Introduction

Recently, operating temperatures in gas turbines have increased to satisfy the need for greater power output and enhanced thermal efficiencies. The leading edges of gas turbine blades are directly subjected to high-temperature combustion gases, reaching temperatures of several thousand degrees. This can cause significant thermal stress and damage to the blades, ultimately leading to reduced performance and lifespan of the turbine. Materials that can withstand higher temperatures are used to mitigate this, and advanced cooling techniques are implemented. Additionally, advanced cooling techniques, such as film and internal cooling, can help protect the blades from hot combustion gases. Recently, there has been an increasing trend in the investigation and development of jet impingement cooling.

Many factors can affect the performance of jet impingement cooling, an internal cooling method that involves directing high-speed coolant jets onto a surface. Some key parameters that can affect the performance of jet impingement cooling include jet Reynolds number (Re), injection hole-to-surface angle [1] and distance [2,3,4], coolant temperature, and impingement plate-target surface distance [5, 6]. Also, elongated jets can directly influence a system’s thermal and hydraulic performance [7,8,9].

Demir et al. [10] investigated the flow field and heat transfer of swirling impinging jets at low nozzle-plate distances, considering three cases with six turbulence models on a flat plate. At distances below G/d < 1.00, negative pressure regions occurred for all Reynolds numbers, with the magnitude increasing as the Reynolds number increased and G/d decreased. In addition, El-Wafa et al. [11] experimentally investigated the impact of a movable air jet on heat transfer characteristics over a flat plate, comparing it with a fixed nozzle. Results show that the movable nozzle is more effective in reducing temperature variation, leading to enhanced heat transfer uniformity more than the fixed nozzle, with maximum improvements of 75%, 42%, 39%, and 43% at different G/d ratios. Fenot et al. [12] examined the high relative curvature effect on heat transfer under an array of impinging jet flow over a curved semi-cylindrical surface by changing parameters such as jet diameter, the distances of adjacent jets, and the jet outlet-target surface. Using a linear regression method, they experimentally determined adiabatic wall temperatures and local heat transfer coefficients. In general, they found that the high relative curvature considerably affects the heat transfer on a concave surface. They also determined that the local heat transfer dissipation on a concave surface exhibits similar characteristics to that on a flat plate. Lyu et al. [13] investigated the variation of heat transfer coefficient and adiabatic wall temperature on a concave surface with four different curvature ratios in a single-row chevron-nozzle impinging jet. They conducted an experimental investigation for several jet-target plate distances under different Re numbers. As a result, they stated that the semi-cylindrical concave surface showed the highest heat transfer performance in the mid-plane. The results indicated that employing a chevron nozzle leads to approximately a 20–30% increase in the longitudinally mean Nusselt number (Nu) compared to a round nozzle. Yang et al. [14] evaluated the effect of designing geometric parameters such as inter-jet distance, jet inclination angle, jet-target surface distance, and jet diameter on the sonic impingement jet’s heat and flow characteristics of a curved surface of a NACA0015 airfoil. According to the results, heat transfer performance could be improved by determining the optimum structural parameters. Also, the flow properties of multi-impinging jets are substantially affected by the inclination angle and the spacing between adjacent jets. Jung et al. [15, 16] conducted a study comparing heat transfer characteristics of two different jet configurations: a standard 3 × 5 jet arrangement and an inclined jet array. According to their findings, the inclined array jet arrangement performed better regarding heat transfer characteristics. Heo et al. [17] performed a study to optimize the placement of inclined impinging jets. They found that the injection hole angle that resulted in the highest cooling effectiveness was around 60 deg on a concave surface. Qui et al. [18] investigated the impact of jet arrangements on fluid flow and heat transfer characteristics for three array jets impinging onto a curved surface with varying jet-to-jet spacing and Reynolds numbers. The findings revealed that both inline and staggered arrangements significantly enhance heat transfer uniformity compared to an array jets case, with improved performance observed when side array jets are positioned further from the middle array.

Another concern of impinging jet cooling is improving the heat transfer uniformity. This is because, in some cases, the heat transfer rate may be non-uniform, leading to hot spots or cold spots on the impingement surface. Various factors, including the non-uniform distribution of the impinging jet or the non-uniform heat flux on the surface, can cause non-uniformity in heat transfer. To enhance heat transfer homogeneity, the impinging jets need to be distributed uniformly on the confinement plate, and the heat transfer dissipation on the target surface needs to be controlled with ribs [19,20,21,22], pins [23, 24], grooves [25], dimples [26,27,28], baffles [29], waveforms [30], etc. In addition to these methods, swirling jets [31] and pulsating jets [32] can be preferred for homogeneous temperature distribution.

When the jet flow hits the surface, it creates a stagnation zone where the fluid velocity is reduced to zero. This region experiences a significantly high local heat transfer coefficient due to the intense turbulence and mixing of the fluid. However, the local heat transfer degrades rapidly away from the stagnation zone as the fluid spreads out and its velocity decreases. This means that the cooling effect of the impinging jet is limited to a relatively small area around the stagnation zone. In the case of a row of jets, the cooling effect is more uniform but still limited to the area directly under the jets. Multiple rows of jets can be used to improve the cooling performance, with each row targeting a different surface area.

