exit. The value of the crater 2 film-cooling hole case increases, continuously comes up to a peak value at X/D = 8 and then lets up while the streamwise length increases. Compared with the cylindrical film-cooling hole model, the crater 1 film-cooling hole model exhibits an enhancement of 28.47% for the area-averaged overall cooling effectiveness, and the crater 2 film-cooling hole model provides an improvement of 38.692%. The improvement in cooling effectiveness is attributed to the change in hole shape from cylindrical to cratered hole. Comparatively, the expansion range in the spanwise direction is greater, resulting in a larger coverage area of coolant and a more significant enhancement in overall cooling effectiveness.

Fig. 15
figure 15

Profiles of ηϕl and ηϕave for composite cooling with different film-cooling holes

Full size image

3.3 Optimal composite cooling at various blowing ratios

As discussed in Sects. 3.1 and 3.2, the impingement cooling layout with dj = 0.5D, tj = 1.0D, and 20° inlet expansion, and the film cooling hole crater 2 yields better overall cooling performance. Hence, the combination of the best impingement cooling layout and film-cooling hole (combined crater 2) is the focus. Two composite cooling models with nominal impingement cooling layouts are compared, including the original model with cylindrical (nominal cylindrical) and crater 2 (nominal crater 2) film-cooling holes. The overall cooling performance for the combined crater 2 model is evaluated at a blowing ratio M = 1.5 and other blowing ratios M = 0.5–1.0, 2.0–2.5.

Figure 16 shows the hc contours over the inner wall surface of the film-cooling plate for different composite cooling models. As the blowing ratio increases, the coolant mass flow rate across the impingement holes increases, resulting in a higher hc. Despite the film-cooling holes in Fig. 16a–b, the internal heat transfer coefficients are almost equal at either blowing ratio. However, the impingement heat transfer performance is improved with a larger jet core and enhanced heat transfer coefficient by the modified impingement cooling layout for the combined crater 2 model, as shown in Fig. 16c.

Fig. 16
figure 16

Contours of hc for different composite cooling configurations at M = 0.5–2.5

Full size image

The distributions of ηϕ are shown in Fig. 17 for the different composite cooling configurations. All three models have an increasing trend with the increase of M from 0.5 to 2.5. Although the coolant departure away from the wall surface occurs for the cylindrical film-cooling hole at M > 1.0, the increment in hc for impingement cooling still yields the increase in ηϕ. Compared with the nominal cylindrical model, when the crater 2 film-cooling hole is adopted, the value of ηϕ is augmented accordingly, as seen in Fig. 17b, due to its higher adiabatic cooling effectiveness. However, the local highest ηϕ occurs in the region 5 ≤ X/D ≤ 20 at M = 1.0 and M = 2.0 for the nominal crater 2 model, meaning that the overall cooling effectiveness depends on the specific coolant coverage over the wall surface. Furthermore, the local overall cooling effectiveness for the nominal crater 2 model is further increased accordingly, benefiting from the enhanced internal heat transfer performance in Fig. 15c. Besides, it is noted that the region upstream is better cooled for the crater 2 film-cooling models than for the cylindrical film-cooling model due to the increased area of the film-cooling hole and, thus, the improved heat conduction capability.

Fig. 17
figure 17

Contours of ηϕ for the different composite cooling configurations at M = 0.5–2.5

Full size image

As shown in Fig. 18, the profiles of ηϕl are compared for three composite cooling models at M = 0.5–2.5. It can be observed that the crater 2 models have much higher ηϕl over the entire streamwise distance at either blowing ratio, especially over the region downstream of the film-cooling hole exit. At M = 0.5, the superior ηϕl for the crater 2 models to the nominal cylindrical model is not remarkable. Nevertheless, the superiority of the crater 2 film-cooling hole models is more obvious with the increasing blowing ratio, owing to its better film-cooling performance at a higher blowing ratio M ≥ 1.0. Although the same film-cooling hole is used, the nominal crater 2 and combined crater 2 models exhibit different profiles of ηϕl versus the streamwise distance. For the nominal crater 2 model, the ηϕ at M = 1.0–2.5 increases sharply at X/D = 0, comes up to a maximal value at nearly X/D = 10, and decreases slightly with a further increase of streamwise distance. In contrast, the combined crater 2 model presents a monotonically increasing trend at any blowing ratio. In the region X/D > 15 at M = 1.0, for example, the combined crater 2 model has higher ηϕl than the nominal crater 2 model, deriving from higher internal impinging cooling performance. However, the higher impinging cooling performance for the combined crater 2 model consumes an additional part of the cooling capacity of the coolants. Hence, the coolant temperature arriving at the film cooling hole is higher, resulting in a lowerηϕl in the region downstream of the film cooling hole exit, X/D < 15 at M = 1.0, for example. The turning location is different depending on the value of the blowing ratio.

