Heat Mass Transfer (2011) 47:1121–1132 DOI 10.1007/s00231-011-0776-x

ORIGINAL

Computational flow and heat transfer of multiple circular jets impinging on a flat surface with effusion M. Ashok kumar • B. V. S. S. S. Prasad

Received: 29 August 2009 / Accepted: 30 November 2009 / Published online: 13 March 2011 Ó Springer-Verlag 2011

Stagnation pressure, N/m2 Jet temperature, K Target plate temperature, K Velocity magnitude in y—direction, m/s Jet exit velocity, m/s Jet plate thickness, m Target plate thickness, m x-co-ordinate, m y-co-ordinate, m z-co-ordinate, m

Abstract Computational investigations are reported on the local flow and heat transfer characteristics from staggered, multiple circular air jets impinging on a flat surface with effusion holes. The geometrical and flow parameters for the computational study are chosen as per the experimental arrangement of Cho and Rhee J Turbomachinery 123:601–608, (14) so as to explain salient features observed in these experiments. The two peaks in the Nusselt number observed in the case of H/D = 6 and three peaks in the case of H/D = 2 are attributed to the flow characteristics such as primary vortices forming an upwash region, followed by secondary vortices resulting in a secondary stagnation zone. The magnitude of local peak in heat transfer increases up to 88% with increasing values of D/d from 0.5 to 1.5 at Re = 10,000.

p0 Tj Ttp v U t1 t2 x y z

List of symbols c Center to center distance of jet holes, m D Diameter of jet hole, m d Diameter of film hole, m H Distance between target surface and jet hole exit, m kf Thermal conductivity of fluid, W/m–K m Mass flow rate, kg/sec Nu Nusselt number q00 Local heat flux, W/m2 p Local static pressure, N/m2 pmax Maximum local static pressure, N/m2 Cp Pressure coefficient (p/p max)

Research during past four decades on jet impingement heat transfer has led to a large body of literature in this area. Reviews by Livingood and Hrycay [1] Martin [2], Down and James in 1987 [3], Jambunathan et al. [4], Viskanta in 1993 [5] are very informative and cover almost all available flow and heat transfer studies and restate useful correlations for single and multiple round jets. Further, extensive studies on single and multiple square jets impinging on heated flat surface are reported by Aldabbagh and Mohamad [6–10]. However, the effect of fluid ejection after impingement through rows of effusion holes has not been given attention in these publications. This topic is of interest in several areas including combined impingement cum film cooling for gas turbine blades, combustor wall cooling, gas-air premixing burner for gas turbines and the like. Hollworth [11, 12] reported perhaps the first investigations on the jets impinging on a flat target surface, with the spent fluid removed through ‘film’ holes. They found an increase in heat transfer rate of about 35 percent by

M. Ashok kumar  B. V. S. S. S. Prasad (&) Thermal Turbo machines Laboratory, Indian Institute of Technology Madras, Chennai 600 036, India e-mail: [email protected] M. Ashok kumar e-mail: [email protected]

1 Introduction

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providing an array of staggered film holes of diameter same as the jet holes (i.e. D/d = 1) and jet hole spacing is twice that of film hole spacing (i.e. Pjet/Pfilm = 2). Ekkad et al. [13] have shown that the effusion holes reduce the cross flow effect of impinging and wall jets and increase the heat transfer locally. The work of Cho and Rhee [14] is of particular interest to us, as they have systematically investigated the effect of jet Reynolds number, height to diameter ratio (H/D), and arrangement of holes (staggered and shifted) in impingement cum effusion system. They conducted mass transfer experiments and made computational investigations. The major conclusions out of their work have been (a) levels of heat/mass transfer increase monotonically with increasing Reynolds numbers, (b) secondary stagnation zone forms a peak in Nusselt number in the up-wash region, (c) secondary vortices are strengthened and flow is accelerated owing to the presence of the effusion holes for the staggered hole arrangement resulting in increases heat transfer and (d) the RSM model used by them with standard wall function for near wall region has not been able to capture the secondary vortices and the secondary stagnation zone. The need for further numerical investigations to explain the reported results e.g. the peaks found in the Nusselt number distributions is thus obvious and is as such recommended in [14]. Therefore, in the present computational study, a physical model which is identical to that used by Cho and Rhee [14] is chosen with the objective of developing the computational model and validate the same by predicting the experimental flow and thermal features reported in [14].