Also, other cooling techniques, such as rib turbulators, can be synchronized with an impinging jet. The use of ribbed surfaces in jet impingement has been extensively studied, both experimentally and numerically. Different types of rib geometries, such as circular, rectangular, and triangular ribs, as well as different rib heights and spacings, have been investigated. Yan et al. [33] experimentally investigated the impact of VSR deg (45, 60, and 90) on local heat transfer distributions on a flat surface using the crystal thermography technique in a conventional jet impingement cooling scheme. In addition, three different normalized injection holes-target surface spacings (H/d = 3.0, 6.0, 9.0) were examined. The best heat transfer performance was achieved using VSR at an angle of 45 deg at H/d = 3.0. Katti and Prabhu [34] experimentally investigated heat transfer dissipation on a flat surface according to parameters such as the height and width of the ribs and the distance between the ribs. The heat transfer in the stagnation region was higher on the roughened surface than on the smooth surface. Caliskan and Baskaya [35] conducted a study to examine the heat transfer and flow characteristics on a flat surface roughened with VSR at different dimensionless jet-target plate ratios (H/d = 3.0 and 12.0). At low H/d ratios, turbulent kinetic energy was higher on VSR-roughened surfaces. In addition, Nu numbers on ribbed surfaces were higher than on smooth surfaces. In a similar study, Caliskan and Baskaya [36] measured heat transfer using a thermal infrared camera on a surface with VSR and convergent-divergent ribs for variable rib heights under a conventional impinging jet. The mean Nu number on the VSR-roughened surface increased from 4% to 26.6% over the smooth surface. Tong et al. [20] numerically examined four different target surface configurations, such as smooth, round pinned, square pinned, and cambered ribbed surfaces, considering the real turbine operating conditions in the multiple array jet impingement model. In particular, they determined that the cambered ribbed elements could improve the average Nu number by up to 62.6% compared to the smooth surface. Rasheed et al. [37] investigated the heat transfer characteristics of a single jet flow over a heated surface with detached ribs. Various rib configurations were explored using the generalized k–ω turbulence model, varying width, height, clearance, pitch, and radial distance. The study revealed that the nozzle’s proximity to the impingement plate is crucial for achieving maximum heat transfer.

Compared to similar studies, the innovation in this work lies in investigating the effects of dimensionless elongated injection holes on heat and flow characteristics over rib-roughened surfaces with inclined impingement jets. For example, in an elongated jet-impinging model, Tepe et al. [38] experimentally examined the heat transfer properties for a constant rib angle (α = 90°) on a flat surface. In the present study, the effects of ribs with variable heights and different angles on a concave surface in accordance with the leading edge of gas turbine blades on both heat and flow characteristics were examined in detail. Moreover, in a numerical study by Tepe [9], dead flow zones were observed on a smooth concave surface, especially in the last jet regions and between elongated adjacent jets. In this study, VSRs were mounted on a concave surface to minimize this negative effect, and the results were compared with smooth surface models. In addition, Yan et al. [33] experimentally investigated the impact of VSR deg on local heat transfer distributions on a flat surface in a conventional (without elongated jet holes) jet impingement cooling scheme. The current study examined the elongated jet effect on a concave surface roughened with VSRs.

The purpose of this study is to investigate the effects of dimensionless elongated injection holes on heat and flow characteristics over rib-roughened surfaces with inclined impingement jets and assess the feasibility of the proposed designs. The study also explores using VSR to address low heat transfer zones and improve the overall heat transfer efficiency on a curved surface. Combining elongated jets and ribs aims to achieve a more uniform heat transfer distribution and enhance the cooling system’s performance. There is no existing literature study that investigates the use of a concave surface in conjunction with elongated jet holes and ribs. Different normalized injection hole-to-target surface distances (0.5 ≤ G/d ≤ 8.0 for H/d = 8.0), normalized VSR heights (Hr/d = 0, 0.2, 0.3, 0.4), rib angles (α = 30°, 45°, 60°, 90°) and Re numbers (5000 ≤ Re ≤ 25,000) are investigated with SST k-w turbulence model in commercial ANSYS Fluent. Also, the mean and local Nu numbers, flow features, and TPC are studied to elucidate the effects of G/d on a rib-roughened semicircular curved surface. Finally, the proposed model is compared to the standard jet-impinging design to determine the influence of the elongated nozzle holes and VSR-roughened surfaces. Furthermore, the results of this investigation will provide insight into whether the combination of elongated jets and ribs can be an effective method for enhancing heat transfer uniformity in gas turbine components.

Model description

Figure 1 provides a detailed view of the proposed model for a rib-roughened surface, including perspective and left-side views. In addition, the figure provides details on the configuration and size of the rib-roughened surface. The semicircular concave duct has a hydraulic inlet diameter measuring 32.5 mm. Airflow exits the jet and impinges the test plate perpendicularly. After impinging on the surface, the airflow discharges from the concave duct in the direction of flow, typically along the x-axis. The jets, which are elongated toward the target surface, have a diameter of d = 5 mm and a wall thickness of t/d = 0.2. The 3 × 5 staggered jet arrangement was created on the confinement plate. The jet rows on the orifice surface are positioned at an angle of 25 deg with respect to the z-axis. Inclined nozzles are assembled at s/d = 0 and ± 2.6. The spacing between consecutive rows of holes in the streamwise direction is 6d. The measurement region on the target surface is determined as ± 60 deg in the s-direction, which is suitable for the experimental study [16].

