1 Introduction

Modern-day gas turbines are expected to produce high power and provide higher efficiency than conventional gas turbines. To achieve improved working performance, a higher turbine inlet temperature is essential. In general, the working fluid temperature is far beyond than the metallurgical limits of the material, and therefore, many contingent cooling techniques have been devised to protect the turbine blades and as well as end walls to extend the component life.

The leading edge and mid-chord region are usually cooled by convection cooling, where, first, heat is exchanged through the blade material by conduction followed by convection through the compressed air flowing inside the cooling passages. Impingement cooling is preferred for cooling at the leading edge because it is the region which is subjected to maximum thermal loads. Due to space constraints, many at times it is difficult to incorporate impingement cooling in mid-chord section and trailing edge region. Relatively higher heat transfer rates can be obtained by impingement cooling, but it has a disadvantage that it could weaken the structural strength of a turbine blade [1].

To enhance heat transfer from the trailing edge region, pin–fin cooling is commonly used. In film cooling, air passes through the bleed cavities and forms a protective layer of cooling around the surface of the blade and protects it against the onslaught of extreme thermal fluxes, thereby reducing blade wall temperature to a considerable extent. Film cooling can act in tandem with convection cooling as well as impingement cooling.

The literature is replete with the experimental and numerical investigation of impingement cooling on curved and flat surfaces. Thermal performance obtained from jet impingement cooling mainly depends on the jet Reynolds number, nozzle-to-target distance, diameter of the jet, nozzle spacing and curvature of target plate, as mentioned in the literature [2]. Earliest investigations of heat transfer from concave surfaces have been done by Chupp et al. [2]. They performed an experimental analysis on a concave surface by testing several impingement tubes with an array of circular cavities. The Nusselt number correlation shows that it varies as the power of 0.7 jet Reynolds number. Metzger and Larsen [3] performed experiments on a rectangular channel with 90° turn and used surface coating to measure heat transfer rate and observed approximately 120% increase in Nusselt number when compared to its value at the inlet due to 90° turn in the flow passage.

An experimental study was performed by Bunker and Metzger [4] to measure localized heat transfer characteristics of the leading edge of a turbine blade by jet impingement cooling through an array of multiple jets. They tested several parameters such as leading edge sharpness, Reynolds number of the flow coming out of the jet, pitch-to-diameter ratio and nozzle-to-target ratio. They showed that heat transfer is proportional to Reynolds number raise to the power of 0.6. The results indicate enhancement in heat transfer with the reduction in nozzle-to-target ratio, decreasing leading edge sharpness and augmentation in heat transfer in the longitudinal direction with a decrease in jet-to-pitch diameter ratio. Tabakoff and Clevenger [5] performed an experimental analysis for heat transfer characteristics for three jet arrangements, viz. slot jet, round jet and an array configuration. Localized heat transfer rate is highest at stagnation region for slot and round jet, whereas array configuration exhibits evenly distributed heat transfer characteristics. Huang et al. [6] experimentally analyzed cross-flow orientation effect on heat transfer distribution for an assortment of jets striking target plate orthogonally. Their result shows that heat transfer variations on target surface significantly depends upon cross-flow exit direction. One of the earliest numerical studies on jet impingement cooling was performed by Kayansayan and Kucuka [7], and they found that concave channel gives a better thermal performance when compared to flat channel due to curvature effect of the concave cooling channel.

Katti et al. [8] performed an experimental analysis on a semicylindrical surface impinged by an array of jets to study the distribution of pressure on the impinging surface. Increasing jet-to-target distance results in reduced pressure coefficients. Moreover, at larger pitches, secondary peaks of pressure coefficients are also noticed between the adjacent jets. D. Singh et al. [9] numerically investigated impingement cooling on a cylindrical surface keeping the target plate at a constant temperature. The results indicate a monotonic reduction in Nusselt number value along both stream-wise and radial directions, and Nusselt number reduces at stagnation with an increase in d/D ratio. Caliskan et al. [10] experimentally and numerically investigated jet impingement cooling by elliptical and rectangular arrays of impinging jets and found elliptical jets provide better thermal performance than rectangular jets. Yang et al. [11] presented an experimental study of jets impinging on the cylindrical target plate and validated it with numerical analysis. Their study indicates that Nusselt number distribution is influenced by cross-flow in the passage, Kelvin–Helmholtz eddy structures and unsteadiness arising because of the above-mentioned phenomenon.