Fig. 18
figure 18

Profiles of ηϕl for different composite cooling configurations

Full size image

Figure 19 compares ηϕave over the external wall surface of the film-cooling plate for the three composite cooling models—the value of ηϕave increases monotonically with the blowing ratio M for either composite cooling model. The highest values occur at M = 2.5 with 0.50, 0.63, and 0.67 for the nominal cylindrical, nominal crater 2, and combined crater 2 models, respectively. Owing to the replacement of the cylindrical film-cooling hole by crater 2 film-cooling hole, the nominal crater 2 model shows a higher ηϕave by 17.90%-36.67% than the nominal cylindrical model. A further 0.42–7.16% increase in ηϕave is achieved for the combined crater 2 model, benefitted from the enhanced impingement heat transfer performance shown in Fig. 16. Meanwhile, the superiority of the combined crater 2 model is more obvious since M > 1.5, due to the higher mass flow rate for the impingement cooling. Overall, using the crater 2 film-cooling hole and highly efficient impingement cooling layout, the enhancement of ηϕave is 20.39–39.14% relative to the composite cooling with cylindrical film-cooling and nominal impingement cooling layout.

Fig. 19
figure 19

Values of ηϕave for different composite cooling configurations

Full size image

4 Conclusions

The conjugate heat transfer simulations are performed in the current study to investigate the overall cooling effectiveness of composite film-impingement cooling. Compared with the original model with nominal impingement cooling layout and cylindrical film-cooling hole, the impingement cooling layouts with different impingement hole diameters, distance, and shapes are altered to enhance the internal heat transfer performance at blowing ratio M = 1.5. Two contoured cratered film-cooling holes with different craters are further adopted to improve the film-cooling performance. Finally, the overall cooling performance is evaluated for enhanced impingement cooling and film-cooling hole at blowing ratios M = 0.5–2.5. The conclusion from the present study can be summarized as follows:

  1. 1.

    At blowing ratio M = 1.5, modifying the impingement cooling hole diameter, distance, and shape could improve the internal heat transfer coefficient and the overall cooling effectiveness. When the impingement cooling hole diameter dj = 0.5D, impingement distance tj = 1.0D, and the impingement cooling hole has a 20° inlet expansion shape, the overall cooling effectiveness is highest in the according variation range, respectively. The corresponding maximum improvements in area-averaged overall cooling effectiveness are 6.7%, less than 0.5%, and 1.5%, compared to the nominal impingement cooling layout.

  2. 2.

    At blowing ratio M = 1.5, the contoured cratered film-cooling holes improve remarkably, with an increase of 28.47% for the crater 1 film-cooling hole and 38.69% for the crater 2 film-cooling hole than the cylindrical film-cooling hole. The main reason for the improvement relies on eliminating the CRVP and strengthening the Anti-CRVP, especially for the crater 2 film-cooling hole.

  3. 3.

    The combined crater 2 composite cooling model achieves the highest overall cooling effectiveness because of the adoption of the crater 2 film cooling hole and the combination of optimal parameters studied for the impingement cooling layout. At a wide blowing ratio M = 0.5–2.5, the enhancement of 20.39–39.14% in area-averaged overall cooling effectiveness holds than the nominal cylindrical composite cooling model.

  4. 4.

    To better understand the impact of changes in the composite cooling structure on overall cooling performance, replacing the flat plate model with an actual blade model is advisable. This will allow us to explore the effects of variations in the internal impingement structure and the external film cooling structure on the overall heat transfer and cooling at different locations. Such an investigation would provide valuable theoretical references for double-wall cooling engineering applications.