Fig. 1
figure 1

Drawings and dimensions of the fluid domain, a 3-D image, b cross-sectional front view, c nozzle and rib configuration of the physical model

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Two different normalized distances of the orifice-to-target surface (H/d = 1.0 and H/d = 8.0) were investigated. While H/d = 1.0 was used only to validate the numerical solution procedure, the H/d = 8.0 ratio was considered constant in all other analyses. Also, the test plate was roughened with VSR. Overall, the combination of the jet diameter, jet angle, rib layout, and distance between the nozzles was chosen to optimize the heat transfer efficiency and temperature uniformity. The geometric parameters, such as the height and angle of the rib, can significantly affect the test plate’s heat transfer distribution and temperature uniformity by influencing the turbulence and fluid flow around the ribs. A study has shown that a sweep angle of around 45 deg results in the best heat transfer enhancement [33]. At this angle, the ribs create vortices that promote mixing and reduce the thickness of the boundary layer, resulting in increased heat transfer. For this reason, 45 deg-VSR was preferred on the relevant surface in this study.

Simulating turbulence flow regimes (5000 ≤ Re ≤ 25,000) was conducted for physical models using computational fluid dynamics (CFD) methods. The values of G/d examined were 0.5, 2.0, 4.0, and 8.0 for H/d = 8.0. Equations (1) and (2) describe relationships between the impingement surface-to-target surface distance (H), jet diameter (d), target surface width (\({W}_{\text{t}}\)), and row spacing (\({G}_{\text{s}}\)). These equations can be used to calculate the \({W}_{\text{t}}\) and \({G}_{\text{s}}\) for a given H and d.

$${W}_{{\text{t}}} = 2.094H+10.472d$$
(1)
$${G}_{{\text{s}}} = 0.436H+2.182d$$
(2)

Governing equations

In this study, the ANSYS Fluent software, which helps optimize the cooling system’s design, was utilized to examine the flow and heat transfer distributions in the physical model and solve the Reynolds-averaged Navier–Stokes (RANS) equations with the SST k-w turbulence model developed by Menter [39, 40]. The impingement jet flow and convection heat transfer characteristics are calculated using governing equations, including the equations of continuity, momentum, and energy in steady-state conditions presented below as Eqs. (3)–(5), respectively.

$$\frac{\partial {(\rho u}_{{\text{i}}})}{\partial {x}_{{\text{i}}}} = 0$$
(3)
$$\frac{\partial (\rho {u}_{{\text{i}}}{u}_{{\text{j}}})}{\partial {x}_{{\text{i}}}} = -\frac{\partial P}{\partial {x}_{{\text{i}}}}+\frac{\partial }{\partial {x}_{{\text{i}}}}\left[\mu \left(\frac{\partial {u}_{{\text{i}}}}{\partial {x}_{{\text{j}}}}\right)+\frac{\partial {u}_{{\text{j}}}}{\partial {{\text{x}}}_{{\text{i}}}}-\frac{2}{3}{\delta }_{ij}\frac{\partial {u}_{{\text{i}}}}{\partial {x}_{{\text{j}}}}\right]+\frac{\partial }{\partial {x}_{{\text{j}}}}\left(-\rho \overline{{{\text{u}} }_{{\text{i}}}^{\mathrm{^{\prime}}}{{\text{u}}}_{{\text{j}}}^{\mathrm{^{\prime}}}}\right)$$
(4)
$$\frac{\partial }{\partial {x}_{{\text{i}}}}\left[{u}_{{\text{i}}}\left(\rho E+P\right)\right] = \frac{\partial }{\partial {x}_{{\text{j}}}}\left[\left(k+\frac{{c}_{{\text{p}}}{\mu }_{{\text{t}}}}{{Pr}_{{\text{t}}}}\right)\frac{\partial T}{\partial {x}_{{\text{j}}}}+{u}_{{\text{i}}}{\left({\tau }_{{\text{ij}}}\right)}_{{\text{eff}}}\right]+{S}_{{\text{h}}}$$
(5)

When dealing with low mass flow rates, laminar flow is often required. However, it is necessary to calculate the turbulence component to predict jet impingement. The boundary layer at the stagnation region is extremely thin, and the flow features within the viscous sub-layer significantly impact the heat transfer on the target wall. To ensure accurate predictions of the impinging jet in this study, the SST k-ω model was utilized because predicting the flow in the viscous sub-layer is well-known due to its diffusion-based calculations. SST k-ω turbulence model uses governing equations, expressed as Eq. (6) and Eq. (7), in a steady state condition.