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input conditions. The velocity is selected so that the jet Reynolds number is maintained at a value of 10,000 for all the cases investigated. Target surface is specified as isothermal wall since mass transfer results reported in [14] can readily be converted to heat transfer data through analogy. On all the other solid surfaces such as the jet and film holes, the adiabatic wall condition is imposed. Periodic boundary condition is specified on all the vertical (imaginary) faces. A computational mesh suitable for finite volume method is generated using the automatic grid generating tool GAMBIT 2.3.16. The important features of the mesh are (1) structured mesh is generated for the entire domain which is composed primarily of hexahedral elements but includes coopered wedge elements where appropriate. (2) fine clustered mesh is generated near all the solid walls, at the jet interface, and in the up-wash region. The computational domain and its details for a typical geometric configuration are shown in Fig. 2. Table 1 shows the mesh sizes and the maximum Y? value for various geometric models chosen for the parametric investigation in this study.

3 Numerical simulation Three dimensional steady state simulations are carried out by using finite volume commercial code Fluent 6.2. Flow is considered incompressible and constant properties are used

2 Physical model and mesh generation The physical model for present numerical investigation mimics the experimental set-up used by Cho and Rhee [14]. Accordingly, experiments are performed with 5 9 5 arrays of jets having jet hole diameter of 15 mm, and jet hole pitch to diameter ratio of 6. For the purpose of numerical study, the entire domain is simplified by considering one effusion hole located at geometric center of target surface (flat plate) surrounded by one quarter domain of a jet hole placed at four corners. This physical model (Fig. 1) enables the choice of the computational domain with periodic boundaries. Thus, although computations are performed for a small part of the domain, it facilitates the simulation of the whole combination of 25 jet holes and an array of 5 9 5 effusion holes. In analogy with experiments a uniform velocity at inlet is simulated in computations by extending the domain to a length of three and six hole diameters (45 and 90 mm) in the upstream and downstream ends respectively (refer Fig. 2). Velocity inlet and outflow at outlet specified as

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Fig. 1 Physical model with jet and effusion holes

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Fig. 3 Comparison of Nusselt number distribution on the target surface (y/D = 0) along line AB (Fig 1)

Fig. 2 Details of computational domain Tables 1 Number of cells and Y? for various geometrical models H/D

D/d

Mesh size (in lacks)

Maximum Y?

2

0.5

14.54

0.81

1.0

10.90

2.41

1.5

10.18

4.04

0.5

24.89

0.76

1.0

18.81

2.29

1.5

17.88

4.03

6

for fluid because of the small variation in temperature and pressure. The governing continuity, momentum and energy equations and turbulence model equations are solved by pressure based solver. Second order upwind scheme is used for discretisation and SIMPLE algorithm is used for Pressure–velocity coupling. The solution is considered converged when the residuals fall below 10-4 for momentum, continuity and turbulence equations and 10-7 for energy equation. In some cases it was difficult to achieve the residual levels with default relaxation parameters, for those cases under-relaxation parameters are increased to get the convergence. Change in Nusselt number along line x/D = 3 is continuously monitored so that there will be no change in the value for consecutive 300 iterations.

the local Nusselt number distribution along the stagnation line z/D = 3 (refer Fig. 1) with the experimental data of Cho and Rhee [14]. The case with H/D = 2, D/d = 1, Re = 10,000 is considered for comparison. Among various turbulence models available in the Fluent code, standard k-e, RSM and k-x -SST models are tested. Figure 3 shows the measured data of [14] and the computed Nusselt number from the present study with these three turbulence models. It is found that standard k-e model over-predicts the Nusselt number at stagnation point by 300% and it matches with experiments only in a part of wall jet region. The RSM model captures the trend in stagnation and wall jet regions, but it also over-predicts the Nusselt number in general, and by over 100% at the stagnation region. The k-x-SST model predicts the overall trend very well and the values match within 3% at stagnation and wall jet regions (x/D = 0.5–3). However it under-predicts the Nusselt number to a maximum extent of 65% at x/D = 0.3. From above discussions it is clear that k-x-SST model predicts the heat/mass transfer better than the other models and hence all the following results are reported with the k-x-SST model. It may be recalled that a similar conclusion was arrived from the author’s previous investigation [15] on a concave surface, but without effusion. Further to the Nusselt number variation, the Sherwood number contours are drawn in Fig. 4 and compared with the similar figure drawn in [14]. The zones with comparison of these Sherwood number contours also indicate the validity of present computations.