Taslim et al. [12] inspected jet impingement cooling on a smooth target surface as well as with corrugated target surface. Maximum heat discharge from the target plate is achieved for notched-horseshoe ribs. Smooth target surface produces maximum heat transfer coefficients. Fregeau et al. [13] presented correlations computed numerically for the mean and maximum Nusselt number of jets impinging on a circular curved surface examining several parameters such as jet-to-target distance, inter-jet distance and Mach number of the flow. Numerical analysis of turbine blade leading edge cooling by combining both impingement and film cooling has been done by Liu et al. [14]. The study indicates that thermal performance of internal surface of leading edge improves with an increase in blowing ratio. Kumar and Prasad [15] performed a numerical analysis of jet striking on a curved surface and studied parameters such as flow Reynolds number, the ratio of nozzle distance to the diameter of a jet and the ratio of target distance to jet diameter. Lower H/D ratios exhibit maximum heat transfer coefficients.

More recently, the effect of jet nozzle diameter and Mach number of the flow was examined by Liu et al. [16] and they concluded that heat transfer from the surface augments with an increase in Mach number of the flow and also with the increase in diameter of the nozzle. Liu and Feng [17] investigated the impingement cooling study on the position of the jet nozzle in the axial direction. By increasing the Mach number of the flow, the Nusselt number values increase and it also increases with a reduction in the distance from the pressure side to the nozzle. Moreover, for better thermal performance side entry of jet should be preferred. Kumar and Prasad [18] conducted numerical investigation on heat transfer on an effused concave surface by impingement method. They found heat transfer to be minimum for the exit configuration with staggered effusion and one edge open. Wang et al. [19] studied the heat transfer effect of slit nozzle impingement on a heated plate surface. Their results show that by increasing jet angle, a significant improvement in heat transfer can be observed for forward-moving inclined slit jet impingement. Chao Ma et al. [24] studied the influence of the non-uniform circumferential flow field generated by radial turbine volutes on the back-disk’s cooling characteristics. Their analyses showed that the circumferential non-uniform distribution of the flow field caused by the volute can significantly impact the relative distribution of cooling efficiency across the surface of the back-disk and can decrease the average cooling efficiency of the back-disk surface by 1.8–6.0% in comparison with when the flow field is uniform.

After a comprehensive review of a host of technical papers available in the literature on jet impingement cooling of a leading edge of a turbine blade model, it is believed that an innovative convergent shape nozzle as impinging jets have not been reported in the literature so far. Hence, based on this gap in the literature the following objective is set forth for the present study. The main objective of present research is to carry out a parametric study to understand the influence of each parameter on gas turbine blade cooling and also to understand the flow and heat transfer characteristics of jet impingement cooling by using convergent shape nozzle for an augmented heat transfer from the blade leading edge using a general CFD code.

2 Numerical modeling

A model of turbine blade consisting of a single array of converging shaped nozzles impinging on the leading edge surface is considered for the purpose of the analysis. When the coolant from the jets strike the curved surface, the fluid stream pattern and the stagnation regions are influenced by the size and shape of the nozzle, cross-flow effect from the air flowing in the span-wise direction, the velocity of the jet, the arrangement of the jets and curvature of the leading edge. These factors influence the cooling and thermal performance of jet impingement cooling.

2.1 Geometric details

The model of the leading edge surface used in the present investigation closely follows the geometry used by Timko [20]. The leading edge of the target surface was stretched by the middle cross section and the flow coming out of a converging nozzle ejected from an array of multiple jets. The length of the leading edge surface is taken at 42 mm. Other parameters such as major (D) and minor (d) diameter of the converging nozzle, pitch distance (C), the distance between nozzle tip to the leading edge of target plate (H) and Mach number (M) of the flow at the inlet are varied. Figure 1 gives the details of the computational domain. Inlet boundary condition is applied at the thirteen converging nozzles and blade tip is given an outlet condition.