$$\frac{\partial (\rho k{u}_{{\text{i}}})}{\partial {x}_{{\text{i}}}} = \frac{\partial }{\partial {x}_{{\text{j}}}}\left({\Gamma }_{{\text{k}}}\frac{\partial k}{\partial {x}_{{\text{j}}}}\right)+{G}_{{\text{k}}}-{Y}_{{\text{k}}}+{S}_{{\text{k}}}$$
(6)
$$\frac{\partial (\rho w{u}_{{\text{i}}})}{\partial {x}_{{\text{i}}}} = \frac{\partial }{\partial {x}_{{\text{j}}}}\left({\Gamma }_{{\text{w}}}\frac{\partial w}{\partial {x}_{{\text{j}}}}\right)+{G}_{{\text{w}}}-{Y}_{{\text{w}}}+{D}_{{\text{w}}}+{S}_{{\text{w}}}$$
(7)

Jet Reynolds number is used to characterize the flow regime of a fluid jet and is defined in Eq. (8) as:

$${\text{Re}} = \frac{\rho Vd}{\mu }$$
(8)

where V represents the average flow velocity within the nozzle, the other variables are the process fluid density (ρ) and dynamic viscosity (µ).

The local heat transfer coefficient (h) is a measure of the rate at which heat is transferred from a fluid to a solid surface and is described in Eq. (9). Also, it is influenced by the flow regime of the fluid jet, the jet’s diameter, the distance between the jet and the surface, and the surface roughness and temperature.

$$ h = \frac{{q^{\prime \prime } }}{{T_{\text{w}} - T_{\infty } }} $$
(9)

The term \(q^{\prime \prime }\) refers to the heat flux, which measures the heat transfer rate per unit area. The temperatures of the fluid (\({T}_{\infty }\)) and the wall (\({T}_{\text{w}}\)) are also important variables in this context. Specifically, \({T}_{\infty }\) represents the temperature of the fluid far away from the wall while \({T}_{\text{w}}\) is the wall’s temperature. Besides, the average Nusselt number is calculated using Eq. (10).

$${Nu}_{{\text{ave}}} = \frac{\overline{h}d}{k }$$
(10)

where \(\overline{h }\) describes the overall heat transfer rate per unit area of the target surface and is calculated by averaging the local heat transfer coefficients over the entire measurement region.

The turbulence intensity (I) describes the ratio of the turbulent velocity fluctuations to the mean velocity of the fluid. It can be calculated at the inlet and outlet sections of a cooling system by Eq. (11).

$$I = 0.16 {{\text{Re}}}^{-1/8}$$
(11)

Numerical analysis

Numerical simulations were conducted using ANSYS Fluent, and the physical model was verified through experimental results obtained by Jung et al. [16]. A comparison is presented between the outcomes of commonly used viscous models to determine the best turbulence model for impingement jets with elongated holes and ribs.

Numerical solution procedure

It is important to note that the choice of boundary conditions can significantly impact the accuracy of the numerical solution and the results. Initial boundary conditions of the computational domain are shown in Table 1. The symmetry plane is defined on the middle axis to reduce computational costs. In addition, the process fluid with a temperature of T∞ = 47 °C enters a circular inlet section. Also, the target surface is kept at a temperature of Tw = 27 °C. The non-slip and adiabatic wall is determined as the boundary condition for the remaining wall portion aside from the 60° target surface. The examination is carried out under three different mass flow rates. The turbulent inlet intensity of the process fluid in the concave channel is set as 4.55%, 3.95%, and 3.73% for Re = 5000, 15,000, and 25,000, respectively. The static pressure of 101.325 kPa is applied for the outlet section of the fluid domain. While the non-slip boundary condition is implemented for the entire calculation area, the adiabatic wall surface boundary condition is defined for all walls except the measurement region (target surface).

Table 1 The computational domain initial boundary conditions
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Based on the specified mass flow rates, Reynolds numbers were determined to be higher than 4 × 103 and developed fully turbulent flow [41] in the concave duct. SIMPLEC algorithm was employed to solve the pressure–velocity coupling. A second-order upwind scheme was utilized for discretizing spatial derivatives in the partial differential equations. When the scaled residuals reached 10−5, the solution was considered converged, and the computations were completed in this study. The thermophysical properties of the coolant air are given in Table 2 [42].

Table 2 Thermophysical properties of coolant air [42]
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Assessment of grid independence

The mesh structure plays a critical role in the accuracy and solution of the numerical model. It is important to ensure the mesh is fine enough in regions where the solution is expected to be highly sensitive to the grid size, such as the target surface, elongated jets, and wall jet region. This is accomplished by creating more dense grid cells in these regions, as shown in Fig. 2. A finer mesh can capture more details of the flow and heat transfer phenomena, which leads to more accurate results. However, a fine mesh also increases the computational cost, so balancing accuracy and computational cost is essential.

Fig. 2
figure 2

Mesh detail of the computational domain for H/d = 8.0 and G/d = 2.0 with ribbed surface

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Figure 2 provides a detailed overview of the grid structure. Polyhedral finite elements, preferred in this study, are a type of mesh structure commonly used in engineering applications. The polyhedral elements can adapt to the model’s geometry, allowing for more accurate results and reduced computational costs. These elements are particularly useful for complex geometries, such as the concave duct with ribs in the proposed physical model. Furthermore, the figure also includes colored representations of named selections.