3.1 Validation for present numerical simulation 4 Data reduction The suitability of solver selection, turbulence model, numerical scheme, discretisation method and convergence criteria used in the present study is validated by comparing

Pressure is presented in dimensionless form as pressure coefficient, Cp. It is defined as the ratio of static pressure at

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Fig. 5 Center line velocity distribution for H/D = 2; D/d = 1; Re = 10,000 Fig. 4 Comparison of Sherwood number deduced from the present study with Cho and Rhee [14]

particular location to maximum stagnation pressure. Heat transfer is presented in dimensionless form as Nusselt number, Nu, based on jet hole diameter. For an isothermal target surface, Nu can be expressed as Nu =

q00wall D : k(Ttp - Tjet Þ

5 Flow characteristics As the air flowing through the holes ejects out in the form of a jet, the mean velocity along the jet centerline remains uniform in the potential core up to a distance of 1.25 D. The potential core is quantified as the region where the center line velocity is within 95% of the jet exit velocity [5]. From this location of the jet (y/D = 0.75) the velocity rapidly reduces to zero, the stagnation point. This zone of falling velocity is referred as stagnation zone, refer Fig. 5. After that, due to the presence of target plate the flow gets deflected, and a wall jet follows the stagnation zone. As the neighboring wall jets interact with each other, large scale primary vortices are formed and an up-wash flow is created. In the up-wash region, small scale secondary counter rotating vortices are generated. Figure 6 shows the secondary vortices which make the flow separate from the target surface and entrain the fluid into the boundary layer of wall jet. This secondary vortex forms a secondary stagnation zone at x/D = 0. A recirculation zone is also formed near the jet hole exit due to the interaction between the main jet and the primary vortices of the upwash flow.

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Figure 7 shows the stream lines plotted in six different planes. At the planes z/D = 3 (plane along stagnation line) the primary vortices (PV) and secondary vortices (SV) are seen clearly. Here the primary vortices appear symmetric about location x/D = 0. As the plane of observation is shifted towards the effusion hole, say z/D = 1.5, one of the primary vortices appear strong and move towards x/D = -3. As the fluid enters into the effusion hole, the streamlines plotted in the plane z/D = 0 show a complete disappearance of the primary vortices due to suction created at the effusion hole. However, it is also observed (in plane z/D = 0) that the up-wash flow is still present at locations x/D = ±3. Some of the features noted in Fig. 7 are reminiscent in the streamlines plotted in the y-plane (Fig. 8). For example, the suction created in the effusion hole may be noticed in Fig. 8a. The velocity vectors, shown in the zoomed view of effusion hole (Fig. 8b) indicate multiple vortex patterns in the effusion hole. Figure 9 shows the velocity contours at planes z/D = 3 and 0. It is observed that the flow accelerates towards the effusion holes and flow velocity is higher near the edges of the effusion holes. Figures 9a, b indicate that higher velocity occurs at the jet hole exit and the effusion hole inlet. The average velocity at jet hole exit and effusion hole inlet are same for D/d = 1. Figure 10 shows a typical distribution of pressure coefficient at the target surface in the plane z/D = 3 for H/D = 2 and D/d = 1. The pressure coefficient (Cp) is symmetric i.e. about x/D = 0. The peak pressure occurs at the stagnation point (x/D = -3) and it reduces as the flow gets accelerated in the wall jet region. The values of pressure coefficient reduce up to 42% of stagnation value at x/D = -2.0 and remain almost constant from x/D = -2.0 to -0.43. Then they start increasing from x/D = -0.43 to 0, wherein the in secondary vortices are