Fig. 1
figure 1

Geometric details of the computational domain

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2.2 Mesh procedure

A commercial software is used to generate the tetrahedral mesh for the computational domain. The low Reynolds SST k–ω model requires y +  < 2. Hence, in this study, y + is considered as less than 1.0 to capture the characteristics of the boundary layer. A mesh density of 4.5 million cells is considered for the analysis purpose. The top view of the meshed computational domain is shown in Fig. 2.

Fig. 2
figure 2

Mesh of the computational domain

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2.3 Grid independence study

A grid independence study is conducted for the model of converging ratio \(\left( \frac{D}{d} \right)\) = 4, minor diameter (d) = 0.5 mm, H = 3 mm, L = 3 mm and Mach number of 0.4. Mesh density of 3.10 million, 3.97 million, 4.66 million and 5.55 million number of elements is considered to validate grid independence results. The SST k–ω turbulence model requires wall y + less than 2.0; hence, for the present grid independence study wall y + of less than 1.0 is considered. The area-weighted average Nusselt number obtained from the simulation results is compared with the Richardson extrapolation method presented in reference [22]. Nusselt number values obtained from the grid size of 4.66 million and 5.55 million elements are used to calculate the extrapolated value. The governing equations are discretized by using a second-order upwind scheme. Hence, according to Roache’s investigation, fourth-order accuracy can be obtained in Nusselt number value from Richardson’s extrapolation. Table 1 shows the Nusselt number obtained from the simulation and extrapolation and relative error between them.

Table 1 Average Nusselt number of the target surface
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Table 1 shows that relative error is 0.291% for 3.97 million number of elements. Value of relative error becomes 0.207% when the number of nodes is increased from 3.97 million to 4.66 million. Further, when the nodes are increased from 4.66 to 5.55 million, percentage relative error marginally increases to 0.211. In view of this, a mesh density of 5.55 million nodes is considered as optimum sized domain for the present numerical analysis. Because of the incorporation of this numerical domain, the analyses take extra computational time, but accurate results are anticipated.

2.4 Boundary conditions

The boundary conditions applied to the coolant (air) and the target surface have been matched with those followed by Zhao Liu [17]. Mass flow inlet boundary condition is given to the jet, uniform temperature of 352 K is given to the air, and turbulent intensity is kept at 5% for the inlet air. Air is considered as compressible in nature. Tip of the blade is assigned as pressure outlet with an atmospheric gage pressure value of zero. The target surface is assumed to be at a fixed temperature of 496.3 K. All other walls are given non-slip adiabatic wall boundary condition. The solution is assumed to reach convergence when residuals of continuity, momentum and turbulent equations are lower than 10−4 and for energy 10−6. Along with residual, surface monitors of Nusselt number are also considered for convergence criteria.

2.5 Governing Equations

ANSYS Fluent, a licensed popular CFD suite, is employed for fluid and thermal simulation study. For the computations carried out in this work, double-precision-based segregated solvers are used. A SIMPLE algorithm is used to couple velocity and pressure [25]. Steady-state compressible Navier–Stokes equation is solved to obtain the solutions and equations are discretized using finite control volume methodology. Second-order upwind scheme is adopted to discretize the continuity, momentum, turbulence and energy equation [25]. Since coolant (air) leaves jet at very high velocity, compressibility effects are important and thus considered in the analysis. The governing mean equations used can be written as:

Continuity equation:

$$ \frac{{\partial \overline{u}}}{\partial x} + \frac{{\partial \overline{v}}}{\partial y} + \frac{{\partial \overline{w}}}{\partial z} = 0 $$
(1)