Mesh independence study typically involves simulations with different mesh structures or grid sizes and comparing the results. Table 3 shows the results of grid structures examined at Re = 25,000 and H/d = 8.0 for G/d = 4.0, Hr/d = 0.2.

Table 3 Grid independence behavior on a ribbed surface
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Grid Convergence Index (GCI) with Richardson extrapolation [43, 44] was used to estimate the discretization error in the numerical calculations. The mean Nu number was quantitatively assessed using three different mesh structures. Relative errors in GCI23 and GCI12 were calculated as 1.07% and 0.19%, respectively. Furthermore, the relative error between the extrapolation and fine grid was only 0.15%. The value of \(\frac{{GCI}_{21}}{{r}^{\text{P}}{GCI}_{32}}\) was found to be 1.0071, indicating that it is in the asymptotic range of convergence \(\left(\frac{{GCI}_{21}}{{r}^{P}{GCI}_{32}} = 1\right)\). Based on these results, it can be concluded that further refinement of the grid is unnecessary, as the reduction in error would be minimal compared to the computational cost of further grid refinement. For this reason, the Fine (1) grid structure was preferred in all computations.

Moreover, the numerical result of the velocity vector field on x/d = 3.0 obtained using Fine (1) under the same conditions (G/d = 1.0 and smooth surface at H/d = 1.0) was compared with the result of Jung et al. [16]. As clearly seen in Fig. 3, primary vortices were formed near the impingement point of the jets (s/d ≅  − 1.5), where the fluid flow is highly turbulent due to the interaction of the fluid flow with the surface and the jet for both studies. Likewise, secondary vortices were formed in almost the same location, farther away from the impingement point (s/d ≅  − 6.0), due to the instability and turbulence of the primary vortices. Therefore, it is also inferred that the mesh procedure agreed well with the result in the literature.

Fig. 3
figure 3

Velocity vector plot comparison of numerical study and Jung et al.’s [16] study

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Verification of computational solution approach

Finding a turbulence model that can accurately analyze every physical problem is difficult. Therefore, it is necessary to carefully compare the results from different solution procedures to select the most appropriate model for a particular problem. In summary, verification of computational solution methodology and turbulence model selection is an essential step in the numerical simulation process to ensure the results are accurate and reliable. Therefore, the local Nu number distribution results along the flow direction of the turbulence models commonly used in the literature in the impinging jet cooling method are presented in Fig. 4a. In addition, the local Nu distribution contour on the smooth concave surface at H/d = 1.0 and Re = 5000 for G/d = 1.0 is also compared in Fig. 4a with the experiment by Jung et al. [16] to validate the numerical solution approach. Moreover, the area-averaged Nu number change for investigated turbulence models depending on the Re numbers is given in Fig. 4b.

Fig. 4
figure 4

Performance evaluation of numerical and experimental results [16] in terms of turbulence models (H/d = 1.0 and G/d = 1.0), a contour plot and variation of local Nu along the axial direction at Re = 5000, and b mean Nu numbers

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Turbulence models tend to overestimate experimental results, especially in impingement regions. On the other hand, the SST k-ω model exhibits a reasonable prediction with the experimental values for axial changing of the local Nu number, as illustrated in Fig. 4a. This is supported by a better agreement of the local Nu distribution on the target surface with the experimental results [16] simultaneously. The figure clearly indicates that the SST k-ω model predicted well the Nu contour of the experimental study. Besides, the area-averaged Nu numbers for the relevant surface were obtained by averaging the local estimations within the − 4.0 ≤ s/d ≤ 4.0 and 0 ≤ x/d ≤ 33. The maximum deviation of 9.4% was calculated in the prediction of the average Nu numbers. The hydraulic performance criterion was also considered to validate the numerical procedure further. Therefore, the pressure drop was also studied. As shown in Fig. 5, the solution procedure reveals Tepe’s numerical pressure drop data [9] with a maximum deviation of 10.6% at H/d = 8.0 under different Re numbers. Consequently, the solution methodology was adopted for all subsequent computations, and the findings were examined in detail.

Fig. 5
figure 5

Comparison of pressure drops at 5000 ≤ Re ≤ 25,000 for H/d = 8.0 between the present study and the numerical study by Tepe [9]

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Result and discussion

V-shaped ribs were applied in the present study to decrease the low heat transfer region on smooth surfaces. However, heat transfer performance significantly varied with the angle of the ribs (α). Therefore, it is better first to discuss how to decide the colocation of impingement holes and VSRs to achieve the best heat transfer performance. Figure 6a shows the mean Nu number increased more on the surface with 45 deg VSRs at Re = 25,000 for Hr/d = 0.2 and H/d = 8.0 and compared to the smooth surface, like in the literature [33]. Also, VSR at an angle of 45 deg helped distribute the flow more evenly and effectively across the surface, preventing areas of low heat transfer, as shown in Fig. 6a. For this reason, 45 deg-VSRs were investigated on the target surface in detail.