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Fig. 6 Velocity vectors predicted by k-x-SST model in plane z/D = 3 for H/D = 2, D/d = 1, Re = 10,000

Fig. 8 Flow behavior on target surface and effusion hole inlet plane

Fig. 7 Streamlines plotted along various planes for H/D = 2; D/d = 1; Re = 10,000

formed. Secondary peak in static pressure is observed at x/d = 0 showing the formation of secondary stagnation. The value of pressure coefficient at secondary stagnation point is about 46.25% of the first stagnation point (x/D = -3). The Cp distribution on target surface (Fig. 11) shows that the pressure distribution appears symmetric about z/D = 0 and x/D = 0. It is also observed that the secondary peak disappears as the flow approaches effusion hole.

Fig. 9 Velocity contours for Re = 10,000; H/D = 2; D/d = 1

5.1 Effect of H/D As the height to diameter ratio is increased from 2 to 6, potential core length of the jet also increases from 1.25 to 5.2 D; refer Fig. 12. This result corroborates with viskanta [5],

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Fig. 10 Distribution of pressure coefficient on target surface along line z/D = 3 for H/D = 2; D/d = 1; Re = 10,000

Fig. 11 Distribution of pressure coefficient on target surface (y = 0) for H/D = 2; D/d = 1; Re = 10,000

who stated that, for the turbulent jet impinging on the flat surface, the maximum value of potential core length varies from 6 to 7 nozzle diameters for a circular nozzle. It is also evident from Fig. 12 that the slope of the curve is larger for H/D = 6 than for H/D = 2, implying that the deceleration of velocity is faster in case of larger height to diameter ratio. In order to understand the effect of height to diameter ratio on the flow characteristics, the streamlines are plotted in Fig. 13. For the case of H/D = 2, as the confinement is more restricted, the primary vortices shown in the plane z/D = 3 extend to the full height from the target surface to the top wall of the confinement region. Further, in plane z/D = 0, the flow pattern is characterized by asymmetric small scale vortices and the degradation of up-wash flow near the edge of effusion holes. However, for the case of H/D = 6, the primary vortices extend only up to half of the height of the

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Fig. 12 Center line velocity distribution for D/d = 1.0

top wall of the confinement region. The streamlines in plane z/D = 0 show no degradation of up wash flow. In addition, two counter rotating vortices are found. Thus, it may be concluded that the suction created is not strong enough to affect the up-wash flow for the higher values of height to diameter ratio. The ratio of height to diameter affects the velocity distribution locally in the gap between the target surface and the top wall of the confinement region. However the average velocity at the jet hole exit and effusion hole inlet remains same for both values of H/D. Figure 14 shows the effect of height to diameter ratio on the distribution of pressure coefficient contours marked on the target surface. Although their distributions are symmetric about x/D = 0 and z/D = 0 for both height to diameter ratios. The values of Cp are high at the stagnation point and they reduce in the direction away from the stagnation point. As discussed earlier, for H/D = 2, there exist a pressure gradient at x/D = -3 and it becomes zero around x/D = -2.0. This value of pressure gradient is marginally larger as H/D decreases. A secondary peak in pressure occurs at x/D = 0, between the two stagnation points for both height to diameter ratios. A negative pressure is created at the edges of effusion holes since flow gets accelerated as it moves towards the effusion holes. The magnitude of negative pressure is independent of the values of H/D, because the velocity through effusion holes remains the same. The Cp values on the target surface in general are lower for higher height to diameter ratios. 5.2 Effect of D/d Figure 15 shows that the center line velocity profile of the jets remains the same for all the diameter ratios. The flow pattern in the confinement region is also not significantly affected by the variation in the diameter ratio in the plane

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Fig. 13 Streamlines and velocity contours for D/d = 1

Fig. 14 Distribution of pressure coefficient on target surface (y = 0) for D/d = 1

z/D = 3. However, as the flow proceeds from the plane z/D = 3 to the plane z/D = 0, the flow pattern exhibits marked changes with the diameter ratio (Fig. 16). For instance, at D/d = 0.5, the flow pattern is symmetric about y-axis. As the diameter ratio is increased, the flow pattern is characterized by asymmetric small scale vortices. The fluid accelerates as it passes through the effusion hole. The velocity at the effusion hole inlet increases with D/d. The velocity contours for H/D = 2 in Fig. 16 show that for D/d = 0.5, higher velocity occurs at the jet potential core (jet exit velocity) than the velocity at effusion hole inlet on the contours, for D/d = 1.5, higher velocity occurs at the effusion hole inlet than that of velocity at jet potential core.