Momentum equation:

$$ \frac{\partial }{{\partial x_{i} }}\left( {\rho \overline{u}\,\overline{u}_{i} } \right) = - \frac{{\partial \overline{p}}}{{\partial \overline{x}_{i} }} + \frac{\partial }{{\partial x_{i} }}\left( {\mu \left( {\frac{{\partial \overline{u}_{i} }}{{\partial x_{i} }} + \frac{{\partial \overline{u}_{j} }}{{\partial x_{i} }} - \frac{2}{3}\delta_{ij} \frac{{\partial \overline{u}_{i} }}{{\partial x_{i} }}} \right)} \right)B_{i} - \frac{\partial }{{\partial x_{i} }}\left( {\overline{\rho }\overline{u}_{i}^{^{\prime}} \overline{u}_{j}^{^{\prime}} } \right) $$
(2)
$$ - \rho \left( {\overline{u}_{i}^{^{\prime}} \overline{u}_{j}^{^{\prime}} } \right) = \mu_{t} \left( {\frac{{\partial u_{i} }}{{\partial x_{j} }} + \frac{{\partial u_{j} }}{{\partial x_{i} }}} \right) - \frac{2}{3}\left( {\rho k + \mu_{t} \frac{{\partial u_{k} }}{{\partial x_{k} }}} \right)\delta_{ij} $$
(3)

Energy equations:

$$ \frac{\partial }{{\partial x_{i} }}\left( {\delta eu_{i} } \right) = - \frac{\partial }{{\partial x_{i} }}\left( {pu_{i} } \right) + \frac{\partial }{{\partial x_{i} }}\left( {k_{eff} \frac{\partial T}{{\partial x_{i} }}} \right) $$
(4)
$$ {\text{where}}\,e = h - \frac{p}{\rho } + \frac{{V^{2} }}{2} $$
(5)

The SST k–ω turbulence model, a two-equation eddy viscosity hypothesis proposed by F. R. Menter [21], which makes use of Wilcox k–ω model in the boundary layer region and in the free stream region, switches to k–ε model. The SST k–ω model equations can be written as [19]:

$$ \frac{D\rho k}{{Dt}} = \tau_{ij} \frac{{\partial u_{i} }}{{\partial x_{j} }} - \beta^{*} \rho \omega k + \frac{\partial }{{\partial x_{j} }}\left[ {\left( {\mu + \sigma_{k1} \mu_{t} } \right)\frac{\partial k}{{\partial x_{j} }}} \right] $$
(6)
$$ \frac{D\rho \omega }{{Dt}} = \frac{{\gamma_{1} }}{{\nu_{t} }}\tau_{ij} \frac{{\partial u_{i} }}{{\partial x_{j} }} - \beta \rho \omega^{2} + \frac{\partial }{{\partial x_{j} }}\left( {\left( {\mu + \sigma_{\omega } \mu_{t} } \right)\frac{\partial \omega }{{\partial x_{j} }}} \right) + 2\left( {1 - F_{1} } \right)\rho \sigma_{w2} \frac{1}{\omega }\frac{\partial k}{{\partial x_{j} }}\frac{\partial \omega }{{\partial x_{j} }} $$
(7)

The constants \(\phi\) of the new model can be treated as a linear combination of two other constants, \(\phi_{1}\) and \(\phi_{2}\), as:

$$ \phi = F_{1} \phi_{1} + \left( {1 - F_{1} } \right)\phi_{2} $$
(8)
$$ {\text{and}}\,\;\sigma_{k1} = 0.85,\,\sigma_{w1} = 0.5,\,\beta_{1} = 0.075,\,a_{1} = 0.31,\,\beta^{*} = 0.09,\,k = 0.41,\,\sigma_{k2} = 1,\,\sigma_{w2} = 0.856,\beta_{2} = 0.0828 $$

Eddy viscosity can be defined as:

$$ \nu_{t} = \frac{{a_{1} k}}{{\max (a_{1} \omega ;\Omega F_{2} )}} $$
(9)
$$ F_{1} = \tanh \left( {\arg_{1}^{4} } \right) $$
(10)
$$ {\text{where}}\,\arg_{1} = \min \left[ {\max \left( {\frac{\sqrt k }{{0.09\omega y}};\frac{500\nu }{{y^{2} \omega }}} \right);\frac{{4\rho \sigma_{w2} k}}{{CD_{k\omega } y^{2} }}} \right] $$
(11)
$$ F_{2} = \tanh \left( {\arg_{2}^{2} } \right) $$
(12)
$$ {\text{where}}\;\arg_{2} = \max \left( {\frac{2\sqrt k }{{0.09\omega y}};\frac{500\nu }{{y^{2} \omega }}} \right) $$
(13)
$$ CD_{k\omega } = \max \left( {2\rho \sigma_{\omega 2} \frac{1}{\omega }\frac{\partial k}{{\partial x_{j} }}\frac{\partial \omega }{{\partial x_{j} }},10^{\, - 20} } \right) $$
(14)