Fig. 6
figure 6

Effects of rib angle at Re = 25,000 on a variation of mean Nu number and b distribution of local Nu number

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Flow features

The fluid’s velocity significantly influences the surface heat transfer rate. Increasing fluid motion on the target plate leads to an enhancement of heat transfer between the fluid and the surface. Also, if the fluid velocity increases, it causes the boundary layer thickness to reduce. This can be achieved using different configurations, such as extended jets and rib-roughened surfaces. Surface roughening is a commonly used technique to improve heat transfer by increasing turbulence and disrupting the boundary layer on the relevant surface. Different normalized rib heights (Hr/d = 0, 0.2, 0.3, and 0.4) at an angle of α = 45° were examined to investigate the impact of VSR height on flow and heat transfer characteristics for various nozzle lengths in this study.

Figure 7 shows the detailed image of the Nu number dissipation and velocity field near the VSR at x/d = 18.5 and Re = 25,000. The results showed that increasing rib height (Hr/d > 0.3) reduced flow velocity on the curved surface-VSR intersection region, resulting in diminished heat transfer. Moreover, elongating the nozzle to the measurement region increased heat transfer and the flow velocity at the surface and near the rib. As shown in the figure, the study also revealed that a relatively low VSR height (Hr/d = 0.2) enhanced convective heat transfer, while an improper designing VSR height (Hr/d = 0.4) diminished heat transfer due to the fluid being trapped in the VSR corner.

Fig. 7
figure 7

Detailed image of the velocity field and Nu distribution near the VSR at x/d = 18.5 and Re = 25,000

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In a jet impingement flow, there are typically three regions. The potential core region is the region closest to the nozzle where the fluid is still accelerating and the velocity is highest. In the developing region, the fluid still adjusts to the surrounding conditions, and the velocity gradually decreases. The fully developed region is the region farthest from the nozzle where the fluid has reached a steady state, and the velocity profile is relatively constant [45]. However, if the distance between the nozzle and the target surface is sufficiently high, the stronger cross-flow (perpendicular to the main flow direction) can develop in the conventional (G/d = 8.0) cooling duct. This causes the stagnation region to move downstream along the streamwise direction of the plate [46,47,48]. In this situation, the flow outside the potential core region impinges the concave surface at a lower velocity. Therefore, increasing the distance between the orifice plate (or nozzle) and the target surface or increasing the G/d ratio significantly in an impinging jet system will decrease the flow rate. It leads to a diminishing in cooling efficiency, accordingly. Moreover, consistent with the literature [6, 38, 49], elongating the injection holes toward the measurement region and locating VSRs to the surface minimized deflection caused by cross-flow, as illustrated in Fig. 7.

In addition, the leading edge of VSRs in the cooling channel had higher local Nu numbers than other parts, such as the top side and trailing edge. The interaction between the elongated impinging jet and VSR can lead to high-velocity gradients at the leading edge of the ribs. Weak backflow regions and strong velocity fluctuations near the VSRs can be formed as a result of these high-velocity gradients. These regions can also enhance heat transfer by creating turbulent mixing, which increases total heat transfer and reduces the boundary layer thickness [35].

Figure 8 compares the flow features for conventional (CJIC) and elongated jet impingement cooling designs at x/d = 24, Hr/d = 0.2, and Re = 25,000, including velocity vector fields and turbulent kinetic energy contour plots. As shown in Fig. 8a, the collision between the adjacent jet stream and the wall in all configurations generated a clockwise vortex close to s/d ≅  − 3.5. Elongated nozzle holes and ribs also contributed to creating more turbulent flow by forming clockwise rotation (CRV) and counter-clockwise rotation (C-CRV) vortexes near the test region, which maintain a stronger attachment of the fluid to the surface. Besides, downward velocity vectors observed in the proposed models (G/d = 0.5 and 4.0 for Hr/d = 0.2) were unavailable in the CJIC. As illustrated in Fig. 8b, elongated nozzles created a stronger flow of fluid that can more effectively carry heat away from the surface being cooled by transferring the turbulent kinetic energy directly to the relevant surface compared to CJIC. VSRs also helped to improve the uniformity of the heat transfer by providing more process fluid to areas that may have been previously under-cooled. This also led to an increase in the area-averaged Nu number.

Fig. 8
figure 8

Comparison of the flow features for conventional (CJIC and elongated jet impingement cooling designs at x/d = 24, Hr/d = 0.2, and Re = 25,000; (a velocity vector field, (b turbulent kinetic energy contour plot

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Average Nu numbers

This study aims to elucidate the combination of elongated nozzles and ribs to augment the heat transfer efficiency of the curved surface and investigate how these modifications affect total heat transfer. The study explains how the Nu number changes on VSR-roughened surfaces for various parameters, including Re number, normalized rib height, and normalized injection holes to target surface distance, as shown in Fig. 9. Numerical results suggested that heat transfer was enhanced by increasing the rib height to Hr/d = 0.2 and extending the injection holes to the target surface. However, the dimensionless rib height above 0.2 for all Re numbers has a negative effect on the mean Nu number, as illustrated in the figure.