Due to increase in velocity near the edges of effusion holes, the magnitude of the negative pressure created near the edge of effusion hole also increases with the diameter ratio, refer Fig. 16. The stream lines plotted in plane z/D = 0 for H/D = 6 show no change in flow pattern with respect to diameter ratio. It is observed from the pressure coefficient contours (Fig. 17) that the D/d ratio plays a major role in pressure distribution. The distribution of pressure coefficient shows that the peak pressure occurs at stagnation point (x/D = 0) and it reduces along stagnation line. In the overall observations, as diameter ratio increases the pressure coefficient also increases. For instance, the values of Cp at a typical location (x/D = -1.5; z/D = 3) are 4, 41,

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6 Heat transfer characteristics

Fig. 15 Centerline velocity profile for H/D = 2

and 74% of pressure coefficient at stagnation point for D/d = 0.5, 1, 1.5 respectively. The magnitude of the secondary peak reduces as the diameter ratio increases. For D/d = 1.5, the magnitude of secondary peak is 63% less than that of D/d = 0.5 and hence more uniform pressure distribution is observed for higher diameter ratios. The magnitude of negative pressure occurring near the edges of effusion hole increases as D/d increases due to increase in flow acceleration.

Fig. 16 Streamlines and velocity contours in plane z/D = 0

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Figure 18 shows the Nusselt number contours on the target surface. The target plate has low and high heat transfer regions and it appears symmetric about x/D = 0 and z/D = 0. It is observed that the Nusselt number increases near the edges of effusion holes due to flow acceleration. The Nusselt number distribution for H/D = 2 and D/d = 1 along the stagnation line (z/D = 3) is shown in Fig. 19. The stagnation point occurs at x/D = -3, the Nusselt number increases initially till it reaches the first peak at x/D = -2.6. The first peak in Nusselt number is about 10% larger than the stagnation value and it is due to high turbulent kinetic energy produced around the nozzle edge, Fig. 20a. After the first peak, Nusselt number exhibits a second peak with a magnitude of about 32% of the first peak at x/D = -1.5. Referring again to Fig. 20a, the formation of second peak may be attributed to the increase in turbulent kinetic energy as the flow starts accelerating in the wall jet region. After reaching second peak the Nusselt number starts reducing form x/D = -0.5 where secondary vortices (refer Fig. 6) start building up, and the Nusselt number reaches a minimum value, at x/D = -0.31. A third peak in Nusselt number whose magnitude is almost same as the second peak is observed at x/D = 0. This peak is due to formation of secondary stagnation because of jet interaction.

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Fig. 17 Distribution of pressure coefficient on target surface (y = 0)

Fig. 19 Nusselt number distribution along line z/D = 3 for H/D = 2 and D/d = 1 Fig. 18 Nusselt number contours on target surface for H/D = 2; D/d = 1

6.1 Effect of H/D Figure 21 shows the Nusselt number contours on the target surface for two height to diameter ratios. The Nusselt

number distribution appears symmetric about x/D = 0 and z/D = 0 for both the values of H/D. Three peaks are observed for H/D = 2 as mentioned earlier. The location of first peak changes when H/D is increased. For H/D = 6 and D/d = 1.0, first peak appears at the stagnation point

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Fig. 20 Turbulent kinetic energy for H/D = 2; D/d = 1