2.6 Comparison with existing experimental data

It is very important that a turbulence model selected for computation should be able to give accurate flow and heat transfer results. Various turbulence models available in FLUENT are SST k–ω model, standard and RNG k–ε model, and BSL k–ω model. Results obtained from these model are compared with the results of Bunker and Metzger [4]. Geometry used for the validation can be found in Ref. [4] and is shown in Fig. 3a. The experimental case with H/B = 24, r* = 1, Re = 6750 and C/D = 4.67 in [4] is chosen for comparison. Air is given a temperature of 347 K and turbulence intensity of 4%. The mass flow boundary condition is given at the inlet. Its value is selected such that the Reynolds number correctly represents the experimental work of Bunker and Metzger [4]. All other boundaries are given non-slip adiabatic wall boundary condition. The temperature of the target surface is kept constant at 296 K. All open boundaries are given outlet condition with zero gage atmospheric pressure and an outlet temperature of 296 K. The wall grid y +  < 1 is chosen for the analysis, and mesh density of 1.2 million elements is used for the present validation case.

Fig. 3
figure 3

a Geometric configurations used by Bunker & Metzger [4] for the experimental work, b comparison of area-weighted average Nusselt number of various turbulence model with experimental results of Bunker & Metzger [4], c relative difference in Nusselt numbers obtained from the experimental results of Bunker & Metzger [4] and CFD analysis using SST k–ω turbulence model

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A validation study has previously been done by Kumar and Prasad [15] and Liu et al. [16]. Nusselt number is plotted along the curve length (S/B) and its value is compared with the experimental results in Reference [4]. Figure 3b shows that all the turbulence model simulates the experimental results well, but results obtained from SST k–ω turbulence model closely match with the experimental results of Ref. [4] with the maximum relative error of 15.23% for S/B ratio of 12. For all other S/B ratio, the relative error is less than 12%. As the experimental error is nearly 10% and the error obtained from the SST k–ω model is least among all the turbulence models, further computations have been done by using SST k–ω model. As per Bunker & Metzger [4], the experimental uncertainty for their study is estimated to be ± 10 percent using the methods for single-sample experiments. The relative difference between the experimental value and the present CFD analysis (using SST k–ω model) is shown in Fig. 3c.

2.7 Parameters Investigated

The converging ratio \(\left( \frac{D}{d} \right)\) of the nozzle, inlet Mach number (M) of the flow, target distance (H) and the pitch distance (C) play an important role in heat transfer augmentation from the target surface in jet impingement cooling. To investigate these parameters, a detailed numerical study is performed to understand the effect of each parameter while keeping other constant. Table 2 shows all the four cases studied in this numerical study.

Table 2 Details of different cases for numerical study
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3 Results and discussions

3.1 Effect of Mach number on flow characteristics for a given jet geometry

Figure 4 shows the velocity contour in span-wise direction for case 1 of Table 2 (for Mach number varying from 0.2 to 0.8). It is observed that impingement effect of the jet shows reduced flow characteristics in the span-wise direction for all the nozzles succeeding the first nozzle as shown in Fig. 4 owing to increase in cross-flow effect due to spent jet air flow.

Fig. 4
figure 4

Velocity vectors along the span-wise direction at the center of the jet nozzle for various Mach number (M)

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The expanded view of Fig. 5 shows that there is a primary stagnation region right beneath the jet and two other prominent complementary secondary stagnation regions on either side of the primary stagnation region, formed due to vortices generated when flow from the contiguous nozzle strikes each other. This effect diminishes in the downstream direction where cross-flow effect dominates the impingement effect and completely disappears at the tip of the blade near the outlet. Similar behavior can be observed for other Mach number flow velocity contours.