Fig. 9
figure 9

Impact of G/d and Hr/d for α = 45° on the variation of mean Nu numbers at a Re = 5000, b Re = 15,000, c Re = 25,000

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It was found that locating the V-shaped ribs on the stagnation regions, which form due to the interaction between adjacent jets, improved overall heat transfer. For instance, on a smooth concave surface at Re = 25,000 and G/d = 4.0, the average Nu number was 59.15. However, on the rib-roughened target surface (Hr/d = 0.2), it increased by 15.70% to 68.44. Also, the mean Nu number was 53.16 in a conventional impinging model with a smooth surface at Re = 25,000. The average Nu number increased by 47.23% to 78.28 for G/d = 0.5 and Hr/d = 0.2 under the same flow conditions on the rib-roughened model. Similar results were observed for the other Re numbers and nozzle lengths investigated. For example, on a smooth curved surface at Re = 15,000 and G/d = 2.0, the mean Nu number was 43.47. On the other hand, it increased on the VSR-roughened surface for Hr/d = 0.2 by 13.90% to 49.52. Furthermore, the mean Nu number was 38.69 in a CJIC scheme with a smooth surface at Re = 15,000. The area-averaged Nu number increased by 38.14% to 53.44 for G/d = 0.5 and Hr/d = 0.2 with identical flow conditions. Because assembling ribs on a concave surface causes turbulence and disrupts the thermal boundary layer, which increases heat transfer in the measurement region.

However, while the average Nu number is 53.16 for Re = 25,000 in the CJIC model, this value is 56.15 on the VSR-roughened surface. In this case, an increase of 5.6% was calculated. This is due to the relatively high distance between the orifice surface and the measurement region (H/d = 8.0) in the absence of elongated jet holes. Overall, the study demonstrated that combining elongated jet holes and ribs can significantly enhance heat transfer performance on a curved surface. It’s worth noting that these findings are specific to the particular study, and the results may vary depending on the specific conditions and geometries of the jet impingement cooling system. However, these dimensionless parameters can be used to optimize the design of jet impingement cooling systems for concave surfaces and improve heat transfer performance by adjusting the rib height and nozzle length.

Local heat transfer

In typical jet-impinging cooling, the flow at high velocity is directed through a hole or slot in the impingement plate. After impinging on the target surface, it changes direction and discharges along the flow direction. Cross-flow created by this situation can affect the system’s heat and flow characteristics. As subsequent jet flows are added, the cross-flow velocity gradually augments toward the main flow direction. This effect is more pronounced in designs with a large spacing between the nozzle and the target surface, as the fluid travels longer before reaching the surface. The longer travel distance means the fluid is more likely to be affected by the cross-flow and lose momentum, leading to a weaker impingement force by tilting the jet flow and lower heat transfer efficiency. Therefore, the combination of elongated jet holes and ribs may mitigate the adverse effects of cross-flow.

Contour plots of Nu number distributions for different G/d and Hr/d ratios for α = 45° at Re = 25,000 are shown in Fig. 10. It can be seen that decreasing G/d significantly contributes to heat transfer, particularly in the stagnation regions. In addition, the contribution of the presence of extended jets was relatively low in the spanwise direction and last jet region (x/d ≥ 27) for the local Nu distribution. Because the orifice surface-target surface distance (H) was too large, the cross-flow effect became dominant, and the system’s overall efficiency was reduced, especially in the last jet region [49]. On the contrary, a significant increase in local heat transfer dissipation was observed in the wall jet regions, particularly in the last region, with the installation of V-shaped ribs on the measurement region. The heat transfer area is more spread toward the wall jet region and streamwise direction on the rib-roughened surfaces than on the smooth surface. However, it is seen in the figure that the relatively high normalized VSR height (Hr/d = 0.4) had an adverse effect on local heat transfer distribution. Overall, the results suggest that using extended jets and VSRs together can effectively enhance heat transfer uniformity on a curved surface.

Fig. 10
figure 10

Contour plots of Nu number distributions for different G/d and Hr/d ratios at Re = 25,000

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Laterally averaged Nu distributions on the target surface at Hr/d = 0.2 are shown for different Re numbers in Fig. 11, where the letters S and R denote smooth and roughened surfaces, respectively. Furthermore, the figure includes the results of the conventional jet impingement cooling scheme to facilitate a comparison with the proposed models. Due to the jet interaction and cross-flow, heat transfer in the final jet region of the CJIC scheme (x/d ≥ 27) has decreased. The impact of elongated jets on heat transfer is evident for smooth and ribbed surfaces across all Re numbers. Additionally, mounting ribs on the surface decreased the uneven local heat transfer distribution along both the flow and lateral directions in the cooling duct. It is found that locating the V-shaped ribs on dead flow zones of the curved surface enhanced heat transfer, particularly for the final region. As given in the figure, the optimal Hr/d and G/d ratios increase convective heat transfer on VSR-roughened concave surfaces and minimize the mutual impact of cross-flow when compared to the CJIC, consistent with the literature [38, 48].

Fig. 11
figure 11

Laterally averaged Nu number distributions on smooth and ribbed surfaces at Hr/d = 0.2 for a Re = 25,000, b Re = 15,000, c Re = 5000

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Furthermore, non-uniform heat transfer distribution can result in uneven thermal expansion and contraction of the material, leading to internal stresses within the material that can cause it to deform or even fail over time. It can also lead to areas of stagnant fluid flow, where the fluid is not effectively removing heat from the surface. This can lead to overheating and accelerated material degradation in those areas. In this regard, ensuring uniform heat transfer at the target surface as much as possible is essential. In other words, the simultaneous use of extended jet holes (especially at decreasing G/d ratios) and ribs reduces the formation of dead flow zones, providing a relatively more uniform dissipation on the curved surface and also increasing the efficiency of the last jets.