Fig. 21 Nusselt number contours on target surface for D/d = 1

(x/D = 0). Nusselt number at the stagnation point for H/D = 6 is 30% more than that of H/D = 2. This is due to reduction of turbulent kinetic energy along the edge of the jet (refer Fig. 20b). The second peak is not observed for H/D = 6, the Nusselt number monotonically decreases from stagnation point to up-wash region, following the behavior of turbulent kinetic energy. For H/D = 6, Nusselt number at the secondary stagnation point reduces by 10.3% as compared to H/D = 2. For both H/D = 2 and 6, the Nusselt number increases steeply near the edge of effusion holes. The magnitude of increase in Nusselt number remains almost the same for both H/D for specific diameter ratios. It is observed from Nusselt number contour (Fig. 21) that at lower height to diameter ratios (H/D = 2), there is a small disturbance in symmetry in the upwash region. This is due to change in flow pattern with respect to diameter ratios which is explained earlier in Figs. 13 and 16. 6.2 Effect of D/d It is discussed with respect to Fig. 17 that D/d ratio plays a major role in the pressure distribution along

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stagnation line (x/D = 3). With reference to heat transfer, there is not much of a difference in local Nusselt number distribution on the target surface. The Nusselt number at stagnation point remains same for lower and higher diameter ratios i.e. D/d = 0.5 and 1.5. For both diameter ratios, with H/D = 2, the first peak appears at x/D = -2.6 and a second peak appears at about x/D = -1.3. Third peak appears between the two stagnation points (x/D = 0). Maximum difference in local Nusselt number distribution occurs at x/D = 1.5 along line z/D = 3. At this point, the case with H/D = 2, D/d = 0.5 shows 6% reduction in Nusselt number than the case with D/d = 1.0 and 1.5. For H/D = 6, the Nusselt number distribution is similar along line z/D = 3 and x/d = 0 for all diameter ratios (refer Fig. 22). Nusselt number contour for H/D = 6 shows that the diameter ratios affect the heat transfer distribution only in the up-wash region. The Nusselt number increases steeply near the effusion holes due to flow acceleration. The increase in Nusselt number near the edges of effusion hole increases with the increase in diameter ratio, as the magnitude of flow acceleration towards the effusion hole increases.

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Fig. 22 Nusselt number contours for various diameter ratios

7 Conclusion Computational study on the local flow and heat transfer characteristics from staggered, multiple circular air jets impinging on a flat surface with effusion holes is performed. The following conclusions are made from the study. 1.

2.

3.

The computations with the k-x SST model seems to predict well the jet potential core, stagnation zone, wall jet region, primary vortices forming the up-wash region, the secondary vortices forming a secondary stagnation zone and the flow pattern through the effusion hole. These are supported by both the pressure coefficient and Nusselt number variations. The heat and mass transfer characteristics from the present estimations, in general, match closely with those of Cho and Rhee [14] except in the up-wash region. For small height to diameter ratio (H/D = 2), three peaks in the Nusselt number are found, corresponding

4.

to the stagnation zone, the wall jet region and the upwash zone. However for larger height to diameter ratios (H/D = 6), the middle peak disappears. For all height to diameter ratios, the Nusselt number increases near the edge of effusion holes due to flow acceleration. This increase in Nusselt number factor increases with decrease in effusion hole size (or) increasing D/d.

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Heat Mass Transfer (2011) 47:1121–1132 11. Hollworth BR, Dangan L (1980) Arrays of impinging jets with spent fluid removal through vent holes on the target surface part I: average heat transfer. Trans ASME J Eng Power 102:994–999 12. Hollworth BR, Lehmann G, Rosiczkowski J (1983) Arrays of impinging jets with spent fluid removal through vent holes on the target surface part II: local heat transfer. Trans ASME J Eng Power 105:393–402 13. Ekkad SV, Huang Y, Han JC (1999) Impingement heat transfer on a target plate with film cooling holes. AIAA J Thermophys Heat Transf 13(4):522–528 14. Cho HH, Rhee DH (2001) Local heat/mass transfer measurement on the effusion plate in impingement/effusion cooling systems. J Turbomachinery 123:601–608 15. Ramakumar BVN, Prasad BVSSS (2008) Computational flow and heat transfer of a circular jets impinging on a concave surface. Heat Mass Transf 44(6):667–678