Fig. 5
figure 5

Zoomed view of the region (i) in Fig. 4a

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Figure 6 shows the pressure coefficient fluctuations along the leading edge of the target surface where flow from the nozzles strike. The upward arrows specify the position of the centerline of the nozzle. This figure also shows that pressure distribution declines as fluid moves in the downstream direction. It is noted that peak values of cp are observed at primary stagnation region beneath the nozzle. Value of cp for all Mach number reaches a peak value of unity at the nozzle, either at No. 2 or at No. 3 and then decreases as the cross-flow effect takes dominance over the impingement effect. The secondary peak in cp values is also observed because of the presence of entrainment formed due to the interaction of impingement flow from the nozzle and the cross-flow from the spent air. Secondary peak is initially observed at the middle of two primary peaks, but then shifts to right as fluid flows to further downstream direction indicating that flow coming out of nozzle does not impinge the target plate directly, but rather deviate from its path because of dominating spent jet air flow.

Fig. 6
figure 6

Pressure distribution along the span-wise direction for various Mach number of the flow for a given jet geometry

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Figure 7 shows the detailing of the contour shape of the flow through each jet vis-à-vis through flow. The parameter used for the contour plot is specific dissipation rate (ω) which throws light on the type of flow dissipation that occurs for the bulk flow in the stream-wise direction. The flow from nozzle No. 1 to No. 13 indicates clearly that in the initial jet impingement on the blade leading edge, there is lower dissipation of ω due to insignificant cross-flow, but as the one proceeds toward the nozzle No. 13, the jet flow mixing occurs along with a stronger cross-flow resulting in a comparatively higher dissipation of ω in the neighborhood of each jet region.

Fig. 7
figure 7

Specific dissipation rate (ω) at various Mach number of the flow

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The cross-flow effect predominates with the increase in Mach number of the flow resulting in the high distortion of the impinging jet flow. However, the heat transfer characteristic for the heat removal is due to both impingement jet and the grade of cross-flow turbulence of the flow. Hence, for higher Mach number flows due to higher turbulence as manifested by larger ω is beneficial to augment cooling of the leading edge in comparison with the lower Mach number flows. This can be clearly brought out in the contour plots of the Nusselt number of the leading edge for various Mach number in Fig. 8.

Fig. 8
figure 8

Nusselt number contour at various Mach number for a given jet geometry

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In Fig. 8a, b, c and d, the bottom left side of the target plate represents hub and the top right side of the plate represent the tip, respectively. Highest Nusselt number values are obtained at the stagnation region as shown in Fig. 8, and each jet covers a certain area where it has its influence. Heat transfer rate also increases with the increase in Mach number of the flow; hence, highest Nusselt number values are obtained for a flow having Mach number as 0.8.

As the fluid moves toward the tip side, cross-flow influence over the jet impingement on the target plate increases; therefore, jet flow does not directly hit the target plate and merges with the bulk flow, and lower Nusselt number values are obtained in that region as evident from the figure. This illustrates that the heat transfer rate obtained from jet impingement cooling is higher than convection cooling. An interesting phenomenon that can be observed that there are significantly lower Nusselt number values at secondary stagnation region. It is mainly because when flow from two consecutive jets meet, the velocity at the secondary stagnation point reduces due to the formation of velocity and thermal boundary layers [23].

Figure 9 shows the Nusselt number plot along the span-wise direction at various Mach number flows for a given jet geometry. Figure 9 shows that the presence of jets causes primary and secondary peak values of Nusselt number almost in a similar fashion to the pressure coefficient peak values observed in Fig. 6. But an important observation that can be made here is that Nusselt number peaks depend upon jet Mach number unlike pressure coefficient plot (Fig. 6).

Fig. 9
figure 9

Nusselt number plot at varying Mach number for a given jet geometry

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It should be noted that Nusselt number peaks are obtained at stagnation region as evident from Nusselt number contour plot of the leading edge of a gas turbine in Fig. 8. Secondary peaks in Nusselt number values can be attributed to the vortex formed due to confluence when flow from contiguous nozzle meet, thus promoting turbulent mixing which increases heat transfer rate with lower secondary stagnation pressure. Maximum thermal performance is obtained for Mach number of 0.8, but it has a disadvantage that it will reduce the combined efficiency of the gas turbine as higher Mach number flow increases the load on the compressor. Hence, a trade-off should be done in jet velocity such that both reasonable thermal performance and less load on a compressor could be achieved.