Thermal performance criterion

TPC analysis is a commonly used method in heat transfer to investigate the effectiveness of impingement cooling on surfaces with varying roughness or topology. However, it is important to consider that these modifications can also adversely affect the pumping power by increasing pressure drop for an optimal cooling duct design. The TPC analysis determines if elongated injection holes are feasible for cooling rib-roughened curved surfaces in this study. As a result of this analysis, if the TPC number is more than 1, the heat transfer enhancement is higher than the increase in compressor work owing to the pressure drop. It can also be expressed, shown in Eq. (12), as a function of the friction factor (f) and the area-averaged Nu number [50, 51].

$$\mathrm{TPC }= \left(\frac{{\overline{Nu} }_{{\text{r}}}}{{\overline{Nu} }_{{\text{s}}}}\right)/{\left(\frac{{f}_{{\text{r}}}}{{f}_{{\text{s}}}}\right)}^{-1/3}$$
(12)

where the variables are related to two different configurations of jet impingement: a staggered conventional jet array with a smooth curved surface (represented by \({f}_{\text{s}}\) and \({\overline{Nu} }_{\text{s}}\)) and a staggered elongated jet array with a rib-roughened curved surface (symbolized by \({f}_{\text{r}}\) and \({\overline{Nu} }_{{\text{r}}}\)).

When designing an impinging jet cooling scheme, it is crucial to consider the balance between heat transfer efficiency and pressure drop. By evaluating TPC for 0.2 \(\le \) Hr/d \(\le \) 0.4 (α = 45°) and 0.5 \(\le \) G/d \(\le \) 8.0, whether the existing together of the elongated nozzles and VSRs is an effective method in jet impingement cooling scheme for the curved surface was determined, as in Fig. 12. The numerical results show that the proposed modifications have negligibly boosted pressure drops as well as the heat transfer efficiency of the system. The figure illustrates that TPC values increase with increasing Re number for suggested models. According to the TPC, the physical models are feasible at Hr/d = 0.2 for enhancing convective heat transfer except for G/d = 8.0 at Re = 5000 on a VSR-roughened surface. However, Hr/d = 0.4 is not a feasible design for enhancement heat transfer for relatively all patterns. Besides, TPCs are maximum at G/d = 2.0 for investigated jet plate-measurement region distance.

Fig. 12
figure 12

Evaluation of TPC on rib-roughened curved surfaces with different injection holes for a Hr/d = 0.2, b Hr/d = 0.3, c Hr/d = 0.4

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Conclusions

The present study aims to elucidate numerically how elongated jet holes on a rib-roughened curved surface affect heat and flow features in a staggered array jet impingement configuration. For this reason, the effect of different normalized injection hole-to-test plate distances (G/d), normalized rib heights (Hr/d), rib angles (α), and Re numbers on smooth and ribbed surfaces’ convective heat transfer performance was examined. With this goal in mind, the current study thoroughly investigated several key parameters, including local and area-averaged Nu numbers, velocity vector plots, velocity contours, turbulence kinetic energies, and TPCs. According to the numerical results, the SST k-w model in ANSYS Fluent could model the complicated flow region with reasonable accuracy and correctly capture the staggered array jet-crossflow interaction. As a result, both local and averaged heat transfer data predicted by the solution procedure agreed well with the experimental and numerical results in the literature. This investigation yielded several important results, as follows.

  • According to the evaluation of TPC, the coexistence of elongated injection holes and ribs in a staggered array of impinging jets on a curved surface was feasible, particularly G/d ≤ 2.0, for enhancing overall heat transfer since the increase in heat transfer was more distinct than the rise in pressure loss.

  • Mounting ribs to the relevant surface led to a significant enhancement in heat transfer performance, especially at a rib angle of α = 45°, compared to the case in the conventional jet-impinging design (without elongated jet holes and ribs). This improvement is attributed to the increased flow interaction and turbulence intensity near the surface. Additionally, the results indicated that the flow field near the surface becomes more complex with ribs due to the formation of secondary vortices, which can further enhance heat transfer. For example, heat transfer has been increased by 15.70% by just locating ribs compared to a smooth surface at G/d = 8.0 and Re = 25,000. Moreover, using a combination of elongated holes and ribs, the highest heat transfer augmentation was determined as 47.23% (at Hr/d = 0.2) at Re = 25,000 when the G/d decreased by 0.5.

  • Numerical results indicate that stagnation regions are formed where the jet stream impinging the surface comes across the adjacent jets while flowing from the wall jet region, adversely affecting the heat transfer correspondingly. However, with the addition of ribs to these areas, stagnation regions are disturbed. As a result, relatively more uniform distribution and an increase in heat transfer are provided.

Although this study provides valuable insights into the effect of extended nozzles on heat transfer and flow characteristics in a staggered array jet impingement configuration on a rib-roughened curved surface, these findings can also aid in the design and optimization of cooling systems for various engineering applications.