3.2 Effect of \(\left( \frac{D}{d} \right)\) on constant M, H and C

In order to study the effect of converging ratio of the nozzle on impingement cooling, a detailed study is performed by varying \(\left( \frac{D}{d} \right)\) and keeping jet-to-target distance as 3 mm, inter-jet distance as 3 mm and Mach number of the flow as 0.4. Figure 10 illustrates the Nusselt number plot showing the effect of \(\left( \frac{D}{d} \right)\) on constant M, H and C. Nusselt number peaks are found to be higher for d = 0.5 mm when compared to d = 0.25 mm keeping constant converging ratio as 4. It is primarily attributed to the fact that with an increase in the diameter of the nozzle, the stagnation region area increases as more fluid can come out of the nozzle and impinge the target surface. The effect of converging ratio at a constant minor diameter of the nozzle as 0.5 mm and 0.25 mm is also shown in Fig. 10.

Fig. 10
figure 10

Nusselt number plot for varying \(\frac{D}{d}\) at constant M, H and C

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It is perceived that the converging ratio has a significant influence on heat dissipation from the target surface. Increasing \(\left( \frac{D}{d} \right)\) from 2 to 4 for d = 0.5 mm and from 4 to 8 for d = 0.25 mm almost doubles the Nusselt number peaks. It is due to the inherent shape of the nozzle. Rise in fluid velocity coming out of the nozzle is more for higher converging ratio, and thus, jet strikes the target plate with much higher velocity. The higher velocity of the flow helps in preventing boundary layer formation at the target surface thereby augmenting heat transfer rate. As discussed in the previous section, the cross-flow effect takes dominance over the impingement effect in the downstream direction, and it results in the reduced heat transfer from the target surface.

Figure 11 shows Nusselt number contours for varying \(\left( \frac{D}{d} \right)\) on constant M, H and C. In Fig. 11a, b, c and d, the bottom left side represents hub and the top right side represents the tip of the leading edge of the turbine blade, respectively. It can be appreciated that converging ratio has a noteworthy influence on the Nusselt number with higher converging ratio providing better heat transfer rate. It also validates impingement cooling is much more effective than convection cooling.

Fig. 11
figure 11

Nusselt number contours for varying \(\left( \frac{D}{d} \right)\) at constant M, H and C

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Secondary peaks in Nusselt number can also be observed between two consecutive primary stagnation regions. It appears because the confluence of jet flow and streamflow produce secondary impingement which may cause the formation of recirculation and vortices which generate swirl and it strikes the target with lower velocity as compared to velocity coming out of the jet. This phenomenon reduces in span-wise direction because jet directly does not impinge on the target plate due to the dominating effect of cross-flow; rather, it deflects from its usual path and turns sideways toward the outlet.

Figure 12 shows pressure coefficient distribution along the span-wise direction of the leading edge showing an effect of \(\left( \frac{D}{d} \right)\) on constant M, H and C. Here also pressure coefficient values reduce as fluid flows in the downstream direction. Highest peak values of cp for all \(\left( \frac{D}{d} \right)\) ratios are obtained for the jet No. 2 as compared to rest of the jets. For constant minor diameter (d = 0.5 mm) at varying \(\left( \frac{D}{d} \right)\) from 2 to 4, we observe that pressure coefficient values increase throughout the span-wise direction. A similar phenomenon can also be noticed when \(\left( \frac{D}{d} \right)\) increases from 4 to 8 at a constant minor diameter (d) of 0.25 mm. It can further be discerned that for constant \(\left( \frac{D}{d} \right)\) and increasing minor diameter from 0.25 mm to 0.5 mm, pressure coefficient values compound.

Fig. 12
figure 12

Pressure coefficient distribution for varying converging ratio at constant M, H and C

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3.3 Effect of H for constant values of \(\left( \frac{D}{d} \right)\), M and C

To analyze the influence of jet-to-target plate distance on heat transfer augmentation from jet impingement cooling, the following section presents the detailed analysis of three different H values of 3 mm, 4 mm and 5 mm for constant values of \(\left( \frac{D}{d} \right)\), M and C. The Nusselt number plot presented in Fig. 13 showing effect of H on constant values of \(\left( \frac{D}{d} \right)\), M and C shows that peak values of Nusselt number reduce with the increase in value of H.

Fig. 13
figure 13

Nusselt number distribution along the span-wise direction for constant values of \(\left( \frac{D}{d} \right)\), M and C

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It can be attributed to the fact that as jet-to-target distance increases, fluid gets more area to mix, and thus, convection effect dominates impingement effect. This can be supported by the Nusselt number contour plot of Fig. 14. An interesting phenomenon can be noticed that with an increase in H, stagnation region widens. This is largely due to higher convection effect than impingement effect as fluid gets more space to mix up which increases the turbulence of the flow. Secondary peaks because of the confluence of jets are also shown in Figs. 13 and 14. Figure 13 shows that peak values in Nusselt number for H = 5 mm deflect a bit toward left away from the centerline of the jet due to dominating cross-flow effect even in the upstream direction as it gets more space to build up and inhibits jet to impinge directly on the target plate. It should be noted that bottom left side for Fig. 14a, b and c represents the hub and top right side represents tip.

Fig. 14
figure 14

Nusselt number contours for varying H at constant values of \(\left( \frac{D}{d} \right)\), M and C

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3.4 Effect of pitch distance on constant values of \(\left( \frac{D}{d} \right)\), M and H

Pitch distance has a substantial influence on the cooling of a turbine blade achieved through jet impingement method. In this section, Nusselt number comparison is presented for varying pitch distance maintaining parameters such as \(\left( \frac{D}{d} \right)\), M and H, as constant. Figure 15 illustrates the effect of pitch distance on Nusselt number distribution along the span-wise direction for constant \(\left( \frac{D}{d} \right)\) = 4, d = 0.25 mm, M = 0.4 and H = 3 mm. Due to the different positioning of the jet nozzle, the Nusselt number peaks are obtained at different locations on the leading edge along the span-wise direction. It is noted that with an increase in pitch distance, the effect of cross-flow diffuses and impingement effect dominates even in the downstream direction. It should be noted from the figure that Nusselt number peaks obtained for varying inter-jet distance in the upstream region are nearly same as the cross-flow effect is not prominent in this region, whereas in downstream region, cross-flow play a significant role for closer spaced jets as compared to farther spaced jets. Therefore, Nusselt number peak subsides for C = 3 mm, whereas for C = 6 mm its value is higher than the other two C values in the downstream region.

Fig. 15
figure 15

Nusselt number distribution along the non-dimensional span-wise distance for constant values of \(\left( \frac{D}{d} \right)\), M and H

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Figure 16 shows Nusselt number contour for varying inter-jet distance. It should be noted that the stagnation area reduces with the increase in nozzle spacing. In Figs. 16a, b and c, the bottom left side represents the hub and top right side represents the tip of a turbine blade.

Fig. 16
figure 16

Nusselt number contour for varying C at constant values of \(\left( \frac{D}{d} \right)\), M and H

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4 Conclusion

From the detailed numerical analysis of the impingement cooling using an innovative convergent nozzle, the following observations can be made:

  • Nusselt number value at the stagnation region is found to be maximum and its values increase with the increase in Mach number of the flow. Secondary peaks in Nusselt number and coefficient of pressure plots are also observed in the upstream region, but this phenomenon reduces in the downstream region.

  • The converging ratio has a significant impact on jet impingement cooling. \(\left( \frac{D}{d} \right)\) ratio of 8 at d = 0.25 mm provides better thermal performance.

  • With the increase in jet-to-target distance (H), Nusselt number reduces due to the dominant nature of the convection effect.

  • Impingement effect dominates cross-flow effect with an increase in pitch distance (C). Nusselt number peaks for C = 6 mm is greater than those obtained when C is kept as 3 mm and 4.5 mm.